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# Uniform distribution
A uniform distribution is a probability distribution where all possible outcomes are equally likely. In other words, the probability of each outcome is the same. Uniform distributions are commonly used to model random events or random variables that have a range of possible values.
To calculate the probability of a value falling within a specific range in a uniform distribution, we can use the formula:
$$P(a \le X \le b) = \frac{b - a}{b - a}$$
where $a$ and $b$ are the lower and upper bounds of the range, and $X$ is the random variable.
Let's say we have a uniform distribution with a range of 0 to 10. To calculate the probability of a value falling within the range of 4 to 7, we can use the formula:
$$P(4 \le X \le 7) = \frac{7 - 4}{10 - 0}$$
$$P(4 \le X \le 7) = \frac{3}{10}$$
So, the probability of a value falling within the range of 4 to 7 in this uniform distribution is 30%.
## Exercise
Calculate the probability of a value falling within the range of 5 to 8 in a uniform distribution with a range of 0 to 10.
### Solution
$$P(5 \le X \le 8) = \frac{8 - 5}{10 - 0}$$
$$P(5 \le X \le 8) = \frac{3}{10}$$
So, the probability of a value falling within the range of 5 to 8 in this uniform distribution is 30%.
# Bernoulli distribution
The Bernoulli distribution is a discrete probability distribution that represents the probability of a random variable taking on one of two possible values, usually 0 and 1. It is named after Swiss mathematician Jacob Bernoulli.
The probability mass function (PMF) of a Bernoulli distribution is given by:
$$P(X = k) = p^k(1-p)^{1-k}$$
where $k$ is the value of the random variable, $p$ is the probability of success (taking the value 1), and $1-p$ is the probability of failure (taking the value 0).
Let's say we have a coin toss, where the probability of getting heads (success) is 0.6 and the probability of getting tails (failure) is 0.4. To calculate the probability of getting heads in a single coin toss, we can use the PMF of a Bernoulli distribution:
$$P(X = 1) = 0.6^1(1-0.6)^{1-1}$$
$$P(X = 1) = 0.6$$
So, the probability of getting heads in a single coin toss is 60%.
## Exercise
Calculate the probability of getting tails in a single coin toss with the given probability of getting heads as 0.6.
### Solution
$$P(X = 0) = 0.4^0(1-0.6)^{1-0}$$
$$P(X = 0) = 0.4$$
So, the probability of getting tails in a single coin toss is 40%.
# Binomial distribution
The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials. It is named after the Italian mathematician Girolamo Cardano and the French mathematician Pierre de Fermat.
The probability mass function (PMF) of a binomial distribution is given by:
$$P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$$
where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success in each trial, and $1-p$ is the probability of failure in each trial.
Let's say we have a coin toss experiment, where the probability of getting heads is 0.6. If we perform 10 coin tosses, the probability of getting exactly 6 heads can be calculated using the PMF of a binomial distribution:
$$P(X = 6) = \binom{10}{6}0.6^6(1-0.6)^{10-6}$$
$$P(X = 6) = \binom{10}{6}0.6^6(0.4)^4$$
So, the probability of getting exactly 6 heads in 10 coin tosses is 0.0792.
## Exercise
Calculate the probability of getting exactly 4 heads in 10 coin tosses with the given probability of getting heads as 0.6.
### Solution
$$P(X = 4) = \binom{10}{4}0.6^4(1-0.6)^{10-4}$$
$$P(X = 4) = \binom{10}{4}0.6^4(0.4)^6$$
So, the probability of getting exactly 4 heads in 10 coin tosses is 0.0792.
# Poisson distribution
The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space. It is named after French mathematician Siméon-Denis Poisson.
The probability mass function (PMF) of a Poisson distribution is given by:
$$P(X = k) = \frac{e^{-\lambda}\lambda^k}{k!}$$
where $k$ is the number of events, $\lambda$ is the average number of events per interval, and $e$ is the base of the natural logarithm.
Let's say we have a telephone switchboard, where the average number of calls per minute is 2. To calculate the probability of receiving exactly 3 calls in a minute, we can use the PMF of a Poisson distribution:
$$P(X = 3) = \frac{e^{-2}\cdot 2^3}{3!}$$
$$P(X = 3) = \frac{e^{-2}\cdot 8}{6}$$
So, the probability of receiving exactly 3 calls in a minute is 0.1881.
## Exercise
Calculate the probability of receiving exactly 4 calls in a minute with the given average number of calls per minute as 2.
### Solution
$$P(X = 4) = \frac{e^{-2}\cdot 2^4}{4!}$$
$$P(X = 4) = \frac{e^{-2}\cdot 16}{24}$$
So, the probability of receiving exactly 4 calls in a minute is 0.0693.
# Normal distribution
The normal distribution, also known as the Gaussian distribution, is a continuous probability distribution that is symmetric and bell-shaped. It is named after the German mathematician and physicist Carl Friedrich Gauss.
The probability density function (PDF) of a normal distribution is given by:
$$f(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$
where $x$ is the random variable, $\mu$ is the mean (average) of the distribution, and $\sigma$ is the standard deviation (measure of dispersion or spread of the distribution).
Let's say we have a standard normal distribution, where the mean is 0 and the standard deviation is 1. To calculate the probability of a value falling within the range of -1 to 1, we can use the PDF of a normal distribution:
$$P(X \le 1) = \int_{-\infty}^1 f(x) dx$$
$$P(X \le 1) = \int_{-\infty}^1 \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} dx$$
So, the probability of a value falling within the range of -1 to 1 in this standard normal distribution is approximately 0.6827.
## Exercise
Calculate the probability of a value falling within the range of -2 to 2 in a standard normal distribution.
### Solution
$$P(X \le 2) = \int_{-\infty}^2 f(x) dx$$
$$P(X \le 2) = \int_{-\infty}^2 \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} dx$$
So, the probability of a value falling within the range of -2 to 2 in a standard normal distribution is approximately 0.9545.
# Expected value and variance
Expected value and variance are two important measures of central tendency and dispersion in probability distributions.
The expected value (mean) of a probability distribution is the value that we would expect the random variable to take on, averaged over all possible outcomes. It is denoted by $\mu$. The formula for the expected value depends on the type of distribution.
The variance of a probability distribution is a measure of how much the random variable deviates from its expected value. It is denoted by $\sigma^2$. The formula for the variance depends on the type of distribution.
Let's say we have a uniform distribution with a range of 0 to 10. The expected value (mean) of this distribution is the average of the lower and upper bounds:
$$\mu = \frac{0 + 10}{2}$$
$$\mu = 5$$
The variance of this distribution is the average of the squared differences between each possible value and the mean:
$$\sigma^2 = \frac{(0 - 5)^2 + (10 - 5)^2}{2}$$
$$\sigma^2 = \frac{25 + 25}{2}$$
$$\sigma^2 = 25$$
So, the expected value of this uniform distribution is 5, and its variance is 25.
## Exercise
Calculate the expected value and variance of a binomial distribution with $n = 10$ and $p = 0.6$.
### Solution
The expected value (mean) of a binomial distribution is given by:
$$\mu = np$$
So, the expected value of this binomial distribution is:
$$\mu = 10 \cdot 0.6$$
$$\mu = 6$$
The variance of a binomial distribution is given by:
$$\sigma^2 = np(1-p)$$
So, the variance of this binomial distribution is:
$$\sigma^2 = 10 \cdot 0.6 \cdot 0.4$$
$$\sigma^2 = 2.4$$
So, the expected value of this binomial distribution is 6, and its variance is 2.4.
# Simulating random variables with C++
To simulate random variables in C++, we can use the random number generation functions provided by the C++ Standard Library. The most commonly used function is `rand()`, which generates a random integer between 0 and RAND_MAX.
To generate a random floating-point number between 0 and 1, we can use the following function:
```cpp
double random_double() {
return static_cast<double>(rand()) / static_cast<double>(RAND_MAX);
}
```
Let's say we want to simulate a random variable from a uniform distribution with a range of 0 to 10. We can use the `random_double()` function to generate a random floating-point number between 0 and 1, and then scale and shift it to the desired range:
```cpp
double random_uniform(double lower, double upper) {
return lower + (upper - lower) * random_double();
}
int main() {
double random_value = random_uniform(0, 10);
cout << "Random value: " << random_value << endl;
return 0;
}
```
## Exercise
Write a C++ function to simulate a random variable from a binomial distribution with a given number of trials `n` and probability of success `p`.
### Solution
```cpp
int random_binomial(int n, double p) {
int k = 0;
for (int i = 0; i < n; i++) {
if (random_double() < p) {
k++;
}
}
return k;
}
int main() {
int random_value = random_binomial(10, 0.6);
cout << "Random value: " << random_value << endl;
return 0;
}
```
# Writing custom C++ functions for probability distributions
To write custom C++ functions for probability distributions, we can use the random number generation functions provided by the C++ Standard Library. The most commonly used function is `rand()`, which generates a random integer between 0 and RAND_MAX.
Let's say we want to write a custom C++ function to calculate the probability mass function (PMF) of a binomial distribution:
```cpp
double binomial_pmf(int n, int k, double p) {
double comb = 1;
for (int i = 1; i <= k; i++) {
comb *= static_cast<double>(n - k + i) / i;
}
return comb * pow(p, k) * pow(1 - p, n - k);
}
int main() {
double probability = binomial_pmf(10, 6, 0.6);
cout << "Probability: " << probability << endl;
return 0;
}
```
## Exercise
Write a C++ function to calculate the probability density function (PDF) of a normal distribution.
### Solution
```cpp
double normal_pdf(double x, double mu, double sigma) {
return 1.0 / (sqrt(2 * M_PI * sigma * sigma)) * exp(-0.5 * pow((x - mu) / sigma, 2));
}
int main() {
double probability = normal_pdf(0, 0, 1);
cout << "Probability: " << probability << endl;
return 0;
}
```
# Applications of probability distributions in C++
Probability distributions can be used to model various real-world phenomena and solve practical problems. Some applications of probability distributions in C++ include:
- Simulating coin toss experiments and calculating the probability of getting heads or tails.
- Simulating dice roll experiments and calculating the probability of getting a specific number.
- Simulating the number of customers arriving at a store in a given time period, assuming a Poisson distribution.
- Simulating the number of defective products in a production line, assuming a binomial distribution.
- Simulating the time taken to complete a task, assuming a normal distribution.
Let's say we want to simulate the number of customers arriving at a store in a given time period, assuming a Poisson distribution with an average number of customers per minute of 2. We can use the `random_binomial()` function to generate a random integer between 0 and 2, and then simulate the arrival of customers:
```cpp
int main() {
int random_customers = random_binomial(10, 0.6);
cout << "Number of customers: " << random_customers << endl;
return 0;
}
```
## Exercise
Write a C++ program to simulate the number of defective products in a production line, assuming a binomial distribution with a given number of trials `n` and probability of success `p`. Calculate the probability of getting a specific number of defective products.
### Solution
```cpp
int main() {
int n = 100;
double p = 0.01;
int k = random_binomial(n, p);
double probability = binomial_pmf(n, k, p);
cout << "Number of defective products: " << k << endl;
cout << "Probability: " << probability << endl;
return 0;
}
```
# Combining probability distributions
Probability distributions can be combined using various mathematical operations to model more complex phenomena. Some common operations include:
- Adding probability distributions to model the sum of independent random variables.
- Multiplying probability distributions to model the product of independent random variables.
- Conditional probability to model the probability of an event given that another event has occurred.
Let's say we have two independent random variables, X and Y, each following a normal distribution with mean 0 and standard deviation 1. We can combine their probability distributions to model the sum of the two random variables:
$$Z = X + Y$$
The probability density function (PDF) of the sum of two normal distributions is given by:
$$f_Z(z) = \frac{1}{\sqrt{2\pi\sigma_X^2}\sqrt{2\pi\sigma_Y^2}}e^{-\frac{(z-\mu_X)^2}{2\sigma_X^2}-\frac{(z-\mu_Y)^2}{2\sigma_Y^2}}$$
where $z$ is the random variable, $\mu_X$ and $\sigma_X$ are the mean and standard deviation of the first distribution, $\mu_Y$ and $\sigma_Y$ are the mean and standard deviation of the second distribution.
## Exercise
Calculate the probability density function (PDF) of the sum of two normal distributions with means 0 and standard deviations 1.
### Solution
```cpp
double normal_pdf(double z, double mu_x, double mu_y, double sigma_x, double sigma_y) {
return 1.0 / (sqrt(2 * M_PI * sigma_x * sigma_x) * sqrt(2 * M_PI * sigma_y * sigma_y)) * exp(-0.5 * pow((z - mu_x), 2) / (sigma_x * sigma_x) - 0.5 * pow((z - mu_y), 2) / (sigma_y * sigma_y));
}
int main() {
double probability = normal_pdf(0, 0, 0, 1, 1);
cout << "Probability: " << probability << endl;
return 0;
}
```
# Advanced topics in probability distributions
There are many advanced topics in probability distributions that are beyond the scope of this textbook. Some of these topics include:
- Multivariate distributions, which involve more than one random variable.
- Mixture distributions, which involve combining multiple probability distributions.
- Markov chains and hidden Markov models, which are used to model sequences of random events.
- Bayesian inference, which involves updating the probability of a hypothesis based on new evidence.
Studying these advanced topics requires a deeper understanding of probability theory and mathematical statistics. | Textbooks |
Leg muscle cross-sectional area measured by ultrasound is highly correlated with MRI
Joshua K. Sponbeck1,
Clint R. Frandsen1,
Sarah T. Ridge1,
Derek A. Swanson1,
Dallin C. Swanson1 &
A. Wayne Johnson1
The leg muscles are important for balance, posture, and movement during static and dynamic activity. Obtaining cross-sectional area measurements (CSA) of the leg muscles helps researchers understand the health and force production capability of individual leg muscles. Therefore, having an easy to use and readily available method to assess leg muscle CSA is needed. Thus, the purpose of this study was to compare the magnitude, repeatability, and validity of CSA measurements of select leg muscles from ultrasound (US) and the current gold standard, magnetic resonance imaging (MRI).
20 healthy volunteers participated in this study. Each participant was imaged via US and MRI. The muscles of interest obtained on each participant consisted of the tibialis anterior at both 30 and 50% of the shank length, tibialis posterior at both 30 and 50% of the shank length, the flexor digitorum longus, the fibularis (peroneus) longus, and the fibularis (peroneus) brevis.
Strong Pearson correlations were seen for all of the muscles when comparing US to MRI with a range from .7840 to .9676. For all measurements, standard error of the measurement ranged from .003 to 0.260 cm2. Minimum detectable difference for muscle measurements ranged from .008 cm2 for MRI fibularis longus and fibularis brevis to .693 cm2 for MRI of tibialis anterior at 30%. US minimum detectable difference ranged from .125 cm2 for the tibialis posterior muscle at 30% to .449 cm2 for the tibialis anterior muscle at 50%.
Based on these results ultrasound is a valid method to obtain CSA of muscles of the leg when compared with MRI.
Imaging and analysis of muscle cross-sectional area (CSA) can give understanding of the health [1] and force production potential of a muscle [2]. This can be particularly useful for assessment of muscles that are hard to isolate during functional testing, for example in the lower leg (knee to ankle, anatomically known as the leg), where several muscles perform the same actions. As there are no commercially readily available devices used to assess strength of specific or isolated leg muscles, anatomical muscle CSA provides the ability to infer force production of these muscles [2].
There are currently limited ways to assess muscle CSA in vivo. These include magnetic resonance imaging (MRI), computed tomography, and ultrasound imaging (US). MRI has been validated and is now largely considered the "gold standard" for comparison of other imaging methods, however MRI is expensive, time consuming, and not always readily available [3,4,5,6]. Computed tomography has limited availability for these purposes in the research and clinical settings as a result of the consequences of repeated radiation exposure, as well as cost [7]. Ultrasound imaging is a relatively low-cost alternative that is becoming readily available in the research and clinical settings [8]; however, validation of US compared to MRI is necessary for specific muscle groups.
The use of US imaging has several advantages for the evaluation of soft tissue. It allows for reliable, high-resolution assessment of soft-tissue under static and dynamic conditions [9,10,11]. Dynamic movement patterns, such as muscle contraction, can be recorded in retrospective video clips (Cine-loops), that have been shown to decrease operator imaging and measurement error [11]. As with other imaging modalities, however, US imaging is operator dependent, requires significant operator training, and has a limited field of view that requires detailed anatomical knowledge of the imaged area [8].
While US measurements of several muscle groups have been validated with MRI, few studies have reported this data from any leg muscles [3, 8, 12, 13]. The specific arrangement and anatomical relationships of leg muscles present unique challenges to image acquisition and measurement [14]. As these leg muscles are crucial during dynamic movement [15] as well as during static posture and balance [16], the ability to assess these muscles' CSA accurately, reliably, and quickly is necessary.
Therefore, the primary purpose of this study was to compare the magnitude, repeatability, and validity leg muscle CSA measurements acquired from US images compared with images taken via MRI. We hypothesized that US imaging and subsequent CSA analysis of selected muscles of the leg would correlate closely with those same muscles analyzed using MRI.
10 males and 10 females completed this study (mean and (SD), age = 34.15 (16.55) years, weight = 80 (4.88) kg, height = 169.72 (34.78) cm). All participants were volunteers, ages 18 years or older, who did not have an extremity injury within the previous one month or leg/foot surgery within the previous year. Each participant read and signed an informed consent approved by the University's Institutional Review Board (study protocol, IRB2019–375). Study participants reported for two visits that consisted of the US session, and the MRI session. On average the two visits were 10 days apart for study participants.
In order to ensure consistency of measurement of each muscle, the linear distances from the lateral knee joint line to the inferior point of the lateral malleolus, as well as the linear distance from the medial knee joint line to the inferior point of the medial malleolus were measured. From these measurements, the 30 and 50% distances from the knee joint line were determined and marked with a soft-tipped marker. These measurement locations were recorded and used in both MRI and US sessions.
MRI protocol
Prior to entering the MRI machine, participants completed a safety screening, in the waiting room of the MRI facility. Upon completion, fish oil tablets attached to a Velcro strap were placed at the previously measured markings of 30 and 50% of the shank length. The fish oil tablets allowed the researchers to consistently locate the appropriate slices to measure at the correct location of the shank. A 3 Tesla magnet (TIM-Trio 3.0 T MRI, Siemens, Erlangen, Germany) was used to scan the left leg first, then the right leg. 30% shank length images were obtained first, followed by the 50% shank length image for each leg. Participants were lying supine and placed feet first into the magnet. The initial localizer scan was centered on the marked location being imaged. T1 weighted MRI images were acquired using a Siemens sequence using an axial orientation, and an acquisition time of 20 s. The resolution was 1.56 mm by 1.95 mm with a slice thickness of 6 mm and a space between slices of 3 mm. The resolution matrix was 256 × 205. An 8-channel knee coil was used to obtain a total of 10 images at each location. Repetition Time (TR)=7.3 ms and Echo Time (TE)=3.6 ms.
All images obtained from the MRI scans were loaded into Osirix (Pixmeo, Geneva, Switzerland) in order to obtain CSA measurements. Two CSA measurements were taken from adjacent slices of the same scan at the location of the fish oil tablets on the MRI.. Measurements were obtained by two researchers (JS and DaS) for each the tibialis anterior, the tibialis posterior, the flexor digitorum longus, the fibularis (peroneus) longus, and the fibularis (peroneus) brevis muscles. Muscles were outlined inside of the muscle fascia (Fig. 1).
MRI (left) and US (right) images of the tibialis anterior muscle with associated segmentation of the muscle
A researcher with 10 years of musculoskeletal US imaging experience gathered images obtained in this study. Participants sat in a relaxed position on a treatment table with an upright, inclined back and had their thigh supported by a bolster so that their calf was uncompressed. The same muscles were imaged via US (LOGIQ S8; GE Healthcare, Chicago, IL) using an ML6–15-D matrix linear transducer. Frequencies ranged between 8 and 12 MHz as determined by the scanner to enhance image clarity. Adjustments to depth, frequency, focal position, and time-gain-compensation were performed as needed to enhance the clarity of the image. Additionally, the Virtual Convex setting was occasionally used to enhance the field of view during scanning.
Short axis images of each muscle starting with the tibialis anterior were obtained at 30 and 50% of the shank length with the lateral border of the tibia serving as an anatomical landmark during imaging. The tibialis posterior was imaged at both the 30 and 50% marks of the shank from the anterior side [10]. The tibialis anterior muscle and interosseous membrane served as anatomical landmarks during imaging. The flexor digitorum longus was imaged at 50% of the shank length on the medial side just posterior to the tibia, with the tibia serving as an anatomical landmark to aid in imaging consistency. The fibularis longus was imaged at 30% of the shank length with the fibula serving as an anatomical landmark for consistency just deep to the fibularis longus. The fibularis brevis was measured at 50% of the shank length with the fibula serving as an anatomical landmark just deep to the fibularis brevis. During imaging, participants were asked to perform muscle contractions causing the imaged muscle to contract and then return to rest. Cine loops were recorded of the contraction cycle to help visualize the fascial borders of the muscles and the conformational changes within the muscle. This allowed the muscles of interest to be distinguished from the adjacent muscles and other leg structures. Two separate recordings of the contraction cycle were taken of each muscle. The transducer was removed from the leg between recordings.
CSA measurements were obtained post imaging session, from a single frame within each of the two separate cine loops.. CSA measurements were obtained using internal software on the LOGIQ S8 machine. Measurements were performed by two members of the research team (DaS and DeS) who have been previously trained to obtain CSA measurements from these specific muscles. All muscles were traced within the facial borders during rest. During US imaging Cine-loops were obtained to aid the researchers performing the measuring by viewing the muscle contraction. This analysis was similar to previously established methods [10, 11] (Fig. 1).
Intraclass-correlation coefficients (ICC3,k) were performed to establish reliability using a CSA measurement from each MRI and US image. As we were interested in our segmentation repeatability, we chose the ICC model with fixed raters and random subjects assessed for absolute agreement. Additionally, the averages of the two measured CSAs for both MRI and US images were calculated. The averages were used in a Pearson product moment correlation to determine the validity of the US estimates of muscles' CSA compared to the MRI. Our interpretation of Pearson's Correlation coefficients will be based upon previous research as follows: 0.9 to 1 is very strong, 0.70 to .89 is strong, 0.5 to 0.69 is moderate, .30 to .49 is low, and 0.0 to 0.29 is negligible [17]. T-tests were performed to determine if any muscle CSA differed significantly between US and MRI and to conclude that a Bland-Altman plot analysis would be appropriate. Bland-Altman plot analysis was used to understand potential systematic error between US and MRI. To represent error inherent to each imaging modality, we calculated the standard error of the measurement (SEM) and minimum detectable difference (MDD) for both MRI and US for each of the muscles imaged using the following equations:
$$ {\displaystyle \begin{array}{c}\mathrm{SEM}=\mathrm{SD}\ \left(\mathrm{Sq}\ \mathrm{rt}\ 1-{\mathrm{r}}_{\mathrm{ICC}}\right)\\ {}95\%\mathrm{CI}\ \mathrm{SEM}=\mathrm{muscle}\ \mathrm{mean}\pm \left(1.96\ast \mathrm{SEM}\right)\\ {}\mathrm{MDD}=\mathrm{SEM}\ast 1.96\ast \mathrm{sqrt}\ 2\end{array}} $$
Statistical analyses were performed using Statistical Analysis Software (JMP pro Version 14.2, SAS Institute, Inc. Cary, NC, USA) for all statistics except for ICC3,k values, which were obtained using SPSS version 26.0 statistical software (IBM Corporation, Armonk, NY). An alpha of 0.05 was used to determine significance of statistical tests.
Table 1 contains all assessed mean muscle CSA values for US and MRI measurements, ICC values, SEM, and MDD. The correlations between MRI and US imaging and segmentation were strong to very strong with a range from 0.784 to 0.968 (Fig. 2) [17]. No muscle CSA means were significantly different between US and MRI measures of CSA with p values ranging from 0.164 to 0.990. Average CSA measured from MRI were slightly larger for all of the muscles (except for fibularis longus, which was nearly equal), as illustrated by the positive biases on the Bland-Altman plots (Fig. 2, Table 2). For all measurements, SEM ranged from 0.003 to 0.260 cm2. MDD for muscle measurements for both US and MRI ranged from 0.008 to 0.693 cm2. Individual reliability was excellent for both MRI and US for each muscle ranging from 0.958 to 0.999.
Table 1 Comparison of muscle CSA by MRI and US. Mean (SD), ICC, and SEM with confidence intervals for muscle CSA
Table 2 Bias, lower limit of agreement (LoA), and upper LoA for each muscle imaged in the leg
(A-F) Correlation data including R2 values, and corresponding Bland-Altman plots for all muscles of interest in the leg. The dotted black line in the correlation plots is the line of best fit for the data. In the Bland Altman plots the blue line represents the bias, and the red lines represent upper and lower limits of agreement (LoA = Bias ± 1.96*SD)
The primary purpose of this study was to establish the validity of leg muscle CSA measurements acquired from US images. The magnitude and measurement repeatability of muscle CSAs obtained from US imaging were compared with those obtained from MRI. The results of our study indicate that US imaging and subsequent segmentation of leg muscles are strongly to very strongly correlated with MRI. We also found excellent intra-rater repeatability for both US and MRI.
Our results support previous research showing muscle CSA when imaged with US is valid and correlated with MRI. Muscle groups that have been previously used to validate and correlate US with MRI include the quadriceps muscles [18, 19], rotator cuff muscles [13], hamstring muscles [20], biceps and triceps brachii muscles [21], and core trunk muscles [8, 22]. Our values fall within the upper range of correlations reported in these studies, and range from 0.53 to 0.99. No studies have reported a comparison of leg muscle CSA between US and MRI, though a single study reported very strong correlation of muscle volume measurements of the tibialis anterior muscle between these imaging modalities [12].
Although not statistically different from MRI, average US muscle measurements were slightly smaller for nearly all muscles measured. Ahtianinen et al. displayed similar findings, with smaller rectus femoris average muscle US CSA when compared to MRI during a training study [18]. Possible explanations for smaller US means include differences in processing algorithms between US and MRI, measuring planes between imaging modalities, or US probe compression of muscle. However, other researchers dispute these possible reasons [23]. Despite these differences, previous intervention studies have shown that changes in muscle size are consistent when measured with US and MRI [18]. These data suggest that either imaging modality can be used to track changes over time.
When using US as an imaging modality, the operator dependence is important to take into account. US offers a limited field of view, is sensitive to operator technique, and requires anatomical knowledge of the imaged area. To address this potential limitation, when multiple clinicians and/or researchers work together, they should practice similar techniques and assess reliability. For example, the use of cine loops has been shown to increase inter-rater reliability [11]. While less operator dependent, MRI is still highly sensitive to participant positioning [24]. Regardless of imaging modality used, it is important to calculate measures such as the MDD when tracking muscle size changes over time. Small MDD provide confidence that true changes occurred, as opposed to error induced by the operator. The repeatability across operators and measures deserves further refining and research. The current study reported similar US leg muscle CSA values compared to previous US research when available. Among direct comparisons of the flexor digitorum longus muscle there was 99% similarity in muscle CSA average with previous research performed by our group [11]. The fibularis brevis muscle had an average mean muscle CSA of 4.18 cm2 as measured by US, which is similar to previous research that measured 4.09 cm2 (9) and 3.5 (24) cm2. Other muscles from the current study were difficult to compare to previous studies. While our research group has previously demonstrated US measurement and segmentation of the tibialis posterior muscle, to our knowledge this is not being performed elsewhere [10]. The fibularis longus and tibialis anterior muscle sizes have been imaged at different locations of the muscle [9, 25, 26] in previous studies, or segmented and measured using width or volume only [11, 12] and not CSA as was used in this current study.
US may provide several advantages to clinicians and researchers for obtaining muscle CSA values, as opposed to MRI. Possible benefits of US may include decreased imaging time, imaging safety, reduced cost, modality availability, visualization of muscle contraction, and potential use to provide biofeedback. During the current study the US imaging session lasted 15 min including participant preparation time, compared with 30 min for the MRI session including participant preparation time. US has virtually no contraindications and very limited side effects making imaging possible to those who may not be indicated for an MRI, such as those with metal implants or a pacemaker [24]. The US unit may be much more readily available, and a fraction of the cost [8]. Additionally, as US measurements are performed in real time, they may be used by clinicians to provide biofeedback for patients. The use of biofeedback has resulted in improved performance and long-term contractile ability of a muscle [22].
One consideration when comparing measurements from different imaging modalities is the amount of day-to-day variability in muscle CSA. Each participant was imaged with both US and MRI one time. Previous research has shown slight muscle CSA variations (ranging from 1 to 4.7%) across days [23]. Although our participants were imaged on different days, participants were imaged at similar times of the day, and physical activity was controlled in an attempt to limit variability.
US is a reliable and valid method of measuring muscle CSA for the tibialis anterior, tibialis posterior, flexor digitorum longus, fibularis longus, and fibularis brevis muscles when compared with MRI. Being able to use US rather than MRI may help researchers and clinicians spend less time completing participant imaging and data analysis, increasing efficiency and lowering cost. Additionally, US allows for dynamic testing and biofeedback. For some researchers and clinicians US is also a more readily available modality and therefore is an important tool when desiring to view and analyze individual muscle CSA of the leg.
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
CSA:
cross-sectional area
MRI:
magnetic resonance image
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Bryce Allen, Victoria Violette, Cole Anderson, Hunter Anderson, Jared Ivan, Jacob Ivan, Christopher Thompson for assistance in data collection processing and coordination of study participants.
No funding was provided for any portion of this study.
Exercise Sciences Department, Brigham Young University, 1000 N University Ave, Provo, UT, 84604, USA
Joshua K. Sponbeck, Clint R. Frandsen, Sarah T. Ridge, Derek A. Swanson, Dallin C. Swanson & A. Wayne Johnson
Joshua K. Sponbeck
Clint R. Frandsen
Sarah T. Ridge
Derek A. Swanson
Dallin C. Swanson
A. Wayne Johnson
JS analyzed and interpreted imaging data, and was a major contributor in writing the manuscript. CF played a major role in data collection including knowledge of the MRI machine. SR contributed in the writing of the manuscript. DaS assisted with data analysis. DeS assisted with data analysis. WJ designed this study, along with assisting in data collection and analysis, and as a contributor in writing the manuscript. All authors have given final approval on the publication of this manuscript.
Correspondence to A. Wayne Johnson.
This study was approved by the institutional review board at Brigham Young University, study protocol, IRB2019–375. All participants read and signed an informed consent prior to participation in this study.
The authors declare they have no competing interests.
Sponbeck, J.K., Frandsen, C.R., Ridge, S.T. et al. Leg muscle cross-sectional area measured by ultrasound is highly correlated with MRI. J Foot Ankle Res 14, 5 (2021). https://doi.org/10.1186/s13047-021-00446-y
Leg muscles | CommonCrawl |
The running time of algorithm is at most $O(n^2)$
The problem is that if an algorithm is $O(n^2)$ then it is also $O(n^3)$ and $O(n^4), O(n^n), \ldots$ and the phrase 'at most' does not make sense in this situation.
For this reason, I am not sure whether this statement is correct or not.
time-complexity asymptotics
edited Nov 2 '20 at 15:59
Derek Allums
jsbcjsbc
$\begingroup$ I'd say the problem with using "at most O(n^2)" is that there isn't a particularly well-known and well-defined meaning of "at most" in this context, which can lead to ambiguity (unless the surrounding context makes the meaning clear). I can think of a few possible things that one might want to imply with that. $\endgroup$ – Bernhard Barker Nov 1 '20 at 21:55
$\begingroup$ Technically it could be stated as "in the worst case, the running time of the algorithm is O(n^2) and higher". $\endgroup$ – Inertial Ignorance Nov 2 '20 at 1:37
The two phrases
The running time is $O(n^2)$
The running time is at most $O(n^2)$
mean the same thing.
This is similar to the following two equivalent claims:
$x = y$ for some $y \leq z$.
$x \leq y$ for some $y \leq z$.
Why would we ever use "at most $O(n^2)$", then? Sometimes we want to stress that the bound $O(n^2)$ is loose, and then it makes sense to use "at most $O(n^2)$". For example, suppose that we have a multi-part algorithm, which we want to show runs in time $O(n^2)$. Suppose that we can bound the running time of the first step by $O(n)$. We could say "the first part runs in $O(n)$, which is at most $O(n^2)$".
Yuval FilmusYuval Filmus
$\begingroup$ "the first part runs in O(n), which is at most O(n^2)" - I wouldn't use "at most" there, but rather just "also" or say just it's O(n^2). "At most" used in that way would to me imply that you can't say it's O(n^3), but of course you can say that. $\endgroup$ – Bernhard Barker Nov 1 '20 at 21:50
$\begingroup$ You wouldn't, some would. It's a matter of style. $\endgroup$ – Yuval Filmus Nov 1 '20 at 22:02
$\begingroup$ I don't agree with this at all. To use a trivial example of Quicksort, you will often hear its running time described as O(n.log(n)), and that is indeed the average running time, but its worst case is O(n²), so it is not at all true to say that "the running time is O(n²)" is the same as saying the running time is at most O(n²)". What a person means by "running time" is contextual. I'd say it usually means average running time not worst case running time. $\endgroup$ – Fraser Orr Nov 2 '20 at 5:28
$\begingroup$ @FraserOrr As far as I'm aware, there are two different meanings in use for "running time" without specification. One is "the metric we care about in the situation where we apply it", which you mention here. The other is "the bound holds in any (worst, best, average, etc.) case", which means it is the worst case running time. I don't think "at most" helps much to disambiguate these two usages. I think it's better to be fully explicit and write "X case running time" if you don't want to be ambiguous. $\endgroup$ – Discrete lizard♦ Nov 2 '20 at 13:51
$\begingroup$ @Discretelizard for sure it would be better to be explicit, but I would say that if someone said that the running time was at most O(n²) it is fairly unambiguous, if not idiomatic, that they mean the worst case runtime. $\endgroup$ – Fraser Orr Nov 3 '20 at 19:33
"At most" might mean "at worst" i.e. that the worst-case time complexity is $O(n^2)$.
For example one might say that "Quicksort is at most $O(n^2)$," meaning that no matter what infinite subset of the inputs you look at, the complexity on that subset is never more than $O(n^2)$.
Theodore NorvellTheodore Norvell
$\begingroup$ This points out the other phrasing you will see, "The runtime is on average $O(n^2)$" $\endgroup$ – Cort Ammon Nov 2 '20 at 1:26
My reading is that it's not necessarily a tight bound, ie. we know the algorithm is $O(n^2)$, but we don't know if it's (for example) $O(n^{1.99})$
BlueRaja - Danny PflughoeftBlueRaja - Danny Pflughoeft
$\begingroup$ I wouldn't assume this is the right interpretation without more context, but it's definitely possible. $\endgroup$ – usul Nov 2 '20 at 2:47
"f(n) is in O(n^2)" means f(n) ≤ cn^2 for all large n and for some c > 0. Clearly if f(n) ≤ cn^2, then f(n) ≤ cn^3, cn^4 etc. So factually, "f(n) is in O(n^4)" is equally true. It just gives you much less information, so it may be less useful.
If someone says "f(n) is at most O(n^2)", I would interpret that as "I proved it is in O(n^2), but I couldn't be bothered to check whether it is possibly in a more narrow class". For example, if your algorithm does Step 1 which takes O(n^3) and then Step 2, and you can prove that Step 2 is in O(n^2), that's good enough for all purposes, and you wouldn't bother checking if it's maybe in O (n^2 / log n) or in O (n^1.5).
There's the class $\Theta(n^2)$ which means $c_1 n^2 ≤ f(n) ≤ c_2 n^2$ for all large n and for some $0 < c_1 < c_2$. Here you can't just substitute n^4 for n^2. And there is "asymptotic O(n^2)" which means f(n) is in O(n^2) and not in o(n^2), which means $c_1 n^2 ≤ f(n) ≤ c_2 n^2$ for infinitely many large n and for some $0 < c_1 < c_2$. Again, here you can't just substitue n^4.
$\begingroup$ +1. I would also interpret it as "I've proven it to be $O(n^2)$ but I haven't checked if it could be e.g. $O(n \log n)$". $\endgroup$ – Fax Nov 2 '20 at 14:27
You can just view $\mathcal{O}(n^2)$ as an anonymous function drawn from the underlying class.
The statement means: The running time of the algorithm is at most quadratic in the input length $n$.
I do not think there is anything controversial or wrong here.
ttnickttnick
I understand it so, that, perhaps, saying "$f$ is at most $O(n^2)$", the speaker wants to emphasize, exaggerate his attitude to upper bound $O(n^2)$ as least one.
Good point anyway.
zkutchzkutch
1,07211 gold badge11 silver badge1010 bronze badges
First, just because it's O(n^2) doesn't mean it's O(n^3) or higher.
And sometimes "at most" is quite relevant. Consider, for example, Quicksort. In the real world it normally runs in very close to O(n log n) time, but for any given implementation you can devise an evil data set that makes it run in O(n^2) time. Certain naive implementations have a big problem with this as the evil data is simply already sorted data--add a few items and resort and it goes slow.
I am sure there are other algorithms that are like this but none come to mind right now.
Loren PechtelLoren Pechtel
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How do you calculate the running time using Big-O notatation? | CommonCrawl |
$G_2$ and Geometry
In a recent question Deane Yang mentioned the beautiful Riemannian geometry that comes up when looking at $G_2$. I am wondering if people could expand on the geometry related to the exceptional Lie Groups. I am not precisely sure what I am looking for, but ostensibly there should be answers forth coming from other who have promised such answers. I understand a bit about how the exceptional Lie groups come up historically, and please correct the following if it is incorrect, but when looking at the possible dynkin diagrams you see that there is no reason for $E_6$,$E_7$,$E_8$,$G_2$, and $F_4$ to not occur as root systems. While root systems are geometric, this is not what I am asking about.
lie-groups riemannian-geometry dg.differential-geometry exceptional-groups
Mariano Suárez-Álvarez
Sean TilsonSean Tilson
$\begingroup$ You might enjoy reading the entry on holonomy in wikipedia: en.wikipedia.org/wiki/Holonomy, and especially the section on Berger's classification of possible holonomy groups. $G_2$ is one of the two possible exceptional holonomy groups. (It can appear as the holonomy group of a 7-dim'l manifold, acting on the tangent bundle through its 7 dim'l irrep. If you google exceptional $G_2$ holonomy you will find a lot of literature on this subject.) $\endgroup$ – Emerton Dec 14 '10 at 6:52
$\begingroup$ This survey of Joyce may also help: people.maths.ox.ac.uk/~joyce/cc.ps $\endgroup$ – Emerton Dec 14 '10 at 6:53
$\begingroup$ The recent question mentioned above: mathoverflow.net/questions/49151/… $\endgroup$ – Unknown Dec 14 '10 at 15:30
I promised Sean a detailed answer, so here it is.
As José has already mentioned, it is only $G_2$ (of the five exceptional Lie groups) which can arise as the holonomy group of a Riemannian manifold. Berger's classification in the 1950's could not rule it out, and neither could he rule out the Lie group $\mathrm{Spin}(7)$, but these were generally believed to not possibly be able to exist. However, in the early 1980's Robert Bryant succeeded in proving the existence of local examples (on open balls in Euclidean spaces). Then in the late 1980's Bryant and Simon Salamon found the first complete, non-compact examples of such manifolds, on total spaces of certain vector bundles, using symmetry (cohomogeneity one) methods. (Since then there are many examples of non-compact cohomogeneity one $G_2$ manifolds found by physicists.) Finally, in 1994 Dominic Joyce stunned the mathematical community by proving the existence of hundreds of compact examples. His proof is non-constructive, using hard analysis involving the existence and uniqueness of solutions to a non-linear elliptic equation, much as Yau's solution of the Calabi conjecture gives a non-constructive proof of the existence and uniqueness of Calabi-Yau metrics (holonomy $\mathrm{SU}(n)$ metrics) on Kahler manifolds satisfying certain conditions. (In 2000 Alexei Kovalev found a new construction of compact $G_2$ manifolds that produced several hundred more non-explicit examples. These are the only two known compact constructions to date.) It is exactly this similarity to Calabi-Yau manifolds (and to Kahler manifolds in general) that I will explain.
When it comes to Riemannian holonomy, the aspect of the group $G_2$ which is important is not really that it is one of the five exceptional Lie groups, but rather that it is the automorphism group of the octonions $\mathbb O$, an $8$-dimensional non-associative real division algebra. The octonions come equipped with a positive definite inner product, and the span of the identity element $1$ is called the real octonions while its orthogonal complement is called the imaginary octonions $\mathrm{Im} \mathbb O \cong \mathbb R^7$. This is entirely analogous to the quaternions $\mathbb H$, except that the non-associativity introduces some new complications. In fact the analogy allows us to define a cross product on $\mathbb R^7$ in the same way, as follows. Let $u, v \in \mathbb R^7 \cong \mathrm{Im} \mathbb O$ and define $u \times v = \mathrm{Im}(uv)$, where $uv$ denotes the octonion product. (In fact the real part of $uv$ is equal to $-\langle u, v \rangle$, just as it is for quaternions.) This cross product satisfies the following relations: \begin{equation} u \times v = - v \times u, \qquad \qquad \langle u \times v , u \rangle = 0, \qquad \qquad {|| u\times v||}^2 = {|| u \wedge v ||}^2, \end{equation} exactly like the cross product on $\mathbb R^3 \cong \mathrm{Im} \mathbb H$. However, there is a difference, unlike the cross product in $\mathbb R^3$, the following expression is not zero: \begin{equation} u \times (v \times w) + \langle u, v \rangle w - \langle u, w \rangle v \end{equation} but is instead a measure of the failure of the associativity $(uv)w - u(vw)$, up to a factor. Note that on $\mathbb R^7$ there can be defined a $3$-form (totally skew-symmetric trilinear form) using the cross product as follows: $\varphi(u,v,w) = \langle u \times v, w \rangle$, which is called the associative $3$-form for reasons that we won't get into here.
Digression: In fact one can show that only on $\mathbb R^3$ and $\mathbb R^7$ can one construct such a cross product, and this is intimately related to the fact that only the spheres $S^2$ and $S^6$ can admit almost complex structures. But I digress...
Getting back to $G_2$ geometry: a $7$-dimensional smooth manifold $M$ is said to admit a $G_2$-structure if there is a reduction of the structure group of its frame bundle from $\mathrm{GL}(7, \mathbb R)$ to the group $G_2$ which can actually be viewed naturally as a subgroup of $\mathrm{SO}(7)$. For those familiar with $G$-structures, this tells you that a $G_2$-structure determines a Riemannian metric and an orientation. In fact, one can show on a manifold with $G_2$-structure, there exists a non-degenerate $3$-form $\varphi$ for which, given a point $p$ on $M$, there exists local coordinates near $p$ such that, in those coordinates, at the point $p$, the form $\varphi$ is exactly the associative $3$-form on $\mathbb R^7$ discussed above. Now one can show that there is a way to canonically determine both a metric and an orientation in a highly non-linear way from this $3$-form $\varphi$. Then one can define a cross product $\times$ by essentially using the metric to ``raise an index'' on $\varphi$. In summary, a manifold $(M, \varphi)$ with $G_2$-structure comes equipped with a metric, cross product, $3$-form, and orientation, which satisfy \begin{equation} \varphi(u,v,w) = \langle u \times v , w \rangle. \end{equation} This is exactly analogous to the data of an almost Hermitian manifold, which comes with a metric, an almost complex structure $J$, a $2$-form $\omega$, and an orientation, which satisfy \begin{equation} \omega(u,v) = \langle Ju , v \rangle. \end{equation} Essentially, a manifold admits a $G_2$-structure if one can identify each of its tangent spaces with the imaginary octonions $\mathrm{Im} \mathbb O \cong \mathbb R^7$ in a smoothly varying way, just as an almost Hermitian manifold is one in which we can identify each of its tangent spaces with $\mathbb C^m$ (together with its Euclidean inner product) in a smoothly varying way.
For a manifold to admit a $G_2$-structure, the necessary and sufficient conditions are that it be orientable and spin. (This is equivalent to the vanishing of the first two Stiefel-Whitney classes.) So there are lots of such $7$-manifolds, just as there are lots of almost Hermitian manifolds. But the story does not end there.
Let $(M, \varphi)$ be a manifold with $G_2$-structure. Since it determines a Riemannian metric $g_{\varphi}$, there is an induced Levi-Civita covariant derivative $\nabla$, and one can ask if $\nabla \varphi = 0$? If this is the case, $(M, \varphi)$ is called a $G_2$-manifold, and one can show that the Riemannian holonomy of $g_{\varphi}$ is contained in the group $G_2 \subset \mathrm{SO}(7)$. These are much harder to find, because it involves solving a fully non-linear partial differential equation for the unknown $3$-form $\varphi$. They are in some ways analogous to Kahler manifolds, which are exactly those almost Hermitian manifolds that satisfy $\nabla \omega = 0$, but those are much easier to find. One reason is because the metric $g$ and the almost complex structure $J$ on an almost Hermitian manifold are essentially independent (they just have to satisfy the mild condition of compatibility) whereas in the $G_2$ case, the metric and the cross product are determined non-linearly from $\varphi$. However, the analogy is not perfect, because one can show that when $\nabla \varphi = 0$, the Ricci curvature of $g_{\varphi}$ necessarily vanishes. So $G_2$-manifolds are always Ricci-flat! (This is one reason that physicists are interested in such manifolds---they play a role as ``compactifications'' in $11$-dimensional $M$-theory analogous to the role of Calabi-Yau $3$-folds in $10$-dimensional string theory.) So in some sense $G_2$-manifolds are more like Ricci-flat Kahler manifolds, which are the Calabi-Yau manifolds.
In fact, if we allow the holonomy to be a proper subgroup of $G_2$, there are many examples of $G_2$-manifolds. For example, the flat torus $T^7$, or the product manifolds $T^3 \times CY2$ and $S^1 \times CY3$, where $CYn$ is a Calabi-Yau $n$-fold, have Riemannian holonomy groups properly contained in $G_2$. These are in some sense ``trivial'' examples because they reduce to lower-dimension constructions. The manifolds with full holonomy $G_2$ are also called irreducible $G_2$-manifolds and it is precisely these manifolds that Bryant, Bryant-Salamon, Joyce, and Kovalev constructed.
We are lacking a ``Calabi-type conjecture'' which would give necessary and sufficient conditions for a compact $7$-manifold which admits $G_2$-structures to admit a $G_2$-structure which is parallel ($\nabla \varphi = 0$.) Indeed, we don't even know what the conjecture should be. There are topological obstructions which are known, but we are far from knowing sufficient conditions. In fact, this question is more similar to the following: suppose $M^{2n}$ is a compact, smooth, $2n$-dimensional manifold that admits almost complex structures. What are necessary and sufficient conditions for $M$ to admit Kahler metrics? We certainly know many necessary topological conditions, but (as far as I know, and correct me if I am wrong) we are nowhere near knowing sufficient conditions.
What makes the Calabi conjecture tractable (I almost said easy, of course it is anything but easy) is the fact that we already start with a Kahler manifold (holonomy $\mathrm{U}(m)$ metric) and want to reduce the holonomy by only $1$ dimension, to $\mathrm{SU}(m)$. Then the $\partial \bar \partial$-lemma in Kahler geometry allows us to reduce the Calabi conjecture to a (albeit fully non-linear) elliptic PDE for a single scalar function. Any analogous ``conjecture'' in either the Kahler or the $G_2$ cases would have to involve a system of PDEs, which are much more difficult to deal with.
That's my not-so-short crash course in $G_2$-geometry. I hope some people read all the way to the end of this...
Spiro KarigiannisSpiro Karigiannis
$\begingroup$ To echo emerton, many thanks. Great stuff. $\endgroup$ – Deane Yang Dec 14 '10 at 16:33
$\begingroup$ +1!!! Thanks, Spiro. I knew this question was way the heck up your alley and was hoping (and expecting) you would answer, but this is beyond my wildest dreams. I will read this all the way through, and although it will take me some time and effort to do so, it will definitely be worth it. It seems to me that there is potential here to parlay this answer into a "What is...a $G_2$-manifold?" article. $\endgroup$ – Pete L. Clark Dec 14 '10 at 17:30
$\begingroup$ "We certainly know many necessary topological conditions" -- are there any as simple as the "odd Betti numbers must be even" one for Kahler manifolds? $\endgroup$ – Allen Knutson Dec 14 '10 at 18:53
$\begingroup$ @Allen: We know $b_3$ (and by Poincaré duality also $b_4$) must be $\geq 1$, because the $3$-form (and its dual $4$-form) both represent non-trivial cohomology classes. This is for any $G_2$-manifold. If you insist that the holonomy be strictly $G_2$, and not a propers subgroup, there are 3 more conditions: the fundamental group $\pi_1(M)$ must be finite; there is a symmetric bilinear form on $H_2 (M, \mathbb R)$ (like the intersection form on $4$-manifolds) which must be definite (positive or negative depends on your sign convention), and the first Pontrjagin class $p_1(M)$ must be non-zero. $\endgroup$ – Spiro Karigiannis Dec 14 '10 at 19:03
$\begingroup$ An expanded version of this answer has just appeared in the April 2011 issue of the Notices of the AMS as a "What is ...?" article. ams.org/notices/201104/rtx110400580p.pdf $\endgroup$ – Scott Morrison♦ Mar 30 '11 at 17:14
If you are willing to move away from Riemannian geometry, but still within differential geometry, then $G_2$ arises in a number of surprising and simple geometric circumstances, but it is not the compact $G_2$ of the answers above, but rather the `split' $G_2$. (Each simple Lie algebra has a compact real form and a ``maximally non-compact real form, called its split real form.) Consider the problem of rolling a sphere of radius 1 about on a sphere of radius R. The resulting configuration space is a 5-manifold $M^5$ which is a circle bundle over the product of the two spheres, the circle encoding the relative orientation of the two spheres at their points of tangency. This rolling problem defines a rank 2 distribution (linear sub-bundle of the tangent bundle) on $M^5$.
Theorem: the local symmetry algebra of this distribution is that of the split $G_2$ if and only if $R =3$ or $R =1/3$. If you move to a double cover of $M^5$ then the split $G_2$ acts by symmetries of the lifted distribution, so that this double cover is a homogeneous space for $G_2$.
Gil Bor and I give some details of this theorem, and the action, and how it relates to Cartan's thesis construction of $G_2$ in the paper $G_2$ and the Rolling Distribution'', L'Enseignement Mathematique,
vol. 55, 157-196 (2009), or (arXiv:math/0612469). R. Bryant and L. Hsu detailed several other surprising realizations of $G_2$ including one on the space of all space curves having constant torsion 1 , in their Inventiones paper.
Richard MontgomeryRichard Montgomery
The comments have so far addressed the issue of $G_2$ holonomy, which might be what was at the heart of Deane's answer to the question mentioned by the OP. Alas, of the exceptional Lie groups, only $G_2$ is part of the holonomy fellowship; namely, the possible holonomy groups of (simply-connected, complete) irreducible non-symmetric riemannian manifolds. To see the geometry associated to the other exceptional Lie groups, one needs to look elsewhere.
One kind of geometry associated to the exceptional Lie groups is the geometry of Riemannian symmetric spaces surrounding the Freudenthal-Tits magic square.
Closely related to this is a geometric realisation of the Lie algebras (not the groups, but then again more than just the root systems) of some of the exceptional Lie groups; namely $F_4$ and $E_8$ in some work of mine. (Sorry for the self-promotion.)
The idea was to find a geometrical interpretation to the spinorial construction of $F_4$ and $E_8$ which you can find described in Frank Adams's posthumous book Lectures on exceptional Lie groups by a procedure familiar to practitioners of supergravity: namely, the construction of a natural $\mathbb{Z}/2$-graded Lie (super)algebra from some geometric data, called the KIlling (super)algebra, as it is generated by Killing spinor fields.
The result is that $F_4$ and $E_8$ are the Killing Lie algebras of the round 8- and 15-sphere, respectively.
It is possible that the similar spinorial constructions of $E_6$ and $E_7$ can also be geometrised in this way, but I have not yet worked out the details.
José Figueroa-O'FarrillJosé Figueroa-O'Farrill
$\begingroup$ @José: Very interesting. I did not know about this relation of $F_4$ to $S^8$. I will definitely read this paper. $\endgroup$ – Spiro Karigiannis Dec 14 '10 at 13:56
The compact group $F_4$ is the group of isometries of the octonionic projective plane $\mathbb{OP}^2$ endowed with an analog of Fubini-Study metric. I suspect the other real groups of type $F_4$ are the isometries groups of the octonionic hyperbolic plane and of the analogous objects built from split octonions. (Related question on mathoverflow.) One of the noncompact real forms of $E_6$ is the group of projective transformations (collineations) of $\mathbb{OP}^2$. The groups of type $F_4$ and $E_6$ arise in this context because of their close relationship to the exceptional Jordan algebras of hermitian three by three matrices over octonions. Indeed -- the group $E_6$ preserves the determinant of these matrices and $F_4$ preserves the determinant and the trace.
The most geometric approach to the exceptional groups that I am aware of (and which goes in this direction) is that of Rosenfeld. Unfortunately I don't have that book. He interprets groups of type $E_7$ and $E_8$ in a similar manner for $(\mathbb{C}\otimes\mathbb{O} ) \mathbb{P}^2$ and $(\mathbb{H}\otimes\mathbb{O} ) \mathbb{P}^2$. Some details and introduction to the subject is in Baez.
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$\begingroup$ The notes by Baez are certainly great. I had never heard of this book by Rosenfeld. It sounds fantastic! Thanks. $\endgroup$ – Spiro Karigiannis Jan 10 '11 at 18:04
$\begingroup$ $E_7$ is related to $\mathbb H\otimes\mathbb OP^2$, $E_8$ is related to $\mathbb O \otimes \mathbb OP^2$. See e.g. Baez article on octonions. However there are some doubts whether these are projective spaces. For sure $F_{II}, E_{III}, E_{VI}, E_{VIII}$ of dimensions $16,32,64,128$ are Riemmanian symmetric spaces. $\endgroup$ – Marek Mitros Jan 30 at 11:49
A review on the history of $G_2$ and its relation with the 7-dimensional geometry, is given in the following article of AMS Notices:
314159.314159.
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Irreducibility of root-height generating polynomial | CommonCrawl |
Local zeta function
In number theory, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as
$Z(V,s)=\exp \left(\sum _{m=1}^{\infty }{\frac {N_{m}}{m}}(q^{-s})^{m}\right)$
where V is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements and Nm is the number of points of V defined over the finite field extension Fqm of Fq.[1]
Making the variable transformation u = q−s, gives
${\mathit {Z}}(V,u)=\exp \left(\sum _{m=1}^{\infty }N_{m}{\frac {u^{m}}{m}}\right)$
as the formal power series in the variable $u$.
Equivalently, the local zeta function is sometimes defined as follows:
$(1)\ \ {\mathit {Z}}(V,0)=1\,$
$(2)\ \ {\frac {d}{du}}\log {\mathit {Z}}(V,u)=\sum _{m=1}^{\infty }N_{m}u^{m-1}\ .$
In other words, the local zeta function Z(V, u) with coefficients in the finite field Fq is defined as a function whose logarithmic derivative generates the number Nm of solutions of the equation defining V in the degree m extension Fqm.
Formulation
Given a finite field F, there is, up to isomorphism, only one field Fk with
$[F_{k}:F]=k\,$,
for k = 1, 2, ... . Given a set of polynomial equations — or an algebraic variety V — defined over F, we can count the number
$N_{k}\,$
of solutions in Fk and create the generating function
$G(t)=N_{1}t+N_{2}t^{2}/2+N_{3}t^{3}/3+\cdots \,$.
The correct definition for Z(t) is to set log Z equal to G, so
$Z=\exp(G(t))\,$
and Z(0) = 1, since G(0) = 0, and Z(t) is a priori a formal power series.
The logarithmic derivative
$Z'(t)/Z(t)\,$
equals the generating function
$G'(t)=N_{1}+N_{2}t^{1}+N_{3}t^{2}+\cdots \,$.
Examples
For example, assume all the Nk are 1; this happens for example if we start with an equation like X = 0, so that geometrically we are taking V to be a point. Then
$G(t)=-\log(1-t)$
is the expansion of a logarithm (for |t| < 1). In this case we have
$Z(t)={\frac {1}{(1-t)}}\ .$
To take something more interesting, let V be the projective line over F. If F has q elements, then this has q + 1 points, including the one point at infinity. Therefore, we have
$N_{k}=q^{k}+1$
and
$G(t)=-\log(1-t)-\log(1-qt)$
for |t| small enough, and therefore
$Z(t)={\frac {1}{(1-t)(1-qt)}}\ .$
The first study of these functions was in the 1923 dissertation of Emil Artin. He obtained results for the case of a hyperelliptic curve, and conjectured the further main points of the theory as applied to curves. The theory was then developed by F. K. Schmidt and Helmut Hasse.[2] The earliest known nontrivial cases of local zeta functions were implicit in Carl Friedrich Gauss's Disquisitiones Arithmeticae, article 358. There, certain particular examples of elliptic curves over finite fields having complex multiplication have their points counted by means of cyclotomy.[3]
For the definition and some examples, see also.[4]
Motivations
The relationship between the definitions of G and Z can be explained in a number of ways. (See for example the infinite product formula for Z below.) In practice it makes Z a rational function of t, something that is interesting even in the case of V an elliptic curve over finite field.
The local Z zeta functions are multiplied to get global $\zeta $ zeta functions,
$\zeta =\prod Z$
These generally involve different finite fields (for example the whole family of fields Z/pZ as p runs over all prime numbers).
In these fields, the variable t is substituted by p−s, where s is the complex variable traditionally used in Dirichlet series. (For details see Hasse–Weil zeta function.)
The global products of Z in the two cases used as examples in the previous section therefore come out as $\zeta (s)$ and $\zeta (s)\zeta (s-1)$ after letting $q=p$.
Riemann hypothesis for curves over finite fields
For projective curves C over F that are non-singular, it can be shown that
$Z(t)={\frac {P(t)}{(1-t)(1-qt)}}\ ,$
with P(t) a polynomial, of degree 2g, where g is the genus of C. Rewriting
$P(t)=\prod _{i=1}^{2g}(1-\omega _{i}t)\ ,$
the Riemann hypothesis for curves over finite fields states
$|\omega _{i}|=q^{1/2}\ .$
For example, for the elliptic curve case there are two roots, and it is easy to show the absolute values of the roots are q1/2. Hasse's theorem is that they have the same absolute value; and this has immediate consequences for the number of points.
André Weil proved this for the general case, around 1940 (Comptes Rendus note, April 1940): he spent much time in the years after that writing up the algebraic geometry involved. This led him to the general Weil conjectures. Alexander Grothendieck developed scheme theory for the purpose of resolving these. A generation later Pierre Deligne completed the proof. (See étale cohomology for the basic formulae of the general theory.)
General formulas for the zeta function
It is a consequence of the Lefschetz trace formula for the Frobenius morphism that
$Z(X,t)=\prod _{i=0}^{2\dim X}\det {\big (}1-t{\mbox{Frob}}_{q}|H_{c}^{i}({\overline {X}},{\mathbb {Q} }_{\ell }){\big )}^{(-1)^{i+1}}.$
Here $X$ is a separated scheme of finite type over the finite field F with $q$ elements, and Frobq is the geometric Frobenius acting on $\ell $-adic étale cohomology with compact supports of ${\overline {X}}$, the lift of $X$ to the algebraic closure of the field F. This shows that the zeta function is a rational function of $t$.
An infinite product formula for $Z(X,t)$ is
$Z(X,t)=\prod \ (1-t^{\deg(x)})^{-1}.$
Here, the product ranges over all closed points x of X and deg(x) is the degree of x. The local zeta function Z(X, t) is viewed as a function of the complex variable s via the change of variables q−s.
In the case where X is the variety V discussed above, the closed points are the equivalence classes x=[P] of points P on ${\overline {V}}$, where two points are equivalent if they are conjugates over F. The degree of x is the degree of the field extension of F generated by the coordinates of P. The logarithmic derivative of the infinite product Z(X, t) is easily seen to be the generating function discussed above, namely
$N_{1}+N_{2}t^{1}+N_{3}t^{2}+\cdots \,$.
See also
• List of zeta functions
• Weil conjectures
• Elliptic curve
References
1. Section V.2 of Silverman, Joseph H. (1992), The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, New York: Springer-Verlag, ISBN 978-0-387-96203-0, MR 1329092
2. Daniel Bump, Algebraic Geometry (1998), p. 195.
3. Barry Mazur, Eigenvalues of Frobenius, p. 244 in Algebraic Geometry, Arcata 1974: Proceedings American Mathematical Society (1974).
4. Robin Hartshorne, Algebraic Geometry, p. 449 Springer 1977 APPENDIX C "The Weil Conjectures"
Bernhard Riemann
• Cauchy–Riemann equations
• Generalized Riemann hypothesis
• Grand Riemann hypothesis
• Grothendieck–Hirzebruch–Riemann–Roch theorem
• Hirzebruch–Riemann–Roch theorem
• Local zeta function
• Measurable Riemann mapping theorem
• Riemann (crater)
• Riemann Xi function
• Riemann curvature tensor
• Riemann hypothesis
• Riemann integral
• Riemann invariant
• Riemann mapping theorem
• Riemann form
• Riemann problem
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• Riemann–Hilbert problems
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| Wikipedia |
\begin{document}
\title{Irreversibility of Entanglement Concentration for Pure State}
\author{Wataru Kumagai$^{1,2}$~~~Masahito Hayashi$^{2,3}$\\ \textit{${}^1$Graduate School of Information Sciences, Tohoku University, Japan \\${}^2$Graduate School of Mathematics, Nagoya University, Japan,\\${}^3$Centre for Quantum Technologies, National University of Singapore, Singapore}} \date{\today}
\begin{abstract} For a pure state $\psi$ on a composite system $\mathcal{H}_A\otimes\mathcal{H}_B$, both the entanglement cost $E_C(\psi)$ and the distillable entanglement $E_D(\psi)$ coincide with the von Neumann entropy $H(\mathrm{Tr}_{B}\psi)$. Therefore, the entanglement concentration from the multiple state $\psi^{\otimes n}$ of a pure state $\psi$ to the multiple state $\Phi^{\otimes L_n}$ of the EPR state $\Phi$ seems to be able to be reversibly performed with an asymptotically infinitesimal error when the rate ${L_n}/{n}$ goes to $H(\mathrm{Tr}_{B}\psi)$. In this paper, we show that it is impossible to reversibly perform the entanglement concentration for a multiple pure state even in asymptotic situation. In addition, in the case when we recover the multiple state $\psi^{\otimes M_n}$ after the concentration for $\psi^{\otimes n}$, we evaluate the asymptotic behavior of the loss number $n-M_n$ of $\psi$. This evaluation is thought to be closely related to the entanglement compression in distant parties.
\end{abstract}
\pacs{03.65.Wj, 03.65.Ud} \maketitle
The entanglement is an essential resource to apply important quantum processing such as the quantum teleportation and the superdense coding. Then, since those protocols often require a suitable entangled quantum state between distant parties, we need a method transforming a given entangled state to a target entangled state. As typical methods, we focus on the entanglement concentration and dilution in this paper. For a pure state $\psi$ on a composite system $\mathcal{H}_A\otimes\mathcal{H}_B$, the optimal rate $E_D(\psi)$ of the entanglement concentration is called the distillable entanglement, that is, $E_D(\psi)$ is determined by the supremum of the limit of the ratio $\mathrm{lim}\frac{L_n}{n}$ when the multiple state $\Phi^{\otimes L_n}$ of the EPR state $\Phi$ on ${\mathbb{C}}^2\otimes{\mathbb{C}}^2$ can be produced from a multiple pure state $\psi^{\otimes n}$ with an asymptotically infinitesimal error. Similarly, the optimal rate $E_C(\psi)$ of the entanglement dilution is called the entanglement cost, that is, $E_C(\psi)$ is determined by the infimum of the limit of the ratio $\mathrm{lim}\frac{L_n}{n}$ when a multiple pure state $\psi^{\otimes n}$ can be produced from the multiple EPR state $\Phi^{\otimes L_n}$ with an asymptotically infinitesimal error. Those values coincide with each other and are characterized by the von Neumann entropy $H$ as $E_D(\psi)=E_C(\psi)=H(\mathrm{Tr}_{B}\psi)$\cite{BBPS, HHT}. Especially, it is known that the entanglement concentration and dilution with the optimal rate $\lim {L_n}/{n}=H(\mathrm{Tr}_B \psi)$ are realizable with an asymptotically infinitesimal error, respectively \cite{BBPS}. Thus, by using those entanglement concentration and dilution, it seems that we can perform the entanglement concentration and recover the initial state with an asymptotically infinitesimal error, and thus, the entanglement concentration is reversible in the asymptotic situation. In general, for an arbitrary entangled state $\rho$ which is not necessarily pure on a composite system, the asymptotic reversibility of the entanglement distillation for a multiple state $\rho^{\otimes n}$ is defined by the agreement of the distillable entanglement $E_D(\rho)$ and the entanglement cost $E_C(\rho)$ \cite{VC, YHHS}.
However, it has not been studied whether both the entanglement concentration and the subsequent recovery operation can be performed with an asymptotically infinitesimal error when the distillable entanglement and the entanglement cost coincide for an entangled state. As is well known, in the non-asymptotic case, the general LOCC transformation has been studied intensively \cite{Nie, JP, LS, Vid}, and as the special case, if a pure state which is different from EPR states can be transformed to EPR states by LOCC without a error, then the initial state can not be exactly recovered by LOCC. Similarly, in the asymptotic case, when the distillable entanglement and the entanglement cost differ for a mixed entangled state $\rho$, it is known that we can not perform both the entanglement distillation and the recovery operation for a multiple state $\rho^{\otimes n}$ with an asymptotically infinitesimal error \cite{VC, YHHS}. On the other hand, in the asymptotic case, it had not been properly discussed until now whether we can perform those operations for a multiple state $\psi^{\otimes n}$ of a pure state $\psi$ with an asymptotically infinitesimal error. To investigate the point, we precisely treat the recovery operation that means the operation to reconstitute the initial state $\psi^{\otimes n}$ (or more generally, a multiple state $\psi^{\otimes M_n}$) with some error after the entanglement concentration, and consider the errors for the entanglement concentration and the recovery operation in this paper. In particular, we focus on the asymptotic behavior of the sum of the errors for a pure entangled state and show the incompatibility between the entanglement concentration and the recovery operation, which implies the irreversibility of the entanglement concentration even in the asymptotic case.
As an application, we evaluate the loss of entanglement for the initial state when the entanglement concentration is used as the entanglement compression. In this setting, the entanglement concentration compresses a multiple entangled state $\psi^{\otimes n}$ into a less dimensional quantum system, and the recovery operation decompresses a multiple state $\psi^{\otimes M_n}$ of $\psi$ with a slight error for large $n$ as is shown in FIG.1. As stated above, the entanglement concentration is irreversible, and hence we can not completely reconstitute the initial state $\psi^{\otimes n}$ after the entanglement concentration with an asymptotically infinitesimal error. Then, we investigate how many copies of $\psi$ vanish at the recovery operation after the concentration when small error $\epsilon$ is permitted. In other words, when we reproduce the multiple state $\psi^{\otimes M_n}$ after the concentration for $\psi^{\otimes n}$, we evaluate the asymptotic behavior of the minimal loss number $n-M_n$ of $\psi$ depending on a permissible error $\epsilon$.
\begin{figure}
\caption{The entanglement concentration $C_n$ for a multiple pure state $\psi^{\otimes n}$ and the recovery operation $D_n$ of the initial state from the concentrated state $C_n(\psi^{\otimes n})$. $e_C$ represents the concentration error between the target e-bits $\Phi^{\otimes L_n}$ and the concentrated state $C_n(\psi^{\otimes n})$, and $e_R$ does the recovery error between the multiple state $\psi^{\otimes M_n}$ and the restored state $D_n\circ C_n(\psi^{\otimes n})$, respectively.}
\label{figure2}
\end{figure}
This paper is organized as follows. At first, we introduce an error concerning the entanglement concentration and the recovery operation. Next, we prove the asymptotic incompatibility between the concentration and the recovery, in particular, which implies the irreversibility of the entanglement concentration for an arbitrary pure state $\psi$ except maximally entangled states in the asymptotic situation. Next, we evaluate the asymptotic behavior of the minimum loss number $n-M_n$ of $\psi$ at the recovery. Finally, we summarize our results and give some remarks. For all theorems in this paper, we give the proofs in Supplemental Material.
~
\noindent{\it {Minimum Concentration-Recovery Error:}}~ In the non-asymptotic case, it is known that the entanglement concentration is not reversible \cite{Vid}. On the other hand, in the asymptotic case, it had been thought that the entanglement concentration for a multiple pure state $\psi^{\otimes n}$ is reversible because there exists a pair of the entanglement concentration and dilution protocol with an asymptotically infinitesimal error and with the common optimal rate \cite{BPRST, VC, YHHS,JP}. However, the argument is actually not precise. To clarify the defect of the argument and show the irreversibility of the entanglement concentration in the asymptotic case, we review the operations to implement the entanglement concentration and dilution.
Let $\mathcal{S}(\mathcal{H})$ be the set of all quantum states on a quantum system $\mathcal{H}$. For LOCC transformations $C_n:\mathcal{S}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})^{\otimes n}\to\mathcal{S}({\mathbb{C}}^2\otimes{\mathbb{C}}^2)^{\otimes L_n}$ and $D_n:\mathcal{S}({\mathbb{C}}^2\otimes{\mathbb{C}}^2)^{\otimes L'_n}\to\mathcal{S}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})^{\otimes M_n}$, we call $(L_n,C_n)$ and $(L'_n,M_n,D_n)$ the concentration and dilution map, respectively. In a concentration map, $L_n$ means the number of the e-bits which we generate from the initial state $\psi^{\otimes n}$. Similarly, in a dilution map, $L'_n$ means the number of the e-bits which we want to generate the target state $\psi^{\otimes M_n}$ from. In particular, given a concentration map $(L_n,C_n)$ and a dilution map $(L'_n,M_n,D_n)$, only if $L_n$ equals $L'_n$, we can provide a recovery operation $D_n$ after the entanglement concentration $C_n$. In the situation, we call a quadruplet $(L_n,M_n,C_n,D_n)$ a concentration-recovery map. At the existing discussion about the reversibility of the entanglement concentration \cite{BPRST, VC, YHHS,JP}, only the consistency of the asymptotic rates for the concentration and dilution has been focused and the difference between $L_n$ and $L'_n$ has been missed. In other words, the numbers $L_n$ and $L'_n$ of e-bits have been separately taken under the condition $\lim L_n/n=\lim L'_n/n=H(\mathrm{Tr}_B\psi)$ in the entanglement concentration $C_n$ and dilution $D_n$ although $D_n\circ C_n$ can not be defined. In the following, we treat concentration-recovery maps as pairs of concentration and dilution map with the common number $L_n=L'_n$.
We prepare to introduce an error criteria for a concentration-recovery map. By using the Fidelity $F(\rho,\sigma):=\mathrm{Tr\sqrt{\sqrt{\rho}{\sigma}\sqrt{\rho}}}$, we define the error function $d(\rho,\sigma):={1-F^2(\rho,\sigma)}$. Then we can qualify the entanglement concentration error and the recovery error by \begin{eqnarray} &e_C^n(\psi,L_n,C_n):=d(C_n(\psi^{\otimes n}),\Phi^{\otimes L_n}),&\nonumber\\ &e_R^n(\psi,L_n,M_n,C_n,D_n):=d(\psi^{\otimes M_n},D_n\circ C_n(\psi^{\otimes n})),&\nonumber \end{eqnarray} respectively. In the following, we focus on the condition \begin{eqnarray}\label{error.condition1} &\lim e_C^n(\psi,L_n,C_n)=0,& \end{eqnarray} for a sequence of concentration maps $\{(L_n,C_n)\}_n$. Here, a sequence satisfying $(\ref{error.condition1})$ means the entanglement concentration with an asymptotic infinitesimal error. Similarly, we also focus on the condition \begin{eqnarray}\label{error.condition2} &\lim e_R^n(\psi,L_n,n,C_n,D_n)=0& \end{eqnarray} for a sequence of concentration-recovery maps $\{(L_n,n,C_n,D_n)\}_n$ when $M_n=n$, and a sequence satisfying $(\ref{error.condition2})$ means the recovery operation with an asymptotic infinitesimal error. Our concern is the compatibility between the entanglement concentration and the recovery operation. In other words, we want to know whether we can carry out both the entanglement concentration and the recovery operation, that is, whether there exists a sequence of concentration-recovery maps satisfying both conditions (\ref{error.condition1}) and (\ref{error.condition2}). Here, let us introduce the significant indicator concerning the compatibility, and call it the minimum concentration-recovery error (MCRE): \begin{eqnarray} &&\delta_n(\psi,M_n) \nonumber\\ &&:=\displaystyle\min_{L_n,C_n,D_n} e_C^n(\psi,L_n,C_n)+e_R^n(\psi,L_n,M_n,C_n,D_n)\nonumber \end{eqnarray} where $L_n$ runs over positive integers, and $C_n:\mathcal{S}(\mathcal{H}_A\otimes\mathcal{H}_B)^{\otimes n}\to\mathcal{S}({\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2})^{\otimes L_n}$ and $D_n:\mathcal{S}({\mathbb{C}}^{2}\otimes{\mathbb{C}}^{2})^{\otimes L_n}\to\mathcal{S}(\mathcal{H}_A\otimes\mathcal{H}_B)^{\otimes M_n}$ run over LOCC operations. Then, if there exist the entanglement concentration and the recovery operation simultaneously satisfying $(\ref{error.condition1})$ and $(\ref{error.condition2})$, $\delta_n(\psi,n)$ goes to $0$. Obviously, the converse is correct. Thus, the equation $\lim\delta_n(\psi,n)=0$ corresponds to the compatibility of the entanglement concentration and the recovery with an asymptotically infinitesimal error. In the following, we treat the case of $M_n=n$ and denote $\delta_n(\psi,n)$ simply by $\delta_n(\psi)$.
To evaluate MCRE, we focus on the minimal error concerning the LOCC transformation between $\rho\in\mathcal{S}(\mathcal{H}_A\otimes\mathcal{H}_B)$ and $\sigma\in\mathcal{S}(\mathcal{H}'_{A}\otimes\mathcal{H}'_{B})$: \begin{eqnarray} &d(\rho\to\sigma) :=\displaystyle\min_{E}d(E(\rho),\sigma)) =1-\displaystyle\max_{E}F^2(E(\rho),\sigma),& \nonumber \end{eqnarray} where $E:\mathcal{S}(\mathcal{H}_A\otimes\mathcal{H}_B)\to\mathcal{S}(\mathcal{H}'_A\otimes\mathcal{H}'_B)$ runs over LOCC transformation. Then, we get the following equation for $M_n\le n$. \begin{eqnarray}\label{eq} &&\delta_n(\psi,M_n)\nonumber\\ &&= \min_{L_n\in{\mathbb{Z}}_{>0}} d(\psi^{\otimes n}\to\Phi^{\otimes {L_n}}) + d(\Phi^{\otimes {L_n}}\to\psi^{\otimes M_n}). \end{eqnarray} The proof is given in Supplemental Material. The equality (\ref{eq}) associates MCRE with the minimal error of the concentration and the dilution, and plays an essential role to show the incompatibility of the entanglement concentration and the recovery operation in the asymptotic situation. To evaluate MCRE, we focus on the right hand side in (\ref{eq}).
~
\noindent{\it {Incompatibility between Entanglement Concentration and Recovery Operation:}}~The entanglement concentration for a multiple pure state $\psi^{\otimes n}$ can be performed with an asymptotically infinitesimal error if the rate $\lim{L_n}/{n}$ of the concentration is less than the distillable entanglement $E_D(\psi)=H(\mathrm{Tr}_B \psi)$ \cite{BBPS}. Moreover, Hayashi et al. \cite{HKMMW} showed the strong converse of the entanglement concentration, that is, the error $d(\psi^{\otimes n}\to\Phi^{\otimes L_n})$ goes to $1$ if the rate $\lim{L_n}/{n}$ is strictly greater than $H(\mathrm{Tr}_B \psi)$. Therefore, the asymptotic behavior of the error $d(\psi^{\otimes n}\to\Phi^{\otimes L_n})$ in (\ref{eq}) is completely analyzed if the rate $\lim{L_n}/{n}$ is not $H(\mathrm{Tr}_B \psi)$. On the other hand, when the rate $\lim{L_n}/{n}$ strictly equals $H(\mathrm{Tr}_B \psi)$, the error $d(\psi^{\otimes n}\to\Phi^{\otimes L_n})$ has not been studied sufficiently. To investigate it, we treat $L_n$ which can be expanded as $L_n=an+b\sqrt{n}+o(\sqrt{n})$, and focus on the coefficients $a$ and $b$, which are called the first and second order rates in information theory, respectively. For $L_n=an+b\sqrt{n}$ and $M_n=n+b'\sqrt{n}$, we get the following theorem.
\begin{thm}\label{2-order} The equations \begin{eqnarray} &&{\lim}d(\psi^{\otimes n} \to \Phi^{\otimes an+b\sqrt{n}})\nonumber\\ &&=\left\{ \begin{array}{ll} 0 & \mathrm{if}~a<H(\mathrm{Tr}_B{\psi}) \\ G\left(\frac{b}{\sqrt{V(\mathrm{Tr}_B{\psi})}}\right) & \mathrm{if}~a=H(\mathrm{Tr}_B{\psi}) \\ 1 & \mathrm{if}~a>H(\mathrm{Tr}_B{\psi}), \end{array} \right.\\ &&{\lim}d(\Phi^{\otimes an+b\sqrt{n}} \to \psi^{\otimes n+b'\sqrt{n}})\nonumber\\ &&=\left\{ \begin{array}{ll} 1 & \mathrm{if}~a<H(\mathrm{Tr}_B{\psi}) \\ 1-G\left(\frac{b-H(\mathrm{Tr}_B\psi)b'}{\sqrt{V(\mathrm{Tr}_B{\psi})}}\right) & \mathrm{if}~a=H(\mathrm{Tr}_B{\psi}) \\ 0 & \mathrm{if}~a>H(\mathrm{Tr}_B{\psi}) \end{array} \right. \end{eqnarray} hold for any pure state $\psi\in\mathcal{H}_A\otimes\mathcal{H}_B$ except maximally entangled states, where $G(x):=\displaystyle\int_{-\infty}^{x}\frac{1}{\sqrt{2\pi}}\mathrm{e}^{-\frac{x^2}{2}}dx$ and $V(\rho):=\mathrm{Tr}\rho(-\mathrm{log}\rho-H(\rho))^2$. \end{thm} When $\psi$ is a maximally entangled state, $V(\mathrm{Tr}_B{\psi})$ equals $0$ and ${x}/{\sqrt{V(\mathrm{Tr}_B{\psi})}}$ can not be defined in ${\mathbb{R}}$. However, we can extend Theorem \ref{2-order} for a maximally entangled state by replacing $G(x/\sqrt{V(\mathrm{Tr}_B{\psi})})$ by $0$ or $1$ when $x$ is non-positive or positive, respectively. Theorem \ref{2-order} describes the asymptotic behavior of the errors for the entanglement concentration and dilution. As you can see from the proof, even if $L_n$ has lower order term as $L_n=an+b\sqrt{n}+o(\sqrt{n})$ (e.g. $o(\sqrt{n})=\log n$), the order does not affect the above errors. Hence, when we want to analyze the errors of the entanglement concentration and dilution, we only have to treat the first and second order rate. From Theorem \ref{2-order}, we get the following theorem.
\begin{thm}\label{MCREineq}
$\displaystyle\underline{\lim}\delta_n(\psi) =1$ holds for any pure state $\psi\in\mathcal{H}_A\otimes\mathcal{H}_B$ except maximally entangled states. \end{thm}
By Theorem \ref{MCREineq}, far from satisfying $\lim\delta_n(\psi)=0$, MCRE converges to $1$. Therefore, we can not perform both the entanglement concentration and the recovery even if we permit some error $0<\epsilon<1$ for the limit of MCRE as $\overline{\lim}\delta_n(\psi)\le\epsilon$. Here, it turned out that there does not exist a sequence of concentration-recovery maps satisfying both (\ref{error.condition1}) and (\ref{error.condition2}) although there exist a sequence of concentration maps $(L_n,C_n)$ satisfying the condition (\ref{error.condition1}) and a sequence of concentration-recovery maps $(L'_n,n,C'_n,D'_n)$ satisfying the condition (\ref{error.condition2}) with the common first order rates $\lim L_n/n=\lim L'_n/n=H(\mathrm{Tr_B \psi})$. The fact may look strange, but can be comprehended by the argument of the second order rates. That is, those $L_n$ and $L'_n$ actually have different second order rates. The proof at Theorem \ref{MCREineq} is given by using the argument of the second order rate of Theorem \ref{2-order}.
~
\noindent{\it {Loss Evaluation for Recovery Operation:}}~Let us consider the entanglement concentration for the initial state $\psi^{\otimes n}$ and the subsequent recovery operation of the multiple state $\psi^{\otimes M_n}$ satisfying the condition $\overline{\lim}\delta_n(\psi,M_n)\le\epsilon$ for $0<\epsilon<1$. Then, we can not take $M_n$ as $n$ due to Theorem \ref{MCREineq}. If we use the entanglement concentration to compress an entanglement state $\psi^{\otimes n}$ into a less dimensional quantum system, it is significant to know how many copies $n-M_n$ are inevitably lost in the concentration and recovery process. Thus, let us evaluate the rate of loss $n-M_n$ in the asymptotic situation. We focus on the following value for $0<\epsilon<1$ and call it the recovery rate for the entanglement concentration: \begin{eqnarray}
R(\psi,\epsilon):=\inf_{\{M_n\}}\left\{ \overline{\lim}\frac{n-M_n}{\sqrt{n}}\Big| \overline{\lim}\delta_n(\psi,M_n)\le\epsilon \right\}\nonumber \end{eqnarray} The recovery rate means the minimum coefficient of the order of $\sqrt{n}$ of the loss $n-M_n$. \begin{thm}\label{2nd.rate} \begin{eqnarray} R(\psi,\epsilon)=\frac{2\sqrt{V(\mathrm{Tr}_B\psi)}}{H(\mathrm{Tr}_B\psi)}G^{-1}\left(1-\frac{\epsilon}{2}\right) \end{eqnarray} \end{thm} Since $R(\psi,\epsilon)$ is a finite real number for $0<\epsilon<1$, the minimum loss $n-M_n$ of copies can be approximated by $R(\psi,\epsilon)\sqrt{n}$ for large enough $n$ when we perform a suitable entanglement concentration for the initial state $\psi^{\otimes n}$ and recovery operation into the multiple state $\psi^{\otimes M_n}$ with some error $\epsilon$. On the other hand, $R(\psi,\epsilon)$ diverges as $\epsilon$ goes to $0$. Therefore, unlike the case $0<\epsilon<1$, the loss $n-M_n$ increases faster than the order of $\sqrt{n}$ for $\epsilon=0$ in the concentration and recovery process.
~
\noindent{\it {Conclusion:}}~In this paper, we treated the entanglement concentration for a pure state. In existing researches, it has been thought that the initial state can be recovered after the concentration if we perform the concentration with the optimal rate in the asymptotic case. In the argument, the entanglement concentration and dilution have been separately considered although we can not independently perform the concentration and dilution. By simultaneously treating those operations and analyzing the error induced from the concentration and the recovery of the initial state, it was shown that the sum of the errors is greater than or equal to $1$ as is represented in Theorem \ref{MCREineq}. In particular, when entanglement concentration with an asymptotically infinitesimal error is performed, the recovery error goes to $1$ and it is concluded that the initial state can not be recovered. When we use the entanglement concentration to compress a multiple entangled state $\psi^{\otimes n}$ into a less dimensional quantum system, we derived the asymptotic minimum loss of copies of $\psi$ depending on the permissible error $\epsilon$. As the research relating to the irreversibility of the entanglement concentration, it is conjectured that the LOCC transformation between multiple states of general (pure) states $\psi$ and $\phi$ is irreversible in the asymptotic situation. But it is still an open problem.
We note that the mathematical structures of the entanglement concentration and the recovery operation in quantum information theory are similar to uniform random number generation and source coding in classical information theory. Uniform random number generation treats the way to generate uniform distribution $P^U_{L_n}$ whose support has size $L_n$ by transforming i.i.d. random distribution $P^{\otimes n}$, and source coding does the way to compress data to be able to recover the initial data, respectively. For those problems, it is known that the initial distribution $P^{\otimes n}$ can not be recovered from the transformed distribution after $P^U_{L_n}$ was asymptotically generated by a suitable transformation for $P^{\otimes n}$ when the error is measured by the variational distance \cite{Hay1}. In other words, uniform random number generation is incompatible with source coding. Thus, it can be said that Theorem \ref{MCREineq} corresponds to the incompatibility between uniform random number generation and source coding in classical information theory.
\noindent{\it { Acknowledgment:}}~
WK acknowledges support from Grant-in-Aid for JSPS Fellows No. 233283. MH is partially supported by a MEXT Grant-in-Aid for Scientific Research (A) No. 23246071. The Center for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation as part of the Research Centres of Excellence programme.
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\end{thebibliography}
\begin{center}{\textbf{SUPPLEMENTAL MATERIAL}} \end{center}
Let us prepare to show the equation (\ref{eq}). For an arbitrary pure state $\psi\in\mathcal{H}_A\otimes\mathcal{H}_B$, we denote the Schmidt coefficients of $\psi$ by $p_{\psi}=(p_{\psi,1},\cdot\cdot\cdot,p_{\psi,M})$. Let $\Phi_{L}=\sum_{i=1}^L \sqrt{1/L}|i\rangle |i\rangle$ be a maximally entangled state with the size $L$ on a quantum system $\mathcal{H}'_A\otimes\mathcal{H}'_B$, which satisfies $L=\min\{{{\rm dim}}\mathcal{H}'_A,{{\rm dim}}\mathcal{H}'_B\}$. When $p^{\downarrow}$ represents the probability distribution which is sorted for the components of a probability distribution $p$ in decreasing order, we define the pure state $\eta_{\psi,L}$ in $\mathcal{H}'_A\otimes\mathcal{H}'_B$ as \begin{eqnarray} \eta_{\psi,L}=
\displaystyle\sum_{i=1}^{J_{\psi,L}-1}\sqrt{p_{\psi,i}^{\downarrow}}|i\rangle |i\rangle
+ \sqrt{\frac{\sum_{j=J_{\psi,L}}^M p_{\psi,j}^{\downarrow}}{L+1-J_{\psi,L}}}\displaystyle\sum_{i=J_{\psi,L}}^L|i\rangle |i\rangle\nonumber \end{eqnarray} by using \begin{eqnarray}
J_{\psi,L}:=\max\{1\}\cup\left\{2\le j \le L \Big|\frac{\sum_{i=j}^M p_{\psi,i}^{\downarrow}}{L+1-j}<p_{\psi,j-1}^{\downarrow}\right\}.\nonumber \end{eqnarray} Then, there exists a suitable LOCC map to transform $\psi$ to $\eta_{\psi,L}$, and we can get the following equation: \begin{eqnarray}\label{Ceq} \max_{C:LOCC} F(C(\psi),\Phi_L)=F(\eta_{\psi,L},\Phi_L) \end{eqnarray} where $C:\mathcal{S}(\mathcal{H}_A\otimes\mathcal{H}_B)\to\mathcal{S}(\mathcal{H}'_A\otimes\mathcal{H}'_B)$ runs over LOCC maps. Similarly, when we define the pure state $\zeta_{\psi,L}$ in $\mathcal{H}_A\otimes\mathcal{H}_B$ as \begin{eqnarray} \zeta_{\psi,L}=
\sqrt{\sum_{i=1}^{L}p_{\psi,i}^{\downarrow}}^{~-1}\displaystyle\sum_{i=1}^{L}\sqrt{p_{\psi,i}^{\downarrow}}|i\rangle |i\rangle, \nonumber \end{eqnarray}
there exists a suitable LOCC map to transform $\Phi_L$ to $\zeta_{\psi,L}$, and the following equation holds as is shown in \cite{VJN}: \begin{eqnarray}\label{Deq} \displaystyle\max_{D:LOCC} F(\psi,D(\Phi_L)) =F(\psi,\zeta_{\psi,L})=\sqrt{\sum_{i=1}^{L}p_{\psi,i}^{\downarrow}} \end{eqnarray} where $D:\mathcal{S}(\mathcal{H}'_A\otimes\mathcal{H}'_B)\to\mathcal{S}(\mathcal{H}_A\otimes\mathcal{H}_B)$ run over LOCC maps. Moreover, we easily get the equation \begin{eqnarray}\label{minequation} \displaystyle\max_{C,D:LOCC}F(\psi, D\circ C(\psi)) =\displaystyle\max_{D:LOCC} F(\psi,D(\Phi_L)), \end{eqnarray} where $C:\mathcal{S}(\mathcal{H}_A\otimes\mathcal{H}_B)\to\mathcal{S}(\mathcal{H}'_A\otimes\mathcal{H}'_B)$ and $D:\mathcal{S}(\mathcal{H}'_A\otimes\mathcal{H}'_B)\to\mathcal{S}(\mathcal{H}_A\otimes\mathcal{H}_B)$ run over LOCC maps.
~
\noindent[Proof of (\ref{eq})]~Due to (\ref{minequation}), \begin{eqnarray} \delta_n(\psi,M_n) \ge \min_{L_n\in{\mathbb{Z}}_{>0}} d(\psi^{\otimes n}\to\Phi^{\otimes {L_n}})+d(\Phi^{\otimes {L_n}}\to\psi^{\otimes M_n})\nonumber \end{eqnarray} holds without any condition for $M_n$. Next, we prove the converse inequality for $M_n\le n$. Let us fix an arbitrary $L_n\in{\mathbb{Z}}_{>0}$. Since there exists a suitable LOCC map from $\eta_{\psi^{\otimes n},2^{L_n}}$ to $\zeta_{\psi^{\otimes M_n},2^{L_n}}$ when $M_n$ is less than or equal to $n$, we get \begin{eqnarray}\label{inequality} &&\delta_n(\psi,M_n)\nonumber\\ &&\le d(\eta_{\psi^{\otimes n},2^{L_n}},\Phi^{\otimes L_n})+d(\psi^{\otimes M_n},\zeta_{\psi^{\otimes M_n},2^{L_n}})\nonumber\\ &&\le d(\psi^{\otimes n}\to\Phi^{\otimes {L_n}})+d(\Phi^{\otimes {L_n}}\to\psi^{\otimes M_n}). \end{eqnarray} Here, we used (\ref{Ceq}) and (\ref{Deq}) to show the inequality (\ref{inequality}). $\blacksquare$
~
\noindent[Proof of Theorem \ref{2-order}]~ We introduce the following values for a sequence $\overline{\rho}=\{\rho_n\}_n$ of general quantum states. \begin{eqnarray}
&\overline{K}(a,b|\overline{\rho}):=\overline{\rm lim}{\rm Tr}\rho_n\{-{\rm log}\rho_n\le an+b\sqrt{n}\}&\nonumber\\
&\underline{K}(a,b|\overline{\rho}):=\underline{\rm lim}{\rm Tr}\rho_n\{-{\rm log}\rho_n\le an+b\sqrt{n}\}&\nonumber \end{eqnarray} When $\psi_n$ is an arbitrary entangled pure state on a composite system $\mathcal{H}_{A,n}\otimes\mathcal{H}_{B,n}$, the following inequalities hold for a sequence $\mathrm{Tr}_B\overline{\psi}:=\{\mathrm{Tr}_B\psi_n\}_n$. \begin{eqnarray}
1-\hspace{-0.3em}\displaystyle\lim_{\gamma\to+0}\overline{K}(a,b+\gamma|{\rm Tr}_{B}\overline{\psi}) \le\underline{\rm lim}d(\Phi^{\otimes an+b\sqrt{n}} \to \psi_n)~~\label{r-1} \end{eqnarray}
\begin{eqnarray} \overline{\rm lim}d(\Phi^{\otimes an+b\sqrt{n}} \to \psi_n)
\le1-\underline{K}(a,b|{\rm Tr}_{B}\overline{\psi})\label{r-2} \end{eqnarray}
\begin{eqnarray}
\hspace{0em}\displaystyle\lim_{\gamma\to+0}\underline{K}(a,b-\gamma|{\rm Tr}_{B}\overline{\psi}) \le\underline{\rm lim}d(\psi_n \to \Phi^{\otimes an+b\sqrt{n}})\label{s-1} \end{eqnarray}
\begin{eqnarray} \overline{\rm lim}d(\psi_n \to \Phi^{\otimes an+b\sqrt{n}})
\le\displaystyle\lim_{\gamma\to+0}\overline{K}(a,b+\gamma|{\rm Tr}_{B}\overline{\psi})\label{s-2} \end{eqnarray}
At first, we prove (\ref{r-1}). By (\ref{Deq}), for an arbitrary state $\psi_n$, arbitrary positive integers ${L_n},{L'_n}$, and an arbitrary entanglement dilution transformation $({L_n},D_n)$, the inequality $F^2(\psi_n,D_n(\Phi^{\otimes L_n}))\le\mathrm{Tr}\rho\{\rho\ge\frac{1}{2^{L'_n}}\}+\frac{2^{L_n}}{2^{L'_n}}$ holds. When ${L_n}={an+b\sqrt{n}}$, ${L'_n}={an+(b+\gamma)\sqrt{n}}$ in the inequality, we get (\ref{r-1}) by taking $\underline{\rm lim}$ and ${\rm lim}_{\gamma\to+0}$.
Next, we prove (\ref{r-2}). By (\ref{Deq}), for an arbitrary state $\psi_n$ and an arbitrary positive integer ${L_n}$, there is an entanglement dilution transformation $({L_n},D_n)$ satisfying the inequality $\mathrm{Tr}\rho\{\rho\ge\frac{1}{2^{L_n}}\}\le F^2(\psi_n,D_n(\Phi^{\otimes L_n}))$. When ${L_n}={an+b\sqrt{n}}$ in the inequality, we get (\ref{r-2}) by taking $\underline{\rm lim}$.
Next, we prove (\ref{s-1}). By Lemmas 4 and 5 in \cite{Hay2}, for an arbitrary state $\psi_n$, arbitrary positive integers ${L_n}\ge {L'_n}$, and an arbitrary concentration map $({L_n},C_n)$, the inequality \begin{eqnarray} &&F^2(C_n(\psi_n),\Phi^{\otimes L_n})\nonumber\\ &\le&\frac{1}{2^{L_n}}\Big(\sqrt{\mathrm{Tr}\{\rho_n\ge{1}/{2^{L'_n}}\}}\sqrt{\mathrm{Tr}\rho\{\rho_n\ge{1}/{2^{L'_n}}\}}\\ &&\hspace{-0.5em}+\sqrt{M-\mathrm{Tr}\{\rho_n\ge{1}/{2^{L'_n}}\}} \sqrt{1-\mathrm{Tr}\rho\{\rho_n\ge{1}/{2^{L'_n}}\}}\Big)^2 \nonumber \end{eqnarray} holds. When ${L_n}={an+b\sqrt{n}}$, ${L'_n}={an+(b-\gamma)\sqrt{n}}$ in the inequality, we get (\ref{s-1}) by taking $\underline{\rm lim}$ and ${\rm lim}_{\gamma\to+0}$.
Finally, we prove (\ref{s-2}). It is enough to prove $\overline{\rm lim}d(\psi_n \to \Phi^{\otimes an+b\sqrt{n}}) \le \overline{K}(a,b+\gamma|{\rm Tr}_{B}\overline{\psi})$ for an arbitrary positive real number $\gamma$. When $\overline{K}(a,b+\gamma|{\rm Tr}_{B}\overline{\psi})=1$, the inequality is obvious. Thus, we assume $\overline{K}(a,b+\gamma|{\rm Tr}_{B}\overline{\psi})<1$. By Lemma 9 and (1) in \cite{Hay2}, for an arbitrary state $\psi_n$ and an arbitrary positive integer ${L_n}$, there is a concentration map $({L_n},C_n)$ satisfying the inequality $1-\mathrm{Tr}\rho_n\{\rho_n\ge x_{2^{L_n}}\}\le F^2(C_n(\psi_n),\Phi^{\otimes L_n})$ where let $h_n(x)$ be $\mathrm{Tr}(\rho_n-x)\{\rho_n-x\ge0\}$ and $x_{{L_n}}$ satisfy $\left\lfloor\frac{1}{x_{{L_n}}}(1-h_n(x_{{L_n}}))\right\rfloor=2^{L_n}$. For ${L_n}:={an+(b+\gamma)\sqrt{n}}+\mathrm{log}(1-h_n(2^{-an-(b+\gamma)\sqrt{n}}))$ and ${L'_n}:={an+b\sqrt{n}}$, we can take as $x_{{L_n}}=2^{-an-(b+\gamma)\sqrt{n}}$. Since $\overline{\lim}h_n(2^{-an-(b+\gamma)\sqrt{n}})\le\overline{K}(a,b+\gamma|\overline{\rho})<1$, ${L'_n}<{L_n}$ holds for enough large integer $n$. Therefore, \begin{eqnarray}\label{inequality10} 1-\mathrm{Tr}\rho_n\{\rho_n\ge x_{{L_n}}\} &\le& F^2(C_n(\psi_n),\Phi^{\otimes L_n}) \nonumber\\ &\le& F^2(E'_n(\psi_n),\Phi^{\otimes{L'_n}}) \end{eqnarray}
holds for suitable $({L_n},C_n)$ and $({L'_n},E'_n)$ for enough large integer $n$. By taking $\underline{\lim}$ in the inequality (\ref{inequality10}), we get $\overline{\rm lim}d(\psi^{\otimes n} \to \Phi^{\otimes an+b\sqrt{n}}) \le \overline{K}(a,b+\gamma|{\rm Tr}_{B}\overline{\psi})$.
Let $\psi$ be a pure state in $\mathcal{H}_A\otimes\mathcal{H}_B$ except maximally entangled state. Then, $V(\mathrm{Tr}_B \psi)$ is not $0$. When $\mathcal{H}_{A,n}=\mathcal{H}_{A}^{\otimes n}$, $\mathcal{H}_{B,n}=\mathcal{H}_{B}^{\otimes n}$, and $\psi_n=\psi^{\otimes n+b'\sqrt{n}}$, \begin{eqnarray}
&&\overline{K}(a,b|{\rm Tr}_{B}\overline{\psi})=\underline{K}(a,b|{\rm Tr}_{B}\overline{\psi})\nonumber\\ &&=\left\{ \begin{array}{ll} 0 & \mathrm{if}~a<H(\mathrm{Tr}_B{\psi}) \\ G\left(\frac{b-H(\mathrm{Tr}_B{\psi})b'}{\sqrt{V(\mathrm{Tr}_B{\psi})}}\right) & \mathrm{if}~a=H(\mathrm{Tr}_B{\psi}) \\ 1 & \mathrm{if}~a>H(\mathrm{Tr}_B{\psi}) \end{array} \right. \end{eqnarray} due to the (classical) central limit theorem. Thus, Theorem \ref{2-order} holds. $\blacksquare$
~
\noindent[Proof of Theorem \ref{MCREineq}]~ We only have to show that $\displaystyle\underline{\lim}\delta_n(\psi) \ge1$ holds for any pure state $\psi$ except maximally entangled states. Let $\delta(\{L_n\})$ be $\underbar{\rm lim}d(\psi^{\otimes n} \to \Phi^{\otimes L_n})+d(\Phi^{\otimes L_n} \to \psi^{\otimes n})$. Then, $\underline{\rm lim}\delta_n(\psi)=\min_{\{L_n\}}\delta(\{L_n\})$ holds by (\ref{eq}). Moreover, we get $\delta(\{L_n\})\ge1$ as follows. We can take a subsequence $n_k$ of $n$ satisfying that $\delta(\{L_n\})={\rm lim}_k d(\psi^{\otimes n_k} \to \Phi^{\otimes L_{n_k}})+d(\Phi^{\otimes L_{n_k}} \to \psi^{\otimes n_k})$ holds, and $a:={\rm lim}_k {L_{n_k}}/{n_k}$ and $b:={\rm lim}_k \sqrt{n_k}({L_{n_k}}/{n_k}-a)$ exist in $[-\infty,+\infty]$ by repeatedly taking the subsequence, if necessary.
When $a> H(\mathrm{Tr}_B{\psi})$, $\delta(\{L_n\})\ge1$ because $\underline{\rm lim}_k d(\psi^{\otimes n_k} \to \Phi^{\otimes L_{n_k}})=1$ by Theorem \ref{2-order}. Similarly, when $a< H(\mathrm{Tr}_B{\psi})$, $\delta(\{L_n\})\ge1$ because $\underline{\rm lim}_k d(\Phi^{\otimes L_{n_k}} \to \psi^{\otimes n_k})=1$ by Theorem \ref{2-order}. We treat the case when $a=H(\mathrm{Tr}_B{\psi})$. Let $\epsilon$ be an arbitrary positive real number. By the definition of $b$, $H(\mathrm{Tr}_B{\psi})n_k+(b-\epsilon)\sqrt{n_k} < L_{n_k} <H(\mathrm{Tr}_B{\psi})n_k+(b+\epsilon)\sqrt{n_k}$ holds for large enough $k$. Then, $d(\Phi^{\otimes L_{n_k}}\to\psi^{\otimes n_k})\ge d(\Phi^{\otimes H(\mathrm{Tr}_B{\psi})n_k+(b+\epsilon)\sqrt{n_k}}\to\psi^{\otimes n_k})$ and $d(\psi^{\otimes n_k}\to\Phi^{\otimes L_{n_k}})\ge d(\psi^{\otimes n_k}\to\Phi^{\otimes H(\mathrm{Tr}_B{\psi})n_k+(b-\epsilon)\sqrt{n_k}})$ hold. Therefor, we get $\underline{\lim}_k d(\Phi^{\otimes L_{n_k}}\to\psi^{\otimes n_k})\ge G((b+\epsilon)/\sqrt{V_{\mathrm{Tr}_B\psi}})$ and $\underline{\lim}_k d(\psi^{\otimes n_k}\to\Phi^{\otimes L_{n_k}})\ge 1-G((b-\epsilon)/\sqrt{V_{\mathrm{Tr}_B\psi}})$ by Theorem \ref{2-order}. Thus, $\delta(\{L_n\})\ge1-G((b-\epsilon)/\sqrt{V_{\mathrm{Tr}_B\psi}}) +G((b+\epsilon)/\sqrt{V_{\mathrm{Tr}_B\psi}})$ is derived. Since $\epsilon$ is arbitrary, we get $\delta(\{L_n\})\ge1$. $\blacksquare$
~
\noindent[Proof of Theorem \ref{2nd.rate}]~ We represent $H(\mathrm{Tr}_B\psi)$ and $V(\mathrm{Tr}_B\psi)$ as $H$ and $V$ in this proof. We can restrict $M_n$ to the form ${\alpha n+\beta\sqrt{n}}$ in the same way as Proof of Theorem \ref{MCREineq}. By the condition $\overline{\lim}\delta_n(\psi,{\alpha n+\beta\sqrt{n}})\le\epsilon<1$ and Theorem \ref{2-order}, the first order rate $\alpha$ of $M_n$ is restricted to $H$. Then, \begin{eqnarray} &&\lim \delta_n(\psi,{ n+\beta\sqrt{n}})\nonumber\\ &&=\min_{\{L_n\}}\lim d(\psi^{\otimes n}\to\Phi^{\otimes L_n})+d(\Phi^{\otimes L_n}\to\psi^{\otimes n+\beta\sqrt{n}})\nonumber\\ &&=\min_{b\in{\mathbb{R}}}\lim d(\psi^{\otimes n}\to\Phi^{\otimes Hn+b\sqrt{n}})\nonumber\\ &&~~~~~~~~~~+d(\Phi^{\otimes Hn+b\sqrt{n}}\to\psi^{\otimes n+\beta\sqrt{n}})\nonumber\\ &&=\min_{b\in{\mathbb{R}}} G\left(\frac{b}{\sqrt{V}}\right)+1-G\left(\frac{b-H\beta}{\sqrt{V}}\right)\nonumber\\ &&=G\left(\frac{H\beta}{2\sqrt{V}}\right)+1- G\left(\frac{-H\beta}{2\sqrt{V}}\right).\nonumber \end{eqnarray} Therefore, we get the following equation. \begin{eqnarray} &&R(\psi,\epsilon)\nonumber\\
&&=\inf_{\beta}\left\{{\lim}\frac{n-(n+\beta\sqrt{n})}{\sqrt{n}}\Big| {\lim}\delta_n(\psi,n+\beta\sqrt{n})\le\epsilon\right\}\nonumber\\
&&=\inf_{\beta}\Big\{-\beta \Big| G\left(\frac{H\beta}{2\sqrt{V}}\right)+1- G\left(\frac{-H\beta}{2\sqrt{V}}\right)\le\epsilon \Big\}\nonumber\\ &&=\frac{2\sqrt{V}}{H}G^{-1}\left(1-\frac{\epsilon}{2}\right),\nonumber \end{eqnarray} where $G^{-1}$ means the inverse function of the Gaussian distribution function $G$.~$\blacksquare$
\end{document}
Let us proceed to consider the LOCC transformation from a pure state $\psi$ to a maximally entangled state $\Phi^{\otimes L}$. When we set as \begin{eqnarray}
J:=\max\{1\}\cup\left\{2\le j \le M \Big|\frac{\sum_{i=j}^L p_{\psi,j}^{\downarrow}}{M+1-j}<p_{\psi,j-1}^{\downarrow}\right\},\nonumber \end{eqnarray} the following equation holds. \begin{thm} \label{concentration}[concentration] \begin{eqnarray} &\displaystyle\max_{E:LOCC} F(E(\psi), \Phi^{\otimes L})&\nonumber\\ &\hspace{-0.5em}=\sqrt{\frac{1}{M}}\left( \displaystyle\sum_{i=1}^{J-1}\sqrt{p_{\psi,i}^{\downarrow}} + (M+1-J)\sqrt{\frac{\sum_{i=J}^L p_{\psi,i}^{\downarrow}}{M+1-J}} \right).& \end{eqnarray} \end{thm} Thus, the following representation is obtained. \begin{eqnarray} &d (\psi \to \Phi^{\otimes L})&\nonumber\\ &\hspace{-0.5em}=1-\frac{1}{M}\left( \displaystyle\sum_{i=1}^{J-1}\sqrt{p_{\psi,i}^{\downarrow}} + (M+1-J)\sqrt{\frac{\sum_{i=J}^L p_{\psi,i}^{\downarrow}}{M+1-J}} \right)^2&\nonumber \end{eqnarray}
~
\hspace{-1em}\textit{\textbf{proof of Theorem \ref{concentration}}}
証明に必要な定理を述べるために準備をする．合成系上の純粋状態$\psi$に対し，その0でないSchmidt係数$p=(p_1,\cdot\cdot\cdot,p_M)$を用いて，$E_l^{\psi}:=\sum_{i=l}^M p_i^{\downarrow}$とおく．ここで純粋状態$\psi\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$及び$|\phi\rangle\in\mathcal{H}_{A'}\otimes\mathcal{H}_{B'}$に対し以下のような量を定義する．$|\phi\rangle$のSchmidt数(i.e. 0でないSchmidt係数の数)を$N$とし，$l_1$を \begin{eqnarray} r_1:=\displaystyle\min_{l\in[1,N]} \frac{E_{l}^{\psi}}{E_{l}^{\phi}}=\frac{E_{l_1}^{\psi}}{E_{l_1}^{\phi}} \end{eqnarray} をみたす最小の整数$\in[1,N]$とする．$l_1=1$でないとき，$l_2$を \begin{eqnarray} r_2:=\displaystyle\min_{l\in[1,l_1-1]} \frac{E_{l}^{\psi}-E_{l_1}^{\psi}}{E_{l}^{\phi}-E_{l_1}^{\phi}} =\frac{E_{l_2}^{\psi}-E_{l_1}^{\psi}}{E_{l_2}^{\phi}-E_{l_1}^{\phi}} \end{eqnarray} をみたす最小の整数$\in[1,l_1-1]$とする．この手順を繰り返すとある$K$に対し$l_K=1$となる．このとき数列$N+1=:l_0>l_1>\cdot\cdot\cdot>l_K$及び$0<r_1<\cdot\cdot\cdot<r_K$を得る．$j=1,\cdot\cdot\cdot,K$に対し \begin{eqnarray} A_j:=E_{l_j}^{\psi}-E_{l_{j-1}}^{\psi},~~~B_j:=E_{l_j}^{\phi}-E_{l_{j-1}}^{\phi} \end{eqnarray}
と定義する．ただし$E_{l_0}^{\psi}=E_{l_0}^{\phi}:=0$とおく．またターゲットの純粋状態$|\phi\rangle$のSchmidt分解を$|\phi\rangle=\sum_{i=1}^N \sqrt{p_{\phi,i}}|i_{A'},i_{B'}\rangle$とするとき，
純粋状態$|\xi_{\psi,\phi}\rangle\in\mathcal{H}_{A'}\otimes\mathcal{H}_{B'}$を \begin{eqnarray}\label{xi}[xi]
|\xi_{\psi,\phi}\rangle:=\displaystyle\sum_{j=1}^K \displaystyle\sum_{i=l_j}^{l_{j-1}-1}\sqrt{r_j p_{\phi,i}^{\downarrow}} |i_{A'},i_{B'}\rangle \end{eqnarray}
と定める．このときLOCC操作を用いて$\psi\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$から$|\phi\rangle\in\mathcal{H}_{A'}\otimes\mathcal{H}_{B'}$を生成することに関して以下の定理が成り立つ．
\begin{thm} (Vidal, Jonathan, Nielsen\cite{VJN})
あるLOCC操作が存在し$\psi$は$|\xi_{\psi,\phi}\rangle$に変換可能である．さらに以下の等式が成り立つ． \begin{eqnarray} \displaystyle\max_{E:LOCC} F(E(\psi),\phi) =F(\xi_{\psi,\phi},\phi) =\displaystyle\sum_{j=1}^K \sqrt{A_j B_j} \end{eqnarray} \end{thm}
(\ref{xi})における$|\xi_{\psi,\Phi^{\otimes L}}\rangle$が \begin{eqnarray}
\displaystyle\sum_{i=1}^{J-1} \sqrt{p_{\psi,i}^{\downarrow}}|i_{A'},i_{B'}\rangle
+\displaystyle\sum_{i=J}^{M} \sqrt{\frac{\sum_{i=J}^L p_{\psi,i}^{\downarrow}}{M+1-J}}|i_{A'},i_{B'}\rangle \end{eqnarray} で与えられることを示せば十分である．
$l_1=J$かつ$r_1=\frac{E_{J}^{\psi}}{E_{J}^{\Phi^{\otimes L}}}=M \sum_{i=J}^L p_{\psi,i}^{\downarrow}/(M+1-J)$を示す．$1\le j<J$に対し \begin{eqnarray} \frac{\sum_{i=j}^L p_{\psi,i}^{\downarrow}}{M+1-j} &\ge& \frac{(J-j)p_{\psi,J-1}^{\downarrow} + \sum_{i=J}^L p_{\psi,i}^{\downarrow}} {M+1-j} \\ \label{inequality}[{inequality}]&>& \frac{(J-j) \frac{\sum_{i=J}^L p_{\psi,i}^{\downarrow}}{M+1-J} + \sum_{i=J}^L p_{\psi,i}^{\downarrow}} {M+1-j} \\ &=& \frac{\sum_{i=J}^L p_{\psi,i}^{\downarrow}}{M+1-J} \end{eqnarray} となる．(\ref{inequality})の不等式では$J$の定義を用いている．これより$1\le j<J$に対し${E_{l_1}^{\psi}}\big/{E_{l_1}^{\Phi^{\otimes L}}} < {E_{j}^{\psi}}\big/{E_j^{\Phi^{\otimes L}}}$が成り立つので，$J\le l_1$がわかる．$J<j\le M$に対しては$J$の定義より$\frac{\sum_{i=j}^L p_{\psi,i}^{\downarrow}}{M+1-j}<p_{\psi,j-1}^{\downarrow}$となる．従って \begin{eqnarray} \frac{\sum_{i=j}^L p_{\psi,i}^{\downarrow}}{M+1-j} \ge \frac{\sum_{i=j-1}^L p_{\psi,i}^{\downarrow}}{M+1-(j-1)} \ge \cdot\cdot\cdot \ge \frac{\sum_{i=J}^L p_{\psi,i}^{\downarrow}}{M+1-J} \end{eqnarray} となる．${E_{j}^{\psi}}\big/{E_{j}^{\Phi^{\otimes L}}} = M \sum_{i=j}^L p_{\psi,i}^{\downarrow}/(M+1-j)$に注意すれば \begin{eqnarray} \frac{E_{J}^{\psi}}{E_{J}^{\Phi^{\otimes L}}} = \min_{1\le j\le M} \frac{E_{j}^{\psi}}{E_{j}^{\Phi^{\otimes L}}} = \frac{E_{l_1}^{\psi}}{E_{l_1}^{\Phi^{\otimes L}}}=r_1 \end{eqnarray} を得る．また，$l_1\le J$となるので，$l_1=J$がわかる．
次に$j\ge 2$に対し$p_{l_j}=p_{l_j+1}=\cdot\cdot\cdot=p_{l_{j-1}-1}$かつ$r_j=Mp_{l_j}$となることを示す．$l_j=l_{j-1}-1$のときは自明なので $l_j\le l_{j-1}-2$とする．ここで$a,b,c,d,>0$に対し \begin{eqnarray} \frac{a}{b}<\frac{c}{d} &\Longleftrightarrow& \frac{a}{b}<\frac{a+c}{b+d} \end{eqnarray} が成り立つことに注意する． \begin{eqnarray} &&\frac{E_{l_j+1}^{\psi}-E_{l_{j-1}}^{\psi}}{E_{l_j+1}^{\Phi^{\otimes L}}-E_{l_{j-1}}^{\Phi^{\otimes L}}} \ge \frac{E_{l_j}^{\psi}-E_{l_{j-1}}^{\psi}}{E_{l_j}^{\Phi^{\otimes L}}-E_{l_{j-1}}^{\Phi^{\otimes L}}} \\ &\Longleftrightarrow& \frac{E_{l_j+1}^{\psi}-E_{l_{j-1}}^{\psi}}{l_{j-1}-(l_j+1)} \ge \frac{E_{l_j}^{\psi}-E_{l_{j-1}}^{\psi}}{l_{j-1}-l_j} \\ &\Longleftrightarrow& \frac{a+E_{l_j+1}^{\psi}-E_{l_{j-1}}^{\psi}}{b+l_{j-1}-(l_j+1)} \ge \frac{E_{l_j}^{\psi}-E_{l_{j-1}}^{\psi}}{l_{j-1}-l_j} \\ &\Longleftrightarrow& 1\ge b \\ &\Longleftrightarrow& \frac{E_{l_j}^{\psi}-E_{l_{j-1}}^{\psi}}{l_{j-1}-l_j} \ge E_{l_j}^{\psi}-E_{l_j+1}^{\psi} \\ &\Longleftrightarrow& \displaystyle\sum_{i=l_j}^{l_{j-1}-1}p_{\psi,i}^{\downarrow} \ge (l_{j-1}-l_j)p_{\psi,l_j}^{\downarrow} \\ &\Longleftrightarrow& \displaystyle\sum_{i=l_j}^{l_{j-1}-1}(p_{\psi,i}^{\downarrow}-p_{\psi,l_j}^{\downarrow}) \ge 0 \end{eqnarray} である．ただし \begin{eqnarray} &a:=E_{l_j}^{\psi}-E_{l_j+1}^{\psi}=p_{\psi,l_j}^{\downarrow},&\\ &b:=(E_{l_j}^{\psi}-E_{l_j+1}^{\psi})\frac{l_{j-1}-l_j}{E_{l_j}^{\psi}-E_{l_{j-1}}^{\psi}}& \end{eqnarray} とおいた．$i\ge l_j$に対し$p_{\psi,i}^{\downarrow}\le p_{\psi,l_j}^{\downarrow}$なので \begin{eqnarray} p_{l_j}=p_{l_j+1}=\cdot\cdot\cdot=p_{l_{j-1}-1} \end{eqnarray} を得る．また定義より$r_j=Mp_{l_j}$となる．
以上より \begin{eqnarray}
|\xi_{\psi,\phi}\rangle\nonumber
&:=&\displaystyle\sum_{j=1}^K \displaystyle\sum_{i=l_j}^{l_{j-1}-1}\sqrt{r_j p_{\Phi^{\otimes L},i}^{\downarrow}} |i_{A'},i_{B'}\rangle \\
&=&\displaystyle\sum_{j=2}^K \displaystyle\sum_{i=l_j}^{l_{j-1}-1}\sqrt{ p_{\psi,l_j}^{\downarrow}} |i_{A'},i_{B'}\rangle \\
&&+ \displaystyle\sum_{i=l_1}^{M}\sqrt{\frac{\sum_{i=J}^L p_{\psi,i}^{\downarrow}}{M+1-J}} |i_{A'},i_{B'}\rangle \\
&=& \displaystyle\sum_{j=2}^K \displaystyle\sum_{i=l_j}^{l_{j-1}-1}\sqrt{ p_{\psi,i}^{\downarrow}} |i_{A'},i_{B'}\rangle \\
&&+ \displaystyle\sum_{i=J}^{M}\sqrt{\frac{\sum_{i=J}^L p_{\psi,i}^{\downarrow}}{M+1-J}} |i_{A'},i_{B'}\rangle \\
&=& \displaystyle\sum_{i=1}^{J-1} \sqrt{p_{\psi,i}^{\downarrow}}|i_{A'},i_{B'}\rangle\\
&&+\displaystyle\sum_{i=1}^{J-1} \sqrt{\frac{\sum_{i=J}^L p_{\psi,i}^{\downarrow}}{M+1-J}}|i_{A'},i_{B'}\rangle \end{eqnarray} を得る．$\Box$
~
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\section{Information Spectrum Methods}
\subsection{Information Spectrum Definitions}
$\bar{\mathcal{H}}:=\{\mathcal{H}_n\}_{n=1}^{\infty}$を量子系の列, $\bar{\rho}:=\{\rho_n\}_{n=1}^{\infty}$を$\bar{\mathcal{H}}$上の量子状態の列, $\epsilon\in[0,1]$を定数とする．本論文では$\pm\infty$も込みで等号、不等号を考えることにする．また空集合の$\inf$及び$\sup$は$\infty$及び$-\infty$と定義する．ここで
\begin{eqnarray}
&\overline{K}(a|\bar{\rho}):=\overline{\lim}\mathrm{Tr}\rho_n \{\rho_n\ge\mathrm{e}^{-na}\},&\\
&\overline{K}(a,b|\bar{\rho}):=\overline{\lim}\mathrm{Tr}\rho_n \{\rho_n\ge\mathrm{e}^{-na-\sqrt{n}b}\}& \end{eqnarray} とおく．これらをそれぞれ一次及び二次の情報スペクトル量と呼ぶことにする．
\subsection{First Order Asymptotics}
$s_n,r_n$は以下の不等式をみたす． \begin{eqnarray}
\displaystyle\lim_{\gamma\to+0}\underline{K}(b+\gamma|\mathrm{Tr}_B\underline{\psi})
&\le& \underline{\lim}s_n(bn|\psi_n)\\
&\le& \underline{K}(b|\mathrm{Tr}_B\overline{\psi}),\\
\displaystyle\lim_{\gamma\to+0}\overline{K}(b+\gamma|\mathrm{Tr}_B\overline{\psi})
&\le& \overline{\lim}s_n(bn|\psi_n)\\
&\le& \overline{K}(b|\mathrm{Tr}_B\overline{\psi}) \end{eqnarray} \begin{eqnarray}
1-\displaystyle\lim_{\gamma\to+0}\overline{K}(b+\gamma|\mathrm{Tr}_B\overline{\psi})
&\le& \underline{\lim}r_n(bn|\psi_n)\\
&\le& 1-\overline{K}(b|\mathrm{Tr}_B\overline{\psi}),\\
1-\displaystyle\lim_{\gamma\to+0}\underline{K}(b+\gamma|\mathrm{Tr}_B\overline{\psi})
&\le& \overline{\lim}r_n(bn|\psi_n)\\
&\le& 1-\underline{K}(b|\mathrm{Tr}_B\overline{\psi}) \end{eqnarray}
さらに \begin{eqnarray}\label{condition1}[condition1]
\displaystyle\lim_{\gamma\to+0}\underline{K}(b+\gamma|\mathrm{Tr}_B\overline{\psi})
=\overline{K}(b|\mathrm{Tr}_B\overline{\psi}) \end{eqnarray} が成り立つような$b$に対して，$s_n$に関する不等式に現れる量は全て一致し \begin{eqnarray}\label{lim1}[lim1]
{\lim}s_n(bn|\psi_n)
= \underline{K}(b|\mathrm{Tr}_B\overline{\psi}) \end{eqnarray} となる．同様に \begin{eqnarray}\label{condition3}[condition3]
\displaystyle\lim_{\gamma\to+0}\overline{K}(b+\gamma|\mathrm{Tr}_B\overline{\psi})=\underline{K}(b|\mathrm{Tr}_B\overline{\psi}) \end{eqnarray} が成り立つような$b$に対して，$r_n$に関する不等式に現れる量は全て一致し \begin{eqnarray}\label{lim3}[lim3]
{\lim}r_n(bn|\psi_n)
= 1-\underline{K}(b|\mathrm{Tr}_B\overline{\psi}) \end{eqnarray} となる．
\subsection{Second Order Asymptotics}
上記の議論は二次のオーダーに着目した場合も同様に \begin{eqnarray}
\displaystyle\lim_{\gamma\to+0}\underline{K}(b_1,b_2+\gamma|\mathrm{Tr}_B\overline{\psi})
&\le& \underline{\lim}s_n(b_1n+b_2\sqrt{n}|\psi_n)\nonumber\\
&\le& \underline{K}(b_1,b_2|\mathrm{Tr}_B\overline{\psi}),\\
\displaystyle\lim_{\gamma\to+0}\overline{K}(b_1,b_2+\gamma|\mathrm{Tr}_B\overline{\psi})
&\le& \overline{\lim}s_n(b_1n+b_2\sqrt{n}|\psi_n)\nonumber\\
&\le& \overline{K}(b_1,b_2|\mathrm{Tr}_B\overline{\psi}) \end{eqnarray} 及び \begin{eqnarray}
&&1-\displaystyle\lim_{\gamma\to+0}\overline{K}(b_1,b_2+\gamma|\mathrm{Tr}_B\overline{\psi})\nonumber\\
&\le& \underline{\lim}r_n(b_1n+b_2\sqrt{n}|\psi_n)\\
&\le& 1-\overline{K}(b_1,b_2|\mathrm{Tr}_B\overline{\psi}),\\
&&1-\displaystyle\lim_{\gamma\to+0}\underline{K}(b_1,b_2+\gamma|\mathrm{Tr}_B\overline{\psi})\nonumber\\
&\le& \overline{\lim}r_n(b_1n+b_2\sqrt{n}|\psi_n)\\
&\le& 1-\underline{K}(b_1,b_2|\mathrm{Tr}_B\overline{\psi}) \end{eqnarray} なる不等式を導く．さらに \begin{eqnarray}\label{condition2}[condition2]
\displaystyle\lim_{\gamma\to+0}\underline{K}(b_1,b_2+\gamma|\mathrm{Tr}_B\overline{\psi})
=\overline{K}(b_1,b_2|\mathrm{Tr}_B\overline{\psi}) \end{eqnarray} が成り立つような$b_1,b_2$に対して，$s_n$の不等式に現れる量は全て一致し \begin{eqnarray}\label{lim2}[lim2]
{\lim}s_n(b_1n+b_2\sqrt{n}|\psi_n)
= \underline{K}(b_1,b_2|\mathrm{Tr}_B\overline{\psi}) \end{eqnarray} となる．同様に \begin{eqnarray}\label{condition4}[condition4]
\displaystyle\lim_{\gamma\to+0}\overline{K}(b_1,b_2+\gamma|\mathrm{Tr}_B\overline{\psi})=\underline{K}(b_1,b_2|\mathrm{Tr}_B\overline{\psi}) \end{eqnarray} が成り立つような$b_1,b_2$に対して，$r_n$の不等式に現れる量は全て一致し \begin{eqnarray}\label{lim4}[lim4]
{\lim}r_n(b_1n+b_2\sqrt{n}|\psi_n)
= 1-\underline{K}(b_1,b_2|\mathrm{Tr}_B\overline{\psi}) \end{eqnarray} となる．
\section{General formula for Entanglement Dillution}
The efficiency of the concentration map is described by the number ${L_n}$ of e-bits that we want to produce and the error $e(\psi^{\otimes n},L_n,C_n):=d(C_n(\psi^{\otimes n}), \Phi^{\otimes L_n})$ between $C_n(\psi^{\otimes n})$ and $\Phi^{\otimes L_n}$. Similarly, the efficiency of the entanglement dilution transformation is described by ${L_n}$ and $\tilde{e}(\psi^{\otimes n},L_n,D_n):=d(\psi^{\otimes n}, D_n(\Phi^{\otimes L_n}))$.
\begin{lem}\label{dilution-lemma1}[{dilution-lemma1}]
任意の純粋状態$|\psi\rangle\in\mathcal{H_{A}\otimes\mathcal{H}_{B}}$，最大エンタングルド状態$\Phi^{\otimes L}\in\mathcal{H_{A'}\otimes\mathcal{H}_{B'}}$，正整数$M'$及びLOCC写像$D:\mathcal{S}(\mathcal{H}_{A'}\otimes\mathcal{H}_{B'})\to\mathcal{S}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$は以下をみたす． \begin{eqnarray} F^2(\psi,D(\Phi^{\otimes L})) \le \mathrm{Tr}\rho\left\{\rho\ge\frac{1}{M'}\right\}+\frac{M}{M'} \end{eqnarray}
ただし$\rho:=\mathrm{Tr}_B|\psi\rangle\langle\psi|$とおいた． \end{lem}
\begin{proof} (\ref{dilution1})より \begin{eqnarray} &&F^2(\psi,D(\Phi^{\otimes L})) \\ &\le& \displaystyle\max_{D:LOCC} F^2(\psi,D(\Phi^{\otimes L}))\\ &\le& \displaystyle\sum_{1\le i\le M:p_{\psi,i}^{\downarrow}\ge M'^{-1}} p_{\psi,i}^{\downarrow}+\displaystyle\sum_{1\le i\le M:p_{\psi,i}^{\downarrow}<M'^{-1}} p_{\psi,i}^{\downarrow} \nonumber\\ &\le& \mathrm{Tr}\rho\left\{\rho\ge\frac{1}{M'}\right\}+\frac{M}{M'} \end{eqnarray} となる． \end{proof}
\begin{lem}\label{dilution-lemma2}[{dilution-lemma2}]
任意の純粋状態$|\psi\rangle\in\mathcal{H_{A}\otimes\mathcal{H}_{B}}$，最大エンタングルド状態$\Phi^{\otimes L}\in\mathcal{H_{A'}\otimes\mathcal{H}_{B'}}$及び正整数$M'\le M$に対し，あるLOCC写像$D:\mathcal{S}(\mathcal{H}_{A'}\otimes\mathcal{H}_{B'})\to\mathcal{S}(\mathcal{H}_{A}\otimes\mathcal{H}_{B})$が存在し以下をみたす． \begin{eqnarray} \mathrm{Tr}\rho\left\{\rho\ge\frac{1}{M'}\right\} \le F^2(\psi,D(\Phi^{\otimes L})) \end{eqnarray}
ただし$\rho:=\mathrm{Tr}_B|\psi\rangle\langle\psi|$とおいた． \end{lem}
\begin{proof} $\psi$のSchmidt係数を$p_{\psi}$と表す．このとき \begin{eqnarray} \mathrm{Tr}\rho\left\{\rho\ge\frac{1}{M'}\right\} \le\mathrm{Tr}\rho\left\{\rho\ge\frac{1}{M}\right\} \le\displaystyle\sum_{i=1}^M p_{\psi,i}^{\downarrow} \end{eqnarray} となる．(\ref{dilution1})より \begin{eqnarray} \displaystyle\max_{D:LOCC} F^2(\psi,D(\Phi^{\otimes L}))=\displaystyle\sum_{i=1}^M p_{\psi,i}^{\downarrow} \end{eqnarray} なので所望の不等式を得る． \end{proof}
以下の量に着目する． \begin{eqnarray}
&&\hspace{-1.5em}T_{+}(\epsilon|\overline{\psi})
:=\displaystyle\inf_{\Upsilon_n}\left\{\overline{\lim}\frac{1}{n}\mathrm{log}|\Upsilon_n|\Big|\overline{\lim}c(\psi_n,\Upsilon_n)< \epsilon \right\}\nonumber\\
&&\hspace{-1.5em}T_{+}(\epsilon,a|\overline{\psi}) \nonumber\\
&&\hspace{-1.5em}:=\displaystyle\inf_{\Upsilon_n}\left\{\overline{\lim}\sqrt{n}\left(\frac{1}{n}\mathrm{log}|\Upsilon_n|-a\right)\Big|\overline{\lim}c(\psi_n,\Upsilon_n)< \epsilon \right\} \nonumber \end{eqnarray}
ここで \begin{eqnarray}
&\underline{F}(a|\overline{\rho}) :=\underline{\lim}\mathrm{Tr}\rho_n\{\rho_n\ge\mathrm{e}^{-na}\}&\\
&\underline{F}(a,b|\overline{\rho}) :=\underline{\lim}\mathrm{Tr}\rho_n\{\rho_n\ge\mathrm{e}^{-na-\sqrt{n}b}\}& \end{eqnarray} とおくとき，一次の情報スペクトル量及び二次の情報スペクトル量を以下のように定める． \begin{eqnarray}
&\overline{H}_+(\epsilon|\overline{\rho})
:=\inf\left\{ a\in{\mathbb{R}}\Big| \underline{F}(a|\overline{\rho})>\epsilon \right\}&\\
&\overline{H}_+(\epsilon,a|\overline{\rho})
:=\inf\left\{ b\in{\mathbb{R}}\Big| \underline{F}(a,b|\overline{\rho})>\epsilon \right\}& \end{eqnarray}
\begin{thm} \begin{eqnarray}
&T_{+}(\epsilon|\overline{\psi})=\overline{H}_+(1-\epsilon|\overline{\rho}),&\label{dilution3}\\
&T_{+}(\epsilon,a|\overline{\psi})=\overline{H}_+(1-\epsilon,a|\overline{\rho})&\label{dilution4} \end{eqnarray} \end{thm}
\begin{proof}
始めに(\ref{dilution3})の$T_{+}(\epsilon|\overline{\psi})\ge\overline{H}_+(\epsilon|\overline{\rho})$を示す．エンタングルメント希釈符号$\{\Upsilon_n=(2^{L_n},D_n)\}$で \begin{eqnarray} 1-\epsilon < \underline{\lim}F^2({\varphi}_n,D_n(\Phi^{\otimes L_n})) \end{eqnarray} をみたすものを任意にとる．任意の$a>\overline{\lim}\frac{1}{n}\mathrm{log}2^{L_n}$に対し，補題\ref{dilution-lemma1}において$M':=\mathrm{e}^{na}$とおくと \begin{eqnarray} 1-\epsilon &<& \underline{\lim}F^2({\varphi}_n,D_n(\Phi^{\otimes L_n}))\\ &\le& \underline{\lim}\left(\mathrm{Tr}\rho_n\left\{\rho_n\ge\frac{1}{\mathrm{e}^{na}}\right\}+\frac{2^{L_n}}{\mathrm{e}^{na}} \right)\\ &=& \underline{\lim}\mathrm{Tr}\rho_n\left\{\rho_n\ge{\mathrm{e}^{-na}}\right\}\\
&=& \underline{F}(a|\overline{\rho}) \end{eqnarray}
となる．よって$a\in\left\{ a\in{\mathbb{R}}\Big| \underline{F}(a|\overline{\rho})>1-\epsilon \right\}$となるので，$a\ge\overline{H}_+(\epsilon|\overline{\rho})$である．$a$の任意性より$\overline{\lim}\frac{1}{n}\mathrm{log}2^{L_n}\ge\overline{H}_+(\epsilon|\overline{\rho})$となり，従って$T_{+}(\epsilon|\overline{\psi})\ge\overline{H}_+(\epsilon|\overline{\rho})$が得られた．
(\ref{dilution4})の$T_{+}(\epsilon,a|\overline{\psi})\ge\overline{H}_+(1-\epsilon,a|\overline{\rho})$については，任意の$b>\overline{\lim}\sqrt{n}\left(\frac{1}{n}\mathrm{log}2^{L_n}-a\right)$に対し$M'=\mathrm{e}^{na+\sqrt{n}b}$とおき上記と同様の議論を行えばよい．
次に(\ref{dilution3})の$T_{+}(\epsilon|\overline{\psi})\le\overline{H}_+(\epsilon|\overline{\rho})$を示す．$\underline{F}(a|\overline{\rho})>1-\epsilon$をみたす任意の$a$をとる．補題\ref{dilution-lemma2}において$2^{L_n}=\mathrm{e}^{na}$とおくと，あるLOCC写像$\{D_n\}$が存在し \begin{eqnarray} \mathrm{Tr}\rho_n\left\{\rho_n\ge\mathrm{e}^{-na}\right\} \le F^2(\psi_n,D_n(\Phi^{\otimes L_n})) \end{eqnarray} をみたす．これより \begin{eqnarray} 1-\epsilon
<\underline{F}(a|\overline{\rho}) \le \underline{\lim}F^2(\psi_n,D_n(\Phi^{\otimes L_n})) \end{eqnarray} かつ \begin{eqnarray} \overline{\lim}\frac{1}{n}\mathrm{log}2^{L_n}=a \end{eqnarray}
となる．従って$T_{+}(\epsilon|\overline{\psi})\le a$となるが，$a$の任意性より$T_{+}(\epsilon|\overline{\psi})\le\overline{H}_+(\epsilon|\overline{\rho})$を得る．
(\ref{dilution4})の$T_{+}(\epsilon,a|\overline{\psi})\le\overline{H}_+(1-\epsilon,a|\overline{\rho})$については，$\underline{F}(a,b|\overline{\rho})>1-\epsilon$をみたす任意の$b$をとり，$M'=\mathrm{e}^{na+\sqrt{n}b}$とおき上記と同様の議論を行えばよい． \end{proof}
\section{エンタングルメント抽出の漸近可逆性を主張する論文}
pure stateのとき，漸近的には可逆な変換であると書いてある論文
~
"Concentrating Partial Entanglement by Local Operations"
Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu, Benjamin Schumacher (1996)
この論文が親玉?．以下の論文は上記の論文を引用している．おそらく純粋状態の抽出と希釈の可逆性に言及した最初期の論文．
~
"Entanglement irreversibility from quantum discord and quantum deficit"
Marcio F. Cornelio, Marcos C. de Oliveiray and Felipe F. Fanchini,(2011)
mixedのときは非可逆であることを主張しているが，純粋状態のときは漸近的に可逆と述べている．quantum discord と quantum deficitという量で議論している．比較的新しめの論文なので近年でも純粋状態の漸近可逆性は信じられていると考えられる．
~
"Irreversibility in asymptotic manipulations of entanglement"
G. Vidal and J. I. Cirac (2001)
mixedのときに漸近的な非可逆性を論じている．束縛エンタングルメントとエンタングルメントコストで議論している．
~
"Exact and Asymptotic Measures of Multipartite Pure State Entanglement"
Charles H. Bennett, Sandu Popescu, Daniel Rohrlich, John A. Smolin, and Ashish V. Thapliyal (2000)
${\mathbb{C}}^4\otimes{\mathbb{C}}^4$上の cat state $\sqrt{\frac{1}{2}}(|0\rangle^{\otimes 4}+|1\rangle^{\otimes 4})$など特別な状態に関しては最大エンタングルド状態から希釈できるが漸近的に最大エンタングルド状態が生成できない（すなわち変換が非可逆であること）を主張している．マルチパーティーに関する考察をしている．Catalystの使用やLOCCの拡張に関しても考察している．
~
ほかにも複数あるようです．
\end{document}
~
\textbf{\textrm{Conjecture}}
命題(\ref{equation})の設定のもと以下の不等式が成り立つ． \begin{eqnarray}\label{inequality2} \min_{E,D}d(\psi,D\circ E(\psi))+d(E(\psi),\Phi^{\otimes L}) \ge \min_{\Phi_{l}} d(\psi,\Phi_{l}) = \left(1-\max_{l=1,\cdot\cdot\cdot,L}\displaystyle\sum_{i=1}^{l}\sqrt{\frac{p^{\downarrow}_i}{l}}\right)^{\frac{1}{2}}. \end{eqnarray} ただし中辺の$\Phi_{l}$は$\mathcal{H}_A\otimes\mathcal{H}_B$内の最大エンタングルド状態全体を動く．また$L:=\min\{{{\rm dim}}\mathcal{H}_A,{{\rm dim}}\mathcal{H}_B\}$とおいた．
~
\cite{Hay1}の定理4では初期確率分布$p_n$からの一様分布の近似誤差と，その後$p_n$の復元誤差の間にはトレードオフがあることを示している． 上記の予想は\cite{Hay1}の定理4の量子系への拡張とみなせる．すなわち初期状態$\psi$からの最大エンタングルド状態の近似誤差と，その後の$\psi$の復元誤差の間にはトレードオフがあることを主張している．
-----------------------------------------------------------------
前節では一般の純粋状態を扱った．本節では前節の応用として特に，$\psi_n$が$\psi$の独立同一な拡張である場合(i.e. $\psi_n=\psi^{\otimes n}$)を扱う．一次の情報スペクトル量を計算すると \begin{eqnarray}
\overline{K}(b|\mathrm{Tr}_B\overline{\psi})
=\underline{K}(b|\mathrm{Tr}_B\overline{\psi}) =\left\{ \begin{array}{lc} 0&\mathrm{if}~ b<H(\mathrm{Tr}_B{\psi})\\ 1&\mathrm{if}~ b>H(\mathrm{Tr}_B{\psi}) \end{array} \right. \end{eqnarray} となる．従って， \begin{eqnarray}
{\lim}s_n(bn|\psi_n) =\left\{ \begin{array}{lc} 0&\mathrm{if}~ b<H(\mathrm{Tr}_B{\psi})\\ 1&\mathrm{if}~ b>H(\mathrm{Tr}_B{\psi}) \end{array} \right.\\
{\lim}r_n(bn|\psi_n) =\left\{ \begin{array}{lc} 1&\mathrm{if}~ b<H(\mathrm{Tr}_B{\psi})\\ 0&\mathrm{if}~ b>H(\mathrm{Tr}_B{\psi}) \end{array} \right. \end{eqnarray} を得る．しかし上の式からは$b=H(\mathrm{Tr}_B{\psi})$のときの漸近的な振舞いは読みとることができない．
次に二次の情報スペクトル量を計算すると \begin{eqnarray}
&&\overline{K}(b_1,b_2|\mathrm{Tr}_B\overline{\psi})
=\underline{K}(b_1,b_2|\mathrm{Tr}_B\overline{\psi})\\ &&=\left\{ \begin{array}{lc} 0&\mathrm{if}~ b_1<H(\mathrm{Tr}_B{\psi})\\ G(b_2)/\sqrt{V(\mathrm{Tr}_B{\psi})}&\mathrm{if}~ b_1=H(\mathrm{Tr}_B{\psi})\\ 1&\mathrm{if}~ b_1>H(\mathrm{Tr}_B{\psi}) \end{array} \right. \end{eqnarray} となる．従って， \begin{eqnarray}
&&{\lim}s_n(b_1n+b_2\sqrt{n}|\psi_n)\nonumber\\ &&=\left\{ \begin{array}{lc} 0&\mathrm{if}~ b_1<H(\mathrm{Tr}_B{\psi})\\ G(b_2)/\sqrt{V(\mathrm{Tr}_B{\psi})}&\mathrm{if}~ b_1=H(\mathrm{Tr}_B{\psi})\\ 1&\mathrm{if}~ b_1>H(\mathrm{Tr}_B{\psi}) \end{array} \right. \end{eqnarray} \begin{eqnarray}
&&{\lim}r_n(b_1n+b_2\sqrt{n}|\psi_n)\nonumber\\ &&=\left\{ \begin{array}{lc} 1&\mathrm{if}~ b<H(\mathrm{Tr}_B{\psi})\\ 1-G(b_2)/\sqrt{V(\mathrm{Tr}_B{\psi})}&\mathrm{if}~ b_1=H(\mathrm{Tr}_B{\psi})\\ 0&\mathrm{if}~ b>H(\mathrm{Tr}_B{\psi}) \end{array} \right. \end{eqnarray} を得る．ここで(\ref{eq})より，任意の純粋状態$\psi\in\mathcal{H}_A\otimes\mathcal{H}_B$に対し, 以下が成り立つ． \begin{eqnarray} &&\displaystyle\underline{\lim}\displaystyle\min_{C_n,D_n:LOCC} \{d(\psi^{\otimes n}, D\circ E(\psi))\\ && ~~+ d(E(\psi^{\otimes n}),\Phi^{\otimes L_n})\} \nonumber\\ && \ge\displaystyle\underline{\lim} \{d(\Phi^{\otimes L_n}\to\psi^{\otimes n})\\ &&~~ + d(\psi^{\otimes n}\to\Phi^{\otimes L_n})\}\\ &&=1. \end{eqnarray}
\end{document} | arXiv |
\begin{document}
\title[Equilibrium States and Entropy Densities] {Equilibrium states and their entropy densities \\ in gauge-invariant $C^*$-systems} \author[N. Akiho]{Nobuyuki Akiho} \address{Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan} \author[F. Hiai]{Fumio Hiai$\,^1$} \address{Graduate School of Information Sciences, Tohoku University, Aoba-ku, Sendai 980-8579, Japan} \author[D. Petz]{D\'enes Petz$\,^2$} \address{Alfr\'ed R\'enyi Institute of Mathematics, Hungarian Academy of Sciences, H-1053 Budapest, Re\'altanoda u.\ 13-15, Hungary} \thanks{$^1\,$Supported in part by Japan-Hungary Joint Research Project (JSPS) and by the program ``R\&D support scheme for funding selected IT proposals" of the Ministry of Public Management, Home Affairs, Posts and Telecommunications.} \thanks{$^2\,$Supported in part by MTA-JSPS project (Quantum Probability and Information Theory) and by OTKA T032662.}
\maketitle
\begin{abstract} A gauge-invariant $C^*$-system is obtained as the fixed point subalgebra of the infinite tensor product of full matrix algebras under the tensor product unitary action of a compact group. In the paper, thermodynamics is studied in such systems and the chemical potential theory developed by Araki, Haag, Kastler and Takesaki is used. As a generalization of quantum spin system, the equivalence of the KMS condition, the Gibbs condition and the variational principle is shown for translation-invariant states. The entropy density of extremal equilibrium states is also investigated in relation to macroscopic uniformity. \end{abstract}
\section*{Introduction}
The rigorous treatment of the statistical mechanics of quantum lattice (or spin) systems has been one of the major successes of the $C^*$-algebraic approach to quantum physics. The main results are due to many people but a detailed overview is presented in the monograph \cite{BR}. (Chapter 15 of \cite{OP} is a concise summary, see also \cite{Se}.) The usual quantum spin system is described on the infinite tensor product $C^*$-algebra of full matrix algebras. Given an interaction $\Phi$, the local Hamiltonian induces the local dynamics and the local equilibrium state. The global dynamics and the global equilibrium states are obtained by a limiting procedure. The equivalence of the KMS condition, the Gibbs condition and the variational principle for translation-invariant states is the main essence in the theory; they were established around 1970 (\cite{Ar0,LR,Ro}). The above mentioned concepts are used to describe equilibrium states. Recently Araki and Moriya extended the ideas to fermionic lattice systems \cite{AM}.
An attempt to extend quantum statistical mechanics from the setting of spin systems to some approximately finite $C^*$-algebras was made by Kishimoto \cite{Ki3,Ki4}. Motivated by the chemical potential theory due to Araki, Haag, Kastler and Takesaki \cite{AHKT}, in our previous paper \cite{HP2} we study the equivalence of the KMS condition, the Gibbs condition and the variational principle on approximately finite $C^*$-algebras as a natural extension of the thermodynamics of one-dimensional quantum lattice systems. It turned out that equation (2.8) in the proof of \cite[Theorem 2.2]{HP2} does not hold and the equivalence formulated in that theorem is recovered here under stronger conditions. (The error in the proof was pointed out to the authors by E.~St\o rmer and S.~Neshveyev some years ago.)
A gauge-invariant $C^*$-system is obtained as the fixed point subalgebra of the infinite tensor product of full matrix algebras under the tensor product unitary action of a compact group. This situation is a typical example of the chemical potential theory. The primary aim of the present paper is to recover the main results in \cite{HP2} in the restrictive setup of such gauge-invariant $C^*$-systems. The second aim is to discuss entropy densities and macroscopic uniformity for extremal equilibrium states in such $C^*$-systems and to extend the arguments in \cite{HP1}.
\section{Equilibrium states with chemical potentials }
We begin by fixing basic notations and terminologies. Let $M_d({\Bbb C})$ be the algebra of $d\times d$ complex matrices. Let ${\cal F}$ denote a one-dimensional spin (or UHF) $C^*$-algebra $\bigotimes_{k\in{\Bbb Z}}{\cal F}_k$ with ${\cal F}_k:=M_d({\Bbb C})$, and $\theta$ the right shift on ${\cal F}$. Let $G$ be a separable compact group and $\sigma$ a continuous unitary representation of $G$ on ${\Bbb C}^d$ so that a product action $\gamma$ of $G$ on ${\cal F}$ is defined by $\gamma_g:=\bigotimes_{\Bbb Z}{\rm Ad}\,\sigma_g$, $g\in G$. Let ${\cal A}:={\cal F}^\gamma$, the fixed point subalgebra of ${\cal F}$ for the action $\gamma$ of $G$. For a finite subset $\Lambda\subset{\Bbb Z}$ let ${\cal F}_\Lambda:=\bigotimes_{k\in\Lambda}{\cal F}_k$ and
${\cal A}_\Lambda:={\cal A}\cap{\cal F}_\Lambda={\cal F}_\Lambda^\gamma$, the fixed point subalgebra for $\gamma|_{{\cal F}_\Lambda}$. Then ${\cal A}$ is an AF $C^*$-algebra generated by $\{{\cal A}_\Lambda\}_{\Lambda\subset{\Bbb Z}}$ (\cite[Proposition 2.1]{Pr}). The algebra ${\cal A}$ is called the {\it observable algebra} while ${\cal F}$ is called the {\it field algebra}. Let ${\cal S}({\cal A})$ denote the state space of ${\cal A}$ and ${\cal S}_\theta({\cal A})$ the set of all $\theta$-invariant states of ${\cal A}$.
An {\it interaction} $\Phi$ is a mapping from the finite subsets of ${\Bbb Z}$ into ${\cal A}$ such that $\Phi(\emptyset)=0$ and $\Phi(X)=\Phi(X)^*\in{\cal A}_X$ for each finite $X\subset{\Bbb Z}$. Given an interaction $\Phi$ and a finite subset $\Lambda\subset{\Bbb Z}$, define the {\it local Hamiltonian} $H_\Lambda$ by $$ H_\Lambda:=\sum_{X\subset\Lambda}\Phi(X), $$ and the {\it surface energy} $W_\Lambda$ by $$ W_\Lambda:=\sum\{\Phi(X): X\cap\Lambda\ne\emptyset,\,X\cap\Lambda^c\ne\emptyset\}, $$ whenever the sum converges in norm.
Throughout the paper we assume that an interaction $\Phi$ is {\it $\theta$-invariant} and has {\it relatively short range}\,; namely, $\theta(\Phi(X))=\Phi(X+1)$, where $X+1:=\{k+1:k\in X\}$, for every finite $X\subset{\Bbb Z}$ and $$
|||\Phi|||:=\sum_{X\ni0}{\|\Phi(X)\|\over|X|}<\infty, $$
where $|X|$ means the cardinality of $X$. Let ${\cal B}({\cal A})$ denote the set of all such interactions, which is a real Banach space with the usual linear operations and the norm $|||\Phi|||$. Moreover, let ${\cal B}_0({\cal A})$ denote the set of all $\Phi\in{\cal B}({\cal A})$ such that $$
\sum_{X\ni0}\|\Phi(X)\|<\infty\quad\mbox{and}\quad
\sup_{n\ge1}\|W_{[1,n]}\|<\infty. $$ Then ${\cal B}_0({\cal A})$ is a real Banach space with the norm $$
\|\Phi\|_0:=\sum_{X\ni0}\|\Phi(X)\|+\sup_{n\ge1}\|W_{[1,n]}\|
\ \ (\ge|||\Phi|||). $$ We define the real Banach space ${\cal B}_0({\cal F})$ in a similar manner.
When $\Phi\in{\cal B}_0({\cal A})$ we have a strongly continuous one-parameter automorphism group $\alpha^\Phi$ of ${\cal F}$ such that $$
\lim_{l,m\to\infty}\|\alpha^\Phi_t(a)-
e^{itH_{[-l,m]}}ae^{-itH_{[-l,m]}}\|=0 $$ for all $a\in{\cal F}$ uniformly for $t$ in finite intervals (see \cite[Theorem 8]{Ki1} and also \cite[6.2.6]{BR}). It is straightforward to see that $\alpha^\Phi_t\theta=\theta\alpha^\Phi_t$ and $\alpha^\Phi_t\gamma_g=\gamma_g\alpha^\Phi_t$ for all $t\in{\Bbb R}$ and $g\in G$ so that $\alpha^\Phi_t({\cal A})={\cal A}$, $t\in{\Bbb R}$. The sextuple $({\cal F},{\cal A},G,\alpha^\Phi,\gamma,\theta)$ is a so-called {\it field system} in the chemical potential theory (\cite{AHKT}, \cite[\S5.4.3]{BR}). The most general notion of equilibrium states is described by the KMS condition in a general one-parameter $C^*$-dynamical system (see \cite[\S5.3.1]{BR} for example). In this paper we consider only $(\alpha^\Phi,\beta)$-KMS states with $\beta=1$; so we refer to those states as just {\it $\alpha^\Phi$-KMS states}. The next proposition says that the $\alpha^\Phi$-KMS states are automatically $\theta$-invariant. This was stated in \cite[Proposition 4.2]{HP2} but the proof there was given in a wrong way.
\begin{prop}\label{P-1.1} Let $\Phi\in{\cal B}_0({\cal A})$, and let $K({\cal A},\Phi)$ denote the set of all $\alpha^\Phi$-KMS states of ${\cal A}$. Then $K({\cal A},\Phi)\subset{\cal S}_\theta({\cal A})$, and $\omega\in K({\cal A},\Phi)$ is extremal in $K({\cal A},\Phi)$ if and only if $\omega$ is extremal in ${\cal S}_\theta({\cal A})$. \end{prop}
\noindent{\it Proof.}\enspace The proof below is essentially same as in \cite[\S III]{FVV}. Recall that the generator of $\alpha^\Phi$ is the closure of the derivation $\delta_0$ with domain $D(\delta_0)=\bigcup_\Lambda{\cal A}_\Lambda$ (over the finite intervals $\Lambda\subset{\Bbb Z}$) given by $$ \delta_0(a):=i\sum_{X\cap\Lambda\ne\emptyset}[\Phi(X),a], \qquad a\in{\cal A}_\Lambda. $$ For each $n\in{\Bbb N}$ let $u_n\in{\cal F}_{[-n,n]}$ be a unitary implementing the cyclic permutation of ${\cal F}_{[-n,n]}=\bigotimes_{-n}^nM_d({\Bbb C})$, i.e., $$ {\rm Ad}\, u_n(a_{-n}\otimes a_{-n+1}\otimes\cdots\otimes a_{n-1}\otimes a_n) =a_n\otimes a_{-n}\otimes a_{-n+1}\otimes\cdots\otimes a_{n-1} $$ for $a_k\in M_d({\Bbb C})$. Since $[u_n,\bigotimes_{-n}^n\sigma_g]=0$, we get $\gamma_g(u_n)=u_n$ for all $g\in G$ so that $u_n\in{\cal A}$. Moreover, since ${\rm Ad}\, u_n(a)=\theta(a)$ whenever $a\in{\cal A}_{[-n,n-1]}$, it is immediate to see that $\theta(a)=\lim_{n\to\infty}{\rm Ad}\, u_n(a)$ for all $a\in{\cal A}$. Hence, one can apply \cite[Corollary II.3]{FVV} (or \cite[5.3.33A]{BR}) to obtain $K({\cal A},\Phi)\subset{\cal S}_\theta({\cal A})$, and it suffices to show that
$\sup_{n\ge1}\|\delta_0(u_n)\|<\infty$. This indeed follows because \begin{eqnarray*}
\|\delta_0(u_n)\|
&=&\Bigg\|\sum_{X\cap[-n,n]\ne\emptyset}[\Phi(X),u_n]\Bigg\| \\
&=&\Bigg\|\sum_{X\cap[-n,n]\ne\emptyset}(\Phi(X)-u_n\Phi(X)u_n^*)\Bigg\| \\
&\le&\Bigg\|\sum_{X\subset[-n,n-1]}(\Phi(X)-\theta(\Phi(X)))\Bigg\|
+\Bigg\|\sum_{X\cap[-n,n]\ne\emptyset\atop X\not\subset[-n,n-1]}
(\Phi(X)-u_n\Phi(X)u_n^*)\Bigg\| \\
&\le&\Bigg\|\sum_{X\subset[-n,n-1]}(\Phi(X)-\Phi(X+1))\Bigg\|
+2\Bigg\|\sum_{X\cap[-n,n]\ne\emptyset\atop X\not\subset[-n,n-1]}
\Phi(X)\Bigg\| \\
&\le&\sum_{X\ni-n}\|\Phi(X)\|+\sum_{X\ni n}\|\Phi(X)\|
+2\sum_{X\ni n}\|\Phi(X)\|
+2\Bigg\|\sum_{X\cap[-n,n]\ne\emptyset\atop X\not\subset[-n,n]}
\Phi(X)\Bigg\| \\
&\le&4\sum_{X\ni0}\|\Phi(X)\|+2\|W_{[-n,n]}\| \\
&\le&4\|\Phi\|_0<\infty. \end{eqnarray*}
For each $\omega\in{\cal S}_\theta({\cal A})$ let $(\pi_\omega,{\cal H}_\omega)$ be the GNS cyclic representation of ${\cal A}$ associated with $\omega$ and $U_\theta$ be a unitary implementing $\theta$ so that $\pi_\omega(\theta(a))=U_\theta\pi_\omega(a)U_\theta^*$ for $a\in{\cal A}$. Since $({\cal A},\theta)$ is asymptotically abelian in the norm sense, i.e.,
$\lim_{|n|\to\infty}\|[a,\theta^n(b)]\|=0$ for all $a,b,\in{\cal A}$, it is well known \cite[4.3.14]{BR} that \begin{equation}\label{F-1.1} \pi_\omega({\cal A})'\cap\{U_\theta\}' \subset\pi_\omega({\cal A})'\cap\pi_\omega({\cal A})''. \end{equation} According to \cite[Lemma 4.7]{TW}, the second assertion is a consequence of this together with the first assertion (see also \cite[4.3.17 and 5.3.30\,(3)]{BR} for extremal points of ${\cal S}_\theta({\cal A})$ and of $K({\cal A},\Phi)$).\qed
\begin{remark}{\rm Since $({\cal A},\theta)$ is asymptotically abelian as mentioned in the above proof, ${\cal S}_\theta({\cal A})$ becomes a simplex. It is also well known that $K({\cal A},\Phi)$ is a simplex. These were shown in \cite[\S4]{TW}, where the lattice (or simplex) structure of state spaces was discussed in a rather general setting. (See also \cite[4.3.11 and 5.3.30\,(2)]{BR}). Moreover, it is seen from \eqref{F-1.1} \cite[Lemma 4.7$'$]{TW} that $K({\cal A},\Phi)$ is a face of ${\cal S}_\theta({\cal A})$. }\end{remark}
It is known \cite[Lemma 4.1]{HP2} that any tracial state $\phi$ of ${\cal A}$ is $\theta$-invariant and $\phi$ is extremal if and only if it is multiplicative in the sense that $\phi(ab)=\phi(a)\phi(b)$ for all $a\in{\cal A}_{[i,j]}$ and $b\in{\cal A}_{[j+1,k]}$, $i\le j<k$. The $\theta$-invariance of any tracial state of ${\cal A}$ is a particular case of Proposition \ref{P-1.1} where $\Phi$ is identically zero. We denote by $\ET^{\rm f}({\cal A})$ the set of all faithful and extremal tracial states of ${\cal A}$. On the other hand, we denote by $\Xi(G,\sigma)$ the set of all continuous one-parameter subgroups $t\mapsto\xi_t$ of $G$. Two elements $\xi,\xi'$ in $\Xi(G,\sigma)$ are identified if there exists $g\in G$ such that ${\rm Ad}\,\sigma_{g^{-1}\xi_tg}={\rm Ad}\,\sigma_{\xi_t'}$, $t\in{\Bbb R}$. In fact, this defines an equivalence relation and we redefine $\Xi(G,\sigma)$ as the set of equivalence classes. Then, \cite[Proposition 4.3]{HP2} says
\begin{prop}\label{P-1.3} There is a bijective correspondence $\phi\leftrightarrow\xi$ between $\ET^{\rm f}({\cal A})$ and $\Xi(G,\sigma)$ under the condition that $\phi$ extends to a $\gamma_\xi$-KMS state of ${\cal F}$. \end{prop}
Let $\tau_0$ be the normalized trace on $M_d({\Bbb C})$. Let $\phi$ and $\xi$ be as in the above proposition. Then there exists a unique selfadjoint $h\in{\cal F}_{\{0\}}=M_d({\Bbb C})$ such that $\tau_0(e^{-h})=1$ and ${\rm Ad}\,\sigma_{\xi_t}={\rm Ad}\, e^{ith}$ for all $t\in{\Bbb R}$. We call this $h$ the {\it generator} of $\xi$. Note that $\tau_0(e^{-h}\,\cdot)$ is a unique KMS state of $M_d({\Bbb C})$ with respect to ${\rm Ad}\, e^{ith}$ and thus $\hat\phi:=\bigotimes_{\Bbb Z}\tau_0(e^{-h}\,\cdot)$ is a unique KMS state of ${\cal F}$ with respect to $\gamma_{\xi_t}=\bigotimes_{\Bbb Z}{\rm Ad}\, e^{ith}$; so
$\phi=\hat\phi|_{\cal A}$.
Let $\Phi\in{\cal B}_0({\cal A})$ and $\xi\in\Xi(G,\sigma)$, and let $\omega$ be an $\alpha^\Phi$-KMS state of ${\cal A}$. We say that $\xi$ is the {\it chemical potential} of $\omega$ if there exists an extension $\hat\omega$ of $\omega$ to ${\cal F}$ which is a KMS state with respect to $\alpha^\Phi_t\gamma_{\xi_t}$. Let $h$ be the generator of $\xi$, and define a $\theta$-invariant interaction $\Phi^h$ in ${\cal F}$ by \begin{equation}\label{F-1.2} \Phi^h(X):= \begin{cases} \Phi(\{j\})+\theta^j(h) & \text{if $X=\{j\}$,\ $j\in{\Bbb Z}$}, \\ \Phi(X) & \text{otherwise}. \end{cases} \end{equation} Since $\Phi^h\in{\cal B}_0({\cal F})$, it generates a one-parameter automorphism group $\alpha^{\Phi^h}$ on ${\cal F}$. Then, we have $\alpha^{\Phi^h}_t=\alpha^\Phi_t\gamma_{\xi_t}$, $t\in{\Bbb R}$, and
$\alpha^\Phi|_{\cal A}=\alpha^{\Phi^h}|_{\cal A}$ (\cite[Lemma 4.4]{HP2}). Due to the uniqueness of an $\alpha^{\Phi^h}$-KMS state of ${\cal F}$ (\cite{Ar1,Ki2}), we notice that there is a unique $\alpha^\Phi$-KMS state with chemical potential $\xi$, which is automatically $\theta$-invariant and faithful. On the other hand, a consequence of the celebrated chemical potential theory in \cite[\S II]{AHKT} together with Proposition \ref{P-1.1} is the following: If $\omega$ is a faithful and extremal $\alpha^\Phi$-KMS state of ${\cal A}$, then $\omega$ enjoys the chemical potential. A complete conclusion in this direction will be given in Theorem \ref{T-1.6} below, and Proposition \ref{P-1.3} is its special case.
To introduce the Gibbs condition, one needs the notion of perturbations of states of ${\cal A}$. Let $\omega,\psi\in{\cal S}({\cal A})$. For each finite interval $\Lambda\subset{\Bbb Z}$, the {\it relative entropy} of
$\psi_\Lambda:=\psi|_{{\cal A}_\Lambda}$ with respect to
$\omega_\Lambda:=\omega|_{{\cal A}_\Lambda}$ is given by $$ S(\psi_\Lambda,\omega_\Lambda) :={\rm Tr}_\Lambda\biggl({d\psi_\Lambda\over d{\rm Tr}_\Lambda}\biggl( \log{d\psi_\Lambda\over d{\rm Tr}_\Lambda}-\log{d\omega_\Lambda\over d{\rm Tr}_\Lambda} \biggr)\biggr). $$ Here, ${\rm Tr}_\Lambda$ denotes the canonical trace on ${\cal A}_\Lambda$ such that ${\rm Tr}_\Lambda(e)=1$ for any minimal projection $e$ in ${\cal A}_\Lambda$. Then the relative entropy $S(\psi,\omega)$ is defined by $$ S(\psi,\omega):=\sup_{\Lambda\subset{\Bbb Z}}S(\psi_\Lambda,\omega_\Lambda) =\lim_{n\to\infty}S(\psi_{[-n,n]},\omega_{[-n,n]}). $$ (See \cite{OP} for details on the relative entropy for states of a $C^*$-algebra.) For each $\omega\in{\cal S}({\cal A})$ and $Q=Q^*\in{\cal A}$, since $\psi\mapsto S(\psi,\omega)+\psi(Q)$ is weakly* lower semicontinuous and strictly convex on ${\cal S}({\cal A})$, the {\it perturbed state} $[\omega^Q]$ by $Q$ is defined as a unique minimizer of this functional (\cite{Do,OP}). Recall \cite{Ar2,Do} that \begin{equation}\label{F-1.3}
|S(\psi,\omega)-S(\psi,[\omega^Q])|\le2\|Q\| \end{equation} for every $\psi,\omega\in{\cal S}({\cal A})$ and $Q=Q^*\in{\cal A}$.
Let $\Phi$ be an interaction in ${\cal A}$ and $\phi$ a tracial state of ${\cal A}$. For each finite $\Lambda\subset{\Bbb Z}$, the {\it local Gibbs state} $\phi^G_\Lambda$ of ${\cal A}_\Lambda$ with respect to $\Phi$ and $\phi$ is defined by $$ \phi^G_\Lambda(a):={\phi(e^{-H_\Lambda}a)\over\phi(e^{-H_\Lambda})}, \qquad a\in{\cal A}_\Lambda. $$ Let $\omega\in{\cal S}({\cal A})$ and $(\pi_\omega,{\cal H}_\omega,\Omega_\omega)$ be the cyclic representation of ${\cal A}$ associated with $\omega$. We say that $\omega$ satisfies the {\it strong Gibbs condition} if $\Omega_\omega$ is separating for $\pi_\omega({\cal A})''$ and if, for each finite $\Lambda\subset{\Bbb Z}$, there exists a conditional expectation from $\pi_\omega({\cal A})''$ onto $\pi_\omega({\cal A}_\Lambda)\vee\pi_\omega({\cal A}_{\Lambda^c})''$ with respect to $[\omega^{-W_\Lambda}]\,\tilde{}\,$ and \begin{equation}\label{F-1.4} [\omega^{-W_\Lambda}](ab)=\phi^G_\Lambda(a)[\omega^{-W_\Lambda}](b), \qquad a\in{\cal A}_\Lambda,\ b\in{\cal A}_{\Lambda^c}. \end{equation} Here, $[\omega^{-W_\Lambda}]\,\tilde{}\,$ is the normal extension of the perturbed state $[\omega^{-W_\Lambda}]$ to $\pi_\omega({\cal A})''$ (see \cite[p.~826]{HP2}). Furthermore, we say that $\omega$ satisfies the {\it weak Gibbs condition} with respect to $\Phi$ and $\phi$ if
$[\omega^{-W_\Lambda}]|_{{\cal A}_\Lambda}=\phi^G_\Lambda$ for any finite $\Lambda\subset{\Bbb Z}$.
Now, let $\Phi\in{\cal B}({\cal A})$, $\phi\in\ET^{\rm f}({\cal A})$ and $\omega\in{\cal S}_\theta({\cal A})$. From now on, for simplicity we write ${\cal A}_n:={\cal A}_{[1,n]}$, $H_n:=H_{[1,n]}$,
$\phi_n:=\phi|_{{\cal A}_n}$, $\omega_n:=\omega|_{{\cal A}_n}$, etc.\ for each $n\in{\Bbb N}$. The {\it mean relative entropy} of $\omega$ with respect to $\phi$ is defined by $$ S_{\rm M}(\omega,\phi):=\lim_{n\to\infty}{1\over n}S(\omega_n,\phi_n) =\sup_{n\ge1}{1\over n}S(\omega_n,\phi_n). $$ (See \cite[Lemma 3.1]{HP2} for justification of the definition.) Define the {\it mean energy} $A_\Phi$ of $\Phi$ by $$
A_\Phi:=\sum_{X\ni0}{\Phi(X)\over|X|}\ \ (\in{\cal A}). $$ Furthermore, it is known \cite[Theorem 3.5]{HP2} that $\lim_{n\to\infty}{1\over n}\log\phi(e^{-H_n})$ exists and $$ \lim_{n\to\infty}{1\over n}\log\phi(e^{-H_n}) =\sup\{-S_{\rm M}(\omega,\phi)-\omega(A_\Phi):\omega\in{\cal S}_\theta({\cal A})\}. $$ The {\it pressure} of $\Phi$ with respect to $\phi$ is thus defined by $$ p(\Phi,\phi):=\lim_{n\to\infty}{1\over n}\log\phi(e^{-H_n}). $$ We have the variational expressions of $p(\Phi,\phi)$ and $S_{\rm M}(\omega,\phi)$ as follows.
\begin{prop}\label{P-1.4} Let $\phi\in\ET^{\rm f}({\cal A})$. If $\Phi\in{\cal B}({\cal A})$, then \begin{equation}\label{F-1.5} p(\Phi,\phi) =\sup\{-S_{\rm M}(\omega,\phi)-\omega(A_\Phi):\omega\in{\cal S}_\theta({\cal A})\}. \end{equation} If $\omega\in{\cal S}_\theta({\cal A})$, then \begin{equation}\label{F-1.6} -S_{\rm M}(\omega,\phi)=\inf\{p(\Phi,\phi)+\omega(A_\Phi):\Phi\in{\cal B}({\cal A})\}. \end{equation} \end{prop}
\noindent{\it Proof.}\enspace The expression \eqref{F-1.5} was given in \cite[Theorem 3.5]{HP2} as mentioned above. We further can transform \eqref{F-1.5} into \eqref{F-1.6} by a simple duality argument. In fact, for each $\omega\in{\cal S}_\theta({\cal A})$ define $f_\omega\in{\cal B}({\cal A})^*$, the dual Banach space of ${\cal B}({\cal A})$, by $f_\omega(\Phi):=-\omega(A_\Phi)$, and set $\Gamma:=\{f_\omega:\omega\in{\cal S}_\theta({\cal A})\}$. Then, it is immediate to see that $\omega\in{\cal S}_\theta({\cal A})\mapsto f_\omega\in\Gamma$ is an affine homeomorphism in the weak* topologies so that $\Gamma$ is a weakly* compact convex subset of ${\cal B}({\cal A})^*$. Define $F:{\cal B}({\cal A})^*\to[0,+\infty]$ by $$ \begin{cases} F(f_\omega):=S_{\rm M}(\omega,\phi) & \text{for $\omega\in{\cal S}_\theta({\cal A})$}, \\ F(g):=+\infty & \text{if $g\in{\cal B}({\cal A})^*\setminus\Gamma$}. \end{cases} $$ Then $F$ is a weakly* lower semicontinuous and convex function on ${\cal B}({\cal A})^*$ (see \cite[Proposition 3.2]{HP2}). Since \eqref{F-1.5} means that $$ p(\Phi,\phi)=\sup\{g(\Phi)-F(g):g\in{\cal B}({\cal A})^*\}, \quad\Phi\in{\cal B}({\cal A}), $$ it follows by duality (see \cite[Proposition I.4.1]{ET} for example) that $$ F(g)=\sup\{g(\Phi)-p(\Phi,\phi):\Phi\in{\cal B}({\cal A})\}, \quad g\in{\cal B}({\cal A})^*. $$ Hence, for every $\omega\in{\cal S}_\theta({\cal A})$, \begin{eqnarray*} S_{\rm M}(\omega,\phi)&=&\sup\{f_\omega(\Phi)-p(\Phi,\phi):\Phi\in{\cal B}({\cal A})\} \\ &=&-\inf\{p(\Phi,\phi)+\omega(A_\Phi):\Phi\in{\cal B}({\cal A})\}, \end{eqnarray*} giving \eqref{F-1.6}.\qed
We say that $\omega$ satisfies the {\it variational principle} with respect to $\Phi$ and $\phi$ if \begin{equation}\label{F-1.7} p(\Phi,\phi)=-S_{\rm M}(\omega,\phi)-\omega(A_\Phi). \end{equation}
With the above definitions in mind we have the next theorem, recovering main results of \cite{HP2} (Corollary 3.11 and Theorem 4.5) in the special setup of gauge-invariant $C^*$-systems.
\begin{thm}\label{T-1.5} Let $\Phi\in{\cal B}_0({\cal A})$, $\phi\in\ET^{\rm f}({\cal A})$ and $\xi\in\Xi(G,\phi)$ with $\phi\leftrightarrow\xi$ in the sense of Proposition $\ref{P-1.3}$. Then the following conditions for $\omega\in{\cal S}({\cal A})$ are equivalent: \begin{itemize} \item[(i)] $\omega$ is an $\alpha^\Phi$-KMS state with chemical potential $\xi$; \item[(ii)] $\omega$ satisfies the strong Gibbs condition with respect to $\Phi$ and $\phi$; \item[(iii)] $\omega\in{\cal S}_\theta({\cal A})$ and $\omega$ satisfies the weak Gibbs condition with respect to $\Phi$ and $\phi$; \item[(iv)] $\omega\in{\cal S}_\theta({\cal A})$ and $\omega$ satisfies the variational principle with respect to $\Phi$ and $\phi$. \end{itemize}
Furthermore, there exists a unique $\omega\in{\cal S}({\cal A})$ satisfying one $($hence all\,$)$ of the above conditions. \end{thm}
\noindent{\it Proof.}\enspace (i) $\Rightarrow$ (ii). Let $\omega$ be an $\alpha^\Phi$-KMS state with chemical potential $\xi$ and $(\pi_\omega,{\cal H}_\omega,\Omega_\omega)$ be the associated cyclic representation of ${\cal A}$. It is well known that $\Omega_\omega$ is separating for $\pi_\omega({\cal A})''$ (see \cite[5.3.9]{BR} for example). According to the proof of \cite[Theorem 2.2, (i) $\Rightarrow$ (ii)]{HP2}, we see that for any finite $\Lambda\subset{\Bbb Z}$ there exists a conditional expectation from $\pi_\omega({\cal A})''$ onto $\pi_\omega({\cal A}_\Lambda)\vee\pi_\omega({\cal A}_{\Lambda^c})''$ with respect to $[\omega^{-W_\Lambda}]\,\tilde{}\,$. (Note that this part of the proof of \cite[Theorem 2.2, (i) $\Rightarrow$ (ii)]{HP2} is valid.) Moreover, the proof of \cite[Theorem 4.5]{HP2} shows that \eqref{F-1.4} holds for any finite $\Lambda\subset{\Bbb Z}$. Hence we obtain (ii).
(ii) $\Rightarrow$ (iii). The proof of \cite[Theorem 2.2, (ii) $\Rightarrow$ (i)]{HP2} guarantees that (ii) implies $\omega\in K({\cal A},\Phi)$. Hence Proposition \ref{P-1.1} gives the $\theta$-invariance of $\omega$.
(iii) $\Rightarrow$ (iv) is contained in \cite[Proposition 3.9]{HP2} proven in a more general setting.
(iv) $\Rightarrow$ (i). To prove this as well as the last assertion, it suffices to show that a state $\omega\in{\cal S}({\cal A})$ satisfying (iv) is unique. First, note that the variational principle \eqref{F-1.7} means that $\Psi\mapsto-\omega(A_\Psi)$ is a tangent functional to the graph of $p(\cdot,\phi)$ on ${\cal B}_0({\cal A})$ at $\Phi$. Let $h\in M_d({\Bbb C})$ be the generator of $\xi$ and $\Phi^h$ a $\theta$-invariant interaction in ${\cal F}$ defined by \eqref{F-1.2}. Since $\Phi^h\in{\cal B}_0({\cal F})$, there is a unique $\alpha^{\Phi^h}$-KMS state $\hat\omega$ of ${\cal F}$. Equivalently, there is a unique $\theta$-invariant state $\hat\omega$ of ${\cal F}$ satisfying the variational principle with respect to $\Phi^h$, i.e., $$ P_{\cal F}(\Phi^h)=s_{\cal F}(\hat\omega)-\hat\omega(A_{\Phi^h}). $$ Recall here that the pressure $P_{\cal F}(\Psi)$ of $\Psi\in{\cal B}_0({\cal F})$ and the mean entropy $s_{\cal F}(\psi)$ of $\psi\in{\cal S}_\theta({\cal F})$ are $$ P_{\cal F}(\Psi):=\lim_{n\to\infty}{1\over n}\log{\rm Tr}_{{\cal F}_n}(e^{-H_n(\Psi)}), \quad s_{\cal F}(\psi):=\lim_{n\to\infty}{1\over n}S(\psi_n), $$ where ${\rm Tr}_{{\cal F}_n}$ is the usual trace on ${\cal F}_n$ and $H_n(\Psi)$ is the local Hamiltonian of $\Psi$ inside the interval $[1,n]$. The uniqueness property above means (see \cite[Proposition I.5.3]{ET} for example) that the pressure function $P_{\cal F}(\cdot)$ on ${\cal B}_0({\cal F})$ is differentiable at $\Phi^h$. We have (see \cite[(4.11)]{HP2}) \begin{equation}\label{F-1.8} p(\Phi,\phi)=P_{\cal F}(\Phi^h)-\log d,\qquad\Phi\in{\cal B}_0({\cal A}). \end{equation} By this and \eqref{F-1.2} we obtain $$p(\Phi+\Psi,\phi)=P_{\cal F}(\Phi^h+\Psi)-\log d,\qquad\Psi\in{\cal B}_0({\cal A}), $$ which implies that $\Psi\in{\cal B}_0({\cal A})\mapsto p(\Psi,\phi)$ is differentiable at $\Phi$. Hence the required implication follows.\qed
The next theorem is a right formulation of what we wanted to show in \cite{HP2}, though in the restricted setup of gauge-invariant $C^*$-systems.
\begin{thm}\label{T-1.6} If $\Phi\in{\cal B}_0({\cal A})$ and $\omega\in{\cal S}({\cal A})$, then the following conditions are equivalent: \begin{itemize} \item[(i)] $\omega$ is a faithful and extremal $\alpha^\Phi$-KMS state; \item[(ii)] $\omega$ is $\alpha^\Phi$-KMS with some chemical potential $\xi\in\Xi(G,\sigma)$; \item[(iii)] $\omega$ satisfies the strong Gibbs condition with respect to $\Phi$ and some $\phi\in\ET^{\rm f}({\cal A})$; \item[(iv)] $\omega\in{\cal S}_\theta({\cal A})$ and $\omega$ satisfies the weak Gibbs condition with respect to $\Phi$ and some $\phi\in\ET^{\rm f}({\cal A})$; \item[(v)] $\omega\in{\cal S}_\theta({\cal A})$ and $\omega$ satisfies the variational principle with respect to $\Phi$ and some $\phi\in\ET^{\rm f}({\cal A})$. \end{itemize} \end{thm}
\noindent{\it Proof.}\enspace In view of Theorem \ref{T-1.5} we only need to prove the equivalence between (i) and (ii). (i) $\Rightarrow$ (ii) is a consequence of the chemical potential theory in \cite[\S II]{AHKT} and Proposition \ref{P-1.1} as mentioned above (after Proposition \ref{P-1.3}). Conversely, suppose (ii) and let $\hat\omega$ be a (unique) KMS state of ${\cal F}$ with respect to
$\alpha^\Phi\gamma_{\xi_t}=\alpha^{\Phi^h}$ so that $\omega=\hat\omega|_{\cal A}$. Since $\hat\omega$ is obviously faithful, so is $\omega$. Moreover, the extremality of $\omega$ in ${\cal S}_\theta({\cal A})$ follows from that of $\hat\omega$ in ${\cal S}_\theta({\cal F})$. This may be well known but we sketch the proof for convenience. Let $(\hat\pi,\hat{\cal H},\hat\Omega,\hat U_\theta)$ be the cyclic representation of ${\cal F}$ associated with $\hat\omega$, where $\hat U_\theta$ is a unitary implementing $\theta$ so that $\hat U_\theta\hat\Omega=\hat\Omega$ and $\hat\pi(\theta(a))=\hat U_\theta\hat\pi(a)\hat U_\theta^*$ for $a\in{\cal F}$. Then the cyclic representation of ${\cal A}$ associated with $\omega$ is given by ${\cal H}_\omega:=\overline{\hat\pi({\cal A})\hat\Omega}$ and
$\pi_\omega(a):=\hat\pi(a)|_{{\cal H}_\omega}$ for $a\in{\cal A}$ with $\Omega_\omega:=\hat\Omega$. Let $P:\hat{\cal H}\to{\cal H}_\omega$ be the orthogonal projection. Since $\hat U_\theta P=P\hat U_\theta$,
$U_\theta^\omega:=\hat U_\theta|_{{\cal H}_\omega}$ is a unitary implementing
$\theta|_{\cal A}$. Let $\hat\sigma$ denote the modular automorphism group of $\hat\pi({\cal F})''$ associated with $\hat\Omega$. Since $\hat\sigma_t(\hat\pi(a))=\hat\pi(\alpha^\Phi_t(a))\in\hat\pi({\cal A})$ for all $a\in{\cal A}$, there exists the conditional expectation $E:\hat\pi({\cal F})''\to\hat\pi({\cal A})''$ with respect to the state $\langle\,\cdot\,\hat\Omega,\hat\Omega\rangle$ (\cite{Ta}). Notice that $E$ is $\theta$-covariant, i.e., $E(\hat U_\theta x\hat U_\theta^*)=\hat U_\theta E(x)\hat U_\theta^*$ for all $x\in\hat\pi({\cal F})''$. Now, assume that $\omega_1\in{\cal S}_\theta({\cal A})$ and $\omega_1\le\lambda\omega$ for some $\lambda>0$; hence there exists $T_1\in\pi_\omega({\cal A})'$ with $0\le T_1\le\lambda$ such that $\omega_1(a)=\<T_1\pi_\omega(a)\Omega_\omega,\Omega_\omega\rangle$ for $a\in{\cal A}$, and $T_1U_\theta^\omega=U_\theta^\omega T_1$. Define $T:=T_1P+({\bf 1}-P)$ on $\hat{\cal H}$. Then it is easy to check that $0\le T\le\lambda$, $T\in\hat\pi({\cal A})'$ and $T\hat U_\theta=\hat U_\theta T$. Define $$ \hat\omega_1(a):=\<TE(\hat\pi(a))\hat\Omega,\hat\Omega\rangle,\qquad a\in{\cal F}, $$
which is a state of ${\cal F}$ with $\hat\omega_1|_{\cal A}=\omega_1$ and $\hat\omega_1\le\lambda\hat\omega$. For any $a\in{\cal F}$ we get $$ \hat\omega_1(\theta(a)) =\<TE(\hat U_\theta\hat\pi(a)\hat U_\theta^*)\hat\Omega,\hat\Omega\rangle =\<TE(\hat\pi(a))\hat\Omega,\hat\Omega\rangle=\hat\omega_1(a) $$ so that the extremality of $\hat\omega$ implies $\hat\omega_1=\hat\omega$ and so $\omega_1=\omega$. Hence $\omega$ is extremal in ${\cal S}_\theta({\cal A})$ (hence in $K({\cal A},\Phi)$), and (ii) $\Rightarrow$ (i) is shown.\qed
\section{More about variational principle} \setcounter{equation}{0}
In this section we consider the variational principle for $\omega\in{\cal S}_\theta({\cal A})$ in terms of the mean entropy and the pressure which are defined by use of canonical traces on local algebras (not with respect to a tracial state in $\ET^{\rm f}({\cal A})$). Let $\nu$ be the restriction of $\bigotimes_{\Bbb Z}\tau_0$ to ${\cal A}$, which is an element of $\ET^{\rm f}({\cal A})$ corresponding to the trivial chemical potential $\xi=1$. For each $n\in{\Bbb N}$ the $n$-fold tensor product $\bigotimes_1^n\sigma$ of the unitary representation $\sigma$ is decomposed as $$ \bigotimes_1^n\sigma =m_1\sigma_1\oplus m_2\sigma_2\oplus\cdots\oplus m_{K_n}\sigma_{K_n}, $$ where $\sigma_i\in\widehat G$, $1\le i\le K_n$, are contained in $\bigotimes_1^n\sigma$ with multiplicities $m_i$. For $1\le i\le K_n$ let $d_i$ be the dimension of $\sigma_i$. Then, we have $\sum_{i=1}^{K_n}m_id_i=d^n$ and \begin{eqnarray} {\cal A}_n&=&\bigoplus_{i=1}^{K_n}(M_{m_i}({\Bbb C})\otimes{\bf 1}_{d_i}) \cong\bigoplus_{i=1}^{K_n}M_{m_i}({\Bbb C}), \label{F-2.1} \\ {\cal F}_n\cap{\cal A}_n'&=&\bigoplus_{i=1}^{K_n}({\bf 1}_{m_i}\otimes M_{d_i}({\Bbb C})) \cong\bigoplus_{i=1}^{K_n}M_{d_i}({\Bbb C}). \label{F-2.2} \end{eqnarray} The canonical traces ${\rm Tr}_{{\cal A}_n}$ on ${\cal A}_n$ and ${\rm Tr}_{{\cal A}_n'}$ on ${\cal F}_n\cap{\cal A}_n'$ are written as \begin{eqnarray*} &&{\rm Tr}_{{\cal A}_n}\Biggl(\sum_ia_i\otimes{\bf 1}_{d_i}\Biggr) =\sum_i{\rm Tr}_{m_i}(a_i),\qquad a_i\in M_{m_i}({\Bbb C}),\ 1\le i\le K_n, \\ &&{\rm Tr}_{{\cal A}_n'}\Biggl(\sum_i{\bf 1}_{m_i}\otimes b_i\Biggr) =\sum_i{\rm Tr}_{d_i}(b_i),\qquad b_i\in M_{d_i}({\Bbb C}),\ 1\le i\le K_n, \end{eqnarray*} where ${\rm Tr}_m$ denotes the usual trace on $M_m({\Bbb C})$.
\begin{lemma}\label{L-2.1} $(1)$\enspace If $\omega\in{\cal S}_\theta({\cal A})$, then $\lim_{n\to\infty}{1\over n}S(\omega_n)$ exists and $$ \lim_{n\to\infty}{1\over n}S(\omega_n)=-S_{\rm M}(\omega,\nu)+\log d, $$ where $S(\omega_n)$ is the von Neumann entropy of $\omega_n$ with respect to ${\rm Tr}_{{\cal A}_n}$, i.e., \begin{equation}\label{F-2.3} S(\omega_n):=-{\rm Tr}_{{\cal A}_n}\biggl({d\omega_n\over d{\rm Tr}_{{\cal A}_n}} \log{d\omega_n\over d{\rm Tr}_{{\cal A}_n}}\biggr) =-\omega_n\biggl(\log{d\omega_n\over d{\rm Tr}_{{\cal A}_n}}\biggr). \end{equation}
$(2)$\enspace If $\Phi\in{\cal B}({\cal A})$, then $\lim_{n\to\infty}{1\over n}\log{\rm Tr}_{{\cal A}_n}(e^{-H_n})$ exists and $$ \lim_{n\to\infty}{1\over n}\log{\rm Tr}_{{\cal A}_n}(e^{-H_n}) =p(\Phi,\nu)+\log d. $$ \end{lemma}
\noindent{\it Proof.}\enspace (1)\enspace Notice that $$ S(\omega_n) =-S(\omega_n,\nu_n)-\omega_n\biggl(\log{d\nu_n\over d{\rm Tr}_{{\cal A}_n}}\biggr). $$ Representing ${\cal A}_n=\bigoplus_{i=1}^{K_n}(M_{m_i}({\Bbb C})\otimes{\bf 1}_{d_i})$ as in \eqref{F-2.1}, we have $$ d^n{d\nu_n\over d{\rm Tr}_{{\cal A}_n}}=\sum_{i=1}^{K_n}d_i{\bf 1}_{m_i}\otimes{\bf 1}_{d_i}, $$ because $$ d^n\nu_n\Biggl(\sum_ia_i\otimes{\bf 1}_{d_i}\Biggr) =\sum_i{\rm Tr}_{m_i}(a_i){\rm Tr}_{d_i}({\bf 1}_{d_i}) =\sum_id_i{\rm Tr}_{m_i}(a_i) $$ for $a_i\in M_{m_i}({\Bbb C})$, $1\le i\le K_n$. Therefore, \begin{equation}\label{F-2.4} {\bf 1}_{{\cal A}_n}\le d^n{d\nu_n\over d{\rm Tr}_{{\cal A}_n}} \le\biggl(\max_{1\le i\le K_n}d_i\biggr){\bf 1}_{{\cal A}_n}. \end{equation} This implies that $$ 0\le\omega_n\biggl(\log{d\nu_n\over d{\rm Tr}_{{\cal A}_n}}\biggr)+n\log d \le\log\biggl(\max_{1\le i\le K_n}d_i\biggr). $$ As is well known (see a brief explanation in \cite[p.~844]{HP2} for example), the representation ring of any compact group has polynomial growth; so we have \begin{equation}\label{F-2.5} \lim_{n\to\infty}{1\over n}\log\biggl(\max_{1\le i\le K_n}d_i\biggr)=0. \end{equation} This implies the desired conclusion.
(2)\enspace By \eqref{F-2.4} we get $$ {\rm Tr}_{{\cal A}_n}(e^{-H_n})\le d^n\nu_n(e^{-H_n}) \le\biggl(\max_{1\le i\le K_n}d_i\biggr){\rm Tr}_{{\cal A}_n}(e^{-H_n}), $$ implying the result.\qed
In view of the above lemma we define the {\it mean entropy} of $\omega\in{\cal S}_\theta({\cal A})$ by $$ s_{\cal A}(\omega):=\lim_{n\to\infty}{1\over n}S(\omega_n) \ \ (=-S_{\rm M}(\omega,\nu)+\log d), $$ and the {\it pressure} of $\Phi\in{\cal B}({\cal A})$ by $$ P_{\cal A}(\Phi):=\lim_{n\to\infty}{1\over n}\log{\rm Tr}_{{\cal A}_n}(e^{-H_n}) \ \ (=p(\Phi,\nu)+\log d). $$ The variational expression \eqref{F-1.5} in case of $\phi=\nu$ is rewritten as $$ P_{\cal A}(\Phi)=\sup\{s_{\cal A}(\omega)-\omega(A_\Phi):\omega\in{\cal S}_\theta({\cal A})\}. $$
\begin{prop}\label{P-2.2} Let $\Phi\in{\cal B}_0({\cal A})$ and $\xi\in\Xi(G,\sigma)$ with the generator $h$. Assume that $\xi$ is central, i.e., $\xi_t$ belongs to the center of $G$ for any $t$ $($this is the case if $G$ is abelian\,$)$. Then $\Phi^h$ defined by \eqref{F-1.2} is an interaction in ${\cal A}$, and $\omega\in{\cal S}_\theta({\cal A})$ is $\alpha^\Phi$-KMS with chemical potential $\xi$ if and only if it satisfies the variational principle \begin{equation}\label{F-2.6} P_{\cal A}(\Phi^h)=s_{\cal A}(\omega)-\omega(A_{\Phi^h}). \end{equation} In particular, $\omega$ is $\alpha^\Phi$-KMS with trivial chemical potential if and only if it satisfies $$ P_{\cal A}(\Phi)=s_{\cal A}(\omega)-\omega(A_\Phi). $$ \end{prop}
\noindent{\it Proof.}\enspace The assumption of $\xi$ being central implies that ${\rm Ad}\,\sigma_g(\sigma_{\xi_t})=\sigma_{\xi_t}$ for all $g\in G$ and $t\in{\Bbb R}$. Hence, it is immediate to see that $\bigotimes_\Lambda e^{-h}=\exp\bigl(-\sum_{j\in\Lambda}\theta^j(h)\bigr)$ is in ${\cal A}_\Lambda$ for any finite $\Lambda\subset{\Bbb Z}$ and so the interaction $\Phi^h$ is in ${\cal A}$. Let $\phi$ be an element of $\ET^{\rm f}({\cal A})$ corresponding to $\xi$ as in Proposition \ref{P-1.3}. We may show that \eqref{F-2.6} is equivalent to the variational principle \eqref{F-1.7} with respect to $\phi$. Since $A_{\Phi^h}=A_\Phi+h$, it suffices to prove the following two expressions: \begin{equation}\label{F-2.7} p(\Phi,\phi)=P_{\cal A}(\Phi^h)-\log d \end{equation} and for every $\omega\in{\cal S}_\theta({\cal A})$ \begin{equation}\label{F-2.8} -S_{\rm M}(\omega,\phi)=s_{\cal A}(\omega)-\omega(h)-\log d. \end{equation} Let $H_n(\Phi^h)$ be the local Hamiltonian of $\Phi^h$ inside the interval $[1,n]$. Since $$ \phi_n(e^{-H_n}) =\nu_n\Biggl(\Biggl(\bigotimes_{j=1}^ne^{-h}\Biggr)e^{-H_n}\Biggr) =\nu_n(e^{-H_n(\Phi^h)}) ={\rm Tr}_{{\cal A}_n}\biggl({d\nu_n\over d{\rm Tr}_{{\cal A}_n}}e^{-H_n(\Phi^h)}\biggr), $$ we obtain \eqref{F-2.7} thanks to \eqref{F-2.4} and \eqref{F-2.5}. On the other hand, since \begin{eqnarray} -S(\omega_n,\phi_n) &=&S(\omega_n)+\omega_n\biggl(\log{d\phi_n\over d{\rm Tr}_{{\cal A}_n}}\biggr) \nonumber\\ &=&S(\omega_n)+\omega_n\Biggl(\log\Biggl({d\nu_n\over d{\rm Tr}_{{\cal A}_n}} \bigotimes_{j=1}^ne^{-h}\Biggr)\Biggr) \label{F-2.9}\\ &=&S(\omega_n)-n\omega(h) +\omega_n\biggl(\log{d\nu_n\over d{\rm Tr}_{{\cal A}_n}}\biggr), \nonumber \end{eqnarray} the expression \eqref{F-2.8} follows.\qed
\section{Entropy densities} \setcounter{equation}{0}
From now on let ${\cal F}$, $G$, $\sigma$, $\gamma$, ${\cal A}$, $\theta$, etc.\ be as in the previous sections. Let $\Phi\in{\cal B}_0({\cal A})$ be given and $\alpha^\Phi$ be the associated one-parameter automorphism group. Furthermore, let $\phi\in\ET^{\rm f}({\cal A})$ and the corresponding $\xi\in\Xi(G,\sigma)$ with generator $h$ be given as in Proposition \ref{P-1.3}; hence $\phi$ extends to the $\gamma_\xi$-KMS state $\hat\phi$ of ${\cal F}$. For each $n\in{\Bbb N}$ we then have the local Gibbs state of ${\cal A}_n$ with respect to $\Phi$ and $\phi$ given by $$ \phi_n^G(a):={\phi(e^{-H_n}a)\over\phi(e^{-H_n})}, \qquad a\in{\cal A}_n, $$ and the local Gibbs state of ${\cal F}_n$ with respect to $\Phi^h$ given by $$ \hat\phi_n^G(a) :={{\rm Tr}_{{\cal F}_n}(e^{-H_n(\Phi^h)}a)\over{\rm Tr}_{{\cal F}_n}(e^{-H_n(\Phi^h)})}. \qquad a\in{\cal F}_n. $$ The notation $\hat\phi_n^G$ is justified as follows: Since $\bigotimes_1^ne^{-h}$ and $e^{-H_n}$ commute (see the proof of \cite[Proposition 4.3]{HP2}), $\hat\phi_n^G$ is written as \begin{equation}\label{F-3.1} \hat\phi_n^G(a) ={{\rm Tr}_{{\cal F}_n}\bigl((\bigotimes_1^ne^{-h})e^{-H_n}a\bigr) \over{\rm Tr}_{{\cal F}_n}\bigl((\bigotimes_1^ne^{-h})e^{-H_n}\bigr)} ={\hat\phi(e^{-H_n}a)\over\hat\phi(e^{-H_n})}, \qquad a\in{\cal F}_n. \end{equation} With these notations we have
\begin{thm}\label{T-3.1} Let $\omega$ be an $\alpha^\Phi$-KMS state of ${\cal A}$ with chemical potential $\xi$ and $\hat\omega$ be the $\alpha^\Phi\gamma_\xi$-KMS state of ${\cal F}$ extending $\omega$. Then \begin{eqnarray*} S_{\rm M}(\omega,\phi)&=&\lim_{n\to\infty}{1\over n}S(\phi_n^G,\phi_n) =\lim_{n\to\infty}{1\over n}S(\hat\phi_n^G,\hat\phi_n) \\ &=&S_{\rm M}(\hat\omega,\hat\phi) =-s_{\cal F}(\hat\omega)+\hat\omega(h)+\log d \end{eqnarray*} and $$ s_{\cal F}(\hat\omega)=\lim_{n\to\infty}{1\over n}S(\hat\phi_n^G) =\lim_{n\to\infty}{1\over n}S(\phi_n^G), $$ where $s_{\cal F}(\hat\omega):=\lim_{n\to\infty}{1\over n}S(\hat\omega_n)$, the mean entropy of $\hat\omega$. In particular, if $\xi$ is central, then $s_{\cal A}(\omega)=s_{\cal F}(\hat\omega)$. \end{thm}
\noindent{\it Proof.}\enspace The following proof of $S_{\rm M}(\omega,\phi)=\lim_{n\to\infty}\frac{1}{n}S(\phi^G_n,\phi_n)$ is a slight modification of \cite[Theorem 2.1]{MvE}. The proof of Theorem \ref{T-1.5} says that $\Psi\in{\cal B}_0({\cal A})\mapsto p(\Psi,\phi)$ is differentiable at $\Phi$ with the tangent functional $\Psi\in{\cal B}_0({\cal A})\mapsto-\omega(A_\Psi)$. Hence we have \begin{equation}\label{F-3.2}
\frac{d}{d\beta}\biggl|_{\beta=1}p(\beta \Phi,\phi) =-\omega(A_\Phi). \end{equation} Furthermore, we obtain \begin{equation}\label{F-3.3}
\frac{d}{d\beta}\biggl|_{\beta=1}\frac{1}{n}\log\phi(e^{-H_n(\beta\Phi)}) =\frac{1}{n}\,\frac{\phi(e^{-H_n}(-H_n))}{\phi(e^{-H_n})} =-\frac{1}{n}\phi^G_n(H_n), \end{equation} and as in \cite{MvE} \begin{equation}\label{F-3.4}
\lim_{n\to\infty}\frac{d}{d\beta}\biggl|_{\beta=1} \frac{1}{n}\log\phi(e^{-H_n(\beta\Phi)})
=\frac{d}{d\beta}\biggl|_{\beta=1}p(\beta\Phi,\phi). \end{equation} Combining \eqref{F-3.2}--\eqref{F-3.4} yields $\lim_{n\to\infty}\frac{1}{n}\phi^G_n(H_n)=\omega(A_\Phi)$. Therefore, Theorem \ref{T-1.5} implies \begin{eqnarray*} S_{\rm M}(\omega,\phi) &=&-p(\Phi,\phi)-\omega(A_\Phi) \\ &=&\lim_{n\to\infty}\frac{1}{n} \bigl(-\log\phi(e^{-H_n})-\phi^G_n(H_n)\bigr) \\ &=&\lim_{n\to\infty}\frac{1}{n} \phi^G_n\biggl(\log\frac{d\phi^G_n}{d\phi_n}\biggr) \\ &=&\lim_{n\to\infty}\frac{1}{n}S(\phi^G_n,\phi_n). \end{eqnarray*} On the other hand, $\hat\omega$ satisfies the variational principle with respect to $\Phi^h$, i.e., $$ P_{\cal F}(\Phi^h)=s_{\cal F}(\hat\omega)-\hat\omega(A_{\Phi^h}). $$ Since $A_{\Phi^h}=A_\Phi+h$, this and \eqref{F-1.8} imply \begin{eqnarray} S_{\rm M}(\omega,\phi) &=&-p(\Phi,\phi)-\omega(A_\Phi) \nonumber\\ &=&-s_{\cal F}(\hat\omega)+\hat\omega(A_\Phi+h)+\log d-\omega(A_\Phi) \nonumber\\ &=&-s_{\cal F}(\hat\omega)+\hat\omega(h)+\log d. \label{F-3.5} \end{eqnarray}
Since $d\hat\phi_n/d{\rm Tr}_{{\cal F}_n}=d^{-n}\bigotimes_{j=1}^ne^{-h}$, we have \begin{eqnarray*} S(\hat\omega_n,\hat\phi_n) &=&-S(\hat\omega_n)-\hat\omega_n\biggl( \log\frac{d\hat\phi_n}{d{\rm Tr}_{{\cal F}_n}}\biggr) \\ &=&-S(\hat\omega_n)+\hat\omega\Biggl( \sum_{j=1}^n\theta^j(h)\Biggr)+n\log d \\ &=&-S(\hat\omega_n)+n\hat\omega(h)+n\log d \end{eqnarray*} so that $$ S_{\rm M}(\hat\omega,\hat\phi) =-s_{\cal F}(\hat\omega)+\hat\omega(h)+\log d. $$ Furthermore, \begin{eqnarray*} S(\hat\phi^G_n,\hat\phi_n) &=&-S(\hat\phi^G_n)+\sum_{j=1}^n\hat\phi^G_n(\theta^j(h))+n\log d \\ &=&-S(\hat\phi^G_n)+\sum_{j=1}^n\hat\phi^G_{[1-j,\,n-j]}(h)+n\log d. \end{eqnarray*} By \cite{MvE} we have $s_{\cal F}(\hat\omega)=\lim_{n\to\infty}\frac{1}{n}S(\hat\phi^G_n)$. The uniqueness of $\alpha^\Phi\gamma_\xi$ ($=\alpha^{\Phi^h}$)-KMS state implies that $\hat\phi^G_{[-\ell,m]}\to\hat\omega$ weakly* as $\ell,m\to\infty$. For each $\varepsilon>0$ one can choose $n_0\in{\Bbb N}$ such that
$\big|\hat\phi^G_{[-\ell,\,m]}(h)-\hat\omega(h)\big|\le\varepsilon$ for all $\ell,m\ge n_0$. If $n>2n_0$ and $n_0<j\le n-n_0$, then $j-1\ge n_0$ and $n-j\ge n_0$ so that
$\big|\hat\phi^G_{[1-j,\,n-j]}(h)-\hat\omega(h)\big|\le\varepsilon$. Hence we have $$
\Bigg|\frac{1}{n}\sum_{j=1}^n\hat\phi^G_{[1-j,\,n-j]}(h)-\hat\omega(h)\Bigg|
\le\frac{4\|h\|n_0}{n}+\varepsilon. $$ This shows that $$ \lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^n \hat\phi^G_{[1-j,\,n-j]}(h)=\hat\omega(h). $$ Therefore, $$ \lim_{n\to\infty}\frac{1}{n}S(\hat\phi^G_n,\hat\phi_n) =-s_{\cal F}(\hat\omega)+\hat\omega(h)+\log d, $$ and the proof of the first part is completed.
The last assertion follows from \eqref{F-2.6} and \eqref{F-3.5}. It remains to prove \begin{equation}\label{F-3.6} \lim_{n\to\infty}{1\over n}S(\phi_n^G) =\lim_{n\to\infty}{1\over n}S(\hat\phi_n^G). \end{equation} To prove this we give a lemma.
\begin{lemma}\label{L-3.2} Under \eqref{F-2.1} and \eqref{F-2.2} let $$ D^0=\sum_{i=1}^{K_n}D^0_i\otimes{\bf 1}_{d_i}\in{\cal A}_n, \quad D'=\sum_{i=1}^{K_n}{\bf 1}_{m_i}\otimes D'_i\in{\cal F}_n\cap{\cal A}_n' $$ with positive semidefinite matrices $D^0_i\in M_{m_i}({\Bbb C})$ and $D'_i\in M_{d_i}({\Bbb C})$ such that ${\rm Tr}_{{\cal F}_n}(D^0D')=1$. Then $D:=D^0D'$ is a density matrix with respect to
${\rm Tr}_{{\cal F}_n}$. If $D|_{{\cal A}_n}$ is the density matrix of
${\rm Tr}_{{\cal F}_n}(D\,\cdot)|_{{\cal A}_n}$ with respect to ${\rm Tr}_{{\cal A}_n}$, then $$
|S(D|_{{\cal A}_n})-S(D)|\le\log\biggl(\max_{1\le i\le K_n}d_i\biggr), $$ where $S(D)$ is the von Neumann entropy of $D$ with respect to ${\rm Tr}_{{\cal F}_n}$
and $S(D|_{{\cal A}_n})$ is that of $D|_{{\cal A}_n}$ with respect to ${\rm Tr}_{{\cal A}_n}$ $($see \eqref{F-2.3}$)$. \end{lemma}
\noindent{\it Proof.}\enspace The first assertion is obvious. Let $E_{{\cal A}_n}$ denote the conditional expectation from ${\cal F}_n$ onto ${\cal A}_n$ with respect to ${\rm Tr}_{{\cal F}_n}$. Notice that \begin{eqnarray*} S(E_{{\cal A}_n}(D))-S(D) &=&{\rm Tr}_{{\cal F}_n}(D\log D-E_{{\cal A}_n}(D)\log E_{{\cal A}_n}(D)) \\ &=&S(D,E_{{\cal A}_n}(D)), \end{eqnarray*} the relative entropy of the densities $D$ and $E_{{\cal A}_n}(D)$ in ${\cal F}_n$. Set $H^0_i:=D^0_i/{\rm Tr}_{m_i}(D^0_i)$, $H'_i:=D'_i/{\rm Tr}_{d_i}(D'_i)$ and $D_i:=H^0_i\otimes H'_i$. The joint convexity of relative entropy implies $$ S(D,E_{{\cal A}_n}(D))\le\sum_{i=1}^{K_n} {\rm Tr}_{m_i}(D^0_i){\rm Tr}_{d_i}(D'_i)S(D_i,E_{{\cal A}_n}(D_i)). $$ Since $E_{{\cal A}_n}(D_i)=d_i^{-1}H^0_i\otimes{\bf 1}_{d_i}$, we get \begin{eqnarray*} &&S(D_i,E_{{\cal A}_n}(D_i)) \\ &&\quad={\rm Tr}_{{\cal F}_n}\Bigl(D_i\Bigl(\log H^0_i\otimes{\bf 1}_{d_i} +{\bf 1}_{m_i}\otimes\log H'_i-\log H^0_i\otimes{\bf 1}_{d_i} +(\log d_i){\bf 1}_{m_i}\otimes{\bf 1}_{d_i}\Bigr)\Bigr) \\ &&\quad={\rm Tr}_{d_i}(H'_i\log H'_i)+\log d_i \\ &&\quad\le\log d_i. \end{eqnarray*} Therefore, \begin{equation}\label{F-3.7} 0\le S(E_{{\cal A}_n}(D))-S(D)\le\log\biggl(\max_{1\le i\le K_n}d_i\biggr). \end{equation}
Next, since for $a=\sum_ia_i\otimes{\bf 1}_{d_i}\in{\cal A}_n$ \begin{eqnarray*} {\rm Tr}_{{\cal F}_n}(aD') &=&{\rm Tr}_{{\cal F}_n}\Biggl(\sum_{i=1}^{K_n}a_i\otimes D'_i\Biggr) =\sum_{i=1}^{K_n}{\rm Tr}_{m_i}(a_i){\rm Tr}_{d_i}(D'_i) \\ &=&{\rm Tr}_{{\cal F}_n}\Biggl(\Biggl(\sum_{i=1}^{K_n}{{\rm Tr}_{d_i}(D'_i)\over d_i} {\bf 1}_{m_i}\otimes{\bf 1}_{d_i}\Biggr)a\Biggr), \end{eqnarray*} we get $$ E_{{\cal A}_n}(D')=\sum_{i=1}^{K_n}{{\rm Tr}_{d_i}(D'_i)\over d_i} {\bf 1}_{m_i}\otimes{\bf 1}_{d_i} $$ so that $$ E_{{\cal A}_n}(D)=D^0E_{{\cal A}_n}(D') =\sum_{i=1}^{K_n}{{\rm Tr}_{d_i}(D'_i)\over d_i} D^0_i\otimes{\bf 1}_{d_i}. $$ Hence we have \begin{eqnarray*} &&S(E_{{\cal A}_n}(D)) \\ &&\quad=-{\rm Tr}_{{\cal F}_n}\Biggl(\sum_{i=1}^{K_n}{{\rm Tr}_{d_i}(D'_i)\over d_i} D^0_i\otimes{\bf 1}_{d_i}\Bigl(\log D^0_i\otimes{\bf 1}_{d_i} +\Bigl(\log{\rm Tr}_{d_i}(D'_i)-\log d_i\Bigr) {\bf 1}_{m_i}\otimes{\bf 1}_{d_i}\Biggr) \\ &&\quad=-\sum_{i=1}^{K_n}{\rm Tr}_{d_i}(D'_i) {\rm Tr}_{m_i}(D^0_i\log D^0_i) -\sum_{i=1}^{K_n}{\rm Tr}_{m_i}(D^0_i){\rm Tr}_{d_i}(D'_i) \Bigl(\log{\rm Tr}_{d_i}(D'_i)-\log d_i\Bigr). \end{eqnarray*}
On the other hand, since $D|_{{\cal A}_n}$ is $\sum_{i=1}^{K_n}{\rm Tr}_{d_i}(D'_i)D^0_i$ as an element of $\bigoplus_{i=1}^{K_n}M_{m_i}({\Bbb C})$, we have \begin{eqnarray*}
S(D|_{{\cal A}_n}) &=&-\sum_{i=1}^{K_n}{\rm Tr}_{m_i}\Bigl({\rm Tr}_{d_i}(D'_i)D^0_i \Bigl(\log D^0_i+\log{\rm Tr}_{d_i}(D'_i)\Bigr)\Bigr) \\ &=&-\sum_{i=1}^{K_n}{\rm Tr}_{d_i}(D'_i) {\rm Tr}_{m_i}(D^0_i\log D^0_i) -\sum_{i=1}^{K_n}{\rm Tr}_{m_i}(D^0_i) {\rm Tr}_{d_i}(D'_i)\log{\rm Tr}_{d_i}(D'_i). \end{eqnarray*} Therefore, $$
S(E_{{\cal A}_n}(D))-S(D|_{{\cal A}_n}) =\sum_{i=1}^{K_n}{\rm Tr}_{m_i}(D^0_i){\rm Tr}_{d_i}(D'_i)\log d_i $$ so that \begin{equation}\label{F-3.8}
0\le S(E_{{\cal A}_n}(D))-S(D|_{{\cal A}_n}) \le\log\biggl(\max_{1\le i\le K_n}d_i\biggr). \end{equation} Combining \eqref{F-3.7} and \eqref{F-3.8} gives the conclusion.\qed
\noindent {\it Proof of \eqref{F-3.6}.}\enspace Let $\hat D_n^G$ be the density of the local Gibbs state $\hat\phi_n^G$ with respect to ${\rm Tr}_{{\cal F}_n}$, which is written as \begin{equation}\label{F-3.9} \hat D_n^G={(\bigotimes_1^ne^{-h})e^{-H_n}\over {\rm Tr}_{{\cal F}_n}\bigl((\bigotimes_1^ne^{-h})e^{-H_n}\bigr)}. \end{equation} This is obviously of the form of $D$ in Lemma 3.2, i.e., the product of an element of ${\cal A}_n$ and an element of ${\cal F}_n\cap{\cal A}_n'$. Furthermore, since
${\rm Tr}_{{\cal F}_n}(\hat D_n^G\,\cdot)|_{{\cal A}_n}=\hat\phi_n^G|_{{\cal A}_n}=\phi_n^G$
thanks to \eqref{F-3.1}, it follows that the density of $\phi_n^G$ with respect to ${\rm Tr}_{{\cal A}_n}$ is $\hat D_n^G|_{{\cal A}_n}$ (in the notation of Lemma \ref{L-3.2}). Hence, Lemma \ref{L-3.2} implies $$
\big|S(\phi_n^G)-S(\hat\phi_n^G)\big| \le\log\biggl(\max_{1\le i\le K_n}d_i\biggr) $$ so that we obtain \eqref{F-3.6} thanks to \eqref{F-2.5}.\qed
\section{Macroscopic uniformity} \setcounter{equation}{0}
Let $\phi\in\ET^{\rm f}({\cal A})$ and $0<\varepsilon<1$. For each $n\in{\Bbb N}$ and for each state $\psi$ of ${\cal A}_n$ we define the two quantities $$ \beta_\varepsilon(\psi):=\min\{{\rm Tr}_{{\cal A}_n}(q): q\in{\cal A}_n\ \mbox{is a projection with}\ \psi(q)\ge1-\varepsilon\}, $$ $$ \beta_\varepsilon(\psi,\phi_n):=\min\{\phi_n(q): q\in{\cal A}_n\ \mbox{is a projection with}\ \psi(q)\ge1-\varepsilon\}. $$ For each state $\psi'$ of ${\cal F}_n$ the quantities $\beta_\varepsilon(\psi')$ and $\beta_\varepsilon(\psi',\hat\phi_n)$ are defined in a similar way with ${\cal F}_n$ instead of ${\cal A}_n$. The aim of this section is to prove
\begin{thm}\label{T-4.1} Let $\Phi$, $\phi$, $\xi$ and $h$ be as in Theorem $\ref{T-1.5}$, and let $\omega$ be an $\alpha^\Phi$-KMS state of ${\cal A}$ with chemical potential $\xi$. Then, for every $0<\varepsilon<1$, \begin{eqnarray} -S_{\rm M}(\omega,\phi) &=&\lim_{n\to\infty}{1\over n}\log\beta_\varepsilon(\omega_n,\phi_n) \label{F-4.1}\\ &=&\lim_{n\to\infty}{1\over n}\log\beta_\varepsilon(\phi_n^G,\phi_n) =\lim_{n\to\infty}{1\over n}\log\beta_\varepsilon(\hat\phi_n^G,\hat\phi_n). \label{F-4.2} \end{eqnarray} Moreover, if $\xi$ is central, then for every $0<\varepsilon<1$, \begin{eqnarray} s_{\cal A}(\omega) &=&\lim_{n\to\infty}{1\over n}\log\beta_\varepsilon(\omega_n) =\lim_{n\to\infty}{1\over n}\log\beta_\varepsilon(\phi_n^G) \label{F-4.3}\\ &=&\lim_{n\to\infty}{1\over n}\log\beta_\varepsilon(\hat\omega_n) =\lim_{n\to\infty}{1\over n}\log\beta_\varepsilon(\hat\phi_n^G). \label{F-4.4} \end{eqnarray} \end{thm}
To prove the theorem, we modify the proofs of \cite[Theorems 3.1 and 3.3]{HP1}. Let $\omega$ be as in the theorem and $(\pi_\omega,{\cal H}_\omega,\Omega_\omega)$ be the cyclic representation of ${\cal A}$ associated with $\omega$. For each $n\in{\Bbb N}$ set $$ D_n:={d\omega_n\over d\phi_n}\quad\mbox{and}\quad D_n^G:={d\phi_n^G\over d\phi_n}={e^{-H_n}\over\phi(e^{-H_n})}. $$
\begin{lemma}\label{L-4.2} For every $n\in{\Bbb N}$, $$
\log D_n^G-\log D_n\le2\|W_n\|. $$ \end{lemma}
\noindent{\it Proof.}\enspace For every state $\psi$ of ${\cal A}_n$ let $\tilde\psi$ be the state of
$\pi_\omega({\cal A}_n)$ such that $\psi=\tilde\psi\circ\pi_\omega|_{{\cal A}_n}$; in particular, let $\tilde\phi_n^G$ be that for $\phi_n^G$. Moreover, let
$\tilde\omega$ be the normal extension of $\omega$ to $\pi_\omega({\cal A})''$; so $\tilde\omega_n=\tilde\omega|_{\pi_\omega({\cal A}_n)}$. Note (see \cite[p.~826]{HP2}) that the normal extension $[\omega^{-W_n}]\,\tilde{}\,$ of $[\omega^{-W_n}]$ coincides with the perturbed state $[\tilde\omega^{-\pi_\omega(W_n)}]$. There exists the conditional expectation $E_n$ from $\pi_\omega({\cal A})''$ onto $\pi_\omega({\cal A}_n)$ with respect to $[\omega^{-W_n}]\,\tilde{}\,$ because $\pi_\omega({\cal A}_n)$ is globally invariant under the modular automorphism associated with this state. (See the proof of \cite[Theorem 2.2, (i) $\Rightarrow$ (ii)]{HP2}; this part of the proof of \cite[Theorem 2.2]{HP2} is valid.) Then, we successively estimate \begin{eqnarray} S(\psi,\omega_n)&=&S(\tilde\psi,\tilde\omega_n) \le S(\tilde\psi\circ E_n,\tilde\omega) \nonumber\\
&\le&S(\tilde\psi\circ E_n,[\omega^{-W_n}]\,\tilde{}\,)+2\|W_n\| \nonumber\\
&=&S(\tilde\psi\circ E_n,\tilde\phi_n^G\circ E_n)+2\|W_n\| \nonumber\\
&=&S(\psi,\phi_n^G)+2\|W_n\|. \label{F-4.5} \end{eqnarray} Here, the first inequality is the monotonicity of relative entropy (\cite[5.12\,(iii)]{OP}) under the restriction of the states of $\pi_\omega({\cal A})''$ to its subalgebra $\pi_\omega({\cal A}_n)$, and the second is due to \eqref{F-1.3}. The second equality follows because Theorem \ref{T-1.5} ((ii) or (iii)) gives $[\omega^{-W_n}]\,\tilde{}\,=\tilde\phi_n^G\circ E_n$. The last equality is seen by applying the monotonicity of relative entropy in two ways (or by \cite[5.15]{OP}). We now obtain $$ \psi(\log D_n^G-\log D_n)
=S(\psi,\omega_n)-S(\psi,\phi_n^G)\le2\|W_n\| $$ for all states $\psi$ of ${\cal A}_n$, which implies the conclusion.\qed
\begin{lemma}\label{L-4.3} For the densities $D_n$ and $D_n^G$, $$ \lim_{n\to\infty}{1\over n}\pi_\omega(-\log D_n) =\lim_{n\to\infty}{1\over n}\pi_\omega(-\log D_n^G) =-S_{\rm M}(\omega,\phi){\bf 1}\ \ \mbox{strongly}. $$ \end{lemma}
\noindent{\it Proof.}\enspace Since $\omega$ is extremal in ${\cal S}_\theta({\cal A})$, the mean ergodic theorem says that $$ \lim_{n\to\infty}{1\over n} \pi_\omega\Biggl(\sum_{j=1}^n\theta^j(A_\Phi)\Biggr) =\omega(A_\Phi){\bf 1}\ \ \mbox{strongly}. $$ Since it follows as in \cite{HP1} that $$ \lim_{n\to\infty}{1\over n}
\Bigg\|\sum_{j=1}^n\theta^j(A_\Phi)-H_n\Bigg\|=0, $$ we have \begin{equation}\label{F-4.6} \lim_{n\to\infty}{1\over n}\pi_\omega(H_n)=\omega(A_\Phi){\bf 1} \ \ \mbox{strongly}. \end{equation} Therefore, we obtain the strong convergence \begin{eqnarray} {1\over n}\pi_\omega(-\log D_n^G) &=&{1\over n}\pi_\omega(H_n)+{1\over n}\bigl(\log\phi(e^{-H_n})\bigr){\bf 1} \nonumber\\ &\longrightarrow&\bigl(\omega(A_\Phi)+p(\Phi,\phi)\bigr){\bf 1} =-S_{\rm M}(\omega,\phi){\bf 1} \label{F-4.7} \end{eqnarray} due to the variational principle of $\omega$ in Theorem \ref{T-1.5}.
Next, let $a_n:=-{1\over n}\log D_n$ and
$b_n:=-{1\over n}\log D_n^G+{2\over n}\|W_n\|$; so $\pi_\omega(b_n)\to-S_{\rm M}(\omega,\phi){\bf 1}$ strongly by what is already shown. We get $a_n\le b_n$ by Lemma \ref{L-4.2}, and moreover \begin{eqnarray*} a_n&=&-{1\over n}\log{d\omega_n\over d{\rm Tr}_{{\cal A}_n}} +{1\over n}\log{d\phi_n\over d{\rm Tr}_{{\cal A}_n}} \ge{1\over n}\log{d\phi_n\over d{\rm Tr}_{{\cal A}_n}} \\ &=&-{1\over n}\sum_{j=1}^n\theta^j(h) +{1\over n}\log{d\nu_n\over d{\rm Tr}_{{\cal A}_n}}
\ge-\|h\|-\log d \end{eqnarray*} (see \eqref{F-2.9} and \eqref{F-2.4}). Hence $\{b_n-a_n\}$ is uniformly bounded. Since \begin{eqnarray*}
\|\pi_\omega(b_n-a_n)\Omega_\omega\|^2
&\le&\Bigl(\sup_m\|b_m-a_m\|\Bigr)\omega(b_n-a_n) \\
&\longrightarrow&\Bigl(\sup_m\|b_m-a_m\|\Bigr) \bigl(-S_{\rm M}(\omega,\phi)+S_{\rm M}(\omega,\phi)\bigr)=0, \end{eqnarray*} we have $\pi_\omega(b_n-a_n)\to0$ strongly because $\Omega_\omega$ is separating for $\pi_\omega({\cal A})''$. Hence $\pi_\omega(a_n)\to-S_{\rm M}(\omega,\phi){\bf 1}$ strongly.\qed
\begin{lemma}\label{L-4.4} Let $n(1)<n(2)<\cdots$ be positive integers, and let $a_k\in{\cal A}_{n(k)}$ be a positive contraction for each $k\in{\Bbb N}$. \begin{itemize} \item[(i)] If $\inf_k\omega(a_k)>0$, then $$ \lim_{k\to\infty}{1\over n(k)}\log\phi_{n(k)}^G(a_k)=0. $$ \item[(ii)] If $\inf_k\phi_{n(k)}^G(a_k)>0$, then $\inf_k\omega(a_k)>0$. \item[(iii)] If $\lim_{k\to\infty}\omega(a_k)=1$, then $\lim_{k\to\infty}\phi_{n(k)}^G(a_k)=1$. \end{itemize} The above assertions {\rm(i)--(iii)} hold also for ${\cal F}_{n(k)}$, $\hat\omega$ and $\hat\phi_{n(k)}^G$ instead of ${\cal A}_{n(k)}$, $\omega$ and $\phi_{n(k)}^G$, respectively. \end{lemma}
\noindent{\it Proof.}\enspace The last assertion is contained in \cite[Lemma 3.2]{HP1}.
Let $$ F(s_1,s_2):=s_1\log{s_1\over s_2}+(1-s_1)\log{1-s_1\over1-s_2}, \qquad0\le s_1,s_2\le1. $$ If the conclusion of (i) does not hold, then one may assume by taking a subsequence that $\phi_{n(k)}^G(a_k)\le e^{-n(k)\eta}$, $k\in{\Bbb N}$, for some $\eta>0$. Using the monotonicity of relative entropy (\cite[5.12\,(iii)]{OP}) applied to the map $\alpha:{\Bbb C}^2\to{\cal A}_{n(k)}$, $\alpha(t_1,t_2):=t_1a_k+t_2({\bf 1}-a_k)$, we have \begin{eqnarray*} S(\omega_{n(k)},\phi_{n(k)}^G) &\ge&S(\omega_{n(k)}\circ\alpha,\phi_{n(k)}^G\circ\alpha)= F(\omega_{n(k)}(a_k),\phi_{n(k)}^G(a_k)) \\ &\ge&-\log2-\omega(a_k)\log\phi_{n(k)}^G(a_k) -(1-\omega(a_k))\log(1-\phi_{n(k)}^G(a_k)) \\ &\ge&-\log2+n(k)\eta\omega(a_k) \end{eqnarray*} and hence $$ \liminf_{k\to\infty}{1\over n(k)}S(\omega_{n(k)},\phi_{n(k)}^G) \ge\eta\inf_k\omega(a_k)>0. $$ This contradicts the equality $$ \lim_{n\to\infty}{1\over n}S(\omega_n,\phi_n^G) =S_{\rm M}(\omega,\phi)+\omega(A_\Phi)+p(\Phi,\phi)=0, $$ which is seen from $S(\omega_n,\phi_n^G)=S(\omega_n,\phi_n)+\omega(H_n)+\log\phi(e^{-H_n})$ and \eqref{F-4.6}. Hence (i) follows.
Furthermore, thanks to the monotonicity of relative entropy as above and \eqref{F-4.5}, we have \begin{eqnarray*} F(\phi_{n(k)}^G(a_k),\omega(a_k)) &\le&S(\phi_{n(k)}^G,\omega_{n(k)}) \\
&\le&S(\phi_{n(k)}^G,\phi_{n(k)}^G)+2\|W_{n(k)}\|=2\|W_{n(k)}\|. \end{eqnarray*} This shows the boundedness of $F(\phi_{n(k)}^G(a_k),\omega(a_k))$, from which (ii) and (iii) are easily verified.\qed
\noindent {\it Proof of \eqref{F-4.1}.}\enspace For each $\delta>0$ and $n\in{\Bbb N}$, let $p_n$ be the spectral projection of $-{1\over n}\log D_n$ corresponding to the interval $(-S_{\rm M}(\omega,\phi)-\delta,-S_{\rm M}(\omega,\phi)+\delta)$. Then we have \begin{equation}\label{F-4.8} \exp\bigl(n(-S_{\rm M}(\omega,\phi)-\delta)\bigr)D_np_n \le p_n\le\exp\bigl(n(-S_{\rm M}(\omega,\phi)+\delta)\bigr)D_np_n, \end{equation} and Lemma \ref{L-4.3} implies that $\pi_\omega(p_n)\to{\bf 1}$ strongly as $n\to\infty$. Choose a sequence $n(1)<n(2)<\cdots$ such that \begin{equation}\label{F-4.9} \lim_{k\to\infty}{1\over n(k)}\log\beta_\varepsilon(\omega_{n(k)},\phi_{n(k)}) =\liminf_{n\to\infty}{1\over n}\log\beta_\varepsilon(\omega_n,\phi_n). \end{equation} For each $k$ choose a projection $q_k\in{\cal A}_{n(k)}$ such that $\omega(q_k)\ge1-\varepsilon$ and \begin{equation}\label{F-4.10} \log\phi_{n(k)}(q_k)\le\log\beta_\varepsilon(\omega_{n(k)},\phi_{n(k)})+1. \end{equation} We may assume that $\pi_\omega(q_k)$ converges to some $y\in\pi_\omega({\cal A})''$ weakly. Since $\pi_\omega(p_{n(k)}q_k)\to y$ weakly, we get $$ \lim_{k\to\infty}\omega(p_{n(k)}q_k)=\<y\Omega_\omega,\Omega_\omega\rangle =\lim_{k\to\infty}\omega(q_k)\ge1-\varepsilon $$ and by \eqref{F-4.8} $$ \phi(q_k)\ge\phi(p_{n(k)}q_k) \ge\exp\bigl(n(k)(-S_{\rm M}(\omega,\phi)-\delta)\bigr)\omega(p_{n(k)}q_k). $$ These give \begin{equation}\label{F-4.11} \liminf_{k\to\infty}{1\over n(k)}\log\phi(q_k) \ge-S_{\rm M}(\omega,\phi)-\delta. \end{equation} Combining \eqref{F-4.9}--\eqref{F-4.11} yields $$ \liminf_{n\to\infty}{1\over n}\log\beta_\varepsilon(\omega_n,\phi_n) \ge-S_{\rm M}(\omega,\phi)-\delta. $$ On the other hand, we obtain $$ \limsup_{n\to\infty}{1\over n}\log\beta_\varepsilon(\omega_n,\phi_n) \le-S_{\rm M}(\omega,\phi)+\delta, $$ because by \eqref{F-4.8} \begin{eqnarray*} {1\over n}\log\beta_\varepsilon(\omega_n,\phi_n) &\le&{1\over n}\log\phi(p_n) \le-S_{\rm M}(\omega,\phi)+\delta+{1\over n}\log\omega(p_n) \\ &\le&-S_{\rm M}(\omega,\phi)+\delta \end{eqnarray*} if $n$ is so large that $\omega(p_n)\ge1-\varepsilon$. Thus, the proof of \eqref{F-4.1} is completed.\qed
\noindent {\it Proof of \eqref{F-4.2}.}\enspace This can be proven by use of (i)--(iii) of Lemma \ref{L-4.4} similarly to the proof of \cite[Theorem 3.3]{HP1}. Since the proof of the second inequality is a bit more involved than the first, we only prove the second.
Let $(\hat\pi,\hat{\cal H},\hat\Omega)$ be the cyclic representation of ${\cal F}$ associated with $\hat\omega$. For each $\delta>0$ and $n\in{\Bbb N}$, let $p_n$ be the spectral projection of $-{1\over n}\log D_n^G$ to $(-S_{\rm M}(\omega,\phi)-\delta,S_{\rm M}(\omega,\phi)+\delta)$. Since ${1\over n}\hat\pi(H_n)\to\hat\omega(A_\Phi){\bf 1}=\omega(A_\Phi){\bf 1}$ and hence ${1\over n}\hat\pi(-\log D_n^G)\to-S_{\rm M}(\omega,\phi){\bf 1}$ strongly as \eqref{F-4.6} and \eqref{F-4.7}, it follows that $\hat\pi(p_n)\to{\bf 1}$ strongly as $n\to\infty$. Furthermore, we have \begin{equation}\label{F-4.12} \exp\bigl(n(-S_{\rm M}(\omega,\phi)-\delta)\bigr) {e^{-H_n}p_n\over\phi(e^{-H_n})} \le p_n\le\exp\bigl(n(-S_{\rm M}(\omega,\phi)+\delta)\bigr) {e^{-H_n}p_n\over\phi(e^{-H_n})}. \end{equation} Choose $n(1)<n(2)<\cdots$ such that \begin{equation}\label{F-4.13} \lim_{k\to\infty}{1\over n(k)} \log\beta_\varepsilon(\hat\phi_{n(k)}^G,\hat\phi_{n(k)}) =\liminf_{n\to\infty}{1\over n}\log\beta_\varepsilon(\hat\phi_n^G,\hat\phi_n). \end{equation} For each $k$ there is a projection $q_k\in{\cal F}_{n(k)}$ such that $\hat\phi_{n(k)}^G(q_k)\ge1-\varepsilon$ and \begin{equation}\label{F-4.14} \log\hat\phi_{n(k)}(q_k) \le\log\beta_\varepsilon(\hat\phi_{n(k)}^G,\hat\phi_{n(k)})+1. \end{equation} Here, we may assume that $\hat\pi(q_k)$ converges to some $y\in\hat\pi({\cal F})''$ weakly. Then we obtain $$ \lim_{k\to\infty}\hat\omega(p_{n(k)}q_kp_{n(k)}) =\<y\hat\Omega,\hat\Omega\rangle=\lim_{k\to\infty}\hat\omega(q_k)>0 $$ by Lemma \ref{L-4.4}\,(ii) (for $\hat\omega$ and $\hat\phi_{n(k)}^G$ with $a_k=q_k$), and hence \begin{equation}\label{F-4.15} \lim_{k\to\infty}{1\over n(k)} \log\hat\phi_{n(k)}^G(p_{n(k)}q_kp_{n(k)})=0 \end{equation} by Lemma \ref{L-4.4}\,(i) (for $\hat\omega$ and $\hat\phi_{n(k)}^G$ with $a_k=p_{n(k)}q_kp_{n(k)}$). Furthermore, since $p_n$ commutes with $e^{-H_n}$ and $\bigotimes_1^ne^{-h}$, we obtain \begin{eqnarray*} \hat\phi_{n(k)}(q_k) &=&d^{-n(k)}{\rm Tr}_{{\cal F}_{n(k)}}\Biggl(\Biggl(\bigotimes_1^{n(k)}e^{-h} \Biggr)q_k\Biggr) \\ &\ge&d^{-n(k)}{\rm Tr}_{{\cal F}_{n(k)}}\Biggl(\Biggl(\bigotimes_1^{n(k)}e^{-h} \Biggr)p_{n(k)}q_k\Biggr) \\ &\ge&\exp\bigl(n(k)(-S_{\rm M}(\omega,\phi)-\delta)\bigr) {d^{-n(k)}{\rm Tr}_{{\cal F}_{n(k)}}\bigl(\bigl(\bigotimes_1^{n(k)}e^{-h} \bigr)e^{-H_{n(k)}}p_{n(k)}q_k\bigr)\over\phi(e^{-H_{n(k)}})} \\ &=&\exp\bigl(n(k)(-S_{\rm M}(\omega,\phi)-\delta)\bigr) {\hat\phi(e^{-H_{n(k)}}p_{n(k)}q_kp_{n(k)})\over\phi(e^{-H_{n(k)}})} \\ &=&\exp\bigl(n(k)(-S_{\rm M}(\omega,\phi)-\delta)\bigr) \hat\phi_{n(k)}^G(p_{n(k)}q_kp_{n(k)}) \end{eqnarray*} using \eqref{F-4.12} and \eqref{F-3.1}. This together with \eqref{F-4.13}--\eqref{F-4.15} yields $$ \liminf_{n\to\infty}{1\over n}\log\beta_\varepsilon(\hat\phi_n^G,\hat\phi_n) \ge-S_{\rm M}(\omega,\phi)-\delta. $$ On the other hand, since $\hat\phi_n^G(p_n)\to1$ by Lemma \ref{L-4.4}\,(iii) (for $\hat\omega$ and $\hat\phi_n^G$), we have $\hat\phi_n^G(p_n)\ge1-\varepsilon$ for large $n$, and for such $n$ $$ {1\over n}\log\beta_\varepsilon(\hat\phi_n^G,\hat\phi_n) \le{1\over n}\log\hat\phi_n(p_n)\le-S_{\rm M}(\omega,\phi)+\delta $$ thanks to \eqref{F-4.12}. Therefore, $$ \limsup_{n\to\infty}{1\over n}\log\beta_\varepsilon(\hat\phi_n^G,\hat\phi) \le-S_{\rm M}(\omega,\phi)+\delta, $$ completing the proof of \eqref{F-4.2}.\qed
\noindent {\it Proof of \eqref{F-4.3} and \eqref{F-4.4}.}\enspace Assume that $\xi$ is central. Since $s_{\cal A}(\omega)=s_{\cal F}(\hat\omega)$ by Theorem \ref{T-3.1}, the assertion \eqref{F-4.4} is contained in \cite[Theorem 3.3]{HP1}. To prove \eqref{F-4.3}, we first assume that $\xi$ is trivial. Then, by Lemma \ref{L-2.1}\,(1) and \eqref{F-4.1} (in case of $\phi=\nu$) we have \begin{eqnarray*} s_{\cal A}(\omega)&=&-S_{\rm M}(\omega,\nu)+\log d \\ &=&\lim_{n\to\infty}{1\over n}\log\beta_\varepsilon(\omega_n,\nu_n)+\log d \\ &=&\lim_{n\to\infty}{1\over n}\log\beta_\varepsilon(\omega_n). \end{eqnarray*} The latter equality in the above is readily verified from \eqref{F-2.4} and \eqref{F-2.5}. The other equality in \eqref{F-4.3} when $\phi=\nu$ is similarly shown from the first equality in \eqref{F-4.2}. When $\xi$ is not trivial, we consider $\Phi^h$ belonging to ${\cal B}_0({\cal A})$ instead of $\Phi$. Note that $\omega$ is an $\alpha^{\Phi^h}$-KMS state with trivial chemical potential and $\phi_n^G$ is the local Gibbs state with respect to $\Phi^h$ and $\nu$. Hence, the above special case gives the conclusion.\qed
\section{Remarks and problems} \setcounter{equation}{0}
Some problems as well as related known results are in order.
\noindent {\bf 5.1.}\enspace It is known \cite{Ha,Pr} that the weak*-closure of $\ET^{\rm f}({\cal A})$ coincides with the set $\cE\cT({\cal A})$ of all extremal tracial states of ${\cal A}$ as far as $G$ is a compact connected Lie group. For $\Phi\in{\cal B}_0({\cal A})$ let ${\cal E} K({\cal A},\Phi)$ denote the set of all extremal $\alpha^\Phi$-KMS states of ${\cal A}$ (see Proposition \ref{P-1.1}) and ${\cal E} K^{\rm f}({\cal A},\Phi)$ the set of all faithful $\omega\in{\cal E} K({\cal A},\Phi)$. Theorems \ref{T-1.5} and \ref{T-1.6} say that there is a bijective correspondence $\phi\leftrightarrow\omega$ between $\ET^{\rm f}({\cal A})$ and ${\cal E} K^{\rm f}({\cal A},\Phi)$. We further know (see \cite[Theorem 4.6]{HP2}) that the correspondence $\phi\mapsto\omega$ is a weak*-homeomorphism from $\ET^{\rm f}({\cal A})$ onto ${\cal E} K^{\rm f}({\cal A},\Phi)$. Upon these considerations we are interested in the following problems: \begin{itemize} \item[(1)] Does the weak*-closure of ${\cal E} K^{\rm f}({\cal A},\Phi)$ coincide with ${\cal E} K({\cal A},\Phi)$ (as far as $G$ is a compact connected Lie group)? \item[(2)] Does the above $\phi\mapsto\omega$ extend to a weak*-homeomorphism from $\cE\cT({\cal A})$ onto ${\cal E} K({\cal A},\Phi)$? \end{itemize}
\noindent {\bf 5.2.}\enspace In the situation of Theorem \ref{T-3.1} it seems that the equality $s_{\cal A}(\omega)=s_{\cal F}(\hat\omega)$ holds without the assumption of $\xi$ being central. This is equivalent to the equality $s_{\cal A}(\omega)=\lim_{n\to\infty}{1\over n}S(\phi_n^G)$, which is the only missing point in Theorem \ref{T-3.1}.
\noindent {\bf 5.3.}\enspace The equality $-S_{\rm M}(\omega,\phi)=\lim_{n\to\infty} {1\over n}\beta_\varepsilon(\hat\omega_n,\hat\phi_n)$ is missing in Theorem 4.1, which is equivalent to \begin{equation}\label{F-5.1} -S_{\rm M}(\hat\omega,\hat\phi)=\lim_{n\to\infty} {1\over n}\log\beta_\varepsilon(\hat\omega_n,\hat\phi_n) \end{equation} due to Theorem \ref{T-3.1}. Note that $\hat\phi$ is a product state of ${\cal F}$ and $\hat\omega$ is completely ergodic, i.e., extremal for all $\theta^n$, $n\ge1$. Thus, the equality \eqref{F-5.1} is an old open problem from the viewpoint of quantum hypothesis testing in \cite{HP0}, where the weaker result was proven: \begin{eqnarray*} -S_{\rm M}(\hat\omega,\hat\phi)&\ge&\limsup_{n\to\infty} {1\over n}\log\beta_\varepsilon(\hat\omega_n,\hat\phi_n), \\ -{1\over1-\varepsilon}S_{\rm M}(\hat\omega,\hat\phi)&\le&\liminf_{n\to\infty} {1\over n}\log\beta_\varepsilon(\hat\omega_n,\hat\phi_n). \end{eqnarray*} In this connection, it is worthwhile to note that T.~Ogawa and H.~Nagaoka established in \cite{ON} the equality $$ -S(\varphi,\psi)=\lim_{n\to\infty}{1\over n} \log\beta_\varepsilon(\varphi_n,\psi_n) $$ when $\varphi,\psi$ are states of $M_d({\Bbb C})$ and $\varphi_n,\psi_n$ are the $n$-fold tensor products of $\varphi,\psi$. The problem of macroscopic uniformity for states of spin $C^*$-algebras was completely solved in a recent paper by I.~Bjelakovi\'c et al.\ as follows: If $\varphi$ is an extremal translation-invariant state of the $\nu$-dimensional spin algebra $\bigotimes_{{\Bbb Z}^\nu}M_d({\Bbb C})$, then $$
s(\varphi)=\lim_{\Lambda\to{\Bbb Z}^\nu}{1\over|\Lambda|}\log\beta_\varepsilon(\varphi) $$ for any $0<\varepsilon<1$. See \cite{BKSS} for details.
\noindent {\bf 5.4.}\enspace Although many arguments in this paper as well as in \cite{HP2} work also in gauge-invariant $C^*$-systems over the multi-dimensional lattice ${\Bbb Z}^\nu$, some difficulties arise when we would extend our whole arguments to the multi-dimensional case. For instance, it does not seem that Proposition \ref{P-1.1} holds in multi-dimensional gauge-invariant $C^*$-systems. The proposition is crucial when we use the chemical potential theory as in the proof of Theorem \ref{T-1.6}. Moreover, the assumption of uniformly bounded surface energies is sometimes useful in our discussions. In the multi-dimensional case, the assumption is obviously too strong and, if it is not assumed, the non-uniqueness of KMS states (or the phase transition) can occur. Indeed, the uniqueness of $\alpha^{\Phi^h}$-KMS state of ${\cal F}$ is essential in the proof of Theorem \ref{T-1.5}. Consequently, some new ideas must be needed to extend the theory to the multi-dimensional setting.
\noindent {\bf Acknowledgments.} The authors are grateful to Professors E.~St\o rmer and S.~Neshveyev who pointed out a mistake in our previous paper \cite{HP2} in 2000, and also thank the referees for their useful suggestions.
\end{document} | arXiv |
\begin{document}
\begin{titlepage} \begin{flushright} TU- 704\\ quant-ph/0401187 \end{flushright} \ \\ \begin{center} \LARGE {\bf Quantum Estimation by Local Observables } \end{center} \ \\ \begin{center} \Large{ Masahiro Hotta${}^\ast$ and Masanao Ozawa${}^\dagger$ }\\ {\it ${}^\ast$ Department of Physics, Faculty of Science, Tohoku University,\\ Sendai, 980-8578,Japan\\ [email protected] \\
\ \\
${}^\dagger$ Graduate School of Information Sciences, Tohoku University,\\ Sendai, 980-8579, Japan \\ [email protected] } \end{center}
\begin{abstract} Quantum estimation theory provides optimal observations for various estimation problems for unknown parameters in the state of the system under investigation. However, the theory has been developed under the assumption that every observable is available for experimenters. Here, we generalize the theory to problems in which the experimenter can use only locally accessible observables. For such problems, we establish a Cram{\'e}r-Rao type inequality by obtaining an explicit form of the Fisher information as a reciprocal lower bound for the mean square errors of estimations by locally accessible observables.
Furthermore, we explore various local quantum estimation problems for composite systems, where non-trivial combinatorics is needed for obtaining the Fisher information. \end{abstract}
\end{titlepage}
\section{Introduction} \ \\
In many experimental situations, we are not allowed to have a large number of
data enough to determine unknown parameters such as coupling constants of hypothetical interactions.
In some cases, the number may be fairly small and it is crucial to theoretically explore the best estimator for the parameter from the small number of our available data.
The problem becomes prominent for quantum systems, since optimal estimation must be well reconciled with inevitable quantum uncertainty arisen from available observables and unknown parameters.
In such situations,
the quantum estimation theory can play a significant role; for detailed reviews, we refer to
Helstrom \cite{Hel76} and Holevo\cite{Holevo}. The theory provides the best observation on the system
for the estimation with the minimum value of the estimate error.
Although the ordinary quantum estimation theory is certainly powerful
for many estimation problems, the theory
includes an implicit assumption, which is not
realistic in some of the practical experiments. The assumption is that
every observable of the system is available for
the observer or the experimenter.
Contrary to the assumption practically available observables are often restricted.
For instance, it is a common situation in experiments that a particle is contained inside the laboratory at the origin of the time and that the experimenter can only use measuring devices inside the laboratory. However, according to the time evolution the particle may go out of the laboratory, so that the ability of estimating the state parameter is restricted to measuring devices inside the laboratory for the later time.
Another example is found in elementary particle physics.
It usually happens due to
the limit of the present technology of measurement
that our apparatus can probe only low-energy portions of the total Hilbert space with visible signals. Thus the observables are certainly restricted.
In such situations, the observable provided by the ordinary theory for the best estimate may in general not be available . Then, the question becomes relevant as to
what is the best estimate among those which are accessible {\em only by use of restricted observables}.
Let us generally call such estimations
local quantum estimations.
In this paper, we elaborate the formulation of quantum estimation theory for local quantum estimation problems on an unknown parameter $g$. For the restricted density operators measured by our apparatus,
a Fisher information is introduced.
Then, we prove the quantum Cram{\'e}r-Rao type inequality
for the local quantum estimation for $g$.
The observable is specified that attains the equality and
yields the best local quantum estimate for $g$ by its measurement. It is also pointed out that there exist nontrivial aspects in the analysis of the local quantum estimation
for the composite system of identical subsystems.
In that case we have two natural estimations
and corresponding two Fisher informations for the unknown parameter $g$. The first alternative
takes a simple form to apply, but may give a smaller value of the Fisher information. The second alternative is able to give a larger value of
the Fisher information and generates a better estimate for $g$,
but have a pretty complicated form to deal with,
compared to the first alternative. Especially,
calculation of the second Fisher information
requires solving independently evolutions of many descendant operators.
In section 2, a brief review on the standard quantum estimation theory
is given.
In section 3, we discuss more physically the quantum estimation problem,
including the biased-estimator case.
Several expected advantages of the quantum estimation are also reviewed. In section 4, we introduce the notion of local quantum estimation problems.
In section 5, a quantum Cram{\'e}r-Rao inequality for
local quantum estimations is established. In section 6, Fisher information is discussed for unnormalized pure states.
In section 7, we reveal nontrivial aspects of local quantum estimations for composite systems. In section 8, two general formulations are proposed for
local quantum estimations for the composite system.
In section 9, a formula which is useful for the
evaluation of one of two sorts of Fisher information
for composite systems is given.
In section 10, we apply our formulation for local quantum estimations
to a decaying two-level system with a small unknown parameter.
In the final section, we summarize our results of this paper.
\section{Cram\'{e}r-Rao bound for quantum estimators} \ \\
Let us briefly review quantum estimation theory
in this section. For a detailed review we refer the reader to
Helstrom \cite{Hel76} and Holevo\cite{Holevo}.
Let ${\bf S}$ be a closed quantum system described by a Hilbert space ${\cal H}$. We assume that the Hamiltonian $H_{tot}$ has a constant real parameter $g\inG$, i.e., $H_{tot}=H_{tot}(g)$, where $G$ is the set of possible values of $g$. The evolution equation for the density operator $\rho_{tot}$ of ${\bf S}$ is given by \begin{equation} i\hbar \partial_{t}\rho_{tot} = [H_{tot},\ \rho_{tot}]. \end{equation} Then, the density operator of ${\bf S}$ at a given time $t$ depends on the time $t$ and the parameter $g$, i.e., \begin{equation}\label{eq:tot} \rho_{tot}=\rho_{tot}(g,t). \end{equation}
We shall consider the following quantum estimation problem for the parameter $g$. In order to estimate the parameter $g$ in $H_{tot}(g)$, we assume that one measures an observable $A$ at time $t$, and the output $\overline{g}$ is taken as the estimate for $g$. Thus, the observable $A$ plays the role of the estimator of this statistical estimation problem.
By the Born statistical formula, the expectation value of the measurement output $\overline{g}$ in the state $\rho_{tot}(g)=\rho_{tot}(g,t)$ is given by \begin{eqnarray} E_{g}[A]=\mbox{\rm Tr}[A\rho_{tot}(g)].\label{1e} \end{eqnarray} Then, $E_{g}[A]$ is the mean of one's estimate $\overline{g}$ for the given true value $g (\inG)$. The variance of the estimate $\overline{g}$ for the true value $g (\inG)$ is given by \begin{eqnarray} V_{g}[A]=(\Delta_{g} A)^{2}=\mbox{\rm Tr}[A^{2}\rho_{tot}(g)] -\mbox{\rm Tr}[A\rho_{tot}(g)]^{2}, \end{eqnarray} where $\Delta_{g} A$ is the uncertainty of observable $A$ in the state $\rho_{tot}(g)$. The mean-square error $\epsilon^2_g [A]$ of the estimate $\overline{g}$ for the true value $g$ is defined by \begin{equation} \epsilon^2_g [A] := E_g [(A-g)^2]. \end{equation} By a simple manipulation, we obtain the relation \begin{equation} \epsilon^2_g [A] =V_g [A] +(E_g [A] -g)^2. \label{mse}. \end{equation}
The estimator $A$ is called {\em unbiased}, if the mean estimate is correct for any possible values $g (\inG)$, i.e., \begin{equation} E_{g}[A]=g \label{ub} \end{equation} for any $g (\inG)$. In this section, we shall confine our attention to unbiased estimators $A$. As seen in eqn(\ref{mse}), for the unbiased $A$, the variance $V_{g}[A]$ represents the mean-square error of the estimate $\overline{g}$ for the true value $g (\inG)$: \begin{equation} \epsilon^2_g [A]=V_g [A]. \end{equation}
The lower bound for $V_{g}[A]$ is given by the well-known quantum Cram\'{e}r-Rao inequality as follows. The symmetric logarithmic derivative (SLD) $L(g)$ for $\rho_{tot}(g)$ is defined as a self-adjoint operator satisfying \begin{eqnarray}\label{eq:SLD} \partial_{g}\rho_{tot}(g)= \frac{1}{2}[\rho_{tot}(g)L(g)+L(g)\rho_{tot}(g)]. \end{eqnarray} Note here that \begin{equation} E_g [L(g)]= \mbox{\rm Tr}[\rho_{tot} (g) L(g)] =0,\label{2e} \end{equation} due to the normalization condition of $\rho_{tot}$. By the above relation, the SLD may not be determined uniquely; however, any two solutions $L_{1}(g)$ and $L_{2}(g)$ satisfy the the relation \cite{FN95} \begin{eqnarray} L_{1}(g)\rho_{tot}(g)=L_{2}(g)\rho_{tot}(g). \end{eqnarray} Thus, the operator $L(g)\rho_{tot}(g)$ is uniquely determined. The Fisher information $J_{g}$ of the parameter $g$ in $\rho_{tot}(g)$ is uniquely defined by \begin{eqnarray} J_{g}=\mbox{\rm Tr}[L^{2}(g)\rho_{tot}(g)]. \label{fi} \end{eqnarray} Then, every unbiased estimator $A$ satisfies the quantum Cram\'{e}r-Rao inequality \cite{Hel76,Holevo} \begin{eqnarray} \epsilon^2_g [A] =V_{g}[A]\ge\frac{1}{J_{g}}.\label{crie} \end{eqnarray} A simple proof is given in the appendix.
\section{More Physical Review of Quantum Estimation} \ \\
In this section, we shall discuss real experimental procedures for estimating the unknown parameter $g$ in the framework of quantum estimation theory given above. In real experiments, rigorous unbiased estimators that satisfy
eqn(\ref{ub}) globally in the parameter space of $g$ are usually not available for experimenters from technical reasons.
Instead, only biased estimators are available. Even in such real situations, as we shall show in the following, the quantum estimation theory described in the previous section plays an active role. We also discuss some advantages of quantum estimation theory to provide more efficient methods in several estimation problems in physics.
We shall first consider a quantum estimation
for a parameter $g$ in $H_{tot}(g)$ performed by a measurement of
a general observable $A$ for a single sample at time $t$. Here $A$ is not
assumed unbiased.
Assume that we get an outcome $\bar{a}$ in the measurement and we make the estimate $\bar{g}$ for $g$ as a function of the outcome $\bar{a}$. Usually, this function $\bar{g}=\phi(\bar{a})$ is obtained by the following way. The relation \begin{equation} E_{g}[A]=a \label{e5} \end{equation} between the true value $g$ and the mean output $a$ from the measurement can be often solved theoretically as a function \begin{equation}\label{phi} g=\phi(a). \end{equation} Usually, sensible experiments are designed to possess a suitable domain
${\cal G } (\subseteq G)$, which includes interesting values of $g$, and
to allow the relation \begin{equation} \bar{g}=\phi(\bar{a}), \end{equation} applying the above function $\phi$ to the measurement output $\bar{a}$, gives a good estimate $\bar{g}$ for the given true value $g (\in {\cal G})$ from the output of the single $A$ measurement.
Note that, due to the quantum nature, the observable $A$ generally possesses
nonzero values of the variance $V_g [A]$.
The variance has a close relationship to the problem;
to what extent the estimate $\bar{g}$ can be trusted.
For example, if $V_g[A]$ is extremely small,
then, even in the single measurement, the observed value $\bar{a}$ must be almost equal
to the expectation value $E_g [A]$ for the correct value of $g$.
Consequently, the estimate $\bar{g}$ has to almost coincide with
the correct $g$ value. From the viewpoint of the dimensional analysis,
it is rather straightforward to introduce an expected error $\delta g_g [A]$ of the estimate $\overline{g}_{\bar{a}} [A]$ for the true value of $g$ as follows. \begin{eqnarray} (\delta g_g [A])^2 :=\frac{V_g [A]}{(\partial_g E_g [A])^2}. \label{er1} \end{eqnarray}
Now, we shall give a justification of the above estimate and error evaluation from the viewpoint of quantum estimation theory. When $V_g [A]$ is small enough,
a domain ${\cal G}$ in the space $G$ of possible values
can be chosen so narrow
that $E_g [A]$ is linearly expanded around a physically interesting
value $g_o (\in {\cal G})$: \begin{equation}
E_g [A] = E_{g_o} [A]+\partial_g E_{g}[A]|_{g=g_o}
(g-g_o) +O\left((g-g_o)^2 \right). \end{equation} Here it is quite useful to remind that a single measurement of the observable $A$ simultaneously implies a single measurement of an observable $f(A)$, where $f(x)$ is an arbitrary real function of $x$.
If an output $\bar{a}$ of the observable $A$ is obtained,
it is interpreted that
an output $f(\bar{a})$ is
observed for the observable $f(A)$ in the same measurement.
In what follows, in order to make a useful choice of $f$, let us impose on the function $f$
the locally unbiased condition:
\begin{equation} E_g [f(A)] = g +O\left((g-g_o)^2 \right). \label{luc} \end{equation}
Then it is noticed that eqn(\ref{luc}) is satisfied
for $g\in {\cal G}$ by a linearized function $f(x)$ such that \begin{equation}\label{f(x)}
f(x) = \frac{x}{\partial_g E_g [f(A)] |_{g=g_o}}
+g_o -\frac{E_{g_o} [f(A)]}{\partial_g E_g [f(A)] |_{g=g_o}}. \end{equation} By linearity of $f$, for the mean output $a=E_{g}[A]$ we have \begin{equation} f(a)=f(E_{g}[A])=E_g [f(A)], \end{equation} so that from eqn(\ref{phi}) we have \begin{equation} \phi(\bar{a})=f(\bar{a}) +O\left((g-g_o)^2 \right). \end{equation} Thus, $\bar{g}=\phi(\bar{a})$ is now reproduced
by substituting the output $\bar{a}$ into the function $f(x)$. By use of eqn(\ref{mse}) and eqn(\ref{luc}), the mean-square error of the estimate $\bar{g}$ as the output $f(\bar{a})$ of the $f(A)$ measurement is evaluated as \begin{equation} \epsilon^2_g [f(A)]=V_g [f(A)] +O\left( (g-g_o)^4 \right). \label{mv} \end{equation} From eqn(\ref{f(x)}), the variance of $f(A)$ is evaluated as \begin{equation}
V_g [f(A)] =\frac{V_g [A]}{(\partial_g E_g |_{g=g_o} [A])^2} =(\delta g_g [A])^2 \left(1 +O\left( g-g_o \right) \right),\label{ve} \end{equation} thus, it is verified due to eqn(\ref{mv}) and eqn(\ref{ve}) that \begin{equation} \epsilon^2_g [f(A)] =(\delta g_g [A])^2 \left(1 +O\left( g-g_o \right)\right). \end{equation}
Hence the validity of the error evaluation by $(\delta g_g [A])^2$ definition has been shown.
It is a significant result from quantum estimation theory
that even for the biased observable $A$, the quantum Cram\'{e}r-Rao inequality can be proven: \begin{equation} (\delta g_g [A] )^2 \geq \frac{1}{J_g}.\label{e111} \end{equation}
Here the Fisher information is defined by eqn(\ref{fi}) and
the equality can be achieved by taking $A\propto L(g)$ for each value of $g$.
This can be shown by adopting not $A$ but
the local unbiased operator $f(A)$, and
returning to the general argument for the unbiased case
in section 2. It is also possible to
prove by using the biased observable $A$ straightforwardly.
The proof can be seen in the appendix.
Next we shall discuss physically relevant cases, the N-samples systems,
by naturally extending the single-sample argument. Let us take a
composite system
which consist of N identical {{\bf S}} subsystems. Assume here that the density operator
is independent and identically distributed; \begin{equation} \rho_{tot}^{(N)} (g,t)= \rho(g,t)^{\otimes N}. \end{equation} Now the average estimators $\bar{A}^{(N)}$ defined by \begin{equation} \bar{A}^{(N)} := \frac{1}{N}\sum^N {\bf 1}\otimes \cdots \otimes A \otimes {\bf 1} \cdots \otimes {\bf 1} \end{equation}
are available.
According to the quantum law of large numbers \cite{LargeN},
the measurement data for $\bar{A}^{(N)}$
are going to be normally distributed
with the average $E_g[A]$ and the standard deviation
$\left(\frac{V_g[A]}{N}\right)^{1/2}$ when the number $N$ becomes large. Since the expected error is solely
a pull back of the quantum deviation of
the observable, we can trust the estimate, \begin{equation} g = \bar{g} \pm
\delta g_{\bar{g}} [\bar{A}^{(N)}], \end{equation}
in the 1-$\sigma$ precision for the large number cases.
Now let us discuss the Fisher information for the N-samples cases.
The SLD for the composite system $L^{(N)}$ are
defined for general density
operators $\rho_{tot}^{(N)}$ by \begin{equation} \partial_{g}\rho_{tot}^{(N)}= \frac{1}{2}[\rho_{tot}^{(N)}L^{(N)}+L^{(N)}\rho_{tot}^{(N)}]. \end{equation} For independent and identically distributed (i.i.d.)
density operators,
it is easily derived that the composite SLD $L^{(N)}$
is given by \begin{equation} L^{(N)} =\sum^N {\bf 1}\otimes \cdot \otimes{\bf 1} \otimes L\otimes {\bf 1} \otimes \cdots \otimes {\bf 1}, \label{e12} \end{equation} where ${\bf 1}$ is the identity operation and $L$ is the SLD for the subsystem defined by eqn(\ref{eq:SLD}). This result yields the following simple relation for the Fisher information of the composite system
$J^{(N)}_g =\mbox{\rm Tr} [(L^{(N) })^2 \rho^{(N)}]$: \begin{equation} J^{(N)}_g =NJ_g^{(1)}. \end{equation}
By virtue of the Cram\'er-Rao inequality, it is easily noticed that the optimized estimation for $g$ in a single measurement of the composite system is achieved by adopting the average estimator $\bar{L}^{(N)} =\frac{1}{N} L^{(N)}$.
The expected error is given by \begin{equation} \delta g_{g} [\bar{L}^{(N)}] =\frac{1}{\sqrt{N J_g^{(1)}}}. \end{equation} This coincides with the usual error of the estimation for $g$ based upon N independent data of measurements of $L^{(1)}$ for the N subsystems. However, stress that there is no need to
measure N times the estimator $L^{(1)}$ for each subsystem ${\bf S}$
to achieve the estimate. In the quantum estimation, just
one measurement of the single observable $\bar{L}^{(N)}$
yields the best estimate. Other
relative-difference components like $L\otimes{\bf 1} \cdots \otimes{\bf 1}
-{\bf 1}\otimes L\cdots\otimes{\bf 1}$
remain unmeasured. This saves effectively
the number of processes in the estimation and exposes
an advantage of the quantum
estimation.
When the entanglement between the subsystems is available,
it is possible \cite{Hayashi}
that the large $N$ behavior of $\delta g_g [\bar{L}^{(N)}]$ can be
improved beyond the $1/\sqrt{N}$ factor as \begin{equation} \delta g_g [\bar{L}^{(N)}] \propto \frac{1}{N}. \end{equation} This reveals another advantage of the quantum estimation.
\section{Quantum Estimation by Local Observables} \ \\
We shall now consider the following constraints on the quantum estimation problem discussed above. In the above general formulation, we have assumed that every observable $A$ of the system ${\bf S}$ is available for our measurement to fix the $g$ value. However, in practice the available observables are restricted. For instance, it is a common situation in experiments that a particle described as the system ${\bf S}$ is contained inside the laboratory at the origin of the time and that we can only use measuring devices inside our laboratory. However, according to the time evolution the particle may go out of the experimental apparatus or our laboratory,
so that for the general $t$, our ability of estimating the parameter $g$ is restricted by the measuring devices inside the laboratory.
Let ${\cal M}$ be a subspace of ${\cal H}$. The projection of ${\cal H}$ onto ${\cal M}$ is denoted by $P$. In this paper, we consider the following two constraints.
(i) The initial state is supposed to be supported by $P$, i.e., \begin{eqnarray}\label{eq:local condition} \rho_{tot}(t=0) =P\rho_{tot}(t=0) P. \end{eqnarray}
(ii) The available observables ${\cal O}$ for our measurements
are restricted to those of the form \begin{eqnarray} {\cal O}=PXP+y(I-P), \end{eqnarray} where $X$ is an arbitrary observable on ${\cal H}$ and $y$ is an arbitrary real number.
Let $\{\ket{a}\}$ be the orthonormal basis of ${\cal M}$ and $\{\ket{\alpha}\}$ be the orthonormal basis of ${\cal H}$ extending $\{\ket{a}\}$, i.e, $\{\ket{a}\}\subseteq\{\ket{\alpha}\}$. Then, condition \Eq{local condition} is equivalent to the relation \begin{equation} \rho_{tot}(t=0)
=\sum_{a,a'} |a\rangle \langle a |\rho_{tot}(t=0)|a' \rangle
\langle a'|. \end{equation} Thus, the density operator initially has only matrix elements inside ${\cal M}$, and according to the time evolution, the density operator $\rho_{tot}$ may have matrix elements outside of ${\cal M}$.
In the case of estimating the parameter $g$ by observing a particle ${\bf S}$ initially localized in a box using the measuring devices effective only inside the box, the subspace ${\cal M}$ corresponds to the space of wave functions localized in the box. In this case, the assumption that the particle is initially localized inside the box is represented by condition (i). Since we assume that we know that the particle inside the box at the origin of the time, by measuring later, for instance, the weight of the box , we can measure $I-P$ via a negative result of the measurement. Since the measuring devices are only effective inside the box, the measuring interaction couples only with the observable of the form $PXP$ so that it is natural to assume that they can measure only observables of the form $PXP$. Therefore, the set of available observables are considered to be restricted to those given by condition (ii).
Initially the density operator $\rho_{tot}$ have only matrix elements in $\{\ket{a}\}$. However, in the course of the time evolution, $\rho_{tot}$ can have matrix elements outside of $\{\ket{a}\}$. For any nonnegative time $t\geq 0$, we define the accessible density operator $\rho_\parallel(t)$ for the subspace ${\cal M}$ by \begin{eqnarray} \rho_\parallel(t)=P\rho_{tot}(t)P.\label{e2} \end{eqnarray} Obviously, $\rho_\parallel$ has the matrix representation \begin{equation}
\rho_\parallel(t)=\left[ \langle a |\rho_{tot}(t) | b \rangle \right]. \end{equation} Then, by the corresponding properties of $\rho_{tot}(t)$, the operator $\rho_\parallel(t)$ is positive and satisfies \begin{equation} 0\leq \mbox{\rm Tr} \rho_\parallel(t) \leq 1. \end{equation} In what follows, we shall consider the time domain of $t$ from $t=0$ to the time just before $t=t_*$ such that $\mbox{\rm Tr}\rho_\parallel (t=t_*) =0$, where we allows the case $t_* =\infty$, so that we have \begin{equation} 0<\mbox{\rm Tr} \rho_\parallel(t) \le 1, \end{equation} for $t\in [0,t_*)$.
From condition (ii), the available estimators $A$ on ${\cal M}$ are naturally restricted and satisfy the relation \begin{equation} A=A_{\parallel} +a_{\perp} ({\bf 1}-P)=A^\dagger\label{40}, \end{equation} where \begin{equation} A_\parallel =PA_{\parallel}P.\label{e3} \end{equation} Using the definitions in eqns (\ref{e2}), (\ref{40}) and (\ref{e3}), it can be shown that
the expectation value of the available estimator $A$ is given by \begin{equation} \langle A \rangle =\mbox{\rm Tr}(\rho_{tot}(t) A) =\mbox{\rm Tr}(\rho_\parallel(t)A_{\parallel}) +a_{\perp}(1-\mbox{\rm Tr}\rho_\parallel(t) ).\label{e1} \end{equation}
In order to define rigorously
the notion of the ``local'' estimators $\tilde{A}$ corresponding to the available estimators $A $ in the restricted situation, let us extend the space ${\cal M}$ to a one-dimension-larger
Hilbert space $\tilde{{\cal M}}$ by adding to the basis of ${\cal M}$
a normal vector $|B\rangle$
orthogonal to every $|a\rangle$,i.e.,
$\tilde{{\cal M}}:={\cal M}\oplus {\bf C}|B\rangle$.
Then, the local estimators $\tilde{A}$ acting
on $\tilde{{\cal M}}$, which corresponds to the available estimator in eqn(\ref{40}), are defined by \begin{equation} \tilde{A}=A_{\parallel}
+a_{\perp} |B\rangle \langle B |. \label{70} \end{equation} In particular, note that \begin{eqnarray}
\widetilde{1-P}= |B\rangle \langle B |.\label{e4}
\end{eqnarray} Since the state $|B\rangle$ represents the inaccessible states by our local observation as seen
in eqn(\ref{e4}), we call $|B\rangle$ the blank state.
Further let us introduce the local density operator $\rho$ acting on $\tilde{{\cal M}}$ and corresponding to $ \rho_\parallel (=P\rho_{tot}P) $
by \begin{eqnarray} \rho=\rho_\parallel
+(1-\mbox{\rm Tr}\rho_\parallel ) |B\rangle \langle B |.\label{e20} \end{eqnarray} It is easily seen that $\rho$ is positive and of unit trace. By a simple manipulation, the expectation value of the available estimator $A$ in eqn(\ref{e1})
can be
reexpressed by use of the local estimator $\tilde{A}$
and the local density operator $\rho$
as \begin{equation} \langle A \rangle =\mbox{\rm Tr}(\rho \tilde{A} ). \end{equation}
\section{Cram\'{e}r-Rao Bound for Local Quantum Estimators} \ \\
In what follows, we shall consider the quantum Cram\'er-Rao inequality for the quantum estimation problem for the coupling constant $g$ in the Hamiltonian $H_{tot}(g)$ by using {\em only} local measuring devices. By the time evolution, the local density operator $\rho=\rho(t,g)$ introduced
in the previous section depends on the time $t$ and the parameter $g$. Now we assume that one measures a local estimator $\tilde{A}$ at time $t$, and the output $\bar{a}$ determines the estimate $\bar{g}_{\bar{a}} [A]$
for $g$ via the relation: \begin{eqnarray*} E_{\bar{g}_{\bar{a}} [A]}[A]=\mbox{\rm Tr}[\tilde{A} \rho(t,\bar{g}_{\bar{a}} [A]) ]=\bar{a}. \end{eqnarray*} Stress that, in this estimating process of $g$,
we are allowed to use only local estimators $\tilde{A}$ instead of arbitrary observables in the theory.
The variance of the local observable $\tilde{A}$ for the correct $g$
value is certainly given by \begin{eqnarray} V_{g}[\tilde{A}]=\mbox{\rm Tr} [\tilde{A}^{2}\rho(t,g)]-\left(\mbox{\rm Tr}[\tilde{A}\rho(t,g)]\right)^{2}. \end{eqnarray} Then it is required at a given time $t$
to find the minimum value of the expected error defined by \begin{equation} \delta g_g [\tilde{A}] : =\sqrt{\frac{V_g [\tilde{A}]}{(\partial_g E_g[\tilde{A}])^2}}. \end{equation} It is shown that this problem is resolved by use of a solution of
the problem on the estimate for the parameter $g$
by arbitrary observables $\tilde{{\cal O}}$ on $\tilde{{\cal M}}$ as follows.
We define the local SLD $\tilde{L}(g)$ on $\tilde{{\cal M}}$
for an arbitrary local density operator $\rho(g)$ in eqn(\ref{e20}) as a self-adjoint operator satisfying \begin{equation} \partial_g \rho(g) =\frac{1}{2}(\tilde{L}(g) \rho +\rho
\tilde{L}(g)),\label{e16} \end{equation} \begin{equation} \tilde{L}(g)^\dagger =\tilde{L}(g).\label{e18} \end{equation} Since $\mbox{\rm Tr}\rho(g)=1$ for any $g\in G$, we have \begin{equation} \mbox{\rm Tr}[\tilde{L}(g)\rho(g) ]=0. \end{equation} It is easy to construct a solution of eqn(\ref{e16}) by introducing
a SLD operator $L(g)$ on ${\cal M}$ for the accessible density operator $\rho_\parallel$. The SLD $L(g)$ on ${\cal M}$ is
defined by \begin{equation} \partial_g \rho_\parallel =\frac{1}{2}(L(g) \rho_\parallel +\rho_\parallel
L(g)),\label{e17} \end{equation} \begin{eqnarray} && L(g)^\dagger =L(g), \\ && PL(g)P =L(g). \end{eqnarray} Due to the fact that $P\rho_\parallel P =\rho_\parallel $,
we can find, at least, a solution of eqn(\ref{e17})
for the SLD with $PL(g)P =L(g)$. Once the SLD $L(g)$ is given, then it is proven by a simple algebra that the operator defined by \begin{equation} \tilde{L}(g)=L(g)+ \partial_g
\ln [1-Tr\rho_{\parallel}(g)]|B\rangle \langle B| \label{e19} \end{equation}
satisfies eqns(\ref{e16}) and (\ref{e18}), thus it is a SLD
on $\tilde{{\cal M}}$ for $\rho(g)$.
Here it is carefully noted that we may have \begin{equation} \mbox{\rm Tr}[L(g)\rho_{\parallel}(g) ] \neq 0, \end{equation} since the trace of $\rho_{\parallel}(g)$ is not necessarily normalized.
The operator $\tilde{L}(g)$ is determined uniquely up to the support of $\rho(g)$; any two solutions $\tilde{L}_{1}(g), \tilde{L}_{2}(g)$ satisfy $\tilde{L}_{1}(g)\rho(g)=\tilde{L}_{2}(g)\rho(g)$. The Fisher information $J_{g}$ of the parameter $g$ in $\rho(g)$ is uniquely defined by \begin{equation} J_{g}=\mbox{\rm Tr}[\tilde{L}(g)^2 \rho(g)] =\mbox{\rm Tr}[L(g)^2 \rho_\parallel(g)] + \frac{\left[\mbox{\rm Tr}[ L(g)\rho_\parallel(g)]\right]^2}{1-\mbox{\rm Tr}[\rho_\parallel(g)]}, \label{34} \end{equation} where we have used eqns(\ref{e20}) and (\ref{e19}). Then, for the
arbitrary observables $\tilde{\cal O}$ on $\tilde{{\cal M}}$
we have the quantum Cram\'er-Rao inequality \begin{equation} (\delta g_g [\tilde{{\cal O}}] )^2 :=\frac{V_{g}[\tilde{{\cal O}}]}{(\partial_g E_g [\tilde{{\cal O}}])^2} \geq \frac{1}{J_{g}}, \end{equation} where $J_g$ is given by eqn(\ref{34}). In order to apply the result to our local estimator problem,
it is crucial to
notice that the SLD
in eqn(\ref{e19}) takes the precise form of the local estimator
on $\tilde{{\cal M}}$ in eqn(\ref{70}).
Therefore, the equality can be attained by a local estimator. This indicates that
the following quantum Cram\'er-Rao inequality for arbitrary
local estimators $\tilde{A}$ on $\tilde{{\cal M}}$ really holds
for the local density operators $\rho$ corresponding
to the accessible density operators
$\rho_\parallel (g) (=P\rho_{tot}(g) P)$: \begin{equation} (\delta g_g [\tilde{A}] )^2 =\frac{V_{g}[\tilde{A}]}{(\partial_g E_g [\tilde{A}])^2} \geq \frac{1}{J_{g}},\label{cr} \end{equation} where the Fisher information $J_g$ is given by eqn(\ref{34}). For a given $g\in G$, the equality is attained by a local estimator $\tilde{A}_o (g)$ such that \begin{equation} \tilde{A}_o (g)\propto \tilde{L}(g)=L(g)
-\frac{Tr[L(g)\rho_\parallel(g)]}{1-Tr\rho_\parallel(g)}|B\rangle\langle B|. \end{equation} Note that the local estimators which give the minimum expected error such as $\tilde{A}_o$
are unique only up to a factor and an additive term proportional to the identity operator. For instance, an estimator such that \begin{equation} \tilde{A}_o '(g) \propto L(g) + \frac{Tr[L(g)\rho_\parallel(g)]}{1-Tr\rho_\parallel(g)}P, \end{equation} which has no matrix element for the blank state, also attains the equality.
\section{The Fisher Information for Unnormalized Pure States } \ \\
In physics, it often happens that the measurement device
is able to probe only a small part
of the physical states of the total system.
Even in such situations, non-unitary formulations are sometimes
available. The state vectors $|\Psi(t)\rangle$
are governed by equations of motion with non-Hermitian
Hamiltonians and
evolve deterministically in the subspace ${\cal M}$, which is accessible
by the experimental devices. Such examples are found
in the various fields of physics,
including
the scattering problems with weak absorption of quanta
in the nuclear physics and the quantum optics,
the flavor-oscillation studies
in the elementary particle physics and so on.
The information about
the coupling constant $g$ in the equations of motion is
imprinted on the state vectors $|\Psi(t,g)\rangle$
during the time evolution.
Let us evaluate the Fisher information for the pure state
$|\Psi (t,g)\rangle$. The accessible density operator for the pure state reads \begin{equation} \rho_\parallel
(t,g) =|\Psi(t,g) \rangle \langle \Psi (t,g) |,\label{36} \end{equation} where $\mbox{\rm Tr}[\rho_\parallel (0,g)] =1 $
and at an advanced time $t(>0)$ the following relation holds: \begin{equation} 0 < \mbox{\rm Tr}[\rho_\parallel (t,g)] \leq 1. \end{equation} We define the SLD operator $L$ on ${\cal M}$, in the same way discussed in the previous section,
for the accessible density operator $\rho_\parallel$. Note that
the operator $L$ is not uniquely determined due to the
purity of $\rho_\parallel$, however, the ambiguity is not relevant at all for the Fisher information, as commented in the previous section.
It is shown that we have a SLD, \begin{equation} L=\frac{2}{\mbox{\rm Tr}[\rho_\parallel]} \partial_g \rho_\parallel -\frac{\mbox{\rm Tr}[\partial_g \rho_\parallel]} {(\mbox{\rm Tr}[\rho_\parallel] )^2}\rho_\parallel, \end{equation} as a simple representative and
the Fisher information itself is uniquely evaluated by \begin{equation} J=4\left(
\langle \partial_g \Psi |\partial_g \Psi \rangle -
\frac{\left|Im \langle \Psi |\partial_g \Psi \rangle \right|^2}
{\langle \Psi |\Psi \rangle} \right)
+4\frac{\left|Re \langle \Psi |\partial_g \Psi \rangle\right|^2}
{1-\langle \Psi |\Psi \rangle},\label{unps} \end{equation}
where $|\partial_g \Psi\rangle :=\partial_g |\Psi (t,g)\rangle$. This result is an extension of that in the reference \cite{FN95},
where the normalized pure state theory is analyzed. The relation enables us to evaluate easily the Fisher information
for many unnormalized pure state theories.
In eqn (\ref{unps}), one may worry about the apparent divergence
of the third term at $\langle \Psi |\Psi \rangle=1$, because
the state evolves initially from the normalized state.
However, for ordinary physical systems, the early behavior of the
norm $\langle \Psi |\Psi \rangle$ is given by \begin{equation}
\langle \Psi (t,g)|\Psi (t,g)\rangle \sim 1-\alpha(g) t^2, \end{equation} where $\alpha$ is a positive function of $g$. Thus the third term is evaluated in the early era as \begin{equation}
4\frac{\left|Re \langle \Psi |\partial_g \Psi \rangle\right|^2}
{1-\langle \Psi |\Psi \rangle} \sim \frac{(\partial_g \alpha (g))^2}{\alpha(g)} t^2. \end{equation} Hence, the limit $t\rightarrow +0$ of eqn(\ref{unps}) exists without any problems.
\section{Problems of the Composite System} \ \\
In the local estimation problem, some nontrivial aspects appear in the
composite system analysis.
Suppose a system ${\bf S}$.
Let us assume our measuring device for ${\bf S}$
is able to access only a subspace
${\cal M}$ of the Hilbert space of ${\bf S}$.
Later let $P$ denote the projection operator onto ${\cal M}$,
and$D_{{\cal M}}$ denote the dimension of the subspace ${\cal M}$.
The accessible density operator on ${\cal M}$
is denoted by $\rho_\parallel$.
The operator $\rho_\parallel (t,g)$ evolves in the subspace ${\cal M}$
and becomes dependent on the coupling constant $g$ in the equation of motion.
Let us consider a composite system
${\bf S}^{\otimes N}$ composed of N identical ${\bf S}$ subsystems.
For instance,
suppose that an independent and identically distributed (i.i.d.)
initial condition
is set for the total density operator $\rho_{tot}(0)$
of the composite system. Also assume that the unitary evolution of the total system is factorized, i.e., $U^{(N)}(t) =U(t)^{\otimes N}$.
Even in such a simple situation, it can be pointed out that
we have, at least, two natural
alternatives for the estimation of $g$ as follows.
The first alternative is rather simple.
In the procedure, one firstly calculates the accessible density operator
$\rho^{(N)}_{\parallel }=P^{\otimes N} \rho_{tot}P^{\otimes N}$
for ${\bf S}^{\otimes N}$ which is reduced to a direct product defined by \begin{equation} \rho^{(N)}_{\parallel } (t,g) :=\rho_\parallel^{\otimes N} (t,g). \end{equation} A local density operator for the accessible density operator
$\rho^{(N)}_{\parallel }$
can be defined straightforwardly by \begin{equation} \rho_1^{(N)} := \rho^{(N)}_{\parallel }
+(1-\mbox{\rm Tr}\rho^{(N)}_{\parallel } ) |B\rangle \langle B |, \end{equation}
where $|B\rangle$ is the blank state.
Let $j^{(N)}$ denote the Fisher information
based upon the first local density operator $\rho_1^{(N)}$.
The estimation problem in the composite systems is nontrivial
because we may have a construction of another local density
operator for ${\bf S}^{\otimes N}$. We are able to define at first the local density operator for
each subsystem ${\bf S}$. For the i-th subsystem ${\bf S}_i$,
the local density operator $\rho_i$ corresponding
to $\rho_{\parallel i}$ is written as \begin{eqnarray} \rho_i =\rho_{\parallel i}
+(1-\mbox{\rm Tr}\rho_{\parallel i} ) |B_i \rangle \langle B_i |, \end{eqnarray}
where $|B_i \rangle $ is the blank vector for the i-th subsystem ${\bf S}_i$.
Then we can define naturally
the second local density operator $\rho^{(N)}_2$ for the
composite system ${\bf S}^{\otimes N}$ by a direct product as follows: \begin{equation} \rho^{(N)}_2 :=\prod^N_{i=1}\otimes \rho_i . \end{equation} Let $J^{(N)}$ denote the Fisher information based upon
$\rho^{(N)}_2$. By construction, the Fisher information $J^{(N)}$
for the i.i.d. density operator is calculated as \begin{equation} J^{(N)} =NJ^{(1)}. \end{equation}
As seen above, there exist two independent Fisher informations for the composite system. Then, it is an important question;
which alternative of the formulation
gives us a more precise estimate for $g$, that is, which Fisher
information, $j^{(N)}$ or $J^{(N)}$, is larger than another.
The problem should be addressed for the general initial conditions for the density operators, beyond the above i.i.d. situations. Note first that the operators $\rho_1^{(N)}$ act on the Hilbert space ${{\cal M}}^{\otimes N}
\oplus {{\bf C}}|B\rangle$ and the dimension of ${{\cal M}}^{\otimes N}
\oplus {{\bf C}}|B\rangle$ is given by
$D_1 =(D_{{\cal M}})^N +1$.
On the other hand,
the operators $\rho_2^{(N)}$ act on the Hilbert space
$\tilde{{{\cal M}}}^{\otimes N}$ and the dimension of $\tilde{{\cal M}}^{\otimes N}$ is given by
$D_2 =(D_{{\cal M}} +1)^N$. Since $D_2 > D_1$ always holds, it is naively
expected that the second Fisher information $J^{(N)}$
is not less than the first Fisher information $j^{(N)}$.
This guess can be
proved affirmatively by use of the monotonicity
argument for the Fisher information as will be mentioned later.
The above argument has been limited to the i.i.d.cases. In order to analyze the composite-system estimation generally, we must
extend the above two formulations. Especially, nontrivial
analyses are required to define $J^{(N)}$.
These are formulated in section 8.
In section 9, it is also pointed out
that evaluation of the larger Fisher information $J^{(N)}$
requires solving time evolutions of
various density operators corresponding to different initial conditions.
Such a feature
does not appear in the evaluation of both the standard Fisher information in the usual cases
and the smaller Fisher information $j^{(N)}$
in the local estimation.
\section{General Formulation for the Composite System} \ \\
The available estimators for the composite system
${\bf S}^{\otimes N}$ now reads \begin{equation} A^{(N)} = \sum_{k_1\cdots k_N} \omega_{k_1 \cdots k_N} A_{k_1} \otimes\cdots\otimes A_{k_N},\label{80} \end{equation} which is just a natural extension of eqn(\ref{40}). Here $A_{k}$
denote the available estimators for the subsystem ${\bf S}$,
which take the form in eqn
(\ref{40}) and $\omega_{k_1 \cdots k_N}$ are real coefficients. The corresponding extension of eqn(\ref{70}) is also possible.
The local estimator $\tilde{A}^{(N)}$ corresponding to
the available estimator $A^{(N)}$ is defined by \begin{equation} \tilde{A}^{(N)} = \sum_{k_1\cdots k_N} \omega_{k_1 \cdots k_N} \tilde{A}_{k_1} \otimes\cdots\otimes \tilde{A}_{k_N}, \end{equation} where $\tilde{A}_k$ are the local estimators corresponding to $\tilde{A}$ in eqn(\ref{70}).
In order to define the two Fisher informations $j^{(N)}$ and $J^{(N)}$
beyond the i.i.d. condition, let us consider
the most general local density operator
$\rho_{tot}^{(N)} (0) = P^{\otimes N} \rho_{tot}^{(N)} (0) P^{\otimes N}$
as the initial total density operator. In the unitary time evolution of the total system, \begin{equation} \rho_{tot}^{(N)} (t,g) =U^{(N)} (t,g) \rho_{tot}^{(N)} (0) U^{(N)\dagger} (t,g), \end{equation}
the total density operator becomes to have
matrix elements between the inaccessible states.
Even for the general initial conditions,
the definition of the first Fisher information $j^{(N)}$
is essentially unchanged.
Let us introduce the accessible operators $\rho_{\parallel}^{(N)}$
by reducing the total density operator $\rho_{tot}^{(N)}$ as \begin{equation} \rho_{\parallel}^{(N)} =P^{\otimes N} \rho_{tot}^{(N)} P^{\otimes N}. \end{equation} For the accessible density operator $\rho^{(N)}_\parallel$,
a SLD operator $L^{(N)}$ is defined by \begin{equation} \partial_g \rho_\parallel^{(N)} =\frac{1}{2}\left[ L^{(N)} \rho^{(N)}_\parallel +\rho^{(N)}_\parallel
L^{(N)}
\right], \end{equation} \begin{eqnarray} && (L^{(N)})^{\dagger} =L^{(N)} , \\ && P^{\otimes N} L^{(N)}P^{\otimes N} =L^{(N)}. \end{eqnarray} According to eqn(\ref{34}),
the Fisher information $j^{(N)}$ is defined straightforwardly as follows. \begin{equation} j^{(N)} :=\mbox{\rm Tr}\left[\left(L^{(N)}\right)^2 \rho_\parallel^{(N)}\right] + \frac{\left(\mbox{\rm Tr}\left[ L^{(N)} \rho_\parallel^{(N)}\right]\right)^2}{1-\mbox{\rm Tr}\left[\rho_\parallel^{(N)}\right]}. \end{equation}
Next, in order to define the second Fisher information $J^{(N)}$,
what we want is a proper definition of a
local density operator
$\rho^{(N)}$ acting on $\tilde{{{\cal M}}}^{\otimes N}$ such that
the total density operator $\rho_{tot}^{(N)}$
is reduced into $\rho^{(N)}$.
Here, it is quite natural to impose that expectation
values of all the available observables $A^{(N)}$ for $\rho_{tot}^{(N)}$
are equivalent to
those of the corresponding local observables $\tilde{A}^{(N)}$
for $\rho^{(N)}$: \begin{equation} \mbox{\rm Tr}[ A^{(N)} \rho_{tot}] =\mbox{\rm Tr}[ \tilde{A}^{(N)} \rho^{(N)} ].\label{ee2} \end{equation}
By some manipulations it is soon noticed that the above constraint is really
satisfied by defining the local density operator $\rho^{(N)}$ as follows. Let index $\alpha_{j}$ for $j=1,\ldots,N$ below
take index $a_{j}$ for states in ${\cal M}$
or the index $B$ for the blank state.
Then the matrix elements of $\rho^{(N)}$ on
$\tilde{{{\cal M}}}^{\otimes N}$ are given by \begin{eqnarray} &&
\langle \alpha_1 \alpha_2 \cdots \alpha_N |
\rho^{(N)}|\alpha_1 ' \alpha_2 '\cdots \alpha_N '\rangle \nonumber\\ &=& \prod_{j=1}^N \left[ \delta_{\alpha_j B} \delta_{\alpha_j ' B} +(1-\delta_{\alpha_j B} )(1-\delta_{\alpha_j ' B} ) \right] \nonumber\\ &&\times \sum_{x_1 \cdots x_N} \prod^N_{k=1} \left[ \delta_{\alpha_k B}\delta_{x_k 1} + (1-\delta_{\alpha_k B} )\delta_{x_k 0} \right] \prod^N_{k'=1} \left[ \delta_{\alpha_{k'} ' B}\delta_{x_{k'} 1} + (1-\delta_{\alpha_{k'} ' B} )\delta_{x_{k'} 0} \right] \nonumber\\ &&\times \mbox{\rm Tr}[(
P_{1,x_1} \otimes P_{2,x_2} \otimes \cdots \otimes P_{N,x_N}) \rho_{tot}], \label{91} \end{eqnarray} where $\mbox{\rm Tr}$ stands for the trace operation on ${\cal H}^{\otimes n}$. For $m=1,\ldots,N$, the subscript $x_m$ takes $0$ or $1$ and the operator $P_{m,x_{m}}$ is defined by \begin{eqnarray} && P_{m,x_{m}} =\ket{a'_{m}}\bra{a_{m}},\quad\mbox{if $x_{m}=0$} \\ && P_{m,x_{m}}={\bf 1}-P,\quad\mbox{if $x_{m}=1$}. \end{eqnarray} By construction the Hermicity of the operator $\rho^{(N)}$ is trivial. Further, taking $A^{(N)} ={\bf 1}^{\otimes N}$ in eqn(\ref{ee2}) yields the normalization condition: \begin{equation} \mbox{\rm Tr}[\rho^{(N)}] =1.\label{ee1} \end{equation}
The positivity of $\rho^{(N)}$ is also proven as follows. Suppose an arbitrary vector $|\Psi\rangle$ on $\tilde{{{\cal M}}}^{\otimes N}$: \begin{eqnarray}
|\Psi\rangle &=& \sum_{\alpha_1\cdots \alpha_N} C_{\alpha_1 \cdots \alpha_N}
|\alpha_1 \cdots \alpha_N \rangle \nonumber\\ &=&
\sum_{(i_1 \cdots i_k)} |\Psi_{[i_1 \cdots i_k]} \rangle, \end{eqnarray} where \begin{equation}
|\Psi_{[\o]}\rangle = \sum_{a_1\cdots a_N} C_{a_1 \cdots a_N}
| a_1 \cdots a_N \rangle, \end{equation} \begin{equation}
|\Psi_{[1]}\rangle = \sum_{a_2\cdots a_N} \ C_{Ba_2 \cdots a_N}
| B a_2 \cdots a_N \rangle \end{equation} and so on.
Then, using the definition of $\rho^{(N)}$
in eqn (\ref{91}),
the expectation values of $\rho^{(N)}$ for the arbitrary
state vectors $|\Psi\rangle$ are evaluated as follows. \begin{eqnarray}
\langle \Psi |\rho^{(N)}|\Psi \rangle &=&
\sum_{i_1 \cdots i_k} \langle \Psi_{[i_1 \cdots i_k]}|
\rho^{(N)} |\Psi_{[i_1 \cdots i_k] }\rangle \nonumber\\ &=& \sum_{i_1 \cdots i_k} \mbox{\rm Tr}[ \tilde{P}_{[i_1 \cdots i_k]} \rho^{(N)} ] \nonumber\\ &=& \sum_{i_1 \cdots i_k} \mbox{\rm Tr}[ P_{[i_1 \cdots i_k] } \rho_{tot} ] , \end{eqnarray} where $\tilde{P}_{[i_1 \cdots i_k]}=
|\Psi_{[i_1 \cdots i_k]} \rangle\langle\Psi_{[i_1 \cdots i_k]}|$ and
$P_{[i_1 \cdots i_k]}$ are defined by replacing $|B\rangle\langle B|$'s
in the operator $\tilde{P}_{[i_1 \cdots i_k]}$ to ${\bf 1} -P$.
Noting that the operators $P_{[i_1 \cdots i_k]}$
can be expressed as $P_{[i_1 \cdots i_k]}
=\sum_\beta
|\Phi_{\beta,[i_1 \cdots i_k]} \rangle\langle\Phi_{\beta,[i_1 \cdots i_k]}|$
by use of vectors $|\Phi_{\beta,[i_1 \cdots i_k]} \rangle$
in the total Hilbert
space, it is proven that \begin{eqnarray}
\langle \Psi |\rho^{(N)}|\Psi \rangle =
\sum_{\beta, i_1, \cdots i_k} \langle \Phi_{\beta,[i_1 \cdots i_k]}| \rho_{tot}^{(N)}
|\Phi_{\beta,[i_1 \cdots i_k]} \rangle
\geq 0. \end{eqnarray} Taking account of the normalization condition in eqn(\ref{ee1}),
this implies the positivity of the operator $\rho^{(N)}$.
Since $\mbox{\rm Tr} [\rho^{(N)}] =1$, we can define in the usual way
a SLD operator ${\cal L}$ for the local density operator
$\rho^{(N)}$: \begin{equation} \partial_g \rho^{(N)} =\frac{1}{2} ({\cal L} \rho^{(N)}
+\rho^{(N)}
{\cal L} ). \end{equation} Then, the Fisher information $J^{(N)}$ is defined by \begin{equation} J^{(N)}=\mbox{\rm Tr}\left[\rho^{(N)} {\cal L}^2 \right]. \end{equation}
Now let us comment on the inequality $J^{(N)} \geq j^{(N)}$,
using the monotonicity of the Fisher information.
The point is that there exists a mapping $R$
of the density operators defined in $\tilde{{\cal M}}^{\otimes N}$ onto
the density operators defined in ${{\cal M}}^{\otimes N}\oplus {{\bf C}}|B\rangle$. Let us denote $P_\parallel$
a projection operator onto the subspace of vectors
in $\tilde{{\cal M}}^{\otimes N}$ that
do not include the blank states at all. Denote $P_\perp$
a projection operator onto the subspace of vectors that include more than one
sub-blanck vectors $|B_i\rangle$. It should be noted that \begin{equation} P_\parallel \rho^{(N)} P_\parallel = \rho^{(N)}_\parallel. \end{equation} Let us define the mapping $R$ as follows. \begin{eqnarray} R[\rho^{(N)} ]&=& P_\parallel \rho^{(N)} P_\parallel +\mbox{\rm Tr}\left[ P_\perp \rho^{(N)}\right]
|B\rangle\langle B| \nonumber\\ &=&\rho^{(N)}_{\parallel} +\mbox{\rm Tr}\left[ P_\perp \rho^{(N)}\right]
|B\rangle\langle B|. \end{eqnarray} By definition, it is clear that the mapping is linear and of unit trace: \begin{equation} \mbox{\rm Tr}\left[ R[\rho^{(N)} ] \right] =\mbox{\rm Tr}[P_\parallel \rho^{(N)}] +\mbox{\rm Tr}[P_\perp \rho^{(N)}] =1. \end{equation} It is also easily seen that this mapping is completely positive, since so are $\rho^{(N)}\mapsto P_\parallel \rho^{(N)} P_\parallel $ and $\rho^{(N)}\mapsto\mbox{\rm Tr}\left[ P_\perp \rho^{(N)}\right]$.
Using the relation $\mbox{\rm Tr}[\rho^{(N)}] =1$, we obtain \begin{eqnarray} \varrho &:=& R[\rho^{(N)} ] \nonumber\\ &=& \rho^{(N)}_\parallel +(1-\mbox{\rm Tr}[\rho^{(N)}_\parallel ])
|B\rangle\langle B|. \end{eqnarray}
Then, the first Fisher information $j^{(N)}$ is given by $\mbox{\rm Tr}[ \varrho^{(N)} (\tilde{L})^2 ]$, where
$\tilde{L}$ is the SLD operator corresponding to $\varrho^{(N)}$.
According to the monotonicity theorem for the Fisher information
\cite{Monotone},
it must be satisfied under the projective mapping $R$ that
$J^{(N)} \geq j^{(N)}$. This result does not depend on whether the total density operators $\rho^{(N)}_{tot}$ are factorized or entangled.
It is worth noting that the information $J^{(N)}$ possesses a decomposition
representation. Let us consider an arbitrary subsequence $(i_1 ,i_2 ,\cdots ,i_n)$ of
the sequence $(1,2,3,\cdots,N)$. Define that $\rho_{[i_1 ,i_2 ,\cdots ,i_n]}$
is a $(dim {\cal M})^{N-n}\times (dim {\cal M} )^{N-n}$ matrix which is composed of components of $\rho^{(N)}$
with $\alpha_{i_m} =\alpha_{i_m}'=B$ for $m=1,\cdots, n$.
The followings are examples. \begin{equation}
\langle a_2 a_3 \cdots a_N | \rho_{[1]}
|a_2 ' a_3 ' \cdots a_N ' \rangle :=
\langle B a_2 a_3 \cdots a_N | \rho^{(N)}
|B a_2 ' a_3 ' \cdots a_N ' \rangle, \label{90} \end{equation} \begin{equation}
\langle a_2 a_4 \cdots a_N | \rho_{[1,3]}
| a_2 ' a_4 '\cdots a_N ' \rangle :=
\langle B a_2 B a_4 \cdots a_N | \rho^{(N)}
|B a_2 ' B a_4 ' \cdots a_N ' \rangle, \end{equation} \begin{equation}
\langle a_1 | \rho_{[2,3,\cdots,N]}
|a_1 ' \rangle :=
\langle a_1 B B \cdots B | \rho^{(N)}
|a_1 ' B B \cdots B \rangle. \end{equation} Note that the empty subsequence $\o$ corresponds to the accessible
density operator: \begin{eqnarray}
\langle a_1 a_2 \cdots a_N | \rho_{[\o]}
|a_1 ' a_2 ' \cdots a_N ' \rangle =
\langle a_1 a_2 \cdots a_N | \rho_\parallel^{(N)}
|a_1 ' a_2 ' \cdots a_N ' \rangle. \end{eqnarray} By definitions the local density operators $\rho_{[i_1 ,i_2 ,\cdots ,i_n]}$
are non-negative, i.e., $\rho_{[i_1 ,i_2 ,\cdots ,i_n]}\geq 0$. For each $\rho_{[i_1 ,i_2 ,\cdots ,i_n]}$,
we can introduce a partial SLD operator
${\cal L}_{[i_1 ,i_2 ,\cdots ,i_n]}$
as \begin{equation} \partial_g \rho_{[i_1 ,i_2 ,\cdots ,i_n]} =\frac{1}{2} ({\cal L}_{[i_1 ,i_2 ,\cdots ,i_n]} \rho_{[i_1 ,i_2 ,\cdots ,i_n]}
+\rho_{[i_1 ,i_2 ,\cdots ,i_n]}
{\cal L}_{[i_1 ,i_2 ,\cdots ,i_n]} ). \end{equation} Then it is possible to rewrite the second information as \begin{eqnarray} J^{(N)}&=&\sum_{(i_1 ,i_2 ,\cdots ,i_n)} J^{(N)}_{[i_1 ,i_2 ,\cdots ,i_n]}, \label{300}\\ J^{(N)}_{[i_1 ,i_2 ,\cdots ,i_n]} &=& Tr\left[ \rho_{[i_1 ,i_2 ,\cdots ,i_n]} {\cal L}^2_{[i_1 ,i_2 ,\cdots ,i_n]} \right]. \end{eqnarray} Here $\sum_{(i_1 ,i_2 ,\cdots ,i_n)}$ means the sum over all the subsequences $(i_1 ,i_2 ,\cdots ,i_n)$ of $(1,2,3,\cdots,N)$, including the empty subsequence $\o$. The decomposition representation makes the evaluation of $J^{(N)}$
easier in many practical applications by using a
useful formula for the operators
$\rho_{[i_1 ,i_2 ,\cdots ,i_n]}$ in the next section.
\section{Evaluation of the Local Density Operator } \ \\
The accessible density operators $\rho^{(N)}_\parallel$
can be followed by our apparatus,
since the operators $\rho_\parallel^{(N)}$
are completely local by definition. Meanwhile, the local density operator $\rho^{(N)}$ has been so far defined based upon the total density operator $\rho_{tot}^{(N)}$
in the previous section.
We must say that the definition is too formal
from the the practical viewpoint,
because we seldom know global information about the
total density $\rho_{tot}^{(N)}$
due to the limitation
of our ability to measure the system.
For the realistic evaluation of $J^{(N)}$,
it is convenient to write down $\rho^{(N)}$ explicitly
in terms of locally accessible quantities
just as the operator $\rho^{(N)}_\parallel$. Such a reformulation can be realized for the cases with factorized evolutions, i.e.,
$U^{(N)}(t) = [U(t)]^{\otimes N}$ as follows. It
should be emphasized that we do {\em not} need to assume the i.i.d. condition for the initial density operator.
Suppose that a composite system ${\bf S}^{\otimes N}$ of N identical ${\bf S}$ subsystems is governed by a unitary evolution and that the evolution is factorized for each subsystem ${\bf S}$: \begin{equation} \rho_{tot}^{(N)} (t) =[U(t)]^{\otimes N} \rho^{(N)}_{tot} (0) [U(t)^\dagger ]^{\otimes N}, \end{equation} where $U(t)$ is the unitary time evolution operator for ${\bf S}$ and $\rho_{tot}(0)$ is arbitrary initial density operator, which may have
entanglement between the subsystems.
Let ${\cal O}_{{\cal M}}:=\{e_a |e_a^\dagger =e_a,\ P e_a P =e_a \}$
denote the complete basis of the available observables acting on ${\cal M}$
for each subsystem ${\bf S}$. Even in our local experiments,
we are able to define and measure the projective evolutions
for the available observables $e_a$.
The projective
evolutions are given by stochastic mappings $\Gamma(g,t)[e_a]$ which are
defined by \begin{equation} \Gamma(g,t)[e_a] :=PU(t) e_a U(t)^\dagger P. \label{203} \end{equation} In various physical systems,
the dynamics is first given by not $\Gamma(g,t)$ but
a Lindblad differential form given by \begin{equation} \partial_t \rho_\parallel =T_g [\rho_\parallel] \end{equation} for arbitrary density operators $\rho_\parallel$
on ${\cal M}$. Here $T_g$ is a time-independent Lindblad super-operator. Then the super-operator $T_g$
is related formally to the stochastic mapping $\Gamma(g,t)$
via \begin{equation} T_g = \partial_t \Gamma(g,t=0). \end{equation} By integrating formally as $\Gamma(g,t) =e^{tT_g}$,
we can recover the stochastic super-operators $\Gamma(g,t)$.
Stress that the operators $\Gamma(g,t)[e_a]$ are completely local quantities we can observe. Moreover, the projective evolutions for the composite available observables are also
completely local quantities, which are written as \begin{eqnarray} \Gamma(g,t)^{\otimes k} [e_{a_1}\otimes \cdots \otimes e_{a_k}] =P^{\otimes k} U(t)^{\otimes k} (e_{a_1}\otimes \cdots \otimes e_{a_k}) (U(t)^{\otimes k} )^\dagger P^{\otimes k}. \end{eqnarray} Our aim in this section is to express the operators $ \rho_{[i_1 ,i_2 ,\cdots ,i_n]}$ of the local density operator
$\rho^{(N)}$ only in terms of
the accessible operators like $\Gamma(g,t)^{\otimes k} [e_{a_1}\otimes \cdots \otimes e_{a_k}]$.
Since the initial density operator satisfies $\rho_{tot}^{(N)}(0)
=P^{(N)} \rho_{tot}^{(N)}(0) P^{(N)}$,
the operator $\rho_{tot}^{(N)} (0)$ can be
expanded
using the basis $\{e_a\}$: \begin{equation} \rho_{tot}^{(N)} (0)=\sum_{a_1}\sum_{a_2}\cdots \sum_{a_N} C_{a_1 a_2 \cdots a_N} e_{a_1} \otimes e_{a_2} \otimes\cdots \otimes e_{a_N}, \end{equation} where the real coefficients $C_{a_1 a_2 \cdots a_N}$ is uniquely determined
by $\rho_{tot}^{(N)} (0)$. After rather straightforward calculations, we argue that the following
relations really hold: \begin{eqnarray} \rho_{[i_1 ,i_2 ,\cdots ,i_n]}(t) &=& \sum^n_{m=0}(-1)^{n-m} \sum_{(j_1,\cdots,j_m )\subseteq(i_1 ,\cdots ,i_n) } \nonumber\\ &&\times \mbox{\rm Tr}_{(i_1 ,i_2 ,\cdots ,i_n)-(j_1,\cdots,j_m )} \left[ \Gamma^{\otimes (N-m) }_{[j_1,\cdots,j_m ]}(g,t) \left[ \mbox{\rm Tr}_{(j_1,\cdots,j_m )} [\rho_{tot}^{(N)}(0)] \right] \right].\nonumber\\ &&\ \label{100} \end{eqnarray} Here $\sum_{(j_1,\cdots,j_m )\subseteq(i_1 ,\cdots ,i_n)}$ means
the sum over all the subsequences $(j_1,\cdots,j_m )$
of the sequence $(i_1 ,\cdots ,i_n)$. $Tr_{(j_1,\cdots,j_m )}$ is a trace operation in terms of the $(j_1,\cdots,j_m )$ degrees of freedom. $Tr_{(i_1 ,i_2 ,\cdots ,i_n)-(j_1,\cdots,j_m )}$ means a trace operation in terms of the complementary subsequence to the subsequence $(j_1,\cdots,j_m )$ of $(i_1 ,\cdots ,i_n)$. When $m=0$, $Tr_{(j_1,\cdots,j_m )}$ is reduced into the identical operation. The operation $\Gamma^{\otimes (N-m) }_{[j_1,\cdots,j_m ]}$ is the time-evolution
operator for all degrees of freedom removing the
$(j_1,\cdots,j_m )$ part. Here it is better to note again that
even though the formula include subtractions,
all the operators $\rho_{[i_1 ,i_2 ,\cdots ,i_n]}$ are
non-negative $\rho_{[i_1 ,i_2 ,\cdots ,i_n]}\geq 0$.
In eqn(\ref{100}), note that the operators $\rho_{[i_1 ,i_2 ,\cdots ,i_n]}(t)$
cannot be evaluated only from the knowledge about the accessible density operator $\rho^{(N)}_\parallel (t)$. It is required to solve independently
time evolutions of many descendant operators, $$\Gamma^{\otimes (N-m) }_{[j_1,\cdots,j_m ]}(g,t) \left[ \mbox{\rm Tr}_{(j_1,\cdots,j_m )} [\rho_{tot}^{(N)}(0)] \right].$$
Here let us just draw the outline of the proof using the $\rho_{[1]}$ case. Substituting eqns(\ref{90}) and (\ref{91}), the following manipulation is possible: \begin{eqnarray} &&
\langle a_2 a_3 \cdots a_N | \rho_{[1]}
|a_2 ' a_3 ' \cdots a_N ' \rangle \nonumber\\ &=&
\langle B a_2 a_3 \cdots a_N | \rho^{(N)}
|B a_2 ' a_3 ' \cdots a_N ' \rangle \nonumber\\ &=&
\langle a_2 a_3 \cdots a_N | \mbox{\rm Tr}_1 \left[ ({\bf 1}-P)\otimes P^{\otimes (N-1)} \rho_{tot}^{(N)} ({\bf 1}-P)\otimes P^{\otimes (N-1)} \right]
|a_2 ' a_3 ' \cdots a_N ' \rangle. \nonumber \end{eqnarray} Moreover we can rewrite the equation as follows: \begin{eqnarray} \rho_{[1]} = \mbox{\rm Tr}_1 \left[ {\bf 1}\otimes P^{\otimes (N-1)} \rho_{tot}^{(N)} (t) {\bf 1}\otimes P^{\otimes (N-1)} \right] - \mbox{\rm Tr}_1\left[ \Gamma^{\otimes N} (g,t)[\rho_{tot}^{(N)} (0)] \right].\nonumber \end{eqnarray} It is noticed that the first term of the r.h.s. is reduced using the expansion of $\rho_{tot}^{(N)} (0)$ as follows. \begin{eqnarray} && \mbox{\rm Tr}_1 \left[ {\bf 1}\otimes P^{\otimes (N-1)} \rho_{tot}^{(N)} (t) {\bf 1}\otimes P^{\otimes (N-1)} \right] \nonumber\\ &=& \sum_{a_1\cdots a_N} C_{a_1 a_2 \cdots a_N} \mbox{\rm Tr}_1 [U(t)e_{a_1} U^\dagger (t)] (PU(t)e_{a_2}U^\dagger(t) P) \otimes \cdots \otimes (PU(t)e_{a_N}U^\dagger (t)P) \nonumber\\ &=& \Gamma^{\otimes {(N-1)}}(g,t)\left[ \sum_{a_1\cdots a_N} C_{a_1 a_2 \cdots a_N} \mbox{\rm Tr}_1 [e_{a_1}]e_{a_2}\otimes \cdots \otimes e_{a_N} \right] \nonumber\\ &=& \Gamma^{\otimes {(N-1)}}(g,t)[\mbox{\rm Tr}_1[\rho^{(N)}_{tot} (0)]]. \end{eqnarray} Consequently we arrive at the relation for $\rho_{[1]}$
in eqn(\ref{100}): \begin{equation} \rho_{[1]} = \Gamma^{\otimes {(N-1)}}(g,t)[\mbox{\rm Tr}_1[\rho^{(N)}_{tot}(0)]] - \mbox{\rm Tr}_1\left[ \Gamma^{\otimes N} (g,t)[\rho_{tot}^{(N)} (0)] \right]. \end{equation} The proofs for the other components in eqn(\ref{100}) can be achieved
in the similar ways.
The relation in eqn(\ref{100}) makes the evaluation of $J^{(N)}$ possible,
only based upon our local knowledge.
\section{ A Decaying Two-Level Model with a Small Unknown Parameter} \ \\
In order to demonstrate our formulation explicitly,
let us consider a system including
a small unknown parameter $g$. In many physical systems,
the estimation of such a small parameter often
provides significant physical information. For example,
tiny coupling constants in the elementary particle interactions
produce only quite rare processes, however, the analyses give a lot of important constraints of high energy
features beyond the today's accelerator technology.
For simplicity suppose a decaying two-level system including the small
coupling $g$. The model has been frequently
used in physics, for instance, to analyze the flavor-oscillating
phenomena in the $K_0 -\bar{K}_0$ system \cite{P}. The Hamiltonian of the example is given as \begin{equation} H=-i\hbar \left[ \begin{array}{cc} \Gamma_+ & 0 \\ 0 & \Gamma_- \end{array} \right] +g\hbar \left[ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right], \end{equation} where $\Gamma_\pm >0 $, $\Gamma_+ \neq \Gamma_-$ and
$|g| \ll \Gamma_\pm, \ |\Gamma_+ -\Gamma_-| $. In the two-level subspace, time evolution of the density matrix $\rho_\parallel$
is governed by the following equation of motion: \begin{equation} i\hbar \partial_t \rho_\parallel =H\rho_\parallel -\rho_\parallel H^\dagger.\label{304} \end{equation}
Define the states $|\pm\rangle$ as \begin{equation}
\sigma_z |\pm\rangle =\pm |\pm\rangle. \end{equation} Here $\sigma_z$ is the z-component of the Pauli matrix.
Let us estimate the time evolution of $|\pm\rangle$ in the first order of $g$. It is solved as \begin{eqnarray}
|\pm (g,t) \rangle =
e^{-\Gamma_\pm t} |\pm\rangle
+ig d(t)|\mp\rangle
+O(g^2) , \end{eqnarray} where the functions $d(t)$ is given as \begin{equation} d(t)=\frac{e^{-\Gamma_+ t} -e^{-\Gamma_- t}}{\Gamma_+ -\Gamma_-}. \end{equation}
Then the information $J_\pm$ for the state $|\pm (t)\rangle$ is evaluated as \begin{eqnarray} J_\pm(t) = 4d(t)^2+O(g). \label{301} \end{eqnarray} The Cram\'er-Rao bound is always achieved by an observable \begin{equation} A(t_\ast ) =\sigma_y +O(g), \end{equation} where $\sigma_y$ is the y-component of the Pauli matrix.
In this simple model we are able to optimize the measurement time $t$. The Fisher information takes its maximum value in the lowest order \begin{equation} J_{max} =\frac{4}{(\Gamma_+ -\Gamma_- )^2} \left( \left(\frac{\Gamma_-}{\Gamma_+}\right)^{\frac{\Gamma_+}{\Gamma_+ -\Gamma_-}} - \left(\frac{\Gamma_+}{\Gamma_-}\right)^{\frac{\Gamma_-}{\Gamma_- -\Gamma_+}} \right)^2 +O(g), \end{equation} at \begin{equation} t_\ast= \frac{\ln \Gamma_+ -\ln \Gamma_-}{\Gamma_+ -\Gamma_-}+O(g).\label{50} \end{equation}
In eqn(\ref{50}), when $\Gamma_-$ is much smaller than $\Gamma_+$: \begin{equation} \Gamma_- \ll \Gamma_+, \end{equation} $t_\ast \sim -\frac{1}{\Gamma_+}\ln\frac{\Gamma_-}{\Gamma_+} $ becomes larger logarithmically. In fact,
the time $t_\ast$ can be late until the first order estimation breaks down, the time $t_g \sim -\frac{1}{\Gamma_+}\ln\frac{g}{\Gamma_+}$. Interestingly, at $t=t_\ast$,
the survival probability for the state $|+(g, t)\rangle$
in the two-level subspace is estimated as \begin{equation}
\langle +(g, t_\ast) |+ (g, t_\ast) \rangle \sim \left(\frac{\Gamma_-}{\Gamma_+} \right)^2 \ll 1. \end{equation} Against a naive expectation, this indicates that the best quantum estimation can be achieved at the
time after the state has almost escaped from the two-level subspace.
Analysis of a composite system of the two subsystems may be also instructive. Let us first take the initial state as an i.i.d. state, \begin{equation}
|\Psi (0) \rangle = |++\rangle. \end{equation} For the density matrix $\rho^{(2)}_\parallel (t)=
|++(t)\rangle\langle ++ (t) |$, the fisher information $j^{(2)}$
defined by
\begin{equation} j^{(2)}: =\mbox{\rm Tr}[ \rho_{\parallel}^{(2)} (L^{(2)})^2] + \frac{ [ \mbox{\rm Tr} [ \rho_{\parallel}^{(2)} L^{(2)} ] ]^2} {1-\mbox{\rm Tr}\rho_{\parallel}^{(2)} } \end{equation}
is evaluated as \begin{equation} j^{(2)} (t) =8d(t)^2 e^{-2\Gamma_+ t} +O(g). \end{equation} Because $j^{(1)} =4d^2 +O(g)$, the relation $j^{(2)}(t) =2j^{(1)}$ is not satisfied due to the $e^{-2\Gamma_+ t}$ factor and $j^{(2)}$ is exponentially
smaller than $2j^{(1)}$ at $t>0$.
Let us compare the result with $J^{(2)}$ defined by eqn(\ref{300}). In this case each component of the local density operator is defined as \begin{eqnarray} \rho_{[\o]}(t)&=& \Gamma^{\otimes 2}(g,t)[\rho(0)]=\rho_\parallel^{(2)} (t),\\ \rho_{[1]}(t)&=& \Gamma(g,t)[\mbox{\rm Tr}_1[\rho(0)]] - \mbox{\rm Tr}_1\left[ \Gamma^{\otimes 2} (g,t)[\rho(0)] \right],\\ \rho_{[2]}(t)&=& \Gamma(g,t)[\mbox{\rm Tr}_2 [\rho(0)]] - \mbox{\rm Tr}_2\left[ \Gamma^{\otimes 2} (g,t)[\rho(0)] \right],\\ \rho_{[1,2]}(t)&=&1- \mbox{\rm Tr}_2\left[ \Gamma(g,t)[\mbox{\rm Tr}_1[\rho(0)]] \right] - \mbox{\rm Tr}_1\left[ \Gamma(g,t)[\mbox{\rm Tr}_2[\rho(0)]] \right] \nonumber\\ && + \mbox{\rm Tr}_{1,2} \left[ \Gamma^{\otimes 2} (g,t)[\rho(0)] \right]. \end{eqnarray} As seen above, to calculate $\rho_{[1]}$, $\rho_{[2]}$ and $\rho_{[1,2]}$, we need the time evolution of the partial density matrices $\Gamma(g,t)[\mbox{\rm Tr}_1[\rho(0)]]$ and $\Gamma(g,t)[\mbox{\rm Tr}_2[\rho(0)]]$. It should be stressed that these evolutions cannot be obtained only from knowledge of $\rho_{\parallel}^{(2)}(t)$, for instance,
by taking any traces for$\rho^{(2)}_\parallel$.
They must be calculated independently by solving eqn(\ref{304}) for the
initial density matrices $\mbox{\rm Tr}_1[\rho(0)]$
and $\mbox{\rm Tr}_2[\rho(0)]$. For the initial i.i.d. density matrix, each $J^{(2)}$ component is calculated as \begin{eqnarray} J^{(2)}_{[\o]} &=&j^{(2)}=8d(t)^2 e^{-2\Gamma_+ t} +O(g), \\ J^{(2)}_{[1]} &=& 4d(t)^2 (1-e^{-2\Gamma_+ t}) +O(g), \\ J^{(2)}_{[2]} &=& 4d(t)^2 (1-e^{-2\Gamma_+ t}) +O(g), \\ J^{(2)}_{[1,2]} &=& O(g). \end{eqnarray} Thus total information $J^{(2)}$ is precisely equal to twice of $J^{(1)}$: \begin{equation} J^{(2)} =8d(t)^2 +O(g) =2J^{(1)}. \end{equation}
Next let us discuss an entangled case. Initially we take a state as \begin{equation}
|\Phi (0) \rangle = \frac{1}{\sqrt{2}}
[|+-\rangle + |-+\rangle]. \end{equation} Calculation of $j^{(2)}(t)$ is easy and the results are as follows. \begin{eqnarray} j^{(2)} (t) = 8d(t)^2 [e^{-2\Gamma_+ t} +e^{-2\Gamma_- t}] +O(g). \end{eqnarray} Note that $j^{(2)} (t)/j^{(1)} (t)$ vanishes exponentially in time
just as in the i.i.d. case.
Evaluation of $J^{(2)} (t)$ needs not only the density matrix \begin{equation}
\rho_{\parallel } (t) =|\Phi (t) \rangle\langle \Phi (t)| \end{equation}
but also another density matrix \begin{eqnarray} \Gamma(g,t)[\mbox{\rm Tr}_1[\rho(0)]]=\Gamma(g,t)[\mbox{\rm Tr}_2[\rho(0)]] =\Gamma(g,t)\left[\frac{1}{2}{\bf 1} \right]. \end{eqnarray} After some manipulations the form of $J^{(2)} (t)$ results in \begin{eqnarray} && J^{(2)} (t) \nonumber\\ &=& 8d(t)^2 [e^{-2\Gamma_+ t} +e^{-2\Gamma_- t}] \nonumber\\ && +4d(t)^2 [1+2e^{-(\Gamma_+ +\Gamma_-)t}]^2 \frac{[e^{-\Gamma_+ t} -e^{-\Gamma_- t}]^2} {e^{-2\Gamma_+ t} (1-e^{-2\Gamma_- t}) + e^{-2\Gamma_- t} (1-e^{-2\Gamma_+ t}) } \nonumber\\ &&+O(g). \end{eqnarray}
Note that at the early era ($t\sim 0$), both $j^{(2)} (t)$ and $J^{(2)} (t)$ have four-times information compared with the single system: \begin{eqnarray} && j^{(2)} (t\sim 0 )\sim 4j^{(1)}, \\ && J^{(2)} (t\sim 0 )\sim 4J^{(1)} . \end{eqnarray} Thus the information is twice larger than the above i.i.d. case.
Obviously this advantage arises due to the entanglement between subsystems.
For the entangled case, $J^{(2)} /J^{(1)}$ becomes smaller than the value of the i.i.d. case (equal to two ) in the late time. Hence, the i.i.d. density operator becomes more relevant than the entangled
density operator for the estimation of $g$. In the limit of $t\rightarrow \infty$, the value of $J^{(2)} /J^{(1)}$ for the entangled case
approaches to the single-system value: \begin{equation} \lim_{t\rightarrow\infty} \frac{J^{(2)} (t)}{J^{(1)} (t)} \sim 1 . \end{equation} This is due to contributions of the one-blank states
($|B\pm\rangle$ and $|\pm B\rangle$). Consequently, it can be said that
the measurement should be at the early times
in order to utilize
enhancement of the Fisher information by the entanglement.
So far we have discussed only systems with small numbers of samples. For the practical estimation of the small parameter $g$, the many-samples estimation is
inevitable beyond the above simple examples. For instance, the minimized expected error $\delta g$ is given by \begin{equation} \delta g=\frac{1}{\sqrt{NJ^{(1)}}} \end{equation} for the i.i.d. cases of $N$-samples systems. Then, in order to get a meaningful estimate,
the number of the samples must be, at least,
$O\left(1/(g^2 J^{(1)}) \right)$ for the correct value $g$.
It is expected that large entanglement between
many samples may extremely improve the estimation for $g$
and make the number of the samples enough for the estimation much smaller.
\ \\ \ \\ \ \\
\section{Summary} \ \\
We have investigated
deeply the local quantum estimation problem of an unknown parameter.
The practical restriction of experimental observables
takes place in various situations of the physical experiments. For a typical example, in particle physics we can probe only low-energy visible sectors
of the whole system by our present devices. Such obstacles
appear because of the limit of the present technology
and so on. Moreover,
observation of quantum phenomena, which
happen only at quite small rates,
often
becomes the crucial target of experiments, which may derive
some profound results of physics like, for instance, CPT violation \cite{HP}.
In such situations,
it is generally difficult to take a large number of data as one wishes,
at least, in the first stage of the experimental studies. Hence the local quantum estimation becomes really important when
the experimental arrangements are designed, because the estimation theory
provides among our available probes the optimized observable
which quantum fluctuation is most suppressed
in the estimation based upon
a limited number of the data.
In spite of such relevance of the local quantum estimation,
the problem
has never been discussed in detail, as far as the authors know. In this paper, the detailed analysis and formulations based upon
the Fisher informations have been completed. After a brief review on the standard quantum estimation theory,
the local quantum estimator for the local estimation has been
defined
by eqn(\ref{40}).
The notion of the local density operators
was clearly introduced in eqn(\ref{e20}),
and the Cram{\'e}r-Rao inequality in the local quantum estimate theory
(eqn(\ref{cr})) has been proven by taking
the local Fisher information defined by eqn(\ref{34}).
The inequality is a fundamental tool in the theory and
will play a significant role in the local estimation
in various physical applications.
In section 6, the Fisher
information for the unnormalized pure state was commented.
The formula in eqn(\ref{unps}) is an extension of that derived by
Fujiwara and Nagaoka, who discussed the Fisher information for
normalized pure states. It is known that in many physical systems
non-unitary theories of pure states also are available and
that the validity is well verified by the experiments.
In such systems with non-unitary evolution,
eqn(\ref{unps}) is quite useful to
evaluate the Fisher information for an unknown parameter.
In section 7, it was pointed out that
the local quantum estimation in the composite system has two
independent formulations, using the i.i.d.cases.
In section 8, two general formulations of
the local quantum estimation for the composite system were proposed. For the composite system of $N$ identical subsystems, we have two Fisher informations, $j^{(N)}$ and $J^{(N)}$.
The information $j^{(N)}$
takes a simple form to define,
but gives, in general, much smaller values than $J^{(N)}$.
The theory of information $J^{(N)}$
can generate a more precise estimate for $g$, but has a pretty complicated
form to deal with, compared to the $j^{(N)}$ case.
In order to avoid the troublesome procedures in evaluation of $J^{(N)}$,
we showed in section 9 the formula in eqn(\ref{100}),
which makes the evaluation tractable.
As seen in eqn(\ref{100}),
calculation of the Fisher information $J^{(N)}$ requires
solving evolutions of many descendant operators,
$\Gamma^{\otimes (N-m) }_{[j_1,\cdots,j_m ]}(g,t) \left[ \mbox{\rm Tr}_{(j_1,\cdots,j_m )} [\rho_{tot}^{(N)}(0)] \right]$, independently of solving the accessible density operator
$\rho^{(N)}_\parallel (t)$ itself. Such
processes never appear in the ordinary quantum estimation theory,
where the Fisher information can be evaluated by using only
a time-evolved density operator.
In section 10, we demonstrated explicitly our formulation
of the local quantum estimation by applying
to a decaying two-level system with a small unknown parameter.
We hope that
the analysis in this paper enables the quantum estimation theory to
take a more active part in the real experimental studies, which
suffer from the restriction of available observables and the practical
limitation of the number of the data.
\section*{Appendix}
In this appendix, the Cram{\'e}r-Rao inequality is proved. Let us write the triangular inequality relation as \begin{equation}
\mbox{\rm Tr} (X^\dagger X)\mbox{\rm Tr}(Y^\dagger Y) \geq |\mbox{\rm Tr}(X^\dagger Y)|^2,\label{e6} \end{equation} where $X$ and $Y$ are arbitrary operators acting on the Hilbert space. Decomposing the operator $X^\dagger Y$ into the sum of the real and imaginary parts as \begin{equation} X^\dagger Y = \frac{1}{2} (X^\dagger Y +Y^\dagger X) +\frac{1}{2}(X^\dagger Y -Y^\dagger X), \end{equation} another inequality relation arises: \begin{eqnarray} && \mbox{\rm Tr} (X^\dagger X)\mbox{\rm Tr}(Y^\dagger Y) \geq
\frac{1}{4}|\mbox{\rm Tr}(X^\dagger Y +Y^\dagger X)|^2 +
\frac{1}{4}|\mbox{\rm Tr}(X^\dagger Y -Y^\dagger X)|^2 \nonumber\\ &&
\geq \frac{1}{4}|\mbox{\rm Tr}(X^\dagger Y +Y^\dagger X)|^2.\label{e7} \end{eqnarray} Here let us take \begin{eqnarray} && X=L(g)\sqrt{\rho_{tot} (g)},\label{e13} \\ && Y=(A -E_g [A] )\sqrt{\rho_{tot} (g)}.\label{e14} \end{eqnarray} Then, from the inequality (\ref{e7}), we can derive that \begin{eqnarray} && \mbox{\rm Tr}[\rho_{tot} (g)L(g)^2] \mbox{\rm Tr}[\rho_{tot} (g) (A -E_g [A] )^2] \nonumber\\ && \geq \frac{1}{4}
\left| \mbox{\rm Tr} \left[\rho_{tot} (g) \left(L(g) (A-E_g[A]) +(A-E_g [A] ) L(g) \right) \right]
\right|^2.\label{e10} \end{eqnarray} The right-hand-side term in the above inequality is able to be calculated
using eqns(\ref{2e}), (\ref{eq:SLD}) and (\ref{1e}) successively as follows. \begin{eqnarray} && \frac{1}{4}
\left|\mbox{\rm Tr} \left[\rho_{tot} (g) \left(L(g) (A-E_g[A]) +(A-E_g [A] ) L(g) \right) \right]
\right|^2 \nonumber\\ &=& \frac{1}{4}
\left|\mbox{\rm Tr} \left[\rho_{tot} (g) \left(L(g) A +A L(g) \right) \right]
\right|^2 \nonumber\\ &=& \frac{1}{4}
\left|\mbox{\rm Tr} \left[ A\left(\rho_{tot} (g) L(g) + L(g) \rho_{tot} (g)\right) \right]
\right|^2 \nonumber\\ &=& \left( \mbox{\rm Tr}[A \partial_g \rho_{tot} (g) ] \right)^2 = \left( \partial_g E_g [A] \right)^2. \end{eqnarray} Consequently the relation (\ref{e10})
implies the following inequality: \begin{equation}
\frac{V_g [A]}{\left( \partial_g E_g [A] \right)^2} \geq \frac{1}{J_g},
\label{e15} \end{equation}
thus, the inequality (\ref{e111}) is proved. For the unbiased case with
$E_g [A] =g$, the inequality(\ref{e15}) is reduced to
(\ref{crie}). The equality is trivially
attained when $X\propto Y$ in eqn(\ref{e7}) and the relation $X\propto Y$
holds in eqns (\ref{e13}) and (\ref{e14})
when we set $A\propto L(g)$, because $E_g [L(g)]=0$.
\end{document} | arXiv |
On global well-posedness of the modified KdV equation in modulation spaces
An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth
June 2021, 41(6): 2947-2969. doi: 10.3934/dcds.2020392
Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems
Jose Anderson Cardoso 1, , Patricio Cerda 2, , Denilson Pereira 3, and Pedro Ubilla 2,,
Departamento de Matemática, Universidade Federal de Sergipe, São Cristóvão-SE, 49100-000, Brazil
Departamento de Matematica y C. C., Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande 58429-900, Brazil
Received December 2019 Revised October 2020 Published June 2021 Early access December 2020
Fund Project: The first author is partially supported by FAPITEC/CAPES and by CNPq - Universal.
The second author was partially supported by Proyecto código 042033CL, Dirección de Investigación, Científica y Tecnológica, DICYT.
The third author was partially supported by Proyecto código 041933UL POSTDOC, Dirección de Investigación, Científica y Tecnológica, DICYT.
The fourth author was partially supported by FONDECYT grant 1181125, 1161635, 1171691
We prove the existence of a bounded positive solution for the following stationary Schrödinger equation
$ \begin{equation*} -\Delta u+V(x)u = f(x,u),\,\,\, x\in\mathbb{R}^n,\,\, n\geq 3, \end{equation*} $
$ V $
is a vanishing potential and
$ f $
has a sublinear growth at the origin (for example if
$ f(x,u) $
is a concave function near the origen). For this purpose we use a Brezis-Kamin argument included in [6]. In addition, if
has a superlinear growth at infinity, besides the first solution, we obtain a second solution. For this we introduce an auxiliar equation which is variational, however new difficulties appear when handling the compactness. For instance, our approach can be applied for nonlinearities of the type
$ \rho(x)f(u) $
is a concave-convex function and
$ \rho $
satisfies the
$ \mathrm{(H)} $
property introduced in [6]. We also note that we do not impose any integrability assumptions on the function
, which is imposed in most works.
Keywords: Concave-convex nonlinearities, upper and lower solutions, variational methods, Schrödinger equation, bounded solutions.
Mathematics Subject Classification: 35J20, 35J10, 35J91, 35J15, 35B09.
Citation: Jose Anderson Cardoso, Patricio Cerda, Denilson Pereira, Pedro Ubilla. Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2947-2969. doi: 10.3934/dcds.2020392
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Jose Anderson Cardoso Patricio Cerda Denilson Pereira Pedro Ubilla | CommonCrawl |
Graded Betti Numbers of Balanced Simplicial Complexes
Martina Juhnke-Kubitzke1 &
Lorenzo Venturello ORCID: orcid.org/0000-0002-6797-52701
Acta Mathematica Vietnamica volume 46, pages 839–871 (2021)Cite this article
We prove upper bounds for the graded Betti numbers of Stanley-Reisner rings of balanced simplicial complexes. Along the way we show bounds for Cohen-Macaulay graded rings S/I, where S is a polynomial ring and \(I\subseteq S\) is a homogeneous ideal containing a certain number of generators in degree 2, including the squares of the variables. Using similar techniques we provide upper bounds for the number of linear syzygies for Stanley-Reisner rings of balanced normal pseudomanifolds. Moreover, we compute explicitly the graded Betti numbers of cross-polytopal stacked spheres, and show that they only depend on the dimension and the number of vertices, rather than also the combinatorial type.
Avoid the most common mistakes and prepare your manuscript for journal editors.
In the last decades tremendous connections between combinatorics, topology and commutative algebra have been established. The theory of Cohen-Macaulay rings led to the proof of celebrated conjectures such as the upper bound theorem for spheres and the g-theorem for simplicial polytopes (see [34] as a comprehensive reference). Since these results rely on algebraic properties of the Stanley-Reisner ring of simplicial complex, it is natural to investigate classical invariants of this ring, such as its minimal graded free resolution as a module over the polynomial ring. Our starting point are mainly two papers: In [26], Migliore and Nagel showed upper bounds for the graded Betti numbers of simplicial polytopes. More recently, building on those results, Murai [28] established a connection between a specific property of a triangulation, so-called tightness and the graded Betti numbers of its Stanley-Reisner ring. Moreover, he employs upper bounds for graded Betti numbers to obtain a lower bound for the minimum number of vertices needed to triangulate a pseudomanifold with a given first (topological) Betti number. It is conceivable that for more specific classes of simplicial complexes, better bounds (for the graded Betti numbers) hold, which then can be turned again into lower bounds for the minimal number of vertices of such a simplicial complex. This serves as the motivation for this article, where we will focus on so-called balanced simplicial complexes.
Those were originally introduced by Stanley [33] under the name completely balanced as pure (d − 1)-dimensional simplicial complexes whose vertex sets can be partitioned into d classes, such that each class meets every face in at most (and hence exactly) one element. Following more recent papers, we will drop the word "completely" and we will not require balanced complexes to be pure. Notable examples are Coxeter complexes, Tits buildings as well as the order complex of a graded poset, with the vertex set partition given by the rank function. This last observation shows that the barycentric subdivision of any simplicial complex is balanced, which gives a constructive way of obtaining balanced triangulations of any topological space and shows that balancedness is a combinatorial rather than a topological property. In recent years, balanced simplicial complexes have been studied intensively and many classical results in face enumeration and combinatorial topology have been proven to possess a balanced analog (see, e.g., [11, 17,18,19,20, 22, 29, 37]).
The aim of this article is to continue with this line of research by studying graded Betti numbers of balanced simplicial complexes. Our main results establish upper bounds for different cases, including arbitrary balanced simplicial complexes, balanced Cohen-Macaulay complexes and balanced normal pseudomanifolds. Along the way, we derive upper bounds on the graded Betti numbers of homogeneous ideals with a high concentration of generators in degree 2.
The structure of this paper is the following:∙ Section 2 is devoted to the basic notions and definitions.∙ In Section 3, we use Hochster's formula to prove a first upper bound for the graded Betti numbers of an arbitrary balanced simplicial complex (see Theorem 1).∙ We next restrict ourselves to the Cohen-Macaulay case, and provide two different upper bounds in this setting. The first approach provides a bound for graded Betti numbers of ideals with a high concentration of generators in degree 2, which immediately specializes to Stanley-Reisner ideals of balanced Cohen-Macaulay complexes (see Theorem 2). This is the content of Section 4.∙ The second approach, presented is Section 5, employs the theory of lex-plus-squares ideals to bound the Betti numbers of ideals containing many generators in degree 2, including the squares of the variables. Again the result on balanced complexes given in Theorem 5 follows as an immediate application.∙ In Section 6, we focus on balanced normal pseudomanifolds. We use a result by Fogelsanger [8] to derive upper bounds for the graded Betti numbers in the first strand of the graded minimal free resolution in this setting (see Theorem 8).∙ In [22] cross-polytopal stacked spheres were introduced as the balanced analog of stacked spheres, in the sense that they minimize the face vector among all balanced spheres with a given number of vertices. In Section 7 (Theorem 10), we compute the graded Betti numbers of those spheres, and show that they only depend on the number of vertices and on the dimension. The same behavior is known to occur for stacked spheres [36]. Moreover, we conjecture that the graded Betti numbers in the linear strand of their resolution provide upper bounds for the ones of any balanced normal pseudomanifold.
As a service to the reader, in particular to help them compare the different bounds, we use the same example to illustrate the predicted upper bounds: Namely, the toy example is a 3-dimensional balanced simplicial complex on 12 vertices with each color class being of cardinality 3. All computations and experiments have been carried out with the help of the computer algebra system Macaulay2 [13].
Algebraic Background
Let \(S=\mathbb {F}[x_{1},\dots ,x_{n}]\) denote the polynomial ring in n variables over an arbitrary field \(\mathbb {F}\) and let \(\mathfrak {m}\) be its maximal homogeneous ideal, i.e., \(\mathfrak {m}=(x_{1},\ldots ,x_{n})\). Denote with Moni(S) the set of monomials of degree i in S, and for u ∈Moni(S) and a term order <, we let Moni(S)<u be the set of monomials of degree i that are smaller than u with respect to <. For a graded S-module R we use Ri to denote its graded component of degree i (including 0), where we use the standard \(\mathbb {N}\)-grading of S. The Hilbert function of a quotient S/I, where \(I\subseteq S\) is a homogeneous ideal is the function from \(\mathbb {N}\to \mathbb {N}\) that maps i to \( \dim _{\mathbb {F}}(R_{i})\). A finer invariant can be obtained from the minimal graded free resolution of S/I. The graded Betti number \(\beta _{i,i+j}^{S}(S/I)\) is the non-negative integer
$$\beta^{S}_{i,i+j}(R):=\dim_{\mathbb{F}} \text{Tor}_{i}^{S}(R,\mathbb{F})_{i+j}.$$
We often omit the superscript S, when the coefficient ring is clear from the context. We refer to any commutative algebra book (e.g., [3]) for further properties of the graded minimal free resolution of S/I.
Let \(I\subseteq S\) be a homogeneous ideal and let R = S/I be of Krull dimension d. Let \({{\varTheta }}=\{\theta _{1},\dots ,\theta _{d}\}\subseteq S_{1}\). Then (i) Θ is a linear system of parameters (l.s.o.p.) for R if \(\dim (R/(\theta _{1},\dots ,\theta _{i})R)=\dim (R)-i\) for all 1 ≤ i ≤ d. (ii) Θ is a regular sequence for R if 𝜃i is not a zero divisor of \(R/(\theta _{1},\dots ,\theta _{i-1})R\) for all 1 ≤ i ≤ d.
We remark that due to the Noether normalization lemma, an l.s.o.p. for R = S/I always exists, if \(\mathbb {F}\) is an infinite field. Moreover, if Θ is a regular sequence, then Θ is an l.s.o.p., but the converse is far from being true in general. The class of rings for which the converse holds is of particular interest.
A graded ring R is Cohen-Macaulay if every l.s.o.p. is a regular sequence for R.
The theory of Cohen-Macaulay rings plays a key role in combinatorial commutative algebra and its importance cannot be overstated (see, e.g., [3, 34]).
The next two statements will be useful for providing upper bounds for graded Betti numbers.
Lemma 1
Let R = S/I with I a homogeneous ideal and 𝜃 ∈ S1. (i) [26, Corollary 8.5] If the multiplication map × 𝜃 : Rk→Rk+ 1 is injective for every k ≤ j, then
$$\beta^{S}_{i,i+k}(R)\leq \beta^{S/\theta S}_{i,i+k}(R/\theta R)$$
for every i ≥ 0 and k ≤ j. (ii) [3, Proposition 1.1.5] If 𝜃 is not a zero divisor of M, then
$$\beta^{S}_{i,i+j}(R)= \beta^{S/\theta S}_{i,i+j}(R/\theta R)$$
for every i,j ≥ 0.
From Lemma 1 it immediately follows that modding out by a regular sequence does not affect the graded Betti numbers.
Lex Ideals
In order to show upper bounds for the graded Betti numbers we will make use of lexicographic ideals. As above, we let \(S=\mathbb {F}[x_{1},\dots ,x_{n}]\). Given a monomial ideal \(I\subseteq S\) we denote by G(I) its unique set of minimal monomial generators and we use G(I)j to denote those monomials in G(I) of degree j. Let >lex be the lexicographic order on S with \(x_{1}>_{\text {lex}}\dots >_{\text {lex}}x_{n}\). I.e., we have \(x_{1}^{a_{1}}x_{2}^{a_{2}}\cdots x_{n}^{a_{n}}>_{\text {lex}}x_{1}^{b_{1}}x_{2}^{b_{2}}{\cdots } x_{n}^{b_{n}}\) if the leftmost non-zero entry of (a1 − b1,…,an − bn) is positive. A monomial ideal \(L\subseteq S\) is called a lexicographic ideal (or lex ideal for short) if for any monomials u ∈ L and v ∈ S of the same degree, with v >lexu it follows that v ∈ L. Macaulay [23] showed that for any graded homogeneous ideal \(I\subseteq S\) there exists a unique lex ideal, denoted with Ilex, such that S/I and S/Ilex have the same Hilbert function. In particular, the \(\mathbb {F}\)-vector space Ilex ∩ Si is spanned by the first \(\dim _{\mathbb {F}}S_{i}-\dim _{\mathbb {F}}(S/I)_{i}\) largest monomials of degree i in S. Note that the correspondence between I and Ilex is far from being one to one, since Ilex only depends on the Hilbert function of I. We conclude this section with two fundamental results on the graded Betti numbers of lex ideals.
Lemma 2 (Bigatti [1], Hulett [16], Pardue [31])
For any homogeneous ideal \(I\subseteq S\) it holds that
$$\beta_{i,i+j}^{S}(S/I)\leq\beta_{i,i+j}^{S}(S/I^{\text{lex}})$$
for all i,j ≥ 0.
Lemma 2 states that among all graded rings with the same Hilbert functions, the quotient with respect to the lex ideal maximizes all graded Betti number simultaneously. Another peculiar property of lex ideals is that their graded Betti numbers are determined just by the combinatorics of their minimal generating set G(Ilex). For a monomial u ∈ S denote with \(\max \limits (u)=\max \limits \left \lbrace i : x_{i}|u \right \rbrace \).
Lemma 3 (Eliahou-Kervaire 6)
Let \(I^{\text {lex}}\subseteq S\) be a lexicographic ideal. Then,
$$\beta_{i,i+j}^{S}(S/I^{\text{lex}})=\sum\limits_{u\in G(I^{\text{lex}})\cap S_{j+1}}\binom{\max(u)-1}{i-1}$$
for all i ≥ 1, j ≥ 0.
Simplicial Complexes
An (abstract) simplicial complexΔ on a (finite) vertex set V (Δ) is any collection of subsets of V (Δ) closed under inclusion. The elements of Δ are called faces, and a face that is maximal with respect to inclusion is called a facet. The dimension of a face F is the number \(\dim (F):=| F|-1\), and the dimension of Δ is \(\dim ({{\varDelta }}):=\max \limits \left \lbrace \dim (F) : F\in {{\varDelta }} \right \rbrace \). In particular \(\dim (\emptyset )=-1\). If all facets of Δ have the same dimension, Δ is said to be pure. One of the most natural combinatorial invariants of a (d − 1)-dimensional simplicial complex to consider is its f-vector\(f({{\varDelta }})=(f_{-1}({{\varDelta }}),f_{0}({{\varDelta }}),\dots ,f_{d-1}({{\varDelta }}))\), defined by \(f_{i}({{\varDelta }}):=|\left \lbrace F\in {{\varDelta }} : \dim (F)=i \right \rbrace |\) for − 1 ≤ i ≤ d − 1. However, for algebraic and combinatorial reasons it is often more convenient to consider a specific invertible linear transformation of f(Δ); namely
$$h_{j}({{\varDelta}})=\sum\limits_{i=0}^{j}(-1)^{j-i}\binom{d-i}{d-j}f_{i-1}({{\varDelta}})$$
for 0 ≤ j ≤ d. The vector \(h({{\varDelta }})=(h_{0}({{\varDelta }}),h_{1}({{\varDelta }}),\dots ,h_{d}({{\varDelta }}))\) is called the h-vector of Δ.
Given a subset \(W\subseteq V({{\varDelta }})\) we define the subcomplex
$${{\varDelta}}_{W}:=\left\lbrace F\in{{\varDelta}}: F\subseteq W \right\rbrace,$$
and we call a subcomplex induced if it is of this form. Another subcomplex associated to Δ is its j-skeleton
$$\text{Skel}_{j}({{\varDelta}}):=\left\lbrace F\in {{\varDelta}} : \dim(F)\leq j \right\rbrace,$$
consisting of all faces of dimension at most j (for 0 ≤ j ≤ d − 1). For two simplicial complexes Δ and Γ with \(\dim ({{\varDelta }})=d-1\) and \(\dim ({{\varGamma }})=e-1\) we define the join of Δ and Γ to be the (d + e − 1)-dimensional complex defined by
$${{\varDelta}}*{{\varGamma}}=\left\lbrace F\cup G : F\in{{\varDelta}}, G\in{{\varGamma}} \right\rbrace.$$
The link lkΔ(F) of a face F ∈Δ describes Δ locally around F:
$$ \text{lk}_{{{\varDelta}}}(F):=\{G\in {{\varDelta}} : G\cup F\in {{\varDelta}}, G\cap F=\emptyset\}. $$
Simplicial complexes are in one-to-one correspondence to squarefree monomial ideals: Given a simplicial complex Δ with V (Δ) = [n] := {1,2,…,n} its Stanley-Reisner ideal is the squarefree monomial ideal \(I_{{{\varDelta }}}\subseteq S\) defined by
$$I_{{{\varDelta}}}:=\left( x_{F} : F\notin{{\varDelta}} \right)\subseteq S:=\mathbb{F}[x_{1},\ldots,x_{n}],$$
where \(x_{F}={\prod }_{i\in F}x_{i}\). The quotient \(\mathbb {F}[{{\varDelta }}]:=S/I_{{{\varDelta }}}\) is called the Stanley-Reisner ring of Δ. It is well-known that \(\dim (\mathbb {F}[{{\varDelta }}])=\dim ({{\varDelta }})+1\).
This correspondence is extremely useful to study how algebraic invariants of the Stanley-Reisner rings reflect combinatorial and topological properties of the corresponding simplicial complex, and vice versa. A special instance for this is provided by Hochster's formula (see [3, Theorem 5.5.1]):
Lemma 4 (Hochster's formula)
$$\beta_{i,i+j}(\mathbb{F}[{{\varDelta}}])=\underset{\underset{| W| =i+j}{W\subseteq V({{\varDelta}})}}{\sum\limits} \dim_{\mathbb{F}}\widetilde{H}_{j-1}({{\varDelta}}_{W};\mathbb{F}).$$
A simplicial complex Δ is called Cohen-Macaulay over \(\mathbb {F}\) if \(\mathbb {F}[{{\varDelta }}]\) is a Cohen-Macaulay ring. As Cohen-Macaulayness (over a fixed field \(\mathbb {F}\)) only depends on the geometric realization of Δ, Cohen-Macaulayness is a topological property (see, e.g., [27]). In particular, triangulations of spheres and balls are Cohen-Macaulay over any field. Another crucial property of Cohen-Macaulay complexes is the following (see, e.g., [34]).
Let Δ be a (d − 1)-dimensional Cohen-Macaulay simplicial complex and let \({{\varTheta }}=(\theta _{1},\dots ,\theta _{d})\) be an l.s.o.p. for \(\mathbb {F}[{{\varDelta }}]\). Then,
$$h_{i}({{\varDelta}})=\dim_{\mathbb{F}}\left( \mathbb{F}[{{\varDelta}}]/{{\varTheta}}\mathbb{F}[{{\varDelta}}]\right)_{i}.$$
In Section 6, we will be interested in another class of simplicial complexes, called normal pseudomanifolds. We call a connected pure (d − 1)-dimensional simplicial complex Δ a normal pseudomanifold if every (d − 2)-face of Δ is contained in exactly two facets and if the link of every face of Δ of dimension ≤ d − 3 is connected.
We finally provide the definition of balanced simplicial complexes.
A (d − 1)-dimensional simplicial complex Δ is balanced if there is a partition of its vertex set such that |F ∩ Vi|≤ 1, for every \(i=1,\dots ,d\) and for every face F ∈Δ.
We often refer to the sets Vi as color classes. Another way to phrase this definition is to observe that Δ is balanced if and only if its 1-skeleton is d-colorable, in the classical graph-theoretic sense. Note that, without extra assumptions on its structure, a balanced simplicial complex does not uniquely determine the partition in color classes, nor their sizes, as shown by the middle and right complexes in Fig. 1. However, in this article, we will always assume the vertex partition to be part of the data contained in Δ.
From left to right: a simplicial complex that is not balanced. Two balanced complexes with different partitions in color classes
The class of pure balanced simplicial complexes agrees with the class of so-called completely balanced complexes, originally introduced by Stanley in [33]. However, a balanced simplicial complex in the sense of Definition 3 does not need to be pure. We want to point out that a balanced simplicial complex cannot have too many edges, since all monochromatic edges are forbidden. This idea will be made more precise and used intensively in the following sections.
General Balanced Simplicial Complexes
In the following, we consider arbitrary balanced simplicial complexes without assuming any further algebraic or combinatorial properties. Our aim is to prove explicit upper bounds for the graded Betti numbers of the Stanley-Reisner rings of those simplicial complexes. This will be achieved by exhibiting (non-balanced) simplicial complexes (one for each strand in the linear resolution), whose graded Betti numbers are larger than those of all balanced complexes on a fixed vertex partition.
We first need to introduce some notation. Recall that the clique complex of a graph G = (V,E) on vertex set V and edge set E is the simplicial complex Δ(G) on vertex set V, whose faces correspond to cliques of G, i.e.,
$$ {{\varDelta}}(G):=\{F\subseteq V : \{i,j\}\in E \text{ for all } \{i,j\}\subseteq F \text{ with }i\neq j\}. $$
Let Δ be a (d − 1)-dimensional balanced simplicial complex with vertex partition . Let ni := |Vi| denote the sizes of the color classes of V (Δ). Throughout this section, we denote with \(K_{n_{1},\ldots ,n_{d}}\) the complete d-partite graph on vertex set . Note that the 1-skeleton of Δ, considered as a graph, is clearly a subgraph of \(K_{n_{1},\ldots ,n_{d}}\) and that, by the definition of a clique complex, we have \({{\varDelta }}\subseteq {{\varDelta }}(K_{n_{1},\dots ,n_{d}})\).
We can now state our first bound, though not yet in an explicit form.
Theorem 1
Let Δ be a (d − 1)-dimensional balanced simplicial complex on with ni := |Vi|. Then
$$\beta_{i,i+j}\left( \mathbb{F}\left[{{\varDelta}}\right] \right)\leq \beta_{i,i+j}\left( \mathbb{F}\left[ \text{Skel}_{j-1}\left( {{\varDelta}}(K_{n_{1},\ldots,n_{d}})\right)\right] \right)$$
The proof relies on Hochster's formula. We fix j ≥ 0. To simplify notation we set \({{\varSigma }}= \text {Skel}_{j-1}({{\varDelta }}(K_{n_{1},\ldots ,n_{d}}))\). Given a simplicial complex Γ, we denote by \((C_{\bullet }({{\varGamma }}),\partial _{j}^{{{\varGamma }}})\) the chain complex which computes its simplicial homology over \(\mathbb {F}\).
Let \(W\subseteq V\). As \(\dim {{\varSigma }}= j-1\), we have \(\dim \left ({{\varSigma }}_{W}\right )\leq j-1\) and hence \(C_{j}\left ({{\varSigma }}_{W}\right )=0\). This implies
$$ \widetilde{H}_{j-1}\left( {{\varSigma}}_{W};\mathbb{F}\right)= \ker \partial_{j-1}^{{{\varSigma}}_{W}}. $$
As \({{\varDelta }}(K_{n_{1},\ldots ,n_{d}})\) is the "maximal" balanced simplicial complex with vertex partition , it follows that \(\text {Skel}_{j-1}({{\varDelta }})\subseteq {{\varSigma }}\) and thus \(C_{j-1}\left ({{\varDelta }}_{W}\right )\subseteq C_{j-1}\left ({{\varSigma }}_{W}\right )\). In particular, we conclude
$$ \ker \partial_{j-1}^{{{\varDelta}}_{W}} \subseteq \ker \partial_{j-1}^{{{\varSigma}}_{W}} $$
and, using (1), we obtain
$$\dim_{\mathbb{F}}\widetilde{H}_{j-1}\left( {{\varDelta}}_{W};\mathbb{F}\right)\leq \dim_{\mathbb{F}}\widetilde{H}_{j-1}\left( {{\varSigma}}_{W};\mathbb{F}\right).$$
The claim follows from Hochster's formula (Lemma 4). □
We now provide a specific example of the bounds in Theorem 1.
The graded Betti numbers of any 3-dimensional balanced simplicial complex on 12 vertices with 3 vertices in each color class can be bounded by the graded Betti numbers of the skeleta of Γ := Δ(K3,3,3,3). More precisely, we can bound \(\beta _{i,i+j}(\mathbb {F}[{{\varDelta }}])\) by the corresponding Betti number of the (j − 1)-skeleton of Γ. We record those numbers in Table 1.
Table 1 Graded Betti numbers of the skeleta of Γ = Δ(K3,3,3,3)
Observe that the (j − 1)-skeleton of the clique complex \({{\varDelta }}(K_{n_{1},\ldots ,n_{d}})\) is balanced if and only if j = d (or, less interestingly, if j = 1). It follows that the upper bounds for the graded Betti numbers of a (d − 1)-dimensional balanced simplicial complex, given in Theorem 1, are attained for the d th (and trivially, the 0th) row of the Betti table. However, they are not necessarily sharp for the other rows of the Betti table and we do not expect them to be.
In order to turn the upper bounds from Theorem 1 into explicit ones, we devote the rest of this section to the computation of the graded Betti numbers of the skeleta of \({{\varDelta }}(K_{n_{1},\dots ,n_{d}})\). We first consider \({{\varDelta }}(K_{n_{1},\dots ,n_{d}})\). As a preparation we determine the homology of induced subcomplexes of \({{\varDelta }}(K_{n_{1},\dots ,n_{d}})\).
Let \({{\varGamma }}={{\varDelta }}(K_{n_{1},\dots ,n_{d}})\) with vertex partition . For \(W\subseteq V\), set Wi := W ∩ Vi, for 1 ≤ i ≤ d and {i1,…,ik} := {i : Wi≠∅}. Then,
$$ \begin{array}{@{}rcl@{}} \widetilde{H}_{j-1}\left( {{\varGamma}}_{W}; \mathbb{F}\right)= \left\{\begin{array}{llll} \mathbb{F}^{| W_{i_{1}}|-1 }\otimes_{\mathbb{F}}\dots\otimes_{\mathbb{F}}\mathbb{F}^{| W_{i_{k}}|-1 } & \text{if }k=j,\\ 0 &\text{if } k\neq j. \end{array}\right. \end{array} $$
In particular, \(\widetilde {H}_{j-1}\left ({{\varGamma }}_{W}; \mathbb {F}\right )\neq 0\) if and only if k = j and \(|W_{i_{\ell }}|\geq 2\) for 1 ≤ ℓ ≤ k.
Denoting by \(\overline {V_{i}}\) the simplicial complex consisting of the isolated vertices in Vi, we can write Γ as the join of those \(\overline {V_{i}}\)
$$ {{\varGamma}}=\overline{V_{1}}*\dots*\overline{V_{d}}. $$
In particular, we have
$${{\varGamma}}_{W}=\overline{W_{i_{1}}}*\dots*\overline{W_{i_{k}}}.$$
Using the Künneth formula for the homology of a join (see, e.g., [27, Section 58]) and the fact that
$$\widetilde{H}_{j}\left( \overline{W_{i}};\mathbb{F}\right)=\begin{cases} \mathbb{F}^{| W_{i}|-1 } &\text{if } j =0,\\ 0 &\text{if } j\neq 0, \end{cases}$$
we deduce the desired formula for the homology. The "In particular"-part follows directly from this formula. □
Since Cohen-Macaulayness is preserved under taking joins and since every 0-dimensional simplicial complex is Cohen-Macaulay, it follows directly from (2) that the clique complex \({{\varDelta }}(K_{n_{1},\dots ,n_{d}})\) is a Cohen-Macaulay complex. Accordingly, the same is true for the skeleta of \({{\varDelta }}(K_{n_{1},\ldots ,n_{d}})\).
Lemma 6 enables us to compute the graded Betti numbers of \({{\varDelta }}(K_{n_{1},\ldots ,n_{d}})\).
Let d,n1,…,nd be positive integers. Then,
$$ \beta_{i,i+j}\left( \mathbb{F}\left[ {{\varDelta}}(K_{n_{1},\dots,n_{d}})\right] \right)=\underset{\underset{I=\left\lbrace i_{1},\dots, i_{j} \right\rbrace}{I\subseteq[d]}}{\sum\limits}\left( \underset{\underset{c_{\ell}\geq 1, \forall \ell\in\left[ 1,j\right]}{c_{1}+{\dots} +c_{j}=i}}{\sum\limits}\left( \prod\limits_{\ell=1}^{j} c_{\ell}\cdot\binom{n_{i_{\ell}}}{c_{\ell}-1} \right)\right) $$
for i,j ≥ 0. In particular, if n1 = ⋯ = nd = k, then
$$\beta_{i,i+j}\left( \mathbb{F}\left[ {{\varDelta}}(K_{k,\ldots, k})\right] \right)=\binom{d}{j}\left( \underset{\underset{c_{\ell}\geq 1, \forall \ell\in\left[ 1,j\right]}{c_{1}+{\dots} +c_{j}=i}}{\sum\limits}\left( \prod\limits_{\ell=1}^{j} c_{\ell}\cdot\binom{k}{c_{\ell}-1} \right)\right) $$
for i,j ≥ 0.
We prove the statement by a direct application of Hochster's formula. Fix i,j ≥ 0. By Lemma 6 and Lemma 4, to compute \(\beta _{i,i+j}({{\varDelta }}(K_{n_{1},\ldots ,n_{d}}))\), we need to count subsets such that |{ℓ : W ∩ Vℓ≠∅}| = j and |W ∩ Vℓ|≠ 1 for 1 ≤ ℓ ≤ d. To construct such a set, we proceed as follows∙ We first choose i1 < ⋯ < ij such that \(W\cap V_{i_{\ell }}\neq \emptyset \) for 1 ≤ ℓ ≤ j.∙ Next, for each iℓ we pick an integer ct ≥ 1, with the property that c1 + ⋯ + cj = i + j.∙ Finally, there are \(\binom {n_{i_{\ell }}}{c_{\ell }}\) ways to choose cℓ vertices among the \(n_{i_{\ell }}\) vertices of \(V_{i_{\ell }}\).
By Lemma 6 the dimension of the (j − 1)st homology of such a subset W equals \({\prod }_{\ell =1}^{j}\left (c_{\ell }-1\right )\). Combining the previous argument, we deduce the required formula (3). The second statement now is immediate. □
We illustrate (3) with an example.
Consider the clique complex Δ(K3,3,2) of K3,3,2. In order to compute \(\beta _{3,5}(\mathbb {F}[{{\varDelta }}(K_{3,3,2})])\), we need to consider the 2-element subsets of [3].
For the set {1,2} the inner sum in (3) equals
$$ \underset{\underset{c_{1},c_{2}\geq 1}{c_{1}+c_{2}=3}}{\sum\limits}c_{1}\cdot c_{2}\cdot \binom{3}{c_{1}-1}\cdot\binom{3}{c_{2}-1}=12, $$
since the sum has two summands (corresponding to (c1,c2) ∈{(1,2),(2,1)}), each contributing with 6.
Similarly, for {1,3} and {2,3}, we obtain 2 for the value of the inner sum. In total, this yields
$$\beta_{3,5}(\mathbb{F}[{{\varDelta}}])=12+2+2=16.$$
We now turn our attention to the computation of the graded Betti numbers of the skeleta of \({{\varDelta }}(K_{n_{1},\dots ,n_{d}})\). The following result, which is a special case of [32, Theorem 3.1] by Roksvold and Verdure, is crucial for this aim.
Let Δ be a (d − 1)-dimensional Cohen-Macaulay complex with f0(Δ) = n. Set Σ = Skeld− 2(Δ). Then
$$\beta_{i,i+j}\left( \mathbb{F}\left[ {{\varSigma}}\right] \right)= \begin{cases} \beta_{i,i+j}\left( \mathbb{F}\left[ {{\varDelta}}\right] \right) & \text{ if }j<d-1,\\ \beta_{i,i+d-1}\left( \mathbb{F}\left[ {{\varDelta}}\right] \right)-\beta_{i-1,i+d-1}\left( \mathbb{F}\left[ {{\varDelta}}\right] \right)\\ +\binom{n-d}{i-1}f_{d-1}\left( {{\varDelta}}\right) & \text{ if }j=d-1,\\ 0 &\text{ if } j\geq d \end{cases}$$
for 0 ≤ i ≤ n − d + 1.
Applying Lemma 8 iteratively, we obtain the following recursive formula for the graded Betti numbers of general skeleta of a Cohen-Macaulay complex:
Corollary 1
Let s be a positive integer and let Δ be a (d − 1)-dimensional Cohen-Macaulay complex with f0(Δ) = n. Set Σ = Skeld−s− 1(Δ). Then,
$$\beta_{i,i+j}\left( \mathbb{F}\left[ {{\varSigma}}\right] \right)= \begin{cases} \beta_{i,i+j}\left( \mathbb{F}\left[ {{\varDelta}}\right] \right) &\text{ if } j<d-s,\\ {\sum}_{k=0}^{s}(-1)^{k}\beta_{i-k,i+d-s}\left( \mathbb{F}\left[ {{\varDelta}}\right] \right)\\ +{\sum}_{t=0}^{s-1}(-1)^{t-s+1}\binom{n-d+t}{i-s+t}f_{d-t-1}\left( {{\varDelta}}\right) & \text{ if }j=d-s,\\ 0 & \text{ if }j\geq d-s+1 \end{cases}$$
for 0 ≤ i ≤ n − d + s.
Since the clique complex \({{\varDelta }}(K_{n_{1},\ldots ,n_{d}})\) is Cohen-Macaulay (see Remark 2), we can use Corollary 1 to compute the graded Betti numbers of its skeleta. Combining this with Theorem 1, we obtain the following bounds for the graded Betti numbers of an arbitrary balanced simplicial complex.
Let Δ be a (d − 1)-dimensional balanced simplicial complex on vertex set , with n := |V | and ni := |Vi|. Let \({{\varGamma }}={{\varDelta }}(K_{n_{1},\dots ,n_{d}})\). Then
$$ \beta_{i,i+j}\left( \mathbb{F}\left[ {{\varDelta}} \right] \right)\leq \sum\limits_{k=0}^{d-j}(-1)^{k}\beta_{i-k,i+j}\left( \mathbb{F}\left[ {{\varGamma}}\right] \right) +\sum\limits_{t=0}^{d-j-1}(-1)^{t-d+j+1}\binom{n-d+t}{i-d+j+t}f_{d-t-1}\left( {{\varGamma}}\right). $$
Note that the graded Betti numbers of \({{\varGamma }}:={{\varDelta }}(K_{n_{1},\ldots ,n_{d}})\) are given in Lemma 7 and that the f -vector of Γ is given by
$$ f_{i}({{\varGamma}})=\sum\limits_{I\subseteq [d],|I|=i+1}{\prod}_{\ell\in I}n_{\ell} $$
for 0 ≤ i ≤ d − 1. Therefore, the previous corollary really provides explicit bounds for the graded Betti numbers of a balanced simplicial complex.
A First Bound in the Cohen-Macaulay Case
We let \(S=\mathbb {F}[x_{1},\ldots ,x_{n}]\) denote the polynomial ring in n variables over an arbitrary field \(\mathbb {F}\). The ultimate aim of this section is to show upper bounds for the graded Betti numbers of the Stanley-Reisner rings of balanced Cohen-Macaulay complexes. On the way, more generally, we will prove upper bounds for the graded Betti numbers of Artinian quotients S/I, where \(I \subseteq S\) is a homogeneous ideal having many generators in degree 2.
Ideals with Many Generators in Degree 2
Throughout this section, we let \(I\subsetneq S\) be a homogeneous ideal that has no generators in degree 1, i.e., \(I\subseteq \mathfrak {m}^{2}\).
First assume that S/I is of dimension 0. It is well-known and essentially follows from Lemma 2 by passing to the lex ideal Ilex that we can bound βi,i+j(S/I) by the corresponding Betti number \(\beta _{i,i+j}(S/\mathfrak {m}^{j+1})\) of the quotient of S with the (j + 1)st power of the maximal homogeneous ideal \(\mathfrak {m}\subseteq S\). Lemma 3 then yields
$$ \beta_{i,i+j}(S/I)\leq \binom{i-1+j}{j}\binom{n+j}{i+j} $$
for all i ≥ 1, j ≥ 0. Moreover, if S/I is Cohen-Macaulay of dimension d, then, by modding out a linear system of parameters \({{\varTheta }}\subseteq S\) (which is a regular sequence by assumption) and using Lemma 1, we can reduce to the 0-dimensional case, which yields the well-known upper bound (see, e.g., [28, Lemma 3.4 (i)])
$$ \beta_{i,i+j}(S/I)\leq \binom{i-1+j}{j}\binom{n-d+j}{i+j}$$
for all i ≥ 1, j ≥ 0. In particular, those bounds apply to Stanley-Reisner rings of Cohen-Macaulay complexes. Moreover, if equality holds in the j th strand, then I has (j + 1)-linear resolution (see, e.g., [15] for the precise definition).
In the following, assume that S/I is Artinian and that there exists a positive integer b such that
$$ \dim_{\mathbb F}(S/I)_{2}\leq \binom{n+1}{2}-b. $$
In other words, I has at least b generators in degree 2. Our goal is to prove upper bounds for βi,i+j(S/I) in this setting. This will be achieved using similar arguments as the ones we just recalled that are used in the general setting. First, we need some preparations.
As, by assumption, I does not contain polynomials of degree 1, neither does its lex ideal \(I^{\text {lex}}\subseteq S\). In particular, we have
$$ |G(I^{\text{lex}})\cap S_{2}|\geq b $$
and Ilex contains at least the b largest monomials of degree 2 in lexicographic order. The next lemma describes this set of monomials explicitly.
Let \(n\in \mathbb N\) be a positive integer and let \(b<\binom {n+1}{2}\). Let xpxq be the b th largest monomial in the lexicographic order of degree 2 monomials in variables x1,…,xn and assume p ≤ q. Then,
$$p=n-\left\lfloor\frac{\sqrt{-8b+4n(n+1)+1}}{2}-\frac{1}{2}\right\rfloor,$$
$$q=b+\frac{(p-1)(p-2n)}{2}.$$
Since the proof of this lemma is technical and since the precise statement is not used later, we defer its proof to the A.
Intuitively, if a lex ideal \(J\subseteq S\) has many generators in degree 2, then there can only exist relatively few generators of higher degree. More precisely, the next lemma provides a necessary condition for a monomial u to lie in G(J)j for j > 2 and thus enables us to bound the number of generators of J of degree j.
Lemma 10
Let j > 2 be an integer and let \(J\subseteq S\) be a lex ideal. Let xpxq be the lexicographically smallest monomial of degree 2 that is contained in J. If u ∈ G(J)j is a minimal generator of J of degree j, then \(u<_{\text {lex}} x_{p}x_{q}x_{n}^{j-2}\). In other words,
$$ G(J)_{j}\subseteq \textup{Mon}_{j}(S)_{<_{\text{lex}} x_{p}x_{q}x_{n}^{j-2}}. $$
To simplify the notation, we set \(w=x_{p}x_{q}x_{n}^{j-2}\). First note that any monomial of degree j that is divisible by xpxq ∈ G(J) cannot be a minimal generator of J. Let u be a monomial of degree j with u >lexw, that is not divisible by xpxq. Then, there exists ℓ < p such that xℓ divides u or u is divisible by xp and there exists p ≤ r < q − 1 such that xpxr divides u. In the first case, let xr be such that xℓxr divides u. Then, xℓxr >lexxpxq and hence xℓxr ∈ J, since J is a lex ideal. This implies u∉G(J). Similarly, in the second case, we have xpxr >lexxpxq ∈ G(J) and hence u∉G(J). The claim follows. □
Recall that a homogeneous ideal \(I\subseteq S\), which is generated in degree d, is called Gotzmann ideal if the number of generators of \(\mathfrak {m}I\) is the smallest possible. More generally, a graded ideal \(I\subseteq S\) is called Gotzmann ideal if all components I〈j〉 are Gotzmann ideals. Here, I〈j〉 denotes the ideal generated by all the elements in I of degree j. By Gotzmann's persistence theorem [12], a graded ideal \(I\subseteq S\) is Gotzmann if and only if I and (Ilex)〈d〉 have the same Hilbert function. Moreover, as shown in [14, Corollary 1.4], this is equivalent to S/I and S/Ilex having the same graded Betti numbers, i.e.,
$$ \beta_{i,i+j}(S/I)=\beta_{i,i+j}(S/I^{\text{lex}}) $$
for all i,j ≥ 0. We state an easy lemma, which will be helpful to prove the main result of this section.
Let j ≥ d be a positive integer and let \(J\subseteq S\) be a Gotzmann ideal that is generated in degree d. Let \(I=J+\mathfrak {m}^{j+1}\). Then
$$ \beta_{i,i+\ell}(S/I)=\beta_{i,i+\ell}(S/I^{\text{lex}}) $$
for all i,ℓ ≥ 0.
We first note that, as J is Gotzmann, so are its graded components J〈j〉. Moreover, as any power of \(\mathfrak {m}\) is Gotzmann, it follows from the definition of a Gotzmann ideal that I has to be Gotzmann as well. The claim now follows from [14, Corollary 1.4]. □
We can now state the main result of this section.
Let \(I\subseteq S \) be a homogeneous ideal, that does not contain linear forms. Let \(\dim _{\mathbb F} (S/I)_{2}\leq \binom {n+1}{2}-b\) for some positive integer b. Let xpxq, where p ≤ q, be the b th largest monomial of degree 2 in lexicographic order on S. Then,
$$ \beta_{i,i+j}(S/I)\leq \sum\limits_{\ell=p+1}^{n}\binom{\ell-p+j-1}{j}\binom{\ell-1}{i-1}+\sum\limits_{\ell=q+1}^{n}\binom{\ell-q+j-2}{j-1}\binom{\ell-1}{i-1} $$
for any i ≥ 0 and j ≥ 2. Moreover if \(I=J+\mathfrak {m}^{j+1}\), where \(J\subseteq S\) is a Gotzmann ideal that is generated by b elements of degree 2, then equality is attained for a fixed j ≥ 2 and all i ≥ 0.
We fix j ≥ 2 and we set \(w:=x_{p}x_{q}x_{n}^{j-1}\). By Lemma 2 we can use the graded Betti numbers of the lex ideal \(I^{\text {lex}}\subseteq S\) of I to bound the ones of I. Using Lemma 3 we infer
$$ \begin{array}{@{}rcl@{}} &&\beta_{i,i+j}(S/I)\leq \beta_{i,i+j}(S/I^{\text{lex}})\\ &=&\sum\limits_{u\in G(I^{\text{lex}})_{j+1}}\binom{\max(u)-1}{i-1}\\ & \overset{(\text{Lemma 10})}{\leq}& \sum\limits_{u\in \text{Mon}_{j+1}(S)_{<w}}\binom{\max(u)-1}{i-1} \cr & =&\underset{\underset{x_{p}|u}{u\in \text{Mon}_{j+1}(S)_{<w}}}{\sum\limits}\binom{\max(u)-1}{i-1} +\underset{\underset{x_{p}\nmid u}{u\in \text{Mon}_{j+1}(S)_{<w}}}{\sum\limits}\binom{\max(u)-1}{i-1}. \end{array} $$
Let u be a monomial of degree j + 1, such that u <lexw. If xp|u, then \(\max \limits (u)\geq q+1\) and u is of the form \(x_{p}x_{\max \limits (u)}\cdot v\), where v is a monomial in \(\mathbb F[x_{q+1},\ldots ,x_{\max \limits (u)}]\) of degree j − 1. In particular, there are \(\binom {(\ell -q)+(j-1)-1}{j-1}\) many such monomials with \(\max \limits (u)=\ell \). Similarly, if u is not divisible by xp, then \(\max \limits (u)\geq p+1\) and u is of the form \(x_{\max \limits (u)}\cdot v\), where v is a monomial of degree j in \(\mathbb F[x_{p+1},\ldots ,x_{\max \limits (u)}]\). There are \(\binom {(\ell -p)+j-1}{j}\) many such monomials with \(\max \limits (u)=\ell \). The desired inequality follows.
For the equality case first note that if \(I=J+\mathfrak {m}^{j+1}\), where J is a Gotzmann ideal generated in degree d, then it follows from Lemma 11 that βi,i+j(S/I) = βi,i+j(S/Ilex) for all i. Moreover, as \(I^{\text {lex}}=\text {Lex}(b)+\mathfrak {m}^{j+1}\), where Lex(b) denotes the lex ideal generated by the b lexicographically largest monomials of degree 2, the lex ideal Ilex attains equality in (6). □
It is worth remarking that if an ideal I attains equality in (6) for a fixed j, then the ideal J (where \(I=J+\mathfrak {m}^{j+1}\) as above) is not necessarily a monomial ideal. E.g., for n = 2 and b = 2 the ideals
$$ ({x_{1}^{2}},x_{1}x_{2})+(x_{1},x_{2})^{3} \quad \text{and}\quad ({x_{1}^{2}}+x_{1}x_{2},{x_{2}^{2}}+x_{1}x_{2})+(x_{1},x_{2})^{3} $$
both maximize βi,i+ 2 for any i. The maximal Betti numbers in this case are β1,3 = β2,4 = 1.
Application: Balanced Cohen-Macaulay Complexes
The aim of this section is to use the results from the previous section in order to derive upper bounds for the graded Betti numbers of balanced Cohen-Macaulay complexes.
In the following, let Δ be a balanced Cohen-Macaulay simplicial complex and let \({{\varTheta }}\subseteq \mathbb {F}[{{\varDelta }}]\) be a linear system of parameters for \(\mathbb {F}[{{\varDelta }}]\). In order to apply Theorem 2 we need to bound the Hilbert function of the Artinian reduction \(\mathbb {F}[{{\varDelta }}]/{{\varTheta }} \mathbb {F}[{{\varDelta }}]\) in degree 2 from above. As Δ is Cohen-Macaulay, it follows from Lemma 5 that
$$ \dim_{\mathbb{F}}\left( \mathbb{F}[{{\varDelta}}]/{{\varTheta}} \mathbb{F}[{{\varDelta}}]\right)_{2}=h_{2}({{\varDelta}}), $$
which implies that we need to find an upper bound for h2(Δ) or, equivalently, for the number of edges f1(Δ).
Let Δ be a (d − 1)-dimensional balanced simplicial complex with vertex partition . Let n := |V | and ni := |Vi|. Then,
$$ h_{2}({{\varDelta}})\leq \binom{n-d+1}{2}-\sum\limits_{i=1}^{d}\binom{n_{i}}{2}. $$
As Δ is balanced, it does not have monochromatic edges, i.e., we have \(\left \lbrace v,w\right \rbrace \notin {{\varDelta }}\), if v and w belong to the same color class Vi (1 ≤ i ≤ d). As there are \(\binom {n_{i}}{2}\) monochromatic non-edges of color i, this gives the following upper bound for f1(Δ)
$$f_{1}({{\varDelta}})\leq\binom{n}{2}-\sum\limits_{i=1}^{d}\binom{n_{i}}{2}.$$
The claim now directly follows from the relation
$$ h_{2}({{\varDelta}})=\binom{d}{2}-(d-1)f_{0}({{\varDelta}})+f_{1}({{\varDelta}}). $$
A direct application of Theorem 2 combined with Lemma 12 finally yields:
Let Δ be a (d − 1)-dimensional balanced Cohen-Macaulay complex with vertex partition and \(b:={\sum }_{i=1}^{d}\binom {n_{i}}{2}\). Let xpxq be the b th largest degree 2 monomial of \(\mathbb {F}[x_{1},\ldots ,x_{n-d}]\) in lexicographic order with p ≤ q. Then
$$\beta_{i,i+j}(\mathbb{F}\left[ {{\varDelta}}\right] )\leq \sum\limits_{\ell=p+1}^{n-d}\binom{\ell-p+j-1}{j}\binom{\ell-1}{i-1}+\sum\limits_{\ell=q+1}^{n-d}\binom{\ell-q+j-2}{j-1}\binom{\ell-1}{i-1}$$
for any i ≥ 0 and 2 ≤ j ≤ d.
The above statement is trivially true also for j > d. However, as the Castelnuovo-Mumford regularity of \(\mathbb {F}[{{\varDelta }}]\) is at most d, we have \(\beta _{i,i+j}(\mathbb {F}[{{\varDelta }}])=0\) for any i ≥ 0 and j > d.
Let \(S=\mathbb {F}[x_{1},\ldots ,x_{n}]\). Let Θ be an l.s.o.p. for \(\mathbb {F}[{{\varDelta }}]\). It follows from Lemma 1 that
$$\beta_{i,i+j}^{S}(\mathbb{F}[{{\varDelta}}])=\beta_{i,i+j}^{S/{{\varTheta}} S}(S/(I_{{{\varDelta}}}+({{\varTheta}}))).$$
Moreover, \(S/ {{\varTheta }} S\cong \mathbb {F}[x_{1},\ldots ,x_{n-d}]=:R\) as rings and there exists a homogeneous ideal \(J\subseteq R\) with \(\mathbb {F}[{{\varDelta }}]/{{\varTheta }}\mathbb {F}[{{\varDelta }}]\cong R/J\) and \(\beta _{i,i+j}^{R}(R/J)=\beta _{i,i+j}^{S/{{\varTheta }} S}(S/(I_{{{\varDelta }}}+({{\varTheta }})))\). In particular, as Δ is Cohen-Macaulay, \(\dim _{\mathbb {F}} (R/J)_{2}=h_{2}({{\varDelta }})\) satisfies the bound from Lemma 12. As \(h_{1}({{\varDelta }})=\dim _{\mathbb {F}} (R/J)_{1}\), the ideal J does not contain any linear form and the result now follows from Theorem 2. □
Whereas we have seen that the bounds in Theorem 2 are tight, the ones in Theorem 3 are not. For example, consider the case that n1 = n2 = 2 and d = 2. In this situation, we have \(b:={\sum }_{i=1}^{d} \binom {n_{i}}{2}=2\) and x1x2 is the second largest degree 2 monomial in the lexicographic order. Theorem 3 gives β1,3 ≤ 1. However, by Hochster's formula, if Δ is a 1-dimensional simplicial complex with \(\beta _{1,3}(\mathbb {F}[{{\varDelta }}])=1\), then Δ must contain an induced 3-cycle. But this means that Δ cannot be balanced.
Let Δ be a 3-dimensional balanced Cohen-Macaulay complex with 3 vertices in each color class, i.e., ni = 3 for 1 ≤ i ≤ 4. We have \(b:={\sum }_{i=1}^{4}\binom {3}{2}=12\) and x2x5 is the 12th largest monomial of degree 2 in variables x1,…,x8. The bounds from Theorem 3 are recorded in the following table:
j ∖ i
2 0 62 360 915 1317 1156 617 185 24
3 0 136 821 2155 3184 2855 1551 472 62
4 0 267 1653 4432 6665 6065 3336 1026 136
We set \(S=\mathbb {F}[x_{1},\ldots ,x_{8}]\) and we let \(I\subseteq S\) be the lex ideal generated by the 12 largest monomials of degree 2 in variables x1,…,x8. It follows from Theorem 2 that \(\beta _{i,i+j}(S/(I+\mathfrak {m}^{j+1}))\) equals the entry of the above table in the row, labeled i and the column, labeled j. Moreover, it is shown in the proof of Theorem 2 that \(\beta _{i,i+\ell }(S/(I+\mathfrak {m}^{j+1}))=0\) if ℓ∉{1,j}. One can easily compute that for any j the first row of the Betti table of \(S/(I+\mathfrak {m}^{j+1})\) is given by
1 0 12 38 66 75 57 28 8 1
Finally, we compare the bounds from the upper table with the numbers \(\beta _{i,i+j}(S/\mathfrak {m}^{j})\), for general 3-dimensional Cohen-Macaulay complexes on 12 vertices. Those are displayed in the next table
2 0 120 630 1512 2100 1800 945 280 36
3 0 330 1848 4620 6600 5775 3080 924 120
4 0 792 4620 11880 17325 15400 8316 2520 330
We point out that while Theorem 3 provides bounds for \(\beta _{i,i+j}(\mathbb F[{{\varDelta }}])\) for all i and all j ≥ 2, it does not give bounds for the graded Betti numbers of the linear strand (i.e., for j = 1). This seems a natural drawback of our approach, since our key ingredient is the concentration of monomials of degree 2 in the lex ideal of IΔ + (Θ) (cf., (7)). However, it follows from the next lemma, that there is no better bound in terms of the total number of vertices n and the dimension d − 1 than in the standard (non-balanced) Cohen-Macaulay case. More precisely, for any n and any d we construct a balanced Cohen-Macaulay complex whose graded Betti numbers equal \(\beta _{i,i+j}(S/\mathfrak {m}^{j})\) for j = 1 and for every i > 0, where \(S=\mathbb F[x_{1},\dots ,x_{n-d}]\).
Let n and d be positive integers. Let Γn−d+ 1 denote the simplicial complex consisting of the isolated vertices 1,2,…,n − d + 1 and let Δd− 2 be the (d − 2)-simplex with vertices {n − d + 2,…,n}. Then, Δd− 2 ∗Γn−d+ 1 is a balanced (d − 1)-dimensional Cohen-Macaulay complex. Moreover
$$\beta_{i,i+1}(\mathbb{F}[{{\varDelta}}_{d-2}\ast{{\varGamma}}_{n-d+1}])= i\binom{n-d+1}{i+1} \quad \text{for all } i.$$
We set Δ = Δd− 2 ∗Γn−d+ 1. As Δ is the join of a (d − 2)-dimensional and a 0-dimensional Cohen-Macaulay complex, it is Cohen-Macaulay of dimension d − 1. Moreover, coloring the vertices of Δd− 2 with the colors 1,…,d − 1 and assigning color d to all vertices of Γn−d+ 1 gives a proper d-coloring of Δ, i.e., Δ is balanced.
By Hochster's formula (Lemma 4), the graded Betti numbers \(\beta _{i,i+1}(\mathbb {F}[{{\varDelta }}])\) are given by
$$ \beta_{i,i+1}(\mathbb{F}[{{\varDelta}}])=\sum\limits_{W\subseteq [n]:| W| =i+1} \dim_{\mathbb{F}}\widetilde{H}_{0}({{\varDelta}}_{W};\mathbb{F}). $$
As ΔW = (Δd− 2)W ∗ (Γn−d+ 1)W, the induced complex ΔW is connected whenever W ∩{n − d + 2,…,n}≠∅. Hence, the only non-trivial contributions to (8) come from (i + 1)-element subsets of [n − d + 1]. For such a subset W, the complex ΔW consists of i connected components and since there are \(\binom {n-d+1}{i+1}\) many such sets, the claim follows. □
Though we have just seen that Betti numbers (in the linear strand) of balanced Cohen-Macaulay complexes can be as big as the ones for general Cohen-Macaulay complexes, it should also be noted that the simplicial complex Δd− 2 ∗Γn−d+ 1 is special, in the sense that all but one "big" color classes are singletons. It is therefore natural to ask, if there are better bounds than those for the general Cohen-Macaulay situation, that take into account the size of the color classes.
A Second Bound in the Cohen-Macaulay Case via Lex-plus-squares Ideals
The aim of this section is to provide further upper bounds for the graded Betti numbers of balanced Cohen-Macaulay complexes. On the one hand, those bounds will be a further improvement of the ones from Theorem 3. On the other hand, however, they are slightly more complicated to state. Our approach is similar to the one used in Theorem 3 with lex-plus-squares ideals as an additional ingredient. More precisely, we will prove upper bounds for the graded Betti numbers of Artinian quotients S/I, where \(I\subseteq S\) is a homogeneous ideal having many generators in degree 2, including the squares of the variables \({x_{1}^{2}},\ldots ,{x_{n}^{2}}\). The desired bound for balanced Cohen-Macaulay complexes is then merely an easy application of those more general results.
Ideals Containing the Squares \({x_{1}^{2}},\ldots ,{x_{n}^{2}}\) with Many Degree 2 Generators
We recall some necessary definitions and results. As in the previous sections, we let \(S=\mathbb F[x_{1},\ldots ,x_{n}]\). We further let \(P:=({x_{1}^{2}},\ldots ,{x_{n}^{2}})\subseteq S\). A monomial ideal \(L\subseteq S\) is called squarefree lex ideal if for every squarefree monomial u ∈ L and every monomial v ∈ S with \(\deg (u)=\deg (v)\) and v >lexu it follows that v ∈ L. For homogeneous ideals containing the squares of the variables the following analog of Lemma 2 was shown by Mermin, Peeva, and Stillman [25] in characteristic 0 and by Mermin and Murai [24] in arbitrary characteristic:
Let \(I\subseteq S=\mathbb F[x_{1},\dots ,x_{n}]\) be a homogeneous ideal containing P. Let \(I^{\text {sqlex}}\subseteq S\) be the squarefree lex ideal such that I and Isqlex + P have the same Hilbert function. Then,
$$ \beta_{i,i+j}^{S}(S/I)\leq\beta_{i,i+j}^{S}(S/(I^{\text{sqlex}}+P)) $$
The existence of a squarefree lex ideal Isqlex as in the previous theorem is a straight-forward consequence of the Clements-Lindström Theorem [5]. Moreover, Theorem 4 provides an instance for which the so-called lex-plus-powers Conjecture is known to be true (see [7, 9, 10] for more details on this topic).
An ideal of the form Isqlex + P is called lex-plus-squares ideal. It was shown in [25, Theorem 2.1 and Lemma 3.1 (2)] that the graded Betti numbers of ideals of the form \(I+P\subseteq S\), where \(I\subseteq S\) is a squarefree monomial ideal can be computed via the Betti numbers of smaller squarefree monomial ideals, via iterated mapping cones. In the next result, we use \(\binom {[n]}{k}\) to denote the set of k-element subsets of [n].
Proposition 1
Let \(I\subseteq S\) be a squarefree monomial ideal. Then, (i)
$$\beta_{i,i+j}^{S}(S/(I+P))=\sum\limits_{k=0}^{j}\left( \sum\limits_{F\in\binom{[n]}{k}}\beta_{i-k,i+j-2k}^{S}(S/(I:x_{F}))\right),$$
where \(x_{F}={\prod }_{f\in F}x_{f}\). (ii) If I is squarefree lex, then the ideal (Isqlex : xF) is a squarefree lex ideal in SF = S/(xf : f ∈ F) for any \(F\in \binom {[n]}{k}\).
We have the following analog of Lemma 9 in the squarefree setting.
Let \(n\in \mathbb N\) be a positive integer and let \(b<\binom {n}{2}\). Let xpxq be the b th largest monomial in the lexicographic order of degree 2 squarefree monomials in variables x1,…,xn and assume p < q. Then,
$$p=n-1+\left\lfloor\frac{1}{2}-\frac{\sqrt{4n(n-1)-8b+1}}{2}\right\rfloor,$$
$$q=b+\binom{p+1}{2}-(p-1)n.$$
The proof is deferred to the A since it is technical and the precise statement is not needed during the remaining part of this article.
For squarefree lex ideals (or more generally squarefree stable ideals) the following analog of the Eliahou-Kervaire formula Lemma 3 is well-known.
[15, Corollary 7.4.2] Let \(I\subseteq S\) be a squarefree lex ideal. Then,
$$ \beta_{i,i+j}^{S}(S/I)=\sum\limits_{u\in G(I)_{j+1}}\binom{\max(u)-j-1}{i-1} $$
for every i ≥ 1, j ≥ 0.
We can now formulate the main result of this section.
Let \(I\subseteq S\) be a homogeneous ideal not containing any linear form. Let \(\dim _{\mathbb F}(S/(I+P))_{2}\leq \binom {n}{2}-b\) for some positive integer b. Let xpxq, where p < q, be the b th largest squarefree monomial in S of degree 2 in lexicographic order. Then,
$$ \begin{array}{@{}rcl@{}} \beta_{i,i+j}(S/(I+P))&\leq& \sum\limits_{k=0}^{j-1}\Bigg[\binom{n-p}{k}\sum\limits_{\ell=p+j-k+1}^{n-k}\binom{\ell-p-1}{j-k}\binom{\ell-j+k-1}{i-k-1}\\ &+&\binom{n-q}{k}\sum\limits_{\ell=q+j-k}^{n-k}\binom{\ell-q-1}{j-k-1}\binom{\ell-j+k-1}{i-k-1}\\ &+&\binom{n-q}{k-1}\sum\limits_{\ell=q+j-k}^{n-k}\binom{\ell-q}{j-k}\binom{\ell-j+k-1}{i-k-1}\Bigg]\\ &+&\binom{n-j}{i-j}\left( \binom{n-p}{j}+\binom{n-q}{j-1}\right) \end{array} $$
for all i > 0, j ≥ 2.
By Theorem 4 we have βi,i+j(S/(I + P)) ≤ βi,i+j(S/(L + P)), where \(L\subseteq S\) is the squarefree lex ideal such that L + P and I + P have the same Hilbert function. By assumption, L does not contain variables and \(\dim _{\mathbb F} L_{2}\geq b\). Hence, L contains all squarefree degree 2 monomials that are lexicographically larger or equal than xpxq. We can further compute βi,i+j(S/(L + P)) using Proposition 1. For this, we need to analyze the ideals (L : xF), where \(F\in \binom {[n]}{k}\). We distinguish four cases (having several subcases).
Assume that F = {f} for 1 ≤ f < p. In particular, we have p > 1. Since L is squarefree lex and xpxq ∈ L, it holds that xfxℓ ∈ L for all ℓ ∈ [n] ∖{f}. This implies \((x_{i} : i\in [n]\setminus \{f\})\subseteq (L:x_{F})\). As, by Proposition 1 (ii) (L : xF) can be considered as an ideal in SF and hence no minimal generator is divisible by xf, we infer that (L : xF) = (xi : i ∈ [n] ∖{f}). As (L : xF) and (x1,…,xn− 1) have the same graded Betti numbers, it follows from Lemma 15 that F only contributes to βi,i+j(S/(L + P)) if j = 1, a case which we do not consider.
Assume that there exist 1 ≤ s < t ≤ n such that \(\{s,t\}\subseteq F\) and xsxt ≥lexxpxq. As L is squarefree lex and xpxq ∈ L, we infer that xsxt ∈ L and hence 1 ∈ (L : xF), i.e., (L : xF) = S. In particular, such F never contributes to βi,i+j(S/(L + P)).
Suppose that there do not exist s,t ∈ F (s≠t) with xsxt ≥lexxpxq. We then have to consider the following two subcases:
C ase 3.1: f > p for all f ∈ F.
C ase 3.2: p ∈ F and f > q for all f ∈ F ∖{p}. C ase 3.1 (a): Assume in addition that there exists f ∈ F with p < f ≤ q. As xpxq ∈ L, xℓxf ≥lexxpxq for 1 ≤ ℓ ≤ p and since L is squarefree lex, we infer that \((x_{1},\ldots ,x_{p})\subseteq (L:x_{F})\). Moreover, by Proposition 1 (ii) (L : xF) is squarefree lex as an ideal in SF. If we reorder (and relabel) the variables x1,…,xn by first ordering {xi : i∉F} from largest to smallest by increasing indices and then adding {xf : f ∈ F} in any order, the ideal (L : xF) will be a squarefree lex ideal in S with respect to this ordering of the variables. If j≠k, then, using Lemma 15, we conclude
$$ \begin{array}{@{}rcl@{}} \beta_{i-k,i+j-2k}(S/(L:x_{F}))&=&\sum\limits_{\ell=p+j-k+1}^{n-k}\left( \sum\limits_{u\in G(L:x_{F})_{j-k+1}}\binom{\ell-(j-k)-1}{i-k-1}\right)\\ &\leq &\sum\limits_{\ell=p+j-k+1}^{n-k} \binom{\ell-p-1}{j-k}\binom{\ell-j+k-1}{i-k-1}, \end{array} $$
where the last inequality follows from the fact that the inclusion \(G(L:x_{F})_{j-k+1}\subseteq G((x_{p+1},\ldots ,\) xn−k)j−k+ 1) holds. For j = k, we note that (after relabeling) we have \(G(L:x_{F})_{1}\subseteq (x_{1},\ldots ,x_{n-k})\), from which it follows that F contributes to βi,i+j(S/(L + P)) with at most
$$ \sum\limits_{\ell=1}^{n-j}\binom{\ell-1}{i-j-1}=\binom{n-j}{i-j}. $$
C ase 3.1 (b): Now suppose that f > q for all f ∈ F. As F≠∅, such f exists. If p > 1, then, as L is squarefree lex and xpxq ∈ L, we have xℓxf ∈ L for all 1 ≤ ℓ ≤ p − 1. It follows that xF ⋅ xℓ = xF∖{f}⋅ (xℓ ⋅ xf) ∈ L for 1 ≤ ℓ ≤ p − 1, which implies \((x_{1},\ldots ,x_{p-1})\subseteq (L:x_{F})\). Moreover, for any p, as xpxq ∈ L, we also have xpxℓ ∈ (L : xF) for p + 1 ≤ ℓ ≤ q. Similar as in Case 3.1 (a) we can assume that, after reordering (and relabeling) the variables, (L : xF) is a squarefree lex ideal in S. As the order of x1,…,xq is not affected by this reordering, the previous discussion implies
$$ \begin{array}{@{}rcl@{}} G(L:x_{F})_{j-k+1}&\subseteq &\{u\in\text{Mon}_{j-k+1}(x_{p+1},\ldots,x_{n-k}) : u \text{ squarefree}\}\cup\\ && \{x_{p}u : u\in \text{Mon}_{j-k}(x_{q+1},\ldots,x_{n-k}), u\text{ squarefree}\} \end{array} $$
if j≠k. Using Lemma 15, we thus obtain
$$ \begin{array}{@{}rcl@{}} \beta_{i-k,i+j-2k}(S/(L:x_{F}))&\leq& \sum\limits_{\ell=p+1+j-k}^{n-k}\binom{\ell-1-p}{j-k}\binom{\ell-j+k-1}{i-k-1}\cr &&+\sum\limits_{\ell=q+j-k}^{n-k}\binom{\ell-1-q}{j-k-1}\binom{\ell-j+k-1}{i-k-1} \end{array} $$
if j≠k. For j = k, a similar computation as in Case 3.1 (a) shows that F contributes to βi,i+j(S/(L + P)) with at most \(\binom {n-j}{i-j}\).
C ase 3.2: Consider \(F\in \binom {[n]}{k}\) such that p ∈ F and f > q for all f ∈ F ∖{p}. As xpxq ∈ L and L is squarefree lex, it holds that \((x_{1},\ldots ,x_{p-1},x_{p+1},\ldots ,x_{q})\subseteq (L:x_{F})\). As in Case 3.1, we can assume that after a suitable reordering (and relabeling) of the variables (L : xF) is a squarefree lex ideal in S. (Note that after relabeling (L : xF) contains x1,…,xq− 1.) We infer that
$$ G(I:x_{F})_{j-k+1}\subseteq \{u\in\text{Mon}_{j-k+1}(x_{q},\ldots,x_{n-k}) : u\text{ squarefree}\}, $$
if j≠k and it hence follows from Lemma 15 that
$$ \beta_{i-k,i+j-2k}(S/(L:x_{F}))\leq \sum\limits_{\ell=q+j-k}^{n-k}\binom{\ell-q}{j-k}\binom{\ell-j+k-1}{i-k} $$
if j≠k. For j = k, it follows from the same arguments as in Case 3.1 (a) that the set F contributes to βi,i+j(S/(L + P)) with at most \(\binom {n-j}{i-j}\).
If F = ∅, then clearly (L : xF) = L. As xpxq ∈ L, we obtain that
$$ \begin{array}{@{}rcl@{}} G(L)_{j+1}\subseteq &\{u\in\text{Mon}_{j+1}(x_{p+1},\ldots,x_{n}) : u \text{ squarefree}\}\cup\\ & \{x_{p}u : u\in \text{Mon}_{j}(x_{q+1},\ldots,x_{n}), u\text{ squarefree}\} \end{array} $$
for j ≥ 2. The same computation as in Case 3.1 (b) now yields that
$$ \begin{array}{@{}rcl@{}} \beta_{i,i+j}(S/(L:x_{F}))&\leq & \sum\limits_{\ell=p+1+j}^{n}\binom{\ell-1-p}{j}\binom{\ell-j-1}{i-1}\\ &+&\sum\limits_{\ell=q+j}^{n}\binom{\ell-1-q}{j-1}\binom{\ell-j-1}{i-1}. \end{array} $$
Combining Cases 1–4, we finally obtain for i > 0 and j > 1
$$ \begin{array}{@{}rcl@{}} &&\beta_{i,i+j}(S/(I+P))\leq \beta_{i,i+j}(S/(L+P))\\ &=& \underbrace{\binom{n-j}{i-j}\left( \binom{n-p}{j}-\binom{n-q}{j}\right)}_{\text{Case 3.1(a), }j=k}+\underbrace{\binom{n-j}{i-j}\binom{n-q}{j}}_{\text{Case 3.1 (b), } j=k}+\underbrace{\binom{n-j}{i-j}\binom{n-q}{j-1}}_{\text{Case 3.2, } j=k}\\ &&+\sum\limits_{k=1}^{j-1}\Bigg[\underbrace{\left( \binom{n-p}{k}-\binom{n-q}{k}\right)\sum\limits_{\ell=p+j-k+1}^{n-k}\binom{\ell-p-1}{j-k}\binom{\ell-j+k-1}{i-k-1}}_{\text{Case 3.1 (a)}}\\ &&+\underbrace{\binom{n-q}{k}\Bigg(\sum\limits_{\ell=p+j-k+1}^{n-k}\binom{\ell-1-p}{j-k}\binom{\ell-j+k-1}{i-k-1}}_{\text{Case 3.1 (b)}}\\ &&+\underbrace{\sum\limits_{\ell=q+j-k}^{n-k}\binom{\ell-q-1}{j-k-1}\binom{\ell-j+k-1}{i-k-1}\Bigg)}_{\text{Case 3.1 (b)}}\cr &&+\underbrace{\binom{n-q}{k-1}\sum\limits_{\ell=q+j-k}^{n-k}\binom{\ell-q}{j-k}\binom{\ell-j+k-1}{i-k-1}}_{\text{Cases 3.2}}\Bigg]\\ &=&\sum\limits_{k=0}^{j-1}\Bigg[\binom{n-p}{k}\sum\limits_{\ell=p+j-k+1}^{n-k}\binom{\ell-p-1}{j-k}\binom{\ell-j+k-1}{i-k-1}\\ &&+\binom{n-q}{k}\sum\limits_{\ell=q+j-k}^{n-k}\binom{\ell-q-1}{j-k-1}\binom{\ell-j+k-1}{i-k-1}\\ &&+\binom{n-q}{k-1}\sum\limits_{\ell=q+j-k}^{n-k}\binom{\ell-q}{j-k}\binom{\ell-j+k-1}{i-k-1}\Bigg]\\ &&+\binom{n-j}{i-j}\left( \binom{n-p}{j}-\binom{n-q}{j}+\binom{n-q+1}{j}\right). \end{array} $$
This completes the proof. □
There might be several ways to simplify the bound of Theorem 5 by losing tightness. However, we decided to state it in the best possible form.
Application: Balanced Cohen-Macaulay Complexes Revisited
The aim of this section is to use Theorem 5 in order to get bounds for the graded Betti numbers of balanced Cohen-Macaulay complexes.
Our starting point is the following result due to Stanley (see [34, Chapter III, Proposition 4.3] or [33]).
Let Δ be a (d − 1)-dimensional balanced simplicial complex with vertex partition and let \(\theta _{i}:={\sum }_{v\in V_{i}}x_{v}\) for 1 ≤ i ≤ d. Then, (i) 𝜃1,…,𝜃d is an l.s.o.p. for \(\mathbb F[{{\varDelta }}]\). (ii) \({x_{v}^{2}}\in I_{{{\varDelta }}}+(\theta _{1},\ldots ,\theta _{d})\subseteq \mathbb F[x_{v} : v\in V]\) for all v ∈ V.
An l.s.o.p. as in the previous lemma is also referred to as a colored l.s.o.p. of \(\mathbb F[{{\varDelta }}]\). If Δ is strongly connected, which is in particular true if Δ is Cohen-Macaulay, then a coloring is unique up to permutation and there is just one colored l.s.o.p. of \(\mathbb F[{{\varDelta }}]\).
An almost immediate application of Theorem 5, combined with Lemma 16 (ii) yields the desired bound for the graded Betti numbers of a balanced Cohen-Macaulay complex.
Let Δ be a (d − 1)-dimensional balanced Cohen-Macaulay complex with vertex partition . Let n := |V |, ni := |Vi| and \(b:={\sum }_{i=1}^{d}\binom {n_{i}-1}{2}\). Let xpxq be the bth largest squarefree degree 2 monomial of \(\mathbb F[x_{1},\ldots ,x_{n-d}]\) in lexicographic order with p ≤ q. Then
$$ \begin{array}{@{}rcl@{}} \beta_{i,i+j}(\mathbb F[{{\varDelta}}])&\leq& \sum\limits_{k=0}^{j-1}\Bigg[\binom{n-d-p}{k}\sum\limits_{\ell=p+j-k+1}^{n-d-k}\binom{\ell-p-1}{j-k}\binom{\ell-j+k-1}{i-k-1}\\ &&+\binom{n-d-q}{k}\sum\limits_{\ell=q+j-k}^{n-d-k}\binom{\ell-q-1}{j-k-1}\binom{\ell-j+k-1}{i-k-1}\\ &&+\binom{n-d-q}{k-1}\sum\limits_{\ell=q+j-k}^{n-d-k}\binom{\ell-q}{j-k}\binom{\ell-j+k-1}{i-k-1}\Bigg]\cr &&+\binom{n-d-j}{i-j}\left( \binom{n-d-p}{j}+\binom{n-d-q}{j-1}\right) \end{array} $$
for all i > 0, j > 1.
The proof follows exactly along the same arguments as the one of Theorem 3, using the colored l.s.o.p. of \(\mathbb F[{{\varDelta }}]\). By Lemma 16 it then holds that the ideal (Θ) + IΔ contains the squares of the variables. It remains to observe that under the isomorphism \(\mathbb F[x_{1},\ldots ,x_{n}]/({{\varTheta }})\cong R\), the ideal \(P=({x_{1}^{2}},\ldots ,{x_{n}^{2}})\subseteq \mathbb F[x_{1},\ldots ,x_{n}]\) is mapped to a homogeneous ideal containing \(({x_{1}^{2}},\ldots ,x_{n-d}^{2})\) and thus \(\mathbb F[{{\varDelta }}]/{{\varTheta }}\cong R/(I+P)\) for a homogeneous ideal \(I\subseteq R\) (not containing linear forms). We further observe that
$$ \dim_{\mathbb F} (R/(I+P))_{2}=h_{2}({{\varDelta}}) \leq \binom{n-d+1}{2}-\sum\limits_{i=1}^{d}\binom{n_{i}}{2}=\binom{n-d}{2}-\sum\limits_{i=1}^{d}\binom{n_{i}-1}{2}. $$
The claim now follows from Theorem 5. □
We consider 3-dimensional balanced Cohen-Macaulay complexes with 3 vertices in each color class, i.e., ni = 3 for 1 ≤ i ≤ 4, as in Example 3. We have \(b:={\sum }_{i=1}^{4}\binom {3}{2}-8=4\) and x1x5 is the 4th largest monomial of degree 2 in variables x1,…,x8. The bounds from Theorem 6 are recorded in the following table:
Comparing those bounds with the ones from Example 3, we see that the lex-plus-squares approach gives better bounds for all entries of the Betti table. The improvement is more significant in the lower rows of the Betti tables.
Consider again a 3-dimensional balanced Cohen-Macaulay complex Δ on 12 vertices, but with a different color partition, namely n1 = 1, n2 = 3, and n3 = n4 = 4. Then since every facet must contain the unique vertex of color 1, Δ is a cone, hence contractible. Theorem 6 yields \(\beta _{8,12}(\mathbb F[{{\varDelta }}])=\dim _{\mathbb F}\widetilde {H}_{3}({{\varDelta }};\mathbb F)\leq 35\). This shows that the bound is not necessarily tight.
The Linear Strand for Balanced Pseudomanifolds
The aim of this section is to study the linear strand of the minimal graded free resolution of the Stanley-Reisner ring of a balanced normal pseudomanifold. In particular, we will provide upper bounds for the graded Betti numbers in the linear strand. Previously, such bounds have been shown for general (not necessarily balanced) pseudomanifolds by Murai [28, Lemma 5.6 (ii)] and it follows from a result by Hibi and Terai [36, Corollary 2.3.2] that they are tight for stacked spheres. We start by recalling those results and by introducing some notation.
Let Δ and Γ be (d − 1)-dimensional pure simplicial complexes and let F ∈Δ and G ∈Γ be facets, together with a bijection φ : F → G. The connected sum of Δ and Γ is the simplicial complex obtained from Δ ∖{F}∪Γ∖{G} by identifying v with φ(v) for all v ∈ F. A stacked (d − 1)-sphere on n vertices is a (d − 1)-dimensional simplicial complex Δ obtained via the connected sum of n − d copies of the boundary of the d-simplex. The mentioned results of Murai [28, Lemma 5.6 (ii)] and Hibi and Terai [36, Corollary 2.3.2] can be summarized as follows:
Let d ≥ 3. Let Δ be a (d − 1)-dimensional normal pseudomanifold with n vertices. Then
$$ \beta_{i,i+1}(\mathbb{F}[{{\varDelta}}])\leq i\binom{n-d}{i+1} \quad \text{ for all } i\geq 0. $$
Moreover, those bounds are attained if Δ is a stacked sphere.
We remark that, in [36], the authors provide explicit formulas not only for the Betti numbers of the linear strand but for all graded Betti numbers of a stacked sphere. In particular, it is shown that these numbers only depend on the number of vertices n and the dimension d − 1.
In order to prove a balanced analog of the first statement of Lemma 17, the following result due to Fogelsanger [8] will be crucial (see also [30, Section 5]).
Let d ≥ 3. Let Δ be a (d − 1)-dimensional normal pseudomanifold. Then, there exist linear forms \(\theta _{1},\dots ,\theta _{d+1}\) such that the multiplication map
$$\times \theta_{i}: \left( \mathbb{F}[{{\varDelta}}]/(\theta_{1},\dots,\theta_{i-1})\mathbb{F}[{{\varDelta}}]\right)_{1}\longrightarrow\left( \mathbb{F}[{{\varDelta}}]/(\theta_{1},\dots,\theta_{i-1})\mathbb{F}[{{\varDelta}}]\right)_{2}$$
is injective for all 1 ≤ i ≤ d + 1.
Intuitively, the previous result compensates the lack of a regular sequence for normal pseudomanifolds in small degrees, since those need not to be Cohen-Macaulay.
Recall that a key step for the proofs of Theorem 2 and Theorem 5 was to find upper bounds for the number of generators of the lex ideal and the lex-plus-squares ideal, respectively, of degree ≥ 3. For the proof of our main result in this section we will use a similar strategy, but since we are interested in the linear strand of the minimal free resolution, we rather need to bound the number of degree 2 generators in a certain lex-ideal. This will be accomplished via the lower bound theorem for balanced normal pseudomanifolds, which was shown by Klee and Novik [22, Theorem 3.4] (see also [11, Theorem 5.3] and [2, Theorem 4.1] for the corresponding result for balanced spheres respectively manifolds and Buchsbaum* complexes).
Let d ≥ 3 and let Δ be a (d − 1)-dimensional balanced normal pseudomanifold. Then
$$h_{2}({{\varDelta}})\geq\frac{d-1}{2}h_{1}({{\varDelta}}).$$
Let d ≥ 3 and let Δ be a (d − 1)-dimensional balanced normal pseudomanifold on n vertices. Let \(b:= \frac {(n-d)(n-2d+2)}{2}\) and let xpxq (where p ≤ q) be the b th largest degree 2 monomial of \(\mathbb F[x_{1},\ldots ,x_{n-d-1}]\) in lexicographic order. Then,
$$ \beta_{i,i+1}(\mathbb{F}[{{\varDelta}}])\leq (p-1)\binom{n-d-1}{i}-\binom{p}{i+1}+\binom{q}{i}. $$
Let \(R^{\prime }:=\mathbb F[x_{1},\ldots ,x_{n-d-1}]\) and let Θ = {𝜃1,…,𝜃d+ 1} be linear forms given by Lemma 18. Then, as in the proof of Theorem 3, we let \(J\subseteq R\) be the homogeneous ideal with \(\mathbb {F}[{{\varDelta }}]/{{\varTheta }}\mathbb {F}[{{\varDelta }}]\cong R/J\) and we let \(J^{\text {lex}}\subseteq R\) be the lex ideal of J. Using Lemma 18, Lemmas 1 and 2 we conclude
$$ \beta_{i,i+1}(\mathbb{F}[{{\varDelta}}])\leq\beta_{i,i+1}^{S/{{\varTheta}} S }(\mathbb{F}[{{\varDelta}}]/{{\varTheta}} \mathbb{F}[{{\varDelta}}])= \beta_{i,i+1}^{R}(R/J)\leq \beta_{i,i+1}^{R}(R/J^{\text{lex}}). $$
To prove inequality (11), we will compute upper bounds for \(\beta _{i,i+1}^{R}(R/J^{\text {lex}})\) using Lemma 3. For those we need an upper bound for the number of generators of degree 2 in Jlex. More precisely, we will prove the following claim: C laim: \(\dim _{\mathbb F}(J^{\text {lex}})_{2}\leq b\).
By the definition of the ideals J and Jlex we have
$$ \begin{array}{@{}rcl@{}} \dim_{\mathbb{F}}(R/J^{\text{lex}})_{2}&=&\dim_{\mathbb{F}}(\mathbb{F}[{{\varDelta}}]/{{\varTheta}}\mathbb{F}[{{\varDelta}}])_{2} =h_{2}({{\varDelta}})-h_{1}({{\varDelta}})\\ &\geq& \frac{d-1}{2}h_{1}({{\varDelta}})-h_{1}({{\varDelta}}) =\frac{d-3}{2}(n-d). \end{array} $$
Here, the second equality follows from the injectivity of the multiplication maps in Lemma 18 and the inequality holds by Theorem 7. We conclude
$$\dim_{\mathbb{F}}(J^{\text{lex}})_{2}\leq \binom{n-d}{2}-\frac{d-3}{2}(n-d)=\frac{(n-d)(n-2d+2)}{2}=b,$$
which shows the claim.
Since \(\dim _{\mathbb {F}}(R/J^{\text {lex}})_{1}=n-d-1=\dim _{\mathbb {F}}(R)_{1}\), the ideal Jlex does not contain variables. Using the just proven claim, we conclude that G(Jlex)2 contains at most the b lexicographically largest degree 2 monomials of R, i.e.,
$$ G(J^{\text{lex}})_{2}\subseteq \{u\in \text{Mon}_{2}(R) : u\geq_{\text{lex}} x_{p}x_{q}\}. $$
To simplify the notation, we set M := {u ∈Mon2(R) : u ≥lexxpxq}. Using Lemma 3, we infer
$$ \begin{array}{@{}rcl@{}} \beta_{i,i+1}^{R}(R/J^{\text{lex}})&\leq&\sum\limits_{u\in M}\binom{\max(u)-1}{i-1}\\ &=&\sum\limits_{\ell=1}^{p}\underset{\underset{\max(u)=\ell}{u\in M}}{\sum\limits}\binom{\ell-1}{i-1}+\sum\limits_{\ell=p+1}^{q}\underset{\underset{\max(u)=\ell}{u\in M}}{\sum\limits}\binom{\ell-1}{i-1} +\sum\limits_{\ell=q+1}^{n-d-1}\underset{\underset{\max(u)=\ell}{u\in M}}{\sum\limits}\binom{\ell-1}{i-1}\\ &=& \sum\limits_{\ell=1}^{p}\ell\binom{\ell-1}{i-1}+p\sum\limits_{\ell=p+1}^{q}\binom{\ell-1}{i-1}+(p-1)\sum\limits_{\ell=q+1}^{n-d-1}\binom{\ell-1}{i-1}\\ &=& i\binom{p+1}{i+1}+(p-1)\binom{n-d-1}{i}-p\binom{p}{i}+\binom{q}{i}\\ &=& (p-1)\binom{n-d-1}{i}-\binom{p}{i+1}+\binom{q}{i} \end{array} $$
for all i ≥ 0. This finishes the proof. □
Note that, unlike the bounds from Theorems 2 and 6, the bounds from Theorem 8 do not depend on the sizes of the color classes.
Let Δ be a 3-dimensional balanced pseudomanifold on 12 vertices, with an arbitrary partition of the vertices into color classes. We have \(b=\frac {(n-d)(n-2d+2)}{2}=24\) and x5x6 is the 24th largest degree 2 monomial in variables x1,…,x7. The bounds for \(\beta _{i,i+1}(\mathbb F[{{\varDelta }}])\) provided by Theorem 8 are recorded in the following table.
1 0 24 89 155 154 90 29 4 0
One should compare those with the bounds provided by Lemma 17 for arbitrary (not necessarily balanced) pseudomanifolds
1 0 28 112 210 224 140 48 7 0
While the bounds in the previous table are realized by any stacked 3-sphere on 12 vertices, we do not know if the ones for the balanced case, shown in the upper table, are attained. In the next section, we will see that they are not attained by the balanced analog of stacked spheres.
In view of Theorem 8 a natural question that arises is if one can also bound the entries of the j th row of the Betti table of a balanced pseudomanifold for j ≥ 2. In order for our approach to work, this would require the multiplication maps from Lemma 18 to be injective also for higher degrees; a property that is closely related to Lefschetz properties.
Betti Numbers of Stacked Cross-polytopal Spheres
The aim of this section is to compute the graded Betti numbers of stacked cross-polytopal spheres explicitly. Stacked cross-polytopal spheres can be considered as the balanced analog of stacked spheres, in the sense that both minimize the h-vector among the class of balanced normal pseudomanifolds respectively all normal pseudomanifolds (see [22, Theorem 4.1] and e.g., [8, 21, 35]). For stacked spheres, explicit formulas for their graded Betti numbers were provided by Hibi and Terai [36] and it was shown that they only depend on the number of vertices and the dimension but not on the combinatorial type of the stacked sphere (see also Lemma 17).
We start by introducing some necessary definitions. We denote the boundary complex of the d-dimensional cross-polytope by \(\mathcal C_{d}\). Combinatorially, \(\mathcal C_{d}\) is given as the join of d pairs of disconnected vertices, i.e.,
$$ \mathcal C_{d}:=\{v_{1},w_{1}\}\ast{\cdots} \ast \{v_{d},w_{d}\}. $$
Let n = kd for some integer k ≥ 2. A stacked cross-polytopal (d − 1)-sphere on n vertices is a simplicial complex obtained via the connected sum of k − 1 copies of \(\mathcal {C}_{d}\). We denote by \(\mathcal {S}\mathcal {T}^{\times }(n,d)\) the set of all stacked cross-polytopal (d − 1)-spheres on n vertices.
Observe that \(\mathcal {S}\mathcal {T}^{\times }(2d,d)=\{\mathcal C_{d}\}\), and, as \(\mathcal C_{d}\) is balanced, so is any stacked cross-polytopal sphere. In analogy with the non-balanced setting, for k ≥ 4, there exist stacked cross-polytopal spheres in \(\mathcal {S}\mathcal {T}^{\times }(kd,d)\) of different combinatorial types, as depicted in Fig. 2. Nevertheless, it is easily seen that the f -vector of a stacked cross-polytopal sphere only depends on n and d. In this section, we will show the same behavior for their graded Betti numbers.
Three non simplicially isomorphic spheres in \(\mathcal {S}\mathcal {T}^{\times }(12,3)\)
As a warm-up, we compute the Betti numbers of the boundary complex of the cross-polytope.
Let d ≥ 1. Then, \(\beta _{i,i+j}(\mathbb F[\mathcal C_{d}])=0\) for all i ≥ 0 and j≠i. Moreover,
$$ \beta_{i,2i}(\mathbb F[\mathcal C_{d}])=\binom{d}{i} $$
for all i.
Being generated by d pairwise coprime monomials, the Stanley-Reisner ideal of \(\mathcal C_{d}\) is a complete intersection, and hence it is minimally resolved by the Koszul complex. □
The following immediate lemma will be very useful, in order to derive a recursive formula for the graded Betti numbers of stacked cross-polytopal spheres.
Let d ≥ 3. Let \({{\varDelta }}\in \mathcal {S}\mathcal {T}^{\times }(kd,d)\) be a stacked cross-polytopal sphere on vertex set V and let F be a facet of Δ. Then, for any \(W\subseteq V\),
$$ \widetilde{H}_{j}({{\varDelta}}_{W};\mathbb F)=\widetilde{H}_{j}(({{\varDelta}}\setminus\{F\})_{W};\mathbb F) \quad \text{for all } 0\leq j\leq d-3. $$
The statement is immediate since Δ and Δ ∖{F} share the same skeleta up to dimension d − 2. □
Consider \({{\varDelta }}\in \mathcal {S}\mathcal {T}^{\times }(kd,d)\) and let ◇1,…,◇k− 1 denote the copies of \(\mathcal C_{d}\) from which Δ was constructed. We call a facet F ∈Δ∩◇i extremal if V (◇i) ∖ F∉Δ, and the facet V (◇i) ∖ F is called the opposite of F. Intuitively a facet F of Δ is extremal if removing all the vertices in F from Δ yields a complex Γ ∖{G}, where \({{\varGamma }}\in \mathcal {S}\mathcal {T}^{\times }((k-1)d,d)\) and G is the opposite of F (Fig. 3).
An extremal facet and its opposite
We have the following recursive formulas for Betti numbers of stacked cross-polytopal spheres.
Note that for the case j = 1 the following formula can be deduced from [4, Corollary 3.4]. We report its proof anyway, as the idea is analogous to the case j ≥ 2.
Let n ≥ 3d and \({{\varDelta }}\in \mathcal {S}\mathcal {T}^{\times } (n,d)\). Then,
$$ \beta_{i,i+j}(\mathbb F[{{\varDelta}}])=\left\{\begin{array}{ll} \sum\limits_{\ell=0}^{d}\binom{d}{\ell}\beta_{i-\ell,i-\ell+1}(\mathbb F[{{\varGamma}}])+d\binom{n-2d}{i-1}\\ +\sum\limits_{\ell=1}^{\min\{i,d\}}\binom{d}{\ell}\binom{n-2d}{i+1-\ell} &\text{ if } j=1,\\ \sum\limits_{\ell=0}^{d}\binom{d}{\ell}\beta_{i-\ell,i-\ell+j}(\mathbb F[{{\varGamma}}])+\binom{d}{j}\binom{n-2d}{i-j} &\text{ if } 2\leq j\leq d-2, \end{array} \right. $$
with \({{\varGamma }}\in \mathcal {S}\mathcal {T}^{\times } (n-d,d)\). In particular, the graded Betti numbers of Δ only depend on n and d.
We will compute the graded Betti numbers using Hochster's formula. Let V be the vertex set of Δ and let F be an extremal facet of Δ with opposite G. Then, we can write Δ = (Γ ∖{G}) ∪ (◇∖{G}), where \({{\varGamma }}\in \mathcal {S}\mathcal {T}^{\times }(n-d,d)\) and ◇ is the boundary complex of the d-dimensional cross-polytope on vertex set F ∪ G. In particular, (Γ ∖{G}) ∩ (◇∖{G}) = ∂(G). We now distinguish two cases.
j = 1. Let \(W\subseteq V\). We have several cases:
(a) If \(W\subseteq V({{\varGamma }})\), then ΔW = (Γ ∖{G})W. By Lemma 20, (Γ ∖{G})W (thus ΔW) and ΓW have the same number of connected components and hence \(\widetilde {H}_{0}({{\varDelta }}_{W};\mathbb F)=\widetilde {H}_{0}({{\varGamma }}_{W};\mathbb F)\).
(b) If \(W\subseteq V(\Diamond )\), then it follows as in (b) that \(\widetilde {H}_{0}({{\varDelta }}_{W};\mathbb F)=\widetilde {H}_{0}(\Diamond _{W};\mathbb F)\).
(c) Assume that W ∩ (V (Γ) ∖ G)≠∅ and W ∩ (V (◇) ∖ G)≠∅. Then, ΔW = (Γ ∖{G})W ∪ (◇∖{G})W. If, in addition, W ∩ G = ∅, then this union is disjoint and, using Lemma 20 we conclude that the number of connected components of ΔW equals the sum of the number of connected components of ΓW and ◇W. Thus, as neither ΓW nor ◇W is the empty complex,
$$ \dim_{\mathbb F} \widetilde{H}_{0}({{\varDelta}}_{W};\mathbb F)=\dim_{\mathbb F}\widetilde{H}_{0}({{\varGamma}}_{W};\mathbb F)+\dim_{\mathbb F}\widetilde{H}_{0}(\Diamond_{W};\mathbb F)+1. $$
If W ∩ G≠∅, then the number of connected components of ΔW is one less than the sum of the number of connected components of (Γ ∖{G})W and (◇∖{G})W. In particular, using Lemma 20, we infer
$$ \dim_{\mathbb F} \widetilde{H}_{0}({{\varDelta}}_{W};\mathbb F)=\dim_{\mathbb F}\widetilde{H}_{0}({{\varGamma}}_{W};\mathbb F)+\dim_{\mathbb F }\widetilde{H}_{0}(\Diamond_{W};\mathbb F). $$
Using Hochster's formula, we obtain
$$ \begin{array}{@{}rcl@{}} \beta_{i,i+1}(\mathbb F[{{\varDelta}}])&=&\sum\limits_{W\subseteq V; |W|=i+1}\dim_{\mathbb F }\widetilde{H}_{i-1}({{\varDelta}}_{W};\mathbb F)\\ &=&\underset{\underset{W\cap G\neq\emptyset}{W\subseteq V; |W|=i+1}}{\sum\limits}\left( \dim_{\mathbb F }\widetilde{H}_{0}({{\varGamma}}_{W};\mathbb F)+\dim_{\mathbb F}\widetilde{H}_{0}(\Diamond_{W};\mathbb F)\right)\\ &&+\underset{\underset{W\cap V({{\varGamma}})\neq \emptyset; W\cap V(\Diamond)\neq \emptyset}{W\subseteq V\setminus G; |W|=i+1}}{\sum\limits}\left( \dim_{\mathbb F }\widetilde{H}_{0}({{\varGamma}}_{W};\mathbb F)+\dim_{\mathbb F}\widetilde{H}_{0}(\Diamond_{W};\mathbb F)+1\right)\\ &&+\underset{\underset{|W|=i+1}{W\subseteq V({{\varGamma}})\setminus G}}{\sum\limits}\dim_{\mathbb F}\widetilde{H}_{0}({{\varGamma}}_{W};\mathbb F)+\underset{\underset{|W|=i+1}{W\subseteq V(\Diamond)\setminus G}}{\sum\limits}\dim_{\mathbb F}\widetilde{H}_{0}(\Diamond_{W};\mathbb F). \end{array} $$
For \(W\subseteq V({{\varGamma }})\) (respectively \(W\subseteq V(\Diamond )\)) the term \(\dim _{\mathbb F}\widetilde {H}_{0}({{\varGamma }}_{W};\mathbb F)\) (respectively \(\dim _{\mathbb F}\widetilde {H}_{0}(\Diamond _{W};\mathbb F)\)) appears \(\binom {d}{i+1-|W|}\) (respectively \(\binom {n-2d}{i+1-|W|}\)) times in the previous expression. Moreover, there are \({\sum }_{\ell =1}^{\min \limits \{i,d\}}\binom {d}{\ell }\binom {n-2d}{i+1-\ell }\) (i + 1)-subsets W of V ∖ G with W ∩ V (Γ)≠∅ and W ∩ V (◇)≠∅. This implies
$$ \begin{array}{@{}rcl@{}} \beta_{i,i+1}(\mathbb F[{{\varDelta}}])&=&\sum\limits_{\ell=1}^{i+1}\binom{d}{i+1-\ell}\left( \sum\limits_{W\subseteq V({{\varGamma}}), |W|=\ell}\dim_{\mathbb F}\widetilde{H}_{0}({{\varGamma}}_{W};\mathbb F)\right)\\ &&+\sum\limits_{\ell=1}^{2d}\binom{n-2d}{i+1-\ell}\left( \sum\limits_{W\subseteq V(\Diamond), |W|=\ell}\dim_{\mathbb F}\widetilde{H}_{0}(\Diamond;\mathbb F)\right) \\ &&+\sum\limits_{\ell=1}^{\min\{i,d\}}\binom{d}{\ell}\binom{n-2d}{i+1-\ell}\\ &=&\sum\limits_{\ell=i+1-d}^{i+1}\binom{d}{i+1-\ell}\beta_{\ell-1,\ell}(\mathbb F[{{\varGamma}}])+\sum\limits_{\ell=1}^{\min\{i,d\}}\binom{d}{\ell}\binom{n-2d}{i+1-\ell}\\ &&+\sum\limits_{\ell=1}^{2d}\binom{n-2d}{i+1-\ell}\beta_{\ell-1,\ell}(\mathbb F[\Diamond]), \end{array} $$
where the last equality holds by Hochster's formula. The desired recursion for \(\beta _{i,i+1}(\mathbb F[{{\varDelta }}])\) now follows from a simple index shift.
2 ≤ j ≤ d − 2. Let \(W\subseteq V\). We consider two cases.
(a) If \(W\subseteq V({{\varGamma }})\), then it follows from Lemma 20 that
$$ \widetilde{H}_{j}({{\varDelta}}_{W};\mathbb F)=\widetilde{H}_{j}({{\varGamma}}_{W};\mathbb F) \text{ for } 0\leq j\leq d-3. $$
(b) If \(W\subseteq V(\Diamond )\), then it follows as in (a) that
$$ \widetilde{H}_{j}({{\varDelta}}_{W};\mathbb F)=\widetilde{H}_{j}(\Diamond_{W};\mathbb F) \text{ for } 0\leq j\leq d-3. $$
(c) Assume that W ∩ (V (Γ) ∖ G)≠∅ and W ∩ (V (◇) ∖ G)≠∅. Then, ΔW = (Γ ∖{G})W ∪ (◇∖{G})W. Let 1 ≤ j ≤ d − 3. We have the following Mayer-Vietoris exact sequence
$$ \begin{array}{@{}rcl@{}} {\ldots} \to \underbrace{\widetilde{H}_{j}(\partial(G)_{W};\mathbb F)}_{=0}&\to& \widetilde{H}_{j}(({{\varGamma}}\setminus \{G\})_{W};\mathbb F)\oplus \widetilde{H}_{j}((\Diamond\setminus \{G\})_{W};\mathbb F) \\ &\to& \widetilde{H}_{j}({{\varDelta}}_{W};\mathbb F)\to \underbrace{\widetilde{H}_{j-1}(\partial(G)_{W};\mathbb F)}_{=0}\to \ldots, \end{array} $$
where we use that (Γ ∖{G})W ∩ (◇∖{G})W = (∂(G))W, which has always trivial homology in dimension ≤ d − 3. It follows from (12) combined with Lemma 20 that
$$ \widetilde{H}_{j}({{\varDelta}}_{W};\mathbb F)\cong\widetilde{H}_{j}({{\varGamma}}_{W};\mathbb F)\oplus \widetilde{H}_{j}(\Diamond_{W};\mathbb F) \quad \text{ for } 1\leq j\leq d-3. $$
Using Hochster's formula, we conclude
$$ \begin{array}{@{}rcl@{}} \beta_{i,i+j}(\mathbb F[{{\varDelta}}])&=&\sum\limits_{W\subseteq V, |W|=i+1}\left( \dim_{\mathbb F}\widetilde{H}_{j-1}({{\varGamma}}_{W};\mathbb F)+ \dim_{\mathbb F}\widetilde{H}_{j-1}(\Diamond_{W};\mathbb F)\right)\\ &=&\sum\limits_{\ell=i+j-d}^{i+j}\binom{d}{i+j-\ell}\left( \sum\limits_{W\subseteq V({{\varGamma}}), |W|=\ell}\dim_{\mathbb F}\widetilde{H}_{j-1}({{\varGamma}}_{W};\mathbb F)\right)\\ &&+\sum\limits_{\ell=1}^{2d}\binom{n-2d}{i+j-\ell}\left( \sum\limits_{W\subseteq V(\Diamond), |W|=\ell}\dim_{\mathbb F}\widetilde{H}_{j-1}(\Diamond;\mathbb F)\right)\cr &=&\sum\limits_{\ell=i+j-d}^{i+j}\binom{d}{i+j-\ell}\beta_{\ell-j,\ell}(\mathbb F[{{\varGamma}}]) +\sum\limits_{\ell=1}^{2d}\binom{n-2d}{i+j-\ell}\beta_{\ell-j,\ell}(\mathbb F[\Diamond])\\ &=&\sum\limits_{\ell=0}^{d}\binom{d}{\ell}\beta_{i-d+\ell,i-d+\ell+j}(\mathbb F[{{\varGamma}}])+\binom{n-2d}{i-j}\binom{d}{j}, \end{array} $$
where the second equality follows, as in Case 1, by a simple counting argument and the last equality follows from Lemma 19.
The statement of the "In particular"-part follows directly by applying the recursion iteratively, and from \(\mathcal {S}\mathcal {T}^{\times }(2d,d)=\{\mathcal C_{d}\}\). □
We remark that due to graded Poincaré duality the graded Betti numbers of any stacked cross-polytopal sphere \({{\varDelta }}\in \mathcal {S}\mathcal {T}^{\times }(n,d)\) exhibit the following symmetry:
$$ \beta_{i,i+j}(\mathbb{F}[{{\varDelta}}])=\beta_{n-d-i,n-i-j}(\mathbb{F}[{{\varDelta}}]). $$
This in particular implies \(\beta _{n-d,n}(\mathbb {F}[{{\varDelta }}])=1\) and \(\beta _{i,i+d}(\mathbb {F}[{{\varDelta }}])=0\) for 0 ≤ i < n − d. Moreover, also \(\beta _{i,i+d-1}(\mathbb {F}[{{\varDelta }}])\) can be computed using the recursion from Theorem 9 (for the linear strand).
In order to derive explicit formulas for the graded Betti numbers of a stacked cross-polytopal sphere, we need to convert the recursive formula of Theorem 9 into a closed expression.
Theorem 10
Let d ≥ 3, k ≥ 2 and let \({{\varDelta }}\in \mathcal {S}\mathcal {T}^{\times }(kd,d)\) be a stacked cross-polytopal sphere. Then, \(\beta _{0,0}(\mathbb {F}[{{\varDelta }}])=\beta _{(k-1)d,kd}(\mathbb {F}[{{\varDelta }}])=1\) and for i ≥ 0
$$\beta_{i,i+j}(\mathbb{F}[{{\varDelta}}])=\left\{\begin{array}{ll} (k-2)\binom{d(k-1)}{i+1}-(k-1)\binom{d(k-2)}{i+1}+ d(k-1)\binom{d(k-2)}{i-1}, & j=1,\\ (k-1)\binom{d}{j}\binom{d(k-2)}{i-j}, & 2\leq j\leq d-2,\\ (k-2)\binom{d(k-1)}{i-1}-(k-1)\binom{d(k-2)}{i-d-1} + d(k-1)\binom{d(k-2)}{i-d+1}, & j=d-1. \end{array}\right.$$
We proof the claim by induction on k.
For k = 2, the first line above equals d if i = 1 and 0 otherwise. Similarly, the second line equals \(\binom {d}{i}\) if j = i and 0 otherwise. The claim for k = 2 now follows from Lemma 19.
Let k ≥ 3 and let \({{\varDelta }} \in \mathcal {S}\mathcal {T}^{\times }(kd,d)\). We first show the case j = 1.
Using Theorem 9 and then the induction hypothesis, we conclude
$$ \begin{array}{@{}rcl@{}} \beta_{i,i+1}(\mathbb{F}[{{\varDelta}}])&=&\sum\limits_{\ell=0}^{\min\{i,d\}}\binom{d}{\ell}\beta_{i-\ell,i-\ell+1}(\mathbb{F}[{{\varGamma}}])+d\binom{n-2d}{i-1} +\sum\limits_{\ell=1}^{\min\{i,d\}}\binom{d}{\ell}\binom{n-2d}{i+1-\ell}\\ &=& (k-3)\sum\limits_{\ell=0}^{\min\{i,d\}}\binom{d}{\ell}\binom{d(k-2)}{(i+1)-\ell} -(k-2)\sum\limits_{\ell=0}^{\min\{i,d\}}\binom{d}{\ell}\binom{d(k-3)}{(i+1)-\ell} \\ &&+d(k-2)\sum\limits_{\ell=0}^{\min\{i,d\}}\binom{d}{\ell}\binom{d(k-3)}{(i-1)-\ell}+d\binom{d(k-2)}{i-1}\\ &&+\sum\limits_{\ell=1}^{\min\{i,d\}}\binom{d}{\ell}\binom{d(k-2)}{(i+1)-\ell}, \end{array} $$
where \({{\varGamma }}\in \mathcal {S}\mathcal {T}\times ((k-1)d,d)\). We now assume that \(\min \limits \{i,d\}=d\). We notice that in (14), we can shift the upper summation indices to i + 1 in the first 2 sums and to i − 1 in the third sum. Using Vandermonde identity we obtain
$$ \begin{array}{@{}rcl@{}} \beta_{i,i+1}(\mathbb{F}[{{\varDelta}}])&=&(k-3)\binom{d(k-1)}{i+1}-(k-2)\binom{d(k-2)}{i+1}+d(k-2)\binom{d(k-2)}{i-1}\\ &&+d\binom{d(k-2)}{i-1}+\left( \binom{d(k-1)}{i+1}-\binom{d(k-2)}{i+1}\right)\\ &=&(k-2)\binom{d(k-1)}{i+1}-(k-1)\binom{d(k-2)}{i+1}+d(k-1)\binom{d(k-2)}{i-1}. \end{array} $$
If i < d (thus \(\min \limits \{i,d\}=i\)), then the same computation as above with an additional summand of \(-(k-3)\binom {d}{i+1}\), \((k-2)\binom {d}{i+1}\) and \(-\binom {d}{i+1}\) for the first, second and fourth sum, respectively, shows the formula for the first line.
We now show the case 1 < j ≤ d − 2.
Applying Theorem 9 and the induction hypothesis, we obtain
$$ \begin{array}{@{}rcl@{}} \beta_{i,i+j}(\mathbb{F}[{{\varDelta}}])&=&\sum\limits_{\ell=0}^{\min\{i,d\}}\binom{d}{\ell}\beta_{i-\ell,i-\ell+j}(\mathbb{F}[{{\varGamma}}])+\binom{d}{j}\binom{d(k-2)}{i-j}\\ &=&\sum\limits_{\ell=0}^{\min\{i,d\}}\binom{d}{\ell}(k-2)\binom{d}{j}\binom{d(k-3)}{i-j-\ell}+\binom{d}{j}\binom{d(k-2)}{i-j}\\ &=&(k-2)\binom{d}{j}\sum\limits_{\ell=0}^{\min\{i-j,d\}}\binom{d}{\ell}\binom{d(k-3)}{i-j-\ell}+\binom{d}{j}\binom{d(k-2)}{i-j}\\ &=&(k-2)\binom{d}{j}\binom{d(k-2)}{i-j}+\binom{d}{j}\binom{d(k-2)}{i-j}\\ &=&(k-1)\binom{d}{j}\binom{d(k-2)}{i-j}, \end{array} $$
where \({{\varGamma }}\in \mathcal {S}\mathcal {T}^{\times }((k-1)d,d) \) and the fourth equality follows from Vandermonde's identity after observing that shifting the upper index of the sum to i − j does not change the sum.
The statement in the last line (j = d − 1) follows from graded Poincaré duality (see (13)). □
For stacked cross-polytopal 3-spheres on 12 vertices Theorem 10 yields the following Betti numbers for the linear strand:
1 0 24 80 116 88 36 8 1 0
If we compare them with the bounds for the Betti numbers of a 3-dimensional balanced normal pseudomanifold on 12 vertices from Theorem 8, displayed in the next table, we see that they are smaller in almost all places.
In light of Lemma 17 and the analogy between stacked and cross-polytopal stacked spheres, the previous example suggests the following conjecture:
Conjecture 1
Let Δ be a (d − 1)-dimensional balanced normal pseudomanifold, with d ≥ 4 and let f0(Δ) = kd, for some integer k ≥ 2. Then
$$\beta_{i,i+1}(\mathbb F[{{\varDelta}}])\leq \beta_{i,i+1}(\mathbb F[{{\varGamma}}])$$
for \({{\varGamma }}\in \mathcal {S}\mathcal {T}^{\times }(kd,d)\), and for every i ≥ 0.
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We would like to thank Giulio Caviglia for directing us to the results on lex-plus-squares ideals by Mermin, Peeva and Stillman. This led to the content of Section 5. We are also grateful to Satoshi Murai for pointing out [14, Corollary 1.4], which simplified and shortened the proof of Lemma 11.
Open Access funding enabled and organized by Projekt DEAL. Both authors were supported by the German Research Council DFG GRK-1916.
Universität Osnabrück, Fakultät für Mathematik, Albrechtstraße 28a, 49076, Osnabrück, Germany
Martina Juhnke-Kubitzke & Lorenzo Venturello
Martina Juhnke-Kubitzke
Lorenzo Venturello
Correspondence to Lorenzo Venturello.
Proof Proof of Lemma 9
Let M be the n × n upper triangular matrix obtained by listing the degree 2 monomials in variables x1,…,xn in decreasing lexicographic order from left to right and top to bottom
$$ M=\begin{bmatrix} {x_{1}^{2}} & x_{1}x_{2} & {\dots} & x_{1}x_{n} \\ 0 & {x_{2}^{2}} & {\dots} & x_{2}x_{n} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ 0 & 0 & {\dots} & {x_{n}^{2}} \end{bmatrix}.$$
From this ordering, it is easily seen, that, if xpxq (with p < q) is the b th largest degree 2 monomial in lexicographic order, then
$$ n-p=\max\left\{s\in \mathbb N : {\sum}_{\ell=1}^{s}\ell\leq\binom{n+1}{2}-b\right\}. $$
As \(s= -\frac {1}{2}+\frac {\sqrt {4n(n+1)+1-8b}}{2}\) is the unique non-negative solution to the equation
$$ (s+1)s/2=(n+1)n/2 -b, $$
$$ p=n-\left\lfloor -\frac{1}{2}+\frac{\sqrt{4n(n+1)+1-8b}}{2}\right\rfloor. $$
Looking at the matrix M, we deduce that the index q, (i.e., the column index of xpxq in M) is given by
$$ \begin{array}{@{}rcl@{}} q&=&b-\sum\limits_{\ell=1}^{p-1}(n+1-\ell) +(p-1) = b-(p-1)(n+1)+\frac{p(p-1)}{2}+(p-1)\\ &=&b+ \frac{(p-1)(-2-2n+p+2)}{2}=b+\frac{(p-1)(p-2n)}{2}. \end{array} $$
The claim follows. □
Proof Proof of Lemma 14
As in the proof of Lemma 9 it is easy to see that, if xpxq (with p < q) is the b th largest squarefree degree 2 monomial, then
$$ n-p=\max\left\{s\in \mathbb N : \sum\limits_{\ell=1}^{s}\ell\leq \binom{n}{2}-b\right\}+1. $$
Since \(s=-\frac {1}{2}+\frac {\sqrt {4n(n-1)-8b+1}}{2}\) is the unique non-negative solution to the equation
$$ (s+1)s/2=n(n-1)/2-b, $$
we infer that \(p= n-1+\left \lfloor \frac {1}{2}-\frac {\sqrt {4n(n-1)-8b+1}}{2}\right \rfloor \). As \(q=b-{\sum }_{\ell =1}^{p-1}(n-\ell )+p\), the claim follows from a straightforward computation. □
Juhnke-Kubitzke, M., Venturello, L. Graded Betti Numbers of Balanced Simplicial Complexes. Acta Math Vietnam 46, 839–871 (2021). https://doi.org/10.1007/s40306-021-00449-8
Issue Date: December 2021
Simplicial complex
Stanley-Reisner ring
Graded Betti numbers
Lex (plus powers) ideals
Mathematics Subject Classification (2010) | CommonCrawl |
On the measurement operator for scattering in layered media
On the set of metrics without local limiting Carleman weights
February 2017, 11(1): 65-85. doi: 10.3934/ipi.2017004
Reducing spatially varying out-of-focus blur from natural image
Faming Fang 1, , Fang Li 2, and Tieyong Zeng 3,,
Shanghai Key Laboratory of Multidimensional Information Processing, Department of Computer Science, East China Normal University, Shanghai, China
Department of Mathematics, East China Normal University, Shanghai, China
Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China
* Corresponding author: Tieyong Zeng
Received December 2015 Revised September 2016 Published January 2017
Figure(11) / Table(2)
In this paper, we focus on the challenging problem of removing the spatially varying out-of-focus blur from a single natural image. We first propose an effective method to estimate the blur map by the total variation refinement on Hölder coefficient, then discuss the properties of the corresponding kernel matrix. A tight-frame based energy functional, whose minimizer is related to the optimal defocus result, is thus built. For tackling functional more efficiently, we describe the numerical procedure based on an accelerated primal-dual scheme. To verify the effectiveness of our method, we compare it with some state-of-the-art schemes using both synthesized and natural images. Experimental results demonstrate that the proposed method performs better than the compared methods.
Keywords: Blur map, debluring, framelet, out-of-focus, spatially-varying, variational method.
Mathematics Subject Classification: Primary: 68U10, 65K10, 35A15, 91-08; Secondary: 62H35.
Citation: Faming Fang, Fang Li, Tieyong Zeng. Reducing spatially varying out-of-focus blur from natural image. Inverse Problems & Imaging, 2017, 11 (1) : 65-85. doi: 10.3934/ipi.2017004
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Figure 1. The spatially-varying blurring images.
Figure 2. The distributions of the NHC for in-focus (red) and blur images (blue).
Figure 1(a) ($800\times 600$ pixels): (a) the rough blur map $\boldsymbol{\tilde{\sigma}}$; (b) the refined blur map $\boldsymbol{{\sigma}}$.">Figure 3. The blur maps of Figure 1(a) ($800\times 600$ pixels): (a) the rough blur map $\boldsymbol{\tilde{\sigma}}$; (b) the refined blur map $\boldsymbol{{\sigma}}$.
52], SHP [39] and proposed methods; (f) the ground truth of blur map; (g) corresponding profiles of 1-st line for (c)-(f).">Figure 4. (a) a sharp image (size:100$\times$ 400); (b) a blur version of (a) (the blur levels are 1 and 4 for 171th-230th and 271th-330th columns, respectively); (c)-(e) the blur maps genrated by ZS [52], SHP [39] and proposed methods; (f) the ground truth of blur map; (g) corresponding profiles of 1-st line for (c)-(f).
Figure 5. (a) and (b) are original images; (c) and (d) are two blur maps (range: 1-5); (e): the blurred version of (a) using (c) as blur map; (f): the blurred version of (a) using (d) as blur map; (g): the blurred version of (b) using (c) as blur map; (h): the blurred version of (b) using (d) as blur map.
Figure 5(e)/(f) by Matlab, XJ [49], CJLS [8], SHP [39], and proposed methods, respectively.">Figure 6. Synthesised experiments: (A1-A5)/(B1-B5) are the deblurred results (with zoomed regions) of Figure 5(e)/(f) by Matlab, XJ [49], CJLS [8], SHP [39], and proposed methods, respectively.
Figure 5(g)/(h) by Matlab, XJ [49], CJLS [8], SHP [39], and proposed methods, respectively.">Figure 7. Synthesised experiments: (A1-A5)/(B1-B5) are the deblurred results (with zoomed regions) of Figure 5(g)/(h) by Matlab, XJ [49], CJLS [8], SHP [39], and proposed methods, respectively.
Figure 8. PSNR, SSIM, and SI values corresponding to each figure.
49], CJLS [8], SHP [39], and the proposed methods.">Figure 9. Original image and its deblurred results with zoomed regions. (a) The original image (RGB, $296\times 877$ pixels); (b)-(f) the results of Matlab, XJ [49], CJLS [8], SHP [39], and the proposed methods.
49], CJLS [8], SHP [39], and the proposed methods.">Figure 10. Original image and its deblurred results with zoomed regions. (a) The original image (RGB, $800\times 600$ pixels); (b)-(f) the results of Matlab, XJ [49], CJLS [8], SHP [39], and the proposed methods.
Table 1. The MAE value of each blur map generated by ZS [52], SHP [39] and Proposed methods.
ZS [52] SHP [39] Proposed
Figure 4(b) 1.3799 0.6495 0.5308
Figure 5(e) 0.4043 0.3753 0.3751
Figure 5(f) 0.3959 0.3950 0.3684
Figure 5(g) 0.4811 0.4455 0.3064
Figure 5(h) 0.4823 0.4756 0.3418
Table 2. Time comparison with other methods (second).
Figure 5(e) Figure 5(f) Figure 5(g) Figure 5(h) Figure 9(a) Figure 10(a) Figure 11(a)
image size 300 × 286 300 × 286 265 × 300 265 × 300 200 × 300 800 × 600 534 × 800
Matlab 1.18 1.22 1.00 1.17 2.03 6.70 6.00
XJ [49](C) 6.45 6.55 6.36 6.35 11.43 18.01 16.95
CJLS [8] 86.59 83.74 88.27 89.84 194.95 780.30 693.77
SHP [39] 70.21 69.40 72.26 71.55 126.37 187.01 168.68
Proposed 38.33 37.60 34.98 36.46 93.45 137.93 141.32
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\begin{document}
\title{On the spectral decomposition of affine Hecke algebras}
\author{Eric M. Opdam} \address{Korteweg-De Vries Institute for Mathematics, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands.} \thanks{The author would like to thank Erik van den Ban, Patrick Delorme, Gerrit Heckman, Mark Reeder, Klaas Slooten and Atsuko Yamamoto for their helpful comments and remarks} \email{[email protected]} \date{\today} \subjclass{20C08, 22D25, 22E35, 43A32}
\begin{abstract} An affine Hecke algebra $\mathcal{H}$ contains a large abelian subalgebra $\mathcal{A}$ spanned by the Bernstein-Zelevinski-Lusztig basis elements $\theta_x$, where $x$ runs over (an extension of) the root lattice. The center $\mathcal{Z}$ of $\mathcal{H}$ is the subalgebra of Weyl group invariant elements in $\mathcal{A}$. The natural trace (``evaluation at the identity'') of the affine Hecke algebra can be written as integral of a certain rational $n$-form (with values in the linear dual of $\mathcal{H}$) over a cycle in the algebraic torus $T=spec(\mathcal{A})$. This cycle is homologous to a union of ``local cycles''. We show that this gives rise to a decomposition of the trace as an integral of positive local traces against an explicit probability measure on the spectrum $W_0\backslash T$ of $\mathcal{Z}$. From this result we derive the Plancherel formula of the affine Hecke algebra. \end{abstract} \maketitle \tableofcontents \section{Introduction} In this paper we will discuss the spectral decomposition of an affine Hecke algebra $\mathcal{H}$ defined over $\mathbb{C}$ or, more precisely, of a natural positive trace $\tau$ defined on $\mathcal{H}$. In the standard basis of $\mathcal{H}$, $\tau$ is simply defined by $\tau(T_e)=1$ and $\tau(T_w)=0$ if $w\not=e$. In addition, $\mathcal{H}$ comes equipped with the natural $*$-operator $T_w^*=T_{w^{-1}}$. This defines a pre-Hilbert structure on $\mathcal{H}$ by $(x,y):=\tau(x^*y)$. The regular representation $\lambda\times\rho$ extends to the Hilbert completion $\mathfrak{H}$ of $\mathcal{H}$, and by the spectral decomposition of $\tau$ we mean the decomposition of $\mathfrak{H}$ in irreducible $*$-representations of $\mathcal{H}\times\mathcal{H}$. By classical results on the decomposition of traces on $C^*$-algebras of type I (see for example \cite{dix2}), this is equivalent to the problem of decomposing the trace $\tau$ as a superposition of irreducible characters of $*$-representations of $\mathcal{H}$. We will call this decomposition the Plancherel decomposition of $\mathcal{H}$, and the associated positive measure on the spectrum $\hat\mathcal{H}$ will be called the Plancherel measure.
In the case of a Hecke algebra of finite type we have the well known decomposition formula \begin{equation}\label{eq:finite} \tau=\frac{1}{P}\sum \chi_\pi d_\pi, \end{equation} where $P$ denotes the Poincar\'e polynomial of $\mathcal{H}$ (we assume that $P\not=0$), $\pi$ runs over the finite set of irreducible representations of $\mathcal{H}$, $\chi_\pi$ denotes the corresponding character of $\pi$, and $d_\pi$ is the generic degree of $\pi$. The formula we are going to discuss in the present paper is the affine analog of equation (\ref{eq:finite}).
This paper is the sequel to \cite{EO}, where we made a basic study of the Eisenstein functionals of an affine Hecke algebra $\mathcal{H}$. These Eisenstein functionals are holomorphic functions of a spectral parameter $t\in T$, where $T$ is a complex $n$-dimensional algebraic torus naturally associated to $\mathcal{H}$. In \cite{EO}, we derived a representation of $\tau$ as the integral of the normalized Eisenstein functional times the holomorphic extension of the Haar measure of the compact form of $T$, against a certain ``global $n$-cycle'' (a coset of the compact form of $T$) in $T$. The kernel of this integral is a meromorphic $(n,0)$-form on $T$.
The present paper takes off from that starting point, and refines step-by-step the above basic complex function theoretic representation formula for $\tau$ until we reach the level of the spectral decomposition of $\tau$, extended to a tracial state on the $C^*$-algebra hull $\mathfrak{C}$ of the regular representation of $\mathcal{H}$ (Main Theorem \ref{thm:mainp}). On the simpler level of the spherical or the anti-spherical subalgebra, a similar approach can be found in \cite{Mat} and \cite{HOH}. In the case of the spherical algebra one should of course also mention the classical work \cite{Ma}, although the point of view is different there, and based on analysis on a reductive $p$-adic group.
\subsubsection{Motivation} There are various motivations for the study of the spectral resolution of $\tau$. A natural application of such a decomposition is the p-adic analog of the Howlett-Lehrer theory for finite reductive groups, see for instance \cite{M}, \cite{Lu3}, \cite{HOH}, \cite{Re0} and \cite{Re}. Here one considers an affine Hecke algebra which arises as the centralizer algebra of a certain induced representation of a p-adic reductive group $G$. The Plancherel measure of $\mathcal{H}$ can be interpreted as the Plancherel measure of $G$ on a part of $\hat{G}$ in this situation. In view of this application it is important that we obtain an (almost) explicit product formula for the Plancherel measure (see Main Theorem \ref{thm:mainp}). In addition we characterize exactly which characters $W_0r\in W_0\backslash T$ of the spectrum of the center $\mathcal{Z}$ of $\mathcal{H}$ support a discrete series representation of $\mathcal{H}$ (see Theorem \ref{thm:support}). These are the so-called ``residual points'' (see Appendix \ref{sub:defn}). This result was recently applied to the representation theory of reductive p-adic groups, see \cite{Hei}.
Another motivation for this approach is that it sets the stage for the definition of a Schwartz-completion $\mathfrak{S}$ of $\mathcal{H}$ (see Subsection \ref{sub:normunif}), and for the subsequent study of the Fourier transform and its inversion on the level of this Fr\'echet algebra (joint work with Patrick Delorme, to appear). This is related to the study of the K-theory and the cyclic homology of $\mathcal{H}$ and its reduced $C^*$-algebra $\mathfrak{C}$, in the spirit of \cite{was1}, \cite{was2}. This point of view is particularly interesting for non-simply laced cases, since it is natural to expect that the K-theory does not depend on the parameters $q(s)$ of the Hecke algebra. On the other hand, in the ``generic case'' these matters seem to be considerably easier to understand than in the ``natural cases'', where the logarithms of the parameters have rational relations. In view of this, it is important that we allow the parameters $q(s)$ of the affine Hecke algebra to assume any real value $>0$. \subsubsection{Outline}\label{subsub:out} It may be helpful to give the reader a rough outline of this paper, and an indication of the guiding principles in the various stages. We also refer the reader to Subsection \ref{sub:out} for a more detailed outline and formulation of the main results (see in particular \ref{sl2}, \ref{sl3} for the results on the Plancherel measure).
(0). The starting point of the present paper is the definition of the Eisenstein functional of the affine Hecke algebra, in \cite{EO}. These functionals are matrix coefficients of minimal principal series modules. The study of their intertwining operators led to a representation of the trace $\tau$ of $\mathcal{H}$, as an integral of a certain rational kernel over a ``global'' cycle (see formula (\ref{eq:basic})).
(1). In section \ref{sect:pre} we recall the definition and first properties of (extended) affine Hecke algebras, we collect some basic facts from the theory of $C^*$-algebras, and we adapt certain classical results from the representation theory of reductive groups to our context. We conclude this section with a discussion of the properties of the natural map $p_z:\hat\mathfrak{C}\to\operatorname{Spec}(\mathcal{Z})$, the spectrum of the center $\mathcal{Z}$ of $\mathcal{H}$, in view of the main results of this paper.
(2). The study of the residues of the rational kernel for $\tau$ as in formula (\ref{eq:basic}), in Section \ref{sec:loctau}. This involves a general (but basic) scheme for the calculation of multivariable residues. After symmetrization over the Weyl group, the result is a decomposition of $\tau$ as an integral of local tracial states against an explicit probability measure on the spectrum $\operatorname{Spec}(Z)=W_0\backslash T$. The main tools in this process are the {\it positivity} of $\tau$, and the geometric properties of the collection of residual cosets (Appendix \ref{sub:defn}). This step is called the ``localization of the trace $\tau$''.
(3). The local trace (as was mentioned in (2)) defined at an orbit $W_0t\subset T$, arises as an integral of the Eisenstein kernel over a ``local cycle'' which is defined in an arbitrarily small neighborhood of the orbit $W_0t$. This gives a natural extension of the local trace to localizations of the Hecke algebra itself (localization as a module over the sheaf of analytic functions on $W_0\backslash T$).
The analytic localization of the Hecke algebra has a remarkable structure discovered by Lusztig in \cite{Lu}. This part of the paper is not self-contained, but draws heavily on the paper \cite{Lu}. By Lusztig's wonderful structure theorem we can now investigate the local traces. We find in this way that everything is organized in accordance with Harish-Chandra parabolic induction (the philosophy of cusp forms). The local traces at residual cosets give rise to finite dimensional Hilbert algebras which we call ``residual algebras''. Their generic structure reduces (via parabolic induction) to the case of the residual algebras at ``residual points'' of certain semisimple subquotients of the Hecke algebra (see Subsection \ref{sub:gene}).
These matters concerning the localization of $\mathcal{H}$ are studied in Section \ref{sect:loc}, leading to the main result Theorem \ref{thm:mainp}. The support of the Plancherel measure and the Plancherel density are expressed in terms of the discrete series of Levi subquotient algebras of $\mathcal{H}$, and their Plancherel masses (formal dimensions).
(4). At this point, two essential problems remain: The classification of the discrete series representations, and the determination of their formal dimensions. Regarding the first problem, we have determined the orbits $W_0r\in W_0\backslash T$ which arise as the central character of a discrete series representation of $\mathcal{H}$ in Theorem \ref{thm:support}. We have no further information to offer on this problem in this paper.
Section \ref{sec:inv} is devoted to the second of these problems. In order to explain our approach, let $\nu(\{W_0r\})$ denote the Plancherel mass of the central character $W_0r$ with respect to the restriction of the tracial state $\tau$ to the center $\mathcal{Z}$ of $\mathcal{H}$. In Subsection \ref{sub:chiA} we find that the formal dimension of an irreducible discrete series representation $\delta$ whose central character is a certain residual point $W_0r$, is equal to the product of $\nu(\{W_0r\})$ and a certain positive real number $d_\delta>0$ (called ``the residual degree'' of $\delta$) depending on $\delta$ (see Corollary \ref{cor:fdim} and Theorem \ref{thm:nu}). (These residual degrees are normalized such that $\sum\operatorname{dim}(\delta)d_\delta=1$, where the sum runs over all square integrable $\delta$ whose central character is $W_0r$).
{\it The factor $\nu(\{W_0r\})$ is a certain explicit product (explicit up to a nonzero rational multiple) of rational functions evaluated at the central character $W_0r$.} The problem that arises here is that we have not much information about the behaviour of the individual ``residual degrees'' $d_\delta$ as functions of the parameters $q(s)$.
In Section \ref{sec:inv} we resolve this matter. If we write the labels $q(s)>0$ in the form $q(s)={\bf q}^{f_s}$ for certain real numbers $f_s$ and ${\bf q}>1$, we prove that the residual Hilbert algebras are {\it independent} of the base ${\bf q}>1$. In other words, the constants $d_\delta>0$ are {independent} of ${\bf q}>1$. This proves that {\it all irreducible discrete series representations of $\mathcal{H}$ associated with a central character $W_0r$, have a formal dimension which is proportional to the mass $\nu(\{W_0r\})$, with a positive real ratio of proportionality which is independent of ${\bf q}>1$}.
In addition we conjecture that the positive reals $d_\delta$ are actually rational numbers (cf. Conjecture \ref{rem:ell}). This conjecture is subject of joint work in progress with Mark Reeder and Antony Wasserman.
(5). In Appendix \ref{sub:defn} we study the geometry of the set of singularities with maximal pole order of the rational $n$-form \begin{equation}\label{eq:simple} \frac{dt}{c(t)c(t^{-1})} \end{equation} on $T$. This leads to the notion ``residual coset'', which is crucially important for the understanding of the residues of the kernel for $\tau$ in (\ref{eq:basic}). It is analogous to the notion of residual subspace which was introduced in \cite{HOH0}. The collection of these cosets can be classified, and from this classification we verify certain important geometric properties of this collection. These geometric facts are used in Section \ref{sec:loctau} (especially in Subsection \ref{sub:chiA}) to establish regularity properties of the residues to be considered in this paper. \subsubsection{Residue calculus} Let us make some remarks about the ``residue calculus'' on which much of this paper is ultimately based. At the heart of it lies the elementary Lemma \ref{thm:resbasic}, which is an adapted version of Lemma 3.1 of \cite{HOH0}. This lemma roughly states that on a complex torus $T$, any linear functional $\tau$ on the ring of Laurent polynomials $\mathbb{C}[T]$ of the form \begin{equation} \tau(f)=\int_{t_0T_u}f\omega \end{equation} where $\omega$ is a rational $(n,0)$-form whose pole set is a union of cosets of codimension $1$ subtori of $T$, can be represented by a unique collection of ``local distributions'' living on certain cosets of the compact form $T_u$ of $T$, and satisfying certain support conditions.
In the context of the representation theory of $\mathcal{H}$, this lemma becomes remarkably efficient. We apply the lemma to linear functionals of the form $a\to\tau(ah)$, where $h\in \mathcal{H}$ and $a\in\mathcal{A}$, a maximal abelian subalgebra of $\mathcal{H}$, using formula (\ref{eq:basic}). At this stage we symmetrize the ``local distributions'' for the action of $W_0$. Using the elementary notion of ``approximating sequence'' (see Lemma \ref{lem:approx}) it is not hard to show that the symmetrized local distributions inherit the positivity of $\tau$. This implies easily that these symmetrized distributions are in fact compactly supported measures on the spectrum $W_0\backslash T$ of the center $\mathcal{Z}=\mathcal{A}^{W_0}$ of $\mathcal{H}$, with values in the positive traces on $\mathcal{H}$ (see Corollary \ref{cor:exten}). This means that all higher order terms in the local distributions cancel out by the symmetrization by $W_0$.
In addition it follows by positivity that all measures are absolutely continuous with respect to a scalar measure $\nu$, the Plancherel measure of the center $\mathcal{Z}$ of $\mathcal{H}$. In fact $\nu$ is obtained by evaluation of the symmetrized local distributions at $1\in\mathcal{H}$. Fortunately the poles of formula (\ref{eq:basic}) simplify to the poles of expression (\ref{eq:simple}) by this evaluation. In this way we see that the contributions at non-residual, quasi-residual cosets must cancel. We can bring into play the geometric properties of the residual subspaces now, established in Appendix \ref{sub:defn}, to prove that the Plancherel measure is smooth on its support, that the local traces are tempered, and that the local traces at discrete mass points of $\nu$ are finite linear combinations of discrete series characters.
We get in this way a decomposition of $\tau$ as a superposition of positive ``local traces'', which is an important step towards the Plancherel decomposition of $\mathcal{H}$.
At the time of the writing of the paper \cite{HOH0}, working on the quantum theory of a certain exactly solvable $n$-particle systems, we were not aware of the already existing results in the spirit of the above lemma on existence and uniqueness of residue distributions. But we should certainly mention here the basic work of Langlands \cite{L}, where residues of Eisenstein series are studied in the theory of automorphic forms for reductive groupes. We also mention the work of Arthur \cite{A}, \cite{A2} in this direction. Langlands' work \cite{L} was elucidated by Moeglin and Waldspurger in \cite{MW}. Langlands' result on existence of ``residue data'' can be found in Theorem V.2.2, and on uniqueness of ``residue data'' in the formulation of Theorem V.3.13(i) of \cite{MW}. It should be pointed out however that these results are of a different nature than our basic Lemma \ref{thm:resbasic}. Lemma \ref{thm:resbasic} is a (very elementary) general result in distribution theory, which has nothing to do with group theory. On the other hand, the above results in \cite{MW} are formulated with already symmetrized ``residue data'', using intertwining operators. In order to even formulate a uniqueness property in this setting, one first needs to show rather deep statements on the holomorphic continuation of certain residue sums of Eisenstein series (see V.3.2 of \cite{MW}).
More recently, inspired by the approach in \cite{HOH0}, Van den Ban and Schlichtkrull \cite{BS1} extended the method by allowing for so-called residue weights. In this generality they applied the residue calculus in their proof of the Plancherel formula for semisimple symmetric spaces. \section{Preliminaries and description of results}\label{sect:pre} The algebraic background for our analysis was discussed in the paper \cite{EO}. The main result of that paper is an inversion formula (see equation (\ref{eq:basic})) which will be the starting point in this paper. The purpose of this section is to define the affine Hecke algebra $\mathcal{H}$ and to review the relevant notations and concepts involved in the above result. Moreover we introduce a $C^*$-algebra hull $\mathfrak{C}$ of $\mathcal{H}$, which will be the main object of study in this paper. Finally we will give a more precise outline of the results in the paper. We refer the reader to \cite{Lu} and \cite{EO} for a more systematic introduction of the basic algebraic notions. \subsection{The affine Weyl group and its root datum} A reduced root datum is a 5-tuple ${\mathcal R}=(X,Y,R_0,R_0^\vee,F_0)$ \index{R@$\mathcal R$, root datum}, where $X$\index{X@$X,Y$, lattices} and $Y$ are free abelian groups with perfect pairing over ${\mathbb Z}$, $R_0\subset X$\index{R2@$R_0\subset X$, reduced integral root system} is a reduced integral root system, $R_0^\vee\subset Y$\index{R2@$R_0^\vee\subset Y$, coroot system} is the dual root system of coroots of $R_0$, and $F_0\subset R_0$\index{F@$F_0\subset R_0$, simple roots of $R_0$} is a basis of simple roots. Each element $\alpha\in R_0$ determines a reflection $s_\alpha\in\operatorname{GL}(X)$ by \begin{equation}\index{s@$s_\alpha$, reflection in $\alpha$} s_\alpha(x)=x-x(\alpha^\vee)\alpha. \end{equation} The group $W_0$\index{W2@$W_0$, Weyl group of $R_0$} in $\operatorname{GL}(X)$ generated by the $s_\alpha$ is called the Weyl group. As is well known, this group is in fact generated by the set $S_0$\index{S@$S_0$, simple reflections of $W_0$} consisting of the reflections $s_\alpha$ with $\alpha\in F_0$. The set $S_0$ is called the set of simple reflections in $W_0$.
By definition the affine Weyl group $W$ \index{W@$W$, affine Weyl group} associated with a reduced root datum $\mathcal R$ is the group $W=W_0\ltimes X$. This group $W$ naturally acts on the set $X$.
We choose once and for all a rational, symmetric, positive definite, $W_0$-invariant pairing $\langle\cdot,\cdot\rangle$ \index{<@$\langle\cdot,\cdot\rangle$!a@rational inner product on $X$,$Y$} on $\mathbb{Q}\otimes Y$. This defines a $W_0$ pairing on the Euclidean spaces $\mathfrak{t}:=\mathbb{R}\otimes Y$\index{T4@$\mathfrak{t}$, Lie algebra $\operatorname{Lie}(T_{rs})=\mathbb{R}\otimes Y$} and its dual $\mathfrak{t}^*=\mathbb{R}\otimes Y$. The action of $W$ on $X$ extends to an action of $W$ on $\mathfrak{t^*}$ by means of isometries.
We can identify the set of integral affine linear functions on $X$ with $Y\times{\mathbb Z}$ via $(y,k)(x):=(x,y)+k$. It is clear that $w\cdot f(x):=f(w^{-1}x)$ defines an action of $W$ on $Y\times {\mathbb Z}$. The affine root system is by definition the subset $R^{\mathrm{aff}}=R_0^\vee\times{\mathbb Z}\subset Y\times {\mathbb Z}$ \index{R5@$R^{\mathrm{aff}}$, affine root system}. Notice that $R^{\mathrm{aff}}$ is a $W$-invariant set in $Y\times {\mathbb Z}$ containing the set of coroots $R_0^\vee$. Every element $a=(\alpha^\vee,k)\in R^{\mathrm{aff}}$ \index{a5@$a=(\alpha^\vee,k)$, affine root} defines an affine reflection $s_a\in W$ \index{s@$s_a$, affine reflection in $a$} , acting on $X$ by \begin{equation} s_a(x)=x-a(x)\alpha. \end{equation} The reflections $s_a$ with $a\in R^{\mathrm{aff}}$ generate a normal subgroup $W^{\mathrm{aff}}=W_0\ltimes Q$ \index{W2@$W^{\mathrm{aff}}=W_0\ltimes Q\subset W$} of $W$, where $Q\subset X$ \index{Q@$Q$, root lattice} denotes the root lattice $Q=\mathbb{Z}R_0$. We can choose a basis of simple affine roots $F^{\mathrm{aff}}$ \index{F@$F^{\mathrm{aff}}$, affine simple roots} by \begin{equation} F^{\mathrm{aff}}:=\{(\alpha^\vee,1)\mid\alpha\in S^m\}\cup \{(\alpha^\vee,0)\mid \alpha\in F_0\}, \end{equation} where $S^m$ consists of the set of minimal coroots with respect to the dominance ordering on $Y$. It is easy to see that every affine root is an integral linear combination of elements from $F^{\mathrm{aff}}$ with either all nonnegative or all nonpositive coefficients. The set $R^{\mathrm{aff}}$ of affine roots is thus a disjoint union of the set of positive affine roots $R^{\mathrm{aff}}_+$ and the set of negative affine roots $R^{\mathrm{aff}}_-$. The set $S^{\mathrm{aff}}$\index{S@$S^{\mathrm{aff}}$, simple reflections of $W^{\mathrm{aff}}$} of simple reflections in $W$ is by definition the set of reflections in $W$ associated with the fundamental affine roots. They constitute a set of Coxeter generators for the normal subgroup $W^{\mathrm{aff}}\subset W$.
There exists an Abelian complement to $W^{\mathrm{aff}}$ in $W$. This is best understood by introducing the important length function $l$ on $W$. The splitting $R^{\mathrm{aff}}=R^{\mathrm{aff}}_+\cup R^{\mathrm{aff}}_-$ \index{R5@$R^{\mathrm{aff}}_\pm$, positive (negative) affine roots} described above implies that $R^{\mathrm{aff}}_+ \cap s_a(R^{\mathrm{aff}}_-)=\{a\}$ when $a\in F^{\mathrm{aff}}$. Define, as usual, the length of an element $w\in W$ by \[
l(w):=|R^{\mathrm{aff}}_+\cap w^{-1}(R^{\mathrm{aff}}_-)|. \] \index{l1@$l$, length function on $W$} It follows that, when $a\in F^{\mathrm{aff}}$, \begin{equation} \label{eqn:simlength} {l}(s_aw)= \begin{cases} {l}(w)+1\text{ if }w^{-1}(a)\in R^{\mathrm{aff}}_+.\\ {l}(w)-1\text{ if }w^{-1}(a)\in R^{\mathrm{aff}}_-.\\ \end{cases} \end{equation} For any $w\in W$ we may therefore write $w=\omega \tilde w$ with $\tilde w\inW^{\mathrm{aff}}$ and with $l(\omega)=0$ (or equivalently, $\omega(F^{\mathrm{aff}})=F^{\mathrm{aff}}$). This shows that the set $\Omega$ \index{0Y@$\Omega$, length $0$ elements in $W$} of elements of length $0$ is a subgroup of $W$ which is complementary to the normal subgroup $W^{\mathrm{aff}}$, so that we have the decomposition \[ W=\Omega\ltimes W^{\mathrm{aff}}. \] Hence $\Omega\simeq W/W^{\mathrm{aff}}\simeq X/Q$ is a finitely generated Abelian group.
Let $m:X\to P$ (where $P$\index{P@$P$, weight lattice} denotes the weight lattice) denote the homomorphism that is adjoint to the inclusion $Q^\vee\to Y$. If we write $Z_X\subset X$ \index{Z@$Z_X$, length $0$ translations in $W$} for its kernel, then $Z_X\subset\Omega$. We have $\Omega/{Z_X}=\Omega_f$ \index{0Y@$\Omega_f=\Omega/Z_X$} where $Z_X$ is free and $\Omega_f=m(X)/Q\subset P/Q$ is finite. It is easy to see that $Z_X$ is the subgroup of elements in $X$ that are central in $W$. The finite group $\Omega_f$ acts faithfully on $S^{\mathrm{aff}}$ by diagram automorphisms.
The dual cone $X^+$ \index{X2@$X^+\subset X$, dominant cone} of the cone $Q_+$ spanned by the positive roots is called the cone of dominant elements of $X$. Thus $x\in X$ belongs to $X^+$ if and only if $\langle x,\alpha^\vee\rangle\geq 0$ for all positive roots $\alpha\in R_{0,+}$. Notice that $X^+\cap X^-$ equals the sublattice $Z_X\subset X$ of translations of length $0$.
Write $v=v_0+v^0$ for the splitting of $v\in\mathfrak{t^*}$ according to the orthogonal decomposition $\mathfrak{t^*}= \mathfrak{t^*}_0+\mathfrak{t^*}^0$, where $\mathfrak{t^*}_0=\mathbb{R}\otimes Q$. We define a norm \begin{equation}\label{eq:bigL}
\mathcal{N}(w)=l(w)+\|w(0)^0\| \end{equation} \index{N@$\mathcal{N}$, norm function on $W$} for $w\in W$. Notice that for all $w,w^\prime\in W$, $ww^\prime(0)^0=w(0)^0+w^\prime(0)^0$. Thus $\mathcal{N}(\omega w)=\mathcal{N}(w\omega)= l(w)+\mathcal{N}(\omega)$ if $w\inW^{\mathrm{aff}}$ and $\omega\in\Omega$. We also see that for all $\omega\in\Omega$, $\mathcal{N}(\omega^k)=k\mathcal{N}(\omega)$ for $k\in\mathbb{N}$. It follows easily that $\omega\in \Omega$ has finite order if and only if $\mathcal{N}(\omega)=0$. Finally notice that it also follows that \begin{equation} \mathcal{N}(ww^\prime)\leq\mathcal{N}(w)+\mathcal{N}(w^\prime) \end{equation} \subsection{Parabolic subsystems}\label{sub:par}
\index{X1a@$X_L\supset R_L$, lattice of $\mathcal{R}_L$, character lattice of $T_L$|(}
\index{Y@$Y_L\supset R_L^\vee$, lattice of $\mathcal{R}_L$, cocharacter lattice of $T_L$|(}
\index{W4@$W_P$, Weyl group of $R_P$, parabolic subgroup $W_0$|(}
\index{R1@$\mathcal{R}_L$, semisimple root datum associated to $L$|(}
\index{R1@$\mathcal{R}^L$, root datum associated to $L$|(}
\index{R3@$R_P\subset R_0$, parabolic subsystem, root system of $\mathcal{R}_P$|(}
\index{W4a@$W^P=W_0/W_P$, set of left cosets $wW_P$. If $P\subset F_0$, identified with shortest length representatives|(} An important role will be played by parabolic subgroups of a Weyl group. A root subsystem $R^\prime\subset R_0$ is called parabolic if $R^\prime=R_0\cap \mathbb{Q}R^\prime$. Let $P\subset R_{0,+}\cap R^\prime$ be the basis of simple roots. We then often write $R_P$ instead of $R^\prime$. The subgroup $W_P:=W(R_P)$ is called the associated parabolic subgroup. If $P\subset F_0$, we call $R_P$ and $W_P$ standard parabolic. Every parabolic subgroup is conjugate to a standard parabolic subgroup. We denote by $W^P$ the set of left cosets $W_0/W_P$. If $W_P$ is standard, we identify this quotient with the set of distinguished coset representatives of minimal length.
In many instances we obtain a parabolic subsystem $R^\prime$ as the set of roots orthogonal to some subspace $V^L\subset\mathfrak{t}:=\mathbb{R}\otimes Y$ which has the property that $V^L=\cap\operatorname{ker}(\alpha)$ where we take the intersection over all the roots $\alpha$ such that $\alpha(V^L)=0$. By abuse of notation we usually denote this parabolic subsystem by $R_L$. Similarly we write $W_L$ and $W^L$. We now denote the basis of $R_{L,+}$ by $F_L$.
To a parabolic subsystem $R_P\subset R_0$ we associate a root datum $\mathcal{R}^P:=(X,Y,R_P,R_P^\vee,P)$ and a root datum $\mathcal{R}_P:=(X_P,Y_P,R_P,R_P^\vee,P)$ where $Y_P:=Y\cap\mathbb{Q}R_P^\vee$ and $X_P:=X/(X\cap(R_P^\vee)^\perp)$.
\index{X1a@$X_L\supset R_L$, lattice of $\mathcal{R}_L$, character lattice of $T_L$|)}
\index{Y@$Y_L\supset R_L^\vee$, lattice of $\mathcal{R}_L$, cocharacter lattice of $T_L$|)}
\index{W4@$W_P$, Weyl group of $R_P$, parabolic subgroup $W_0$|)}
\index{R1@$\mathcal{R}_L$, semisimple root datum associated to $L$|)}
\index{R1@$\mathcal{R}^L$, root datum associated to $L$|)}
\index{R3@$R_P\subset R_0$, parabolic subsystem, root system of $\mathcal{R}_P$|)}
\index{W4a@$W^P=W_0/W_P$, set of left cosets $wW_P$. If $P\subset F_0$, identified with shortest length representatives|)} \subsection{Root labels}\label{sub:rl} The second ingredient in the definition of $\mathcal{H}$ is a function $q$\index{q@$q$, $l$-multiplicative function on $W$} on $S^{\mathrm{aff}}$ with values in the group of invertible elements of a commutative ring, such that \begin{equation}\label{cond1} q(s)=q(s^\prime) {\mathrm{\ if\ }} s\ {\mathrm{and}}\ s^\prime\ {\mathrm{are\ conjugate\ in\ }}W. \end{equation} A function $q$ on $S^{\mathrm{aff}}$, satisfying \ref{cond1}, can clearly be extended uniquely to a length-multiplicative function on $W$\index{Length multiplicative function}, also denoted by $q$. By this we mean that the extension satisfies \begin{equation} q({ww^\prime})=q(w)q({w^\prime}) \end{equation} whenever \begin{equation} {l}(ww^\prime)={l}(w)+ {l}(w^\prime), \end{equation} and in addition, \begin{equation} \forall \omega\in\Omega: q(\omega)=1. \end{equation} Conversely, every length multiplicative function on $W$ restricts to a function on $S^{\mathrm{aff}}$ that satisfies \ref{cond1}. Another way to capture the same information is by assigning labels $q_a$\index{q@$q_a$, affine root label} to the affine roots $a\in R^{\mathrm{aff}}$. These labels are uniquely determined by the rules \begin{gather}\label{eqn:afflab} \begin{split} (i)&\ q_{wa}=q_a\ \forall w\in W,\mathrm{\ and\ }\\ (ii)&\ q(s_a)=q_{a+1}\ \forall a\in F^{\mathrm{aff}}.\\ \end{split} \end{gather} Note that a translation $t_x$ acts on an affine root $a=(\alpha^\vee,k)$ by $t_xa=a-\alpha^\vee(x)$. Hence by $(i)$, $q_{a}=q_{\alpha^\vee}$, except when $\alpha^\vee\in 2Y$, in which case $q_{a}=q_{(\alpha^\vee,k(\mathrm{mod}2))}$. This last case occurs iff $W$ contains direct factors which are isomorphic to the affine Coxeter group whose diagram equals $C_n^\mathrm{aff}$.
Yet another manner of labeling will play an important role. It involves a possibly non-reduced root system $R_{\mathrm{nr}}$\index{R4@$R_{\mathrm{nr}}$, non reduced root system}, which is defined by: \begin{equation} R_{\mathrm{nr}}:= R_0\cup\{2\alpha\mid \alpha^\vee\in R_0^\vee\cap 2Y\}. \end{equation} Now define labels for the roots $\alpha^\vee/2$ in $R_{\mathrm{nr}}^\vee\backslash R_0^\vee$ by: \[ q_{\alpha^\vee/2}:= \frac{q_{1+\alpha^\vee}}{q_{\alpha^\vee}}. \] \index{q@$q_{\alpha^\vee}$, label for $\alpha^\vee\inR_{\mathrm{nr}}^\vee$} This choice is natural, because it implies the formula \begin{equation} q(w)=\prod_{\alpha\in R_{\mathrm{nr},+}\cap w^{-1}R_{\mathrm{nr,-}}} q_{\alpha^\vee}, \end{equation} for all $w\in W_0$.
Let $R_L\subset R_0$ be a parabolic root subsystem. With respect to the root datum $\mathcal{R}_L$ we have $R_{L,\mathrm{nr}}=\mathbb{Q}R_L\cap R_{\mathrm{nr}} \subset R_{\mathrm{nr}}$. In this sense we can define a label function denoted by $q_L$ \index{q@$q_L$, restriction of $q$ to $\mathcal{R}_L$} for the root datum $\mathcal{R}_L$, by restriction from $R_{\mathrm{nr}}^\vee$ to $R_{L,\mathrm{nr}}^\vee$. Similarly, we define $q^L$ by restriction of $q$ to $\mathcal{R}^L$. \index{q@$q^L$, restriction of $q$ to $\mathcal{R}^L$}
We denote by $R_1$\index{R4@$R_1$, system of long roots in $R_{\mathrm{nr}}$} the root system of long roots in $R_\mathrm{nr}$. In other words \begin{equation} R_{1}:= \{\alpha\in R_\mathrm{nr}\mid 2\alpha\not\in R_\mathrm{nr}\}. \end{equation} \subsection{The Iwahori-Hecke algebra as a Hilbert algebra} \label{sub:iwhehil} Many of the results of this subsection are well known, see \cite{Mat}. Let $\mathcal{R}$ be a root datum, and let ${\bf q}$\index{q@${\bf q}$, base for the labels $q(s)$} be a real number with ${\bf q}>1$. We assume that for all $s\in S^{\mathrm{aff}}$ we are given a {\it real} number $f_s$\index{f@$f_s=\log_{{\bf q}}(q(s))$}. Throughout this paper we use the convention that the labels as discussed in the previous subsection are defined by: \begin{conv}\label{eq:scale} The labels are of the form \begin{equation} q(s)={\bf q}^{f_s}\ \forall s\in S^{\mathrm{aff}}. \end{equation} \end{conv}
We write $q:=(q(s))_{s\in S^{\mathrm{aff}}}$ for the corresponding label function on $S^{\mathrm{aff}}$. The following theorem is well known. \begin{thm} There exists a unique complex associative algebra $\mathcal{H}=\mathcal{H}({\mathcal R},q)$\index{H@$\mathcal{H}$, affine Hecke algebra} with $\mathbb{C}$-basis $(T_w)_{w\in W}$\index{T4@$T_w$, basis elements of $\mathcal{H}$} which satisfy the following relations: \begin{enumerate}\label{eq:case} \item[(a)] If ${l}(ww^\prime)={l}(w)+ {l}(w^\prime)$ then $T_wT_{w^\prime} =T_{ww^\prime}$. \item[(b)] If $s\in S^{\mathrm{aff}}$ then $(T_s+1)(T_s-q(s))=0$. \end{enumerate} The algebra $\mathcal{H}=\mathcal{H}(\mathcal{R},q)$ is called the affine Hecke algebra (or Iwahori-Hecke algebra) associated to $(\mathcal{R},q)$. \end{thm} We equip the Hecke algebra $\mathcal{H}$ with an anti-linear anti-involutive $*$ \index{*@$*$!$h\to h^*$, conjugate linear anti-involution of $\mathcal{H}$} operator defined by $$ T_w^*=T_{w^{-1}}. $$ In addition, we define a trace functional $\tau$ \index{0t@$\tau$, trace functional of $\mathcal{H}$} on $\mathcal{H}$, by means of $\tau(T_w)=\delta_{w,e}$. It is a well known basic fact that \[ \tau(T_w^*T_{w^\prime})=\delta_{w,{w^\prime}}q(w), \] implying that $\tau$ is positive and central. Hence the formula \[ (h_1,h_2):=\tau(h_1^*h_2) \index{$(\cdot,\cdot)$!inner product on $\mathcal{H}$}, \] defines an Hermitian inner product satisfying the following rules: \begin{gather}\label{eqn:inprod} \begin{split}
(i)&\ (h_1,h_2)=(h_2^*,h_1^*).\\ (ii)&\ (h_1h_2,h_3)=(h_2,h_1^*h_3).\\
\end{split} \end{gather} The basis $T_w$ is orthogonal for $(\cdot, \cdot)$. We put \begin{equation}\label{eq:norm} N_w:=q(w)^{-1/2}T_w \end{equation}\index{N@$N_w$, normalized basis elements of $\mathcal{H}$} for the orthonormal basis of $\mathcal{H}$ that is obtained from the orthogonal basis $T_w$ by scaling. Let us denote by $\lambda(h)$ \index{0l@$\lambda(h)$, left multiplication by $h\in\mathcal{H}$} and $\rho(h)$\index{0r@$\rho(h)$, right multiplication by $h\in\mathcal{H}$} the left and right multiplication operators on $\mathcal{H}$ by an element $h\in\mathcal{H}$. Let $\mathfrak{H}$\index{H8@$\mathfrak{H}$, Hilbert completion of $\mathcal{H}$} be the Hilbert space obtained from $\mathcal{H}$ by completion; in other words, $\mathfrak{H}$ is the Hilbert space with Hilbert basis $N_w$. The operator $*$ extends to an isometric involution on $\mathfrak{H}$. Let $B(\mathfrak{H})$ \index{B@$B(\mathfrak{H})$, bounded linear operators on $\mathfrak{H}$} denote the space of bounded operators on the Hilbert space $\mathfrak{H}$. \begin{lem} For all $h\in\mathcal{H}$, both $\lambda(h)$ and $\rho(h)$ extend to $\mathfrak{H}$ as bounded operators (elements of $B(\mathfrak{H})$), with $\Vert\lambda(h)\Vert=\Vert\rho(h)\Vert$. For a simple reflection $s\in S^{\mathrm{aff}}$, $\Vert\lambda(N_s)\Vert =\operatorname{max}\{q(s)^{\pm 1/2}\}$. \end{lem} \begin{proof} We first prove the formula for the norm of $\Vert\lambda(N_s)\Vert$ ($s\in S^{\mathrm{aff}}$). For every $w$ such that $l(sw)>l(w)$, $\lambda(N_s)$ acts on the two-dimensional subspace $V_{w}$ of $\mathcal{H}$ spanned by $N_w$ and $N_{sw}$ as a self-adjoint operator with eigenvalues $q(s)^{1/2}$ and $-q(s)^{-1/2}$. Since $\mathfrak{H}$ is the Hilbert sum of the subspaces $V_{w}$, we see that $\lambda(N_s)$ extends to $\mathfrak{H}$ as a self-adjoint operator with operator norm equal to $q(s)^{\pm 1/2}$. Hence for any $h\in\mathcal{H}$, $\lambda(h)$ extends as a bounded operator on $\mathfrak{H}$. Finally notice that $(\lambda(h)^*(x))^*=\rho(h)(x)$, proving the equality $\Vert\lambda(h)\Vert=\Vert\rho(h)\Vert$. \end{proof} The above lemma shows that $\mathcal{H}$ has the structure of a Hilbert algebra in the sense of Dixmier \cite{dix1}. Moreover, this Hilbert algebra is unital, and the Hermitian product is defined with respect to the trace $\tau$.
We define the operator norm $\Vert\cdot\Vert_o$ \index{$\Vert\cdot\Vert_o$, operator norm on $\mathcal{H}$} on $\mathcal{H}$ by $\Vert h\Vert_o:=\Vert\lambda(h)\Vert=\Vert\rho(h)\Vert$. The closure of $\mathcal{H}$ with respect to the operator norm $\Vert\cdot\Vert_o$ is denoted by $\mathfrak{C}$\index{C@$\mathfrak{C}$, the reduced $C^*$ algebra of $\mathcal{H}$}. The map $\lambda$ ($\rho$) extends to an isometry from $\mathfrak{C}$ to the $C^*$-subalgebra $\lambda(\mathfrak{C})\subset B(\mathfrak{H})$ ($\rho(\mathfrak{C})\subset B(\mathfrak{H})$ resp.), the norm closure of $\lambda(\mathcal{H})$ ($\rho(\mathcal{H})$ resp.).
We identify $\mathfrak{C}$ with a subset of $\mathfrak{H}$ via the continuous injection $c\to \lambda(c)(1)$. We equip $\mathfrak{C}$ with the structure of a unital $C^*$-algebra by the product $c_1c_2:=\lambda(c_1)(c_2)=\rho(c_2)(c_1)$ and the $*$-operator coming from $\mathfrak{H}$. Then $\lambda$ ($\rho$) is a faithful left (right) representation of $\mathfrak{C}$ in the Hilbert space $\mathfrak{H}$ (note that we consider $\rho$ as a {\it right} representation on $\mathfrak{H}$). \begin{dfn} We call $\mathfrak{C}$ the reduced $C^*$-algebra of $\mathcal{H}$, and $\lambda$ ($\rho$) is called the left (right) regular representation of $\mathfrak{C}$ on $\mathfrak{H}$. \end{dfn} An element $a\in\mathfrak{H}$ is called {\it bounded} if there exists an element $\lambda(a)\in B(\mathfrak{H})$ such that for all $h\in\mathcal{H}$, \begin{equation} \lambda(a)(h)=\rho(h)(a). \end{equation} By continuity we see that $\lambda(a)$ is uniquely determined by $a=\lambda(a)(1)$.
When $a\in\mathfrak{H}$ is bounded, there also exists a unique $\rho(a)$ such that for all $h\in\mathcal{H}$, \begin{equation} \rho(a)(h)=\lambda(h)(a). \end{equation} It is obvious that the elements of $\mathfrak{C}$ are bounded. Let us denote by $\mathfrak{N}\subset\mathfrak{H}$ \index{N@$\mathfrak{N}$, von Neumann algebra completion of $\mathcal{H}$} the subspace of bounded elements. We equip $\mathfrak{N}$ with the involutive algebra structure defined by the product $n_1n_2:=\lambda(n_1)(n_2)=\rho(n_2)(n_1)$ and the $*$ operator as before. \begin{prop} The subspace $\lambda(\mathfrak{N}):= \{\lambda(a)\mid a\in\mathfrak{N}\}\subset B(\mathfrak{H})$ is the von Neumann algebra completion of $\lambda(\mathcal{H})$. In other words, $\lambda(\mathfrak{N})$ is the closure of $\lambda(\mathcal{H})$ in $ B(\mathfrak{H})$ with respect to the strong topology (defined by the semi-norms $T\to\Vert T(x)\Vert$ with $x\in\mathfrak{H}$). The analogous statements hold when we replace $\lambda$ by $\rho$. The centralizing algebra of $\lambda(\mathfrak{N})$ is $\rho(\mathfrak{N})$. \end{prop} \begin{proof} All this can be found in \cite{dix1}, Chapitre I, paragraphe 5. In general, $\lambda(\mathfrak{N})$ is a two-sided ideal of the von Neumann algebra hull of $\lambda(\mathcal{H})$, but in the presence of the unit $1\in\mathcal{H}$ the two spaces coincide. In fact, when $A\in B(\mathfrak{H})$ and $A$ is in the strong closure of $\lambda(\mathcal{H})$, it is simple to see that $A(1)\in\mathfrak{H}$ is bounded. \end{proof} The pre-Hilbert structure coming from $\mathfrak{H}$ gives $\mathfrak{N}$ itself the structure of a unital Hilbert algebra. The algebra $\mathfrak{N}$ can and will be identified with its associated standard von Neumann algebra $\lambda(\mathfrak{N})$. In this situation, $\mathfrak{N}$ is said to be a saturated Hilbert algebra (with unit element).
Let $\mathcal{H}^*$\index{H1@$\mathcal{H}^*$, algebraic dual of $\mathcal{H}$} denote the algebraic dual of $\mathcal{H}$, equipped with its weak topology. Notice that $\tau$ extends to $\mathcal{H}^*$ by the formula $\tau(\phi):=\phi(1)$. The $*$-operator can be extended to $\mathcal{H}^*$ by $\phi^*(h):=\overline{\phi(h^*)}$. We have the following chain of inclusions: \begin{equation} \mathcal{H}\subset\mathfrak{C}\subset\mathfrak{N}\subset\mathfrak{H}\subset\mathfrak{C}^\prime\subset\mathcal{H}^*. \end{equation} (where $\mathfrak{C}^\prime$\index{C@$\mathfrak{C}^\prime$, dual of $\mathfrak{C}$ as topological vector space} denotes the space of continuous linear functionals on $\mathfrak{C}$). \begin{prop} The restriction of $\tau$ to $\mathfrak{N}$ is central, positive and finite. It is the natural trace of the Hilbert algebra $\mathfrak{N}$, in the sense that \begin{equation} \tau(a)=(b,b) \end{equation} for every positive $a\in\mathfrak{N}$, and $b\in\mathfrak{N}$ such that $a=b^2$. \end{prop} \begin{proof} A square root $b$ is in $\mathfrak{N}$ and is Hermitian (i.e. $b^*=b$). Then $(b,b)=(1,a)=\tau(a)$. \end{proof} \begin{cor} The Hilbert algebra $\mathfrak{N}$ is finite. \end{cor} \begin{cor}\label{cor:regrep} The tracial state $\tau$ on $\mathfrak{C}$ is finite, and we have $\lambda=\lambda_\tau$ and $\rho=\rho_\tau$, where $\lambda_\tau$ and $\rho_\tau$ are the representations of $\mathfrak{C}$ naturally associated with the state $\tau$ (the classical GNS-construction). \end{cor} \begin{proof} This is immediate from the definitions, see \cite{dix2}, Paragraphe 6.7. \end{proof} \subsection{Bernstein's description of the center $\mathcal{Z}$}\label{sub:bern} By a well known (unpublished) result of J.~Bernstein (see \cite{Lu}), $\mathcal{H}$ can be viewed as the product $\mathcal{H}_0\mathcal{A}$ or $\mathcal{A}\mathcal{H}_0$ of an abelian subalgebra $\mathcal{A}$ (isomorphic to the group algebra of the lattice $X$), and the Hecke algebra
$\mathcal{H}_0=\mathcal{H}(W_0,q|_{S_0})$ \index{H2@$\mathcal{H}_0=\mathcal{H}(W_0,q\mid_{S_0})\subset\mathcal{H}$} of the finite Weyl group $W_0$. Both product decompositions $\mathcal{H}_0\mathcal{A}$ and $\mathcal{A}\mathcal{H}_0$ give a linear isomorphism of $\mathcal{H}$ with the tensor product $\mathcal{H}_0\otimes\mathcal{A}$. The relations between products in $\mathcal{H}_0\mathcal{A}$ and in $\mathcal{A}\mathcal{H}_0$ are described by the {\it Bernstein-Zelevinski-Lusztig relations} (see for example \cite{EO}, Theorem 1.10), and with the above additional description of the structure of $\mathcal{A}$ and $\mathcal{H}_0$ these give a complete presentation of $\mathcal{H}$.
The algebra $\mathcal{A}$\index{A@$\mathcal{A}$, abelian subalgebra of $\mathcal{H}$} has a ${\mathbb C}$-basis of invertible elements $\theta_x$\index{0i@$\theta_x$, basis elements of $\mathcal{A}$} (with $x\in X$) such that $x\to\theta_x$ is a monomorphism of $X$ into the group of invertible elements of $\mathcal{H}$. This monomorphism is uniquely determined by the property that $\theta_x=N_{t_x}$ (see (\ref{eq:norm})) when $x\in X^+$. As an important corollary of this presentation of $\mathcal{H}$, Bernstein identified the center $\mathcal{Z}$\index{Z1@$\mathcal{Z}$, the center of $\mathcal{H}$} of $\mathcal{H}$ as the space $\mathcal{Z}=\mathcal{A}^{W_0}$ of $W_0$-invariant elements in $\mathcal{A}$ (see \cite{EO}, Theorem 1.11). The following Proposition is well known and easy (cf. \cite{EO}, Proposition 1.12): \begin{prop}\label{prop:thetastar} Let $w_0\in W_0$ \index{wa@$w_0$, longest element of $W_0$} denote the longest element of $W_0$. Then we have for all $x\in X$: \begin{equation} \theta_x^*=T_{w_0}\theta_{-w_0(x)}T_{w_0}^{-1}. \end{equation} In particular, $\mathcal{A}\subset\mathcal{H}$ is not a $*$-subalgebra in general. The center $\mathcal{Z}\subset\mathcal{H}$ is a Hilbert subalgebra. \end{prop} Let $T=\operatorname{Spec}(\mathcal{A})=\operatorname{Hom}_{\mathbb Z}(X,{\mathbb C}^\times)$ \index{T@$T=\operatorname{Hom}_{\mathbb Z}(X,{\mathbb C}^\times)$, complex algebraic torus}. This algebraic torus of complex characters of $X$ has a natural $W_0$-action, and we have $\operatorname{Spec}(\mathcal{Z})\simeq W_0\backslash T$. \begin{prop}\label{prop:rk} The Hecke algebra $\mathcal{H}$ is finitely generated over its center
$\mathcal{Z}$. At a maximal ideal $m=m_t$ (with $t\in T$) of $\mathcal{Z}$, the local rank equals $|W_0|^2$ if and only if the stabilizer group $W_t\subset W_0$ \index{W3d@$W_t$, stabilizer in $W_0$ of $t\in T$} is generated by reflections. \end{prop} \begin{proof} It is clear that $\mathcal{H}\simeq \mathcal{H}_0\otimes \mathcal{A}$ is finitely generated over $\mathcal{Z}=\mathcal{A}^{W_0}$. When $W_t$ is generated by reflections, it is easy to see that the rank of $m$-adic completion $\hat\mathcal{A}_m$ over
$\hat\mathcal{Z}_m$ is exactly $|W_0|$ (see Proposition 2.23(4) of \cite{EO}). \end{proof} This fact plays a predominant role in the representation theory of $\mathfrak{C}$. Let us look at some basic consequences. \begin{cor} \begin{enumerate}
\item[(i)] Let $\pi$ be a finite dimensional irreducible representation of $\mathcal{H}$ with representation space $V$. The dimension of $V$ is less than or equal to $|W_0|$. \item[(ii)] In addition, the center $\mathcal{Z}$ of $\mathcal{H}$ acts by scalars on $V$. Thus $\pi$ determines a ``central character'' $t_\pi\in\operatorname{Spec}(\mathcal{Z})$ such that for all $z\in\mathcal{Z}$, $\pi(z)=t_\pi(z)\operatorname{Id}_V$. \item[(iii)] The characters of any finite set of inequivalent finite dimensional irreducible representations of $\mathcal{H}$ are linearly independent. \item[(iv)] A topologically irreducible $*$-representation $\pi$ of the involutive algebra $\mathcal{H}$ is finite dimensional. \end{enumerate} \end{cor} \begin{proof} Elementary and well known. Use the Frobenius-Schur theorem for (iii), Dixmier's version of Schur's lemma for (iv), and Proposition \ref{prop:rk}. \end{proof}
\begin{cor}\label{cor:typeI} (See also \cite{Mat}) Restriction to $\mathcal{H}$ induces an injection of the set $\hat\mathfrak{C}$\index{C@$\hat\mathfrak{C}$, dual (spectrum) of $\mathfrak{C}$} into the space $\hat{\mathcal{H}}$\index{H8@$\hat{\mathcal{H}}$, space of irreducible $*$-representations of $\mathcal{H}$} of finite dimensional irreducible $*$-representations of $\mathcal{H}$. Consequently, the $C^*$-algebra $\mathfrak{C}$ is of finite type I. \end{cor} \begin{proof} Because $\mathcal{H}\subset\mathfrak{C}$ is dense, it is clear that a representation of $\mathfrak{C}$ is determined by its restriction to $\mathcal{H}$ and that (topological) irreducibility is preserved. Hence by the previous Corollary, all irreducible representations of $\mathfrak{C}$ have finite dimension. \end{proof} We equip $T$ and $W_0\backslash T$ with the {\it analytic} topology. Given $\pi\in\hat\mathfrak{C}$ we denote by $W_0t_\pi\inW_0\backslash T$ the character of $\mathcal{Z}$ such that $\chi_\pi(z)=\operatorname{dim}(\pi)z(t_\pi)$ (note that $\pi(\mathcal{Z})$ can not vanish identically since $1\in\mathcal{Z}$).
By Proposition \ref{prop:thetastar}, the $*$-operator on $\mathcal{Z}$ is such that $z^*(t)=\overline{z(\overline{t^{-1}})}$. When $\pi\in\hat\mathfrak{C}$, we have $\chi_\pi(x^*)=\overline{\chi_\pi(x)}$. It follows that $\overline{t_\pi^{-1}}\in W_0t_\pi$ for all $\pi\in\hat\mathfrak{C}$. Let us denote by $W_0\backslash T^{herm}$ the closed subset $\{W_0t\inW_0\backslash T\mid \overline{t_\pi^{-1}}\in W_0t_\pi\}$ of $W_0\backslash T$. \begin{prop}\label{prop:im} The map $p_z:\hat\mathfrak{C}\toW_0\backslash T$ \index{p9@$p_z:\hat\mathfrak{C}\to\operatorname{Spec(\mathcal{Z})}$, projection} defined by $p_z(\pi)=W_0t_\pi$ is continuous and finite. Its image $S=p_z(\hat\mathfrak{C})\subsetW_0\backslash T^{herm}$ \index{S@$S\subsetW_0\backslash T$, image of $p_z$} is the spectrum $\hat{\overline{\mathcal{Z}}}$ of the closure ${\overline{\mathcal{Z}}}$ \index{Z1@$\overline{\mathcal{Z}}$, closure of $\mathcal{Z}$ in $\mathfrak{C}$} of $\mathcal{Z}$ in $\mathfrak{C}$. The map $p_z:\hat\mathfrak{C}\to S$ is closed. \end{prop} \begin{proof} It is clear that the image is in $W_0\backslash T^{herm}$ and that the map is finite (by Proposition \ref{prop:rk}). Since $W_0\backslash T$ is Hausdorff and $\hat\mathfrak{C}$ is compact, the map $p_z$ is closed if it is continuous.
So it remains to show that $p_z$ is continuous. The closure ${\overline{\mathcal{Z}}}\subset \mathfrak{C}$ is a unital commutative $C^*$-subalgebra of $\mathfrak{C}$. By the Gelfand transform it is isomorphic to the algebra of continuous functions $C(\hat{\overline{\mathcal{Z}}})$ on the compact Hausdorff space $\hat{\overline{\mathcal{Z}}}$. Denote by $\alpha$ the map $\alpha:\hat\mathfrak{C}\to\hat{\overline{\mathcal{Z}}}$ defined by the condition
$\chi_\pi|_{{\overline{\mathcal{Z}}}}=\operatorname{dim}(\pi)\alpha(\pi)$. By Proposition 2.10.2 of \cite{dix2}, $\alpha$ is surjective. In other words, every primitive ideal $M$ of $\mathfrak{C}$ intersects ${\overline{\mathcal{Z}}}$ in a maximal ideal $m$ of ${\overline{\mathcal{Z}}}$, and all maximal ideals of ${\overline{\mathcal{Z}}}$ are of this form. The corresponding surjective map from the set $\operatorname{Prim}(\mathfrak{C})$ of primitive ideals of $\mathfrak{C}$ to the set of maximal ideals $\operatorname{Max}({\overline{\mathcal{Z}}})$ is also denoted by $\alpha$. Next we claim that $\alpha$ is continuous. The topologies of $\hat\mathfrak{C}$ and $\hat{\overline{\mathcal{Z}}}$ are defined by the Jacobson topologies on $\operatorname{Prim}(\mathfrak{C})$ and $\operatorname{Max}({\overline{\mathcal{Z}}})$. This means that $U\subset\operatorname{Prim}(\mathfrak{C})$ is closed if and only if every $M\in\operatorname{Prim}(\mathfrak{C})$ which contains $I(U)=\cap_{u\in U}u$ is in $U$. Let $V\subset \operatorname{Max}({\overline{\mathcal{Z}}})$ be closed, and put $U=\alpha^{-1}(V)$. By the surjectivity of $\alpha$ we have $I(U)\cap{\overline{\mathcal{Z}}}=I(V)$. Hence if $M\in\operatorname{Prim}(\mathfrak{C})$ contains $I(U)$, then $\alpha(M)=M\cap{\overline{\mathcal{Z}}}$ contains $I(V)$, implying that $\alpha(M)\in V$. Therefore $M\in U$, proving that $U$ is closed as desired.
Next, we consider the injective map $\beta:\hat{\overline{\mathcal{Z}}}\toW_0\backslash T$ defined by restriction to $\mathcal{Z}\subset {\overline{\mathcal{Z}}}$. Its image $S\subsetW_0\backslash T^{herm}$ is bounded because for every $z\in\mathcal{Z}$,
$\Vert z\Vert_o=\max_{\chi\in\hat{\overline{\mathcal{Z}}}}|\chi(z)|=\max_{s\in S}|z(s)|$, showing that each $|z|$ with $z\in\mathcal{Z}$ has a maximum on $S$. Because $\overline{S}\subset W_0\backslash T^{herm}$, we see that $z^*(s)=\overline{z(s)}$ for each $z\in\mathcal{Z}$ and $s\in \overline{S}$. By the Stone-Weierstrass theorem, the restriction to $S$ of a continuous function $f\in C(W_0\backslash T)$ can be uniformly approximated by elements in $\mathcal{Z}$ considered as functions on $S$. In other words, there exists a $z\in{\overline{\mathcal{Z}}}$ such that $f(\beta(\chi))=\chi(z)$ for all $\chi\in\hat{\overline{\mathcal{Z}}}$. Hence $f\circ \beta$ is continuous on $\hat{\overline{\mathcal{Z}}}$ for all $f\in C(W_0\backslash T)$, showing that $\beta$ is continuous and $S=\overline{S}$. Since $\hat{\overline{\mathcal{Z}}}$ is compact and $S$ is Hausdorff it follows that $\beta:\hat{\overline{\mathcal{Z}}}\to S$ is a homeomorphism. The proposition now follows from the remark that $p_z=\beta\circ\alpha$. \end{proof} \subsection{Positive elements and positive functionals} \begin{dfn} We denote by $\mathcal{H}_+$ \index{H3@$\mathcal{H}_+$, positive elements in $\mathcal{H}$} the set of Hermitian elements $h\in\mathcal{H}$ such that $\forall x\in \mathcal{H}: (hx,x)\geq 0$. We call this the set of positive elements of $\mathcal{H}$. \end{dfn} By spectral theory in the Hilbert completion $\mathfrak{H}\supset\mathcal{H}$, this is equivalent to saying that $\lambda(h)\in B(\mathfrak{H})$ is Hermitian and has its spectrum in ${\mathbb R}_{\geq 0}$. Thus $\mathcal{H}_+$ is the intersection of $\mathcal{H}$ with the usual positive cone $\mathfrak{C}_+$ of the completion $\mathfrak{C}$. It is clear that for all $x\in\mathcal{H}$, $x^*x\in\mathcal{H}_+$ but not every positive element is of this form. We write $\mathcal{H}^{re}$ \index{H3@$\mathcal{H}^{re}$, Hermitian (or real) elements in $\mathcal{H}$} for the real subspace of Hermitian (or real) elements, i.e. $h\in\mathcal{H}$ such that $h^*=h$. \begin{lem}\label{lem:easy} \begin{enumerate} \item[(i)] If $z\in\mathcal{Z}_+,\ h\in\mathcal{H}_+$ then $zh\in\mathcal{H}_+$. \item[(ii)] If $h\in\mathcal{H}^{re}$ and $A\in{\mathbb R}_+$ such that $A\geq\Vert h\Vert_o$, then $A.1+h\in\mathcal{H}_+$. \end{enumerate} \end{lem} \begin{proof} A square root $\sqrt{z}\in{\overline{\mathcal{Z}}}$, the closure of $\mathcal{Z}$ in $\mathfrak{C}$, of $z$ has obviously the property that $\lambda(\sqrt{z})=\rho(\sqrt{z})$. Hence for every $x\in\mathcal{H}$, $h\in\mathcal{H}_+$: \[ (zhx,x)=(h\lambda(\sqrt{z})x,\lambda(\sqrt{z})x)\geq 0. \] The second assertion follows since $\operatorname{Spec}(\lambda(h))\subset [-\Vert h\Vert_o,\Vert h\Vert_o]$. \end{proof} \begin{dfn} We call a linear functional $\chi\in\mathcal{H}^*$ {\it positive} if $\chi(x)\geq 0$ for all $x\in\mathcal{H}_+$. \end{dfn} \begin{cor}\label{cor:contrace} \begin{enumerate} \item[(i)] A positive linear functional $\chi\in\mathcal{H}^*$ extends uniquely to a continuous functional $\chi\in\mathfrak{C}^\prime$ with norm $\Vert\chi\Vert=\chi(1)$. \item[(ii)] The character $\chi_\pi$ of an irreducible representation $\pi$ of $\mathfrak{C}$ is a positive functional $\chi_\pi\in\mathfrak{C}^\prime$. \item[(iii)] An irreducible $*$-representation $\pi$ of $\mathcal{H}$ extends to $\mathfrak{C}$ if and only if its character is a positive functional. \end{enumerate} \end{cor}
\begin{proof} (i). By the above Lemma \ref{lem:easy}, $|\chi(x)|\leq \chi(1)\Vert x\Vert_o$ for all Hermitian $x\in \mathcal{H}$. In addition, the bitrace $(x,y):=\chi(x^*y)$ is a positive semi-definite Hermitian form, and thus satisfies the Schwarz inequality. Hence for arbitrary $x\in\mathcal{H}$ we have
$|\chi(x)|^2\leq \chi(x^*x)\chi(1)\leq \chi(1)^2\Vert x^*x\Vert_o= \chi(1)^2\Vert x\Vert_o^2$, proving the continuity of $\chi$.
(ii). Because every irreducible representation $\pi$ of $\mathfrak{C}$ is finite dimensional (Corollary \ref{cor:typeI}), it is clear that the character $\chi_\pi(x)$ is a well defined positive functional on $\mathfrak{C}$. It is continuous by (i).
(iii). If the character $\chi_\pi$ is positive, we have by (i) that $\chi_\pi$ extends to a finite continuous character of $\mathfrak{C}$. Because $\mathfrak{C}$ is of finite type I (and thus liminal), the standard construction in \cite{dix2}, paragraphe 6.7 shows that there is up to equivalence a unique irreducible representation $\tilde\pi$ of $\mathfrak{C}$ whose character is $\chi_\pi$. The converse statement follows by (ii). \end{proof} \begin{rem} In general not all irreducible $*$-representations of $\mathcal{H}$ extend to $\mathfrak{C}$. See for instance Corollary \ref{cor:onedimlext}. \end{rem} \subsection{Casselman's criteria}\label{subsect:cass} For later use, we discuss in this subsection a suitable version of Casselman's criteria (see \cite{cas}, Lemma 4.4.1) to decide whether a representation of $\mathcal{H}$ is tempered (see the definition below) or is a subrepresentation of $\mathfrak{H}$. See also \cite{Mat}.
Recall the norm function $\mathcal{N}$ on $W$ as was introduced in Section \ref{sect:pre}. \begin{dfn} A functional $f\in\mathcal{H}^*$ is called tempered if there exists an $N\in\mathbb{N}$ and constant $C>0$ such that for all $w\in W$, \begin{equation}
|f(N_w)|\leq C(1+\mathcal{N}(w))^N. \end{equation} Here $N_w=q(w)^{-1/2}T_w$ are the orthonormal basis elements of $\mathcal{H}$ introduced in (\ref{eq:norm}). \end{dfn} Let $(V,\pi)$ be a finite dimensional module over $\mathcal{H}$, and let $t\in T$. We define $V_t:=\{v\in V\mid \forall a\in\mathcal{A}\ \exists n\in{\mathbb N}:(a-a(t))^n(v)=0\}$. The nonzero subspaces of the form $V_t\subset V$ are called the generalized $\mathcal{A}$-weight spaces of $V$. We call the corresponding elements $t\in T$ the $\mathcal{A}$-weights of $V$. \begin{lem}\label{lem:cas}(Casselman's criterion for temperedness). The following statements are equivalent: \begin{enumerate} \item[(i)] All matrix coefficients of $\pi$ are tempered. \item[(ii)] The character $\chi$ of $\pi$ is tempered.
\item[(iii)] The weights $t$ of the generalized $\mathcal{A}$-weight spaces of $V$ satisfy $|x(t)|\leq 1$, for all $x\in X^+$. \end{enumerate} \end{lem} \begin{proof} (i)$\Rightarrow$(ii). This is trivial.
(ii)$\Rightarrow$(iii). If there exists a weight $t$ of $V$ violating the condition, then there exists a $x\in X^+$ such that $|x(t)|>1$. We may assume that
$|x(t)|\geq|x(t^\prime)|$ for all weights $t^\prime$ of $V$. By Lemma 4.4.1 of \cite{cas}, the function \begin{equation}
f_x(n)=|x(t)|^{-n} \sum_{t^\prime}\operatorname{dim}(V_{t^\prime})x(t^\prime)^n=
|x(t)|^{-n}\chi(\theta_{nx}) \end{equation}
is not summable on $\mathbb{N}$. Hence for all $\epsilon>0$, $\chi(\theta_{nx})$ can not be bounded by a constant times $|x(t)|^{n(1-\epsilon)}$. On the other hand, suppose that $\chi$ is tempered. Since $\theta_{nx}=N_{nx}$ and $\mathcal{N}(nx)=n\mathcal{N}(x)$, we obtain that $\chi(\theta_{nx})$ is bounded by a polynomial in $n$, a contradiction.
(iii)$\Rightarrow$(i) Recall that the elements $T_u \theta_x T_v$ with $x\in X^+$, $u,v\in W_0$ span the subspace of $\mathcal{H}$ with basis $N_w$, where $w$ runs over the double coset $W_0xW_0\subset W$ (see the proof of Lemma 3.1 of \cite{EO}). It is not difficult to see that in fact we can write, for $w=uxv\in W_0xW_0$ with $x\in X^+$, \begin{equation} N_w=\sum_{v^\prime,u^\prime\in W_0}c_{w,(u^\prime, v^\prime)}T_{u^\prime}\theta_xT_{v^\prime}, \end{equation} such that the coefficients $c_{uxv,(u^\prime, v^\prime)}$ and $c_{uyv,(u^\prime, v^\prime)}$ are equal if $x$ and $y$ belong to the same facet of the cone $X^+$. Moreover, by the length formula \cite{EO}, equation 1.1, we have \begin{equation}
l(x)-|R_{0,+}|\leq l(uxv)\leq l(x)+|R_{0,+}| \end{equation} with $l(x)=x(2\rho^\vee)$. Thus we also have \begin{equation}
\mathcal{N}(x)-|R_{0,+}|\leq \mathcal{N}(uxv)\leq \mathcal{N}(x)+|R_{0,+}|. \end{equation}
It therefore suffices to show that the matrix entries of $\pi(\theta_x)$ with $x\in X^+$ are polynomially bounded in $\mathcal{N}(x)=x(2\rho^\vee)+\|x^0\|$. Since $V$ is a direct sum of generalized $\mathcal{A}$-weight spaces $V_t$, it is enough to consider the matrix coefficients of a generalized $\mathcal{A}$-weight space $V_t$ with weight $t$, satisfying the condition (iii). Observe finally that it is sufficient to consider the case that $x=x_0+x^0\in Q+Z_X$, a sublattice in $X$ of finite index.
By Lie's Theorem we can put the $\pi(\theta_x)$ simultaneously in upper triangular form. Choose ${\mathbb Z}_+$-generators $x_1, \dots, x_m$ for the cone $Q^+$, and a basis $x_{m+1}, \dots, x_n$ for the lattice $Z_X$. The Jordan decomposition $\pi(\theta_{x_i})=D_iU_i$ gives mutually commuting matrices $D_i$ and $U_i$, with $D_i$ semisimple and $U_i$ unipotent upper triangular. By conjugation in the group of invertible upper triangular matrices we may assume that the commuting semisimple matrices $D_i$ are diagonal. The strictly upper triangular matrices $M_i=\log(U_i)$ are commuting and satisfy \begin{equation}
\pi(\theta_{x_i})|_{V_t}=x_i(t)\exp(M_i)|_{V_t}. \end{equation} Hence for $x=\sum_{i=1}^n l_ix_i$, with $l_i\in\mathbb{Z}_+$ when $1\leq i\leq m$, we have \begin{equation}
\pi(\theta_{x})|_{V_t}=x(t)\exp(\sum_{i=1}^n l_iM_i|_{V_t}). \end{equation}
Since $|x(t)|\leq 1$ by assumption, and the exponential map is polynomial of degree $N:=\max_t\{\dim(V_t)\}-1$ on the space of strictly upper triangular matrices commuting with $\pi(\mathcal{A})$, we see that the matrix entries are bounded by a polynomial of degree $N$ in the coefficients $l_i$. Observe that $x_i(2\rho^\vee)\in{\mathbb Z}_{>0}$ when $1\leq i\leq m$. Since the coefficients $l_i$ are nonnegative this implies that for all $1\leq i\leq m$, $l_i\leq x(2\rho^\vee)$. On the other hand, there exists a constant $d>0$ independent of $x$ such that
$|l_i|\leq d\|x^0\|$ for all $i>m$. Thus there exists a constant $d^\prime$ independent of $x$ such that $|l_i|\leq d^\prime\mathcal{N}(x)$ for all $i$. This gives us the desired estimate of the matrix entries by a polynomial in $\mathcal{N}(x)$, of degree $N$. \end{proof} \begin{dfn} A representation $\pi$ of $\mathcal{H}$ satisfying the above equivalent conditions is called a tempered representation of $\mathcal{H}$. \end{dfn} Along the same lines one proves: \begin{lem}\label{lem:casds} (Casselman's criterion for discrete series representations.) Let $(V,\pi)$ be a finite dimensional representation of $\mathcal{H}$. The following are equivalent: \begin{enumerate} \item[(i)] $(V,\pi)$ is a subrepresentation of $(\mathfrak{H},\lambda)$. \item[(ii)] All matrix coefficients of $\pi$ belong to $\mathfrak{H}$. \item[(iii)] The character $\chi$ of $\pi$ belongs to $\mathfrak{H}$. \item[(iv)] The weights $t\in T$ of the generalized
$\mathcal{A}$-weight spaces of $V$ satisfy: $|x(t)|<1$, for all $0\not=x\in X^+$.
\item[(v)] $Z_X=\{0\}$, and there exist $C>0,\epsilon>0$ such that the inequality $|m(N_w)|<C{\bf q}^{-\epsilon l(x)}$holds for all matrix coefficients $m$ of $\pi$. \end{enumerate} \end{lem} \begin{proof} (i)$\Leftrightarrow$(ii) Let $E$ denote the projector of $\mathfrak{H}$ onto $V$. Then $E=\rho(e)$ for some idempotent of $\mathfrak{N}$, and since $E$ is open we have $V=\mathcal{H} e\subset\mathfrak{N}$. Choose an orthonormal basis $v_i$ of $V$. The corresponding matrix coefficients $(v_i,xv_j)=(v_iv_j^*,x)$ can be identified with the elements $v_jv_i^*\in\mathfrak{H}$. Conversely, suppose that, given a basis $v_i$ of $V$ with dual basis $\phi_j$ of $V^*$, there exist elements $h_{i,j}\in\mathfrak{H}$ such that for all $x\in\mathcal{H}$, $\phi_i(\pi(x)v_j)=(h_{i,j}^*,x)$. It follows that for each $i$, the map $v_j\to h_{i,j}$ defines an embedding of $(V,\pi)$ as a subrepresentation of $(\mathfrak{H},\lambda)$.
(ii)$\Rightarrow$(iii)$\Rightarrow$(iv)$\Rightarrow$(v) This is similar to the proof of Lemma \ref{lem:cas}. For the last implication we first remark that (iv) implies that $X^+$ can not contain $-x$ for any $x\in X^+$. Thus $Z_X=\{0\}$ in this case, hence $\mathcal{N}(x)=l(x)$.
(v)$\Rightarrow$(ii) The number of elements in $W$ with length $l$ grows polynomially in $l$. Thus, by the exponential decay, it is clear that $m=\sum_w\overline{m(N_w^*)}N_w$ is convergent in $\mathfrak{H}$, and moreover $m(x)=(m^*,x)$. \end{proof} \begin{cor} If the $\mathcal{A}$-weights of a finite dimensional representation $(V,\pi)$ of $\mathcal{H}$ satisfy the condition of Lemma \ref{lem:casds}(iv), then $V$ carries a Hilbert structure such that $\pi$ is a $*$-representation of $\mathcal{H}$, and moreover $\pi$ extends to a representation of $\mathfrak{C}$. \end{cor} \begin{dfn} Irreducible representations of $\mathcal{H}$ satisfying the equivalent conditions of Lemma \ref{lem:casds} are called discrete series representations. \end{dfn} \subsection{The Plancherel measure} Since $\mathfrak{C}$ is separable, liminal and unital, the spectrum $\hat\mathfrak{C}$ is a compact $T_1$ space with countable base. Moreover it contains an open dense Hausdorff subset.
The algebra $\mathfrak{C}$ comes equipped with the tracial state $\tau$, defining the representations $\lambda,\rho:\mathfrak{C}\to B(\mathfrak{H})$ of $\mathfrak{C}$ (see Corollary \ref{cor:regrep}). The general theory of the decomposition of a trace on a separable, liminal $C^*$-algebras (see \cite{dix2}, paragraphe 8.8), asserts that there exists a unique positive Borel measure $\mu_{Pl}$ \index{0m@$\mu_{Pl}$, Plancherel measure on $\hat\mathfrak{C}$} on $\hat\mathfrak{C}$ such that \begin{equation}\label{eq:dec} \mathfrak{H}\simeq\int^\oplus_{\hat\mathfrak{C}}\operatorname{End}(V_\pi)d\mu_{Pl}(\pi) \end{equation} and such that \begin{equation}\label{eq:chadec} \tau(h)=\int_{\hat\mathfrak{C}}\chi_\pi(h)d\mu_{Pl}(\pi). \end{equation} The measure $\mu_{Pl}$ is called the {\it Plancherel measure}. \begin{thm}\label{thm:supds} The support of $\mu_{Pl}$ is equal to $\hat\mathfrak{C}$. In addition, an irreducible representation $\pi$ of $\mathfrak{C}$ is a subrepresentation of $(\mathfrak{H},\lambda)$ if and only if $\mu_{Pl}(\pi)>0$ \end{thm} \begin{proof} The representation $\lambda_\tau=\lambda$ associated with the state $\tau$ is faithful (see Subsection \ref{sub:iwhehil}). The results thus follow from Proposition 8.6.8 of \cite{dix2}. \end{proof}
The center $\mathfrak{Z}$ \index{Z3@$\mathfrak{Z}$, the center of $\mathfrak{N}$} of $\mathfrak{N}$ will be mapped onto the algebra of diagonalizable operators $L^\infty(\hat\mathfrak{C}, \mu_{Pl})$. This is an isomorphism of algebras, continuous when we give $\mathfrak{Z}$ the weak operator topology and $L^\infty(\hat\mathfrak{C}, \mu_{Pl})$ the weak topology of the dual of $L^1(\hat\mathfrak{C}, \mu_{Pl})$. It is an isometry. \begin{rem} In many cases there exist non $\mu_{Pl}$-negligible subsets $V$ of
$\Hat\mathfrak{C}$ such that $1<|p_z^{-1}(y)|(<\infty)$ for all $y\in p_z(V)$. For instance, the affine Hecke algebra of type $G_2$ has two discrete series representations $\pi_3$, $\pi_{2,1}$ associated with the subregular unipotent class $G_2(a1)$ (notation as in \cite{C}, Section 13.3) of the complex algebraic group of type $G_2$ (also see Appendix \ref{KL}). Then $p_z(\pi_3)=p_z(\pi_{2,1})$ is equal to the $W_0$-orbit of the weighted Dynkin diagram associated with $G_2(a1)$. According to Lemma \ref{lem:casds} and Theorem \ref{thm:supds}, both $\{\pi_3\}$ and $\{\pi_{2,1}\}$ are non-negligible.
In such case, the above remarks imply in particular that the weak closure of $\mathcal{Z}$ in $\mathfrak{N}$ is strictly smaller than $\mathfrak{Z}$, the center of $\mathfrak{N}$. \end{rem} \subsection{Outline of the main results}\label{sub:out} This subsection is a continuation of the outline given in \ref{subsub:out}.
It is our goal to describe the spectral measure of the tracial state $\tau$ of $\mathfrak{C}$ explicitly. We will not completely succeed, as was explained in \ref{subsub:out}(4), but we will obtain a product formula for the density of $\mu_{Pl}$ on each component of its support, explicit up to a (intractable) positive real constant factor. An important intermediate step is the description of the more easily accessible spectral measure $\nu$ of the restriction of $\tau$ to the closure $\overline{\mathcal{Z}}\subset\mathfrak{C}$. \subsubsection{Plancherel measure $\nu$ of $\mathcal{Z}$}\label{sl}
\index{S@$S\subsetW_0\backslash T$, image of $p_z$|(}
\index{Tempered coset|(}
\index{Residual coset|(} \index{0n@$\nu$, Plancherel measure of $\overline{\mathcal{Z}}$
on $W_0\backslash T$|(}
\index{0n@$\nu_L$, smooth measure on $L^{temp}$ such that $\nu=\sum_L\nu_L$|(}
\index{m@$m_L$, density function of $\nu_L/{\overline \kappa}_{W_LL}$|(}
\index{m@$m^L$, quotient $m_L/k_L\nu_{\mathcal{R}_L,\{r_L\}}(\{r_L\})$|(}
\index{L@$L^{temp}$, tempered residual coset|(}
\index{r9@$r_L$, element of $L\cap T_L$|(}
\index{0kl@${\overline \kappa}_{W_0r}$(=${\overline \kappa}_{\mathcal{R},W_0r}$), rational factor in $\nu(\{r\})$|(} \index{0k@${\overline \kappa}_{W_LL}$, rational factor in
$\nu_L$; average of $\kappa_L$|(} The subalgebra $\mathcal{Z}\subset \mathcal{H}$ is a $*$-subalgebra. The spectral measure $\nu$ of the restriction of $\tau$ to $\overline{\mathcal{Z}}$ is determined in Subsection \ref{sub:chiA} by the use of the residue calculus.
A residual coset $L\subset T$ is a coset of a subtorus of $T$ such that the pole order of the rational function \begin{equation} \frac{1}{c(t,q)c(t^{-1},q)} \end{equation} along $L$ is equal to $\operatorname{codim}(L)$. Here $c(t,q)$ denotes Macdonald's $c$-function, see equation (\ref{eq:defc}). In Appendix \ref{sub:defn} this collection of residual cosets is carefully introduced, classified and studied. It turns out to be a finite, $W_0$-invariant collection of cosets, with various good properties which play an important role in the calculus of residues (see Subsection \ref{sub:resiprop} of Appendix). The residual cosets are of the form (cf. Proposition \ref{prop:red} and \ref{prop:conv}) $L=r_LT^L$, where $T^L$ is the connected component of the unit element $e$ in the fix point set in $T$ of the Weyl group $W_L$ of a parabolic subsystem $R_L$ of $R_0$ (a subtorus of $T$), and where $r_L\in T_L=\operatorname{Hom}(X_L,\mathbb{C}^\times)\subset T$ is a residual {\it point} with respect to the root datum $\mathcal{R}_L$ and the restriction $q_L$ of $q$ (see Subsection \ref{sub:rl}). This reduces the classification of these cosets to the case of the residual points. The {\it tempered form} $L^{temp}$ of $L=r_LT^L$ is the compact form of $L$ defined by $L^{temp}:=r_LT^L_u$.
Using the identification of the space of $W_0$-invariant continuous functions on $T$ and the space of continuous functions on $W_0\backslash T$, $\nu$ can be viewed as a $W_0$-invariant measure on $T$ supported on $\cup_{L}L^{temp}$ (union over the residual cosets). We show (cf. Theorem \ref{thm:nu}, Proposition \ref{prop:par} and Theorem \ref{thm:support}) that $\nu=\sum\nu_L$, where the sum is over all residual cosets $L$, and where $\nu_L$ is a {\it smooth} (with respect to the Haar measure $d^L$ on $L^{temp}$) measure with support equal to $L$, such that for all $w\in W_0$, $\nu_{wL}=w_*\nu_L$ (the push forward of $\nu_L$ along $w:L^{temp}\to (wL)^{temp}$). There is an explicit (up to a certain rational constant factor ${\overline \kappa}_{W_LL}$) product formula for $\nu_L$, compatible with parabolic induction (Proposition \ref{prop:par}(iv)): \begin{equation}\label{eq:firstmL} \nu_L(r_Lt^L)=k_L\nu_{\mathcal{R}_L,\{r_L\}}(\{r_L\})m^L(r_Lt^L)d^Lt^L, \end{equation}
where $k_L=|K_L|$ with $K_L=T_L\cap T^L$, $m^L$ is the rational function (\ref{eq:m^L}), and $\nu_{\mathcal{R}_L,\{r_L\}}(\{r_L\})$ is the the mass of the residual {\it point} $\{r_L\}\subset T_L$ with respect to the $W_L$-invariant spectral measure $\nu_{\mathcal{R}_L}$ on $T_L$ determined by $(\mathcal{R}_L,q_L)$. In the case where $L=\{r\}$ is a residual point we have (cf. Theorem \ref{thm:nu}) \begin{equation}\label{eq:a} \nu_{\{r\}}(\{r\})={\overline \kappa}_{W_0r}m_{\{r\}}(r), \end{equation} where $m_{\{r\}}$ is the given by the product (\ref{eq:m_L}).
The support of $\nu$ is by definition equal to the image $S$ of the map $p_z$ (cf. Proposition \ref{prop:im}). Thus we conclude from the above description of $\nu$ that $S=W_0\backslash\cup_{L}L^{temp}$ (union over all residual cosets $L$ with respect to $\mathcal{R}$ and root labels $q$).
\index{S@$S\subsetW_0\backslash T$, image of $p_z$|)}
\index{Tempered coset|)}
\index{Residual coset|)} \index{0n@$\nu$, Plancherel measure of $\overline{\mathcal{Z}}$
on $W_0\backslash T$|)}
\index{0n@$\nu_L$, smooth measure on $L^{temp}$ such that $\nu=\sum_L\nu_L$|)}
\index{m@$m_L$, density function of $\nu_L/{\overline \kappa}_{W_LL}$|)}
\index{m@$m^L$, quotient $m_L/k_L\nu_{\mathcal{R}_L,\{r_L\}}(\{r_L\})$|)}
\index{L@$L^{temp}$, tempered residual coset|)}
\index{r9@$r_L$, element of $L\cap T_L$|)}
\index{0kl@${\overline \kappa}_{W_0r}$(=${\overline \kappa}_{\mathcal{R},W_0r}$), rational factor in $\nu(\{r\})$|)} \index{0k@${\overline \kappa}_{W_LL}$, rational factor in
$\nu_L$; average of $\kappa_L$|)} \subsubsection{Separation by central character}
\index{0w@$\chi_t$, local trace of $\mathcal{H}$, sum (over $c$) of densities $d(\mathfrak{Y}_c^h)/d\nu$ at $t$|(} Next we deduce the formula (cf. Corollary \ref{cor:exten}) \begin{equation}\label{eq:int:int} \tau(hz)=\int_T z(t)\chi_t(h)d\nu(t) \end{equation} for all $z\in \mathcal{Z}$ and $h\in\mathcal{H}$. The $W_0$-invariant function $t\to\chi_t\in\mathcal{H}^*$ (defined on the support $\cup_L L^{temp}$ of $\nu$, and extended to $T$ by $0$) has values in the positive tracial states of $\mathfrak{C}$. It follows that $\chi_t$ is a finite positive linear combination of irreducible characters of $\mathfrak{C}$, which have central character $W_0t$ (cf. Definition \ref{dfn:resalg} and Theorem \ref{thm:mainind}). In other words, the state $\chi_t$ is a positive linear combination of the irreducible characters of $\mathfrak{C}$ in the (finite) fiber $p_z^{-1}(W_0t)$. This decomposition of $\chi_t$ is the subject of Section 4, and will be described below. \subsubsection{Generic spectrum and residual algebra}\label{sl1}
\index{0D3@$\Delta_{\mathcal{R},W_0r}$, irreducible discrete series representations of $\mathcal{H}(\mathcal{R},q)$ with central character $W_0r$|(}
\index{d@$d_\delta(=d_{\mathcal{R},\delta})$, residual degree; degree of $\delta$ in the residual Hilbert algebra $\overline{\mathcal{H}^r}$|(}
\index{H4@$\mathcal{H}^P=\mathcal{H}(\mathcal{R}^P,q^P)$, parabolic subalgebra of $\mathcal{H}$|(}
\index{H4@$\mathcal{H}_L:=\mathcal{H}(\mathcal{R}_L,q_L)$, semisimple quotient of $\mathcal{H}^L$|(}
\index{0p@$\pi(\mathcal{R}_P,W_Pr,\delta,t)=\operatorname{Ind}_{\mathcal{H}^P}^\mathcal{H}(\delta_t)$, parabolically induced representation|(}
\index{0w@$\chi_{\mathcal{R}_L,W_Lr_L,\delta,t^L}$, character of the induced representation $\pi(\mathcal{R}_L,W_Lr_L,\delta,t^L)$|(} The projection $p_z:\hat\mathfrak{C}\to S=\hat{\overline{\mathcal{Z}}}$ is complicated near non-Hausdorff points of $\hat\mathfrak{C}$. Using a variation of techniques introduced in \cite{Lu} we define an open dense subset $S^{gen}\subset S$ such that the restriction of $p_z$ to $p_z^{-1}(S^{gen})$ is a covering map (cf. Theorem \ref{thm:homeom}).
The absolute continuity of $\nu_L$ with respect to the Haar measure $d^L$ on $L^{temp}$ enables us to disregard a set of positive codimension, so that we can restrict the domain of integration in the above integral to (the pull back to $T$ of) $S^{gen}$.
Given $t\in T$ such that $W_0t\in S^{gen}$, there exists a unique residual coset $L$ such that $t\in L^{temp}$. Choose $r_L\in T_L\cap L$, and write $t=r_Lt^L$ with $t^L\in T^L_u$. The results of \cite{Lu}, suitably adapted, show that in this situation there exists a bijective correspondence between the equivalence classes $[\Delta_{\mathcal{R}_L,W_Lr_L}]$ of irreducible discrete series representations of $\mathcal{H}_L:=\mathcal{H}(\mathcal{R}_L,q_L)$ with central character $W_Lr_L$ and the equivalence classes of irreducible tempered representations of $\mathcal{H}$ with central character $W_0t$. The correspondence is established by an inflation to $\mathcal{H}^L:=\mathcal{H}(\mathcal{R}^L,q^L)$ using the induction parameter $t^L\in T^L_u$, and induction from $\mathcal{H}^L$ to $\mathcal{H}$ (Subsection \ref{sub:lus}, in particular Corollary \ref{cor:short}).
For $t=r_Lt^L\in L^{temp,gen}$ such that $R_L\subset R_0$ is standard parabolic (cf. Theorem \ref{thm:mainind}) we obtain \begin{equation}\label{eq:pointt}
\chi_t=|W^L|^{-1}\sum_{\delta\in\Delta_{\mathcal{R}_L,W_Lr_L}} d_{\mathcal{R}_L,\delta}\chi_{\mathcal{R}_L,W_Lr_L,\delta,t^L}, \end{equation} where $\chi_{\mathcal{R}_L,W_Lr_L,\delta,t^L}$ is the character of the representation $\pi_{\mathcal{R}_L,W_Lr_L,\delta,t^L}$ which is induced from the irreducible discrete series module $\delta$ of $\mathcal{H}_L$ (with central character $W_Lr_L$) with induction parameter $t^L$, and where $d_{\mathcal{R}_L,\delta}$ denotes the coefficient of the character $\chi_\delta$ in the decomposition of the tracial state $\chi_{\mathcal{R}_L,r_L}$ of $\mathcal{H}_L$.
The point is that the coefficients in (\ref{eq:pointt}) are {\it independent} of the induction parameter $t^L$. They are certain positive real constants, which we conjecture to be rational, see Conjecture \ref{rem:ell} below.
The positive real $d_{\mathcal{R},\delta}$ is the degree of $\delta$ with respect to the finite dimensional ``residual Hilbert algebra'' $\overline{\mathcal{H}^r}$, the quotient of $\mathcal{H}$ with respect to the radical of the positive semi-definite form defined by the tracial state $\chi_r$. Hence we have (cf. Corollary \ref{cor:fdim}) \begin{equation}\label{eq:b} \mu_{Pl}(\{\delta\})=d_{\mathcal{R},\delta}\nu(\{W_0r\}). \end{equation} Although Conjecture \ref{rem:ell} is out of reach for the methods used in this paper, there is a weaker statement which is relatively easy to prove within the context of this paper, and which is already useful for certain applications (see Subsection \ref{sub:uni}). We prove in Section 5 that the real constants $d_{\mathcal{R},\delta}$ are {\it independent} of ${\bf q}$, where we assume that $q(s)$ is written in the form Convention \ref{eq:scale}.
\index{0D3@$\Delta_{\mathcal{R},W_0r}$, irreducible discrete series representations of $\mathcal{H}(\mathcal{R},q)$ with central character $W_0r$|)}
\index{d@$d_\delta(=d_{\mathcal{R},\delta})$, residual degree; degree of $\delta$ in the residual Hilbert algebra $\overline{\mathcal{H}^r}$|)}
\index{0w@$\chi_t$, local trace of $\mathcal{H}$, sum (over $c$) of densities $d(\mathfrak{Y}_c^h)/d\nu$ at $t$|)}
\index{H4@$\mathcal{H}^P=\mathcal{H}(\mathcal{R}^P,q^P)$, parabolic subalgebra of $\mathcal{H}$|)}
\index{H4@$\mathcal{H}_L:=\mathcal{H}(\mathcal{R}_L,q_L)$, semisimple quotient of $\mathcal{H}^L$|)}
\index{0p@$\pi(\mathcal{R}_P,W_Pr,\delta,t)=\operatorname{Ind}_{\mathcal{H}^P}^\mathcal{H}(\delta_t)$, parabolically induced representation|)}
\index{0w@$\chi_{\mathcal{R}_L,W_Lr_L,\delta,t^L}$, character of the induced representation $\pi(\mathcal{R}_L,W_Lr_L,\delta,t^L)$|)} \subsubsection{Plancherel measure $\mu_{Pl}$ and Fourier transform}\label{sl2} \index{W8@$\mathcal{W}$, groupoid whose set of objects is $\mathcal{P}$, with morphisms $\operatorname{Hom}_\mathcal{W}(P,Q)=
\mathcal{W}(P,Q):=K_Q\times W(P,Q)$|(} \index{W6a@$W(\mathcal{O}_1,\mathcal{O}_2)=\{n\in W(P_1,P_2)\mid \exists k\in K_{P_2}:(k\times n)
\in\mathcal{W}(\delta_1,\delta_2)\}$|(}
\index{W6b@$W(\mathcal{O})=W(\mathcal{O},\mathcal{O})$|(}
\index{0O@${\Xi}=\Lambda\times_{\Gamma}\Delta$|(}
\index{O@$\tilde\mathcal{O}$, connected component of ${\Xi}$|(}
\index{F@$\mathcal{F}_\mathcal{H}$, Fourier transform on $\mathcal{H}$|(} Equations (\ref{eq:int:int}) and (\ref{eq:pointt}) yield a decomposition of the trace $\tau$ in terms of an integral over $t\in S^{reg}$, where the integration kernel is a sum over the finite fiber $p_z(W_0t)$. This integral can be rewritten more sensibly as an integral over a space of ``standard induction data'' $\Xi$, invariant for the action of a groupoid $\mathcal{W}$ acting on the standard induction data (cf. Subsection \ref{sub:pla}). Next we decompose $\mathcal{W}\backslash\Xi$ in its connected components. This leads to the final formulation of the spectral decomposition of $\mathfrak{H}$ in terms of the Fourier isomorphism $\mathcal{F}$ (cf. Theorem \ref{thm:mainp}). This formulation is parallel to the formulation of the Harish-Chandra Plancherel formula for $p$-adic groups, cf. \cite{W2}, \cite{D}.
Let $R_P\subset R_0$ be standard parabolic root system, and $\delta$ an irreducible discrete series representation of $\mathcal{H}_P=\mathcal{H}(\mathcal{R}_P,q_P)$ with central character $W_Pr$. The space $\mathcal{O}$ of all equivalence classes of ``twists of $\delta$'', representations of $\mathcal{H}^P$ of the form $\delta_{t^P}$ where $t^P$ varies over $T^P_u$, is a compact torus of the form $K_\delta\backslash T^P_u$, where $K_\delta$ is the isotropy subgroup of $[\delta]$ in $K_P=T^P\cap T_P$. There exists a principal fiber bundle $\mathcal{V}_\mathcal{O}$ over $\mathcal{O}$ whose fiber at $\omega=(\mathcal{R}_P,W_Pr_P,\delta,K_\delta t^P)\in\mathcal{O}$ is equal to the common representation space $i(V_\delta)$ of the induced representations $\pi(\mathcal{R}_P,W_Pr_P,\delta,t^P)$. Thanks to the regularity of certain intertwining operators (see Subsection \ref{sub:gene}) there exists a natural action of the group $W(\mathcal{O})=\{w\in W_0\mid w(R_{P,+})= R_{P,+},\ \mathrm{and}\ \exists k\in K_P:\Psi_w(\delta)\simeq\Psi_k(\delta)\}$ (where $\Psi_w(\delta), \Psi_k(\delta)$ denote twists of $\delta$ by automorphisms $\psi_w, \psi_k$ of $\mathcal{H}_P$ induced by $w$ and $k$ respectively) on the smooth sections of $\operatorname{End}(\mathcal{V}_\mathcal{O})$.
The Fourier transform $\mathcal{F}_\mathcal{H}$ is the algebra homomorphism from $\mathcal{H}$ into the direct sum of the algebras of smooth, $W(\mathcal{O})$-equivariant sections of $\operatorname{End}(\mathcal{V}_\mathcal{O})$, defined by $(\mathcal{F}_\mathcal{H}(h))(\omega)=\pi(\mathcal{R}_P,W_Pr_P,\delta,t^P)(h)$ if $\omega=(\mathcal{R}_P,W_Pr_P,\delta,K_\delta t^P)$.
In this terminology the Plancherel measure can be expressed as follows (cf. Theorem \ref{thm:mainp}): \begin{equation} d\mu_{Pl}(\pi(\omega))=
\mu_{\mathcal{R}_P,Pl}(\{\delta\})|K_P\delta|m^P(\omega)d^\mathcal{O}\omega, \end{equation} \index{0m@$\mu_{Pl}$, Plancherel measure on $\hat\mathfrak{C}$} where $K_P\delta$ denotes the set of equivalence classes of discrete series representations of $\mathcal{H}_P$ in the $K_P$-orbit of $\delta$, $d^\mathcal{O}$ is the normalized Haar measure on $\mathcal{O}$, $m^P(\omega)=m^P(r_Pt^P)$ is as in equation (\ref{eq:firstmL}), and $\mu_{\mathcal{R}_P,Pl}(\{\delta\})$ is given by (\ref{eq:b}) (applied to $\mathcal{R}_P$). When we equip the space of smooth sections of $\operatorname{End}(\mathcal{V}_\mathcal{O})^{W(\mathcal{O})}$ with the inner product \begin{equation}
(f_1,f_2)=\sum_{\mathcal{O}/\sim}|W(\mathcal{O})|^{-1}\int_{\mathcal{O}} \operatorname{tr}(f_1(\omega)^*f_2(\omega))d\mu_{Pl}(\pi(\omega)), \end{equation} then the Fourier transform $\mathcal{F}_\mathcal{H}$ is an isometry, which extends uniquely to a unitary isomorphism $\mathcal{F}:\mathfrak{H}\to L^2(\operatorname{End}(\mathcal{V}_\Xi))^{\mathcal{W}}$. \index{W8@$\mathcal{W}$, groupoid whose set of objects is $\mathcal{P}$, with morphisms $\operatorname{Hom}_\mathcal{W}(P,Q)=
\mathcal{W}(P,Q):=K_Q\times W(P,Q)$|)} \index{W6a@$W(\mathcal{O}_1,\mathcal{O}_2)=\{n\in W(P_1,P_2)\mid \exists k\in K_{P_2}:(k\times n)
\in\mathcal{W}(\delta_1,\delta_2)\}$|)}
\index{W6b@$W(\mathcal{O})=W(\mathcal{O},\mathcal{O})$|)}
\index{0O@${\Xi}=\Lambda\times_{\Gamma}\Delta$|)}
\index{O@$\tilde\mathcal{O}$, connected component of ${\Xi}$|)}
\index{F@$\mathcal{F}_\mathcal{H}$, Fourier transform on $\mathcal{H}$|)} \subsubsection{Further remarks}\label{sl3} \index{0l@$\lambda_\delta(=\lambda_{\mathcal{R},\delta}):= \overline{\kappa}_{W_0r}\mid W_0r\mid d_\delta$, constant factor in $\mu_{Pl}(\{\delta\})$,
$\delta\in\Delta_{W_0r}$|(} In \cite{DO} we prove that the Fourier isomorphism restricts to an isomorphism of the Schwartz completion $\mathfrak{S}$ of $\mathcal{H}$ (cf. \ref{subsub:schwartz}) onto $C^\infty(\operatorname{End}(\mathcal{V}_\Xi))^{\mathcal{W}}$. Consequently, $\mathfrak{C}\simeq\mathcal{F}(\mathfrak{C})=C(\operatorname{End}(\mathcal{V}_\Xi))^{\mathcal{W}}$. In particular, the connected components of $\hat\mathfrak{C}$ are the closures $\hat\mathfrak{C}_\mathcal{O}$ of $\pi(\mathcal{O}^{gen})\subset\hat\mathfrak{C}$. We expect that these results will provide an effective approach towards the problem of classification of irreducible tempered modules, using an analog of the analytic R-group (see for example \cite{A1}) in our context, granted the classification of the discrete series.
The methods used in this paper are not suitable to obtain a parametrization of the finite set of discrete series representations $[\Delta_{W_0r}]$ with central character $W_0r$. If all the labels of $\mathcal{H}$ are equal this information is contained in the work of Kazhdan and Lusztig \cite{KL}. They solved this problem using equivariant K-theory in the case when the labels $q_i$ are equal, and $X=P$. This result can be extended to the general equal label case, see \cite{RR}, \cite{Re2}. In the appendix Section \ref{KL} one can find an account of the results of Kazhdan and Lusztig, and the relation with residual cosets.
Let $F$ be a p-adic field and let $\mathcal{G}$ be the group of $F$-rational points in an adjoint semisimple group over $F$ which splits over an unramified extension of $F$. Lusztig \cite{Lu2}, \cite{Lu4} solved the above classification problem in principle for any Hecke algebra which arises as the centralizer algebra of a representation of $\mathcal{G}$ which is induced from a cuspidal unipotent representation of (the Levi quotient of) a parahoric subgroup of $\mathcal{G}$. The Hecke algebras that are not dealt with by Lusztig are ``generic'', and these generic algebras are simpler with respect to this problem of parametrization.
Starting from the generic case, Slooten \cite{Slooten} formulated an interesting combinatorics which (among others) conjecturally parametrizes the irreducible tempered modules with real central character for all classical root systems (a generalized Springer correspondence).
For $\delta\in\Delta_{W_0r}$
we define $\lambda_\delta:={\overline \kappa}_{W_0r}|W_0r|d_{\delta}$, so that \begin{equation} \mu_{Pl}(\{\delta\})=\lambda_\delta m_{\{r\}}(r) \end{equation} (see (\ref{eq:a}) and (\ref{eq:b})). This constant $\lambda_\delta$ has been computed explicitly by Mark Reeder in the cases where the Hecke algebra arises as the endomorphism algebra of a representation of a simple p-adic group of exceptional, split adjoint type which is induced from a cuspidal unipotent representation of a parahoric subgroup \cite{Re}. He conjectured an interpretation (in this situation) of $\lambda_\delta$ (see also \cite{Re0}) in terms of the Kazhdan-Lusztig parameters of $\delta$. In the exceptional cases he verified this conjecture, using a formula of Schneider and Stuhler \cite{schstu} for the formal degree of a discrete series representation of an almost simple $p$-adic group which contains fixed vectors for the pro-unipotent radical $U$ of a maximal compact subgroup $K$. This formula of Schneider and Stuhler however is an alternating sum of terms which does not explain the product structure of the formal dimension. In order to rewrite this sum as a product one needs to resort to a case-by-case analysis (computer aided) in \cite{Re}.
The method of \cite{Re} is likely to extend to the general case (joint work with Mark Reeder and Antony Wasserman, in progress). This would imply the following conjecture: \begin{con}\label{rem:ell} The $d_{\delta}$ (equivalently, the $\lambda_\delta$) are rational numbers. \end{con} \index{0l@$\lambda_\delta(=\lambda_{\mathcal{R},\delta}):= \overline{\kappa}_{W_0r}\mid W_0r\mid d_\delta$, constant factor in $\mu_{Pl}(\{\delta\})$,
$\delta\in\Delta_{W_0r}$|)} \section{Localization of $\tau$ on $\operatorname{Spec}(\mathcal{Z})$}\label{sec:loctau} Recall the decomposition of $\tau$ we derived in \cite{EO}, Theorem 3.7: \begin{gather}\label{eq:basic} \begin{split} \tau=\int_{t\in t_0T_u} \left(\frac{E_t}{q(w_0)\Delta(t)}\right) \omega\\ \end{split} \end{gather} where $\omega$ denotes the rational $(n,0)$-form \begin{equation}\label{eq:omo} \omega:=\frac{dt}{c(t,q)c(t^{-1},q)} \end{equation} \index{0y@$\omega=\frac{dt}{c(t,q)c(t^{-1},q)}$, $(n,0)$-form on $T$} on $T$. Let us briefly review the various ingredients of this formula. First of all, $T_u=\operatorname{Hom}(X,S^1)$ \index{T3@$T_u=\operatorname{Hom}(X,S^1)$, compact form of $T$}, the compact form of the algebraic torus $T=\operatorname{Hom}(X,{\mathbb C}^\times)$, and $t_0\in T_{rs}$ \index{T1@$T_{rs}=\operatorname{Hom}(X,{\mathbb R}^\times_+)$, real split form of $T$}, the real split part of $T$, and should be deep in the negative chamber $T_{rs,-}$ \index{T2@$T_{rs,-}$, negative chamber in $T_{rs}$}. The precise conditions will be formulated below, see equation (\ref{eq:mininf}).
The form $dt$ \index{d@$dt$, holomorphic extension of Haar measure on $T_u$} denotes the holomorphic $(n,0)$-form on $T$ which restricts to the normalized Haar measure on $T_u$. It is given by the formula \[ dt:=(2\pi i)^{-n}(x_1x_2\dots x_n)^{-1}dx_1\wedge dx_2\wedge\dots\wedge dx_n \] if $(x_1,\dots,x_n)$ is a basis of $X$.
The function $\Delta(t):=\prod_{\alpha\in R_{1,+}}\Delta_\alpha(t)$ \index{0D@$\Delta$, Weyl denominator} with \begin{equation}\label{eq:defD} \Delta_\alpha:=1-\theta_{-\alpha}\in\mathcal{A} \end{equation} is the Weyl denominator. Here we use the convention to consider the subalgebra $\mathcal{A}\subset\mathcal{H}$ as the algebra of regular functions on $T$ via $\theta_x(t):=x(t)$.
The function $c(t)=c(t,q)$ \index{c@$c=c(t,q)$, Macdonald's $c$-function} is Macdonald's $c$-function. This $c$-function is introduced as an element of ${}_\mathcal{F}\mathcal{A}$ \index{F@${}_\mathcal{F}\mathcal{A}$, field of fractions of $\mathcal{A}$}, the field of fractions of $\mathcal{A}$, and will be interpreted as a rational function on $T$ (cf. \cite{EO}, Definition 1.13). Explicitly, we put \begin{equation}\label{eq:defc} c:=\prod_{\alpha\in R_{0,+}}c_\alpha=\prod_{\alpha\in R_{1,+}}c_\alpha. \end{equation} Here we define $c_\alpha$ \index{c@$c_\alpha$, rank one $c$-function} for $\alpha\in R_1$ by \begin{equation} c_\alpha:= \frac{(1+q_{\alpha^\vee}^{-1/2}\theta_{-\alpha/2}) (1-q_{\alpha^\vee}^{-1/2}q_{2\alpha^\vee}^{-1}\theta_{-\alpha/2})} {1-\theta_{-\alpha}}\in{}_\mathcal{F}\mathcal{A}. \end{equation} If $\alpha\in R_0\backslash R_1$ then we define $c_\alpha:=c_{2\alpha}$.
\begin{rem}\label{rem:conv} We have thus associated a $c$-function $c_\alpha$ to each root $\alpha\in R_{nr}$, but $c_\alpha$ only depends on the direction of $\alpha$. This convention is different from the one used in \cite{EO}. It is handy to write the formulas for the $c$ functions in the above form, but strictly speaking incorrect if $\alpha\in R_1$ and $\alpha/2\not\in X$. However, we formally put $q_{2\alpha^\vee}=1$ if $\alpha/2\not\in R_0$, and then rewrite the numerator as $(1-q_{\alpha^\vee}^{-1}\theta_{-\alpha})$. Here and below we use this convention. \end{rem}
The exact inequalities which have to be met by $t_0\in T_{rs}$ in order to represent the trace functional $\tau$ are as follows. If $q(s)>1$ for all $s\in S^{\mathrm{aff}}$, then according to \cite{EO}, Definition 1.4 and Corollary 3.2, the representation (\ref{eq:basic}) holds if \begin{equation} \forall \alpha\in F_0:\alpha(t_0)<q_{\alpha^\vee}^{-1}q_{\alpha^\vee/2}^{-1/2}. \end{equation} It is clear that this representation of $\tau$ remains true if we vary the parameters $q$ in a connected open set $U$ such that $\{q\mid \forall s:q(s)>1\}\subset U\subset \{q\mid \forall s:q(s)>0\}$, as long as the poles of the kernel of the integral for any $q\in U$ do not intersect the integration cycle $t_0T_u$. It follows that the representation (\ref{eq:basic}) of $\tau$ holds for any $q$ such that $\forall s\in S^\mathrm{aff}:q(s)>0$, provided that \begin{equation}\label{eq:mininf} \forall \alpha\in F_0:\alpha(t_0)<\operatorname{min} \{(q_{\alpha^\vee}q_{\alpha^\vee/2}^{1/2})^{\pm 1},q_{\alpha^\vee/2}^{\pm 1/2}\}. \end{equation} Observe that \begin{equation} q_{\alpha^\vee}q_{\alpha^\vee/2}^{1/2}=q_{\alpha^\vee}^{1/2}q_{\alpha^\vee+1}^{1/2};\ q_{\alpha^\vee/2}^{1/2}=q_{\alpha^\vee}^{-1/2}q_{\alpha^\vee+1}^{1/2} \end{equation}
The expression $E_t\in\mathcal{H}^*$ \index{E@$E_t$, Eisenstein functional} is the holomorphic {\it Eisenstein series} for $\mathcal{H}$, with the following defining properties (cf. \cite{EO}, Propositions 2.23 and 2.24): \begin{gather}\label{fundeis} \begin{split} (i)&\ \forall h\in\mathcal{H},\mbox{ the map }T\ni t\to E_t(h)\mbox{ is regular}.\\ (ii)&\ \forall x, y\in X,h\in\mathcal{H},\ E_t(\theta_x h\theta_y)=t(x+y)E_t(h).\\ (iii)&\ E_t(1)=q(w_0)\Delta(t).\\ \end{split} \end{gather} We want to rewrite the integral (\ref{eq:basic}) representing the trace functional as an integral over the collection of tempered residual cosets, by a contour shift. It turns out that such a representation exists and is unique. To find it, we need an intermediate step. We will first rewrite the integral as a sum of integrals over a larger set of tempered ``quasi-residual cosets'', and then we will show that if we symmetrize the result over $W_0$, all the contributions of non-residual cosets cancel. \subsection{$\omega$-residual cosets} The basic scheme to compute residues has nothing to do with the properties of root systems. It is therefore convenient to formulate everything in a more general setting first. Later we will consider the consequences that are specific to our context.
Let $T$ be a complex algebraic torus with character lattice $X$. \begin{dfn}\label{dfn:om} Let $\omega=pdt/q$ be a rational $(n,0)$-form on $T$. Assume that $p,q$ are of the form \begin{equation} q(t)=\prod_{m\in\mathcal{M}}(d_m^{-1}x_m(t)-1);\ p(t)=\prod_{m^\prime\in \mathcal{M}^\prime}(d_{m^\prime}^{-1}x_{m^\prime}(t)-1), \end{equation} where the products are taken over finite multisets $\mathcal{M},\mathcal{M}^\prime$. The multisets $\mathcal{M}$ and $\mathcal{M}^\prime$ come equipped with maps $m\to(x_m,d_m)\in X\times{\mathbb C}^\times$. For $m\in\mathcal{M}\cup\mathcal{M}^\prime$ we denote by $L_m\subset T$ the codimension $1$ subvariety $L_m=\{t\mid x_m(t)=d_m\}$, and we denote by $D_\omega$ \index{D@$D_\omega$, minus the divisor of $\omega$ on $T$} the divisor $\sum_{m\in\mathcal{M}}L_m-\sum_{m^\prime\in\mathcal{M}^\prime}L_{m^\prime}$ on $T$ of $q/p$.
An $\omega$-residual coset $L$ is a connected component of $\cap_{m\in J}L_m$ for some $J\subset \mathcal{M}$, such that the pole order $i_L$ of $\omega$ along $L$ satisfies \[
i_L:= |\{m\in \mathcal{M}\mid L\subset L_m\}|-|\{m^\prime\in \mathcal{M}^\prime\mid L\subset L_{m^\prime}\}|\geq\operatorname{codim}(L). \] \index{i@$i_L$, pole order along $L$} The collection of $\omega$-residual cosets is denoted by $\mathcal{L}^{\omega}$ \index{L1@$\mathcal{L}^\omega$, collection of $\omega$-residual cosets}. This is a finite, nonempty collection of cosets of subtori of $T$, which includes by definition $T$ itself (the empty intersection of the cosets $L_m$). \end{dfn} Note that $\omega$ as in the above definition is completely determined by the divisor $D_\omega$ on $T$.
Let $\langle\cdot,\cdot\rangle$ \index{<@$\langle\cdot,\cdot\rangle$!a@rational inner product on $X$,$Y$} be a rational inner product on the vector space ${\mathbb Q}\otimes Y$, where $Y$ is the cocharacter lattice of $T$. This defines an isomorphism between ${\mathbb Q}\otimes X$ and ${\mathbb Q}\otimes Y$, and we also denote by $\langle\cdot,\cdot\rangle$ the corresponding inner product on ${\mathbb Q}\otimes X$. Through the exponential map $\exp:\mathfrak{t}_{\mathbb C}:={\mathbb C}\otimes Y\to T$ we obtain a distance function on $T$. It is defined by taking the distance between $2\pi i Y$-orbits in $\mathfrak{t}_{\mathbb C}$. We denote by
$|t|$
\index{t@$|t|$, distance $d(t,e)$ on $T$} the distance of $t\in T$ to $e\in T$.
Suppose that $L$ is a connected component of the intersection $\cap_{m\in J}L_m$ for some subset $J\subset \mathcal{M}$. Then $L$ is a coset for the connected component of $e$ of the subgroup $\cap_{m\in J}T^m\subset T$, where $T^m:=\{t\in T\mid x_m(t)=1\}$. We denote this connected component by $T^L\subset T$ \index{T@$T^L\subset T$, algebraic subtorus of which $L$ is a coset}. Its character lattice $X^L:=\operatorname{Hom}(T^L,{\mathbb C}^\times)$ \index{X1b@$X^L$, character lattice of $T^L$} is equal to the quotient $X^L=X/((\sum_{m\in J}{\mathbb Q} x_m)\cap X)$. Let $X_L$ \index{X1a@$X_L\supset R_L$, lattice of $\mathcal{R}_L$, character lattice of $T_L$} be the quotient $X_L:=X/(\cap_{m\in J}x_m^{\perp}\cap X)$. Then $T_L:=\operatorname{Hom}(X_L,{\mathbb C}^\times)$ \index{T@$T_L\subset T$, algebraic subtorus orthogonal to $L$} is an algebraic subtorus of $T$, the subtorus ``orthogonal to $T^L$''. The intersection $K_L:=T_L\cap T^L$ \index{K@$K_L$, finite abelian group $T_L\cap T^L$} is a finite abelian group, and is canonically isomorphic to character group of the quotient $X/(X_L+X^L)$. It follows that $L\cap T_L$ is a coset for the finite subgroup $K_L\subset T_u$.
We denote by $\mathcal{M}_L\subset \mathcal{M}$ the subset $\{m\in \mathcal{M}\mid x_m(L)=d_m\}$. We choose an element $r_L=s_Lc_L\in T_L\cap L$ \index{r9@$r_L$, element of $L\cap T_L$} for each $L$ so that we can write $L=r_LT^L$. We call $c_L\in T_{rs}$ \index{c1@$c_L$, center of $L$} the center of $L$, and note that this center is determined uniquely by $L$. We write $c_L=\exp{\gamma_L}$ \index{0c@$\gamma_L\in \mathfrak{t}$, logarithm of $c_L$} with $\gamma_L\in\mathfrak{t}_L$. The set of centers of the $\omega$-residual cosets is denoted by $\mathcal{C}^{\omega}$ \index{C@$\mathcal{C}^{\omega}$, set of centers of $\omega$-residual cosets}. The tempered form of a $\omega$-residual $L=r_LT^L$ is by definition $L^{temp}:=r_LT^L_u$ \index{L@$L^{temp}$, tempered residual coset} (which is independent of the choice of $r_L$), and such a coset will be called an $\omega$-tempered coset.
Basically, the only properties of the collection $\mathcal{L}^{\omega}$ we will need are \begin{prop}\label{prop:triv} \begin{enumerate} \item[(i)] If $c\in \mathcal{C}^{\omega}$ then the union $$S_c:=\cup_{\{L\in \mathcal{L}^{\omega}\mid c_L=c\}}L^{temp}\subset cT_u$$ \index{S@$S_c$, support of $\mathfrak{X}_c$} is a regular support in the sense of \cite{Schw} in $cT_u$. This means that a distribution on $cT_u$ with support in $S_c$ can be written as a sum of derivatives of push forwards of measures on $S_c\subset cT_u$. \item[(ii)]
If $c=\exp\gamma\in T_{rs}$, and $L$ is $\omega$-residual with $|\gamma_L|\geq |\gamma|$ but $\gamma_L\not=\gamma$, then there exists a $m\in \mathcal{M}_L$ such that $f(t)=x_m(t)-d_m$ is non-vanishing on $cT_u$. \end{enumerate} \end{prop} \begin{proof} The set $S_c$ is a finite union of smooth varieties, obviously satisfying the condition of \cite{Schw}, Chapitre III, \S 9 for regularity, proving (i). As for (ii), first note that the assumption implies that $\gamma_L\not=0$, hence that $L\not=T$. Thus the codimension of $L$ is positive, and $\mathcal{M}_L\not=\emptyset$. Clearly $\gamma\not\in \gamma_L+\mathfrak{t}^L=\log(T_{rs}\cap LT_u)$ since $\gamma_L$ is the unique smallest vector in this affine linear space. Because $\{x_m\mid m\in \mathcal{M}_L\}$ spans $\mathfrak{t}_L=(\mathfrak{t}^L)^\perp$, we can find a $m\in \mathcal{M}_L$ such that $x_m(\gamma)\not=x_m(\gamma_L)$. This implies the result. \end{proof} \subsection{The contour shift and the local contributions} The following lemma is essentially the same as Lemma 3.1 of \cite{HOH0}, but because of its basic importance we have included the proof here, adapted to the present context. See also \cite{BS1} for a more general method in the same spirit. \begin{lem}\label{thm:resbasic} Let $\omega$ be as in Definition \ref{dfn:om} and let $t_0\in T_{rs}\backslash \cup (T_{rs}\cap T_uL_m)$. Fix an inner product $\langle\cdot,\cdot\rangle$ on ${\mathbb Q}\otimes Y$. Then there exists a unique collection of distributions $\{\mathfrak{X}_c\in C^{-\infty}(cT_u)\}_{c\in \mathcal{C}^{\omega}}$ \index{X3@$\mathfrak{X}_c$, local contribution to $\int_{t_0T_u}a\omega$} such that the following conditions hold: \begin{enumerate} \item[(i)] The support of $\mathfrak{X}_c$ satisfies $\operatorname{supp}(\mathfrak{X}_c)\subset S_c$ (cf. Proposition \ref{prop:triv}). \item[(ii)] For every $a\in \mathcal{A}^{an}(T)$ (the ring of analytic functions on $T$) we have \begin{equation} \int_{t\in t_0T_u}a(t)\omega(t) =\sum_{c\in\mathcal{C}^{\omega}}
\mathfrak{X}_c(a|_{cT_u}). \end{equation} \end{enumerate} \end{lem} \begin{proof} The existence is proved by induction on the dimension $n$ of $T$, the case of $n=0$ being trivial. Suppose that the result is true for tori of dimension $n-1$. Choose a smooth path in $T_{rs}$ from $t_0$ to the identity $e$ which intersects the real projection $L_{m,r}=T_{rs}\cap T_uL_m$ of the codimension $1$ cosets $L_m$ transversally and in at most one point $t(L_{m,r})$. We may assume that the intersection points are mutually distinct with possible exception for the cases $t(L_{m,r})=e$, i.e. when $e\in L_{m,r}$. When we move $t_0$ along the curve towards $e$, then we pick up residues when we pass at a point $t=t(L_{m,r})\not=e$ on the curve. We may assume that the cosets $L_m$ are connected (by factoring the defining equations, and adapting $\mathcal{M}$ accordingly). Let $L=L_m$ be such that $t\in L_{m,r}$. For simplicity of notation we write $(x,d)$ instead of $(x_m,d_m)$. Recall the decomposition $L=r_LT^L=s_Lc_LT^L$ with $r_L\in T_L$. Let $d^Lt$ \index{d1@$d^Lt$ ($d_Lt$), holomorphic extension of normalized Haar measure on $T^L_u$ ($T_{L,u}$)} denote the holomorphic extension to $T^L$ of the normalized Haar measure on $T^L_u$, and similarly for $d_Lt$ on $T_L$. Let $K_L=T^L\cap T_L$, and let $k_L$ \index{k_L@$k_L$, order of $K_L$} be its order. The product homomorphism $\pi:T^L\times T_L\to T$ has kernel $\{(k,k^{-1})\mid k\in K_L\}$. The residue that is picked up on $L$ when we cross at $t$ can be written as follows: \[ \int_{t^L\in T^L_u}\sum_{k\in K_L}\int_{t_L\in kC} (ap/q)(ts_Lt^L t_L)d_L(t_L)d^L(t^L), \] where $C$ denotes a small circle in $T_L\simeq{\mathbb C}^\times$ around $1$. Using the action of $\operatorname{Ker}(\pi)$ and the invariance of $d^L$ and $d_L$, and in addition using $r_L$ as a base point of $L$, this equals \[ k_L\int_{t^L\in tc_L^{-1}T^L_u}\int_{t_L\in C} (ap/q)(r_Lt^Lt_L)d_L(t_L)d^L(t^L). \]
Let $x_L\in X_L$ be a generator of $X_L$. Let $D$ be the holomorphic constant vector field on $T_L$ which is dual to $x_L$. We extend $D$ to a constant holomorphic vector field on $T$. We define a $k_L$-th root of $d$ by $x_L(r_L)=d^{1/k_L}$, so that the pull back of $d^{-1}x-1$ to $T_L\times T^L$ factors as follows: \[ (d^{-1}x-1)=\prod_{k\in K_L}(x_L(k^{-1})d^{-1/k_L}x_L-1):= \prod_{k\in K_L}(d_k^{-1}x_L-1). \] With these notations, the above residue contribution is of the form \[ \int_{t^\prime\in tc_L^{-1}T^L_u} \left(B_{i_L-1}(D)
(((d_k^{-1}x_L-1)^{i_L}p/q)a)(r_Lt^Lt_L)\right)|_{t_L=1}d^L(t^L), \] where $B_{j}(T)\in{\mathbb Q}[T]$ is a certain polynomial of degree $j$. Note that there may exist other $L_{m^\prime}$ with $t\in L_{m^\prime,r}$. We pick up similar residues with respect to these cosets as well when we cross at $t$.
The above integral can be rearranged as follows \[
\sum_{j=0}^{i_L-1}\int_{ts_LT^L_u}(D^{j}(a)|_L)\omega_{j}, \] where $\omega_{j}$ is itself a rational $(n-1,0)$-form on $L$ which is a linear combination \[ \omega_j=\sum_i f_{j,i}\omega_{j,i} \] on $L$ with regular holomorphic coefficients $f_{j,i}$, and $(n-1,0)$-forms $\omega_{j,i}$ which factor as in Definition \ref{dfn:om}. The forms $\omega_{j,i}$ have poles along the intersections $L^\prime_n=L\cap L_{n}$ (with $n\in \mathcal{M}$) which are of codimension $1$ in $L$. A simple computation shows that we can choose this decomposition of $\omega_j$ such that for every ${j,i}$ and every connected component $H$ of an intersection of cosets of the form $L_n^\prime\subset L$, the index $i_{\omega_{j,i},H}$ of $H\subset L$ satisfies $i_{\omega_{j,i},H}\leq (i_{\omega,H}-1)-j$. It follows that the union over all $j,i$ of the $\omega_{j,i}$-residual cosets in $L$ is contained in the collection of $\omega$-residual cosets of $T$. Moreover, when we take $r_L$ as a base point of $L$, so that we identify $L$ with $T^L$ through the map $t^L\to r_Lt^L$, then the tempered form of a $\omega_{j,i}$-residual coset in $L$ is equal to its tempered form as a $\omega$-residual coset in $T$. By the induction hypotheses we can thus rewrite the residue on $L$ in the desired form, where the role of the identity element in the coset $L$ is now played by $r_L$.
At the identity $e\in T$ itself we have to take a boundary value of $\omega$ towards $T_u$, which defines a distribution on $T_u$. This proves the existence.
The uniqueness is proved as follows. Suppose that we have a collection $\{\mathfrak{Y}_c\in C^{-\infty}(cT_u)\}_{c\in \mathcal{C}^{\omega}}$ of distributions such that \begin{enumerate} \item[(i)] $\operatorname{supp}(\mathfrak{Y}_c)\subset S_c$. \item[(ii)] $\forall a\in {\mathbb C}[T]:
\sum_{c\in \mathcal{C}^{\omega}}\mathfrak{Y}_c(a|_{cT_u})=0$. \end{enumerate}
We show that $\mathfrak{Y}_c=0$ by induction on $|\gamma=\log(c)|$. Choose $c\in \mathcal{C}^{\omega}$ such that $\mathfrak{Y}_{c^\prime}=0$ for all $c^\prime$ with
$|\gamma^\prime|<|\gamma|$. For each $L\in\mathcal{L}^{\omega}$ with $|\gamma_L|\geq|\gamma|$ and $\gamma_L\not=\gamma$ we choose a $l\in \mathcal{M}_L$ such that $x_l(t)-d_l$ does not vanish on $cT_u$ (Proposition \ref{prop:triv}) and we set \[
\nu(t):=\prod_{\{L:|\gamma_L|\geq|\gamma|\text{\ and\ }\gamma_L\not=\gamma\}}(x_l(t)-d_l). \] It is clear that for sufficiently large $N\in {\mathbb N}$, $\mathfrak{Y}_c(\nu^Na)=0$ for all $a\in {\mathbb C}[T]$. On the other hand, by the theory of Fourier series of distributions on $T_u$,
${\mathbb C}[T]|_{cT_u}$ is a dense set of test functions on $cT_u$. Since
$\nu^N$ is nonvanishing on $cT_u$, this function is a unit in the space of test functions in $cT_u$. Thus also $\nu^N{\mathbb C}[T]|_{cT_u}$ is dense in the space of test functions. It follows that $\mathfrak{Y}_c=0$. \end{proof} \subsubsection{Approximating sequences}\label{subsub:approx} There is an ``analytically dual'' formulation of the result on residue distributions that will be useful later on. The idea to deal with the residue distributions in this way was inspired by the approach in \cite{H} to prove the positivity of certain residual spherical functions. \begin{lemma}\label{lem:approx} For all $N\in {\mathbb N}$ there exists a collection of sequences $\{a_n^{N,c}\}_{n\in{\mathbb N}}$ ($c\in \mathcal{C}^\omega$) in $\mathcal{A}$ with the following properties: \begin{enumerate} \item[(i)] For all $n\in {\mathbb N}$, $\sum_{c\in \mathcal{C}^\omega}a^{N,c}_n=1$. \item[(ii)] For every holomorphic constant coefficient differential operator $D$ of order at most $N$ on $T$, $D(a^{N,c}_n)\to D(1)$ uniformly on $S_c$ and $D(a^{N,c}_n)\to 0$ on $S_{c^\prime}$ if $c^\prime\not=c$. \end{enumerate} \end{lemma} \begin{proof} We construct the sequences with induction on the norm
$|\gamma=\log(c)|$. We fix $N$ and suppress it from the notation. Let $c\in\mathcal{C}^\omega$ and assume that we have already constructed such sequences $a^{c^\prime}_n$ satisfying (ii) for all $c^\prime$ with
$|\gamma^\prime|<|\gamma|$. Consider the function $\nu$ constructed in the second part of the proof of Lemma \ref{thm:resbasic}. By Fourier analysis on $cT_u$ it is clear that there exists a sequence $\{\phi_n\}_{n\in {\mathbb N}}$ in ${\mathbb C}[T]$ such that for each holomorphic constant coefficient differential operator $D$ of order at most $N$ there exists a constant $c_D$ such that \[
\Vert (D(\phi_n)-D(\nu^{-(N+1)}))|_{cT_u}\Vert_\infty<c_D/n \] Applying Leibniz' rule to $\nu^{(N+1)}\phi_n-1=\nu^{(N+1)}(\phi_n-\nu^{-(N+1)})$ repeatedly we see that this implies that there exists a constant $c^\prime_D$ for each holomorphic constant coefficient differential operator $D$, such that \[ \Vert
(D(\nu^{(N+1)}\phi_n)-D(1))|_{cT_u}\Vert_\infty<c_D^\prime/n. \] Notice that $D(\nu^{(N+1)}\phi_n)=0$ on all $S_{c^\prime}$ with
$|\gamma^\prime|\geq|\gamma|$ but $\gamma^\prime\not=\gamma$. On the other hand, for each holomorphic constant coefficient differential operator $E$ the function
$E(1-\sum_{\{c^\prime\mid|\gamma^\prime|<|\gamma|\}} a^{c^\prime}_{k})$ converges uniformly to zero on each $S_{c^\prime}$ with
$|\gamma^\prime|<|\gamma|$. Again applying Leibniz' rule repeatedly we see that there exist a $k\in {\mathbb N}$ (depending on $n$) such that the function \[
a^{c}_n:=\nu^{(N+1)}\phi_n(1-\sum_{\{c^\prime\mid|\gamma^\prime|<|\gamma|\}} a^{c^\prime}_{k}) \] has the property that \[
\Vert D(a^{c}_n)|_{\cup S_{c^\prime}}\Vert_\infty<c^\prime_D/n, \] where the union is taken over all $c^\prime$ with
$|\gamma^\prime|<|\gamma|$. It is clear that the sequence $a^c_n$ thus constructed satisfies (ii). We continue this process until we have only one center $c$ left. For this last center we can simply put \[ a_n^c:=1-\sum_{c^\prime\not=c}a_n^{c^\prime}. \] This satisfies the property (ii), and forces (i) to be valid. \end{proof} The use of such collections of sequences is the following: \begin{prop} In the situation of Lemma \ref{thm:resbasic} and given any collection of sequences $\{a_n^c\}$ as constructed in Lemma \ref{lem:approx} we can express the residue distributions as (with $a\in\mathcal{A}$): \[ \mathfrak{X}_c(a)=\lim_{n\to\infty}\tau(a_n^ca), \] provided $N$ (in Lemma \ref{lem:approx}) is chosen sufficiently large. \end{prop} \begin{proof} Because we are working with distributions on compact spaces, the orders of the distributions are finite. Take $N$ larger than the maximum of all orders of the $\mathfrak{X}_c$. By Proposition \ref{prop:triv} we can thus express $\mathfrak{X}_{c^\prime}$ as a sum of derivatives of order at most $N$ of measures supported on $S_{c^\prime}$. The result now follows directly from the defining properties of the sequence $a_n^c$. \end{proof} \subsubsection{Cycles of integration}\label{subsub:cycle} Yet another useful way to express the residue distribution is by means of integration of $a\omega$ over a suitable compact n-cycle. The results of this subsection will be needed later on to compute certain residue distributions at ``generic'' points of their support.
In the proposition below we will use the distance function on $T$ which measures the distance between $2\pi i Y$-orbits in $\mathfrak{t}_{\mathbb C}$. For $\delta>0$ and each $L$ which is a connected component of an intersection of codimension $1$ cosets $L_m\subset T$ with $m\in \mathcal{M}$, we denote by $\mathcal{B}_L(r_L,\delta)$ \index{B@$\mathcal{B}_L(r_L,\delta)$, ball in $T_L$, center $r_L$ and radius $\delta$} a ball in $T_L$ with radius $\delta$ and center $r_L$, and by $\mathcal{B}^L_{rs}(\delta)$ \index{B@$\mathcal{B}^L_{rs}(\delta)$, ball in $T^L_{rs}$, radius $\delta$ and center $e$} a ball with radius $\delta$ and center $e$ in $T^L_{rs}$. We put $\mathcal{M}_L\subset\mathcal{M}$ for the $m\in\mathcal{M}$ such that $L\subset L_m$, and $\mathcal{M}^L\subset \mathcal{M}$ for the $m\in\mathcal{M}$ such that $L_m\cap L$ has codimension $1$ in $L$. We write $T^m=\{t\mid x_m(t)=1\}$.
Let $U^L(\delta)\subset T^L$ be the open set $\{t\in T^L\mid\forall m\in\mathcal{M}^L: t\overline{\mathcal{B}_L(r_L,\delta)}\cap L_m=\emptyset\}$. Note that $U^L(\delta_1)\subset U^L(\delta_2)$ if $\delta_1>\delta_2$, and that the union of these open sets is equal to the complement of union of the codimension $1$ subsets $r_L^{-1}(L\cap L_m)\subset T^L$ with $m\in \mathcal{M}^L$. \begin{prop}\label{prop:cycle} Let $\epsilon>0$ be such that for all $m\in\mathcal{M}$ and $L\in\mathcal{L}^\omega$, $L_m\cap\mathcal{B}_L(r_L,\epsilon)\mathcal{B}^L_{rs}(\epsilon)T_u^L\not=\emptyset$ implies that $L^{temp}\cap L_m\not=\emptyset$. Denote by $\mathcal{M}^{L,temp}$ the set of $m\in \mathcal{M}^L$ such that $L^{temp}\cap L_m\not=\emptyset$. There exist \begin{enumerate} \item[(i)] $\forall L\in\mathcal{L}^\omega$, a point $\epsilon^L\in\mathcal{B}^L_{rs}(\epsilon)\backslash\cup_{m\in\mathcal{M}^{L,temp}}T^m$, \item[(ii)] a $0<\delta<\epsilon$ such that $\forall L\in\mathcal{L}^\omega,\ \epsilon^LT^L_u \subset U^L(\delta)$, and \item[(iii)] $\forall L\in\mathcal{L}^\omega$, a compact cycle $\xi_L\subset\mathcal{B}_L(r_L,\delta)\backslash\cup_{m\in\mathcal{M}_L}L_m$ \index{0o@$\xi_L$, compact cycle in $\mathcal{B}_L(r_L,\delta)\backslash\cup_{m\in\mathcal{M}_L}L_m$} of dimension $\operatorname{dim}_{\mathbb{C}}(T_L)$, \end{enumerate} such that \begin{equation} \forall c\in\mathcal{C}^\omega, \forall\phi\in C^\infty(cT_u):\mathfrak{X}_c(\phi)=\sum_{\{L\mid c_L=c\}}\mathfrak{X}_L(\phi), \end{equation} where $\mathfrak{X}_L$ \index{X4@$\mathfrak{X}_L$, contribution to $\int_{t_0T_u}a\omega$ supported on $L^{temp}$} is the distribution on $cT_u$ with support $L^{temp}$ defined by \begin{equation} \forall a\in\mathcal{A}:\mathfrak{X}_L(a)=\int_{\epsilon^LT_u^L\times\xi_L}a\omega. \end{equation} If $\mathcal{M}^{L,temp}=\emptyset$ we may take $\epsilon^L=e$. \end{prop} \begin{proof} We begin the proof by remarking that (i), (ii) and (iii) imply that the functional $\mathfrak{X}_L$ on $\mathcal{A}$ indeed defines a distribution on $c_LT_u$, supported on $L^{temp}$. Consider for $t\in U^L(\delta)$ the inner integral \begin{equation}\label{eq:inner} \int_{t\xi}a\omega:=i(a,t)d^Lt. \end{equation} Then $i(a,t)$ is a linear combination of (possibly higher order) partial derivatives $D_\kappa a$ of $a$ at $r_Lt$ in the direction of $T_L$, with coefficients in the ring of meromorphic functions on $T^L$ which are regular outside the codimension $1$ intersections $r_L^{-1}(L\cap L_m)$: \begin{equation} i(a,t)=\sum_{\kappa}f_\kappa D_{\kappa} a. \end{equation} Hence $\mathfrak{X}_L(a)$ is equal to the sum of the boundary value distributions $\operatorname{BV}_{\epsilon^L,f_\kappa}$ of the meromorphic coefficient functions, applied to the corresponding partial derivative $D_\kappa a$ of $a$, restricted to $L^{temp}$: \begin{equation}
\mathfrak{X}_L(a)=\sum_\kappa \operatorname{BV}_{\epsilon^L,f_\kappa}(D_\kappa a|_{L^{temp}}). \end{equation} We see that $\mathfrak{X}_L$ is a distribution supported in $L^{temp}\subset c_LT_u$, which only depends on $\xi_L$ and on the component of $\mathcal{B}^L_{rs}(\epsilon)\backslash\cup_{m\in\mathcal{M}^{L,temp}}T^m$ in which $\epsilon^L$ lies.
Hence, by the uniqueness assertion of Lemma \ref{thm:resbasic}, we conclude that it is sufficient to prove that we can choose $\epsilon^L,\ \delta,\ \xi_L$ in such a way that \begin{equation} \forall a\in\mathcal{A}: \int_{t_0T_u}a\omega=\sum_{L\in\mathcal{L}^\omega}\mathfrak{X}_L(a). \end{equation} In order to prove this it is enough to show that we can choose $\epsilon^L,\ \delta,\ \xi_L$ as in (i), (ii) and (iii) for the larger collection $\tilde\mathcal{L}^\omega$ of all the connected components of intersections of the $L_m$ (with $m\in\mathcal{M}$), such that \begin{equation} t_0T_u\sim\cup_{L\in\tilde\mathcal{L}^\omega}\epsilon^LT^L_u\times\xi_L. \end{equation} Here $\sim$ means that the left hand side and the right hand side are homologous cycles in $T\backslash \cup_{m\in\mathcal{M}}L_m$. The desired result follows from this, since the functional $\mathfrak{X}_L$ is equal to $0$ unless $L$ is $\omega$-residual (because the inner integral (\ref{eq:inner}) is identically equal to $0$ for non-residual intersections, by an elementary argument which is given in detail in the proof of Theorem \ref{thm:nu}).
Let $k\in\{0,1,\dots,n-1\}$. Denote by $\tilde\mathcal{L}^\omega(k)$ the collection of connected components of intersections of the $L_m$ ($m\in\mathcal{M}$) such that $\operatorname{codim}(L)<k$. Assume that we already have constructed points $\epsilon^L,\ \delta,\ \xi_L$ satisfying (i), (ii) and (iii) for all $L\in\tilde\mathcal{L}^\omega(k)$ and in addition, for each $L\in\tilde\mathcal{L}^{\omega}$ with $\operatorname{codim}(L)=k$, a finite set of points $\Omega_L\subset T^L_{rs}$ such that $\Omega_L T^L_u\subset U^L(\delta)$ and a cycle $\xi_{L,w}\subset\mathcal{B}_L(r_L,\delta)\backslash\cup_{m\in\mathcal{M}_L}L_m$ for each $w\in \Omega_L$, such that $t_0T_u$ is homologous to \begin{equation}\label{eq:induct} \cup_{L\in\tilde\mathcal{L}^\omega(k)}(\epsilon^LT^L_u\times\xi_L) \cup\cup_{L\in\tilde\mathcal{L}^\omega(k+1)\backslash\tilde\mathcal{L}^\omega(k)} \cup_{w\in\Omega_L}(wT^L_u\times\xi_{L,w}). \end{equation} This equation holds for $k=0$, with $\Omega_T=\{t_0\}$, which is the starting point of the inductive construction to be discussed below. We will construct $\epsilon^L$, $\delta_1$ and $\xi_L$ for $L\in\tilde\mathcal{L}^\omega(k+1)\backslash\tilde\mathcal{L}^\omega(k)$, and finite sets $\Omega_L$ for $L\in\tilde\mathcal{L}^\omega(k+2)\backslash\tilde\mathcal{L}^\omega(k+1)$, with a cycle $\xi_w$ for each $w\in\Omega_L$ such that equation (\ref{eq:induct}) holds with $k$ replaced by $k+1$, and $\delta$ by $\delta_1$.
First of all, notice that we may replace $\delta$ by any $0<\delta^\prime<\delta$ in equation (\ref{eq:induct}), because we can shrink the $\xi_L$ and $\xi_{L,w}$ within their homology class to fit in the smaller sets $\mathcal{B}_L(r_L,\delta^\prime)\backslash\cup_{m\in\mathcal{M}_L}L_m$. Choose $\delta^\prime$ small enough such that for each $L\in \tilde\mathcal{L}^\omega(k+1)\backslash\tilde\mathcal{L}^\omega(k)$ there exists a point $\epsilon^L\in \mathcal{B}^L_{rs}(\epsilon)$ with the property that $\epsilon^LT^L_u\subset U^L(\delta^\prime)$.
The singularities of the inner integral are located at codimension $1$ cosets in $T^L$ of the form $r_L^{-1}N$, where $N$ is a connected component of $L\cap L_m$ for some $m\in\mathcal{M}^L$. We have $r_L^{-1}N=r_L^{-1}r_NT^N\subset T^L$, and thus $c_L^{-1}c_NT^N_{rs}\subset T^L_{rs}$. Choose paths inside $T^L_{rs}$ from $w\in\Omega_L$ to the point $\epsilon^L$. We choose each path such that it intersects the real cosets $c_L^{-1}c_NT^N_{rs}$ transversally and in at most one point, and such that these intersection points are distinct. If $p=\gamma(x_0)$ is the intersection point with the path $\gamma$ from $w\in\Omega_L$ to $\epsilon^L$ then $p$ is of the form $p=c_L^{-1}c_Nw_{L,N,w}\in c_L^{-1}c_NT^N_{rs}$ with $w_{L,N,w}\in T^N_{rs}$. Given $N\in \tilde\mathcal{L}^\omega(k+2)\backslash\tilde\mathcal{L}^\omega(k+1)$ we denote by $\Omega_N$ the set of all $w_{L,N,w}$ arising in this way, for all the $L\in\tilde\mathcal{L}^\omega(k+1)\backslash\tilde\mathcal{L}^\omega(k)$ such that $L\supset N$, and $w\in\Omega_L$.
Notice that if $v=w_{L,N,w}\in\Omega_N$ and $vs\in r_N^{-1}(N\cap L_m)$ for some $m\in\mathcal{M}^N$ and $s\in T_u$, we have that $c_L^{-1}c_Nv\in c_L^{-1}(c_NT^N_{rs}\cap c_{N^\prime}T^{N^\prime}_{rs})$ where $N^\prime=L\cap L_m$. Since $T^{N^\prime}\not=T^N$, this contradicts the assertion that the intersection points of the paths in $T^L_{rs}$ and the cosets $c_L^{-1}c_NT^N_{rs}$ are distinct. We conclude in particular that the compact set $\Omega_NT^N_u$ is contained in the union of the open sets $U^N(\delta^\prime)$. We can thus choose $\delta^\prime$ small enough such that in fact $\Omega_NT^N_u\subset U^N(\delta^\prime)$, as required in equation (\ref{eq:induct}).
Write $T_{N\subset L}$ for the identity component of the $1$-dimensional intersection $T_N\cap T^L$, and decompose the torus $T^L$ as the product $T^N\times T_{N\subset L}$. Let $v=w_{L,N,w}\in\Omega_N$ and put $p=c_L^{-1}c_Nv$ for the corresponding intersection point in $T^L_{rs}$. Notice that for a codimension $1$ coset $r_L^{-1}N^\prime\subset T^L$ with $N^\prime\in\tilde\mathcal{L}^\omega$ we have that \begin{equation} pT_{N\subset L,u}\cap r_L^{-1}N^\prime=\left\{ \begin{aligned}{} &\emptyset\text{\ if\ } c_L^{-1}c_{N^\prime}T^{N^\prime}\not=c_L^{-1}c_NT^{N}, \\ &G_{L,N^\prime,w}\text{\ otherwise}\\ \end{aligned} \right. \end{equation} where $G_{L,N^\prime,w}$ is a coset of the subgroup $T_{N\subset L}\cap T^N$ of the finite group $K_{N^\prime}=K_N=T_N\cap T^N\subset T^N_u$, of the form \begin{equation} G_{L,N^\prime,w}=(T_{N\subset L}\cap T^N)r_L^{-1}r_{N^\prime}v. \end{equation} The cosets $G_{L,N^\prime,w}$ are disjoint. Let $\delta(L,w)$ be the minimum distance of two points in the union of these cosets, and let $\delta(k+1)$ denote the minimum of the positive real numbers $\delta(L,w)$ when we vary over all the $L$ and $w\in\Omega_L$. Choose $\delta_1>0$ smaller than the minimum of $\delta^\prime$ and $\delta(k+1)$. Let $\eta$ be a circle of radius $\delta_1/2$ with center $e$ in $T_{N\subset L}$. Next we make $\delta^\prime$ sufficiently small so that $\cup_{N^\prime}G_{L,N^\prime,w}\eta\subset U^L(\delta^\prime)$. For $x_-,x_+$ suitably close to $x_0$ with $x_-<x_0<x_+$ we have in $U^L(\delta^\prime)$: \begin{equation}\label{eq:point} \gamma(x_-)T_{N\subset L,u}\sim\gamma(x_+)T_{N\subset L,u}\cup \cup_{N^\prime}G_{L,N^\prime,w}\eta, \end{equation} where the union is over all $N^\prime\subset L$ such that $c_L^{-1}c_{N^\prime}T^{N^\prime}=c_L^{-1}c_NT^{N}$. Define \begin{equation} \xi_{L,N^\prime,v}:=r_L^{-1}r_{N^\prime}\eta\times\xi_{L,w}. \end{equation} Observe that $T_{N\subset L,u}\times T^N_u$ is a
$|T_{N\subset L}\cap T^N|$-fold covering of $T^L_u$, and that $g\eta\times vT^N_u\sim g^\prime\eta\times vT^N_u$ if $g,g^\prime\in G_{L,N^\prime,w}$. We thus have \begin{equation}\label{eq:subscycle} \gamma(x_-)T_u^L\times \xi_{L,w}\sim\gamma(x_+)T^L_u\times\xi_{L,w}\cup \cup_{N^\prime}vT_u^N\times \xi_{L,N^\prime,v}. \end{equation} By possibly making $\delta^\prime$ smaller we get that $\xi_{L,N,v}\subset \mathcal{B}_N(r_{N},\delta_1)$ for all possible choices $N, L$ and $w$. If $L_m\supset N$ but $L_m\not\supset L$, then, since $r_L^{-1}r_{N}\eta\subset U^L(\delta^\prime)$ and $\xi_{L,w}\subset \mathcal{B}_L(r_L,\delta^\prime)$, we have $\xi_{L,N,v}\cap L_m=\emptyset$. If on the other hand $L_m\supset L$ then $\xi_{L,N,v}\cap L_m=r_L^{-1}r_{N}\eta\times(\xi_{L,w}\cap L_m)=\emptyset$. Finally we put \begin{equation} \xi_{N,v}:=\cup_{(L,w)}\xi_{L,N,v}, \end{equation} where we take the union over all pairs $(L,w)$ with $L\in\tilde\mathcal{L}^\omega(k+1)\backslash\tilde\mathcal{L}^\omega(k)$ such that $L\supset N$ and $w\in\Omega_L$ such that there is an intersection point $w_{L,N,w}$ with $w_{L,N,w}=v$. We have shown that \begin{equation} \xi_{N,v}\subset\mathcal{B}_N(r_{N},\delta_1)\backslash\cup_{m\in\mathcal{M}^{N}}L_m, \end{equation} as required in equation (\ref{eq:induct}).
Applying equation (\ref{eq:subscycle}) for all the intersections of all the paths we chose, we obtain equation (\ref{eq:induct}) with $k$ replaced by $k+1$ and $\delta$ by $\delta_1$. We thus take $\xi_L=\cup_{w\in\Omega_L}\xi_{L,w}$ for $L\in\tilde\mathcal{L}^\omega(k+1)\backslash\tilde\mathcal{L}^\omega(k)$, and for $N\in\tilde\mathcal{L}^\omega(k+2)\backslash\tilde\mathcal{L}^\omega(k+1)$ we take $\Omega_N$ and $\xi_{N,v}$ as constructed above.
This process continues until we have $k=n-1$ in equation (\ref{eq:induct}). In the next step we proceed in the same way. Notice that for $N\in\tilde\mathcal{L}^\omega(n+1)\backslash\tilde\mathcal{L}^\omega(n)$, either $\Omega_N=\{e\}$ (if we cross $c_L^{-1}c_N$ with some curve from $\Omega_L$ to $\epsilon_L$ in $T^L_{rs}$, for one of the one dimensional residual cosets $L$ containing $N$), or else $\Omega_N=\emptyset$. The process now stops, since also $\epsilon^N=e$. This proves the desired result, with $\delta$ equal to the $\delta_1$ obtained in the last step of the inductive construction. \end{proof} \begin{rem} The homology classes of the cycles $\xi_L$ are not uniquely determined by the above algorithm. The splitting $\mathfrak{X}_c=\sum_{\{L\mid c_L=c\}}\mathfrak{X}_L$ is not unique without further assumptions. However, in our application to spectral theory of $\mathfrak{C}$, we shall see that the decomposition $\mathfrak{X}_c=\sum_{\{L\mid c_L=c\}}\mathfrak{X}_L$ is such that each $\mathfrak{X}_L$ is a regular measure supported on $L^{temp}$, and such a decomposition is of course unique. \end{rem} We list some useful properties of the cycles $\xi_L$. We fix $\omega$, and suppress it from the notation. \begin{dfn} Let $L\in\mathcal{L}$. Denote by $\mathcal{L}^L$ \index{L@$\mathcal{L}^L$, real projections of residual cosets $\supset L$} the configuration of real cosets $M^L:=c_LT^M_{rs}$ where $M\in\mathcal{L}$ such that $M\supset L$, $M\not= T$. The ``dual'' configuration, consisting of the cosets $M_L:=c_LT_{M,rs}\subset T_L$ with $M\in\mathcal{L}$ such that $M\supsetneqq L$, is denoted by $\mathcal{L}_L$ \index{L@$\mathcal{L}_L$, dual configuration of $\mathcal{L}^L$}. Given an (open) chamber $C$ in the complement of $\mathcal{L}^L$, we call $C^d=\{c_L\exp(v)\mid (v,w)<0\forall w\in\overline{\log(c_L^{-1}C)}\backslash\{0\}\}$ the anti-dual cone. This anti-dual cone is the interior of the closure of a union of chambers of the dual configuration $\mathcal{L}_L$ in $T_L$. We denote by $\mathcal{L}(L)$ \index{L@$\mathcal{L}(L)$, intersection of $T_L$ with residual cosets $\supset L$} the residual cosets in $T_L$ with respect to $K_L$-invariant divisor $\sum_{m\in\mathcal{M}_L}(L_m\cap T_L)-\sum_{m^\prime\in\mathcal{M}_L^\prime} (L_{m^\prime}\cap T_L)$ on $T_L$. \end{dfn} \begin{prop}\label{prop:t0} \begin{enumerate} \item[(i)] If $t_0$ is moved inside a chamber of $\mathcal{L}^L$ we can leave $\xi_L$ unchanged. \item[(ii)] Let $t_0(L)=T^L_{rs}t_0\cap T_L$. For each $k\in K_L:=T^L\cap T_L$, we can choose the cycle $\xi_{kr_L}(L)$ (defined with respect to the configuration $\mathcal{L}(L)$ in $T_L$ and initial point $t_0(L)\in T_L$) equal to $k\xi_L$. \end{enumerate} \end{prop} \begin{proof} (i) If $t_0$ is moved within a chamber of $\mathcal{L}^L$, the path from $t_0$ to $e$ can be chosen equal to the original path up to a path which only crosses codimension one cosets of the form $L_mT_u\cap T_{rs}$ which do not contain $c=c_L$. Therefore this does not change $\xi_L$.
(ii) We may replace $\mathcal{M}$ by $\mathcal{M}_L$ and $\mathcal{M}^\prime$ by $\mathcal{M}_L^\prime$ and leave $\xi_L$ is unchanged, because the $L_m\not\supset L$ do not contribute to $\xi_L$ in the procedure of the proof of Proposition \ref{prop:cycle}. By (i) we may also replace $t_0$ by $t_0(L)$ without changing $\xi_L$. We apply Proposition \ref{prop:cycle} in this situation in $T$. Then we intersect with $T_L$ and use the formula $T_L\cap(T^L_u\times\xi_L)=\sum_{k\in K_L}k\xi_L$. \end{proof} \begin{prop}\label{prop:regequiv} Write $L=r_LT^L=c_Ls_LT^L$ as usual, and let $M\in\mathcal{L}$. Then $L^{temp}\subset M^{temp}$ if and only if $L\subset M$ and $e\in M_L$. In particular, $L^{temp}$ is maximal in the collection of $\omega$-tempered cosets if and only if $e$ is regular with respect to the configuration $\mathcal{L}_L$. \end{prop} \begin{proof} If $M\in\mathcal{L}$, then $L^{temp}\subset M^{temp}\Leftrightarrow L\subset M\mathrm{\ and\ }c_L=c_M$ (since then $s_L\in(c_M^{-1}M)\cap T_u=s_MT_u^M$, implying that $r_L\in M^{temp}$). Now $c_L=c_M\Leftrightarrow c_L\in T_M\Leftrightarrow e\in M_L$. \end{proof} \begin{prop}\label{prop:antidual}(cf. \cite{HOH0}, Lemma 3.3.) If $e$ is not in the closure of the anti-dual cone of the chamber of $\mathcal{L}^L$ in which $t_0$ lies, we can take $\xi_L=\emptyset$. \end{prop} \begin{proof} By Proposition \ref{prop:t0} it is sufficient to show this in the case where $L=r_L$ is a residual point.
We identify $T_{rs}$ with the real vector space $\mathfrak{t}$ via the map $t\to \log(c_L^{-1}t)$, and we denote by $\langle\cdot,\cdot\rangle$ \index{<@$\langle\cdot,\cdot\rangle$!Euclidean inner product on $T_{rs}$} the Euclidean inner product thus obtained on $T_{rs}$. Notice that the role of the origin is played by $c_L$. The sets $M^L=c_MT^M_{rs}$ with $M\in\mathcal{L}(L)$ satisfy $c_MT^M_{rs}\ni c_L$, and are equipped with the induced Euclidean inner product.
By the assumption and Proposition \ref{prop:t0} we can choose $t_0$ within its chamber such that $\langle t_0, e\rangle>0$. Assume by induction that in the $k$-th step of the inductive process of Proposition \ref{prop:cycle} we have, $\forall N\in\mathcal{L}(L)$ with $\operatorname{codim}(N)=k$ and $\forall w\in\Omega_N$, that \begin{equation}\label{eq:posi} \langle c_Nw,c_N \rangle>0 \end{equation} (see equation (\ref{eq:induct}) for the meaning of $\Omega_N$). Notice in particular that this implies that $\Omega_N=\emptyset$ if $c_N=c_L$. By choosing $\epsilon$ sufficiently small, we therefore have $\langle c_N\epsilon^N,c_N \rangle>0$ when $\Omega_N\not=\emptyset$. Take the path $\gamma$ in $T^N_{rs}$ from $w\in\Omega_N$ to $\epsilon^N$ equal to $c_N^{-1}[c_Nw,c_N\epsilon^N]$, where $[c_Nw,c_N\epsilon^N]$ denotes the (geodesic) segment from $c_Nw$ to $c_N\epsilon^N$ in the Euclidean space $c_NT_{rs}^N$. Consequently, we have $\langle x,c_N \rangle>0$ for all $x\in[c_Nw,c_N\epsilon^N]$. Let $M\subset\mathcal{L}(L)$ with $\operatorname{codim}(M)=k+1$ and $M\subset N$. If $\gamma$ intersects $c_N^{-1}c_MT^M_{rs}$ in $c_N^{-1}c_Mw_{N,M,w}$, then we have $0<\langle c_Mw_{N,M,w},c_N\rangle=\langle c_Mw_{N,M,w},c_M\rangle$. By induction on $k$ this proves that we can perform the contour shifts in such a way that (\ref{eq:posi}) holds for each $k\in\{0,\dots,\operatorname{codim}(L)\}$. This implies that $\Omega_L=\emptyset$, and thus that $\xi_L=\emptyset$. \end{proof} In the next proposition we view the constants $d_m$ as variables. We choose a continuous path $[0,1]\ni \sigma\to(d_m(\sigma))_{m\in\mathcal{M}}$ from $(d_m)_{m\in\mathcal{M}}$ to $(d_m^\prime)_{m\in\mathcal{M}}$, and consider the resulting deformation of $\omega$ and $\mathcal{L}$. The end point of the path corresponds to the form $\omega^\prime$ and its collection of $\omega^\prime$-residual cosets, denoted by $\mathcal{L}^\prime$. Recall that $\mathcal{M}_L$ denotes the multiset of $m\in\mathcal{M}$ such that $L_m\supset L$. Assume that $\cap_{m\in\mathcal{M}_L}L_m(\sigma)\not=\emptyset$ for all $\sigma$. In this situation there exists a continuous path $\sigma\to r_L(\sigma)$ such that $L(\sigma):=r_L(\sigma)T^L$ is a connected component of $\cap_{m\in\mathcal{M}_L}L_m(\sigma)$. We may take $r_L(\sigma)\in T_L\cap L(\sigma)$. We put $L^\prime=L(1)$. Assume that $\{m\in\mathcal{M}\mid x_m(L^\prime)=d_m^\prime\}=\{m\in\mathcal{M}\mid x_m(L)=d_m\}$. \begin{prop}\label{prop:cyinv} Assume that $e(\sigma):=c_Lc_{L(\sigma)}^{-1}$ stays within a facet of $\mathcal{L}_L$ for all $\sigma$, and $t_0(\sigma):=t_0c_Lc_{L(\sigma)}^{-1}$ stays within a chamber of $\mathcal{L}^L$. With these assumptions we can take $\xi_{L^\prime}=r_L^{-1}r_{L^\prime}\xi_L$. \end{prop} \begin{proof} As above, we may assume that in fact $L=r_L$ is a point. The only contributions to $\xi_L$ come from contour shifts inside residual cosets of the configuration $\mathcal{L}(L)$ as in the proof of Proposition \ref{prop:antidual}. Likewise, for the construction of $\xi_{L^\prime}$ we only need to consider the translated configuration $r_L^{-1}r_{L^\prime}\mathcal{L}(L)$. By the assumption on $t_0$ and Proposition \ref{prop:t0} we can construct $r_Lr_{L^\prime}^{-1}\xi_{L^\prime}$ by working with $\mathcal{L}(L)$ and $t_0$, but with the center $e$ of $T$ replaced by $e^\prime:=e(1)$.
We now follow the deformations of the centers $c_M(\sigma)$ with $\sigma\in [0,1]$ and $M\in\mathcal{L}(L)$. The assumption on $e(\sigma)$ implies that $c_M\not=c_L\Leftrightarrow\forall \sigma:\ c_M(\sigma)\not=c_L$. This implies we can use $\xi_L$ also as the cycle associated with $L$ relative to the center $e^\prime$. \end{proof} \begin{rem} Note that for some $\sigma\in (0,1)$ there may be additional $M\in\mathcal{L}(\sigma)$ such that $L(\sigma)\subset M$. It may also happen that for some values of $\sigma\in [0,1]$, $L(\sigma)^{temp}$ contains smaller tempered cosets. We may need to adjust $\epsilon^{L(\sigma)}$ accordingly. \end{rem} \subsection{Application to the trace functional} We will now apply the above results to the integral (\ref{eq:basic}). We thus use the rational $(n,0)$-form \begin{equation}\label{eq:om} \eta(t):=\frac{dt}{q(w_0)^2\Delta(t)c(t,q)c(t^{-1},q)} \end{equation} and define the notion of {\it quasi}-residual coset \index{Quasi residual coset} as the $\eta$-residual cosets introduced above. We write $\mathcal{L}^{\operatorname{qu}}$ \index{L1@$\mathcal{L}^{\operatorname{qu}}$, collection of quasi-residual cosets} for the collection of these $\eta$-residual spaces, $\mathcal{C}^{\operatorname{qu}}$ \index{C@$\mathcal{C}^{\operatorname{qu}}$, centers of quasi-residual cosets} for their centers etc. Note: the collection of residual cosets of Appendix \ref{sub:defn} is {\it strictly} included in this collection.
Apply Lemma \ref{thm:resbasic} to $\eta$ of equation (\ref{eq:om}), with $t_0$ such that (\ref{eq:mininf}) is satisfied. Denote the resulting local distributions by $\mathfrak{X}_{\eta,c}$. \begin{prop}\label{prop:dfn} The collection $\{\mathfrak{X}_c^h\}_{c\in\mathcal{C}^{\operatorname{qu}},h\in\mathcal{H}}$ \index{X5@$\mathfrak{X}_c^h$, local contribution to $a\to\tau(ah)$ at $c$} of distributions $\mathfrak{X}_c^h\in C^{-\infty}(cT_u)$ defined by $\mathfrak{X}_c^h(a):=\mathfrak{X}_{\eta,c}(\{t\to a(t)E_t(h)\})$ satisfies \begin{enumerate} \item[(i)] $\operatorname{supp}(\mathfrak{X}_c^h)\subset S_c^{\operatorname{qu}}$ \index{S@$S_c^{\operatorname{qu}}$, support of $\mathfrak{X}_c^h$}. \item[(ii)] $\forall a\in \mathcal{A}:\ \tau(ah)=\sum_{c\in\mathcal{C}^{\operatorname{qu}}}\mathfrak{X}_c^h(a)$ (where $\mathfrak{X}_c(a)$ means
$\mathfrak{X}_c(a|_{cT_u})$). \item[(iii)] The application $h\to\mathfrak{X}_c^h$ is ${\mathbb C}$-linear. \item[(iv)] $\forall a,b\in\mathcal{A},h\in\mathcal{H}:\mathfrak{X}_c^{ah}(b)=\mathfrak{X}_c^h(ab)$. \end{enumerate} \end{prop} \begin{proof} These properties are simple consequences of \ref{fundeis}. \end{proof} \subsubsection{Symmetrization and positivity}\label{subsub:pos} The main objects of this section are the $W_0$-symmetric versions of the local distributions $\mathfrak{X}_c^h$. \begin{dfn}\label{dfn:Y} Let $\mathcal{C}_-^{\operatorname{qu}}$ \index{C@$\mathcal{C}_-^{\operatorname{qu}}$, quasi-residual centers in $\overline{T_{rs,-}}$} denote the set of elements in $\mathcal{C}^{\operatorname{qu}}$ which lie in the closure of the negative chamber $T_{rs,-}=\{t\in T_{rs}\mid \forall\alpha\in R_{0,+}:\alpha(t)< 1\}$. For $h\in \mathcal{H}$, $a\in \mathcal{A}$, and $c\in\mathcal{C}_-^{\operatorname{qu}}$ put: \begin{equation} \mathfrak{Y}_c^h(a):=\sum_{c^\prime\in W_0c}\mathfrak{X}^h_{c^\prime}(\bar{a}), \end{equation} \index{Y@$\mathfrak{Y}_c^h$, symmetrized local contribution to $a\to\tau(ah)$ at $c$}
where $\bar{a}:=|W_0|^{-1}\sum_{w\in W_0}a^w$. Then $\mathfrak{Y}_c^h$ is a $W_0$-invariant distribution on $\cup_{c^\prime\in W_0c}c^\prime T_u$, with support in $W_0S_c^{\operatorname{qu}}$, such that for all $z\in \mathcal{Z}$: \begin{equation}\label{eq:restz} \tau(zh)=\sum_{c\in\mathcal{C}_-^{\operatorname{qu}}}\mathfrak{Y}_c^h(z). \end{equation} \end{dfn} It is elementary to compute the distribution $\mathfrak{Y}_c^h$ when $c=e$. Recall that (\cite{EO}, Corollary 2.26) we have the following identity for the character of the minimal principal series $I_t$: \begin{equation} \chi_{I_t}=q(w_0)^{-1}\sum_{w\in W_0}\Delta(wt)^{-1}E_{wt}. \end{equation} Hence we can write for all $z\in \mathcal{Z}$: \begin{equation} \begin{split} \mathfrak{Y}_e^h(z)&=\int_{T_u}z(t)E_t(h)\eta(t)\\ &=\int_{W_0\backslash T_u}z(t)\chi_{I_t}(h)d\mu_T(t), \end{split} \end{equation} where $\mu_T$ is the positive measure on $T_u$ given by \begin{equation}\label{eq:prin} d\mu_T(t):=\frac{dt}{q(w_0)c(t)c(t^{-1})}. \end{equation} Here we used the $W_0$-invariance of $c(t)c(t^{-1})$, and the fact that for $t\in T_u$ we have \begin{equation}
c(t)c(t^{-1})=c(t)c(\bar{t})=|c(t)|^2. \end{equation} We see that $h\to\mathfrak{Y}_e^h(1)$ is the integral of the function $T_u=S_e\ni t\to\chi_{I_t}(h)$ against a positive measure on $T_u$. Moreover, for every $t\in T_u$, the function $h\to\chi_{I_t}(h)$ is positive and central, and is a $\mathcal{Z}$-eigenfunction with character $t$. Our first task will be to prove these properties for arbitrary centers $c\in\mathcal{C}^{\operatorname{qu}}$. The main tools we will employ are the approximating sequences. \subsubsection{Positivity and centrality of the kernel} Let us choose, for a suitably large $N$, approximating sequences $a_n^c$ for the distributions $\mathfrak{X}_{\eta,c}$. We remark that the group $\pm W_0$ acts on the collection of quasi-residual subspaces. In addition, complex conjugation also leaves this collection stable. We define an action $\cdot$ on $\mathcal{A}$ of the group $G$ of automorphisms of $T$ generated by $W_0$, $\operatorname{inv}:t\to t^{-1}$ and $\operatorname{conj}:t\to \bar{t}$. For elements $g\in\pm W_0$ this action is given by $g\cdot a:=a^g$, and $(\operatorname{conj}\cdot a)(t):=\overline{a(\overline{t})}$. \begin{lemma}\label{lem:equiv} We can choose the $a_n^c$ in a $G$-equivariant way, i.e. such that $\forall g\in G:\ a_n^{gc}=g\cdot(a_n^c)$. \end{lemma} \begin{proof} Just notice that for any given collection of approximating sequences $A:=\{a^c_n\}$ and any $g\in G$, $g\cdot A=\{g\cdot a_n^{g^{-1}c}\}$ is also a collection of approximating sequences for the distributions $\mathfrak{X}_{\eta,c}$, and this defines an action of $G$ on the set of collections of approximating sequences for the $\mathfrak{X}_{\eta,c}$. Hence we can take the average over $G$. \end{proof} For $c\in\mathcal{C}^{\operatorname{qu}}_-$ we now define \begin{equation} z^c_n:=\sum_{c^\prime\in W_0c}a_n^{c^\prime}. \end{equation} Then these sequences in the center $\mathcal{Z}$ of $\mathcal{H}$ have the property that for all $c\in \mathcal{C}_-^{\operatorname{qu}}$, $z\in \mathcal{Z}$ and $h\in\mathcal{H}$: \begin{equation} \mathfrak{Y}_c^h(z)=\lim_{n\to\infty}\tau(z_n^czh). \end{equation}
It is easy to see that the map $h\to\mathfrak{Y}_c^h$ is central: \begin{prop}\label{prop:cent} For all $c\in\mathcal{C}_-^{\operatorname{qu}}$, we have $\mathfrak{Y}_c^h=0$ if $h$ is a commutator. \end{prop} \begin{proof} We compute $\mathfrak{Y}_c^h(z)=\lim_{n\to\infty}\tau(z^c_nzh)=0$, because $z^c_nzh$ is also a commutator and $\tau$ is central. \end{proof} We define an anti-holomorphic involutive map $t\to t^*$ \index{*@$*$!$t\to t^*$, anti-holomorphic involution on $T$} on $T$ by $t^*:=\overline{{t}^{-1}}$. In view of the action of conjugation on $\mathcal{A}$, we see that for all $z\in\mathcal{Z}$, $z^*(t)=\overline{z(t^*)}$. By Lemma \ref{lem:equiv} we have, for each $c\in \mathcal{C}^{\operatorname{qu}}_-$, \begin{equation} z_n^{c^*}(t)=z_n^c(t^{-1})=(z_n^c)^*(t). \end{equation}
Now we embark on the proof that the distributions $\mathfrak{Y}_c^h$ are in fact (complex) measures. \begin{lem}\label{lem:supp} \begin{enumerate} \item[(i)] If $c^*\not\in W_0c$ then $\mathfrak{Y}^h_c=0$. \item[(ii)] Let $c^*\in W_0c$, and $cs\in S_c^{\operatorname{qu}}$ such that $(cs)^*=c^{-1}s\not\in W_0(cs)$. Then $cs\not\in\operatorname{Supp}(\mathfrak{Y}_c^h)$. \end{enumerate} \end{lem} \begin{proof} (i). Any $h\in\mathcal{H}$ can be decomposed as $h=h_r+ih_i$ with $h_r^*=h_r$ and $h_i^*=h_i$, so it suffices to prove the assertion for $h\in \mathcal{H}^{re}$. Thus by Lemma \ref{lem:easy} it is sufficient to prove the assertion for a positive element $h\in \mathcal{H}_+$. Similarly $z\in\mathcal{Z}$ is a linear combination of positive central elements, so that it is sufficient to show that $\mathfrak{Y}_c^h(z)=\mathfrak{Y}_c^{zh}(1)=0$ for each positive central element $z$. By Lemma \ref{lem:easy} this reduces our task to proving that $\mathfrak{Y}_c^h(1)=0$ for an arbitrary element $h\in\mathcal{H}_+$. Then \begin{equation}
0\leq\lim_{n\to\infty}\tau(h(z^{c^*}_n+uz^c_n)^*(z^{c^*}_n+uz^c_n))= u\mathfrak{Y}_c^h(1)+\overline{u}\mathfrak{Y}_{c^*}^h(1).
\end{equation} It follows easily that $\mathfrak{Y}_{c^*}^h(1)=\mathfrak{Y}_c^h(1)=0$.
(ii). This is essentially the same argument that we used to prove (i). Since $(cs)^*\not\in W_0(cs)$, we can find an open neighborhood $U\ni cs$ in $cT_u$ such that $W_0U\cap U^*=\emptyset$. Let $\phi\in C_c^{\infty}(W_0U)^{W_0}$. Then $\phi^*\phi=0$, where $\phi^*(x):=\overline{\phi(x^*)}$. We want to prove that $\mathfrak{Y}_c^h(\phi)=0$ for $h\in\mathcal{H}_+$. By Fourier analysis on $cT_u$ we can find a sequence $f_n\in\mathcal{A}^{W_c}$ such that $D(f_n)$ converges uniformly to $D(\phi)$ on $cT_u$ for every holomorphic constant coefficient differential operator $D$ on $T$ of order at most $N$ on $T$. We can then find a sequence $g_n$ of the form $g_n=f_na_{k(n)}^c$ such that $D(g_n)$ converges uniformly to $D(\phi)$ on $S_c^{\operatorname{qu}}$, and to $0$ on $S_{c^\prime}^{\operatorname{qu}}$ for every $c^\prime\not=c$. Hence if we put \[ \phi_n=\sum_{w\in W^c}g_n^{w}\in\mathcal{Z}, \] then for each holomorphic constant coefficient differential operator $D$ on $T$ of order at most $N$, $D(\phi_n)\to D(\phi)$ uniformly on $W_0S_c^{\operatorname{qu}}$, and $D(\phi_n)\to 0$ uniformly on $S_{c^\prime}^{\operatorname{qu}}$ for $c^\prime\not\in W_0c$. Hence $\forall h\in\mathcal{H}_+, u\in{\mathbb C}$, \begin{equation} \begin{split} 0&\leq\lim_{n\to\infty}\tau(h(uz_n^c+\phi_n)^*(uz_n^c+\phi_n))\\&=
|u|^2\mathfrak{Y}^h_c(1)+u\mathfrak{Y}^h_c(\phi^*)+\overline{u}\mathfrak{Y}_c^h(\phi) \end{split} \end{equation}
If we divide this inequality by $|u|$ and send $|u|$ to $0$, we get that
$\forall\epsilon\in{\mathbb C}$ with $|\epsilon|=1$, \begin{equation} 0\leq \epsilon\mathfrak{Y}^h_c(\phi^*)+\overline{\epsilon}\mathfrak{Y}_c^h(\phi) \end{equation} It follows that $\forall h\in\mathcal{H}_+$, $\mathfrak{Y}_c^h(\phi)=\mathfrak{Y}_c^h(\phi^*)=0$. Hence the same is true for arbitrary $h\in \mathcal{H}$. \end{proof} \begin{cor}\label{cor:pos} If $h\in\mathcal{H}_+$, the distribution $\mathfrak{Y}_c^h$ is a $W_0$-invariant positive Radon measure on $W_0cT_u$, supported on $W_0S_c^{\operatorname{qu}}$. \end{cor} \begin{proof} It suffices to show that $\mathfrak{Y}_c^h$ is a positive distribution. Assume that $\phi\in C^\infty(W_0cT_u)^{W_0}$ and that $\phi>0$. Then the positive square root $\sqrt{\phi}$ is also in $C^\infty(W_0cT_u)^{W_0}$. Using the approximating sequences as we did before, we can find a sequence $f_n\in\mathcal{Z}$ such that $D(f_n)\to D(\sqrt{\phi})$, uniformly on $W_0S_c^{\operatorname{qu}}$, and to $0$ on $S_{c^\prime}^{\operatorname{qu}}$ for $c^{\prime}\not=c$. By Lemma \ref{lem:supp}, the support of $\mathfrak{Y}_c^h$ is contained in $W_0S^{\operatorname{herm}}_c:=W_0S_c^{\operatorname{qu}}\cap T^{\operatorname{herm}}$, where $T^{\operatorname{herm}}:=\{t\in T\mid t^*\in W_0t\}$. This is itself a regular support for distributions. On $W_0S^{\operatorname{herm}}_c$, the sequence $\phi_n:=f_n^*f_n\in\mathcal{Z}_+$ converges uniformly to $\phi$ up to derivatives of order $N$. Hence \begin{equation} 0\leq\lim_{n\to\infty}\tau(hf_n^*f_n)=\mathfrak{Y}_c^h(\phi). \end{equation} This proves the desired inequality. \end{proof} \begin{cor}\label{cor:cont} Put $\nu_c:=\mathfrak{Y}_c^1$ \index{0n@$\nu_c=\mathfrak{Y}_c^1$, positive $W_0$-invariant measure on $T$}. This is a positive Radon measure, with support in $W_0S_c^{\operatorname{qu}}$, and for all $h\in\mathcal{H}$, $\mathfrak{Y}^h_c$ is absolutely continuous with respect to $\nu_c$. \end{cor} \begin{proof} It is enough to prove this for $h\in \mathcal{H}$ which are Hermitian, i.e. such that $h^*=h$. By Lemma \ref{lem:easy} and Corollary \ref{cor:pos} we see that for positive functions $\phi\in C^\infty(W_0cT_u)^{W_0}$, \begin{equation}\label{eq:cont} -\Vert h\Vert_o\nu_c(\phi)\leq\mathfrak{Y}_c^h(\phi)\leq\Vert h\Vert_o\nu_c(\phi). \end{equation} \end{proof} \begin{dfn}\label{dfn:chi} Let $\nu:=\sum_{c\in\mathcal{C}^{\operatorname{qu}}_-}\nu_c$ \index{0n@$\nu$, Plancherel measure of $\overline{\mathcal{Z}}$ on $W_0\backslash T$}. By equation (\ref{eq:restz}), this is the spectral measure on $\hat{\overline\mathcal{Z}}$ of the restriction of $\tau$ to $\mathcal{Z}$ (the ``Plancherel measure'' of $\mathcal{Z}$). For $h\in\mathcal{H}$ we define a measurable, essentially bounded, $W_0$-invariant function $t\to\chi_t(h)$ \index{0w@$\chi_t$, local trace of $\mathcal{H}$, sum (over $c$) of densities $d(\mathfrak{Y}_c^h)/d\nu$ at $t$} on $T$ by \begin{equation}
\sum_{c\in\mathcal{C}^{qu}_-}\mathfrak{Y}_c^h(\phi|_{W_0S_c^{\operatorname{qu}}})= \int_T\phi(t)\chi_t(h)d\nu(t) \end{equation} for each $\phi\in C_c(T)^{W_0}$. For $t$ outside the support of $\nu$ we set $\chi_t(h)=0$. \end{dfn} \begin{cor}\label{cor:exten} The function $t\to \chi_t\in\mathcal{H}^*$ \index{0w@$\chi_t$, local trace of $\mathcal{H}$, sum (over $c$) of densities $d(\mathfrak{Y}_c^h)/d\nu$ at $t$} satisfies \begin{enumerate} \item[(i)] The support of $t\to\chi_t$ is the support of $\nu$. \item[(ii)] $\chi_t\in\mathcal{H}^*$ is a positive, central functional such that $\chi_t(1)=1$, $\nu$ almost everywhere on $T$. \item[(iii)] For $h\in\mathcal{H},\ z\in\mathcal{Z}:\chi_t(zh)=z(t)\chi_t(h)$, $\nu$ almost everywhere on $T$. \item[(iv)] $\chi_t$ extends, for $\nu$-almost all $t$, to a continuous tracial state of the $C^*$-algebra $\mathfrak{C}$. \item[(v)] We have the following decomposition of $\tau$ for all $h\in\mathcal{H}$, \begin{equation} \tau(h)=\int_T\chi_t(h)d\nu(t). \end{equation} \end{enumerate} \end{cor} \begin{proof} Everything is clear. Assertion (iv) follows from Corollary \ref{cor:contrace} by (ii). \end{proof} For $t\in\operatorname{Supp}(\nu)$, we define the positive semi-definite Hermitian form $(x,y)_t:=\chi_t(x^*y)$ associated to the tracial state $\chi_t$ of $\mathcal{H}$. It is clear that the maximal ideal $\mathcal{I}_t\subset \mathcal{Z}$ of elements vanishing at $t$ is contained in the radical $\operatorname{Rad}_t$ of $(\cdot,\cdot)_t$. Hence the radical is a cofinite two-sided ideal of $\mathcal{H}$. Consequently the GNS-construction produces a finite dimensional Hilbert algebra associated with $\chi_t$: \begin{dfn}\label{dfn:resalg} The algebra $\overline{\mathcal{H}^t}:=\mathcal{H}/\operatorname{Rad}_t$ \index{H7@$\overline{\mathcal{H}^t}$, residual Hilbert algebra at $t$} is a finite dimensional Hilbert algebra with trace $\chi_t$. We will refer to this Hilbert algebra as the residual algebra at $t$.
Let $\{e_i\}_{i=1}^{l_t}$ \index{e1@$e_i$, minimal central idempotent of $\overline{\mathcal{H}^t}$} denote the set of minimal central idempotents of $\overline{\mathcal{H}^t}$, and $\chi_{t,i}$ \index{0w@$\chi_{t,i}$, irreducible character of $\overline{\mathcal{H}^t}$} the associated irreducible characters given by \begin{equation} \chi_{t,i}(x)=\operatorname{dim}(e_i\overline{\mathcal{H}^t})^{1/2} \chi_t(e_i)^{-1}\chi_t(e_ix) \end{equation} We define $d_{t,i}:=\operatorname{dim}(e_i\overline{\mathcal{H}^t})^{-1/2} \chi_t(e_i)\in{\mathbb R}_+$ \index{d@$d_{t,i}(=d_{W_0t,i})$, residual degree; degree of $\chi_{t,i}$ in $\overline{\mathcal{H}^t}$} \index{d@$d_{t,i}(=d_{W_0t,i})$, residual degree; degree of $\chi_{t,i}$ in
$\overline{\mathcal{H}^t}$|see{$d_\delta$}}, so that \begin{equation} \chi_t=\sum_{i=1}^{l_t}\chi_{t,i}d_{t,i}. \end{equation} Note that everything in sight depends on the orbit $W_0t$ rather than on $t$ itself. We will sometimes use the notation $d_{W_0t,i}$ etc. in order to stress this. (This notation and parametrization for the irreducible characters of $\overline{\mathcal{H}^t}$ is provisional. We return to these matters in a systematic way in Section \ref{sect:loc} (see e.g. Theorem \ref{thm:mainind}).) \end{dfn} \subsection{The Plancherel measure $\nu$ of $\mathcal{Z}$, and the $\mathcal{A}$-weights of $\chi_t$}\label{sub:chiA} The results in this subsection are based on the fact that the Eisenstein kernel of (\ref{eq:basic}) simplifies considerably when restricted to the subalgebra $\mathcal{A}\supset\mathcal{Z}$ of $\mathcal{H}$. This means that the $(n,0)$-form $\eta$ (see (\ref{eq:om})) can be replaced by the better behaved $(n,0)$-form $\omega$ (cf. (\ref{eq:omo}) and subsection \ref{sub:quick}) in the residue calculus. This has as an important consequence (see below) that the support of the measure $\nu$ can be identified as the union of the tempered residual cosets, which is only a small subcollection of the tempered quasi residual cosets, and very well behaved (see Subsection \ref{sub:resiprop} of Appendix \ref{sub:defn}). Since we have derived that $\mathfrak{Y}^h$ is absolutely continuous with respect to $\nu$ for general $h\in\mathcal{H}$(see Corollary \ref{cor:cont}), we conclude that the support of the density function $t\to\chi_t$ is the union of the tempered residual cosets.
The probability measure $\nu$ can be computed almost explicitly, due to the good properties of residual cosets. We will exploit these facts here to study the behaviour of the states $\chi_t$ on $\mathcal{A}$. \begin{thm}\label{thm:nu} The $W_0$-invariant probability measure $\nu$ has a decomposition $\nu=\sum_{L}\nu_L$, where $L$ runs over the collection of residual cosets as defined in Appendix \ref{sub:defn}, and where $\nu_L$ is the push forward to $T$ of a smooth measure on $L^{temp}$. Let $d^L$ denote the normalized Haar measure on $T^L_u$, transported to the coset $L^{temp}$ by translation. The measure $\nu_L$ \index{0n@$\nu_L$, smooth measure on $L^{temp}$ such that $\nu=\sum_L\nu_L$} is given by a density function ${\overline \kappa}_{W_LL}m_L(t):=\frac{d\nu_L(t)}{d^Lt}$ \index{m@$m_L$, density function of $\nu_L/{\overline \kappa}_{W_LL}$}, where ${\overline \kappa}_{W_LL}\in{\mathbb Q}$ is a constant, and where $m_L$ is of the form \begin{equation}\label{eq:m_L} m_L(t)=q(w_0)\frac{\prod^\prime_{\alpha\in R_1}(\alpha(t)-1)}{\prod^\prime_{\alpha\in R_1}(q_{\alpha^\vee}^{1/2}{\alpha(t)^{1/2}}+1) \prod^\prime_{\alpha\in R_1}(q_{\alpha^\vee}^{1/2} q_{2\alpha^\vee}\alpha(t)^{1/2}-1)}. \end{equation} Here we used the convention of Remark \ref{rem:conv}. The constant ${\overline \kappa}_{W_LL}$ is independent of ${\bf q}$ if we assume $q$ to be as in Convention \ref{eq:scale}. The notation $\prod^\prime$ means that we omit the factors which are identically equal to $0$ on $L$. The density $m_L$ is a smooth function on $L^{temp}$. \end{thm} \begin{proof} We know already that $\nu$ is a $W_0$-invariant measure supported on the union of the tempered quasi residual cosets. We apply Proposition \ref{prop:cycle} to the integral \[ \tau(a)=\int_{t_0T_u}a\omega =\sum_{c\in\mathcal{C}^{\operatorname{qu}}}\mathfrak{X}_c^1(a) \] (cf. equations (\ref{eq:basic}), (\ref{eq:omo}) and \ref{fundeis}). Choose $\epsilon>0$. For a suitably small $\delta>0$ we can find, for each quasi residual subspace $L$, an $\epsilon^L\in T_{rs}^L$ in an $\epsilon$ neighborhood of $e$, and a cycle $\xi_L\subset\mathcal{B}_L(r_L,\delta)\backslash\cup_{L_m^\prime\supset L}L_m^\prime$, where $\mathcal{B}_L(r_L,\delta)\subset T_L$ denotes a ball of radius $\delta>0$ centered around $r_L$, such that \begin{equation}\label{eq:roughly} \mathfrak{X}_c^1(a):=\sum_{L:c_L=c} k_L\int_{t\in\epsilon^LT^L_u}\left\{\int_{\xi_L}a(tt^\prime)\frac{d_L(t^\prime)} {q(w_0)c(tt^\prime)c({(tt^\prime)}^{-1})}\right\}d^L(t). \end{equation} Here $d^L(t)$ is the holomorphic extension to $L$ of $d^L$, and $d_L{t^\prime}$ denotes the Haar measure on $T_{L,u}$, also extended as a holomorphic form on $T_L$. We assume that $\delta$ is small enough to assure that $\log$ is well defined on $\mathcal{B}_L(r_L,\delta)$. For the inner integral we use a basis $(x_i)$ of $X\cap{\mathbb Q} R_L$ as coordinates on $\log{(\mathcal{B}_L(r_L,\delta))}$, shifted so that the coordinates are centered at $\log(r_L)$. We can then write the integration kernel as: \begin{equation} t^\prime\to a(tt^\prime)m_L(tt^\prime)(1+f_t(t^\prime))\omega_L(t^\prime) \end{equation} where $\omega_L$ is a rational homogeneous $(l:=\dim(T_L),0)$-form (independent of $t$) in the $x_i$, and $f_t$ is a power series in $x_i$ such that $f_t(0)=0$. In fact, the form $\omega_L$ is easily seen to be (including the factor $k_L$ of (\ref{eq:roughly})) \begin{equation}\label{eq:tochhandig} \omega_L(x)=\frac{\prod_{\alpha\in R_{L}^z}\alpha(x)}{(2\pi i)^l \prod_{\beta\in R_L^p}\beta(x)}dx_1\wedge dx_2\dots\wedge dx_l. \end{equation} By Corollary \ref{cor:simpledefres} it follows that the form $\omega_L$ has homogeneous degree $\geq 0$ if $L$ is residual in the sense of Definition \ref{dfn:ressub}. A homogeneous closed rational form of positive homogeneous degree is exact. Hence the inner integral will be nonzero only if $L$ is in fact a residual coset. In that case the inner integral will have value \begin{equation}\label{eq:inint} \kappa_La(r_Lt)m_L(r_Lt) \end{equation} with \begin{equation} \kappa_L=\int_{\xi_L}\omega_L. \end{equation} \index{0k@$\kappa_L$, rational number $\int_{\xi_L}\omega_L$} We note that $\kappa_L\in{\mathbb Q}$, since $\omega_L$ defines a rational cohomology class. Let us therefore assume that $L$ is residual from now on. Write $r_L=sc$. By Theorem \ref{thm:ster} we know that $r_L^*=sc^{-1}=w_s(r_L)$ with $w_s\in W(R_{L,s,1})$. When $t\in L^{temp}$, the expression $m_L(t)$ can be rewritten as \begin{equation}\label{eq:ml}
q(w^L)m_{\mathcal{R}_L,\{r_L\}}(r_L)\prod_{\alpha\in R_{1,+}\backslash R_{L,1,+}}\frac{|1-\alpha(t)|^2}{|1+q_{\alpha^\vee}^{1/2}\alpha(t)^{1/2}|^2
|1-q_{\alpha^\vee}^{1/2}q_{2\alpha^\vee}\alpha(t)^{1/2}|^2}. \end{equation} Here we used that if $t=cu\in L^{temp}$ with $u\in sT^L_u$, we have $w_sc=c^{-1}$, $w_su=u$, and $w_s(R_{1,+}\backslash R_{L,1,+})=R_{1,+}\backslash R_{L,1,+}$. By the same argument as was used in Theorem 3.13 of \cite{HOH0} we see that this expression is real analytic on $L^{temp}$. This implies that we can in fact take $\epsilon^L=e$ for all residual $L$ in equation (\ref{eq:roughly}) after we evaluate the inner integrals. This leads to \begin{equation} \mathfrak{X}_c^1(a)=\sum_{L:c_L=c}\kappa_L\int_{L^{temp}}a(t)m_L(t)d^L(t). \end{equation} where the sum is taken over residual cosets only. When we combine terms over $W_0$ orbits of residual cosets we find the desired result. Let $W_0L$ denote the set of residual cosets in the orbit of $L$. We have to take \begin{equation}\label{eq:denk}
{\overline \kappa}_{W_LL}=\frac{1}{|W_0L|}\sum_{L^\prime\in W_0L}\kappa_{L^\prime}. \end{equation} \index{0k@${\overline \kappa}_{W_LL}$, rational factor in $\nu_L$; average of $\kappa_L$} When we now define a measure $\nu_L$ on $L^{temp}$ by \begin{equation} \int_{t\in L^{temp}}f(t)d\nu_L(t):={\overline \kappa}_{W_LL}\int_{t^L\in T^L_u}f(r_Lt^L)m_L(r_Lt^L)d^L(t^L) \end{equation} then we have the equality $\nu=\sum_L\nu_L$ (sum over the residual subspaces).
We note in addition that $\kappa_L=k_L\kappa_{\mathcal{R}_{L},\{r_L\}}$
\index{0k@$\kappa_{\mathcal{R}_L,\{r_L\}}$|see{$\kappa_L$}}, because the cycle $\xi_L$ is constructed inside $T_L$, entirely in terms of the root system $R_{L}$ (see Proposition \ref{prop:t0}) (the factor $k_L$ comes from the facorization $dt=k_Ld^Lt^Ld_Lt_L$, see (\ref{eq:roughly})). Also, it is clear that $m_{\mathcal{R}_L,\{r_L\}}(r_L)$ is independent of the choice of $r_L$, because the finite group $K_L=T_L\cap T^L$ is contained in the simultaneous kernel of the roots of $R_{L}$. Finally, the independence of ${\bf q}$ is clear from Proposition \ref{prop:cyinv}. When we apply a scaling transformation ${\bf q}\to{\bf q}^\epsilon$, the point $c_L$ moves such that the facet of the dual configuration containing $e$ does not change. Hence $r_L^{-1}\xi_L$ and $\omega_L$ will be independent of $\epsilon$. \end{proof} \begin{rem}\label{rem:smoothnest} We note that the smoothness of $m_L$ implies Theorem \ref{thm:nonnest}, similar to \cite{HOH0}, Remark 3.14. \end{rem} \begin{prop}\label{prop:par} For $L$ residual consider the root datum $\mathcal{R}_L=(X_L,Y_L,R_L,R^\vee_L,F_L)$ (see Subsection \ref{sub:par}) associated with the parabolic root subsystem $R_L\subset R_0$. Let $q_L$ be the restriction of the label function $q$ to $\mathcal{R}_L$. Then $\{r_L\}\subset T_L$ is a $(\mathcal{R}_L,q_L)$ residual point. Assume that $R_L$ is a standard parabolic subsystem of roots, and thus that $F_L\subset F_0$. Denote by $W_L$ the standard parabolic subgroup $W_L=W(R_L)$ of $W_0$, and let $W^L$ denote the set of minimal length representatives of the left $W_L$ cosets in $W_0$. \begin{enumerate} \item[(i)] When $w\in W^L$, we may take $\xi_{wL}=w(\xi_L)$. Consequently, $\kappa_L=\kappa_{wL}$ if $w\in W^L$. \item[(ii)] Put \begin{align}\label{eq:m^L} m^L(t)&=q(w^L)^{-1}\prod_{\alpha\in R_1\backslash R_{L,1}}c_\alpha(t)^{-1}\\
&=q(w^L)\prod_{\alpha\in R_{1,+}\backslash R_{L,1,+}}\frac{|1-\alpha(t)|^2}{|1+q_{\alpha^\vee}^{1/2}\alpha(t)^{1/2}|^2
|1-q_{\alpha^\vee}^{1/2}q_{2\alpha^\vee}\alpha(t)^{1/2}|^2}. \end{align} \index{m@$m^L$, quotient $m_L/k_L\nu_{\mathcal{R}_L,\{r_L\}}(\{r_L\})$} Then $m^L$ and $m_L$ are $\operatorname{Aut}(W_0)$-equivariant, i.e. $m^L(t)=m^{gL}(gt)$ and $m_L(t)=m_{gL}(gt)$ for every $g\in\operatorname{Aut}(W_0)$. In particular, $m^L$ and $m_L$ are invariant for the stabilizer $N_L$ of $L$ in $W_0$. \item[(iii)] We have $\kappa_L=k_L\kappa_{\mathcal{R}_L,\{r_L\}}$, ${\overline \kappa}_{W_LL}=k_L{\overline \kappa}_{\mathcal{R}_L,W_Lr_L}$. \item[(iv)] For $z\in\mathcal{Z}$, we have \begin{equation}
\frac{1}{|W_0L|}\int_T zd\nu_{W_0L}=k_L\nu_{\mathcal{R}_L,\{r_L\}}(\{r_L\}) \int_{L^{temp}}z(t)m^L(t) d^L(t). \end{equation} \item[(v)] Assuming that $q$ is expressed as in Convention \ref{eq:scale} with $f_s\in2\mathbb{Z}$. Then $\nu_{\mathcal{R}_L,\{r_L\}}(\{r_L\})= {\overline \kappa}_{\mathcal{R}_L,W_Lr_L}m_{\mathcal{R}_L,\{r_L\}}(r_L)$ is of the form $d{\bf q}^nf({\bf q})$, where $d\in\mathbb{Q}$, $n\in{\mathbb Z}$, and where $f$ is a quotient of products of cyclotomic polynomials in ${\bf q}$. \end{enumerate} \end{prop} \begin{proof} (i). We note that for $w\in W^L$, $t_0$ and $w^{-1}t_0$ are in the same chamber of $\mathcal{L}^L$. Hence, by application of Proposition \ref{prop:t0}, we may replace $\xi_{wL}$ by $w(\xi_L)$.
(ii). This is trivial.
(iii). The formula $\kappa_L=k_L\kappa_{\mathcal{R}_L,\{r_L\}}$ was explained in the proof of Theorem \ref{thm:nu}. Let $W_L=W(R_L)$. Let $N_{T^L}$ be the stabilizer of $T^L$ in $W_0$. Observe that $N_L\subset N_{T^L}$ and $W_L\vartriangleleft N_{T^L}$. If we define $\Gamma_L=N_{T^L}\cap W^L$ then $\Gamma_L$ is a complementary subgroup of $W_L$ in $N_{T^L}$. Using (i), (ii) and the remark $\kappa_L=k_L\kappa_{\mathcal{R}_L,\{r_L\}}$ we see that \begin{equation} \begin{split} {\overline \kappa}_{W_LL}&=
\frac{1}{|W_0L|}\sum_{L^\prime\in W_0L}\kappa_{L^\prime}\\
&=\frac{|W_0T_L|}{|W_0L|}\sum_{L^\prime\in N_{T^L}L}\kappa_{L^\prime}\\
&=\frac{|W_0T_L||N_{T^L}L|}{|W_0L|} k_L{\overline \kappa}_{\mathcal{R}_L,W_Lr_L}= k_L{\overline \kappa}_{\mathcal{R}_L,W_Lr_L} \end{split} \end{equation} \index{0kl@${\overline \kappa}_{W_0r}$(=${\overline \kappa}_{\mathcal{R},W_0r}$), rational factor in $\nu(\{r\})$} Using Theorem \ref{thm:nu} and equation (\ref{eq:ml}) the result follows.
(iv). Follows easily from (iii).
(v). Since equation (\ref{eq:m_L}) involves only roots in $R_0$, it is sufficient to prove the statement for $R_0$ indecomposable and $X=Q$. Notice that for all $\alpha\in R_0$, $\alpha(s)$ is a root of unity and, by Theorem \ref{thm:ster}(iii), $\alpha(c)$ is an integral power of ${\bf q}$. Looking at the explicit formula (\ref{eq:m_L}), we see that it remains to show that this expression has rational coefficients if $L=r=sc$ is a residual point. Let $k$ be the extension of $\mathbb{Q}$ by the values of $\alpha(s)$, where $\alpha$ runs over $R_0$. In the case where $\mathcal{R}$ is of type $C_n^{\text{aff}}$ it follows by Lemma \ref{lem:order2} that $k=\mathbb{Q}$, and we are done. For the other classical cases it follows from the result of Borel and de Siebenthal \cite{BS} that the order of $s$ is at most two, and hence that $k=\mathbb{Q}$. Next let $\mathcal{R}$ be of exceptional type, and $\sigma\in\operatorname{Gal}(k/\mathbb{Q})$. Define a character $\sigma(s)$ of $X=Q$ by $Q\ni x\to \sigma(x(s))=:x(\sigma(s))$. By Lemma \ref{lem:conj} we see that there exists a $w_1\in W_0$ such that $\sigma(s)=w_1s$. Moreover, $w_1:R_{s,0}\to R_{s,0}$ acts as an automorphism and $c$ is an $R_{s,0}$-residual point. If $F_{s,0}$ contains isomorphic components then these are of type $A$, which has only one real residual point up to the action of $W(R_{s,0})$. Hence by Theorem \ref{thm:ster}(i), there exists a $w_2\in W(R_{s,0})$ such that $w_1(c)=w_2(c)$. Put $w=w_2^{-1}w_1$, so that $wr=c\sigma(s)$. By the $W_0$-equivariance of $m_{\{r\}}(r)$ we see that (with the action of $\sigma$ being extended to $k[{\bf q},{\bf q}^{-1}]$ by its action on the coefficients) $\sigma(m_{\{r\}}(r))=m_{\{wr\}}(wr)=m_{\{r\}}(r)$, whence the desired rationality. \end{proof} The next proposition is a direct consequence of (the proof of) Theorem \ref{thm:nu} and the definition of $\chi_t$. \begin{prop}\label{chiA} Let $r=sc\in T$ be a residual point, and let $a\in \mathcal{A}$. Then \begin{equation} \nu(W_0r)\chi_r(a)=m_{\{r\}}(r)\sum_{r^\prime\in W_0r}\kappa_{\{r^\prime\}}a(r^\prime). \end{equation} \end{prop} \begin{theorem}\label{thm:support} The support of $\nu$ is exactly equal to the union of the tempered residual cosets. In other words, $S=W_0\backslash\cup_{L\ \mathrm{residual}}L^{temp}$. \end{theorem} \begin{proof} The equality $S=W_0\backslash\operatorname{Supp}(\nu)$ was explained in \ref{sl}, so it suffices to show that the support of $\nu$ is equal to the union of the tempered residual cosets. By Theorem \ref{thm:nu} we know that $\nu$ is supported on this set, so we need only to show that $W_0L^{temp}$ is contained in the support for each tempered residual coset $L$.
By Proposition \ref{prop:par} this reduces to the case of a residual point $r=sc$. By Proposition \ref{chiA} it is enough to show that there exists at least one $r^\prime=wr\in W_0r$ such that $\kappa_{\{r^\prime\}}\not=0$. In other words, using Proposition \ref{chiA} we single out the point residue at $r^\prime$. In particular, we ignore all residues at residual cosets which do not contain $r^\prime$ and thus do not contribute to $\kappa_{\{r^\prime\}}$ in the argument below.
By the $W_0$-invariance of $\omega$, we can formulate the problem as follows. Recall from the proof of Theorem \ref{thm:nu} that \begin{equation} \kappa_{\{r\}}m_{\{r\}}(r)=\int_{\xi}\omega, \end{equation} where $\xi$ is the residue cycle at $r$, which is obtained from Proposition \ref{prop:cycle} if we use the $n$-form \begin{equation} \omega(t)=\frac{dt}{c(t)c(t^{-1})} \end{equation} and a base point $t_0\in T_{rs}$ such that $\forall \alpha_i\in F_0:\ \alpha_i(t_0)<q(s_i)$. By definition, $m_{\{r\}}(r)\not=0$. For $r^\prime=wr$ we have \begin{equation}\label{eq:niet} \kappa_{\{r^\prime\}}m_{\{r\}}(r)=\int_{\xi(w)}\omega, \end{equation} where $\xi(w)$ is the cycle near $r$ which we obtain in Proposition $\ref{prop:cycle}$ when we replace $t_0$ by $w^{-1}t_0$. Hence we have to show that there exists a proper choice for $t_0$ such that when we start the contour shift algorithm from this point, the corresponding point residue at $r$ will be nonzero. The problem we have to surmount is possible cancellation of nonzero contributions to $\kappa_{\{r^\prime\}}$. We will do this by showing that there exists at least one chamber such that the residue at $r$ consists only of one nonzero contribution.
We consider the real arrangement $\mathcal{L}^{\{r\}}$ in $T_{rs}$, and transport the Euclidean structure of $\mathfrak{t}$ to $T_{rs}$ by means of $t\to\log(c^{-1}t)$. Then $\mathcal{L}^{\{r\}}$ is the lattice of intersections of a central arrangement of hyperplanes with center $c$. We assign indices $i_L$ to the elements of $\mathcal{L}^{\{r\}}$ by considering the index of the corresponding complex coset containing $r$, and we note that by Corollary \ref{cor:simpledefres}, $i_{\{r\}}=n:=\operatorname{codim}(\{r\})$. From Corollary \ref{cor:simpledefres} we further obtain the result that there exist full flags of subspaces $c_LT^L_{rs}\in\mathcal{L}^{\{r\}}$ such that $i_L=\operatorname{codim}(L)$. In particular, there exists at least one line $l$ through $r$ with $i_l=n-1$.
By Theorem \ref{thm:nonnest} we see that the centers $c_L$, $c_{L^\prime}$ of two ``residual subspaces'' $c_LT^L\subset c_{L^\prime}T^{L^\prime}$ (i.e. $\operatorname{codim}(T^L)=i_L$ and $\operatorname{codim}(T^{L^\prime})=i_{L^\prime}$) in $\mathcal{L}^{\{r\}}$ satisfy $c_{L^\prime}\not=c_{L}$ unless $c_LT^L= c_{L^\prime}T^{L^\prime}$. Hence $d(e,c_{L^\prime})\leq d(e,c_{L})$ (where $d$ denotes the distance function), with equality only if $c_LT^L= c_{L^\prime}T^{L^\prime}$. In the case of a residual line $l\in \mathcal{L}^{\{r\}}$, $cT^l$ is divided in two half lines by $c$, and $c_l$ lies in one of the two halves (i.e. does not coincide with $c$).
We want to find a chamber for $t_0$ such that the corresponding point residue $\kappa_{\{r^\prime\}}m_{\{r\}}(r)$ at $r$ is nonzero. We argue by induction on the rank. If the rank of $R_0$ is $1$, obviously we get $\kappa_{\{r^\prime\}}\not=0$ if we choose $t_0$ in the half line not containing $e=c_T$, because we then have to pass a simple pole of $\omega$ at $r$ when moving the contour $t_0T_u$ to $T_u$ (since $t_0$ and $e=c_T$ are separated by $c$). Assume by induction that for any residual point $p$ of a rank $n-1$ root system, we can choose a chamber for $t_0$ such that $\kappa_{\{p\}}\not= 0$. Let $S\subset T_{rs}$ be a sphere centered at $r$ through $e$, and consider the configuration of hyperspheres in $\mathcal{L}^{\{r\}}\cap S$. Let us call $e\in S$ the north pole of $S$. If $L_S=c_LT^L_{rs}\cap S$ with $\operatorname{dim}(T^L)>1$, we denote by $c_{L\cap S}$ the intersection of the half line through $c_L$ beginning in $c$ (recall that $c\not=c_L$) and $L_S$. By the above remarks, $c_{L\cap S}$ is in the northern hemisphere for all residual $L\supset r$ of dimension $>1$. We call this point $c_{L\cap S}$ the center of $L_S$.
In the case when $\operatorname{dim}(T^L)=1$, $L_S$ is disconnected and consists of two antipodal points $c_{L\cap S}$ (north) and $\overline{c_{L\cap S}}$ (south), its opposite. In this case of residual lines through $r$, both of these antipodal points are considered as centers of $\mathcal{L}^{\{r\}}\cap S$. We call $c_{L\cap S}$ the northern center, and its opposite is called the southern center. All centers of $\mathcal{L}^{\{r\}}\cap S$ lie in the northern hemisphere, with the exception of the southern centers of the residual lines through $r$.
Consider a closed (spherical) ball $D\subset S$ centered at $e$ such that $D$ contains a southern center $p$ in its boundary, but no southern centers in its interior. Since $e$ is regular with respect to $\mathcal{L}^{\{r\}}$ (a trivial case of Theorem \ref{thm:nonnest}, as $e$ is the center of $T$), we have $D\not=S$.
We take $t_0$ in $S$, and we apply the algorithm as described in the proof of Proposition \ref{prop:cycle}, {\it but now on the sphere $S$, and with respect to the sets $L_S$ and their centers}.
By the induction hypothesis, we can take $t_0\in S$ close to $p$ in a chamber of the configuration $\mathcal{L}^{\{r\}}\cap S$ which contains $p$ in its closure, such that a nonzero residue at $l$ is picked up in $p$. Denote by $\mathcal{L}^p\cap S$ the central subarrangement of elements of $\mathcal{L}^{\{r\}}\cap S$ containing $p$. Consider any alternative ``identity element'' $\tilde{e}$ which belongs to the same chamber of the {\it dual} configuration of $\mathcal{L}^p\cap S$ as the real (original) identity element $e$.
As was explained in (the proof of) Proposition \ref{prop:cyinv}, when we apply the contour shifts as in (the proof of) Proposition \ref{prop:cycle} to $\mathcal{L}^{\{r\}}\cap S$, the residue at $p$ only depends on the dual chamber which contains the identity element. In other words, we may use the new identity $\tilde{e}$ instead of $e$ without changing the residue at $p$. We can and will choose $\tilde{e}$ close to $p$, and in the interior of $D$. By Proposition \ref{prop:cycle} we can replace the integral over $t_0T_u$ by a sum of integrals over cosets of the form $\tilde{c}_{L\cap S}\tilde{s}_LT_u^L$ (for some $\tilde{s}_L\in T_u$) of the residue kernel $\tilde{\kappa}_Lm_L$ (cf. equation (\ref{eq:inint})) on $L$. As was mentioned above, we are only interested in such contributions when $r\in L$, which means that we may take $\tilde{s}_L=s$. The new ``centers'' $\tilde{c}_{L\cap S}$ with respect to the new identity element $\tilde{e}$ are in the interior of $D$.
Next we apply the algorithm of contour shifts as in Proposition \ref{prop:cycle} to move the cycles $\tilde{c}_{L\cap S}sT_u^L$ to ${c}_{L\cap S}sT_u^L$. Since both the new centers $\tilde{c}_{L\cap S}$ and the original centers ${c}_{L\cap S}$ belong to the interior of $D$, and since the intersection of $D$ with $L_S$ is connected if $\operatorname{dim}(L_S)>0$, we can choose every path in the contour shifting algorithm inside the interior of $D$. Thus, the centers $c_{L\cap S}$ of the residual cosets $L$ that arise in addition the one southern center $c_l=p$ in the above process are in the interior of $D$. In particular, with the exception of $psT_u^l$, the one dimensional cosets of integration which show up in this way, all have a {\it northern} center.
Finally, in order to compute the residue $\kappa_{r^\prime}m_{\{r\}}(r)$ at $r^\prime$, we now have to move the center $c_{L\cap S}\in S$ of $L_S$ to the corresponding center $c_L\in T_{rs}$ of $L$, for each residual coset $L$ which contains $r$ and which contributes to $\int_{t_0T_u}\omega$. The only such center of $\mathcal{L}^{\{r\}}\cap S$ which will cross $c$ is the southern center $p$. Since $m_l$ has a simple pole at $r=sc$, we conclude that this gives a nonzero residue at $r$. Hence with the above choice of $t_0$ we get $\kappa_{\{r^\prime\}}\not=0$, which is what we wanted to show. \end{proof} \subsection{Discrete series} In this subsection we show that the irreducible characters $\chi_{r,i}$ (see Definition \ref{dfn:resalg}) associated to a residual point are in fact discrete series characters. \begin{cor}(of Theorem \ref{thm:support})\label{cor:cas} For every residual point $r=sc$, the sum
${\overline \kappa}_{W_0r}|W_0r|=\sum_{r^\prime\in W_0r}\kappa_{r^\prime}\not=0$, and for all $a\in\mathcal{A}$: \begin{equation}
\chi_r(a)=\frac{1}{{\overline \kappa}_{W_0r}|W_0r|}\sum_{r^\prime\in W_0r}\kappa_{\{r^\prime\}}a(r^\prime). \end{equation}
Moreover, $\kappa_{\{r^\prime\}}=0$ unless $\forall x\in X^+\backslash\{0\}:|x(r^\prime)|<1$ (where $X^+$ denotes the set of dominant elements in $X$). \end{cor} \begin{proof} This is immediate from Proposition \ref{chiA} and Theorem \ref{thm:support}, except for the last assertion. This fact follows from Proposition \ref{prop:antidual}. We know that $e$ is regular in $\mathcal{L}_{\{r^\prime\}}$ by Theorem \ref{thm:nonnest}. On the other hand, $t_0$ lies in $c^\prime T_{rs,-}$, which is clearly a subset of a chamber of $\mathcal{L}^{\{r^\prime\}}$. The anti-dual of the chamber of $\mathcal{L}^{\{r^\prime\}}$ containing $t_0$ is thus a subset of $c^\prime T_{rs}^+$, with $T_{rs}^+:=\{t\in T_{rs}\mid \forall x\in X^+\backslash\{0\}:x(t)> 1\}$. Thus when $e$ is contained in the anti-dual chamber we have $c^\prime\in T_{rs}^-$ \index{T2@$T_{rs}^-$, anti-dual of the positive chamber $T_{rs,+}$} as desired. \end{proof} We introduce the notation $\Delta_{\mathcal{R}}(=\Delta_{\mathcal{R},q})$ \index{0D3@$\Delta_{\mathcal{R}}$(=$\Delta_{\mathcal{R},q}$), irreducible discrete series of $\mathcal{H}(\mathcal{R},q)$} for a complete set of representatives of the finite set of equivalence classes of the irreducible discrete series representations of $\mathcal{H}(\mathcal{R},q)$, and $\Delta_{\mathcal{R},W_0r}(=\Delta_{\mathcal{R},W_0r,q})$ \index{0D3@$\Delta_{\mathcal{R},W_0r}$, irreducible discrete series representations of $\mathcal{H}(\mathcal{R},q)$ with central character $W_0r$} for the representatives of the classes of irreducible discrete series of $\mathcal{H}(\mathcal{R},q)$ with central character $W_0r$. (We sometimes drop $\mathcal{R}$ from the notation if no confusion is possible, and write $\Delta_{W_0r}$.) \begin{lem} $\Delta_{W_0r}$ is nonempty if and only if $r$ is residual. If $r$ is residual, $\Delta_{W_0r}$ is in bijective correspondence with the collection $\{\delta_{r,i}\}$ of irreducible characters of $\overline{\mathcal{H}^r}$. In particular, $\mathcal{H}(\mathcal{R},q)$ has at most finitely many discrete series representation. \end{lem} \begin{proof} We have \begin{equation} \chi_{r,i}(a)=\sum_{r^\prime\in W_0r}\dim(V_{r,i}^{r^\prime})a(r^\prime). \end{equation} Hence from $d_{r,i}>0$, \begin{equation}\label{eq:packet} \chi_{r}(a)=\sum_{i}\chi_{r,i}(a)d_{r,i}, \end{equation} and Corollary \ref{cor:cas} we conclude that the generalized weight spaces of $V_{r,i}$ indeed satisfy the Casselman criterion Lemma \ref{lem:casds} for discrete series.
Conversely, if $\delta$ is a discrete series representation, Theorem \ref{thm:supds} implies that $\mu_{Pl}(\delta)>0$. By Corollary \ref{cor:exten}, the central character $W_0r$ of $\delta$ is such that $\nu(\{r\})>0$. Theorem \ref{thm:nu} implies that such points $r$ are necessarily residual. \end{proof} In view of the above, we adapt the notations of Definition \ref{dfn:resalg} accordingly, i.e. we write $d_{\mathcal{R},\delta}$ (or simply $d_{\delta}$) \index{d@$d_\delta(=d_{\mathcal{R},\delta})$, residual degree; degree of $\delta$ in the residual Hilbert algebra $\overline{\mathcal{H}^r}$} instead of $d_{r,i}$ \index{d@$d_{t,i}(=d_{W_0t,i})$, residual degree; degree of $\chi_{t,i}$ in $\overline{\mathcal{H}^t}$} if $\delta\in\Delta_{\mathcal{R},W_0r}$, and its character $\chi_\delta$ \index{0w@$\chi_\delta$, character of $\delta$} descends to $\chi_{r,i}$ \index{0w@$\chi_{t,i}$, irreducible character of $\overline{\mathcal{H}^t}$} on $\overline{\mathcal{H}^r}$ etc. \begin{cor}\label{cor:fdim} Let $\delta\in\Delta_{W_0r}$. The formal dimension $\mu_{Pl}(\delta)$ of $\delta$ equals \begin{equation}\label{eq:fdim} \mu_{Pl}(\delta)=\operatorname{fdim}(\delta)={d_\delta\nu(\{W_0r\})}
=|W_0r|{\overline{\kappa}}_{W_0r} d_\delta m_{\{r\}}(r) \end{equation} \end{cor} \begin{proof} Combine equation (\ref{eq:chadec}), Corollary \ref{cor:exten}, and Theorem \ref{thm:nu}. \end{proof} \begin{cor}\label{rem:cas} For a residual point $r$ there exist constants $C, \epsilon>0$ such that \begin{equation}
|\chi_r(N_w)|\leq C\exp(-\epsilon l(w)). \end{equation} \end{cor} \begin{cor} The residual degrees $d_{\delta}>0$ of the irreducible characters $\chi_{\delta}$ of the residual algebra $\overline{\mathcal{H}^r}$ (with $r$ a residual point) satisfy the following system of linear equations. \begin{equation}\label{eq:rats} \sum_{\delta\in[\Delta_{\mathcal{R},W_0r}]} \dim(V^{r^\prime}_\delta)d_\delta=
\frac{\kappa_{\{r^\prime\}}}{{\overline \kappa}_{W_0r}|W_0r|}. \end{equation} (with $V_\delta^{r^\prime}$ the generalized $r^\prime$-weight space in the space $V_\delta$ of $\delta$). In particular we conclude that the nonzero $\kappa_{\{r^\prime\}}$ all have the same sign (equal to the sign of $m_{\{r\}}(r)$). \end{cor} \begin{rem}
We note in addition that if the restrictions $\chi_\delta|_{\mathcal{A}}$ to $\mathcal{A}$ of the characters $\chi_\delta$ are linearly independent, it follows from the equations (\ref{eq:rats}) that $d_\delta\in{\mathbb Q}$ for all $\delta\in [\Delta_{\mathcal{R}}]$. I did not find any argument in favor of this linear independence. However, we do conjecture that the constants $d_{\delta}$ are rational, see Conjecture \ref{rem:ell}. \end{rem} \subsection{Temperedness of the traces $\chi_t$} In this subsection we discuss the tempered growth behaviour of the $\chi_t$ on the orthonormal basis $N_w$ of $\mathcal{H}$, as a corollary of the analysis of the $\mathcal{A}$-weights of $\chi_t$. \begin{prop}\label{cor:indA} Let $L$ be residual such that $W_L$ is a standard parabolic subgroup of $W_0$. For $t\in L^{temp}$ we write $t=r_Lt^L$, with
$t^L\in T^L_u$. We consider $\chi_t|_{\mathcal{A}}$ as a formal linear combination of elements of $T$. Likewise, let $\mathcal{A}_L={\mathbb C}[X_L]$ be the ring of regular functions on $T_L\subset T$. We consider
$\chi_{\mathcal{R}_L,\{r_L\}}|_{\mathcal{A}_L}$ as a formal linear combination of elements of $T_L$. In this sense we have, $\nu_L$-almost everywhere on $L^{temp}$, \begin{equation}\label{eq:div}
\chi_t|_{\mathcal{A}}=\frac{1}{|W^L|}\sum_{w\in W^L}w(t^L\chi_{\mathcal{R}_L,\{r_L\}}|_{\mathcal{A}_L}). \end{equation} Hence $\nu$-almost everywhere, $\chi_t$ is a nonzero tempered functional on $\mathcal{H}$. \end{prop} \begin{proof} Equation (\ref{eq:div}) follows by a straightforward computation similar to Proposition \ref{chiA}, using Proposition \ref{prop:par} and the definition of $\chi_t$. Since $\chi_t$ is a positive combination of the irreducible characters of the residual algebra $\overline{\mathcal{H}^t}$, it follows that the weights $t^\prime\in W_0t$ of the generalized $\mathcal{A}$-eigenspaces of the irreducible characters of $\overline{\mathcal{H}^t}$ all satisfy the condition
$\forall x\in X^+:\ |x(t^\prime)|\leq 1$. This shows, by Casselman's criterion Lemma \ref{lem:cas}, that $\chi_t$ is a tempered functional on $\mathcal{H}$. \end{proof} \section{Localization of the Hecke algebra}\label{sect:loc} We have obtained thus far a decomposition of the trace $\tau$ as an integral of positive, finite traces $\chi_t$ against an explicit probability measure $\nu$ on $T$, such that each $\chi_t$ is a finite positive linear combination of finite dimensional, irreducible characters of $\mathfrak{C}$. This is an important step towards our goal of finding the Plancherel decomposition, but it is not yet satisfactory because we know virtually nothing about the behavior of the decomposition of $\chi_t$ in irreducible characters at this stage, neither as a function of $t$, nor as a function of ${\bf q}$. In particular, the residual degrees $d_{t,i}\in \mathbb{R}_+$ of the residual algebras are obscure at this point, and these degrees are involved in the Plancherel measure $\mu_{Pl}$.
In the remaining part of the paper we will formulate the Plancherel theorem, and also remedy to some extend the above problems. The support $S$ of $\nu$ (viewed as a $W_0$-invariant measure on $T$) decomposes as a union of the closed sets $L^{temp}$ (see \ref{sl}). For each $L$ we show that, up to isomorphism of Hilbert algebras, the residual algebras $\overline{\mathcal{H}^t}$ are independent of $t$, $\nu$-almost everywhere on $L^{temp}$.
The above is based on ideas of Lusztig \cite{Lu} about completions of the affine Hecke algebra. Lusztig describes the $\mathcal{I}_t$-adic completion of $\mathcal{H}$, where $\mathcal{I}_t$ is a maximal ideal of $\mathcal{Z}$. It is not hard to see that Lusztig's arguments can be adapted to (analytic) localization with respect to suitably small open neighborhoods $U\supset W_0t$ of orbits of points in $T$, and this will be discussed in present section.
When $s=s_\alpha\in S_0$ (with $\alpha\in F_1$), we define an intertwining element $\iota_s$ as follows: \begin{equation} \begin{split} \iota_s&=(1-\theta_{-\alpha})T_s+((1-q_{\alpha^\vee}q_{2\alpha^\vee}) +q_{\alpha^\vee}^{1/2}(1- q_{2\alpha^\vee})\theta_{-\alpha/2})\\ &=T_s(1-\theta_{\alpha})+((q_{\alpha^\vee}q_{2\alpha^\vee}-1) \theta_{\alpha} +q_{\alpha^\vee}^{1/2}(q_{2\alpha^\vee}-1)\theta_{\alpha/2})\\ \end{split} \end{equation} \index{0j@$\iota_s$, intertwining element of $\mathcal{H}$} We remind the reader of the convention of Remark \ref{rem:conv}. These elements are important tools to study the Hecke algebra. We recall from \cite{EO}, Theorem 2.8 that these elements satisfy the braid relations, and they satisfy (for all $x\in X$) \[ \iota_s\theta_x=\theta_{s(x)}\iota_s, \] and finally they satisfy \[ \iota_s^2=(q_{\alpha^\vee}^{1/2}+\theta_{-\alpha/2}) (q_{\alpha^\vee}^{1/2}+\theta_{\alpha/2}) (q_{\alpha^\vee}^{1/2}q_{2\alpha^\vee}-\theta_{-\alpha/2}) (q_{\alpha^\vee}^{1/2}q_{2\alpha^\vee}-\theta_{\alpha/2}). \] (where we have again used the convention of Remark \ref{rem:conv}!). Suitably normalized versions of the $\iota_s$ generate a group isomorphic to the Weyl group $W_0$. In order to normalize the intertwiners, we need to tensor $\mathcal{H}$ by the field of fractions $\mathcal{F}$ of the center $\mathcal{Z}$. So let us introduce the algebra \begin{equation} {}_\mathcal{F}\mathcal{H}:=\mathcal{F}\otimes_\mathcal{Z}\mathcal{H} \end{equation} \index{H6@${}_\mathcal{F}\mathcal{H}:=\mathcal{F}\otimes_\mathcal{Z}\mathcal{H}$, $\mathcal{F}$ field of fractions of $\mathcal{Z}$}
with the multiplication defined by $(f\otimes h) (f^\prime\otimes h^\prime):=ff^\prime\otimes hh^\prime$. Notice that this an algebra over $\mathcal{F}$ of dimension $|W_0|^2$. The subalgebra ${}_\mathcal{F}\mathcal{A}=\mathcal{F}\otimes_\mathcal{Z}\mathcal{A}$ \index{A@${}_\mathcal{F}\mathcal{A}:=\mathcal{F}\otimes_\mathcal{Z}\mathcal{A}$, $\mathcal{F}$ field of fractions of $\mathcal{Z}$} is isomorphic to the field of fractions of $\mathcal{A}$. The field extension $\mathcal{F}\subset{}_\mathcal{F}\mathcal{A}$ has Galois group $W_0$, and we denote by $f\to f^w$ the natural action of $W_0$ on the field of rational functions on $T$. The elements $T_w$ with $w\in W_0$ form a basis for ${}_\mathcal{F}\mathcal{H}$ for multiplication on the left or multiplication on the right by ${}_\mathcal{F}\mathcal{A}$, in the sense that \begin{equation} {}_\mathcal{F}\mathcal{H}=\oplus_{w\in W_0}{}_\mathcal{F}\mathcal{A} T_w=\oplus_{w\in W_0}T_w{}_\mathcal{F}\mathcal{A}. \end{equation} The algebra structure of ${}_\mathcal{F}\mathcal{H}$ is determined by the Bernstein-Zelevinski-Lusztig relations as before: when $f\in{}_\mathcal{F}\mathcal{A}$ and $s=s_{\alpha}$ with $\alpha\in F_1$, we have \begin{equation} fT_s-T_sf^s=((q_{2\alpha^\vee}q_{\alpha^\vee}-1) +q_{\alpha^\vee}^{1/2} (q_{2\alpha^\vee}-1)\theta_{-\alpha/2}) \frac{f-f^s} {1-\theta_{-\alpha}} \end{equation} We have identified $\mathcal{A}$ with the algebra of regular functions on $T$ in the above formula.
Let us introduce \begin{equation} \begin{split} n_\alpha &:= q(s_\alpha)\Delta_\alpha c_\alpha \\ &=(q_{\alpha^\vee}^{1/2}+\theta_{-\alpha/2}) (q_{\alpha^\vee}^{1/2}q_{2\alpha^\vee}-\theta_{-\alpha/2})\in \mathcal{A},\\ \end{split} \end{equation} \index{n@$n_\alpha$, numerator of $c_\alpha$} where we used the Macdonald $c$-function introduced in equation (\ref{eq:defD}) and (\ref{eq:defc}).
The normalized intertwiners are now defined by (with $s=s_\alpha$, $\alpha\in R_1$): \begin{equation}\label{eq:defint} \iota^0_s :=n_\alpha^{-1}\iota_s\in {}_\mathcal{F}\mathcal{H}. \end{equation} \index{0j@$\iota_w^0$, normalized intertwining element of ${}_\mathcal{F}\mathcal{H}$} By the properties of the intertwiners listed above it is clear that $(\iota_s^0)^2=1$. In particular, $\iota_s^0\in{}_\mathcal{F}\mathcal{H}^\times$, the group of invertible elements of ${}_\mathcal{F}\mathcal{H}$. From the above we have the following result: \begin{lemma} The map $S_0\ni s\to \iota^0_s\in {}_\mathcal{F}\mathcal{H}^\times$ extends (uniquely) to a homomorphism $W_0\ni w\to \iota^0_w\in {}_\mathcal{F}\mathcal{H}^\times$. Moreover, for all $f\in {}_\mathcal{F}\mathcal{A}$ we have that $\iota_w^0f\iota_{w^{-1}}^0=f^w$. \end{lemma} Lusztig (\cite{Lu}, Proposition 5.5) proved that in fact \begin{theorem}\label{thm:ind} \begin{equation} {}_\mathcal{F}\mathcal{H}=\oplus_{w\in W_0}\iota^0_w{}_\mathcal{F}\mathcal{A}=\oplus_{w\in W_0}{}_\mathcal{F}\mathcal{A} \iota^0_w \end{equation} \end{theorem} Let $U\subset T$ be a nonempty, open, $W_0$-invariant subset. We denote by $\mathcal{Z}^{an}(U)$ \index{Z2@$\mathcal{Z}^{an}(U)$, ring of $W_0$-invariant holomorphic functions on $U\subset T$} the ring of $W_0$-invariant holomorphic functions of $U$. Consider the algebras $\mathcal{A}^{an}(U):=\mathcal{Z}^{an}(U)\otimes_\mathcal{Z}\mathcal{A}$ \index{A1@$\mathcal{A}^{an}(U):=\mathcal{Z}^{an}(U)\otimes_\mathcal{Z}\mathcal{A}$, ring of holomorphic functions on $U\subset T$} and $\mathcal{H}^{an}(T):=\mathcal{Z}^{an}(T)\otimes_\mathcal{Z}\mathcal{H}$. The algebra structure on $\mathcal{H}^{an}(T)$ is defined by $(f\otimes h) (f^\prime\otimes h^\prime):=ff^\prime\otimes hh^\prime$ (similar to the definition of ${}_\mathcal{F}\mathcal{H}$).
Let us first remark that the finite dimensional representation theory of the ``analytic'' affine Hecke algebra $\mathcal{H}^{an}(T)$ is the same as the finite dimensional representation theory of $\mathcal{H}$. Every finite dimensional representation $\pi$ of $\mathcal{H}$ determines a co-finite ideal $J_\pi\subset \mathcal{Z}$, the ideal of central elements of $\mathcal{H}$ which are annihilated by $\pi$. Denote by $J_\pi^{an}$ the ideal of $\mathcal{Z}^{an}(T)$ generated by $J_\pi$. Because of the co-finiteness we have an isomorphism \begin{equation} \mathcal{Z}/J_\pi \tilde{\rightarrow} \mathcal{Z}^{an}(T)/J^{an}_\pi(T). \end{equation} This shows that $\pi$ can be uniquely lifted to a representation $\pi^{an}$ \index{0p1@$\pi^{an}$, $\pi$ extended to $\mathcal{H}^{an}$} of $\mathcal{H}^{an}(T)$ whose restriction to $\mathcal{H}$ is $\pi$. The functor $\pi\to\pi^{an}$ defines an equivalence between the categories of finite dimensional representations of $\mathcal{H}$ and $\mathcal{H}^{an}(T)$ (with the inverse given by restriction).
For any $W_0$-invariant nonempty open set $U\subset T$ we define the localized affine Hecke algebra \begin{equation} \mathcal{H}^{an}(U):=\mathcal{Z}^{an}(U)\otimes_{\mathcal{Z}}\mathcal{H}. \end{equation} \index{H61@$\mathcal{H}^{an}(U):=\mathcal{Z}^{an}(U)\otimes_\mathcal{Z}\mathcal{H}$, the Hecke algebra with coefficients in $\mathcal{Z}^{an}(U)$} This defines a presheaf of $\mathcal{Z}^{an}$-algebras on $W_0\backslash T$, which is finitely generated over the analytic structure sheaf $\mathcal{Z}^{an}$ of the geometric quotient $W_0\backslash T$.
A similar argument as above shows that \begin{prop}\label{prop:anequiv} The category $\operatorname{Rep}(\mathcal{H}^{an}(U))$ of finite dimensional modules $\pi^{an}_U$ \index{0p2@$\pi^{an}_U$, $\pi$ extended to $\mathcal{H}^{an}(U)$} over $\mathcal{H}^{an}(U)$ is equivalent to the category $\operatorname{Rep}_U(\mathcal{H})$ \index{Rep@$\operatorname{Rep}_U(\mathcal{H})$, category of finite dimensional representations of $\mathcal{H}$ whose $\mathcal{Z}$-spectrum is contained in $U$} of finite dimensional modules $\pi$ over $\mathcal{H}$ whose $\mathcal{Z}$-spectrum is contained in $U$. \end{prop} \begin{lem}\label{lem:chinese} For every $W_0$-invariant nonempty open set $U$ in $T$, we have the isomorphism $\mathcal{A}^{an}(U)\simeq\mathcal{Z}^{an}(U)\otimes_\mathcal{Z} \mathcal{A}$, where $\mathcal{A}^{an}(U)$ denotes the ring of analytic functions on $U$. \end{lem} \begin{proof} Both the left and the right hand side are finitely generated modules over $\mathcal{Z}^{an}(U)$, and we have a natural morphism from the right hand side to the left hand side (product map). In order to prove that this map is an isomorphism it suffices to show this in the stalks of the corresponding sheaves at each point of $W_0\backslash U$. Let $\mathcal{I}_t$ denote the maximal ideal in $\mathcal{Z}$ corresponding to $W_0t$, and let $\hat\mathcal{Z}_{t}$ denote the $\mathcal{I}_t$-adic completion. Because $\hat\mathcal{Z}_{t}$ is faithfully flat over $\mathcal{Z}_t^{an}$ (the stalk at $W_0t$ of the sheaf $\mathcal{Z}^{an}$), it suffices to check that for each $t\in U$, we have \begin{equation}\label{imp} \hat\mathcal{Z}_{t}\otimes_{\mathcal{Z}^{an}_{t}}\mathcal{A}^{an}_{W_0t}\simeq \hat\mathcal{Z}_{t}\otimes_{\mathcal{Z}} \mathcal{A}, \end{equation} where $\mathcal{A}^{an}_{W_0t}=\oplus_{t^\prime}\mathcal{A}^{an}_{t^\prime}$ denotes the space of analytic germs at the set $W_0t$. Let $m_t$ denote the maximal ideal of $\mathcal{A}$ at $t\in T$, and let $\mathcal{I}_t\mathcal{A}=\prod_{t^\prime\in W_0t}j_{t^\prime}$ with $j_{t^\prime}=\mathcal{I}_t\mathcal{A}\cap m_{t^\prime}$. For all $t^\prime\in W_0t$ we have $\widehat{\mathcal{A}^{an}_{t^\prime}}_{j_{t^\prime}\mathcal{A}^{an}_{t^\prime}} =\hat{\mathcal{A}}_{j_{t^\prime}}$. Since $A^{an}_t\cap\mathcal{I}_tA^{an}_{W_0t}=j_t\mathcal{A}^{an}_t$, the left hand side of \ref{imp} is equal to $\oplus_{t^\prime\in W_0t}\hat{\mathcal{A}}_{j_{t^\prime}}$, the sum of the completions of $\mathcal{A}$ with respect to $j_{t^\prime}$. The right hand side of \ref{imp} is equal to the completion $\hat\mathcal{A}_{\mathcal{I}_t\mathcal{A}}$. By the Chinese remainder theorem, $\hat\mathcal{A}_{I_t\mathcal{A}}\simeq \oplus_{t^\prime\in W_0t}\hat\mathcal{A}_{j_{t^\prime}}$, finishing the proof. \end{proof} \begin{prop} The algebra $\mathcal{H}^{an}(U)$ is a free $\mathcal{A}^{an}(U)$ module of rank
$|W_0|$, with basis $T_w\otimes 1$ ($w\in W_0$). When $f\in\mathcal{A}^{an}(U)$ and $s=s_{\alpha}$ with $\alpha\in F_1$ we have again the Bernstein-Zelevinski-Lusztig relation \begin{equation} fT_s-T_sf^s=((q_{2\alpha^\vee}q_{\alpha^\vee}-1) +q_{\alpha^\vee}^{1/2} (q_{2\alpha^\vee}-1)\theta_{-\alpha/2}) \frac{f-f^s} {1-\theta_{-\alpha}}. \end{equation} This describes the multiplication in the algebra $\mathcal{H}^{an}(U)$. The center of $\mathcal{H}^{an}(U)$ is equal to $\mathcal{Z}^{an}(U)$. \end{prop} Similarly we have the localized meromorphic affine Hecke algebra $\mathcal{H}^{me}(U)$, which is defined by \begin{equation} \mathcal{H}^{me}(U):=\mathcal{F}^{me}(U)\otimes_{\mathcal{Z}}\mathcal{H}, \end{equation} \index{H6@$\mathcal{H}^{me}(U):=\mathcal{F}^{me}(U)\otimes_{\mathcal{Z}}\mathcal{H}$, localized Hecke algebra with meromorphic coefficients} where $\mathcal{F}^{me}(U)$ \index{F@$\mathcal{F}^{me}(U)$, quotient field of $\mathcal{Z}^{an}(U)$} it the quotient field of $\mathcal{Z}^{an}(U)$. We write $\mathcal{A}^{me}(U):=\mathcal{F}^{me}(U)\otimes_{\mathcal{Z}}\mathcal{A}$ \index{A3@$\mathcal{A}^{me}(U):=\mathcal{F}^{me}(U)\otimes_{\mathcal{Z}}\mathcal{A}$, ring of meromorphic functions on $U\subset T$}. It is the ring of meromorphic functions on $U$. \begin{theorem}\label{thm:gralg} \begin{equation} \mathcal{H}^{me}(U)=\oplus_{w\in W_0}\mathcal{A}^{me}(U)\iota^0_w=\oplus_{w\in W_0} \iota_w^0\mathcal{A}^{me}(U) \end{equation} \end{theorem} \begin{proof} This is clear from Theorem \ref{thm:ind} by the remark that $\mathcal{H}^{me}$ arises from the $\mathcal{F}$-algebra ${}_\mathcal{F}\mathcal{H}$ by extension of scalars according to \begin{equation} \begin{split} \mathcal{H}^{me}(U)&=\mathcal{F}^{me}(U)\otimes_\mathcal{Z}\mathcal{H}\\ &=\mathcal{F}^{me}(U)\otimes_\mathcal{F}{}_\mathcal{F}\mathcal{H}.\\ \end{split} \end{equation} \end{proof} \subsection{Lusztig's structure theorem and parabolic induction}\label{sub:lus} We shall investigate the structure of the tracial states $\chi_t$, using Lusztig's technique of localization of $\mathcal{H}$ as discussed above. The results in the present subsection are substitutes for the usual techniques of parabolic induction for reductive groups. The results in this subsection are closely related to the results on parabolic induction in the paper \cite{BM}.
We use in fact a slight variation of the results of Lusztig \cite{Lu}. There are two main differences. First of all we work with analytic localization at suitably small neighborhoods, instead of Lusztig's use of adic completion. In addition we have replaced the root system of the localized algebra which Lusztig has defined by something slightly different. Lusztig's construction only works with the additional assumption in Convention \ref{eq:scale} that $f_s\in\mathbb{N}$, and this assumption is not natural in our context. We have therefore adapted the construction.
We define a function \begin{equation} T\ni t\to R_{P(t)}\subset R_0, {\mathrm{\ a\ parabolic\ subsystem}} \end{equation} \index{R6@$R_{P(t)}=R_{P(\varpi)}\subset R_0$, parabolic subsystem associated with $t\in\varpi\subset T$} by putting $R_{P(t)}:=R_0\cap\mathfrak{t^*}_{<t>}$, with $\mathfrak{t^*}_{<t>}\subset\mathfrak{t^*}= \mathbb{R}\otimes_{\mathbb{Z}}X$ the subspace spanned by the roots $\alpha\in R_0$ for which one of the following properties holds \begin{enumerate} \item $c_\alpha\not\in\mathcal{O}_t^\times$ (the invertible holomorphic germs at $t$). \item $\alpha(t)=1$, \item $\alpha(t)=-1$ and $\alpha\not\in 2X$. \end{enumerate} We let $P(t)\subset R_{P(t),+}:=R_{P(t)}\cap R_{0,+}$ \index{P@$P(\varpi)=P(t)$, basis of simple roots in $R_{P(t),+}$ where $\varpi=W_{P(t)}t$} be the basis of simple root for $R_{P(t),+}$.
We have the following easy consequences of the definition: \begin{prop}\label{prop:eltprop} \begin{enumerate}\label{def:two} \item $t\to R_{P(t)}$ is lower semi-continuous with respect to the Zariski-topology of $T$ and the ordering of subsets of $R_0$ by inclusion. \item $t\to R_{P(t)}$ is equivariant: for all $w\in W$ we have $R_{P(wt)}=w(R_{P(t)}$. \end{enumerate} \end{prop} We denote by $W_{P(t)}$ \index{W4@$W_{P(t)}=W(R_{P(t)})$, parabolic subgroup associated with $t\in T$} the parabolic subgroup of $W_0$ generated by the reflections $s_\alpha$ with $\alpha\in R_{P(t)}$. We say that $t_1,t_2\in W_0t$ are equivalent if there exists a $w\in W_{P(t_1)}$ such that $t_2=w(t_1)$. To see that this is actually an equivalence relation, observe that $R_{P(t_2)}=R_{P(t_1)}$ for all $t_2\in W_{P(t_1)}t_1$. The equivalence classes are the orbits $\varpi=W_{P(t)}t$. This gives a partition of $W_0t$ in a collection equivalence classes which are denoted by $\varpi\subset W_0t$. If $t\in \varpi$ we sometimes write $P(\varpi)$, $W_{P(\varpi)}$ etc. instead of $P(t)$, $W_{P(t)}$ etc. Note that $W_0$ acts transitively on the set of equivalence classes and that for each equivalence class $\varpi$, $W_{P(\varpi)}$ acts transitively on $\varpi$.
Let $\varpi\subset W_0t$ be the equivalence class of $t$. We define: \begin{equation} W_\varpi:=\{w\in W_0\mid w(\varpi)=\varpi\}. \end{equation} \index{W4a@$W_\varpi$, stabilizer in $W_0$ of $\varpi=W_{P(t)}t$} By Proposition \ref{prop:eltprop} it is clear that $W_{P(\varpi)}\vartriangleleft W_\varpi$, and that this normal subgroup is complemented by the subgroup \begin{equation} W(\varpi):=\{w\in W_\varpi\mid w(P(\varpi))=P(\varpi)\} \end{equation} \index{W4b@$W(\varpi)= \{w\in W_\varpi\mid w(P(\varpi))=P(\varpi)\}$, complement of $W_{P(\varpi)}$ in $W_\varpi$} \begin{lem}\label{lus:lem8.2b} For $\alpha\in R_0$ we have: $\alpha\in R_{P(\varpi)}\Longleftrightarrow s_\alpha\in W_\varpi$. \end{lem} \begin{proof} We only need to show that $s_\alpha\in W_\varpi$ implies that $\alpha\in R_{P(\varpi)}$ (the other direction being obvious). Notice that if $t\in\varpi$ we have \begin{equation} t^{-1}\varpi\subset\mathbb{Z}R_{P(\varpi)}^\vee\otimes\mathbb{C}^\times. \end{equation} If $s_\alpha\in W_\varpi$ then $s_\alpha(t)\in\varpi$, and thus \begin{equation}\label{eq:lusz} \alpha^\vee\otimes\alpha(t)\in\mathbb{Z} R_{P(\varpi)}^\vee\otimes\mathbb{C}^\times. \end{equation} By Proposition \ref{prop:eltprop}, we have $s_\alpha(R_{P(\varpi)})=R_{P(\varpi)}$. Since $R_{P(\varpi)}$ is parabolic this implies that either $\alpha\in R_{P(\varpi)}$ or that $\alpha(R_{P(\varpi)}^\vee)=0$. In the first case we are done, so let us assume the second case. By (\ref{eq:lusz}) it follows that $1=\alpha(\alpha^\vee\otimes\alpha(t))=\alpha(t)^2$. If $\alpha(t)=1$ we have $\alpha\in R_{P(\varpi)}$ by definition, contradicting the assumption. If $\alpha(t)=-1$ and $\alpha\not\in2X$ then, by definition, $\alpha\in R_{P(\varpi)}$, contrary to the assumption. If $\alpha(t)=-1$ and $\alpha=2x$ for some $x\in X$ then (\ref{eq:lusz}) implies $1=x(\alpha^\vee\otimes\alpha(t))=\alpha(t)=-1$, again a contradiction. We conclude that the second case does not arise altogether, and we are done. \end{proof} Consider the algebra $\mathcal{H}^{P(t)}:=\mathcal{H}(X,Y,R_{P(t)},R_{P(t)}^\vee,P(t))$. Note that $W(\varpi)$ acts by means of automorphisms on $\mathcal{R}^{P(t)}=(X,Y,R_{P(t)},R_{(t)}^\vee,P(t))$, compatible with the root labels $q$. Thus we may define an action of $\gamma\in W(\varpi)$ on $\mathcal{H}^{P(t)}$ by $\gamma(T_w\theta_x)=T_{(\gamma w\gamma^{-1})}\theta_{\gamma x}$. In this way we form the algebra $\mathcal{H}^\varpi:=\mathcal{H}^{P(t)}[W(\varpi)]$ \index{H7@$\mathcal{H}^\varpi$, cross product of $\mathcal{H}^{P(\varpi)}$ by $W(\varpi)$}, with its product being defined by $(h_1\gamma_1)(h_2\gamma_2)=h_1\gamma_1(h_2)\gamma_1\gamma_2$.
By Proposition \ref{prop:eltprop}(i) it is obvious that for any $t\in T$ there exists an open ball $B\subset\mathfrak{t}_{\mathbb C}$ centered around the origin such that the following conditions are satisfied: \begin{cond}\label{cond} \begin{enumerate}
\item[(i)] $\forall \alpha\in R_0,b\in B:|\mbox{\rm Im}\,(\alpha(b))|<\pi$. In particular, the map $\exp:\mathfrak{t}_{\mathbb C}\to T$ restricted to $B$ is an analytic diffeomorphism onto its image $\exp(B)$ in $T$. \item[(ii)] If $w\in W_0$ and $t\exp(B)\cap w(t\exp(B))\not=\emptyset$ then $wt=t$. \item[(iii)] For all $t^\prime\in t\exp(B)$, we have $R_{P(t^\prime)}\subset R_{P(t)}$. \end{enumerate} \end{cond}
Let $t\in T$. We take $B\subset\mathfrak{t}_{\mathbb C}$ as above and we put $U=W_0t\exp(B)$. Concerning the analytic localization $\mathcal{H}^{an}(U)$ we have the following analog of Lusztig's first reduction theorem (see \cite{Lu}): \begin{thm}\label{thm:lusind} For $\varpi\subset W_0t$ an equivalence class, we put $U_\varpi:=\varpi\exp(B)$. We define $1_\varpi\in\mathcal{A}^{an}(U)$ by $1_\varpi(u)=1$ if $u\in U_\varpi$ and $1_\varpi(u)=0$ if $u\not\in U_\varpi$. The elements $1_\varpi$ \index{>@$1_\varpi$, idempotent in $\mathcal{A}^{an}(U)$ with support $U_\varpi$} are mutually orthogonal idempotents. Let $t\in\varpi$. \begin{enumerate} \item[(i)] We have $\mathcal{H}^{\varpi,an}(U_\varpi):=\mathcal{H}^{P(\varpi),an}(U_\varpi)[W(\varpi)]\simeq 1_\varpi\mathcal{H}^{an}(U)1_\varpi$. \item[(ii)] We can define linear isomorphisms \begin{equation} \Delta_{\varpi_1,\varpi_2}:\mathcal{H}^{\varpi,{an}}(U_\varpi)\to 1_{\varpi_1}\mathcal{H}^{an}(U)1_{\varpi_2}. \end{equation} \index{0D4@$\Delta_{\varpi_1,\varpi_2}:\mathcal{H}^{\varpi,{an}}(U_\varpi)\to 1_{\varpi_1}\mathcal{H}^{an}(U)1_{\varpi_2}$, linear isomorphisms} such that $\Delta_{\varpi_1,\varpi_2}(h)\Delta_{\varpi_3,\varpi_4}(h^\prime)= \Delta_{\varpi_1,\varpi_4}(hh^\prime)$ if $\varpi_2=\varpi_3$, and $\Delta_{\varpi_1,\varpi_2}(h)\Delta_{\varpi_3,\varpi_4}(h^\prime)=0$ else. \item[(iii)] The center of $\mathcal{H}^{\varpi,an}(U_\varpi)$ is $\mathcal{Z}^{\varpi,an}(U_\varpi):=(\mathcal{A}^{an}(U_\varpi))^{W_\varpi}$. This algebra is isomorphic to $\mathcal{Z}^{an}(U)$ via the map $z\to 1_\varpi z$, and this gives $\mathcal{H}^{\varpi,an}(U_\varpi)$ the structure of a $\mathcal{Z}^{an}(U)$-algebra. \item[(iv)] Let $N$ denote the number of equivalence classes in $W_0t$. There exists an isomorphism $\mathcal{H}^{an}(U)\simeq(\mathcal{H}^{\varpi,an}(U_\varpi))_N$, the algebra of $N\times N$ matrices with entries in $1_\varpi\mathcal{H}^{an}(U)1_\varpi\simeq\mathcal{H}^{\varpi,an}(U_\varpi)$. It is an isomorphism of $\mathcal{Z}^{an}(U)$-algebras. \end{enumerate} \end{thm} \begin{proof} The difference with Lusztig's approach is that he works with the $\mathcal{I}_t$-adic completions of the algebras instead of the localizations to $U$. In addition, we have a different definition of the root system $R_{P(t)}$.
Using Lemma \ref{lem:chinese} we can copy the arguments of \cite{Lu}, because of the Conditions \ref{cond} for $B$ and because of Lemma \ref{lus:lem8.2b} (which replaces in our situation Lemma 8.2b of \cite{Lu}). By this we see that the function $c_\alpha$ is analytic and invertible on $U_\varpi\cup U_{s_\alpha\varpi}$ for all $\alpha\in R_0$ such that $s_\alpha\not\in W_\varpi$ (compare \cite{Lu}, Lemma 8.9), and this is the crucial point of the construction. \end{proof} \begin{cor} The functor $V\to V_\varpi:=1_\varpi V$ defines an equivalence between the category of finite dimensional representations of $\mathcal{H}^{an}(U)$ and the category of finite dimensional representations of $\mathcal{H}^{\varpi,an}(U_\varpi)=\mathcal{H}^{P(t),an}(U_\varpi)[W(\varpi)]$. We have $\dim(V)=N\dim(V_\varpi)$ where $N$ denotes the number of equivalence classes in $W_0t$.
$\square$\par
\end{cor} \begin{dfn}\label{defn:generic} Let $R_P\subset R_0$ be a parabolic root subsystem, with $P\subset R_{P,+}:=R_P\cap R_{0,+}$ its basis of simple roots. We denote the corresponding parabolic subgroup of $W_0$ by $W_P:=W(R_P)$. We call $t\in T$ an {\it $R_P$-generic point} if $W_\varpi\subset W_P$ for $\varpi=W_{P(t)}t$. \end{dfn} \begin{cor}\label{cor:gen} If $t$ is $R_P$-generic we have $R_{P(t)}\subset R_P$. \end{cor} \begin{proof} This is immediate from the definitions. \end{proof}
We define for any parabolic subsystem $R_P\subset R_0$ with basis $P$ of $R_{P,+}$ the parabolic subalgebra $\mathcal{H}^P=\mathcal{H}(X,Y,R_P,R_P^\vee,P)\subset \mathcal{H}$ with root labels $q^P$\index{H4@$\mathcal{H}^P=\mathcal{H}(\mathcal{R}^P,q^P)$, parabolic subalgebra of $\mathcal{H}$}.
Assume that $B$ satisfies the Conditions \ref{cond}. Notice that if $t$ is $R_P$-generic, then every $t^\prime\in t\exp(B)$ is $R_P$-generic. Indeed, let $\varpi^\prime=W_{P(t^\prime)}t^\prime$ and $\varpi=W_{P(t)}t$. If $w\in W_{\varpi^\prime}$, then there exists a $w^\prime\in W_{P(t^\prime)}\subset W_{P(t)}$ (by condition \ref{cond}(iii)) such that $w^\prime t^\prime= w t^\prime$ (since the equivalence class of $t^\prime$ is a $W_{P(t^\prime)}$-orbit). By condition \ref{cond}(ii), also $w^\prime t=wt$. Hence $w\in W_\varpi\subset W_P$, as required.
We now put $U=W_0t\exp(B)$, $U_P=W_Pt\exp(B)$ \index{U@$U_P$, certain $W_P$-invariant open set in $T$} and consider the localization $\mathcal{H}^{P,an}(U_P)$. \begin{cor}\label{cor:struct} Assume that $t\in T$ is $R_P$-generic. We have
$\mathcal{H}^{an}(U)\simeq(\mathcal{H}^{P,an}(U_P))_{|W^P|}$, where $W^P=W_0/W_P$. Moreover, when we define $1_P:=\sum_{\varpi\subset W_Pt}1_\varpi$ \index{>@$1_P$, idempotent in $\mathcal{A}^{an}(U)$, with support $U_P$} then $\mathcal{H}^{P,an}(U_P)\simeq 1_P\mathcal{H}^{an}(U)1_P$. These are isomorphisms of $\mathcal{Z}^{an}(U)$-algebras. \end{cor} \begin{proof} The fact that $t$ is $R_P$-generic implies that the $W_0$-equivalence classes of the elements of $W_Pt$ are equal to the $W_P$-equivalence classes of these elements. Therefore we have, by the above theorem, $\mathcal{H}^{P,an}(U_P)\simeq(\mathcal{H}^{\varpi,an}(U_\varpi))_{n_P}$, where $n_P$ is the number of equivalence classes $\varpi^\prime$ in the orbit $W_Pt$. And for each $w\in W_0$, $wW_Pt\subset W_0t$ is a union of $n_P$ distinct $W_0$-equivalence classes. The orbit
$W_0t$ is the disjoint union of $|W^P|$ subsets of the form $wW_Pt\subset W_0t$, since the stabilizer of $t$ is contained in $W_P$ (because $t$ is $R_P$-generic). Each subset $wW_{P(t)}t$ in $W_0t$ is partitioned into $n_P$ equivalence classes, and the result follows. \end{proof} Recall that, by Proposition \ref{prop:anequiv}, a finite dimensional representation $(V,\pi)$ of $\mathcal{H}$ with its $\mathcal{Z}$-spectrum contained in $U$ extends uniquely to a representation $(V^{an},\pi^{an})$ of $\mathcal{H}^{an}(U)$. \begin{cor}\label{cor:restr} In the situation of Corollary \ref{cor:struct}, there exists an equivalence $(V,\pi)\to (V_P,\pi_P)$ between $\operatorname{Rep}_U(\mathcal{H})$ and $\operatorname{Rep}_{U_P}(\mathcal{H}^P)$, characterized by $V_P^{an}=1_PV^{an}$ \index{V@$(V_P,\pi_P)$, representation of $\mathcal{H}^P$ with $V_P=1_PV$}. We have
$\dim(V)=|W^P|\dim(V_P)$, and the inverse functor is given by $V_P\to\operatorname{Ind}_{\mathcal{H}^P}^\mathcal{H}(V_P)=\mathcal{H}\otimes_{\mathcal{H}^P}V_P$. The character $\chi^P$ of the module $(V_P,\pi_P)$ of $\mathcal{H}^P$ is given in terms of the character $\chi_{\pi}$ of $(V,\pi)$ by the formula $\chi^P(h)=\chi_{\pi}(1_Ph)$. \end{cor} \begin{proof} We localize both the algebras $\mathcal{H}$ and $\mathcal{H}^P$ and use Proposition \ref{prop:anequiv} and Corollary \ref{cor:struct}. Using Corollary
\ref{cor:struct} we see that the functor $V\to 1_PV^{an}|_{\mathcal{H}^P}$ is the required equivalence. The relation between the dimensions of $V$ and $V_P$ is obvious from this definition. Conversely, again using Corollary \ref{cor:struct}, we have \begin{equation} \begin{split} 1_P(\operatorname{Ind}_{\mathcal{H}^P}^\mathcal{H} V_P)^{an}&= 1_P(\mathcal{H}\otimes_{\mathcal{H}^P}V_P)^{an}\\ &=1_P(\sum_{P^\prime,P^{\prime\prime}}1_{P^\prime} \mathcal{H}^{P,an}(U_P)1_{P^{\prime\prime}})\otimes_{\mathcal{H}^{P,an}(U_P)}1_PV^{an}\\ &=1_PV^{an}=V_P^{an}, \end{split} \end{equation} finishing the proof. \end{proof} \begin{prop}\label{luspardef} Let $P\subset F_0$ be a subset, and let $R_P\subset R_0$ be the corresponding {\it standard} parabolic subsystem. We define the subtori $T_P$, $T^P$ and the lattices $X_P$, $Y_P$ as in Proposition \ref{prop:red}. Put $\mathcal{R}_P=(X_P,Y_P,R_P,R_P^\vee,F_P)$, and let $t\in T^P$. There exists a surjective homomorphism $\phi_{t}:\mathcal{H}^P\to\mathcal{H}_P$ \index{0v@$\phi_t:\mathcal{H}^P\to\mathcal{H}_P$, surjective homomorphism} which is characterized by (1) $\phi_t$ is the identity on the finite dimensional Hecke algebra $\mathcal{H}(W_P)$, and (2) $\phi_{t}(\theta_x)=x(t)\theta_{\overline{x}}$, where $\overline{x}\in X_P$ is the natural image of $x$ in $X_P=X/{}^PX=X/(X\cap{Y_P}^\perp)$. \end{prop} \begin{proof} We have to check that $\phi_{t}$ is compatible with the Bernstein-Zelevinski-Lusztig relations. Let $s=s_\alpha$ with $\alpha\in P\subset F_0$. Then \begin{gather} \begin{split} \theta_x&T_s-T_s\theta_{s(x)}=\\ &=\left\{ \begin{array}{ccc} &(q_{\alpha^\vee}-1)\frac{\theta_x-\theta_{s(x)}} {1-\theta_{-\alpha}}\ &{\rm if}\ 2\alpha\not\inR_{\mathrm{nr}}.\\ &((q_{\alpha^\vee/2}q_{\alpha^\vee}-1) +q_{\alpha^\vee/2}^{1/2} (q_{\alpha^\vee}-1)\theta_{-\alpha}) \frac{\theta_x-\theta_{s(x)}} {1-\theta_{-2\alpha}}\ &{\rm if}\ 2\alpha\inR_{\mathrm{nr}}.\\ \end{array} \right.\\ \end{split} \end{gather} Since $s$ acts trivially on $T^P$, we have $x(t)=sx(t)$. This implies the result. \end{proof} \begin{dfn}\label{dfn:ind} Let $P\subset F_0$ be a subset. In this case we identify the algebra $\mathcal{H}^P=\mathcal{H}(X,Y,R_P,R_P^\vee,P)$ with the subalgebra in $\mathcal{H}$ generated by $\mathcal{H}(W_P)$ and ${\mathbb C}[X]$. Let $(V,\delta)$ be a representation of $\mathcal{H}_P$ with central character $W_Pr\in W_P\backslash T_P$, and let $t\in T^P$. Denote by $\delta_t$ the representation $\delta_t=\delta\circ\phi_t$ of $\mathcal{H}^P$. We define a representation $\pi(\mathcal{R}_P,W_Pr,\delta,t)$ of $\mathcal{H}$ by $\pi(\mathcal{R}_P,W_Pr,\delta,t)=\operatorname{Ind}_{\mathcal{H}^P}^\mathcal{H}(\delta_t)$ \index{0p@$\pi(\mathcal{R}_P,W_Pr,\delta,t)=\operatorname{Ind}_{\mathcal{H}^P}^\mathcal{H}(\delta_t)$, parabolically induced representation}. We refer to such representations as parabolically induced representations. \end{dfn} \begin{cor}\label{cor:short} Let $W_0t\in W_0\backslash T$, and let $R_P$ be a standard parabolic subsystem of $R_0$. Suppose that there exists an $r\in T_P$ and $t^P\in T^P$ such that $rt^P\in W_0t$ is an $R_P$-generic point. The map $\delta\to\pi(\mathcal{R}_P,W_Pr,\delta,t^P)$ gives an equivalence between the representations of $\mathcal{H}$ with central character $W_0t$ and the representations of $\mathcal{H}_P$ with central character $W_Pr$. \end{cor} \begin{proof} By Corollary \ref{cor:restr}, the induction functor from representations of $\mathcal{H}^P$ to $\mathcal{H}$ gives rise to an equivalence between the representations of $\mathcal{H}^P$ with $R_P$-regular central character $W_Pt$ and the representations of $\mathcal{H}$ with central character $W_0t$. If $\pi$ is a representation of $\mathcal{H}^P$ with central character $W_Pt$, then it is easy to see that the annihilator of $\pi$ contains the kernel of the homomorphism $\phi_{t^P}$. Thus $\pi$ is the lift via $\phi_{t^P}$ of a representation $\delta$ of $\mathcal{H}_P$. This gives an equivalence between the category of representations of $\mathcal{H}_P$ with central character $W_Pr$ and the representations of $\mathcal{H}^P$ with central character $W_Pt$. \end{proof} The following proposition describes the induced modules analogous to the ``compact realization'' of parabolically induced representations of real reductive groups. \begin{prop}\label{prop:unit} Let $(V,\delta)$ be an irreducible representation of $\mathcal{H}_P$ with central character $W_Pr\in W_P\backslash T_P$ as before. Suppose that $(V,\delta)$ is unitary with respect to an Hermitian inner product $(\cdot,\cdot)$, and that $t\in T^P_u$. We identify the underlying representation space $V_\pi$ of $\pi:=\pi(\mathcal{R}_P,W_Pr,\delta,t)$ with \begin{equation} V_\pi:=\mathcal{H}(W^P)\otimes V, \end{equation} where $\mathcal{H}(W^P)\subset H(W_0)$ denotes the subspace spanned by the elements $T_w$ with $w\in W^P$. Then $\pi$ is unitary with respect to the Hermitian inner product $\langle\cdot,\cdot\rangle$ \index{<@$\langle\cdot,\cdot\rangle$!inner product on $V_\pi$} defined on $V_\pi$ by (with $x,y\in \mathcal{H}(W^P)$, and $u,v\in V$): \begin{equation} \langle x\otimes u,y\otimes v\rangle:= \tau(x^*y)(u,v). \end{equation} \end{prop} \begin{proof} The above form is clearly Hermitian and positive definite. It remains to show that the inner product satisfies \begin{equation}\label{eq:desire} \langle \pi(h)m_1,m_2\rangle=\langle m_1,\pi(h^*)m_2\rangle \end{equation} for each $m_1,m_2\in V_\pi, h\in\mathcal{H}$. To this end we recall Theorem 2.20 of \cite{EO}. Let $i_s:\mathcal{H}\to\operatorname{End}(\mathcal{H}_0)$ denote the minimal principal series representation induced from $s\in T$. Then the nondegenerate sesquilinear pairing defined on $\mathcal{H}_0\times \mathcal{H}_0$ by \begin{equation}\label{eq:natinprod} (x,y):=\tau(x^*y) \end{equation} satisfies the property \begin{equation}\label{eq:starcomp} (i_{s}(h)x,y)=(x,i_{s^*}(h^*)y) \end{equation} (see Theorem \ref{thm:ster} for the definition of $s^*$). We have $\mathcal{H}_0=\mathcal{H}(W^P)\otimes \mathcal{H}(W_P)$, and the pairing (\ref{eq:natinprod}) on $\mathcal{H}_0$ factors as the tensor product of the pairings on $\mathcal{H}(W^P)$ and on $\mathcal{H}(W_P)$ which are also defined by equation (\ref{eq:natinprod}) but with $x,y$ both in $\mathcal{H}(W^P)$ or both in $\mathcal{H}(W_P)$.
We choose $r\in T_P$ such that $V$ contains a simultaneous eigenvector $v$ for $X_P$ with eigenvalue $r$. Via $\delta_t$, the vector $v\in V$ has eigenvalue $rt\in T$ with respect to $X$. Thus there is a surjective morphism of $\mathcal{H}^P$-modules $\alpha: \mathcal{H}(W_P)\twoheadrightarrow V$, where $\mathcal{H}(W_P)$ is the minimal principal series module $i_{rt}^P$ for $\mathcal{H}^P$, and $V$ is the representation space of $\delta_t$. By the above, applied to $\mathcal{H}^P$, we have the adjoint injective morphism $\alpha^*:V\hookrightarrow \mathcal{H}(W_P)$, where the action on $\mathcal{H}(W_P)$ is via $i^P_{r^*t}$ (since $(rt)^*=r^*t$, because $t\in T^P_u$). By the exactness and the transitivity of induction we get morphisms of $\mathcal{H}$-modules $\operatorname{Ind}(\alpha):i_{rt}\twoheadrightarrow\pi$ and $\operatorname{Ind}(\alpha^*):\pi\hookrightarrow i_{r^*t}$. Notice that $\operatorname{Ind}(\alpha)= \operatorname{Id}_{\mathcal{H}(W^P)}\otimes \alpha$ and similarly, $\operatorname{Ind}(\alpha^*)= \operatorname{Id}_{\mathcal{H}(W^P)}\otimes \alpha^*$. By the factorization of the pairing (\ref{eq:natinprod}) we see that $\operatorname{Ind}(\alpha)$ and $\operatorname{Ind}(\alpha^*)$ are adjoint with respect to the pairings $\langle\cdot,\cdot\rangle$ on $V_\pi$ and (\ref{eq:natinprod}) on $\mathcal{H}_0$. This, the surjectivity of $\operatorname{Ind}(\alpha)$ and (\ref{eq:starcomp}) gives the desired result (\ref{eq:desire}). \end{proof} \begin{prop}\label{prop:indtemp} With the notations as above, assume that $(V,\delta)$ is a tempered representation with central character $W_Pr\in W_P\backslash T_P$ and that $t\in T^P_u$. Then $\pi:=\pi(\mathcal{R}_P,W_Pr,\delta,t)$ is a tempered representation of $\mathcal{H}$. \end{prop} \begin{proof} Recall that we have the identification \begin{equation} V_\pi:=\mathcal{H}(W^P)\otimes V, \end{equation} where $\mathcal{H}(W^P)\subset H(W_0)$ denotes the subspace spanned by the elements $T_w$ with $w\in W^P$. Recall from the proof of Lemma \ref{lem:cas} that we can find a basis $(v_j)$ of $V$ such that $X_P$ acts by upper triangular matrices with respect to this basis. By Casselman's criterion, the diagonal entries are characters $r_{j,j}\in W_Pr$
of $X_P$ which satisfy $|x(r_{i,i})|\leq 1$ for $x\in X_{P}^+$. Let $(w_i)$ denote an ordering of the set $W^P$ such that the length $l(w_i)$ increases with $i$. We take the tensors $T_{w_i}\otimes v_j$, ordered lexicographically, as a basis for the representation space of $\pi$. From a direct application of the Bernstein-Zelevinski-Lusztig relations we see that the $\theta_x$ are simultaneously upper triangular in this basis, and that the diagonal entries are the elements $w_i(tr_{j,j})$. Since $w_i\in W^P$, $t\in T^P_u$ and since the vector part of $r_{j,j}$ is an element of the cone generated by the negative roots of $R_P^\vee$, it follows from a well known characterization of $W^P$ that the vector parts of these diagonal entries are in the antidual of the positive chamber. Again using Lemma \ref{lem:cas} we conclude that $\pi$ is tempered. \end{proof} \subsection{The tracial states $\chi_t$ and parabolic induction}\label{sub:states} In this subsection we will compute the states $\chi_t$ on $W_0$-orbits of tempered residual cosets of positive dimension in terms of the characters of unitary representations which are induced from discrete series characters of parabolic subalgebras, as was discussed in Subsection \ref{sub:lus}.
Let $L$ be a residual coset such that $R_L\subset R_0$ is a standard parabolic subsystem. In other words, $F_L\subset F_0$. Let us denote by $\mathcal{H}_L$ the affine Hecke algebra with root datum $\mathcal{R}_L:=(X_L,Y_L,R_L,R_L^\vee,F_L)$ and root labels $q_L$. \index{H4@$\mathcal{H}_L:=\mathcal{H}(\mathcal{R}_L,q_L)$, semisimple quotient of $\mathcal{H}^L$} (see Proposition \ref{prop:red}). Let $r_L=c_Ls_L\in T_L$ be the corresponding residual point of $\mathcal{R}_L$. \begin{lem}\label{lem:eisext} Let $U\subset T$ be a nonempty $W_0$-invariant open subset. Let $t\in U$. There exists a unique extension of the Eisenstein functional (cf. equations \ref{fundeis}) $E_t$ (which we will also denote by $E_t$) to the localization $\mathcal{H}^{an}(U)$, such that $E_t(fh)=E_t(hf)=f(t)E_t(h)$ for all $f\in\mathcal{A}^{an}(U)$. \end{lem} \begin{proof} The functional $E_t$ factors to a functional of the finite dimensional ${\mathbb C}$-algebra $\mathcal{H}^t:=\mathcal{H}/\mathcal{I}_t\mathcal{H}$ \index{H7@$\mathcal{H}^t=\mathcal{H}/\mathcal{I}_t\mathcal{H}$, where $\mathcal{I}_t$ is the maximal ideal of $W_0t$ in $\mathcal{Z}$}, where $\mathcal{I}_t$ is the maximal ideal in $\mathcal{Z}$ corresponding to $W_0t$. We have $\mathcal{H}/\mathcal{I}_t\mathcal{H}=\mathcal{H}^{an}(U)/\mathcal{I}^{an}_t(U)\mathcal{H}^{an}(U)$ for $t\in U$, and this defines the extension with the required property uniquely. \end{proof} \begin{lem}\label{lem:pareis} Let $L$ be such that $R_L\subset R_0$ is a standard parabolic subset of roots, and let $t_0\in T$ be $R_L$-generic. Set $U=W_0t_0\exp(B)$ with $B$ satisfying the conditions \ref{cond} (i), (ii), and (iii). As before, we put $U_L=W_Lt_0\exp(B)$.
We denote by $E^L_t$ \index{E1@$E^L_t$, Eisenstein functional of $\mathcal{H}^L$} the Eisenstein functional of the subalgebra $\mathcal{H}^L\subset \mathcal{H}$. For $t_L\in T_L$ we write $E_{L,t_L}$ \index{E1@$E_{L,t_L}$, Eisenstein functional of $\mathcal{H}_L$} to denote the Eisenstein functional at $t_L\in T_L=\operatorname{Hom}(X_L,{\mathbb C}^\times)$ of the algebra $\mathcal{H}_L$. Let $t=t^Lt_L\in U$ with $t^L\in T^L$ and $t_L\in T_L$. Recall $1_L$ is the characteristic function of $U_L$. We have, for all $h\in \mathcal{H}^L$: \begin{enumerate} \item[(i)] $E_t(1_Lh1_L)={q(w^L)1_L(t)\Delta^L(t)}E^L_t(h)$. \item[(ii)] $E_t^L(h)=E_{L,t_L}(\phi_{t^L}(h))$. \end{enumerate} \end{lem} \begin{proof} Because these are both equalities of holomorphic functions of $t\in U$ it suffices to check them for $t$ regular, and outside the union of the residual cosets (in other words, $c(t)c(t^{-1})\not=0$).
(i). By the defining properties \ref{fundeis} and \cite{EO}, 2.23(4) we need only to show that the left hand side satisfies the properties $E_t(1_Lxh1_L)= E_t(1_Lhx1_L)=t(x)1_L(t)E_t(h)$ and $E_t(1_L)=q(w_0)1_L(t)\Delta(t)$. These facts follow from Lemma \ref{lem:eisext}.
(ii). We see that \begin{align}\nonumber E_{L,t_L}(\phi_{t^L}(\theta_xh))&=E_{L,t_L}(x(t^L) \theta_{\overline{x}}\phi_{t^L}(h))\\\nonumber &=x(t^L)\overline{x}(t_L)E_{L,t_L}(\phi_{t^L}(h))\\\nonumber &=x(t)E_{L,t_L}(\phi_{t^L}(h)), \end{align} and similarly for $E_{L,t_L}(\phi_{t^L}(h\theta_x))$. The value at $h=1$ is equal to $q(w_L)\Delta_L(t)$ on both the left and the right hand side. As in the proof of (i), this is enough to prove the desired equality. \end{proof} \begin{theorem}\label{thm:mainind} Let $L$ be a residual coset such that $R_L\subset R_0$ is a standard parabolic subset of roots. Let $t^L\in T^L_u$ be such that $t:=r_Lt^L\in L^{temp}$ is $R_L$-generic. Notice that this condition is satisfied outside a finite union of real codimension one subsets in $T^L_u$. Let $\Delta_{\mathcal{R}_L,W_Lr_L}$ be complete set of inequivalent irreducible representations of the residual algebra $\overline{\mathcal{H}_L^{r_L}}$, and let $\chi_{\mathcal{H}_L,r_L}=\sum_{\delta\in\Delta_{\mathcal{R}_L,W_Lr_L}} \chi_{\delta}d_{\mathcal{R}_L,\delta}$ be the corresponding decomposition in irreducible discrete series characters of the tracial state $\chi_{\mathcal{H}_L,r_L}$ of ${\mathcal{H}_L}$. \begin{enumerate} \item[(i)] For all $\delta$, $\pi(\mathcal{R}_L,W_Lr_L,\delta,t^L)$ is irreducible, unitary and tempered with central character $W_0t$. These representations are mutually inequivalent. \item[(ii)] We have
\begin{equation}|W^L|\chi_t= \sum_{\delta\in\Delta_{\mathcal{R}_L,W_Lr_L}}\chi_{\mathcal{R}_L,W_Lr_L,\delta,t^L} d_{\mathcal{R}_L,\delta}, \end{equation} where $\chi_{\mathcal{R}_L,W_Lr_L,\delta,t^L}$ \index{0w@$\chi_{\mathcal{R}_L,W_Lr_L,\delta,t^L}$, character of the induced representation $\pi(\mathcal{R}_L,W_Lr_L,\delta,t^L)$} denotes the character of $\pi(\mathcal{R}_L,W_Lr_L,\delta,t^L)$. In particular, the constants $d_{t,i}$ as in Definition \ref{dfn:resalg} are independent of $t$. \item[(iii)] For all (not necessarily $R_L$-generic) $t=r_Lt^L\in L^{temp}$, the character $\chi_{\mathcal{R}_L,W_Lr_L,\delta,t^L}$ is a positive trace on $\mathcal{H}$. Consequently, the irreducible subrepresentations of $\pi(\mathcal{R}_L,W_Lr_L,\delta,t^L)$ extend to $\mathfrak{C}$. \end{enumerate} \end{theorem} \begin{proof} (i). This is a direct consequence of Corollary \ref{cor:short}, Proposition \ref{prop:unit} and Proposition \ref{prop:indtemp}.
(ii). Recall the definition of the states $\chi_t$. Recall that the support of the measure $\nu$ is the union of the tempered residual cosets. We combine Definition \ref{dfn:Y}, Proposition \ref{prop:dfn}, and Proposition \ref{prop:cycle} to see that (with $N_L$ the stabilizer of $L$ in $W_0$) \begin{align}\label{inteq}
\frac{|W_0|}{|N_L|}\int_{t\in L^{temp}}z(t)&\chi_t(h)d\nu_L(t)\\\nonumber&=\sum_{M\in W_0L}\int_{t^M\in T^M_u}\int_{t^\prime\in t^M\epsilon^M\xi_M}z(t^\prime) E_{t^\prime}(h)\eta(t^\prime)\\\nonumber \end{align} for all $h\in\mathcal{H}$ and $z\in\mathcal{Z}$. Rewrite the right hand side as \begin{equation}\label{eq:av}
\frac{k_L}{|N_L|}\int_{t^L\in T^L_u}\sum_{w\in W_0}J_{w,L}(\epsilon^{wL}r_{wL}w(t^L))d^L(t^L) \end{equation} where the inner integrals equal, with $s\in wT^L$, \begin{equation}\label{eq:J} J_{w,L}(r_{wL}s)=\int_{t^\prime\in s\xi_{wL}}z(t^\prime) \frac{E_{t^\prime}(h)}{q(w_0)\Delta^{wL}(t^\prime)} m^{wL}(t^\prime)\frac{1}{\Delta_{wL}(t^\prime)}\omega_{wL}(t^\prime), \end{equation} where \begin{equation} \omega_{wL}(t):= \frac{d_{wL}(t_{wL})}{q(w_L)c_{R_L}(w^{-1}t)c_{R_L}(w^{-1}t^{-1})}. \end{equation} Hence $J_{w,L}(r_{wL}s)$ is a linear combination of (possibly higher order) partial derivatives (in the direction of $wT_L$) of the kernel \begin{equation}\label{eq:kern} z(t^\prime) \frac{E_{t^\prime}(h)}{q(w_0)\Delta^{wL}(t^\prime)} m^{wL}(t^\prime), \end{equation} evaluated at $r_{wL}s$. The $N_L$-invariant measure on $L^{temp}$ on the left hand side of \ref{inteq} is thus obtained by taking the boundary values $\epsilon^{wL}\to 1$ of the $J_{w,L}(\epsilon^{wL}r_{wL}w(t))$, and then sum over the Weyl group as in equation (\ref{eq:av}). Notice that the collection of $R_L$-generic points in $L$ is the complement of a union of algebraic subsets of $L$ of codimension $\geq 1$. The kernel (\ref{eq:kern}) is regular at such points of $L$. The boundary values at $R_L$-generic points are therefore computed simply by specialization at $\epsilon^{wL}=1$. We thus have \begin{equation}\label{eq:sumJ}
z(t)\chi_t(h){\overline \kappa}_{W_LL}m_L(t)=\frac{k_L}{|W_0|}\sum_{w\in W_0} J_{w,L}(r_{wL}w(t^L)). \end{equation} The expression on the left hand side can be extended uniquely to $z\in \mathcal{Z}^{an}(U)$ and $h\in \mathcal{H}^{an}(U)$. By equation (\ref{eq:J}), each summand in the expression on the right hand can also be extended uniquely to such locally defined analytic $z$ and $h$.
Take $U=W_0t\exp(B)$. We restrict both sides to $1_L\mathcal{H}^L1_L\subset\mathcal{H}^{L,an}(U_L)$. Substitute $h$ by $1_Lh1_L$ with $h\in \mathcal{H}^L$. On the left hand side we get, by Corollary \ref{cor:restr}, \begin{equation}\label{eq:left}
\frac{1}{|W^L|}z(t)\chi_t^L(h){\overline \kappa}_{W_LL}m_L(t) \end{equation} where $\chi_t^L$ is a central functional on $\mathcal{H}^L$, normalized by $\chi_t^L(e)=1$.
On the other hand, by Lemma \ref{lem:pareis}, if $h=1_Lh1_L$ with $h \in \mathcal{H}^L$ then \begin{equation} J_{w,L}(w(r_{L}t^L))=\int_{t^\prime\in w(t^L)\xi_{wL}}z(t^\prime) \frac{E_{L,t^\prime_L}(\phi_{t^{\prime,L}}(h))}{q(w_L)\Delta_L(t^\prime_L)} m^{wL}(t^\prime)\omega_{wL}(t^\prime) \end{equation} if $w(r_Lt^L)\in U_L$, and $J_{w,L}(w(r_{L}t^L))=0$ otherwise.
Observe that, because of condition (ii) for $B$, $wt\in U_L$ implies that there exists a $w^\prime\in W_L$ such that $wt=w^\prime t$. Since $t=r_Lt^L$ is $R_L$-generic, we see in particular that the stabilizer of $t$ is contained in $W_L$. Thus $wt\in U_L$ implies that $w\in W_L$, and hence that $wt=w(r_L)t^L$.
Therefore the sum at the right hand side of equation (\ref{eq:sumJ}) reduces, if $h$ is of the form $1_Lh1_L$ with $h\in \mathcal{H}^L$, to \begin{equation}
\frac{k_L}{|W_0|}\sum_{w\in W_L}\int_{t^\prime\in \xi_{wL}}z(t^Lt^\prime)m^{wL}(t^Lt^\prime) \frac{E_{L,t^\prime}(\phi_{t^{L}}(h))}{q(w_L)\Delta_L(t^\prime)} \omega_{wL}(t^\prime) \end{equation} The function $wr_L\exp(B\cap\mathfrak{t}_L)\ni t^\prime\to m^{wL}(t^Lt^\prime)$ is $W_L$-invariant on $W_Lr_L\exp(B\cap\mathfrak{t}_L)=(t^L)^{-1}U_L\cap T_L$, because $m^L(t)$ is $W_L$-equivariant (i.e. $m^{wL}(wt)=m^L(t)$ when $w\in W_L$). In other words, this function is in the center of $\mathcal{H}_L^{an}((t^L)^{-1}U_L\cap T_L)$. By Definition \ref{dfn:Y}, Corollary \ref{cor:pos}, Definition \ref{dfn:chi}, and Theorem \ref{thm:nu} applied to $\mathcal{H}_L$ therefore, this sum reduces to \begin{equation}
\frac{|W_L|}{|W_0|}z(t)\chi_{\mathcal{H}_L,r_L}(\phi_{t^L}(h)) m^L(t){\overline \kappa}_{W_LL}m_{\mathcal{R}_L,r_L}(r_L), \end{equation} which we can rewrite as \begin{equation}
\frac{|W_L|}{|W_0|}z(t)\chi_{\mathcal{H}_L,r_L}(\phi_{t^L}(h)) {\overline \kappa}_{W_LL}m_L(t). \end{equation} Comparing this with (\ref{eq:left}) we see that, in view of equation (\ref{eq:packet}), this implies that for $h\in \mathcal{H}^L$, \begin{equation}
|W^L|\chi_t(1_Lh)=\sum_{\delta\in\Delta_{\mathcal{R}_L,W_Lr_L}} \chi_{\delta}(\phi_{t^L}(h))d_{\mathcal{R}_L,\delta}=\chi^L_t(h). \end{equation} Applying Corollary \ref{cor:restr}, Definition \ref{dfn:ind}, and Corollary \ref{cor:short} we obtain (ii).
(iii). By Corollary \ref{cor:exten}, $\chi_t$ is $\nu$-almost everywhere a positive trace. On the set of $R_L$-generic points $t\in L^{temp}$, we have expressed $\chi_t$ as a positive linear combination of the irreducible induced characters $\chi_{\mathcal{R}_L,W_Lr_L,\delta,t^L}$. This gives the decomposition of $\chi_t$ in terms of irreducible characters of the finite dimensional algebra $\mathcal{H}^t:=\mathcal{H}/J_t$, where $J_t$ denotes the two-sided ideal of $\mathcal{H}$ generated by the maximal ideal $\mathcal{I}_t$ of the elements of the center $\mathcal{Z}$ which vanish at $W_0t$.
On the other hand, we have the decomposition of $\chi_t$ in irreducible characters of the finite dimensional Hilbert algebra $\overline{\mathcal{H}^t}$, as in Definition \ref{dfn:resalg}. This algebra is a quotient algebra of $\mathcal{H}^t$. Because $\mathcal{H}^t$ is finite dimensional, there is no distinction between topological and algebraic irreducibility. We therefore have two decompositions of $\chi_t$ in terms of irreducible characters of $\mathcal{H}^t$. The irreducible characters of $\mathcal{H}^t$ are linearly independent, and thus the two decompositions are necessarily the same. This implies that the induced characters $\chi_{\mathcal{R}_L,W_Lr_L,\delta,t^L}$ are characters of the Hilbert algebra $\overline{\mathcal{H}^t}$. In particular, the characters are positive traces for all $R_L$-generic $t\in L^{temp}$. These induced characters are regular functions of $t\in L^{temp}$. Hence by continuity, they are positive traces for all $t^L$.
By Corollary \ref{cor:contrace}(i), $\chi(\mathcal{R}_L,W_Lr_L,\delta,t^L)$ extends to a continuous trace of $\mathfrak{C}$ for all $t\in L^{temp}$. According to the construction in \cite{dix2}, Paragraphe 6.6. this character is the trace of a (obviously finite dimensional) representation of $\mathfrak{C}$, quasi-equivalent with $\pi(\mathcal{R}_L,W_Lr_L,\delta,t^L)$ when restricted to $\mathcal{H}$. Hence all subrepresentations of $\pi(\mathcal{R}_L,W_Lr_L,\delta,t^L)$ extend to $\mathfrak{C}$. \end{proof} \begin{cor}\label{cor:xcont} For all $x\in\mathcal{H}$, the $\nu_L$-measurable function $L^{temp}\ni t\to\chi_t(x)$ can be defined by the restriction to $L^{temp}$ of a regular function on $L$. For $x\in\mathfrak{C}$, the function $t\to\chi_t(x)$ is continuous on $L^{temp}$. \end{cor} \begin{proof} The first assertion was shown in the proof of Theorem \ref{thm:mainind}(iii). If $x\in\mathfrak{C}$ there exists a sequence $(x_i)$ with $x_i\to x$ and $x_i\in \mathcal{H}$. By Corollary \ref{cor:contrace}(i), the function $t\to\chi_t(x)$ is a uniform limit of the functions $t\to\chi_t(x_i)$, proving the continuity. \end{proof} The next Theorem basically is the Plancherel decomposition of $\mathfrak{H}$. (In the next subsection we will refine the formula by adding more details about the spectrum of $\mathfrak{C}$.) \begin{thm}\label{thm:thisisathm} We have the following isomorphism of Hilbert spaces: \begin{equation}\label{eq:intdec}
\mathfrak{H}=\int_{W_0\backslash T}^\oplus\overline{\mathcal{H}^t}|W_0t|d\nu(t). \end{equation} The support of the probability measure $\nu$ is the union of the tempered residual cosets. If $t=r_Lt^L\in L^{temp}$ is $R_L$-generic, then the residue algebra $\overline{\mathcal{H}^t}$ has the structure \begin{equation}\label{eq:struct}
\overline{\mathcal{H}^t}\simeq(\overline{\mathcal{H}^{r_L}_L})_{|W^L|}. \end{equation} Finally, the residue algebra $\overline{\mathcal{H}^{r}}$ at a residual point $r\in T$ is of the form \begin{equation} \overline{\mathcal{H}^{r}}=\bigoplus_{\delta\in\Delta_{\mathcal{R},W_0r}}\operatorname{End}(V_{\delta}) \end{equation} with the Hermitian form on the summand $\operatorname{End}(V_{\delta})$ given by \begin{equation}\label{eq:innerp} (A,B)=d_{\delta}\operatorname{trace}(A^*B), \end{equation} where the positive real numbers $d_{\delta}$ are defined as in Definition \ref{dfn:resalg} (with the notational convention that $d_{\delta}=d_{r,i}$ if $\delta=\delta_i$ is the irreducible representation of $\overline{\mathcal{H}^{r}}$ corresponding to the central idempotent $e_i$). \end{thm} \begin{proof} The Hilbert space $\mathfrak{H}$ is the completion of $\mathcal{H}$ with respect to the positive trace $\tau$. In Corollary \ref{cor:exten}(v) we have written $\tau$ as a positive superposition of positive traces $\chi_t$, with $t\in W_0\backslash T$. In Theorem \ref{thm:mainind} we established that, outside a set of measure zero, $\chi_t$ is a finite linear combination of traces of irreducible representations $\pi_{\mathcal{R}_L,W_Lr_L,\delta,t^L}$ which extend to $\mathfrak{C}$. Thus Corollary \ref{cor:exten}(v) is a positive decomposition of $\tau$ in terms of traces of elements of $\hat{\mathfrak{C}}$. This allows us to apply 8.8.5 and 8.8.6 of \cite{dix2} in order to obtain (\ref{eq:intdec}). The statements about the residual algebra of a residual point follow directly from the Definition \ref{dfn:resalg}. Finally, equation (\ref{eq:struct}) follows from Theorem \ref{thm:mainind} in combination with the factorization Proposition \ref{prop:unit} of the inner product $\langle\cdot,\cdot\rangle$ on the induced representations. \end{proof} \subsection{The generic residue of the Hecke algebra}\label{sub:gene} In this subsection we will use Theorem \ref{thm:mainind} in order to compute explicitly the local traces $\chi_t$ when $t=r_Lt^L$ is an $R_L$-generic element of $L^{temp}$. Here we assume that $L$ is a residual coset such that $R_L$ is a standard parabolic subset of $R_0$ with basis $F_L$ of simple roots. Since we assume that $t$ is $R_L$-generic, we have $P(t)=F_L$. As before, we put $W_Lt=\varpi$, the equivalence class of $t$. By the genericity of $t$, $W(\varpi)=1$.
Observe that the residual algebra $\overline{\mathcal{H}^t}$ (see Definition \ref{dfn:resalg}) is of the form $\overline{\mathcal{H}^t}=\mathcal{H}^t/\operatorname{Rad}_t$, where $\mathcal{H}^t$ is the quotient algebra $\mathcal{H}^t:=\mathcal{H}/\mathcal{I}_t\mathcal{H}$ (with $\mathcal{I}_t$ the maximal ideal of $\mathcal{Z}$ corresponding to $t$), and where $\operatorname{Rad}_t$ is the radical of the positive semi-definite form $(x,y)_t:=\chi_t(x^*y)$ \index{$(x,y)_t=\chi_t(x^*y)$, semi-definite Hermitian form on $\mathcal{H}^t$} (viewed as a form on $\mathcal{H}^t$).
By Lusztig's Structure Theorem \ref{thm:lusind}, $\mathcal{H}^t$ has the following decomposition in the case where $t$ is $R_L$-generic: \begin{equation} \mathcal{H}^t=\bigoplus_{u,v\in W^L}\iota^0_ue_\varpi\mathcal{H}^L \iota^0_{v^{-1}} \end{equation} where $e_\varpi=1_\varpi\ \operatorname{mod}(\mathcal{I}_t\mathcal{H}^{an}(U))$, the image of $1_\varpi$ in $\mathcal{H}^t$ \index{e1@$e_\varpi$, image of $1_\varpi$ in $\mathcal{H}^t$} (in the notation of Section \ref{sect:loc}). We remark that this is {\it not} an orthogonal decomposition with respect to $(x,y)_t$.
The subspace $\iota^0_ue_\varpi\mathcal{H}^L \iota^0_{v^{-1}}$ is equal to $e_{u\varpi}\mathcal{H}^t e_{v\varpi}$. If $u=v$ then this is a subalgebra of $\mathcal{H}^t$. If $u=v=e$ then this subalgebra is isomorphic to $\mathcal{H}^{L,t}$ via the isomorphism $\mathcal{H}^{L,t}\ni x\to e_\varpi x\in e_\varpi\mathcal{H}^L$.
Let $P,Q\subset F_0$ be subsets. We denote by $W(P,Q)$ the following set of Weyl group elements: $W(P,Q):=\{w\in W_0\mid w(P)=Q\}$ \index{W5a@$W(P,Q)=\{w\in W_0\mid w(P)=Q\}$, with $P,Q\subset F_0$}. If $P=Q\subset F_0$ then we simply write $W(P)=W(P,P)$ \index{W5b@$W(P)$ for the stabilizer in $W_0$ of $P\subset F_0$}
Let $n\in W_0$ be such that $n(F_L)=F_M\subset F_0$, in other words, let $n\in W(F_L,F_M)$. Then the map \begin{align}\label{eq:delta} \Delta_{n\varpi,n\varpi}: e_\varpi\mathcal{H}^L&\to e_{n\varpi}\mathcal{H}^M\\ \nonumber x&\to \iota^0_n x \iota^0_{n^{-1}} \end{align} \index{0D4@$\Delta_{\varpi_1,\varpi_2}:\mathcal{H}^{\varpi,{an}}(U_\varpi)\to 1_{\varpi_1}\mathcal{H}^{an}(U)1_{\varpi_2}$, linear isomorphisms} is an isomorphism. By the results of Lusztig \cite{Lu}, section 8, it satisfies (with $h\in \mathcal{H}^L$): \begin{equation}\label{eq:psi} \Delta_{n\varpi,n\varpi}(e_\varpi h)=e_{n\varpi} \psi_n(h), \end{equation} where $\psi_n:\mathcal{H}^L\to \mathcal{H}^M$ \index{0x@$\psi_g:\mathcal{H}^L\to\mathcal{H}^M$, isomorphism for $g\in K_M\times W(F_L,F_M)$} is the isomorphism of algebras defined by (with $w\in W_L$ and $x\in X$): \begin{equation} \psi_n(T_w)=T_{nwn^{-1}}, \psi_n(\theta_x)=\theta_{nx}. \end{equation}
Recall that Theorem \ref{thm:mainind}(ii) implies that for all $h\in\mathcal{H}^L$, \begin{equation}\label{eq:bel}
|W^L|\chi_t(e_\varpi h)=\chi_{\mathcal{H}_L,r_L}\bigl(\phi_{t^L}(h)\bigr). \end{equation} \begin{cor}\label{cor:con} Write $n(t)=s=r_M^\prime s^M$. \begin{enumerate} \item \begin{equation} \chi_{\mathcal{H}_M,r_M^\prime}\bigl(\phi_{s^M}(\psi_n(h))\bigr)= \chi_{\mathcal{H}_L,r_L}\bigl(\phi_{t^L}(h)\bigr). \end{equation} \item Let $\Psi_n:\Delta_{\mathcal{R}_L,W_Lr_L}\to\Delta_{\mathcal{R}_M,W_Mr_M^\prime}$ \index{0X@$\Psi_g:\Delta_{\mathcal{R}_L,W_Lr_L}\to\Delta_{\mathcal{R}_M,W_Mr_M^\prime}$, bijection induced by $\psi_g$} be the bijection such that $\Psi_n(\delta)\simeq \delta\circ \psi_n^{-1}$. Then $\Psi_n$ respects the residual degree: $d_{\mathcal{R}_L,\delta}=d_{\mathcal{R}_M,\Psi_n(\delta)}$. \end{enumerate} \end{cor} \begin{proof} (i) This follows from the above text, and the fact that $\chi_t$ is a central functional.
(ii) It is clear from Casselman's criteria that $\Psi_n$ indeed defines a bijection between the sets of discrete series representations $\Delta_{\mathcal{R}_L,W_Lr_L}$ and $\Delta_{\mathcal{R}_M,W_Mr_M^\prime}$. The result follows from (i) and Theorem \ref{thm:mainind}(ii). \end{proof} \begin{prop}\label{prop:constterm} For all $h,g\in \mathcal{H}^L$ we have \begin{equation}
|W^L|m^L(t)\chi_t\bigl((e_\varpi h)^*(e_\varpi g)\bigr) =\chi_{\mathcal{H}_L,r_L}\bigl(\phi_{t^L}(h^\sharp g)\bigr), \end{equation} where $\sharp$ \index{$\sharp$, the $*$-operator of $\mathcal{H}^L$} denotes the $*$-operator of $\mathcal{H}^L$ (thus $T^\sharp_w=T_{w^{-1}}$ if $w\in W_L$, and $\theta_x^\sharp= T_{w_L}\theta_{-w_Lx}T_{w_L}^{-1}$ where $w_L$ \index{wb@$w_P$, longest element of $W_P$} denotes the longest element of $W_L$). \end{prop} \begin{proof} This will be proved by the computation in the proof of Theorem \ref{thm:unitint}. \end{proof} \begin{cor}\label{cor:corr} We equip $e_\varpi\mathcal{H}^t e_\varpi$ with the positive semi-definite sesquilinear pairing obtained by restriction of the pairing
$|W^L|m^L(t)(x,y)_t=|W^L|m^L(t)\chi_t(x^*y)$ defined on $\mathcal{H}^t$. The residual algebra $\overline{\mathcal{H}^{L,t}}$ is isomorphic as a Hilbert algebra to the quotient of $e_\varpi \mathcal{H}^t e_\varpi$ by the radical of this pairing, via the map $x\to e_\varpi x$. \end{cor}
\begin{cor}\label{cor:emb} Recall the notation of Proposition \ref{prop:unit}. We consider $V_\pi$ as a module over $\mathcal{H}^t$. Put $V_{\pi,\varpi}=\pi(e_\varpi)(V_\pi)$. Choose an isometric embedding $\overline{i}:V\to\overline{\mathcal{H}_{L}^{r_L}}$ (as $\mathcal{H}_L$-modules). Let $\overline{j}$ denote the unique module map $\overline{j}:V_\pi\to\overline{\mathcal{H}^t}$ such that $\overline{j}(1\otimes v)=e_\varpi (\overline{\phi}_{t^L})^{-1}(\overline{i}(v))$, where $\overline{\phi}_{t^L}: \overline{\mathcal{H}^{L,t}}\to\overline{\mathcal{H}_L^{r_L}}$ denotes the isometric isomorphism determined by the homomorphism $\phi_{t^L}$ (cf. Proposition \ref{luspardef}). For any $v\in V$ we denote by $i(v)$ any lift of $\overline{i}(v)$, and similarly for $j$. We have: \begin{enumerate} \item $V_{\pi,\varpi}=1\otimes V$, and $\overline{j}:V_{\pi,\varpi} \stackrel{\sim}{\to}e_\varpi (\overline{\phi}_{t^L})^{-1}(\overline{i}(V))$. \item The positive definite Hermitian form $\langle\cdot,\cdot\rangle$ on $V_\pi$ (see Proposition \ref{prop:unit}) can be expressed by: \begin{equation}
\langle v,w\rangle=|W^L|m^L(t)\chi_t(j(v)^*j(w)). \end{equation} \end{enumerate} \end{cor} \begin{proof} (i) is straightforward by observing that $\phi_{t^L}$ descends to $\mathcal{H}^{L,t}$ and so defines an isometric isomorphism $\overline{\phi}_{t^L}$ by (\ref{eq:bel}), applied to
$\mathcal{R}^L$ instead of $\mathcal{R}$ (thus $e_\omega=1$, and $|W^L|=1$). Since $V_\pi$ is irreducible, it is enough to compare the inner products on $V_{\pi,\varpi}$ in order to prove (ii). Apply Proposition \ref{prop:constterm} and Corollary \ref{cor:corr}. \end{proof} Assume that $R_M$ and $R_L$ are associate standard parabolic subsystems. Let $\varpi_1=\varpi$ and $\varpi_2$ be equivalence classes inside $W_0t$ such that $\varpi_1=W_Lt$ and $\varpi_2=W_Ms$. \begin{thm} \label{thm:unitint} Let $n\in W(F_M,F_L)$ be such that $n(\varpi_2)=\varpi_1$. Let $h\in\mathcal{H}^L$ and $h^\prime\in\mathcal{H}^M$ such that $e_{\varpi_2}h^\prime=\iota^0_{n^{-1}}e_{\varpi_1}h \iota^0_n\in e_{\varpi_2}\mathcal{H}^M$. We have \begin{align*} \chi_t\bigl((he_{\varpi_1} \iota^0_n)^*(he_{\varpi_1} \iota^0_n)\bigr)&= \chi_t\bigl((he_{\varpi_1})^*(he_{\varpi_1})\bigr)\\&= \chi_t\bigl((h^\prime e_{\varpi_2})^*(h^\prime e_{\varpi_2})\bigr)\\ \end{align*} \end{thm} \begin{proof} Before we embark on this computation we establish some useful relations. First recall that (Theorem \ref{thm:ster}) $t^*:=\overline{t^{-1}}\in W_Lt$. Also recall Proposition \ref{prop:thetastar}. We see that \begin{align}\label{eq:ms1} e_{\varpi_1}^*&=T_{w_0}e_{w_0\varpi_1}T_{w_0}^{-1}\\ \nonumber&=T_{w^L}e_{w^L\varpi_1}T_{w^L}^{-1}, \end{align} where $w^L$ \index{wc@$w^P$, longest element of $W^P$} denotes the longest element of set of minimal coset representatives $W^L$. Next we observe that for any $w\in W_0$, \begin{equation}\label{eq:ms2} (\iota^0_w)^*=T_{w_0}\left( \prod_{{\alpha>0}\atop{w^\prime(\alpha)<0}} \left(\frac{c_\alpha}{c_{-\alpha}}\right)\iota^0_{{w^\prime}^{-1}}\right)T_{w_0}^{-1}, \end{equation} where ${w^\prime}:=w_0 w w_0$. This formula follows in a straightforward way from Definition (\ref{eq:defint}).
If $s$ is a simple reflection and $\varpi\subset W_0t$ an equivalence class, we check that (recall that $t$ is $R_L$-generic) \begin{equation}\label{eq:ms3} e_{s\varpi}T_se_\varpi= \left\{ \begin{aligned} &e_\varpi T_s \hskip70pt\text{\ if\ }s\varpi=\varpi\\ &e_{s\varpi}q(s)c_\alpha \iota^0_s \hskip32pt\text{\ else.\ }\\ \end{aligned} \right. \end{equation} Since we assume that $t$ is $R_L$-generic, we have $u(\varpi_1)\not=w(\varpi_1)$ for all $w\in W^L$ and all $u\in W_0$ such that $l(u)<l(w)$. From this, (\ref{eq:ms3}) and induction to the length of $l(w)$ we see that \begin{align}\label{eq:ms4} e_{w\varpi_1}T_we_{\varpi_1}&= q(w)\Bigl(\prod_{{\alpha>0}\atop{w^{-1}(\alpha)<0}}c_\alpha\Bigr) \iota^0_w e_{\varpi_1}\\ \nonumber&=e_{w\varpi_1} q(w)\Bigl(\prod_{{\alpha>0}\atop{w^{-1}(\alpha)<0}}c_\alpha\Bigr) \iota^0_w \end{align} for all $w\in W^L$. Observe that we also have \begin{equation}\label{eq:ms5} e_{w\varpi_1}T_{w^{-1}}^{-1}e_{\varpi_1}= \Bigl(\prod_{{\alpha>0}\atop{w^{-1}(\alpha)<0}}c_\alpha\Bigr) \iota^0_w e_{\varpi_1}. \end{equation} We note that $w_0=w^Lw_L$. Since $w_L$ and $w_0$ are involutions, this implies that $(w^L)^{-1}=w^{L^\prime}$, where $R_{L^\prime}=w_0(R_L)$ (also a standard parabolic subsystem).
Let $h=T_w\theta_x\in \mathcal{H}^L$, with $w\in W_L$ and $x\in X$. Keeping in mind the above preliminary remarks, and the fact that $\chi_t$ is central, we now compute (with $x^\prime:=-w_0(x)$ and $w_0\varpi_1=\varpi_1^\prime$): \begin{gather}\label{eq:long} \begin{split} \chi_t&\bigl((he_{\varpi_1}\iota_n^0)^*(he_{\varpi_1}\iota_n^0)\bigr)\\ &=\chi_t\Bigl(T_{w_0} \prod_{{\alpha>0}\atop{n^\prime(\alpha)<0}} \left(\frac{c_\alpha}{c_{-\alpha}}\right)\iota^0_{{n^\prime}^{-1}} e_{\varpi_{1}^{\prime}}\theta_{x^\prime}T_{w_0}^{-1}T_{w^{-1}}T_w \theta_xe_{\varpi_1} \iota^0_n\Bigr)\\ &=\chi_t\Bigl(T_{w^{M^\prime}} \prod_{{\alpha>0}\atop{n^\prime(\alpha)<0}} \left(\frac{c_\alpha}{c_{-\alpha}}\right) \iota^0_{{n^\prime}^{-1}} e_{\varpi_{1}^{\prime}}\theta_{x^\prime}T_{w^{L^\prime}}^{-1} T_{w_L}^{-1}T_{w^{-1}}T_w \theta_x e_{\varpi_1} \iota^0_nT_{w_M}\Bigr)\\ &=\chi_t\Bigl(e_{\varpi_2}T_{w^{M^\prime}} e_{\varpi_{2}^{\prime}} \Bigl({{\prod_{{\alpha>0}\atop{w^{M^\prime}}(\alpha)<0}}{c_{\alpha}}}\Bigr) \Bigl({{\prod_{{\alpha>0}\atop{w^{L^\prime}}(\alpha)<0}} {c_{{n^\prime}^{-1}\alpha}^{-1}}}\Bigr)\\ &\hskip101pt \iota^0_{{n^\prime}^{-1}} \theta_{x^\prime}e_{\varpi_{1}^{\prime}}T_{(w^L)^{-1}}^{-1}e_{\varpi_1} T_{w_L}^{-1}T_{w^{-1}}T_w \theta_x e_{\varpi_1} \iota^0_nT_{w_M}\Bigr)\\ &=q(w^{M^\prime})\chi_t\Bigl(e_{\varpi_2} \iota^0_{w^{M^\prime}} \Bigl(\prod_{{\alpha>0}\atop{w^{M^\prime}\alpha<0}}c_{-\alpha}\Bigr) \Bigl({{\prod_{{\alpha>0}\atop{w^{M^\prime}}(\alpha)<0}}{c_{\alpha}}}\Bigr) \Bigl({{\prod_{{\alpha>0}\atop{w^{L^\prime}}(\alpha)<0}} {c_{{n^\prime}^{-1}\alpha}^{-1}}}\Bigr)\\ &\hskip97pt \iota^0_{{n^\prime}^{-1}} \theta_{x^\prime} \Bigl(\prod_{{\alpha>0}\atop{w^{L^\prime}}\alpha<0}c_\alpha\Bigr) \iota^0_{w^L} T_{w_L}^{-1}T_{w^{-1}}T_w \theta_x e_{\varpi_1} \iota^0_nT_{w_M}\Bigr)\\ &=q(w^L)\chi_t\Bigl(e_{\varpi_2} \iota^0_{n^{-1}}\iota^0_{w^{L^\prime}} \Bigl(\prod_{{\alpha>0}\atop{w^{M^\prime}\alpha<0}}c_{-n^\prime\alpha}\Bigr) \Bigl({{\prod_{{\alpha>0}\atop{w^{M^\prime}}(\alpha)<0}}{c_{n^\prime\alpha}}}\Bigr) \Bigl({{\prod_{{\alpha>0}\atop{w^{L^\prime}}(\alpha)<0}}{c_\alpha^{-1}}}\Bigr)\\ &\hskip124pt \Bigl(\prod_{{\alpha>0}\atop{w^{L^\prime}}\alpha<0}c_\alpha\Bigr) \theta_{x^\prime} \iota^0_{w^L}T_{w_L}^{-1}T_{w^{-1}}T_w \theta_x e_{\varpi_1} \iota^0_nT_{w_M}\Bigr)\\ &=q(w^L)\chi_t\Bigl(e_{\varpi_2} \iota^0_{n^{-1}} \Bigl(\prod_{{\alpha>0}\atop{w^{M^\prime}\alpha<0}}c_{-w^{L^\prime}n^\prime\alpha}\Bigr) \Bigl({{\prod_{{\alpha>0}\atop{w^{M^\prime}}(\alpha)<0}} {c_{w^{L^\prime}n^\prime\alpha}}}\Bigr)\\ &\hskip187pt \theta_{-w_Lx}T_{w_L}^{-1}T_{w^{-1}}T_w \theta_x e_{\varpi_1} \iota^0_nT_{w_M}\Bigr)\\ &=q(w^L)\chi_t\Bigl(e_{\varpi_1} \Bigl(\prod_{\alpha\not\in R_M}c_{n\alpha}\Bigr) e_{\varpi_1}\theta_{-w_Lx} T_{w_L}^{-1}T_{w^{-1}}T_w \theta_xT_{w_L}\Bigr)\\ &=q(w^L) \Bigl(\prod_{\alpha\not\in R_L}c_{\alpha}(t)\Bigr) \chi_t\Bigl(e_{\varpi_1} T_{w_L}\theta_{-w_Lx} T_{w_L}^{-1}T_{w^{-1}}T_w \theta_x\Bigr)\\ &=m^L(t)^{-1} \chi_t\bigl(e_{\varpi_1} h^\sharp h\bigr)\\
&=|W^L|^{-1}m^L(t)^{-1} \chi_{\mathcal{H}_L,r_L} \bigl(\phi_{t^L}(h^\sharp h)\bigr).\\ \end{split} \end{gather} The result is independent of $n$, implying the first equality of the theorem. The second equality follows because $\chi_t$ is central. Indeed, this implies that we have \begin{equation}\label{eq:short} \chi_t(e_{\varpi_1}h^\sharp h) =\chi_t(e_{\varpi_2}(h^\prime)^\sharp h^\prime), \end{equation} where the second $\sharp$ of course refers to the $*$-structure on $\mathcal{H}^M$.
In equation (\ref{eq:long}) we have used the evaluation \begin{equation} \Bigl(\prod_{\alpha\not\in R_L}c_{\alpha}\Bigr)e_{\varpi_1}= \Bigl(\prod_{\alpha\not\in R_L}c_{\alpha}(t)\Bigr)e_{\varpi_1}. \end{equation} This is allowed because the element \begin{equation} \Bigl(\prod_{\alpha\not\in R_L}c_{\alpha}\Bigr)\in\mathcal{A} \end{equation} is $W_L$-invariant, and thus belongs to the center of $\mathcal{H}^L$.
At several places in equations (\ref{eq:long}) and (\ref{eq:short}) we have freely used formulae of Lusztig \cite{Lu} (see Theorem \ref{thm:lusind}) for the structure of $\mathcal{H}^t$. For example, \begin{equation} \iota^0_{n^{-1}} e_{\varpi_1} h \iota^0_{n}= e_{\varpi_2}\psi_{n^{-1}}(h)=e_\varpi h^\prime \end{equation} when $h\in \mathcal{H}^L$. By this we easily see that for all $h\in \mathcal{H}^L$, \begin{equation} \iota^0_{n^{-1}} e_{\varpi_1}h^\sharp \iota^0_{n}=e_{\varpi_2}(h^\prime)^\sharp, \end{equation} and hence we may conclude by equation (\ref{eq:long}) that \begin{equation} \chi_t\bigl((he_{\varpi_1})^*(he_{\varpi_1})\bigr)= \chi_t\bigl((h^\prime e_{\varpi_2})^*(h^\prime e_{\varpi_2})\bigr). \end{equation} The proof is finished. \end{proof} \begin{cor}\label{cor:hehe} Let $L,M_1,M_2$ be residual cosets such that $F_L,F_{M_1}$ and $F_{M_2}$ are associate subsets of $F_0$, and let $n_i\in W(L,M_i)$ ($i=1,2$). The map \begin{align} \Delta_{n_1\varpi,n_2\varpi}:e_\varpi\mathcal{H}^L&\to \iota^0_{n_1} e_\varpi\mathcal{H}^L \iota^0_{n_2^{-1}}=e_{n_1\varpi}\mathcal{H} e_{n_2\varpi}\\ \nonumber x&\to \iota^0_{n_1}x \iota^0_{n_2^{-1}} \end{align} is a partial isometry with respect to the natural positive semi-definite pairing on $\mathcal{H}^t$ given by $(x,y)_t:=\chi_t(x^*y)$. \end{cor} \begin{proof} We have, with $\varpi^\prime:=n_1\varpi$, $\Delta_{n_1\varpi,n_2\varpi}= \Delta^{\varpi^\prime}_{\varpi^\prime,n_2n_1^{-1}\varpi^\prime} \circ\Delta_{n_1\varpi,n_1\varpi}$, where $\Delta^{\varpi^\prime}_{\varpi^\prime,n_2n_1^{-1}\varpi^\prime}$ is defined by \begin{align} \Delta^{\varpi^\prime}_{\varpi^\prime,n_2n_1^{-1}\varpi^\prime} :e_{\varpi^\prime}\mathcal{H}^{M_1}&\to e_{\varpi^\prime}\mathcal{H}^{M_1}\iota^0_{n_1n_2^{-1}}\\ x&\to x\iota^0_{n_1n_2^{-1}} \end{align} Both these respect the pairing $(\cdot,\cdot)_t$, by Theorem \ref{thm:unitint}. \end{proof} \subsection{Unitarity and regularity of intertwining operators} Let $L,M$ be associate residual subspaces such that $R_L, R_M$ are standard parabolic subsystems of $R_0$. Let $n\in W_0$ be such that $n(R_{L,+})=R_{M,+}$. As before we let $\psi_n:\mathcal{H}^L\to\mathcal{H}^M$ denote the isomorphism defined by $\psi_n(T_w)=T_{nwn^{-1}}$ and $\psi_n(\theta_x)=\theta_{nx}$. Let $(V,\delta)$ be an irreducible discrete series representation of $\mathcal{H}_{L}^{r_L}$ and let $t=r_Lt^L$ be an $R_L$-generic point of $r_LT^L$. Let $s=n(t)=r_M^\prime s^M$, and let $(V^\prime,\delta^\prime)$ be a realization of the discrete series representation $\delta^\prime=\Psi_n(\delta)$.
Choose a {\it unitary} isomorphism $\tilde\delta:V\to V^\prime$ such that \begin{equation}\label{eq:inttwist} \tilde\delta(\delta_t(h)v)=\delta^\prime_s(\psi_n(h))(\tilde\delta(v)). \end{equation} Recall that $V_\pi$ with $\pi=\pi(\mathcal{R}_L,W_Lr_L,\delta,t^L)$ is isomorphic to \begin{equation} V_\pi\simeq \mathcal{H}^{an}(U)\otimes_{\mathcal{H}^{L,an}(U_\varpi)}V_{t^L} \end{equation} (see Subsection \ref{sub:lus}), where $V_{t^L}$ denotes the representation space $V$ with $\mathcal{H}^L$ action defined by $h\to\delta(\phi_{t^L}(h))$. Put $\pi^\prime=\pi(\mathcal{R}_M,W_Mr_M^\prime,\delta^\prime,s^M)$. \begin{dfn} For $t^L\in T^L$ such that $r_Lt^L$ is $R_L$-generic, we define an intertwining isomorphism $A(n,\mathcal{R}_L,W_Lr_L,\delta,t^L):V_\pi\to V_{\pi^\prime}$ by \begin{align} A(n,\mathcal{R}_L,W_Lr_L,\delta,t^L):\mathcal{H}\otimes_{\mathcal{H}^L}V_{t^L} &\to\mathcal{H}\otimes_{\mathcal{H}^M}V^\prime_{s^M}\\ \nonumber h\otimes v&\to h \iota^0_{n^{-1}}\otimes \tilde\delta(v) \end{align} \index{A@$A(n,\mathcal{R}_L,W_Lr_L,\delta,t^L)$, unitary intertwining operator ($n\in W(F_L,F_M)$)} \end{dfn} It is easy to check that this is well defined and that this map intertwines the $\mathcal{H}$ actions. \begin{thm}\label{thm:ms} Recall the compact realization $V_\pi=\mathcal{H}(W^L)\otimes V$, with its inner product $\langle\cdot,\cdot\rangle_\pi$ (see Proposition \ref{prop:unit}). \begin{enumerate} \item In the ``compact realization'', the intertwining map \begin{equation} A(n,\mathcal{R}_L,W_Lr_L,\delta,t^L):\mathcal{H}(W^L)\otimes V\to\mathcal{H}(W^M)\otimes V^\prime \end{equation} is rational as a function of induction parameter $t^L$, and regular outside the set of zeroes of the functions $t^L\to\Delta_\alpha c_\alpha(u(r_L)t^L)$, where $\alpha$ runs over the set of positive roots in $R_1$ such that $n(\alpha)<0$, and $u(r_L)$ (with $u\in W_L$) runs over the set of $X_L$-weights in $V$. \item When $t^L\in T^L_u$ and $A(n,\mathcal{R}_L,W_Lr_L,\delta,t^L)$ is regular at $t^L$, then in fact $A(n,\mathcal{R}_L,W_Lr_L,\delta,t^L)$ is unitary with respect to the inner products $\langle\cdot,\cdot\rangle_\pi$ and $\langle\cdot,\cdot\rangle_{\pi^\prime}$. \item With respect to these inner products we have \begin{equation} A^*(n,\mathcal{R}_L,W_Lr_L,\delta,t^L)= A(n^{-1},\mathcal{R}_M,W_Mr_M^\prime,\Psi_n(\delta),n(t^L)). \end{equation} \end{enumerate} \end{thm} \begin{proof} (i) The representation $\pi$ is cyclic and generated by $1\otimes v$, with $v\not=0$ an arbitrary vector in $V$. By the intertwining property it is therefore enough to show that $A(n,\mathcal{R}_L,W_Lr_L,\delta,t^L)(1\otimes v)\in \mathcal{H}(W^L)\otimes V$ is meromorphic in $t^L$, and regular outside the indicated set. Using equation (\ref{eq:defint}), we have \begin{equation} A(n,\mathcal{R}_L,W_Lr_L,\delta,t^L)(1\otimes v)=\pi(\iota_{n^{-1}}) \pi(\prod_{\alpha>0,n(\alpha)<0}\Delta_\alpha c_\alpha)^{-1}(1\otimes \tilde\delta(v)). \end{equation} Since $\pi(h)$ is a regular function on $T^L$ for all $h\in \mathcal{H}$, this is a rational expression. The generalized $X$-weights in $1\otimes V_{t^L}$ are of the form $u(r_L)t^L$. So the inverse of $\pi(\prod_{\alpha>0,n(\alpha)<0}\Delta_\alpha c_\alpha)$ can have poles only at the indicated set.
(ii) In order to see the unitarity, we first note that by Corollary \ref{cor:emb}(ii) and Theorem \ref{thm:unitint}, the statement is equivalent to the unitarity with respect to the inner products on $V_\pi$ and $V_{\pi^\prime}$ defined by the embedding of these spaces in $\overline{\mathcal{H}^t}$ as in Corollary \ref{cor:emb}. Choose an embedding $\overline{i}:V\to\mathcal{H}_{L}^{r_L}$ as in Corollary \ref{cor:emb}. By Theorem \ref{thm:unitint} and Corollary \ref{cor:con}, the map
$\Delta_{n\varpi,n\varpi}|_{\overline{j}(V_{\pi,\varpi})}$ is an isometry. By equation (\ref{eq:psi}) we see that this isometry satisfies, for $h\in H^L$, $\psi_n(h)\cdot\Delta_{n\varpi,n\varpi}(\overline{j}(1\otimes v)) =\Delta_{n\varpi,n\varpi}(\overline{j}(1\otimes\delta_{t^L}(h)(v)))$. Hence if we identify $V_{t^L}$ with $\overline{j}(V_{\pi,\varpi})$, we can define $V^\prime_{n(t^L)}=\Delta_{n\varpi,n\varpi}(V_{t^L})$. Then the map $\tilde\delta=\Delta_{n\varpi,n\varpi}$ defines a unitary map satisfying (\ref{eq:inttwist}).
Now it is clear, in the notation of Corollary \ref{cor:emb}, that we can identify the space $\overline{j^\prime}(V_{\pi^\prime,\varpi^\prime})$ with $\overline{j}(V_{\pi, \varpi})\iota^0_{n^{-1}}$, and the map $A(n,\mathcal{R}_L,W_Lr_L,\delta,t^L)$ is then identified with the right multiplication with $\iota^0_{n^{-1}}$, thus with $\Delta_{\varpi,n\varpi}$. This is unitary on $\overline{j}(V_{\pi, \varpi})$, by Corollary \ref{cor:hehe}. By the irreducibility of $V_\pi$ and $V_{\pi^\prime}$ this concludes the proof of (ii).
(iii) This last assertion of the Theorem is now obvious, since these maps are clearly inverse to each other. \end{proof} The next Corollary is an important classical application of the unitarity of the intertwiners, see \cite{BCD}, Th\'eor\`eme 2. \begin{cor}\label{cor:hol} The intertwining map $t^L\to A(n,\mathcal{R}_L,W_Lr_L,\delta,t^L)$ extends holomorphically to an open neighborhood of $T^L_u$ in $T^L$. \end{cor} \begin{proof} By the unitarity on $T^L_u$, the meromorphic matrix entries of $A(n,\mathcal{R}_L,W_Lr_L,\delta,t^L)$ are uniformly bounded for $t^L\in T^L_u$ in the open set of $T^L_u$ where $A(n,\mathcal{R}_L,W_Lr_L,\delta,t^L)$ is well defined and regular. This is the complement of the collection of real codimension $1$ cosets in $T^L_u$ as described in Theorem \ref{thm:ms}. This implies that the singularities of the matrix entries which meet $T^L_u$ are actually removable. \end{proof} \subsection{The Plancherel decomposition of the trace $\tau$}\label{sub:pla} In this section we rewrite the decomposition Theorem \ref{thm:thisisathm} as a decomposition of $\tau$ in terms of characters of irreducible tempered representations induced from cuspidal representations of the subalgebras $\mathcal{H}^P$.
Using the results of the previous section, we show that the corresponding Fourier homomorphism maps $\mathcal{H}$ into a certain space of smooth sections defined over orbits of irreducible cuspidal representations of the subalgebras $\mathcal{H}^P$, equivariant with respect to the natural actions of intertwining operators.
This final formulation of the results (Theorem \ref{thm:mainp}) is inspired by and parallel to the notations used in the theory of the Harish-Chandra Plancherel formula for p-adic groups, as treated in \cite{W2} and \cite{D}.
We need to develop some notations. Let $P\subset F_0$ \index{P@$\mathcal{P}$, power set of $F_0$} denote the power set of $F_0$, and let $\Gamma$ \index{0C@$\Gamma$, set of all pairs $(\mathcal{R}_P,W_Pr)$ with $P\in\mathcal{P}$ and $W_Pr$ an orbit of residual points in $T_P$} denote the set of all pairs $\gamma=(\mathcal{R}_P,W_Pr)$ with $P\in\mathcal{P}$, $\mathcal{R}_P$ the associated parabolic root datum, and $W_Pr$ an orbit of residual points in $T_P$. We consider the disjoint union of the set of all triples $\Lambda=\{(\mathcal{R}_P,W_Pr,t)\}$ \index{0L@$\Lambda$, disjoint union of all triples of the form $\{(\mathcal{R}_P,W_Pr,t)\}$}, where $(\mathcal{R}_P,W_Pr)\in\Gamma$ and $t\in T^P_u$. Let $\Lambda_{\mathcal{R}_P,W_Pr}=\Lambda_\gamma$ \index{0L2@$\Lambda_{\mathcal{R}_P,W_Pr}=\Lambda_\gamma$, triples in $\Lambda$ with $\gamma=(\mathcal{R}_P,W_Pr)\in\Gamma$ fixed} be the subspace of such triples with $\gamma=(\mathcal{R}_P,W_Pr)\in\Gamma$ fixed. Hence for all $\gamma\in\Gamma$, $\Lambda_\gamma$ is a copy of $T^P_u$ and $\Lambda=\cup\Lambda_\gamma$ (disjoint union). Therefore $\Lambda$ is a disjoint union of finitely many compact tori, which gives $\Lambda$ the structure of a compact Hausdorff space. In addition, each $\Lambda_\gamma$ comes with its (normalized) Haar measure, thus defining a measure on $\Lambda$. We denote by $\Lambda_\gamma^{gen}$ \index{0L3@$\Lambda_\gamma^{gen}$, triples in $\Lambda_\gamma$ with $rt$ $R_P$-generic} the open, dense subset of triples $(\mathcal{R}_P,W_Pr,t)$ such that $(\mathcal{R}_P,W_Pr)=\gamma$ and $rt$ is $R_P$-generic. We put $\Lambda^{gen}=\cup \Lambda_\gamma^{gen}$ \index{0L1@$\Lambda^{gen}=\cup \Lambda_\gamma^{gen}$} (disjoint union over $\gamma\in\Gamma$).
Define a map $m:\Lambda\to S\subset W_0\backslash T$ by \begin{equation} m(\mathcal{R}_P,W_Pr,t)=W_0(rt) \end{equation} \index{m@$m$!$m:\Lambda\to S\subset W_0\backslash T$, projection} By Theorem \ref{thm:support}, $m$ is surjective, and obviously $m$ is continuous and finite.
Let $P,Q\in\mathcal{P}$. Recall the set $W(P,Q)\subset W_0$ defined by $W(P,Q):=\{n\in W_0\mid n(P)=Q\}$. We put $W(P)=W(P,P)$, which is a subgroup of $W_0$ and acts on $R_P$ through diagram automorphisms. Observe that $W(P)\subset N_{W_0}(W_P)$ \index{N@$N_{W_0}(W_P)$, normalizer of $W_P$ in $W_0$} is a subgroup which is complementary to the normal subgroup $W_P\subset N_{W_0}(W_P)$. Moreover, $W(P,Q)$ is a left $W(P)$ coset and a right $W(Q)$ coset.
The action of $n\in W(P,Q)$ on $T$ restricts to isomorphisms $T_P\to T_Q$ and $T^P\to T^Q$. Recall that $K_P=T_P\cap T^P$, so that $n\in W(P,Q)$ gives rise to an isomorphism $n:K_P\to K_Q$.
Consider the groupoid $\mathcal{W}$ \index{W8@$\mathcal{W}$, groupoid whose set of objects is $\mathcal{P}$, with morphisms $\operatorname{Hom}_\mathcal{W}(P,Q)= \mathcal{W}(P,Q):=K_Q\times W(P,Q)$} whose set of objects is $\mathcal{P}$, with morphisms $\operatorname{Hom}_\mathcal{W}(P,Q) =\mathcal{W}(P,Q):=K_Q\times W(P,Q)$ \index{W8a@$\mathcal{W}(P,Q)=\operatorname{Hom}_\mathcal{W}(P,Q)$} and the composition defined by $(k_1\times n_1)\circ(k_2\times n_2)=k_1n_1(k_2)\times (n_1\circ n_2)$. We denote by $\mathcal{W}(P)$ the group $\mathcal{W}(P)=\mathcal{W}(P,P)$. \index{W8b@$\mathcal{W}(P)=\mathcal{W}(P,P)$}
If $k\times n\in\mathcal{W}(P,Q)$ we define for $\gamma=(\mathcal{R}_P,W_Pr)\in\Gamma_P$: \begin{equation} (k\times n)(\gamma)=(k\times n)(\mathcal{R}_P,W_Pr):=(\mathcal{R}_Q,W_Q(k^{-1}n(r))). \end{equation} This defines a left action of $\mathcal{W}$ on $\Gamma$. If $t\in T^P_u$ then $(\gamma,t)\in\Lambda_\gamma$, and we define \begin{equation} (k\times n)(\gamma,t):=((k\times n)(\gamma),kn(t)). \end{equation} This defines a left action of $\mathcal{W}$ on $\Lambda$. With these definitions we obviously have \begin{equation}\label{eq:minv} m(g(\lambda))=m(\lambda) \end{equation} for all $g\in \operatorname{Hom}(\mathcal{W})$ and $\lambda\in\Lambda$ such that $g(\lambda)$ is defined. In other words, $m$ is $\mathcal{W}$-invariant. \begin{lem}\label{lem:free} The action of $\mathcal{W}$ on $\Lambda^{gen}$ is free. \end{lem} \begin{proof} Let $\lambda=(\mathcal{R}_P,W_Pr,t)\in\Lambda_{\mathcal{R}_P,W_Pr}^{gen}$ and let $g=k\times n\in \mathcal{W}(P,Q)$ be such that $g(\lambda)=\lambda$. Then $Q=P$, $g$ fixes $W_Pr$, and $kn(t)=t$. We have $g(W_Pr):=W_P(k^{-1}n(r))$, thus $n(r)=kw(r)$ for some $w\in W_P$. Hence $n(rt)=n(r)n(t)=w(r)(kn(t))=w(rt)$. Since $rt$ is $R_P$-generic this implies that $w^{-1}n\in W_P$. Hence $n=e$, and thus also $k=e$. \end{proof} \begin{lem}\label{lem:orb} Let $L$ be a residual subspace, and let $t=r_Lt^L\in L^{temp}$ be $R_L$ generic. Then $m^{-1}(W_0t)$ is a $\mathcal{W}$-orbit in $\Lambda$. \end{lem} \begin{proof} By making a suitable choice of $t$ in the orbit $W_0t$ we may assume that $R_L=R_P$ for some $P\in\mathcal{P}$. We write $r_P$ instead of $r_L$ and $t^P$ instead of $t^L$. Thus it is assumed that $t=r_Pt^P\in L^{temp}=r_PT^P_u$ is $R_P$-generic. Define $\lambda:=(\mathcal{R}_P,W_Pr_P,t^P)\in\Lambda_{\mathcal{R}_P,W_Pr_P}^{gen}$. Clearly, $\mathcal{W}\cdot\lambda\subset m^{-1}(W_0t)$ by the $\mathcal{W}$-invariance of $m$.
Conversely, suppose that $\mu=(\mathcal{R}_Q,W_Qr_Q,t^Q)\in m^{-1}(W_0t)$. Hence there exists a $w\in W_0$ such that $r_Qt^Q=wt=w(r_P)w(t^P)$. This is an element of the tempered residual subspace $L_Q^{temp}:=r_QT^Q_u$, so that $R_Q\subset R_{P(wt)}$. Since $t\in L^{temp}$ is $R_P$ generic, we have $R_{P(t)} = R_P$ by Corollary \ref{cor:gen}. Because $R_{P(wt)}=w(R_{P(t)})$, we obtain $R_Q\subset w(R_P)$. This implies that $w(L^{temp})=w(t)w(T^P_u)\supset r_QT^Q_u=L_Q^{temp}$. By Theorem \ref{thm:nonnest} we see that $w(L^{temp})=L_Q^{temp}$. Hence we have $w(R_P)=R_Q$, $w(T_P)=T_Q$ and $w(T^P)=T^Q$. We conclude that $r_Q^{-1}w(r_P)=k\in K_Q$. There exists a unique $u\in W_Q$ such that $uw\in W(P,Q)$. One easily checks that $\mu=(k\times uw)(\lambda)$.
Note that it follows that the intersection $m^{-1}(W_0t)\cap \Lambda_{\mathcal{R}_P,W_Pr_P}$ is contained in $\Lambda_{\mathcal{R}_P,W_Pr_P}^{gen}$ (for any choice of $P\in\mathcal{P}$ and $W_Pr_P$). \end{proof} \begin{cor}\label{cor:romeo} We form the quotients $\Sigma=\mathcal{W}\backslash\Lambda$ \index{0S@$\Sigma$, quotient space of $\Lambda$ for the action of $\mathcal{W}$} and $\Sigma^{gen}=\mathcal{W}\backslash\Lambda^{gen}$. The map $m$ factors through $\Sigma$, and defines a homeomorphism (also denoted by $m$) from $\Sigma^{gen}$ onto the open dense set $S^{gen}:=m(\Lambda^{gen})\subset S$. \index{m@$m$!$m:\Sigma^{gen}\to S^{gen}:=m(\Lambda^{gen})\subset S$, homeomorphism} \end{cor} \begin{proof} By equation (\ref{eq:minv}), $m$ is well defined on $\Sigma$, and thus $m(\Sigma)=m(\Lambda)=S$. By the previous lemma, the set $\Lambda^{gen}$ is $m$-saturated. Since $m$ is closed, this implies that $S^{gen}=m(\Lambda^{gen})\subset S$ is open (and obviously dense) in $S=m(\Sigma)$. Finally, again by the previous lemma, $m$ is injective on $\Sigma^{gen}$. Thus, being a closed map, $m$ is homeomorphic onto its image. \end{proof}
$\Sigma$ can be realized as a disjoint union of orbifolds as follows. Choose a complete set $\Gamma_a$ \index{0C@$\Gamma_a$, complete set of representatives of the association classes in $\Gamma$} of representatives for the association classes (the orbits of $\mathcal{W}$) of elements in $\Gamma$. Put \begin{equation} \Sigma_\gamma:=\mathcal{W}(\gamma)\backslash \Lambda_\gamma, \end{equation} where $\mathcal{W}(\gamma)$ \index{W8c@$\mathcal{W}(\gamma)=\{g\in\mathcal{W}\mid g\gamma=\gamma\}$} denotes the isotropy group of $\gamma\in\Gamma$ in $\mathcal{W}$. Then \begin{equation}\label{eq:union} \Sigma\simeq\cup\Sigma_\gamma, \end{equation} where the (disjoint) union is taken over the set of $\gamma\in\Gamma_a$. \subsubsection{Groupoid $\mathcal{W}_{\Xi}$ of standard induction data} Recall the complete set of representatives $\Delta_\gamma$ ($\gamma=(\mathcal{R}_P,W_Pr_P)\in\Gamma$) of the irreducible discrete series representations with central character $W_Pr_P$ of $\mathcal{H}_P$. We denote by $\Delta=\cup\Delta_\gamma$ \index{0D@$\Delta=\cup_{\gamma\in\Gamma}\Delta_\gamma$} the disjoint union of these sets over all $\gamma\in\Gamma$. The composition $\Delta\to\Gamma\to\mathcal{P}$ gives a surjection of $\Delta\to\mathcal{P}$, whose fibers are denoted by $\Delta_P$. \index{0D2@$\Delta_P$, fiber at $P\in\mathcal{P}$ of the surjection $\Delta\to\mathcal{P}$} \index{0D1@$\Delta_\gamma$, fiber at $\gamma\in\Gamma$ of the surjection $\Delta\to\Gamma$}
There is a natural left action $\Psi$ of $\mathcal{W}$ on $\Delta$ as follows: When $k\in K_P=T^P\cap T_P$, we have an automorphism $\psi_k:{\mathcal{H}_P}\to{\mathcal{H}_P}$ defined by $\psi_k(\theta_xT_w)=k(x)\theta_xT_w$. This induces an isomorphism $\psi_k:\overline{\mathcal{H}_P^{r_P}}\to \overline{\mathcal{H}_P^{k^{-1}r_P}}$. We define a bijection $\Psi_k$ from $\Delta_{\mathcal{R}_P,W_Pr_P}$ to $\Delta_{\mathcal{R}_P,k^{-1}W_Pr_P}$ by $\Psi_k(\delta)\simeq \delta\circ \psi_k^{-1}$.
Let $Q\in F_0$ be associate to $P$, and $n\in W(P,Q)$. Then $n$ induces an isomorphism of root data and labels $(\mathcal{R}_P,q)\to(\mathcal{R}_Q,q)$, thus inducing an isomorphism $\psi_n$ on $\mathcal{H}_P\to\mathcal{H}_Q$. Recall that $\psi_n$ induces an isomorphism $\psi_n:\overline{\mathcal{H}_P^{r_P}}\to\overline{\mathcal{H}_Q^{n(r_P)}}$ (Corollary \ref{cor:con}), and thus a bijection $\Psi_n:\Delta_{\mathcal{R}_P,W_Pr_P} \to \Delta_{\mathcal{R}_Q,W_Qn(r_P)}$ by $\Psi_n(\delta)\simeq \delta\circ\psi_n^{-1}$. One easily checks that these definitions combine to define a left action $\Psi$ of $\mathcal{W}$ on $\Delta$ \index{0X@$\Psi$, left action of $\mathcal{W}$ on $\Delta$}, compatible with the surjection $\Delta\to\mathcal{P}$ mentioned above. \index{0x@$\psi_g:\mathcal{H}^L\to\mathcal{H}^M$, isomorphism for $g\in K_M\times W(F_L,F_M)$} \index{0X@$\Psi_g:\Delta_{\mathcal{R}_L,W_Lr_L}\to\Delta_{\mathcal{R}_M,W_Mr_M^\prime}$, bijection induced by $\psi_g$}
Consider the product ${\Xi}:=\Lambda\times_{\Gamma}\Delta$ \index{0O@${\Xi}=\Lambda\times_{\Gamma}\Delta$}. This set comes equipped with a natural surjection ${\Xi}\to\mathcal{P}$ and compatible left action of $\mathcal{W}$ (the diagonal action). We form the cross product $\mathcal{W}_{\Xi}:=\mathcal{W}\times_\mathcal{P}{\Xi}$ \index{W8d@$\mathcal{W}_{\Xi}:=\mathcal{W}\times_\mathcal{P}{\Xi}$, groupoid of standard induction data}, which has itself a natural groupoid structure with $\operatorname{Obj}(\mathcal{W}_{\Xi}):={\Xi}$, and $\operatorname{Hom}_{\mathcal{W}_{\Xi}}(\xi_1,\xi_2):=\{w\in\mathcal{W}\mid w(\xi_1)=\xi_2\}$. The composition maps are defined by the composition in $\mathcal{W}$. We will refer to this structure as the groupoid of standard induction data of $\mathcal{H}$. Its set of objects ${\Xi}$ has the structure of a disjoint union of compact tori, and with this structure $\mathcal{W}_{\Xi}$ is obviously a smooth compact groupoid.
Recall that we associated to each $\xi=\lambda\times\delta=(\mathcal{R}_P,W_Pr_P,t)\times(\mathcal{R}_P,\delta)\in{\Xi}$ (i.e. $\delta$ is an irreducible discrete series representation of $\mathcal{H}(\mathcal{R}_P,q)$ with central character equal to $W_Pr_P$) a tempered, unitary representation $\pi(\xi)=\pi(\mathcal{R}_P,W_Pr_P,\delta,t)$ of $\mathcal{H}$ with central character $m(\lambda)=W_0(r_Pt)$ and representation space $V_{\pi(\xi)}=\mathcal{H}(W^P)\otimes V_\delta$ (the compact realization) (cf. Definition \ref{dfn:ind}, Proposition \ref{prop:unit} and Proposition \ref{prop:indtemp}).
For every $(g,\xi)\in\mathcal{W}_{\Xi}$ with source $\xi=\lambda\times\delta$, we choose a unitary isomorphism $\tilde\delta_g: V_\delta\to V_{\Psi_g(\delta)}$ \index{0dZ@$\tilde\delta_g: V_\delta\to V_{\Psi_g(\delta)}$, unitary isomorphism intertwining $\delta\circ\psi_g$ and $\Psi_g(\delta)$} so that we have \begin{equation} \Psi_g(\delta)(h)\circ\tilde\delta_g=\tilde\delta_g\circ\delta(\psi_g^{-1}h) \end{equation} for $h\in \mathcal{H}_P$ (where $P=P(\delta)$).
This enables us to define intertwining operators (depending on the choices of the isomorphisms $\tilde\delta_g$) \begin{equation} A(g,\xi)\in\operatorname{Hom}_\mathcal{H}(V_{\pi(\xi)},V_{\pi(g(\xi))}) \end{equation} as follows:
For $k\in K_P$ and $h\in\mathcal{H}^P$ we have $\phi_{kt}(h)=\psi_k(\phi_t(h))$, so that if $h\in \mathcal{H}^P$ we have that $\tilde\delta_k\circ\delta(\phi_{t}(h))= \Psi_k(\delta)(\phi_{kt}(h))\circ\tilde\delta_k$. With this notation we have for each $\delta\in\Delta_{\mathcal{R}_P,W_Pr_P}$, in view of Proposition \ref{prop:unit}, a unitary intertwining isomorphism \begin{equation}\label{eq:iso} \operatorname{Id}\otimes\tilde\delta_k:\pi(\xi)\to \pi(k(\xi)). \end{equation} We denote this unitary intertwining operator by $A(k,\mathcal{R}_P,W_Pr,\delta,t)$ or more simply $A(k,\xi)$. Notice that it is constant, i.e. independent of $t$.
For $n\in W(P,Q)$ (with $P,Q\in\mathcal{P}$ associate subsets) we defined (cf. Theorem \ref{thm:ms} and Corollary \ref{cor:hol}) an intertwining isomorphism (depending on the choice of $\tilde\delta_n$) \begin{equation} A(n,\mathcal{R}_P,W_Pr_P,\delta,t):\pi(\mathcal{R}_P,W_Pr_P,\delta,t)\to \pi(\mathcal{R}_Q,W_Qn(r_P),\Psi_n(\delta),n(t)), \end{equation} which is rational in $t$, well defined and regular in a neighborhood of $T^P_u$, and unitary for $t\in T^P_u$. We now denote this isomorphism by $A(n,\xi)$.
The above intertwining isomorphisms combine (as one easily checks directly from the definitions) to a functor \begin{equation} \mathcal{W}_{\Xi}\to {PRep}_{unit,temp}(\mathcal{H}) \end{equation} where ${PRep}_{unit,temp}(\mathcal{H})$ denotes the category of finite dimensional, tempered, unitary modules over $\mathcal{H}$, with morphisms $\operatorname{Hom}_{PRep}(\pi_1,\pi_2)= PU_\mathcal{H}(V_{\pi_1},V_{\pi_2})$ (the space of unitary intertwiners modulo the action of scalars).
Summarizing the above we have: \begin{thm}\label{cor:proact} There exists an induction functor \begin{equation} \pi:\mathcal{W}_{\Xi}\to{PRep}_{unit,temp}(\mathcal{H}) \end{equation} \index{0p@$\pi$, induction functor on $\mathcal{W}_{\Xi}$} such that for $\xi=(\mathcal{R}_P,W_Pr_P,t)\times(\mathcal{R}_P,\delta)\in {\Xi}$ and $(g,\xi)\in\mathcal{W}_{\Xi}$ (thus $g\in\mathcal{W}$ with source $P(\xi)=P$), $\pi(\xi):=\pi(\mathcal{R}_P,W_Pr_P,\delta,t)$ and $\pi(g,\xi):=A(g,\xi)=A(n,\mathcal{R}_P,W_Pr_P,\delta,t)$. \end{thm} \subsubsection{Generic spectrum} Consider the natural projection \begin{equation} p_\Sigma:\mathcal{W}_{\Xi}\backslash{\Xi} =\mathcal{W}\backslash(\Lambda\times_\Gamma\Delta) \to\Sigma=\mathcal{W}\backslash\Lambda. \end{equation}
Since the action of $\mathcal{W}$ is free on the set of generic points $\Lambda^{gen}$, we obtain a finite covering \begin{equation} p_\Sigma:\mathcal{W}_{\Xi}\backslash{\Xi}^{gen}\to\Sigma^{gen}, \end{equation} where ${\Xi}^{gen}:=\Lambda^{gen}\times_\Gamma\Delta$ \index{0O@${\Xi}^{gen}=\Lambda^{gen}\times_\Gamma\Delta$}.
By what was said in the previous subsection and Corollary \ref{cor:short}, it is clear that the map (see Proposition \ref{prop:im} for the definition of $p_z$): \begin{equation} \begin{split} [\pi]: {\Xi}^{gen}&\to p^{-1}_z(S^{gen})\\ \xi&\to[\pi(\xi)] \end{split} \end{equation} factors through the quotient $\mathcal{W}_{\Xi}\backslash{\Xi}^{gen}$ \index{0p1@$[\pi(\xi)]$, class of $\pi(\xi)$ modulo equivalence}. We thus have the following commutative diagram: \begin{equation} \begin{CD}\label{CD?} \mathcal{W}_{\Xi}\backslash{\Xi}^{gen}@>[\pi]>>p^{-1}_z(S^{gen})\\ @V{p_\Sigma}VV @VV{p_z}V\\ \Sigma^{gen}@>>m>S^{gen} \end{CD} \end{equation} \begin{thm}\label{thm:homeom} The map $[\pi]$ in the diagram \ref{CD?} is a homeomorphism. \end{thm} \begin{proof} The topology on $\hat\mathfrak{C}$ is second countable since $\mathfrak{C}$ is separable. Thus, in order to check the continuity of $[\pi]$, it suffices to check that $[\pi]$ maps a converging sequence $\lambda_i\times\delta\to\lambda\times\delta\in\Lambda_{\mathcal{R}_P,W_Pr}$ to a converging sequence in $\hat{\mathfrak{C}}$. We check this using the Fell-topology description of the topology of $\hat{\mathfrak{C}}$ (see \cite{Fell}). By \cite{Fell}, Proposition 1.17, restriction to the dense subalgebra $\mathcal{H}\subset\mathfrak{C}$ is a homeomorphism with respect to the Fell topologies. Let $V_{\lambda\times\delta}=\mathcal{H}(W^P)\otimes V$ be the representation space of $\pi(\lambda\times\delta)$ (with $\lambda\in\Lambda_{\mathcal{R}_P,W_Pr}$). We equip $V_{\lambda\times\delta}$ with the inner product $\langle\cdot,\cdot\rangle$ of Proposition \ref{prop:unit} (which is independent of $\lambda\in\Lambda_{\mathcal{R}_P,W_Pr}$), and we choose an orthonormal basis $(e_i)$ of $V_{\lambda\times\delta}$. In order to check that $\pi(\lambda_i\times\delta)\to\pi(\lambda\times\delta)$ in the Fell topology with respect to $\mathcal{H}$, we need to check that for all $h\in\mathcal{H}$, $\pi(\lambda_i\times\delta)(h)_{k,l}\to\pi(\lambda\times\delta)(h)_{k,l}$ for all matrix coefficients. This is clear since the matrix coefficients are regular functions of the induction parameter.
To see that the map $[\pi]$ is closed, assume that we have a sequence $\rho_i=[\pi](\lambda_i\times\delta_i)$ converging to $\rho\in p_z^{-1}(S^{gen})$. Since $m$ is a homeomorphism and $\Sigma^{gen}$ is a finite quotient of $\Lambda^{gen}$, we may assume that $\lambda_i$ converges, to $\lambda_0\in\Lambda^{gen}$ say, by possibly replacing the sequence by a subsequence. Since $\Delta_{\mathcal{R}_P,W_Pr}$ is finite for each $R_P$ and $W_Pr$, we may assume that $\forall i: \delta_i=\delta$, again by taking a subsequence. Then $d=\operatorname{dim}(\rho_i)$ is independent of $i$, and lower semi-continuity of $\operatorname{dim}$ on $\hat\mathfrak{C}$ implies that $\operatorname{dim}(\rho)\leq d$. Choose an orthonormal basis $B$ for $\rho$. Convergence in the Fell-topology means that there exists for all $i$ an orthonormal subset $B_i$ of size $\operatorname{dim}(\rho)$ in the representation space $V_{\rho_i}=\mathcal{H}(W^P)\otimes V$ of $\rho_i$, such that the matrix coefficients of $\rho_i$ with respect to $B_i$ converge to the matrix coefficients of $\rho$ with respect to $B$. By the independence of the inner product $\langle\cdot,\cdot\rangle$ of the induction parameter (Proposition \ref{prop:unit}) we may assume, by further restricting to a subsequence, that the sets $B_i$ converge in $\mathcal{H}(W^P)\otimes V$ to an orthonormal set $B_0$. It follows that the matrix of $\rho(x)$ with respect to $B$ equals a principal block of the matrix of $\pi(\lambda_0\times\delta)(x)$ with respect to a suitable orthonormal basis $\tilde{B}$ of $\mathcal{H}(W^P)\otimes V$ for $\pi(\lambda_0\times\delta)$. Since $\pi(\lambda_0\times\delta)$ is irreducible it is easy to see that this is impossible unless $\rho\simeq\pi(\lambda_0\times\delta)$.
The map $[\pi]$ is injective by Corollary \ref{cor:short} and Lemma \ref{lem:orb}.
Finally, by Theorem \ref{thm:nu}, Theorem \ref{thm:support}, Theorem \ref{thm:mainind} and Theorem \ref{thm:thisisathm} we see that the complement of $[\pi](\mathcal{W}_{\Xi}\backslash{\Xi}^{reg})$ has measure $0$ in $\hat\mathfrak{C}$ with respect to the Plancherel measure of the representation $\mathfrak{H}$ of $\mathfrak{C}$. The support of the Plancherel measure is equal to $\hat\mathfrak{C}$, since $\mathfrak{H}$ is a faithful representation of $\mathfrak{C}$ (by definition of $\mathfrak{C}$). Thus the closure of $[\pi](\mathcal{W}_{\Xi}\backslash{\Xi}^{reg})$ is $\hat\mathfrak{C}$. But $[\pi](\mathcal{W}_{\Xi}\backslash{\Xi}^{reg})\subset p_z^{-1}(S^{reg})$ is closed as we have seen above, implying that $[\pi]$ is surjective. \end{proof} \begin{cor} The restriction of the map $p_z$ of Corollary \ref{prop:im} to $p_z^{-1}(S^{reg})$ is a covering map. \end{cor} \subsubsection{Fourier transform} Let $\tilde\mathcal{O}\subset{\Xi}$ \index{O@$\tilde\mathcal{O}$, connected component of ${\Xi}$} be a connected component of ${\Xi}$. Thus there exists a $\delta\in\Delta$ such that $\tilde\mathcal{O}=\Lambda_\gamma\times\{\delta\}:=\tilde\mathcal{O}_\delta$, where $\gamma=\gamma(\delta)\in\Gamma$. Explicitly, if $\gamma(\delta)=(\mathcal{R}_P,W_Pr_P)$ then $\tilde\mathcal{O}_\delta$ is a copy of the subtorus $T^P_u\subset T_u$.
The representation space $V_{\pi(\xi)}$ of $\pi(\xi)$ is equal to $V_{\pi(\xi)}=\mathcal{H}(W^P)\otimes V_\delta$ for $\xi\in\tilde\mathcal{O}_\delta$ with $\delta\in\Delta_P$. In particular, $V_{\pi(\xi)}$ depends only on the connected component $\tilde\mathcal{O}_\delta$ of $\Xi$ containing $\xi$, and not on the choice of $\xi\in\tilde\mathcal{O}_\delta$. We will use the notation $i(V_\delta):=\mathcal{H}(W^P)\otimes V_\delta=V_{\pi(\xi)}$ for any choice of $\xi\in\tilde\mathcal{O}_\delta$ (where $P=P(\delta)\in\mathcal{P}$). \index{i@$i(V_{\delta})=\mathcal{H}(W^P)\otimes V_\delta$ if $\delta\in\Delta_P$}
We form the trivial fiber bundle $\mathcal{V}_{\tilde\mathcal{O}}=\tilde\mathcal{O}_\delta\times i(V_\delta)$ \index{V1@$\mathcal{V}_{\tilde\mathcal{O}}=\tilde\mathcal{O}_\delta\times i(V_\delta)$, trivial fiber bundle over $\tilde\mathcal{O}=\tilde\mathcal{O}_\delta$} over $\tilde\mathcal{O}=\tilde\mathcal{O}_\delta$, and put \begin{equation} \mathcal{V}_{\Xi}:=\cup_{\delta\in\Delta}\mathcal{V}_{\tilde\mathcal{O}_\delta}. \end{equation} \index{V3@$\mathcal{V}_{\Xi}$, trivial fiber bundle over $\Xi$} We identify the connected component $\tilde\mathcal{O}_\delta$ of $\Xi$ with the compact torus $T^P_u$ ($P=P(\delta)$). This allows us to define the function spaces $\operatorname{Pol}(\Xi)$ \index{Pol@$\operatorname{Pol}(\Xi)$, space of Laurent polynomials on $\Xi$} (Laurent polynomials on $\Xi$) and $C^\infty(\Xi)$ \index{C@$C^\infty(\Xi)$, space of $C^\infty$-functions on $\Xi$}. We also introduce the space $\operatorname{Rat}^{reg}(\Xi)= \oplus_{\delta\in\Delta}\operatorname{Rat}^{reg}(\tilde\mathcal{O}_d)$, \index{RatX@$\operatorname{Rat}^{reg}(\Xi)$, regular rational functions on $\Xi$} \index{RatO@$\operatorname{Rat}^{reg}(\tilde\mathcal{O})$, rational functions on $\tilde\mathcal{O}$, regular in an open neighborhood of $\tilde\mathcal{O}\simeq T^P_u\subset T^P$} where $\operatorname{Rat}^{reg}(\tilde\mathcal{O}_d)$ denotes the space of restrictions to $T^P_u$ (which we identify with $\tilde\mathcal{O}_d$) of rational functions on $T^P$ which are regular in a open neighborhood of $T^P_u$. The corresponding spaces of (global) sections are denoted by $\operatorname{Pol}(\mathcal{V}_{\Xi})= \operatorname{Pol}(\Xi)\otimes\mathcal{V}_{\Xi}$ \index{PolV@$\operatorname{Pol}(\mathcal{V}_{\Xi})= \operatorname{Pol}(\Xi)\otimes\mathcal{V}_{\Xi}$}, $C^\infty(\mathcal{V}_{\Xi})= C^\infty(\Xi)\otimes\mathcal{V}_{\Xi}$ \index{C@$C^\infty(\mathcal{V}_{\Xi})= C^\infty(\Xi)\otimes\mathcal{V}_{\Xi}$}, and $\operatorname{Rat}^{reg}(\mathcal{V}_{\Xi})= \operatorname{Rat}^{reg}(\Xi)\otimes\mathcal{V}_{\Xi}$ \index{RatV@$\operatorname{Rat}^{reg}(\mathcal{V}_{\Xi})= \operatorname{Rat}^{reg}(\Xi)\otimes\mathcal{V}_{\Xi}$} respectively.
Recall that $\pi(g,\xi)\in PU_\mathcal{H}(i(V_\delta),i(V_{gd}))$ (with $\xi=\lambda\times\delta=(\mathcal{R}_P,W_Pr,t)\allowbreak\times\delta$) is rational and regular for $t\in T^P$ in a neighborhood of $T^P_u$ (Corollary \ref{cor:hol}). We define \begin{align*} \operatorname{Pol}(&\operatorname{End}(\mathcal{V}_\Xi))^\mathcal{W}\\= &\{f\in\operatorname{Pol}(\operatorname{End}(\mathcal{V}_\Xi)) \mid \forall(g,\xi)\in\mathcal{W}_\Xi:\ \pi(g,\xi)f(\xi)=f(g\xi)\pi(g,\xi)\}\\\simeq &\bigoplus_{\tilde\mathcal{O}}\operatorname{Pol}(\mathcal{V}_{\tilde\mathcal{O}})^{\mathcal{W}(\tilde\mathcal{O},\tilde\mathcal{O})} \end{align*} \index{Pol@$\operatorname{Pol}(\operatorname{End}(\mathcal{V}_\Xi))^\mathcal{W}$, space of $\mathcal{W}_\Xi$-equivariant sections in $\operatorname{Pol}(\operatorname{End}(\mathcal{V}_\Xi))$} where the direct sum runs over a complete set of representatives of connected components $\tilde\mathcal{O}$ for the action of $\mathcal{W}$, and $\mathcal{W}(\tilde\mathcal{O}_1,\tilde\mathcal{O}_2)$ denotes the set of $w\in\mathcal{W}$ such that $w(\tilde\mathcal{O}_1)=\tilde\mathcal{O}_2$. We define the space of $\mathcal{W}_\Xi$-equivariant sections in other spaces of sections of $\operatorname{End}(\mathcal{V}_\Xi)$ similarly. \begin{dfn} The Fourier transform is the algebra homomorphism \begin{align*}\label{eq:FT} \mathcal{F}_\mathcal{H}:\mathcal{H}&\to\operatorname{Pol}(\operatorname{End}(\mathcal{V}_\Xi))^\mathcal{W}\\ h&\to\{\xi\to\pi(\xi)(h)\} \end{align*} \end{dfn} \index{F@$\mathcal{F}_\mathcal{H}$, Fourier transform on $\mathcal{H}$}
We would like to replace $\Xi$ by the set of equivalence classes of cuspidal representations of the standard parabolic subalgebras $\mathcal{H}^P$. This can be done as follows. Consider the subgroupoid $\mathcal{K}\subset \mathcal{W}$ \index{K@$\mathcal{K}$, normal subgroupoid of $\mathcal{W}$} of $\mathcal{W}$, with set of objects $\mathcal{P}$, and $\mathcal{K}(P_1,P_2)=\emptyset$ if $P_1\not=P_2$, and $\mathcal{K}(P,P)=K_P$. This subgroupoid is normal in the sense that $gK_Pg^{-1}=K_Q$ if $g\in\mathcal{W}(P,Q)$. The quotient groupoid $\mathcal{W}/\mathcal{K}=\mathcal{W}/\mathcal{K}$ has $\mathcal{P}$ as set of objects, and $\mathcal{W}/\mathcal{K}(P,Q)=W(P,Q)$.
Suppose that $\delta_t\simeq\delta^\prime_s$ with $\delta,\delta^\prime\in\Delta_{\mathcal{R}_P}$ and $s,t\in T^P_u$. Then $W_Prt=W_Pr^\prime s$, and thus $s=kt$ for some $k\in K_P$, and $\delta^\prime_s=\Psi_k(\delta)_{kt}$. Conversely, in view of the text above (\ref{eq:iso}), $\xi\simeq k(\xi)$ for every $k\in K_P$ and $\xi\in\Xi_P$, viewed as representation of $\mathcal{H}^P$.
The connected components $\mathcal{O}$ \index{O@$\mathcal{O}$, orbit of twists of cuspidal representations} of the quotient $\mathcal{K}\backslash\Xi$ are called ``orbits of twists of cuspidal representations'' of the parabolic subalgebras $\mathcal{H}^P$. Such a component can be viewed as the collection of mutually inequivalent representations of $\mathcal{H}^P$ of the form $\delta_t$. It is isomorphic to a smooth quotient $\mathcal{O}\simeq\mathcal{K}(\tilde\mathcal{O},\tilde\mathcal{O})\backslash\tilde\mathcal{O}$, a finite quotient of the subtorus $T^P_u\subset T_u$.
We have $\mathcal{W}\backslash\Xi= (\mathcal{K}\backslash\mathcal{W})\backslash(\mathcal{K}\backslash\Xi)$. For $\mathcal{O}$ a connected component of $\mathcal{K}\backslash\Xi$, we choose a connected component $\Xi\supset\tilde\mathcal{O}\to\mathcal{O}$ covering $\mathcal{O}$. Let $\tilde\mathcal{O}=\Lambda_\gamma\times\delta$ and write $P(\gamma)=P$. The isotropy group $\{k\in K_P\mid k(\tilde\mathcal{O}) =\tilde\mathcal{O}\}$ equals the isotropy group $K_\delta$ \index{K@$K_\delta\subset K_P$, isotropy subgroup of $[\delta]\in [\Delta_P]$}. Notice that $K_\delta$ is independent of the choice of $\tilde\mathcal{O}\to\mathcal{O}$. We define the principal fiber bundle $\mathcal{V}_\mathcal{O}:=\tilde\mathcal{O}\times_{\mathcal{K}_\delta}i(V_\delta)$ \index{V2@$\mathcal{V}_\mathcal{O}:=\tilde\mathcal{O}_\delta\times_{\mathcal{K}_\delta}i(V_\delta)$} over $\mathcal{O}$. This fiber bundle is not necessarily trivial. We put \begin{align} \operatorname{Pol}(\operatorname{End}(\mathcal{V}_\mathcal{O}))& =\left(\bigoplus_{\tilde\mathcal{O}:\tilde\mathcal{O}\to\mathcal{O}}\operatorname{Pol} (\operatorname{End}(\mathcal{V}_{\tilde\mathcal{O}}))\right)^{K_P}\\\nonumber &\simeq\operatorname{Pol} (\operatorname{End}(\mathcal{V}_{\tilde\mathcal{O}}))^{K_\delta} \end{align} \index{Pol@$\operatorname{Pol}(\operatorname{End}(\mathcal{V}_\mathcal{O}))$, polynomial sections in fiber bundle $\operatorname{End}(\mathcal{V}_\mathcal{O})$} and \begin{equation} \operatorname{Pol}(\operatorname{End}(\mathcal{V}_{(\mathcal{K}\backslash\Xi)}))= \bigoplus_{\mathcal{O}\mathrm{\ orbit}}\operatorname{Pol}(\operatorname{End}(\mathcal{V}_\mathcal{O})) \end{equation} \index{Pol@$\operatorname{Pol}(\operatorname{End}(\mathcal{V}_{(\mathcal{K}\backslash\Xi)}))$, polynomial sections in fiber bundle $\operatorname{End}(\mathcal{V}_{(\mathcal{K}\backslash\Xi)})$}
The quotient $\mathcal{W}/\mathcal{K}$ acts on $\mathcal{K}\backslash\Xi$ and thus also on the set of orbits. Given orbits $\mathcal{O}_1,\mathcal{O}_2$ with $P(\mathcal{O}_i):=P_i$ and $O_i=\Lambda_{\gamma_i}\times \delta_i$, we have $\mathcal{W}/\mathcal{K}(\mathcal{O}_1,\mathcal{O}_2)=\{n\in \mathcal{W}/\mathcal{K}(P_1,P_2) \mid n(\mathcal{O}_1)=\mathcal{O}_2\}= \{n\in W(P_1,P_2) \mid \exists k\in K_{P_2}:k\times n \in\mathcal{W}(\delta_1,\delta_2)\}$. We denote this set by $W(\mathcal{O}_1,\mathcal{O}_2)$ \index{W6a@$W(\mathcal{O}_1,\mathcal{O}_2)=\{n\in W(P_1,P_2)\mid \exists k\in K_{P_2}:(k\times n) \in\mathcal{W}(\delta_1,\delta_2)\}$}. We also put $W(\mathcal{O}):=W(\mathcal{O},\mathcal{O})$ \index{W6b@$W(\mathcal{O})=W(\mathcal{O},\mathcal{O})$}.
In this way we get \begin{align}\label{eq:FTbis} \operatorname{Pol}(\operatorname{End}(\mathcal{V}_\Xi))^{\mathcal{W}}&= \operatorname{Pol}(\operatorname{End}(\mathcal{V}_{(\mathcal{K}\backslash\Xi)})) ^{\mathcal{K}\backslash\mathcal{W}}\\\nonumber &\simeq \bigoplus_\mathcal{O}\operatorname{Pol}(\operatorname{End}(\mathcal{V}_\mathcal{O}))^{W(\mathcal{O})}, \end{align} where the direct sum runs over a complete set of representatives of orbits $\mathcal{O}$ modulo the action of $\mathcal{W}/\mathcal{K}$ (association classes of orbits).
We use similar notations for spaces of sections with coefficients in other types of functions spaces (e.g. continuous, $C^\infty$, etc.) in $\operatorname{End}(\mathcal{V}_\Xi)$ and $\operatorname{End}(\mathcal{V}_{(\mathcal{K}\backslash\Xi)})$. \subsubsection{Averaging projections}\label{subsub:av} Consider a function space $F$ on $\Xi$ which is a module over $\operatorname{Rat}^{reg}(\Xi)$. Due to the regularity of the intertwining operators (cf. Corollary \ref{cor:hol}), there exists a natural averaging projection $f\to\overline{f}$ \index{$f\to\overline{f}$, average of a section of $\operatorname{End}(\mathcal{V}_\Xi)$} from $F(\operatorname{End}(\mathcal{V}_\Xi))$ (sections of $\operatorname{End}(\mathcal{V}_\Xi)$ with coefficients in $F$) to $F(\operatorname{End}(\mathcal{V}_\Xi))^\mathcal{W}$. It is defined by (where $\mathcal{W}_\xi=\{g\in\mathcal{W}\mid (g,\xi)\in\mathcal{W}_\Xi\}$) \begin{equation}
\overline{f}(\xi)=|W_\xi|^{-1}\sum_{g\in\mathcal{W}_\xi} \pi(g,\xi)^{-1}f(g(\xi))\pi(g,\xi). \end{equation} Notice that the function space $F=\operatorname{Pol}(\Xi)$ is too small; in general the average of $f\in\operatorname{Pol}(\operatorname{End}(\mathcal{V}_\Xi))$ will be in $\operatorname{Rat}^{reg}(\operatorname{End}(\mathcal{V}_\Xi))^{\mathcal{W}}$.
There is a similar averaging procedure $f\to \overline{f}_{\mathcal{K}}$ which sends the space of sections $F(\operatorname{End}(\mathcal{V}_\Xi))$ to $F(\operatorname{End}(\mathcal{V}_{(\mathcal{K}\backslash\Xi)}))$ (in this case $F$ should be a module over ${\mathbb C}$). \subsubsection{Plancherel formula} We now define the Plancherel measure on $\mathcal{W}\backslash\Xi$. The following proposition says that the natural action of $\mathcal{W}_\Xi$ (via $\psi$) on the residual algebras is through isomorphisms of Hilbert algebras. \begin{prop}\label{prop:impinv} Let $\delta\in\Delta_{\mathcal{R}_P,W_Pr}$ and let $g=(k\times n)\in\mathcal{W}_{\mathcal{R}_P,W_Pr}$. We have (in the notation of Theorem \ref{thm:mainind}) $d_{\mathcal{R}_P,\Psi_g(\delta)}=d_{\mathcal{R}_P,\delta}$. \end{prop} \begin{proof} This is a simple extension of Corollary \ref{cor:con}, with a similar proof. \end{proof} Let $\tilde\mathcal{O}=\Lambda_\gamma\times\delta$ and let $\mathcal{O}=K_\delta\backslash\tilde\mathcal{O}$. If $P=P(\gamma)$ then $\mathcal{O}$ is a copy of the subquotient torus $\mathcal{K}_\delta\backslash T^P_u$. For $\omega\in\mathcal{O}$ we put $d^\mathcal{O}\omega$ for the normalized Haar measure on $\mathcal{O}$. Let $\gamma=(\mathcal{R}_P,W_Pr)$ and let $\omega=(\mathcal{R}_P,W_Pr,\delta,K_\delta t^P)$ be $R_P$-generic. Let $L^{temp}=rT^P_u$ denote a residual subspace underlying $\tilde\mathcal{O}$. We define \begin{equation}\label{eq:plameasure} \begin{split}
d\mu_{Pl}(\pi(\omega))&=|W_0(rt^P)||W^P|^{-1}d_{\mathcal{R}_P,\delta} d\nu_L(rt^P)\\
&=\frac{|W_P|}{|W_P\cap W_{r}|} \nu_{\mathcal{R}_P}(\{r\})d_{\mathcal{R}_P,\delta}k_Pm^P(rt^P)d^P(t^P)\\
&=\mu_{\mathcal{R}_P,Pl}(\{\delta\})|K_P\delta|m^P(\omega)d^\mathcal{O}\omega, \end{split} \end{equation} \index{0m@$\mu_{Pl}$, Plancherel measure on $\hat\mathfrak{C}$} where $d_{\mathcal{R}_P,\delta}>0$ is the residual degree of $\delta$ in the residual algebra $\overline{\mathcal{H}^{r}_L}$, $\mu_{\mathcal{R}_P,Pl}$ is given in Corollary \ref{cor:fdim}, $m^P(\omega)=m^L(\omega)$ is the common value of $m^L$ (as defined in Proposition
\ref{prop:par}) on the $K_\delta$ orbit $\omega$, and $k_P:=|K_P|$. We have used that the isotropy subgroup $W_{rt^P}$ equals $W_P\cap W_r$ if $rt^P$ is $R_P$-generic.
Recall that by definition we have \begin{equation} \sum_{\delta\in \Delta_{\mathcal{R}_P,W_Pr}}\operatorname{dim}(\delta) d_{\mathcal{R}_P,\delta}=1 \end{equation} Recall Conjecture \ref{rem:ell} stating that $d_{\mathcal{R}_P,\delta}\in\mathbb{Q_+}$.
We define an Hermitian inner product on $\operatorname{Pol}(\operatorname{End}(\mathcal{V}_{(\mathcal{K}\backslash\Xi)}))$ as follows: \begin{equation}\label{eq:hermien}
(f_1,f_2)=\sum_{\mathcal{O}}|W(\mathcal{O})|^{-1}\int_{\mathcal{O}} \operatorname{tr}(f_1(\omega)^*f_2(\omega))d\mu_{Pl}(\pi(\omega)), \end{equation} \index{$(\cdot,\cdot)$!inner product on $\operatorname{Pol}(\operatorname{End}(\mathcal{V}_{(\mathcal{K}\backslash\Xi)}))$} where the sum runs over a complete set of representatives for the association classes of orbits $\mathcal{O}$ (an association classes is an orbit under the action of $\mathcal{W}/\mathcal{K}$). Note that $f_1,f_2$ are in fact $K_\delta$-equivariant sections over the covering $\tilde\mathcal{O}\to\mathcal{O}$. The expression $\operatorname{tr}(f_1(\xi)^*f_2(\xi))$ is independent of a choice of $\xi\in\tilde\mathcal{O}$ such that $K_\delta\xi=\omega$. The common value is denoted by $\operatorname{tr}(f_1(\omega)^*f_2(\omega))$. \begin{thm}\label{thm:mainp} (Main Theorem) \begin{enumerate} \item[(i)] Let $\mathcal{O}$ be an orbit (a connected component of $\mathcal{K}\backslash\Xi$). We put $\hat\mathfrak{C}_\mathcal{O}^{gen}= [\pi](W(\mathcal{O})\backslash\mathcal{O}^{gen})\subset\hat\mathfrak{C}$, and we denote its closure by $\mathfrak{C}_\mathcal{O}\subset\hat\mathfrak{C}$ \index{C@$\hat\mathfrak{C}_\mathcal{O}$, component of $\hat\mathfrak{C}$, the closure of $[\pi](W(\mathcal{O})\backslash\mathcal{O}^{gen})\subset\hat\mathfrak{C}$}. Then $\mathfrak{C}_{\mathcal{O}_1}^{gen}\cap\mathfrak{C}_{\mathcal{O}_2}^{gen}=\emptyset$ unless $\mathcal{O}_1$ and $\mathcal{O}_2$ are in the same $\mathcal{W}/\mathcal{K}$-orbit, and \begin{equation} {\hat\mathfrak{C}}^{gen}:=\cup\hat\mathfrak{C}_\mathcal{O}^{gen} \end{equation} (union over a complete set of representatives for the association classes of orbits) is a dense set in $\mathfrak{C}$, whose complement has measure zero. \item[(ii)] The Plancherel measure of $\mathfrak{C}$ (i.e. the measure on $\hat\mathfrak{C}$ determined by the tracial state $\tau$ of $\mathfrak{C}$) is given on $\hat\mathfrak{C}_\mathcal{O}$ by equation (\ref{eq:plameasure}). The decomposition of $\tau$ in irreducible, mutually distinct characters of $\mathfrak{C}$ is given by \begin{equation} \tau=\sum_{\mathcal{O}} \int_{\omega\in W(\mathcal{O})\backslash\mathcal{O}} \chi_{\pi(\omega)}d\mu_{Pl}(\pi(\omega)) \end{equation} (sum over a complete set of representatives for the association classes of orbits). \item[(iii)] Equivalently, the algebra homomorphism $\mathcal{F}_\mathcal{H}$ (see (\ref{eq:FT}) and (\ref{eq:FTbis})) is an isometry with respect to the inner product (\ref{eq:hermien}), and extends uniquely to an isomorphism of $\mathfrak{C}\times\mathfrak{C}$ modules \begin{equation} \mathcal{F}:\mathfrak{H}\stackrel{\sim} {\to}L^2(\operatorname{End}(\mathcal{V}_{\Xi}))^\mathcal{W}\simeq \bigoplus_{\mathcal{O}}L^2(\operatorname{End}(\mathcal{V}_{\mathcal{O}}))^{W(\mathcal{O})} \end{equation} (sum over a complete set of representatives for the association classes of orbits). \end{enumerate} \end{thm} \begin{proof} (i) See Theorem \ref{thm:homeom}. The complement of ${\hat\mathfrak{C}}^{gen}$ has measure zero by the argument in the last part of the proof of that theorem. The density follows since $\hat\mathfrak{C}$ is the support of the Plancherel measure (cf. Theorem \ref{thm:supds}).
(ii) By formula of Proposition \ref{prop:dfn}(v) and Corollary \ref{cor:romeo} we have \begin{equation} \tau=\int_{\mathcal{W}\backslash\Lambda^{reg}}\chi_{m(\lambda)}d\nu(m(\lambda)) \end{equation} We decompose $\chi_{m(\lambda)}$ according to Theorem \ref{thm:mainind}(ii) to obtain \begin{equation} \tau=\int_{\lambda\in\mathcal{W}\backslash\Lambda^{reg}}
|W^{P(\lambda)}|^{-1}\sum_{\delta\in\Delta_{\gamma(\lambda)}}d_{\mathcal{R}(\lambda),\delta} \chi_{\pi(\lambda\times\delta)}d\nu(m(\lambda)) \end{equation} By Corollary \ref{cor:short} and Theorem \ref{thm:homeom} we have $\{[\pi](\lambda\times \delta)\}_{\delta\in\Delta_{\gamma}}= [\pi](p_\Sigma^{-1}(\mathcal{W}\lambda))$. Thus (by Theorem \ref{thm:homeom}) we can rewrite the integral as integral over $\mathcal{W}\backslash\Xi^{reg}$. When we use parameters and notations as explained in equation (\ref{eq:plameasure}), and we express $d\nu$ according to Proposition \ref{prop:par}, we obtain \begin{equation} \tau=\int_{\xi\in\mathcal{W}\backslash\Xi^{reg}}
|W^{P(\lambda)}|^{-1}|W_0(rt^P)|d_{\mathcal{R}(\xi),\delta(\xi)} \chi_{\pi(\xi)} d\nu_L(rt^P) \end{equation} According to our definition of $\mu_{Pl}$ this is equal to \begin{equation} \tau=\sum_{\mathcal{O}}\int_{\omega\in W(\mathcal{O})\backslash \mathcal{O}^{reg}} \chi_{\pi(\omega)}d\mu_{Pl}(\pi(\omega)) \end{equation} This is a decomposition of $\tau$ in characters of inequivalent irreducible representations of $\mathfrak{C}$ (see Theorem \ref{thm:mainind}(iii)). Hence this uniquely determines the Plancherel measure (by \cite{dix2}, Th\'eor\`eme 8.8.6) on $\hat\mathfrak{C}$. We conclude that $\mu_{Pl}$ is equal to the Plancherel measure of $\mathfrak{C}$.
(iii) The equivalence of (ii) and (iii) is well known, see the proof of Theorem \ref{thm:thisisathm}. It is allowed to use the formulation with $W(\mathcal{O})$-equivariant sections because of the unitarity and the regularity of intertwining operators (Theorem \ref{thm:ms}) and by Proposition \ref{prop:impinv}. \end{proof} \begin{rem} In \cite{DO} it is shown that the $\hat\mathfrak{C}_\mathcal{O}$ are the components of $\hat\mathfrak{C}$. Moreover, $\mathcal{F}(\mathfrak{S})$ (see \ref{subsub:schwartz} for the definition of $\mathfrak{S}$) and $\mathcal{F}(\mathfrak{C})$ are determined in \cite{DO}. \end{rem} \begin{cor}\label{cor:av} Let $\mathcal{J}:L^2(\operatorname{End}(\mathcal{V}_{(\mathcal{K}\backslash\Xi)}))\to\mathfrak{H}$ denote the adjoint of $\mathcal{F}$. Then $\mathcal{J}\mathcal{F}=\operatorname{Id}$ and $\mathcal{F}\mathcal{J}(f)=\overline{f}$ (see subsection \ref{subsub:av}). \end{cor} \begin{proof} By the isometry property of $\mathcal{F}$, $(\mathcal{J}\mathcal{F}(x),y)=(x,y)$ for all $x,y\in\mathfrak{H}$. Whence the first assertion. It is clear that $\mathcal{J}(f)=\mathcal{J}(\overline{f})$. If $g\in L^2(\operatorname{End}(\mathcal{V}_\Xi))^\mathcal{W}$ then $g=\mathcal{F}(x)$ for some $x\in\mathfrak{H}$. Thus $\mathcal{F}\mathcal{J}(g)=\mathcal{F}\mathcal{J}\mathcal{F}(x)=\mathcal{F}(x)=g$ for $\mathcal{W}$-equivariant $g$. Hence $\mathcal{F}\mathcal{J}(f)=\mathcal{F}\mathcal{J}(\overline{f})=\overline{f}$. \end{proof} \section{Base change invariance of the residual algebra} \label{sec:inv} Thus far we have found the spectral decomposition for $\mathcal{H}$ in terms of the ``residual degrees'' $d_{\mathcal{R}_L,\delta}$ of the residual algebras $\overline{\mathcal{H}_L^r}$. We prove in this section that the residual algebras are {\it independent} of ${\bf q}$ (using the Convention \ref{eq:scale}), up to isomorphism of Hilbert algebras. \subsection{Scaling of the root labels} Let $r=sc\in T$ be fixed, with $s\in T_u$ and $c=\exp(\gamma)$ with $\gamma\in \mathfrak{t}$. Assume that $B\subset \mathfrak{t}_{\mathbb C}$ is an open ball centered around the origin such that the conditions \ref{cond} (with respect to $r\in T$) are satisfied.
The second condition implies that each connected component of the union $U:=W_0(r\exp(B))$ contains a unique element of the orbit $W_0r$. Given $u\in U$ there is a unique $r^\prime =s^\prime c^\prime\in W_0r$ such that $u\in r^\prime\exp(B)$. By (i) there is a unique $b\in B$ such that $u=s^\prime c^\prime\exp(b)=s^\prime\exp(b+\gamma^\prime)$. Now let $\epsilon\in (0,1]$ be given. We define an analytic map $\sigma_\epsilon$ on $U$ by \begin{equation} \sigma_\epsilon(u):=s^\prime\exp(\epsilon\log((s^\prime)^{-1}u))= s^\prime\exp(\epsilon(b+\gamma^\prime)). \end{equation} \begin{lemma} The map $\sigma_\epsilon$ is an analytic, $W_0$-equivariant diffeomorphism from $U$ onto $U_\epsilon:=W_0(sc^\epsilon\exp(\epsilon B))$. The inverse of $\sigma_\epsilon$ will be denoted by $\sigma_{1/\epsilon}$. \end{lemma} \begin{proof} On the connected component $r^\prime\exp(B)$ the map $\sigma_\epsilon$ is equal to $\sigma_\epsilon=\mu_{s^\prime}\circ \exp\circ M_\epsilon \circ \log\circ\mu_{(s^\prime)^{-1}}$ where $\mu_{s^\prime}$ is the multiplication in $T$ by $s^\prime$, and $M_\epsilon$ is the multiplication in $\mathfrak{t}_{\mathbb C}$ by $\epsilon$. These are all analytic diffeomorphisms, because of condition (i). The $W_0$ equivariance follows from the fact that $\log$ is well defined (and thus equivariant, since $\exp$ is equivariant) from $W_0\exp(B+\gamma)$ to $W_0(B+\gamma)$, and that $M_\epsilon$ is $W_0$-equivariant. This implies that for $w\in W_0$, $w\exp(\epsilon\log((s^\prime)^{-1}u))=\exp(\epsilon\log((ws^\prime)^{-1}wu))$. It follows that \begin{equation} \begin{split} \sigma_\epsilon(wu)&=ws^\prime\exp(\epsilon\log((ws^\prime)^{-1}wu))\\ &=ws^\prime w\exp(\epsilon\log((s^\prime)^{-1}u))\\ &=w(\sigma_\epsilon(u)). \end{split} \end{equation} \end{proof} \begin{lemma}\label{lem:ana} Denote by $q^\epsilon$ the label function $q^\epsilon(s)=q(s)^\epsilon={\bf q}^{\epsilon f_s}$, and denote by $\mathcal{H}_{q^\epsilon}$ the affine Hecke algebra with root datum $\mathcal{R}$ (same as the root datum of the affine Hecke algebra $\mathcal{H}=\mathcal{H}_q$), but with the labels $q$ replaced by $q^\epsilon$. Let $c_{\alpha,\epsilon}\in{}_\mathcal{F}\mathcal{A}_{q^\epsilon}\subset{}_\mathcal{F}\mathcal{H}_{q^\epsilon}$ be the corresponding Macdonald $c$-functions. For every root $\alpha\in R_1$ we have: \begin{equation} U\ni u\to (c_{\alpha,\epsilon}(\sigma_\epsilon(u))c_\alpha(u)^{-1})^{\pm 1}\in \mathcal{A}^{an}(U). \end{equation} \end{lemma} \begin{proof} For $u$ in the connected component $r^\prime\exp(B)$ of $U$ we write $u=s^\prime v$ with $v\in c^\prime\exp(B)$. We have \begin{equation}\label{eq:ana} \begin{split} c_{\alpha,\epsilon}(\sigma_\epsilon(u))&c_\alpha(u)^{-1} =\frac{(1+q_{\alpha^\vee}^{-\epsilon/2}\alpha(v)^{-\epsilon/2}\alpha(s^\prime)^{-1/2})} {(1+q_{\alpha^\vee}^{-1/2}\alpha(v)^{-1/2}\alpha(s^\prime)^{-1/2})}\\ &\times\frac{(1-q_{\alpha^\vee}^{-\epsilon/2}q_{2\alpha^\vee}^{-\epsilon} \alpha(v)^{-\epsilon/2}\alpha(s^\prime)^{-1/2}) (1-\alpha(v)^{-1}\alpha(s^\prime)^{-1})} {(1-q_{\alpha^\vee}^{-1/2}q_{2\alpha^\vee}^{-1}\alpha(v)^{-1/2} \alpha(s^\prime)^{-1/2})(1-\alpha(v)^{-\epsilon}\alpha(s^\prime)^{-1})} \end{split} \end{equation} We remind the reader of the convention Remark \ref{rem:conv}; in particular, the expression $\alpha(s^\prime)^{1/2}$ occurs only if $\alpha/2\in R_0$, in which case this expression stands for $(\alpha/2)(s^\prime)$. If $\alpha/2\not\in R_0$, we should reduce formula (\ref{eq:ana}) to \begin{equation} c_{\alpha,\epsilon}(\sigma_\epsilon(u))c_\alpha(u)^{-1} =\frac{(1-q_{\alpha^\vee}^{-\epsilon}\alpha(v)^{-\epsilon}\alpha(s^\prime)^{-1}) (1-\alpha(v)^{-1}\alpha(s^\prime)^{-1})} {(1-q_{\alpha^\vee}^{-1}\alpha(v)^{-1}\alpha(s^\prime)^{-1}) (1-\alpha(v)^{-\epsilon}\alpha(s^\prime)^{-1})} \end{equation} By conditions (i) and (iii) it is clear that poles and zeroes of these functions will only meet $U$ if $\alpha(s^\prime)=1$ when $\alpha\in R_0\cap R_1$ or $\alpha(s^\prime)=\pm 1$ if $\alpha\in 2R_0$. In these cases the statement we want to prove reduces to the statement that the function \begin{equation} f(x):=\frac{1-\exp(-\epsilon x)}{1-\exp(-x)} \end{equation} is holomorphic and invertible on the domain $x\in p+\alpha(\gamma^\prime+B)$, where $p$ is a real number and $\alpha\in R_0$. By condition (i) both the denominator and the numerator of $f$ have a zero in this domain only at $x=0$ (if this belong to the domain), and this zero is of order $1$ both for the numerator and the denominator. The desired result follows. \end{proof} Recall Theorem \ref{thm:gralg}. This result tells us that the structure of the algebra with coefficients in the locally defined meromorphic functions on $U$ is independent of the root labels. We will now show that the subalgebra with analytic coefficients (defined locally on $U$) is invariant for scaling transformations. \begin{theorem} The map \begin{equation} \begin{split} j_\epsilon:\mathcal{H}^{me}(U)&\mapsto\mathcal{H}_{q^\epsilon}^{me}(U_\epsilon)\\ \sum_{w\in W_0}f_w\iota_w^0&\mapsto \sum_{w\in W_0}(f_w\circ \sigma_{1/\epsilon})\iota_{w,\epsilon}^0 \end{split} \end{equation} \index{j@$j_\epsilon:\mathcal{H}^{me}(U)\mapsto\mathcal{H}_{q^\epsilon}^{me}(U_\epsilon)$, ``scaling'' isomorphism of localized Hecke algebras} defines an isomorphism of ${\mathbb C}$-algebras, with the property that $j_\epsilon(\mathcal{F}^{me}(U))\allowbreak =\mathcal{F}_{q^\epsilon}^{me}(U_\epsilon)$ and $j_\epsilon(\mathcal{A}^{me}(U))=\mathcal{A}^{me}_{q^\epsilon}(U_\epsilon)$. Moreover (and most significantly), $j_\epsilon(\mathcal{H}^{an}(U))=\mathcal{H}_{q^\epsilon}^{an}(U_\epsilon)$. \end{theorem} \begin{proof} The map $j_\epsilon$ as defined above is clearly a ${\mathbb C}$-linear isomorphism by Theorem \ref{thm:gralg}. It is an algebra homomorphism because we have \begin{equation} \begin{split} j_\epsilon(\sum_{u\in W_0}f_u\iota_u^0\sum_{v\in W_0}g_v\iota_v^0)&= j_\epsilon(\sum_{u,v\in W_0}f_ug^u_v\iota_{uv}^0)\\ &= \sum_{u,v\in W_0}(f_u\circ \sigma_{1/\epsilon})(g^u_v\circ \sigma_{1/\epsilon})\iota_{uv,\epsilon}^0\\&= \sum_{u,v\in W_0}(f_u\circ \sigma_{1/\epsilon})(g_v\circ \sigma_{1/\epsilon})^u\iota_{uv,\epsilon}^0\\&= \sum_{u,v\in W_0}(f_u\circ \sigma_{1/\epsilon})\iota_{u,\epsilon}^0(g_v\circ \sigma_{1/\epsilon})\iota_{v,\epsilon}^0\\&= j_\epsilon(\sum_{u\in W_0}f_u\iota_u^0)j_\epsilon(\sum_{v\in W_0}g_v\iota_v^0) \end{split} \end{equation} What remains is the proof that $j_\epsilon(\mathcal{H}^{an}(U))=\mathcal{H}^{an}_{q^\epsilon}(U_\epsilon)$. Notice that $\mathcal{H}^{an}(U)$ is the subalgebra generated by $\mathcal{A}^{an}(U)$ and the elements $T_s$ where $s=s_\alpha$ with $\alpha\in R_1$. The $j_\epsilon$-image of $\mathcal{A}^{an}(U)$ equals $\mathcal{A}^{an}_{q^\epsilon}(U_\epsilon)$ since $\sigma_\epsilon$ is an analytic diffeomorphism. To determine the image of $T_s$ we use formula Lemma 2.27(2) of \cite{EO}, applied to the situation $W_0=\{e,s\}$. This tells us that \begin{equation} (1+T_s)=q_{\alpha^\vee} q_{2\alpha^\vee}c_\alpha(1+\iota_s^0). \end{equation} Hence we see that \begin{equation} \begin{split} j_\epsilon(T_s)&=q_{\alpha^\vee} q_{2\alpha^\vee}(c_\alpha\circ \sigma_{1/\epsilon})(1+\iota_{s,\epsilon}^0)-1\\ &=q^{1-\epsilon}_{\alpha^\vee} q^{1-\epsilon}_{2\alpha^\vee}(c_\alpha\circ \sigma_{1/\epsilon})c_{\alpha,\epsilon}^{-1} (1+T_{s,\epsilon})-1. \end{split} \end{equation} By Lemma \ref{lem:ana} it is clear that this is indeed in $\mathcal{H}^{an}_{q^\epsilon}(U_\epsilon)$, and that these elements together with $\mathcal{A}^{an}_{q^\epsilon}(m_{\epsilon}(U))$ generate $\mathcal{H}^{an}_{q^\epsilon}(U_\epsilon)$. \end{proof} \subsection{Application to the residual algebras} In order to prove that the residual algebras $\overline{\mathcal{H}^t}$ are invariant for the scaling transformation ${\bf q}\to{\bf q}^\epsilon$ it suffices to consider the case $\overline{\mathcal{H}^r}$ for a residual point $r\in T$. This follows from Theorem \ref{thm:mainind}, expressing $\chi_t$ in terms of characters induced from discrete series characters of proper parabolic subalgebras.
When $r=sc\in T$ is a residual point, the state $\chi_r$ has a natural extension to the localized algebras $\mathcal{H}^{an}(U)$ where $U=W_0r\exp(B)$, with $B$ an open ball in $\mathfrak{t}_{\mathbb C}$ satisfying the conditions \ref{cond} with respect to the point $r\in T$. Because the radical $\operatorname{Rad}_r^{an}(U)$ of the bitrace $(x,y)_r:=\chi_r(x^*y)$ on $\mathcal{H}^{an}(U)$ is contained in the maximal ideal $\mathcal{I}_r^{an}(U)$ of functions in the center $\mathcal{Z}^{an}(U)$ which vanish in the orbit $W_0r$, we clearly have \begin{equation} \overline{\mathcal{H}^r}=\mathcal{H}^{an}(U)/\operatorname{Rad}_r^{an}(U). \end{equation} The structure of this algebra as a Hilbert algebra is given by the bitrace defined by $\chi_r$. Therefore, we need to prove independence of $\chi_r$ for the scaling transformation. We start with a simple lemma: \begin{lem} Let $h\in\mathcal{H}^{an}(U)$ We have \begin{equation} \frac{E_{q^\epsilon,\sigma_\epsilon(t)}(j_\epsilon(h))}{q^\epsilon(w_0)\Delta(\sigma_\epsilon(t))} =\frac{E_{t}(h)}{q(w_0)\Delta(t)} \end{equation} \end{lem} \begin{proof} For all $x\in X$ we have \begin{align} E_{q^\epsilon,\sigma_\epsilon(t)}(j_\epsilon(\theta_xh)) &=E_{q^\epsilon,\sigma_\epsilon(t)}(j_\epsilon(\theta_x)j_\epsilon(h))\\\nonumber &=(x\circ \sigma_{1/\epsilon})(\sigma_\epsilon(t)) E_{q^\epsilon,\sigma_\epsilon(t)}(j_\epsilon(h))\\\nonumber &=x(t)E_{q^\epsilon,\sigma_\epsilon(t)}(j_\epsilon(h)), \end{align} showing that the left hand side has the correct eigenvalue for multiplication of $h$ by $\theta_x$ on the left. For the multiplication of $h$ by $\theta_x$ on the right a similar computation holds. This shows, in view of Lemma \ref{lem:eisext} and \cite{EO}, Proposition 2.23(3) that, for regular $t$ and outside the union of all residual cosets, the left and the right hand side are equal up to normalization. But both the left and the right hand side are equal to $1$ if $h=T_e=1$. Hence generically in $t$, we have the desired equality. Since both expressions are holomorphic in $t$, the result extends to all $t\in T$. \end{proof} \begin{lem} Let $\epsilon\in(0,1]$ be given. We have, for all $h\in \mathcal{H}^{an}(U)$, \begin{equation} \chi_{q^\epsilon,\sigma_\epsilon(r)}(j_\epsilon(h))=\chi_r(h). \end{equation} \end{lem} \begin{proof} Take a neighborhood $U=W_0r\exp(B)$ with $B$ satisfying conditions \ref{cond} relative to $r$. Let ${\cup\xi}\in\mathcal{H}_n(U)$ denote the $n$-cycle defined by ${\cup\xi}=\cup_{r^\prime\in W_0r}\xi_{r^\prime}$. In view of Proposition \ref{prop:cycle}, Definition \ref{dfn:Y} and Definition \ref{dfn:chi} we see that, for all $h\in \mathcal{H}$, \begin{equation} \nu(\{W_0r\})\chi_r(h)=\int_{\cup\xi} \left(\frac{E_{t}(h)}{q(w_0)\Delta(t)}\right) \frac{dt}{q(w_0)c(t)c(t^{-1})} \end{equation} Let $r^\prime\in W_0r$. The scaling operation sends the root labels $q$ to $q^\epsilon$, and follows the corresponding path of $\epsilon\to \sigma_\epsilon(r^\prime)$ of the residual point $r^\prime$. Obviously the position of $t_0$ (in equation (\ref{eq:basic})) relative to $\mathcal{L}^{\{\sigma_\epsilon(r^\prime)\}}$ is independent of $\epsilon$. And also, the position of $e$ relative to the facets of the dual configuration $\mathcal{L}_{\{\sigma_\epsilon(r^\prime)\}}$ is independent of $\epsilon$, since the effect of the scaling operation on $\mathcal{L}_{\{r^\prime\}}\subset T_{rs}$ simply amounts to the application of the map $c\to c^\epsilon$. In view of Proposition \ref{prop:t0} and Proposition \ref{prop:cyinv}, we can take the cycle $\sigma_\epsilon({\cup\xi})\in H_n(\sigma_\epsilon(U))$ in order to define the state $\chi_{\sigma_\epsilon(r)}$ of $\mathcal{H}_{q^\epsilon}^{an}(\sigma_\epsilon(U))$. In other words, we have, for $h\in \mathcal{H}^{an}(U)$, \begin{align}\label{eq:start} \nu_{q^\epsilon}(\{&W_0\sigma_\epsilon(r)\})\chi_{q^\epsilon,\sigma_\epsilon(r)}(j_\epsilon(h))\\ \nonumber &=\int_{\sigma_\epsilon({\cup\xi})} \left(\frac{E_{t}(j_\epsilon(h))}{q^\epsilon(w_0)\Delta(t)}\right) \frac{dt}{q^\epsilon(w_0)c_{\epsilon}(t)c_{\epsilon}(t^{-1})}\\\nonumber &=\int_{{\cup\xi}}\left(\frac{E_{\sigma_\epsilon(t)}(j_\epsilon(h))}{q^\epsilon(w_0) \Delta(\sigma_\epsilon(t))}\right) \frac{d(\sigma_\epsilon(t))}{q^\epsilon(w_0)c_{\epsilon} (\sigma_\epsilon(t))c_{\epsilon}(\sigma_\epsilon(t^{-1}))}\\\nonumber &=\int_{{\cup\xi}}\left(\frac{E_{t}(h)}{q(w_0)\Delta(t)}\right) \phi_\epsilon(t)\frac{dt}{q(w_0)c(t)c(t^{-1})}, \end{align} where \begin{equation} \phi_\epsilon(t):=\frac{\epsilon^nq(w_0)c(t)c(t^{-1})} {q^\epsilon(w_0)c_{\epsilon}(\sigma_\epsilon(t))c_{\epsilon}(\sigma_\epsilon(t^{-1}))}. \end{equation} By Lemma \ref{lem:ana}, the function $t\to\phi_\epsilon$ extends, for all $\epsilon\in(0,1]$, to a regular holomorphic function on $U$. Clearly, $\phi_\epsilon$ is $W_0$-invariant. In other words, $\phi_\epsilon$ is an element of $\mathcal{Z}^{an}(U)$. Its value in $W_0r$ can be computed easily, if we keep in mind that the index $i_{\{r^\prime\}}=n$ (by Theorem \ref{thm:equal} applied to the residual coset $r^\prime$). We obtain, by a straightforward computation: \begin{equation} \phi_\epsilon(W_0r)=\frac{m_{q^\epsilon,\{\sigma_\epsilon(r)\}}(r)}{m_r(r)} =\frac{\nu_{q^\epsilon}(\{W_0\sigma_\epsilon(r)\})}{\nu(\{W_0r\})}. \end{equation} We now continue the computation which we began in equation (\ref{eq:start}), using the fact that $\phi_\epsilon\in\mathcal{Z}^{an}(U)$ and the fact that $\chi_r$ extends uniquely to $\mathcal{H}^{an}(U)$ in such a way that for all $\phi\in\mathcal{Z}^{an}(U)$ and $h\in \mathcal{H}^{an}(U)$, $\chi_r(\phi h)=\phi(r)\chi(h)$. We get \begin{align} \nu_{q^\epsilon}(\{W_0\sigma_\epsilon(r)\})\chi_{q^\epsilon,\sigma_\epsilon(r)}(j_\epsilon(h))&= \nu(\{W_0r\})\chi_r(\phi_\epsilon h)\\\nonumber &=\nu_{q^\epsilon}(\{W_0\sigma_\epsilon(r)\})\chi_r(h). \end{align} This gives the desired result. \end{proof} \begin{thm}\label{thm:bch} The ``base change'' isomorphism $j_\epsilon$ induces an isomorphism \begin{equation} \overline{j}_\epsilon:\overline{\mathcal{H}^r} {\tilde{\longrightarrow}}\overline{\mathcal{H}^{\sigma_\epsilon(r)}_{q^\epsilon}} \end{equation} of Hilbert algebras. In particular, the positive constants $d_{\mathcal{R}_P,\delta}$ (in the notation of Theorem \ref{thm:mainind}, see also equation (\ref{eq:plameasure})) (in Corollary \ref{cor:fdim} these constant were denoted by $d_{r,i}$) are independent of ${\bf q}$. \end{thm} \begin{proof} This is an immediate consequence of the previous lemma. \end{proof} \section{Applications and closing remarks}\label{app} \subsection{Formation of L-packets of unipotent representations}\label{sub:uni} Let $F$ be a nonarchimedean local field, and let $G$ be a split simple algebraic group of adjoint type defined over $F$. We denote by $\mathcal{G}$ the group of $F$-rational points in $G$. The finite set of irreducible unipotent discrete series representations of $\mathcal{G}$ is by definition the disjoint union of the discrete series constituents of induced representations of the form $\sigma_{\mathcal{P}}^{\mathcal{G}}$, where $\mathcal{P}$ is a parahoric subgroup of $\mathcal{G}$, $\sigma$ is a cuspidal unipotent representation of the Levi quotient $L:=\mathcal{P}/{\mathcal{U}_\mathcal{P}}$ of $\mathcal{P}$, and the union is taken over a complete set of representatives of conjugacy classes of pairs $(\mathcal{P},\sigma)$.
The formal dimension is an effective tool to partition these unipotent discrete series representations into L-packets. This observation is due to Reeder \cite{Re0}. He conjectured that the formal dimensions of the unipotent discrete series representations of $\mathcal{G}$ within one unipotent discrete series L-packet are proportional, with a rational ratio of proportionality {\it independent} of $F$, and used this in \cite{Re0} to form the unipotent L-packets for groups of small rank.
It is known \cite{M} that the endomorphism algebra $\mathcal{H}(\mathcal{G},\mathcal{P},\sigma)$ of $\sigma_{\mathcal{P}}^{\mathcal{G}}$ has the structure of an affine Hecke algebra, whose root datum and root labels depend only on $\mathcal{P}$. The root labels are integral powers ${\bf q}^{n_a}$ of the cardinality ${\bf q}$ of the residue field of $F$ (cf. \cite{Lu3}), and are explicitly known. Moreover, if we define a trace functional $\operatorname{Tr}$ on $H(\mathcal{G},\mathcal{P},\sigma)$ by $\operatorname{Tr}(f):=\operatorname{Tr}(f(e),V_\sigma)$, this corresponds to the trace $\tau$ studied in this paper by the formula \begin{equation}\label{eq:tr} \operatorname{Tr}=\operatorname{Vol}(\mathcal{P})^{-1} \operatorname{dim}(V_\sigma)\tau. \end{equation} Thus there is a bijection between the set of discrete series representations of $H(\mathcal{G},\mathcal{P},\sigma)$ (in the sense of this paper) and the unipotent discrete series representations arising from the pair $(\mathcal{P},\sigma)$. The formal dimension of such a discrete series representation of $\mathcal{G}$ is then equal to the formal dimension of the corresponding discrete series representation of the affine Hecke algebra $H(\mathcal{G},\mathcal{P},\sigma)$, but with its trace $\operatorname{Tr}$ normalized by equation (\ref{eq:tr}).
In \cite{HOH} we computed the formal dimension of the discrete series representations of the ``anti-spherical'' subalgebra of the affine Hecke algebra, i.e. the commutative subalgebra $e_-\mathcal{H} e_-=e_-\mathcal{Z}$, where $e_-$ denotes the idempotent of $\mathcal{H}_0$ corresponding to the sign representation (see Subsection \ref{sub:range}). The formula we obtained was expressed entirely in terms of the central character of the representation, the root datum and the root labels. We conjectured in \cite{HOH} that our formula would also hold for the full affine Hecke algebra. For the group of type $E_8$, we showed that this, in combination with Reeder's conjecture, leads to a partitioning of the unipotent discrete series L-packets which is in agreement with Lusztig's conjecture \cite{Lu0} for the Langlands parameters of the members of these packets. In other words, the formal dimension seems to be a sufficient criterion to separate the L-packets of unipotent representations in the case $E_8$.
Theorem \ref{thm:mainp} of this paper proves the conjecture in \cite{HOH} mentioned above.
Reeder \cite{Re} proves an exact formula for the formal dimension of the unipotent discrete series representation of all split exceptional groups, based on a result of Schneider and Stuhler \cite{schstu}. In this approach one first represents the formal dimension by an alternating sum of rational functions (depending on the $K$-types (cf. Subsection \ref{sub:types})), rather than the product formula which we have obtained. On the other hand, there are no intractable constants such as the constant $d_\delta$ in our formula. Using his previous work on non-standard intertwining operators for affine Hecke algebra modules and Theorem \ref{thm:nu} of the present paper, Reeder gave the precise partitioning of the unipotent discrete series for exceptional groups into L-packets, in complete agreement with Lusztig's conjecture mentioned above.
Recently, Lusztig \cite{Lu4} established the partitioning of unipotent discrete series representations into L-packets if $G$ is split over an unramified extension of $F$. This is based on a different approach. It is worth mentioning that this classification includes a geometric parametrization of the set $\Delta_{W_0r}$ of discrete series representations with central character $W_0r$ if the affine Hecke algebra arises as an endomorphism algebra of an induced representation of the form $\sigma_{\mathcal{P}}^{\mathcal{G}}$. \subsection{Operator norm estimate and the Schwartz completion}\label{sub:normunif} \subsubsection{Uniform norm estimate} We know that the generators $N_i:=q(s_i)^{-1/2}T_i$ satisfy \begin{equation} \Vert N_i\Vert_o=\operatorname{max}\{q(s_i)^{\pm 1/2}\} \end{equation} Therefore we have the trivial estimate $\Vert N_w\Vert_o\leq \operatorname{max}\{q(w)^{\pm 1/2}\}$. By the spectral decomposition it is easy to see that the operator norm $\Vert N_w\Vert_o$ is actually bounded by a {\it polynomial} in $\mathcal{N}(w)$: \begin{theorem}\label{thm:normunif} Let ${\bf q}>1$ be fixed. \begin{enumerate} \item[(i)] There exist constants $C\in{\mathbb R}_+$ and $d\in{\mathbb N}$ such that for all $w\in W$, \begin{equation} \Vert N_w \Vert_o\leq C(1+\mathcal{N}(w))^d \end{equation} \item[(ii)] For a residual point $r\in T$, let $\Delta_{W_0r}:=\Delta_{\mathcal{R},W_0r}$ denote the collection of all discrete series representations with central character $W_0r$. Denote by $\Vert x \Vert_{ds}$ the operator norm of the left multiplication by $x\in\mathcal{H}$, restricted to the finite dimensional subspace \begin{equation} \mathfrak{H}_{ds}:=\oplus_{W_0r} \oplus_{\pi\in\Delta_{W_0r}}\operatorname{End}(\pi)\subset\mathfrak{H}. \end{equation} Then there exist constants $C,\epsilon>0$ such that \begin{equation} \Vert N_w \Vert_{ds} \leq C{\bf q}^{-\epsilon l(w)}. \end{equation} \end{enumerate} \end{theorem} \begin{proof} (i) As was explained in the proof of Lemma \ref{lem:cas}, it is sufficient to prove this statement for $w=x\in (Z_X+Q)\cap X^+$. Recall that in this case $N_x=\theta_x$. We have \begin{equation} \Vert \theta_x\Vert^2_o=\Vert \theta_x^*\theta_x\Vert_o= \sup\{\sigma(\pi(T_{w_0}\theta_{-w_0x}T_{w_0}^{-1}\theta_x)) \mid \pi\in\hat\mathfrak{C}\}, \end{equation} where $\sigma(A)$ denotes the spectral radius of $A$. According to Theorem \ref{thm:mainp}, the spectrum $\hat\mathfrak{C}$ equals the union of the compact sets $\hat\mathfrak{C}_{\mathcal{R}_P,W_Pr,\delta}$. This set is by definition the closure in $\hat\mathfrak{C}$ of the set $\pi(\Gamma_{\mathcal{R}_P,W_Pr,\delta}^{gen})$. It is well known that for any $h\in\mathcal{H}$, the map $\pi\to\Vert \pi(h)\Vert_o$ is lower semi-continuous as a function of $\pi\in\hat\mathfrak{C}$ (cf. \cite{Fell}, VII, Proposition 1.14). Since there are only finitely many triples $(R_P,W_Pr,\delta)$, it is sufficient to show that there exist constants $C$, $d$ such that the spectral radius of \begin{equation} \pi(\mathcal{R}_P,W_Pr,\delta,t^P)(T_{w_0}\theta_{-w_0x}T_{w_0}^{-1}\theta_x) \end{equation} is bounded by $C(1+\mathcal{N}(x))^{d}$, uniformly in $t^P$.
The roots of a monic polynomial are bounded by the sum of the absolute values of the coefficients of the equation (including the top coefficient $1$). Hence the spectral radius of an $m\times m$ matrix $A$ is bounded by a polynomial of degree $m$ in
$\operatorname{max}(A):=\max\{|a_{i,j}|\mid 1\leq i,j\leq m\}$. Since $m\leq |W_0|$ it is sufficient to show that there exists a suitable basis for the parameter family $\pi(\mathcal{R}_P,W_Pr,\delta,t^P)$ (with $t^P\in T^P_u$) of representations, in which the matrix coefficients of the $\theta_x$ (with $x\in (Z_X+ Q)\cap X^+$) are uniformly bounded by $C(1+\mathcal{N}(x))^{d}$ for suitable constants $C$ and $d$.
As in the proof of Proposition \ref{prop:indtemp}, there exists a basis $T_{w_i}\otimes (v_j)$ of $V_\pi=\mathcal{H}(W^P)\otimes V$, the representation space of $\pi(\mathcal{R}_P,W_Pr,\delta,t^P)$, such that the $\theta_x$ ($x\in X$) simultaneously act by means of upper triangular matrices in this basis. Moreover, by Proposition \ref{prop:indtemp} it is clear that the diagonal elements are bounded in norm by $1$ when $x\in X^+$. By the compactness of $T^P_u$ we conclude that there exists, for each $x\in X^+$, an unipotent upper triangular matrix $U_x$ with positive coefficients such that every matrix coefficient of $M_x(t^P):=\pi(\mathcal{R}_P,W_Pr,\delta,t^P)(\theta_x)$ in the above basis is uniformly (in $t^P$) bounded by the corresponding matrix coefficient of $U_x$.
Let us denote by $P$ the set of $n\times n$ matrices with non-negative entries, and introduce the notation
$|A|=(|A_{i,j}|)_{i,j}$ for complex matrices $A$. Introduce a partial ordering in $P$ by defining $A\leq B$ if and only if $B-A\in P$. Since $P$ is a semigroup for matrix multiplication, it is clear that if $A,B$ and $C$ are in $P$ and $A\leq B$, then
$AC\leq BC$. In addition we have the rule $|AB|\leq |A||B|$ for arbitrary complex matrices $A$ and $B$.
Let $x_1,\dots, x_m$ denote a set of $\mathbb{Z}_+$-generators for the cone $Q^+$, and let moreover $x_{m+1},\dots,x_N$ be a basis of $Z_X$. Put $M_{i,\epsilon}(t^P):=M_{\epsilon x_i}(t^P)$ for $1\leq i\leq N$, $\epsilon=\pm 1$ with $\epsilon=1$ if $i\leq m$. We can thus find an upper triangular unipotent matrix $U\in P$ such that for all $i$, $\epsilon$, $t^P$:
$|M_{i,\epsilon}(t^P)|\leq U$. If we write $x=\sum_il_ix_i$ with $l_i\geq 0$ if $i\leq m$, then \begin{align*} \mathcal{N}(x)&=x(2\rho^\vee)+\Vert \sum_{i>m}l_ix_i\Vert\\ &=\sum_{i\leq m}l_ix_i(2\rho^\vee)+\Vert \sum_{i>m}l_ix_i\Vert\\ \end{align*} with $x_i(2\rho^\vee)\geq 1$ if $i\leq m$. From this we see that there exists a constant $K$ independent of $x$ such that
$\alpha:=\sum |l_i|\leq K\mathcal{N}(x)$. Thus, with $\log(U)$ the nilpotent logarithm of $U$, and $l_i=\epsilon(i)|l_i|$: \begin{equation}\label{eq:est} \begin{split} \operatorname{max}(M_x(t^P))&=
\operatorname{max}(|M_1^{l_1}(t^P)\dots M^{l_N}_N(t^P)|)\\ &\leq
\operatorname{max}(|M_{1,\epsilon(1)}^{|l_1|}(t^P)|
\dots|M^{|l_N|}_{N,\epsilon(N)}(t^P)|)\\ &\leq \operatorname{max}(U^\alpha)\\ &\leq \operatorname{max}(\exp(\alpha\log(U)))\\ &\leq \sum_{i=0}^d \operatorname{max}(\log(U)^i)\alpha^i/i!\\ &\leq c_U(1+\mathcal{N}(x))^d\\ \end{split} \end{equation} where $c_U$ is a constant depending on $U$ only, and $d$ is the degree of the polynomial function $\alpha\to\exp({\alpha\log(U)})$. This finishes the proof.
(ii) As in the proof of (i), but we restrict ourselves to the (finitely many) discrete series representations. This implies that we can find $U\in P$ unipotent and $\epsilon>0$ such that for all $i$,
$|M_i|\leq {\bf q}^{-2\epsilon} U$. Inserting this in the inequalities (\ref{eq:est}) we find that the matrix entries of $M_x$ are bounded by $C{\bf q}^{-\epsilon l(x)}$, with $C$ independent of $x\in X^+$. Hence the spectral radius of ${\bf q}^{2\epsilon l(x)}M_x^*M_x$ is uniformly bounded for $x\in X^+$, proving the desired estimate. \end{proof} \begin{cor}\label{cor:onlytemp} $\hat\mathfrak{C}$ consists only of tempered representations. \end{cor} \begin{proof}
The character $\chi_\pi$ of $\pi\in\hat\mathfrak{C}$ is a positive trace, and thus satisfies the inequality $|\chi_\pi(x)|\leq \chi_\pi(1)\Vert x\Vert_o$ by Corollary \ref{cor:contrace}. By Casselman's criterion Lemma \ref{lem:cas} and Theorem \ref{thm:normunif} this implies that $\pi$ is tempered. \end{proof} \begin{prop}\label{prop:onediml} The trivial representation $\pi_{triv}(T_w)=q(w)$ is tempered if and only if the point $r_{triv}\in T_{rs}$ defined by $\forall\alpha\in F_0:\alpha(r_{triv})=q_{\alpha^\vee/2}^{1/2}q_{\alpha^\vee}$ satisfies $r_{triv}\in\overline{T_{rs}^-}$. It is discrete series if and only if $r_{triv}\in T_{rs}^-$. The Steinberg representation $\pi_{St}(T_w)=(-1)^{l(w)}$ is tempered if and only if $r_{triv}^{-1}:=r_{St}\in\overline{T_{rs}^-}$, and discrete series if and only if $r_{St}\in T_{rs}^-$. \end{prop} \begin{proof} This is well known, and follows easily by the remark that the restriction to $\{\theta_x\mid x\in X\}$ of the trivial representation is equal the square root of the Haar modulus $\delta$ (see e.g. \cite{EO}, Corollary 1.5): $\pi_{triv}(\theta_x)=q(x)^{1/2}=\delta^{1/2}(x):=x(r_{triv})$. Now apply the Casselman criteria Lemma \ref{lem:cas}. Similar remarks apply to the case of the Steinberg representation. \end{proof} \begin{cor}\label{cor:onedimlext} If the trivial representation extends to $\mathfrak{C}$ then $r_{triv}\in \overline{T_{rs}^-}$. If the Steinberg representation extends to $\mathfrak{C}$ then $r_{triv}\in\overline{T_{rs}^+}$. \end{cor} \begin{proof} Use Corollary \ref{cor:onlytemp} and Proposition \ref{prop:onediml}. \end{proof} \subsubsection{The Schwartz completion of $\mathcal{H}$}\label{subsub:schwartz} Using Theorem \ref{thm:normunif} we now define a Fr\'echet completion of $\mathcal{H}$. For all $n\in\mathbb{N}$ we define a norm $p_n$ on $\mathcal{H}$ by \begin{equation}
p_n(h)=\max_{w\in W}|(N_w,h)|(1+\mathcal{N}(w))^n \end{equation} Here $\mathcal{N}$ denotes the norm function on $W$ which was defined by equation (\ref{eq:bigL}). \begin{thm} The functions $\tau$, $*$ and the multiplication $\cdot$ of $\mathcal{H}$ are continuous with respect to the family of norms $p_n$. \end{thm} \begin{proof} The continuity of $\tau$ and $*$ is immediate from the definitions. So let us look at the multiplication. Let us write \begin{equation} N_uN_v=\sum_w c_{u,v}^w N_w \end{equation} It is easy to see that $w(0)^0=u(0)^0+v(0)^0$ and that $l(w)\leq l(u)+l(v)$ if $c_{u,v}^w\not=0$. Therefore \begin{equation}\label{eq:1} c_{u,v}^w\not=0\Rightarrow \mathcal{N}(w)\leq \mathcal{N}(u)+\mathcal{N}(v), \end{equation} and by Theorem \ref{thm:normunif}, there exist constants $C,d$ such that for all $u,v$ and $w$: \begin{equation}\label{eq:2}
|c_{u,v}^w|\leq C\min\{(1+\mathcal{N}(u))^d,(1+\mathcal{N}(v))^d\} \end{equation} We put $D_w=\{(u,v)\in W\times W\mid c_{u,v}^w\not=0\}$. It is easy to see that there exists a $b\in\mathbb{N}$ such that \begin{equation} \sum_{u\in W}\frac{1} {(1+\mathcal{N}(u))^b}=\mu<\infty \end{equation} converges. By (\ref{eq:1}) we have that \begin{equation} (1+\mathcal{N}(u))(1+\mathcal{N}(v))\geq (1+\mathcal{N}(w)) \end{equation} for all $(u,v)\in D_w$. Given $n\in \mathbb{N}$, let $k=\max\{b+d,n\}$. Using these remarks we see that for all $0\not=x=\sum x_u N_u$ and $0\not=y=\sum y_v N_v$ in $\mathcal{H}$ the following holds: \begin{align*} \frac{p_n(xy)}{p_{2k}(x)p_{2k}(y)}
&=\frac{1}{p_{2k}(x)p_{2k}(y)}\max_w |(xy,N_w)|(1+\mathcal{N}(w))^n\\
&\leq\max_w\sum_{u,v\in D_w}\frac{|x_u||y_v|}{p_{2k}(x)p_{2k}(y)}
|c_{u,v}^w|(1+\mathcal{N}(w))^n\\ &\leq C\sup_w\sum_{u,v\in D_w}\frac{\min\{(1+\mathcal{N}(u))^d,(1+\mathcal{N}(v))^d\} (1+\mathcal{N}(w))^n}{(1+\mathcal{N}(u))^{2k}(1+\mathcal{N}(v))^{2k}}\\ &\leq C\sup_w\sum_{u,v\in D_w}\frac{(1+\mathcal{N}(u))^d(1+\mathcal{N}(v))^d (1+\mathcal{N}(w))^{n-k}}{(1+\mathcal{N}(u))^{k}(1+\mathcal{N}(v))^{k}}\\ &\leq C\sum_{u,v} (1+\mathcal{N}(u))^{d-k}(1+\mathcal{N}(v))^{d-k}\\ &\leq \mu^2 C \end{align*} This finishes the proof. \end{proof}
Notice that, by Theorem \ref{thm:normunif}, $\|x\|_o\leq Cp_d(x)$ for all $x\in\mathcal{H}$. Therefore the completion of $\mathcal{H}$ with respect to the family of norms $p_n$ will be a subspace of $\mathfrak{C}$. \begin{dfn}\label{dfn:schw} We define the Schwartz completion $\mathfrak{S}$ of $\mathcal{H}$ by \begin{equation} \mathfrak{S}:=\{x=\sum_w x_w N_w\in\mathcal{H}^*\mid p_n(x)<\infty\ \forall n\in\mathbb{N}\}. \end{equation} We have $\mathcal{H}\subset\mathfrak{S}\subset{\mathfrak{C}}$, and $\mathfrak{S}$ is a $*$-subalgebra of $\mathfrak{C}$. $\mathfrak{S}$ is a nuclear Fr\'echet algebra with respect to the topology defined by the family of norms $p_n$. It comes equipped with continuous trace $\tau$ and anti-involution $*$. \end{dfn} \index{S@$\mathfrak{S}$, the Schwartz completion of $\mathcal{H}$} \begin{cor} (of definition) The topological dual $\mathfrak{S}^\prime$ is the space of tempered linear functionals on $\mathcal{H}$. \end{cor} \subsection{A Hilbert algebra isomorphism; abelian subalgebras} \label{sub:range} There exists a trace preserving $*$-algebra isomorphism \begin{align*} i:\mathcal{H}(\mathcal{R},q)&\to\mathcal{H}(\mathcal{R},q^{-1})\\ N_w&\to(-1)^{l(w)}N_w. \end{align*} (see \cite{HOH}). Clearly this induces a $*$-algebra isomorphism, and $\tau$ is respected. Thus $i$ induces an isomorphism of $C^*$-algebras $i:\mathfrak{C}(\mathcal{R},q)\to\mathfrak{C}(\mathcal{R},q^{-1})$, also respecting the traces. The corresponding homeomorphism $\hat{i}:\hat{\mathfrak{C}}(\mathcal{R},q^{-1})\to\hat{\mathfrak{C}}(\mathcal{R},q)$ is therefore Plancherel measure preserving. Note that $i$ restricts to a (Plancherel measure preserving) $*$-isomorphism from the subalgebra $e_+\mathcal{H}(\mathcal{R},q)e_+$ (the spherical subalgebra) to the subalgebra $e_-\mathcal{H}(\mathcal{R},q^{-1})e_-$ (the anti-spherical subalgebra) (see \cite{HOH}). \subsubsection{The Plancherel measure of the center} The commutative subalgebras $e_\pm\mathcal{H}(\mathcal{R},q)e_\pm$ are both isomorphic as algebras to the center $\mathcal{Z}$ via the Satake isomorphism $\mathcal{Z}\ni z\to e_\pm z\in e_\pm\mathcal{H}(\mathcal{R},q)e_\pm$. These subalgebras are commutative Hilbert subalgebras of $\mathcal{H}(\mathcal{R},q)$. They are in general not isomorphic as Hilbert algebras. The Hilbert algebra isomorphism $i$ restricts to an isomorphism $\mathcal{Z}(\mathcal{R},q)\to\mathcal{Z}(\mathcal{R},q^{-1})$. Recall the $\nu$ is the Plancherel measure of $\overline{\mathcal{Z}}\subset \mathfrak{C}$. The above is reflected by the symmetry \begin{cor} $\nu(t,q)=\nu(t,q^{-1})$, \end{cor} \noindent{which can be verified directly} (see Theorem \ref{thm:nu} and Proposition \ref{prop:par}).
The spherical algebra $e_+\mathcal{H}(\mathcal{R},q)e_+$ with $X=P$ (weight lattice) and $q(s)>1$ has a very important basis, uniquely defined by orthogonality and by a triangularity requirement with respect to the standard monomial basis $e_+m_\lambda$ (with $\lambda\in P^+$ and $m_\lambda=\sum_{\mu\in W_0\lambda}t^\mu\in\mathcal{Z}$). In type $A$ these are the Hall-Littlewood polynomials. It would be interesting to study such orthogonal, triangular bases for the center $\mathcal{Z}$ as well. \subsection{Central idempotents of $\mathfrak{C}$ and $\mathfrak{S}$} \label{sub:idem} \index{C@$\hat\mathfrak{C}_\mathcal{O}$, component of $\hat\mathfrak{C}$, the closure of
$[\pi](W(\mathcal{O})\backslash\mathcal{O}^{gen})\subset\hat\mathfrak{C}$|(} Recall that $\hat\mathfrak{C}=\cup\hat\mathfrak{C}_\mathcal{O}$ (union over a complete set of representatives of the association classes orbits). In Theorem \ref{thm:mainp} we have shown that two distinct closed subsets in $\hat\mathfrak{C}$ of the form $\hat\mathfrak{C}_{\mathcal{O}_i}$ ($i=1,2$) intersect in a subset of measure $0$. In fact more is true: according to \cite{DO}, these closed subsets are the components of $\hat\mathfrak{C}$.
There is a bijection $I\to\hat{I}$ between the closed two-sided ideals of $\mathfrak{C}$ and the open subsets of $\hat\mathfrak{C}$. Hence the decomposition of $\hat\mathfrak{C}$ into components $\hat\mathfrak{C}_\mathcal{O}$ corresponds to the decomposition of $1\in\mathfrak{C}$ as a sum of minimal central orthogonal idempotents $e_\mathcal{O}$ of $\mathfrak{C}$.
If $\mathcal{O}=\mathcal{K}_\delta\backslash(\Lambda_\gamma\times\delta)$, $e_\mathcal{O}\in\mathfrak{C}$ is determined by \begin{equation} \mathcal{F}(e_\mathcal{O})(\pi)= \begin{cases} \operatorname{Id_{i(V_\delta)}}\text{\ if\ }[\pi]\in\hat\mathfrak{C}_\mathcal{O}\\ 0\text{\ else\ },\\ \end{cases} \end{equation} \index{e1@$e_\mathcal{O}\in\mathfrak{S}$, central idempotent associated with $\mathcal{O}$} where $\mathcal{F}$ is the isomorphism of Theorem \ref{thm:mainp}.
In fact, the results of \cite{DO} on smooth wave packets even imply that $e_\mathcal{O}\in\mathfrak{S}$. We thus have the following decomposition of the unit element in central, Hermitian, mutually orthogonal, minimal idempotents of $\mathfrak{S}$: \begin{equation} 1=\sum_{\mathcal{O}} e_\mathcal{O}. \end{equation} \begin{theorem}\label{thm:idemexp} \begin{enumerate} \item[(i)] Let $\mathcal{O}=\mathcal{K}_\delta\backslash(\Lambda_\gamma\times\delta)$. Then \begin{equation} \begin{split}
(e_\mathcal{O},e_\mathcal{O})=|W^P| \dim(\delta)\mu_{Pl}(\hat\mathfrak{C}_\mathcal{O}). \end{split} \end{equation} \item[(ii)] These idempotents have the following expansion with respect to any orthonormal basis $B$ of $\mathfrak{H}$: \begin{equation} e_\mathcal{O}=\sum_{b\in B}\chi_\mathcal{O}(b^*)b \end{equation} with \begin{equation} \chi_\mathcal{O}(b):= \int_{\pi\in\hat\mathfrak{C}_\mathcal{O}}\chi_\pi(b)d\mu_{Pl}(\pi). \end{equation} In particular this holds with respect to the orthonormal basis $(N_w)_{w\in W}$. \end{enumerate} \end{theorem} \begin{proof} (i). The dimension of $\pi\in\hat\mathfrak{C}_\mathcal{O}$ equals
$|W^P|\dim(\delta)$ on an open dense subset, and the measure $\mu_{Pl}$ is absolutely continuous with respect to the Haar measure on $\mathcal{O}$. Hence, using the fact that $\mathcal{F}$ is an isometry, we find \begin{equation} \begin{split} (e_\mathcal{O},e_\mathcal{O})&=\int_{\pi\in\hat\mathfrak{C}} \operatorname{Tr}_{V_\pi}(\mathcal{F}(e_\mathcal{O})(\pi))d\mu_{Pl}(\pi)\\
&=|W^P|\dim(\delta)\mu_{Pl}(\hat\mathfrak{C}_\mathcal{O}).\\ \end{split} \end{equation}
(ii). As in (i) we get \begin{equation} \begin{split} \chi_\mathcal{O}(b)&=(e_\mathcal{O},b)\\ &=\int_{\pi\in\hat\mathfrak{C}}\operatorname{Tr}_{V_\pi} (\mathcal{F}(e_\mathcal{O})(\pi)\mathcal{F}(b)(\pi))d\mu_{Pl}(\pi)\\ &=\int_{\pi\in\hat\mathfrak{C}_\mathcal{O}} \operatorname{Tr}_{V_\pi}(\pi(b))d\mu_{Pl}(\pi)\\ &=\int_{\pi\in\hat\mathfrak{C}_\mathcal{O}}\chi_\pi(b)d\mu_{Pl}(\pi).\\ \end{split} \end{equation} \end{proof} The above depends on the results of \cite{DO}, but in the special case of isolated points in $\hat\mathfrak{C}$ these facts are more elementary: \begin{prop} When $\pi\in\Delta_{\mathcal{R}}$, put $\mathcal{O}_\pi=\{[\pi]\}$ for the corresponding component of $\hat\mathfrak{C}$ (an isolated point). Put $e_\pi$ for the corresponding central idempotent of $\mathfrak{C}$. The expansion \begin{equation} e_\pi=\mu_{Pl}(\{\pi\})\sum_{b\in B}\chi_{\pi}(b^*)b \end{equation} is convergent in $\mathfrak{S}$. \end{prop} \begin{proof} The expansion follows as in the above Theorem. It is convergent in $\mathfrak{S}$ because of Corollary \ref{rem:cas} and Definition \ref{dfn:schw}. \end{proof} \begin{cor}\label{cor:normeq} Let $B$ be a Hilbert basis of $\mathfrak{H}$. For any residual point $r$ and discrete series representation $\pi\in\Delta_{W_0r}$ we have \begin{equation}
\sum_{b\in B}|\chi_\pi(b)|^2=\frac{\dim(\pi)}
{|W_0r|{\overline \kappa}_{W_0r}d_{\pi}m_{\{r\}}(r)} \end{equation} where the constant ${\overline \kappa}_{W_0r}\in\mathbb{Q}$ is defined by (\ref{eq:denk}), the constant $d_{\pi}\in\mathbb{R}_+$ by Definition \ref{dfn:resalg} and $m_{\{r\}}(r)$ by Theorem \ref{thm:nu}. \end{cor} \begin{proof}
This follows from $\mu_{Pl}(\pi)=|W_0r|{\overline \kappa}_{W_0r} d_{\pi}m_{\{r\}}(r)$. Note that $d_{\pi}$ is indeed constant (i.e. independent of ${\bf q}$) by Theorem \ref{thm:bch}. Also note Conjecture \ref{rem:ell}. \end{proof} \index{C@$\hat\mathfrak{C}_\mathcal{O}$, component of $\hat\mathfrak{C}$, the closure of
$[\pi](W(\mathcal{O})\backslash\mathcal{O}^{gen})\subset\hat\mathfrak{C}$|)} \subsection{Some examples}\label{sub:example} \subsubsection{The Steinberg representation} A basic example is the Steinberg representation. We obtain a well known expression for the Poincar\' e series of $W$.
This result was first (for equal labels, using Morse theory) derived by Bott \cite{Bott}, and by elementary means by Steinberg \cite{Stein}. Macdonald \cite{Ma2} observed that the arbitrary parameter case can be obtained by Steinberg's method. Macdonald proved formula (\ref{poin}) below, expressing the Poincar\'e polynomial in terms of the roots, in an elementary way using case-by-case verifications. In \cite{Ma3} Macdonald reproved the formula in a uniform way. Also note that the Steinberg representation is a representation of $e_-\mathcal{H} e_-$ (cf. \ref{sub:range}). Hence its formal degree can also be computed by means of the (simpler) techniques of \cite{HOH}.
We assume that $Q\subset X\subset P$. Let $\pi_{St}$ be the Steinberg representation, which is the representation defined by $\pi_{St}(T_w)=(-1)^{l(w)}$. This is a one dimensional discrete series representation provided that $r_{St}\in T_{rs}^-$ (see Proposition \ref{prop:onediml}). Recall that $r_{St}\in T_{rs}$ is defined by $\forall\alpha\in F_0: \alpha(r_{St})=q_{\alpha^\vee/2}^{-1/2}q_{\alpha^\vee}^{-1}$. Generically this residual point is regular. In this regular case, the residual codimension $1$ cosets containing $wr_{St}$ form a normal crossing divisor $D_w$ locally at $wr_{St}$. By Proposition \ref{prop:antidual} we find that $[\xi_{wr_{St}}]\in H_n(U\backslash D_w)$ (with $U$ a small ball around $wr$) is zero if $w\not=e$, and for $w=e$ it is straightforward to see that $[\xi_{r_{St}}]=(-1)^n.C$, where $C$ is the positive generator of $H_n(U\backslash D_e)$. In view of (\ref{eq:tochhandig}) we find that (still in the regular case) \begin{equation}
\kappa_{wr_{St}}=\frac{(-1)^n\delta_{w,e}}{|X:Q|}, \end{equation}
and thus that ${\overline \kappa}_{W_0r_{St}}=(-1)^n|W_0|^{-1}|X:Q|^{-1}$.
Hence if $\pi$ is a discrete series representation with central character $r_{St}$ then, assuming that $r_{St}$ is regular, equation (\ref{eq:rats}) implies that $\pi$ can only have a nonzero weight space for the weight $r_{St}$. But this weight space is one dimensional, so that $\pi=\pi_{St}$. Hence in the regular case, the Steinberg representation is the only member of $\Delta_{W_0r_{St}}$, and thus $d_{\pi_{St}}=1$ in this case.
Inserting the values of these constants, above identity thus specializes to (using the Hilbert bases $(N_w)_{w\in W}$) Macdonald's product formula for the Poincar\'e series of $W$: \begin{equation}\label{poin}
\sum_{w\in W}q(w)^{-1}=\frac{(-1)^n|X:Q|}{m_{\{r_{St}\}}(r_{St})} \end{equation} By continuity, this formula holds in general provided that $r_{St}\in T_{rs}^-$. When we take $q(s)={\bf q}$ for all $s\in S^{\mathrm{aff}}$, and $X=Q$, then we obtain \begin{equation} \sum_{w\in W}{\bf q}^{-l(w)}= \prod_{i=1}^n\frac{({\bf q}^{m_i+1}-1)}{({\bf q}^{m_i}-1)({\bf q}-1)} \end{equation} where $(m_i)$ is the list of exponents of $W_0$. \subsubsection{The subregular unipotent orbit of $Sp_{2n}$} Let $F$ be a nonarchimedean local field, and let ${\bf q}$ be the cardinality of its residue field. Consider the group $\mathcal{G}=\operatorname{SO}_{2n+1}(F)$ for $n\geq 3$. The Langlands dual group of $\mathcal{G}$ is $\hat{G}=\operatorname{Sp}_{2n}(\mathbb{C})$, whose root datum (with basis) we write as $\mathcal{R}=(C_n,{\mathbb Z}^n,B_n,{\mathbb Z}^n,F_0)$, with $F_0=(e_1-e_2,\dots,e_{n-1}-e_n,2e_n)$. We normalize the Haar measure of $\mathcal{G}$ by $\operatorname{Vol}(\mathcal{I})=1$, where $\mathcal{I}\subset\mathcal{G}$ is an Iwahori subgroup. Let us compute the formal dimensions of the irreducible square integrable, Iwahori-spherical representations of $\mathcal{G}$ whose Kazhdan-Lusztig parameters $(r_u,u,\rho)$ (cf. Appendix \ref{KL}) are such that $u$ is the subregular unipotent orbit of $\hat{G}$. Take $r_u\in T_{rs}$ dominant in its $W_0$-orbit. It follows from the discussion in Appendix \ref{KL} that the value $\alpha(r_u)$ with $\alpha\in F_0$ is given by ${\bf q}^{D_u(\alpha)/2}$, where $D_u(\alpha)$ is the weight of $\alpha$ in the Bala-Carter diagram of $u$ (cf. \cite{C}). In our case, the vector of values $\alpha(r_u)$ with $\alpha\in F_0$ is $({\bf q},\dots,{\bf q},1,{\bf q})$, which is a residual point for $(\mathcal{R},q_1)$, where $q_1$ denotes the length multiplicative function $q_1(w)={\bf q}^{l(w)}$.
The Springer correspondence for all classical types has been computed explicitly in \cite{Sho}. We use the description of \cite{Lu5} (see \cite{C}). In our particular case, the partition $\lambda\vdash 2n$ of elementary divisors of $u$ is $\lambda=(2,2n-2)$. Thus the Springer representation corresponding with $(u,1)$ ($1$ denoting the trivial representation of the component group $A(r_u,u)$ (see Appendix \ref{KL})) is the representation $\phi_{(n-1,1)}$ of $W_0$ labeled by the double partition $(n-1,1)$ of $n$. This is the reflection representation of $W_0$.
The component group is equal to $A(r_u,u)\simeq C_2$ (Chapter 13, loc. cit.). Both representations $\pm 1$ of $A(r_u,u)$ are geometric, and one easily finds that the Springer correspondent of $(u,-1)$ is the representation $\phi_{(-,n)}$ of $W_0$. This is the $1$-dimensional representation in which $s_i$ acts by $1$ for $i=1,\dots,n-1$, and in which $s_n$ acts by $-1$.
Let us denote by $\pi_{\pm 1}$ the irreducible square integrable $\mathcal{I}$-spherical representations of $\mathcal{G}$ with the Kazhdan-Lusztig parameters $(r_u,u,\pm 1)$, and put $\rho_{\pm 1,1}:=\pi_{\pm 1}^\mathcal{I}$. The Kazhdan-Lusztig model \cite{KL}, and the explicit results of \cite{Lu0} imply the following: $\rho_{1,1}$ is an $(n+1)$-dimensional discrete series representation of $\mathcal{H}(\mathcal{R},q_1)$, with central character $W_0r_u$, and with restriction to $\mathcal{H}(W_0,q_1)$ whose limit for ${\bf q}\to 1$ is equal to $\phi_{(-,1^n)}\otimes(\phi_{(n-1,1)}\oplus\phi_{(n,-)})$ (here $\phi_{(-,1^n)}$ is the sign representation, and $\phi_{(n,-)}$ is the trivial representation of $W_0$). The representation $\rho_{-1,1}$ is $1$-dimensional, and has $\mathcal{H}(W_0,q_0)$-type corresponding to $\phi_{(-,1^n)}\otimes\phi_{(-,n)}=\phi_{(1^n,-)}$ in the limit ${\bf q}\to 1$.
According to Corollary \ref{cor:fdim} (also see Subsection \ref{sub:uni}) we have \begin{equation}
\operatorname{fdim}(\pi_{\pm 1})=|W_0r_u|{\overline \kappa}_{W_0r_u} d_{\rho_{\pm 1}}m_{\{r_u\}}(r_u), \end{equation} with $m_{\{r_u\}}(r_u)$ equal to the rational function (\ref{eq:m_L}), ${\overline \kappa}_{W_0r_u}\in\mathbb{Q}^\times$ and $d_{\rho_{\pm 1}}\in\mathbb{R}_+$, subject to the condition $(n+1)d_{\rho_{+1}}+d_{\rho_{-1}}=1$.
In general I do not know how to compute the constants ${\overline{\kappa}}_{W_0r_u}$ and $d_{\rho_{\pm 1}}$ (there is a tedious ``algorithm'' for ${\overline \kappa}_{W_0r_u}$ (analogous to \cite{HOH0}), and for $d_{\rho_{\pm 1}}$ not even that). However, in the case of {\it regular} central characters these constants are easy to determine. In the situation at hand we we are able to determine the constants by slightly deforming $q$, since the orbits of residual points that ``emerge'' from $r_u$ (there are two of them, corresponding to the two representations $\rho_{\pm 1,1}$) under such a deformation are regular. Moreover, one can show in the current example that the formal dimensions are continuous under this deformation.
So let us consider {\it generic} root labels $q_f$ (cf. \cite{HOH0}, \cite{Slooten}) defined by $q_f(s_i)={\bf q}$ ($i=1,\dots,n-1$) and $q_f(s_n)={\bf q}^{f}$, where $0<f<2$, $f\not=1$. There are two generic orbits $W_0r_{\pm 1,f}$ of residual points such that $W_0r_{\pm 1,1}=W_0r_u$. By the generic parametrization of \cite{HOH0} of orbits of residual points of the graded affine Hecke algebra (which, by Theorem \ref{thm:lieres}, can also be used for $(\mathcal{R},q_f)$ residual points) of type $C_n$ such a generic orbit corresponds to a partition of $n$. In this case the partitions are $\xi_1=(n-1,1)$ and $\xi_{-1}=1^n$. The (standard basis) coordinates of these residual points (suitably chosen within their $W_0$-orbits) are (by \cite{HOH0}) $r_{1,f}=({\bf q}^{2-n-f/2},{\bf q}^{3-n-f/2},\dots,{\bf q}^{1-f/2})$ and $r_{-1,f}=({\bf q}^{1-n+f/2},{\bf q}^{2-n+f/2},\dots,{\bf q}^{f/2})$. In particular, these are regular orbits of residual points.
By Theorem \ref{thm:support}, for each of these central characters there exists at least $1$ irreducible square integrable representation of $\mathcal{H}$. In addition, it is not difficult to see (cf. \cite{Slooten}) that the residual Hilbert algebra of a {\it regular} orbit of residual points is in fact simple. Thus for $f\not=1$, we find precisely two irreducible square integrable representations $\rho_{\pm 1,f}$, with central characters $W_0r_{\pm 1,f}$.
One checks directly that $r_{-1,f}$ is the $\mathcal{A}$-weight space of a $1$ dimensional (square integrable, by Casselman's criterion) representation where $T_i$ ($i<n$) acts by $-1$, and $T_n$ by ${\bf q}^f$. This is a continuous family of square integrable representations in the parameter $f$ (if $f$ is in the range $0<f<2$ and $n\geq 3$). We call this parameter family $\rho_{-1,f}$.
The other orbit $W_0r_{1,f}$ also carries a continuous parameter family of square integrable representations $\rho_{1,f}$, the twist by the automorphism $i$ (see Subsection \ref{sub:range}) of the affine reflection representation of $\mathcal{H}$ (a representation of dimension $(n+1)$). To see this, we give the following model for the representation (there are several possible constructions one could invoke here, but none of these is obvious (as far as I know)). Our approach here is based on the simplifying circumstance that the representation contains the sign representation of $\mathcal{H}(W_0)$ (is ``anti-spherical'').
We will use the spherical function $\phi(\mu,k)$ of the Yang system (cf. \cite{HOH0}), with $R=C_n$, $k_\alpha=\log(q(s_\alpha))$, and ${\bf{k}}=\log({\bf q})$. Recall that this function depends {\it analytically} on $(\mu,k)$. First we consider $-{\bf{k}}<0$ (the attractive case in \cite{HOH0}), and we consider the residual point $\mu=\log(w_0r_{1,f})=-{\bf{k}}(f/2+n-2,f/2+n-3,\dots,f/2,f/2-1)$. The list of positive roots $\alpha$ such that $\alpha(\mu)=-k_\alpha$ is $\mathcal{L}=(e_1-e_2,\dots,e_{n-1}-e_n,2e_{n-1})$. In order to compute the dimension (in the regular case $f\not=1$) of the graded Hecke module generated by $\phi(\mu,k)$ we have to count the number of exponentials $e^{w\mu}$ which have a nonzero coefficient in $\phi(\mu,k)$. Assuming that $0<f<2, f\not=1$, we see that $\mu$ satisfies the condition of Lemma 3.3 of \cite{HOH0}, and by Remark 3.4 of \cite{HOH0} this shows that $\mu$ is an exponent of $\phi(\mu,k)$. Then $w^{-1}\mu$ is also an exponent iff $w\delta=w(n,n-1,\dots,1)$ satisfies $w\delta(\alpha)>0$ for all $\alpha\in\mathcal{L}$. One easily verifies that this is satisfied iff $w\delta=(n,n-1,\dots,2,\pm 1)$ or $w\delta=(n,n-1,\dots,\hat{j},\dots,1,-j)$ ($j=n,n-1,\dots,2$). Hence the module generated by $\phi(\mu,-k)$ (with $\mu$ as above, and $0<f<2$) is a spherical discrete series module of the graded Hecke algebra, which is irreducible (this is always true, see the discussion above Section 3, loc. cit.), and of dimension $n+1$ if $f\not=1$. Now apply the involution $i$ (see Section 5, loc. cit., and also Subsection \ref{sub:range}) to replace $-{\bf{k}}$ by ${\bf{k}}$, and then integrate the representation (as in \cite{Lu}, Section 9) so obtained to get a representation of $\mathcal{H}$. We obtain a parameter family (depending on $f$ with $0<f<2$) of irreducible square integrable representations generated by an anti-spherical vector, of dimension $n+1$ if $f\not=1$. Now observe that for $f=1$ this representation has to be irreducible of dimension $n+1$ as well, by the classification of the square integrable representations with central character $W_0r_u$ as described above. We call this $n+1$-dimensional family $\rho_{1,f}$. It follows easily from the above discussion that the characters of $\rho_{\pm 1,f}$ are continuous in $0<f<2$, and uniformly square integrable.
Hence we can compute the formal dimension of both representations by taking the limit for $f\to 1$ of the corresponding generic formal dimensions. It is easy to see that \begin{equation} \lim_{f\to 1}m_{\{r_{\pm 1,f}\}}(r_{\pm 1,f}) =\pm\frac{1}{2}m_{\{r_u\}}(r_u). \end{equation} For $f\not=1$ one obviously has $d_{\rho_{\pm}}=1$
and $\kappa_{wr_{\pm 1,f}}=\pm(-1)^n|X:Q|^{-1}=\pm(-1)^n/2$ for all $w$ such that $wr_{\pm 1,f}$ is a weight in $\rho_{\pm 1,f}$, and $=0$ else.
Combining these facts, we find in the limit $f\to1$ that
$|W_0r_u|{\overline \kappa}_{W_0r_u}=(-1)^n(n+2)/4$, and $d_{\rho_{\pm 1}}=1/(n+2)$. Hence both constants $\lambda_{\rho_{\pm 1}}$ are equal to $(-1)^n/4$, which is in accordance with Reeder's conjectural formula (\cite{Re}, equation (0.5)) for the formal dimension (up to a sign). A computation yields: \begin{equation}\label{eq:fdimsubreg} \operatorname{fdim}(\pi_{\pm 1})=\frac{1}{4} \frac{{\bf q}({\bf q}-1)^{n+2}({\bf q}^{n-2}-1)\prod_{i=1}^{n-2}({\bf q}^{2i+1}-1)} {({\bf q}^2-1)({\bf q}^n-1)\prod_{i=1}^{n-1}({\bf q}^{2i}-1)} \end{equation} \begin{rem} It would be interesting to work out the product formula (\ref{eq:fdim}) for formal dimensions (without the precise analysis of the constants $\lambda_\rho$) for classical root systems in general (for ``special parameters'', see \cite{Slooten}), and to express the answer (in the case of real central characters) in terms of the symbol of the Springer correspondent according to the conjecture in \cite{Slooten}. \end{rem} \subsection{$K$-types}\label{sub:types} We touch superficially upon the analogue of the problem of the ``$K$-type decomposition'' of admissible representations of a reductive group for tempered representations of the affine Hecke algebra $\mathcal{H}$. We refer to \cite{Re} for a deep connection between the ``$K$-types'' of an irreducible discrete series representation, and its formal dimension. We refer to \cite{Slooten} for precise conjectures on the $K$-types of the irreducible tempered modules with real central character for affine Hecke algebras of classical type (and general root labels).
The role of $K$ can be played by any maximal finite type Hecke subalgebra of the form $\mathcal{H}(W_J)\subset\mathcal{H}$, with $J\subset F^{\mathrm{aff}}$ a maximal proper subset. Such a subalgebra is a finite dimensional $*$-subalgebra. The restriction of $\tau$ to $\mathcal{H}(W_J)$ is equal to the usual trace of the finite type Hecke algebra $\mathcal{H}(W_J)$, normalized in such a way that $\tau(T_e)=1$.
For $\sigma\in\hat{W_J}$ we denote by $d_{J,\sigma}(q)$ its generic degree with respect to $\mathcal{H}(W_J)$ with label $q|_{W_J}$. Thus we have (\ref{eq:finite}) \begin{equation}
\tau|_{\mathcal{H}(W_J)}=(P_{W_J}(q))^{-1} \sum_{\sigma\in\hat{W_J}}d_{J,\sigma}(q)\chi_\sigma, \end{equation} where $P_{W_J}(q)$ denotes the Poincar\'e polynomial of $W_J$ with respect to the label function $q$ (restricted to $W_J$).
Now observe that the restriction to $\mathcal{H}(W_J)$ of $\pi(\omega)$ is independent of $\omega\in \mathcal{O}$. We denote the multiplicities by $n_\mathcal{O}(\sigma)$, thus \begin{equation}
\chi_{\pi(\omega)}|_{\mathcal{H}(W_J)}=\sum_{\sigma\in\hat{W_J}} n_\mathcal{O}(\sigma)\chi_\sigma. \end{equation} We introduce for $\gamma=(\mathcal{R}_P,W_Pr_P)\in\Gamma$ the following rational functions of $q$: \begin{equation} M_\gamma:=\int_{t\in T^P_u} m^P(r_Pt)d^Pt. \end{equation} Notice that for all orbits of the form $\mathcal{O}=K_\delta\backslash\Lambda_\gamma\times\delta$, \begin{equation} \int_\mathcal{O} m^P(\omega)d^\mathcal{O}\omega=M_\gamma. \end{equation} From the Plancherel decomposition of $\mathcal{H}$ (Theorem \ref{thm:mainp}) we thus obtain the following identities: For all $J\subset F^{\mathrm{aff}}$ a maximal proper subset, and each $\sigma\in\hat{W_J}$, \begin{equation}\label{eq:gendgs} d_{J,\sigma}(q)=P_{W_J}(q) \sum_{\gamma\in\Gamma_a}M_\gamma \sum_{\delta\in\Delta_{\gamma,a}}
|W(\mathcal{O})|^{-1}n_\mathcal{O}(\sigma)\mu_{\mathcal{R}_P,Pl}(K_{P(\gamma)}\delta), \end{equation} where $\Gamma_a$ is a complete set of representatives for the association classes (=$\mathcal{W}$-orbits) in $\Gamma$, $\Delta_{\gamma,a}$ is a complete set of representatives in $\Delta_\gamma$ for the action of $\mathcal{W}(\gamma)$, and where $\mathcal{O}$ denotes the orbit $\mathcal{O}=K_\delta\backslash\Lambda_\gamma\times\delta$ of cuspidal representations of $\mathcal{H}^{P(\gamma)}$ (for a given pair $(\gamma,\delta)\in\Gamma_a\times\Delta_{\gamma,a}$). \begin{ex} It is instructive to verify (\ref{eq:gendgs}) for $R_0=B_2$ (equal label case), both for $X=Q$ and $X=P$, using the discussion in Example \ref{ex:b2}. The residual point $({\bf q},-1)$ for $X=Q$ (notation of Example \ref{ex:b2}) is the most complicated part. This orbit of residual points carries $2$ one-dimensional discrete series representations which are exchanged by the nontrivial affine diagram automorphism. Their direct sum lifts to the two-dimensional irreducible discrete series representation which is carried by the (regular) orbit $({\bf q}^{1/2},-1)$ of residual points for the extended affine Hecke algebra with $X=P$. Using Corollary \ref{cor:normeq} one concludes that the formal dimension of this two-dimensional representation (which is easily computed, since the underlying central character is regular) is equal to the formal dimension of each of the two one-dimensional discrete series in which it decomposes upon restriction to the case $X=Q$. \end{ex} \subsection{A remark on the residual degrees $d_{\pi}$} \label{sub:remarks} We mention one further consequence of Corollary \ref{cor:normeq} regarding the constants $d_{\pi}\in\mathbb{R}_+$. \begin{cor} Assume that the constants $f_s$ in Convention \ref{eq:scale} are integers. Let $r$ be a residual point, and let $\pi\in\Delta_{W_0r}$. Assume that the character values of $\pi$ on $T_w$ are contained in $k[{\bf q}^{1/2},{\bf q}^{-1/2}]$, where $k$ is a subfield of ${\mathbb C}$. Then $d_{\pi}\in\mathbb{R}_+\cap k$. \end{cor} See also Conjecture \ref{rem:ell}; we expect that the $d_{\pi}\in {\mathbb Q}$. \begin{proof} The main step is to show that Casselman's bound of Corollary \ref{rem:cas} becomes uniform in ${\bf q}$ under the assumption. Let $r=sc$ and choose $n\in{\mathbb N}$ such that $s^n=s$. Let us first fix ${\bf q}>1$. Consider the isomorphism of localized Hecke algebras \begin{equation} j_{1/n}:\mathcal{H}^{an}_{q^n}(U)\to\mathcal{H}^{an}(U_{1/n}), \end{equation} where $U=W_0sc^n\exp{B}=W_0r^n\exp{B}$, with $B$ a suitably small ball around the origin in $\mathfrak{t}_\mathbb{C}$. We have, by the assumption that $s^n=s$: \begin{equation} j_{1/n}(\theta_{x,q^n})=\theta_{nx}. \end{equation} On the other hand, for all $s\in F_0$, the eigenvalues of the self adjoint operator $\pi(j_{1/n}(N_{s,{\bf q}^n}))$ are of the form $\pm{\bf q}^{\pm f_sn/2}$. Hence the operator norms of the operators $\pi(j_{1/n}(N_{s,{\bf q}^n}^{\pm 1}))$ are bounded by ${\bf q}^{Mn}$, for a suitable constant $M$.
Given $w\in W$ we can write $w=uxv$ with $x\in X^+$, $u\in W^x$ and $v\in W_0$, where $W^x$ denotes the set of shortest length representatives of the left cosets of the stabilizer $W_x$ of $x$ in $W_0$. If we write $u=s_{i_1}\dots s_{i_k}$ and $v=s_{j_1}\dots s_{j_l}$, we can thus choose signs $e_i$ and $d_j$ such that \begin{equation} N_w=N_{i_1}^{e_1}\dots N_{i_k}^{e_k}\theta_x N_{j_1}^{d_1}\dots N_{j_l}^{d_l}. \end{equation} Let us simply denote this decomposition by $N_w=N_u^e\theta_x N_v^d$.
Now by Theorem \ref{thm:normunif}(ii), and the remark that $j_{1/n}$ intertwines the action of $\mathcal{H}^{an}_{{\bf q}^n}(U)$ on
$\pi|_{{\bf q}^n}$ with that of $\mathcal{H}^{an}(U_{1/n})$ on $\pi|_{{\bf q}}$, we have ($M$ is a constant, not necessarily the same as above): \begin{equation} \begin{split}
|\chi_{\pi,{\bf q}^n}(N_{w,{\bf q}^n})|^2&=|\chi_{\pi,{\bf q}}(j_{1/n}(N_{w,{\bf q}^{n}}))|^2\\
&=|\chi_{\pi,{\bf q}}(j_{1/n}(N_{u,{\bf q}^n}^e N_{v,{\bf q}^n}^d)\theta_{nx,{\bf q}})|^2\\ &\leq\operatorname{dim}(\pi)^2\Vert\pi(j_{1/n}(N_{u,{\bf q}^n}^e N_{v,{\bf q}^n}^d))\Vert^2_o \Vert \theta_{nx,{\bf q}}\Vert_{ds}^2\\ &\leq C{\bf q}^{2n(M-\epsilon l(x))}\\ \end{split} \end{equation}
where $C$ is independent of $w$ and $n$. In particular, this implies that the highest power of ${\bf q}$ in $|\chi_\pi(N_w)|^2= \chi_\pi(N_w)\chi_\pi(N_{w^{-1}})\in k_\mathbb{R}[{\bf q}^{1/2},{\bf q}^{-1/2}]$ tends to $-\infty$ with $l(w)$ (with $k_\mathbb{R}:=k\cap\mathbb{R}$). Hence the left hand side of the equality Corollary \ref{cor:normeq} is a Laurent series in ${\bf q}^{-1/2}$ with coefficients in $k_\mathbb{R}$.
On the other hand, according to Proposition \ref{prop:par}(iv), $m_{\{r\}}(r)^{-1}$ can be expanded as a Laurent series in ${\bf q}^{-1/2}$ with coefficients in $\mathbb{Q}$. The desired result follows. \end{proof} \section{Appendix: Residual Cosets}\label{sub:defn} \subsection{Introduction and quick guide}\label{sub:quick} Our approach to the spectral resolution is through residues of certain rational $n$-forms on a complex torus $T$. In order for our method to work well, we need to have a certain a priori knowledge on the geometric and combinatorial properties of the set of poles of these rational forms. The present section serves to collect such facts about the set of poles, and to classify the collection of ``residual cosets'', the sets of maximal pole order, which will eventually turn out to constitute the projection of support of the Plancherel measure to ${W_0}\backslash T$.
Recall that we have chosen a rational, positive definite, $W_0$-invariant symmetric form on $X$. This defines an isomorphism between $X\otimes_{\mathbb Z} {\mathbb Q}$ and $Y\otimes_{\mathbb Z} {\mathbb Q}$, and thus also a rational, positive definite symmetric form on $Y$. We extend this form to a positive definite Hermitian form on $\mathfrak{t}_{\mathbb C}:=\operatorname{Lie}(T)=Y\otimes_{\mathbb Z}{\mathbb C}$, where $T$ is the complex torus $T=\operatorname{Hom}(X,{\mathbb C}^\times)$. Via the exponential covering map $\exp:\mathfrak{t}_{\mathbb C}\to T$ this determines a distance function on $T$.
Let $q$ be a set of root labels. If $2\alpha\not\in R_{\mathrm{nr}}$ we formally put $q_{\alpha^\vee/2}=1$, and always $q_{\alpha^\vee/2}^{1/2}$ denotes the positive square root of $q_{\alpha^\vee/2}$. Let $L$ be a coset of a subtorus $T^L\subset T$ of $T$. Put $R_L:=\{\alpha\in R_0\mid \alpha(T^L)=1\}$. This is a parabolic subsystem of $R_0$. The corresponding parabolic subgroup of $W_0$ is denoted by $W_L$. Define \begin{equation} R_L^p:=\{\alpha\in R_L\mid \alpha(L)=-q_{\alpha^\vee/2}^{1/2} \ \mathrm{or} \ \alpha(L)=q_{\alpha^\vee/2}^{1/2}q_{\alpha^\vee}\} \end{equation} and \begin{equation} R_L^z:=\{\alpha\in R_L\mid \alpha(L)=\pm 1\}. \end{equation} We write $R_L^{p,ess}=R_L^p\backslash R_L^z$ and $R_L^{z,ess}=R_L^z\backslash R_L^p$. We define an index $i_L$ by \begin{equation}
i_L:=|R_L^p|-|R_L^z|. \end{equation} As a motivation for the somewhat more technical definition in the next subsection, we remark that this index $i_L$ computes the order of the pole along $L$ of the rational $(n,0)$-form \begin{equation} \omega:=\frac{dt}{c(t,q)c(t^{-1},q)}, \end{equation} \index{0y@$\omega=\frac{dt}{c(t,q)c(t^{-1},q)}$, $(n,0)$-form on $T$} which plays a main role in this paper (cf. equation (\ref{eq:basic})). We will find (cf. Corollary \ref{cor:simpledefres}) that for each coset $L$ of a subtorus of $T$, \begin{equation}\label{eq:eq} i_L\leq\operatorname{codim}(L). \end{equation} Suppose that $L$ is a coset such that $i_L<\operatorname{codim}(L)$, and let $T_L$ denote the subtorus orthogonal to $T^L$. Let $C_L$ be a cycle of dimension $\operatorname{dim}(C_L)=\operatorname{codim}(L)$ in a sufficiently small neighborhood of $e$ in $T_L$, and let $C^L$ be any compact cycle in $L\backslash\cup_{L\not\subset L^\prime}L^\prime$ of dimension $\operatorname{dim}(C^L)=\operatorname{dim}(L)$. Then for every homolorphic function $f$ on $T$, \begin{equation}\label{eq:f} \int_{C^L\times C_L}f\omega=0. \end{equation} We call a coset $L$ {\it residual} if $i_L=\operatorname{codim}(L)$. It will turn out that the support of the spectral measure of the restriction of the trace $\tau$ to the center of the Hecke algebra is precisely equal to the union of all the ``tempered forms'' of the residual cosets (see Theorem \ref{thm:support}). The spectral measure arises as a sum of integrals of the form (\ref{eq:f}).
For technical convenience, the Definition \ref{dfn:ressub} of the notion ``residual coset'' in the next subsection is slightly more complicated. We will define the residual cosets by induction on their codimension in $T$, in such a way that the collection of residual cosets is easily amenable to classification. In the next subsection we discuss their elementary properties and show how the classification can be reduced to the case of residual subspaces in the sense of \cite{HOH0}. These residual subspaces were already classified in the paper \cite{HOH0}. By this classification we verify equation (\ref{eq:eq}) (cf. Corollary \ref{cor:simpledefres}). Using Lemma \ref{lem:ats} this implies that the following are equivalent for a coset $L\subset T$: \begin{enumerate} \item[(i)] $L$ is residual (in the sense of Definition \ref{dfn:ressub}). \item[(ii)] $i_L\geq\operatorname{codim}(L)$. \item[(iii)] $i_L=\operatorname{codim}(L)$. \end{enumerate} \subsection{Definition and Classification of Residual Cosets} We give the following recursive definition of the notion {\it residual coset}\label{sub:dfnser}. \begin{dfn}\label{dfn:ressub} A coset $L$ of a subtorus of $T$ is called residual if either $L=T$, or else if there exists a residual coset $M\supset L$ such that $\mathrm{dim}(M)=\mathrm{dim}(L)+1$ and \begin{equation}\label{eq:tech} i_L\geq i_M+1. \end{equation} \end{dfn} \begin{cor}\label{cor:fininv} The collection of residual cosets is a nonempty, finite collection of cosets of algebraic subtori of $T$, closed for the action of the group of automorphisms of the root system preserving $q$ (in particular the elements of $W_0$, but also for example $-\operatorname{Id}$). \end{cor} \begin{proof} By induction on the codimension. In a residual coset $M$ of codimension $k-1$ we find only finitely many cosets $L\subset M$ of codimension $1$ in $M$ with $i_L>i_M$. The invariance is obvious from the invariance of the index function $i_L$. \end{proof} \begin{prop}\label{prop:red} If $L$ is residual, then \begin{itemize} \item[(i)] $R_L^{p,ess}$ spans a subspace $V_L$ of dimension $\operatorname{codim}(L)$ in the ${\mathbb Q}$ vectorspace $V=X\otimes{\mathbb Q}$. \item[(ii)] We have $R_L=V_L\cap R_0$, and the rank of $R_L$ equals $\operatorname{codim}(L)$. \item[(iii)] Put ${}_LX:=V_L\cap X$ and $X^L:=X/{}_LX$. Then $T^L=\{t\in T\mid x(t)=1 \ \forall x\in{}_LX \}= \operatorname{Hom}(X^L,{\mathbb C}^\times)=(T^{W_L})^0$. \item[(iv)] Put $Y_L:=Y\cap{\mathbb Q} R^\vee_L$ and ${}^LX:=Y_L^\perp\cap X$. Let $X_L:=X/{}^LX$. We identify $R_L$ with its image in $X_L$. Let $F_L$ be the basis of $R_L$ such that $F_L\subset R_{0,+}$. Then $\mathcal{R}_L:=(X_L,Y_L,R_L,R_L^\vee,F_L)$ is a root datum. \item[(v)] Put $T_L:=\operatorname{Hom}(X_L,{\mathbb C}^\times)\subset T$ (we identify $T_L$ with its canonical image in $T$). Then $T_L$ is the subtorus in $T$ orthogonal to $L$. Define $K_L:=T^L\cap T_L= Hom(X/({}_LX+{}^LX),{\mathbb C}^\times)\subset T$, a finite subgroup of $T$. The intersection $L\cap T_L$ is a $K_L$-coset consisting of {\it residual points} in $T_L$ with respect to the root datum $\mathcal{R}_L$ and the root label $q_L$ obtained from $q$ by restriction to $R_{L,\mathrm{nr}}^\vee\subset R_\mathrm{nr}^\vee$. When $r_L\in T_L\cap L$, we have $L=r_LT^L$. Such $r_L$ is determined up to multiplication by elements of $K_L$. \end{itemize} \end{prop} \begin{proof} By induction on $\operatorname{codim}(L)$ we may assume that the assertions of (i) and (ii) hold true for $M$ in (\ref{eq:tech}). From the definition we see that $R_L^{p,ess}\backslash R_M^{p,ess}$ is not the empty set. An element $\alpha$ of $R_L^{p,ess}\backslash R_M^{p,ess}$ can not be constant on $M$, and hence $\alpha\not\in R_M=V_M\cap R_0$. Thus $$\operatorname{dim}(V_L)\geq\operatorname{dim}(V_M)+1 =\operatorname{codim}(M)+1=\operatorname{codim}(L).$$ Since also $$V_L\subset \operatorname{Lie}(T^L)^\perp,$$ equality has to hold. Hence $R_L\subset V_L$ and $R_L$ spans $V_L$. Since $R_L$ is parabolic, we conclude that $R_L= V_L\cap R_0$. This proves (i) and (ii). The subgroup $\{t\in T\mid x(t)=1 \ \forall x\in{}_LX \}\subset T$ is isomorphic to $\operatorname{Hom}(X^L,{\mathbb C}^\times)$, which is a torus because $X^L$ is free. By (ii) then, its dimension equals $\operatorname{dim}(T^L)$. It contains $T^L$, hence is equal to $T^L$. It follows that $T^L$ is the connected component of the group of fixed points for $W_L$, proving (iii). The statements (iv) and (v) are trivial. \end{proof}
For later reference we introduce the following notation. A residual coset $L$ determines a parabolic subsystem $R_L\subset R_0$, and we associated with this a root datum $\mathcal{R}_L$. When $\Sigma\subset R_0$ is any root subsystem, {\it not necessarily parabolic}, we associate to $\Sigma$ two new root data, namely $\mathcal{R}^\Sigma:=(X,Y,\Sigma,\Sigma^\vee,F_\Sigma)$ with $F_\Sigma$ determined by the requirement $F_\Sigma\subset R_{0,+}$, and $\mathcal{R}_\Sigma:=(X_\Sigma,Y_\Sigma,\Sigma,\Sigma^\vee,F_\Sigma)$ where the lattice $X\to X_\Sigma$ is the quotient of $X$ by the sublattice orthogonal to $\Sigma^\vee$, and $Y_\Sigma\subset Y$ is the sublattice of elements of $Y$ which are in the ${\mathbb R}$-linear span of $\Sigma^\vee$.
There is an obvious converse to Proposition \ref{prop:red}: \begin{prop}\label{prop:conv} Let $R^\prime\subset R_0$ be a parabolic subsystem of roots, and let $T^L\subset T$ be the subtorus such that $R^\prime=R_L$. Let $T_L\subset T$ be the subtorus whose Lie algebra $\operatorname{Lie}(T_L)$ is spanned by $R_L^\vee$. Let $r\in T_L$ be a residual point with respect to $(\mathcal{R}_L,q_L)$ as in Proposition \ref{prop:red}(v). Then $L:=rT^L$ is a residual coset for $(\mathcal{R},q)$. \end{prop}
The recursive nature of the definition of residual cosets makes it feasible to give a complete classification of them. By Lemma \ref{prop:red}, this classification problem reduces to the classification of the {\it residual points}. In turn, Lusztig \cite{Lu} indicates how the classification of residual points reduces to the classification of residual points in the sense of \cite{HOH0} for certain graded affine Hecke algebras. This classification is known by the results in \cite{HOH0}. Let us explain this in detail. Following \cite{Lu} we call a root datum $\mathcal{R}=(X,Y,R_0,R_0^\vee,F_0)$ {\it primitive} if one of the following conditions is satisfied: \begin{enumerate} \item[(1)] $\forall\alpha\in R_0:\ \alpha^\vee\not\in 2Y$. \item[(2)] There is a unique $\alpha\in F_0$ with $\alpha^\vee\in 2Y$ and $\{w(\alpha)\mid w\in W_0\}$ generates $X$. \end{enumerate} A primitive root datum $\mathcal{R}$ satisfying (2) is of the type $C_n^{\operatorname{aff}}$ ($n\geq 1$), by which we mean that $$\mathcal{R}=(Q(B_n)={\mathbb Z}^n,P(C_n)={\mathbb Z}^n, B_n,C_n,\{e_1-e_2,\dots,e_{n-1}-e_n,e_n\}).$$ By \cite{Lu} we know that every root datum is a direct sum of primitive summands. \begin{prop}\label{prop:lusz} Let $r\in T$ be a residual point, and write $r=sc\in T_uT_{rs}$ for its polar decomposition (with $T_u=\operatorname{Hom}(X,S^1)$ and $T_{rs}=\operatorname{Hom}(X,{\mathbb R}_+)$). The root system \[ R_{s,1}:=\{\alpha\in R_1\mid \alpha(s)=1\} \] \index{R4@$R_{s,0}$ ($R_{s,1}$), roots of $R_0$ ($R_1$) vanishing in $s\in T$} has rank $\operatorname{dim}(T)$. The system \[ R_{s,0}:= \{\alpha\in R_0\mid k\alpha\in R_{s,1}\ \mathrm{for\ some}\ k\in{\mathbb N}\} \] contains both $R_{\{r\}}^{p,ess}$ and $R_{\{r\}}^{z,ess}$, and $r$ is residual with respect to the affine Hecke subalgebra $\mathcal{H}^s\subset \mathcal{H}$ whose root datum is given by $\mathcal{R}^{s}:=(X,Y,R_{s,0},R_{s,0}^\vee,F_{s,0})$ (with $F_{s,0}$ the basis of $R_{s,0}$ contained in $R_{0,+}$). \end{prop} \begin{proof} It is clear from the definitions that $R_{s,0}$ contains $R_{\{r\}}^{p,ess}$ and $R_{\{r\}}^{z,ess}$, and hence has maximal rank. Given a full flag of $\mathcal{R}$-residual subspaces $\{c\}=L_0\subset L_1\subset \dots\subset L_n=T$, satisfying (\ref{eq:tech}) at each level, we see that the sets $R_{L_i}^p$, $R_{L_i}^z$ are contained in $R_{s,0}$. It follows by reverse induction on $i$ (starting with $L_n=T$) that each element of the flag is $\mathcal{R}^{s}$-residual. \end{proof} \begin{lem}\label{lem:order2} Given a residual point $r=sc$, let $s_0\in T_u=\operatorname{Hom}(X,S^1)$ be the element which coincides with $s$ on each primitive summand of type $C_n^{\operatorname{aff}}$ and is trivial on the complement of these summands. Then $s_0$ has at most order $2$. \end{lem} \begin{proof} To see this we may assume that $\mathcal{R}$ is of type $C_n^{\operatorname{aff}}$. Then $R_1$ is of type $C_n$, $s=s_0$, and $R_{s,1}=\{\alpha\in R_1\mid \alpha(s_0)=1\}$, being of maximal rank in $R_1$, is of type $C_k+C_{n-k}$ for some $k$. In particular, $\pm 2e_i\in R_{s,1}$ for all $i=1,\dots,n$. Moreover the index of ${\mathbb Z} R_{s,1}$ in ${\mathbb Z} R_1$ is at most $2$. Thus $s_0$ takes values in $\{\pm 1\}$ on $R_1$, and is trivial on elements of the form $\pm 2 e_i$. It follows that $s_0$ is of order at most $2$ on $X={\mathbb Z}^n$. \end{proof} Denote by $h\in\operatorname{Hom}(Q,S^1)$ the image of $s_0$ in $\operatorname{Hom}(Q,S^1)$. Choose root labels $k_\alpha=k_{s,\alpha}\in {\mathbb R}$ with $\alpha\in R_{s,0}$ by the requirement ($k_\alpha$ depends on the image of $s$ in $\operatorname{Hom}(Q,S^1)$, but we suppress this in the notation if there is no danger of confusion) \begin{equation}\label{eq:log} \begin{split} \mathrm{e}^{k_\alpha}=&q_{\alpha^\vee}^{h(\alpha)/2}q_{\alpha^\vee+1}^{1/2}\\ =&\begin{cases} q_{\alpha^\vee/2}^{1/2}q_{\alpha^\vee}&\mathrm{\ if\ }h(\alpha)=+1\\ q_{\alpha^\vee/2}^{1/2}&\mathrm{\ if\ }h(\alpha)=-1\\ \end{cases} \end{split} \end{equation} \begin{thm}\label{thm:lieres} Let $r=sc$ be a $(\mathcal{R},q)$-residual point. Then $\gamma:=\log(c)\in\mathfrak{t}:=\operatorname{Lie}(T_{rs})$ is a residual point in the sense of \cite{HOH0} for the graded Hecke algebra $H^{s}={\mathbb C}[W(R_{s,0})]\otimes \operatorname{Sym}(\mathfrak{t})$ with root system $R_{s,0}$ and root labels $k_{s}:=(k_{s,\alpha})_{\alpha\in R_{s,0}}$. This means explicitly that there exists a full flag of affine linear subspaces $\{\gamma\}=\mathfrak{l}_n\subset\mathfrak{l}_{n-1} \subset\dots \subset\mathfrak{l}_0=\mathfrak{t}$ such that the sequence \begin{equation}
i_{s,\mathfrak{l}_i}:=|R_{s,0,i}^p|-|R_{s,0,i}^z| \end{equation} is strictly increasing, where \begin{equation} R_{s,0,i}^p=\{\alpha\in R_{s,0}\mid \alpha(\mathfrak{l}_i)=k_{s,\alpha}\}, \end{equation} and \begin{equation} R_{s,0,i}^z=\{\alpha\in R_{s,0}\mid \alpha(\mathfrak{l}_i)=0\}. \end{equation} Conversely, given a $s\in T_u$ such that $R_{s,1}\subset R_1$ has rank equal to $\operatorname{rank}(X)$, and a residual point $\gamma\in\mathfrak{t}$ for the root system $R_{s,0}$ with labels $(k_{s,\alpha})$ defined by (\ref{eq:log}), the point $r:=s\exp{\gamma}$ is $(\mathcal{R},q)$-residual. This sets up a $1-1$ correspondence between $W_0$-orbits of $(\mathcal{R},q)$- residual points and the collection of pairs $(s,\gamma)$ where $s$ runs over the $W_0$-orbits of elements of $T_u$ such that $R_{s,1}$ has rank equal to $\operatorname{rank}(X)$, and $\gamma\in\mathfrak{t}$ runs over the $W(R_{s,0})$-orbits of residual points (in the sense of \cite{HOH0}) for $R_{s,0}$ with the labels $k_{s}$. \end{thm} \begin{proof} Straightforward from the definitions. \end{proof} For convenience we include the following lemma: \begin{lemma}\label{lem:conj} If the rank of $R_0$ equals the rank of $X$ (a necessary condition for existence of residual points!), the $W_0$-orbits of points $s\in T_u$ such that $R_{s,1}\subset R_1$ has maximal rank correspond $1-1$ to the $\operatorname{Hom}(P(R_1)/X,S^1)\simeq Y/Q(R_1^\vee)$-orbits on the affine Dynkin diagram $R^{(1)}_1$. In particular, $R_{s,1}$ only depends on the corresponding $P(R_1^\vee)/Q(R_1^\vee)$-orbit of vertices of $R^{(1)}_1$. \end{lemma} \begin{proof} In the compact torus $\operatorname{Hom}(P(R_1),S^1)$, the $W_0$-orbits of such points correspond to the vertices of the fundamental alcove for the action of the affine Weyl group $W_0\ltimes 2\pi iQ(R_1^\vee)$ on $Y\otimes 2\pi i{\mathbb R}$. Now we have to restrict to $X\subset P(R_1)$. \end{proof} With the results of this subsection at hand, the classification of residual cosets is now reduced to the classification of residual subspaces as was given in \cite{HOH0}. \begin{ex}\label{ex:b2} Let $R_0=B_2=\{\pm e_1,\pm e_2,\pm e_1\pm e_2\}$ with basis $\alpha_1=e_1-e_2$, $\alpha_2=e_2$, and let $X=Q={\mathbb Z}^2$ (this is $C_2^{\mathrm{aff}}$). Assume that $q(s_i)={\bf q}$ for $i=0,1,2$. Then $R_1=\{\pm 2e_1,\pm 2e_2,\pm e_1\pm e_2\}$ and thus $X=P(R_1)$. We use $(\alpha_1,\alpha_2)$ as a basis of $X$ (so a point $t\in T$ is represented by $(\alpha_1(t),\alpha_2(t))$). The orbits of points $s\in T_u$ such that $R_{s,1}$ has rank 2 are represented by $(1,1),(1,-1)$ and $(-1,1)$. The latter point corresponds to $R_{s,0}=\{\pm e_1,\pm e_2\}$, but since it has value $-1$ on $\pm e_1$, the labels of $\pm e_1$ are equal to $1$ (by (\ref{eq:log})). Therefore there are no residual points associated with $(-1,1)$. The other two points each give one orbit of residual points, namely $({\bf q},{\bf q})$ and $({\bf q},-1)$.
In addition we have $2$ orbits of one-dimensional residual cosets (with $K_{\{1\}}=C_2$ and $K_{\{2\}}=1$), and finally the principal two-dimensional one, $T$.
Let us now consider $R_0=B_2$ with the lattice $X=P$ and again $q(s_i)={\bf q}$ for $i=0,1,2$. We take $(\alpha_1/2,\alpha_2)$ as a basis for $P$. Now $R_1=R_0$, and thus $X=P(R_1)$. So again we have $3$ orbits of points $s$ for which the rank of $R_{s,0}$ is $2$, namely $(1,1),(1,-1)$ and $(-1,1)$. Each corresponds to a (regular) orbit of residual points: $({\bf q}^{1/2},{\bf q}),({\bf q}^{1/2},-1)$, and $(-{\bf q}^{1/2},{\bf q})$.
In addition there are $3$ one-dimensional residual cosets, $2$ associated with $P=\{1\}$ (with $K_{\{1\}}=1$) and $1$ with $P=\{2\}$ (with $K_{\{2\}}=C_2$). Finally we have the principal residual coset $T$. \end{ex} \subsection{Properties of residual and tempered cosets}\label{sub:resiprop} In the derivation of the Plancherel formula of the affine Hecke algebra, the following properties of residual cosets will play a crucial role. \begin{theorem}\label{thm:equal} For each residual coset $L\subset T$ we have \begin{equation} i_L=\operatorname{codim}(L). \end{equation} In other words, for every inclusion $L\subset M$ of residual cosets with $\operatorname{dim}(L)=\operatorname{dim}(M)-1$, the inequality (\ref{eq:tech}) is actually an equality. \end{theorem} \begin{proof} Unfortunately, I have no classification free proof of this fact. With the classification of residual subspaces at hand it can be checked on a case-by-case basis. By the previous subsection (Proposition \ref{prop:red} and Theorem \ref{thm:lieres}) the verification reduces to the case of residual points for graded affine Hecke algebras. In \cite{HOH0} (cf. Theorem 3.9) this matter was verified. \end{proof} Theorem \ref{thm:equal} has important consequences, as we will see later. At this point we show that the definition of residual cosets can be simplified as a consequence of Theorem \ref{thm:equal}. We begin with a simple lemma: \begin{lem}\label{lem:ats} Let $V$ be a complex vector space of dimension $n$, and suppose that $\mathcal{L}$ is the intersection lattice of a set $\mathcal{P}$ of linear hyperplanes in $V$. Assume that each hyperplane $H\in\mathcal{P}$ comes with a multiplicity $m_H\in{\mathbb Z}$, and define the multiplicity $m_L$ for $L\in\mathcal{L}$ by $m_L:=\sum m_H$, where the sum is taken over the hyperplanes $H\in \mathcal{P}$ such that $L\subset H$. Assume that $\{0\}\in\mathcal{L}$ and that $m_{\{0\}}\geq n$. Then there exists a full flag of subspaces $V=V_0\supset V_1\dots\supset V_n=\{0\}$ such that $m_k:=m_{V_k}\geq k$. \end{lem} \begin{proof} We construct the sequence inductively, starting with $V_0$. Suppose we already constructed the flag up to $V_k$, with $k\leq n-2$. Let $\mathcal{P}_k\subset\mathcal{L}$ denote the set of elements of $\mathcal{L}$ of dimension $n-k-1$ contained in $V_k$, and let $N_k$ denote the cardinality of $\mathcal{P}_k$. By assumption, $N_k\geq n-k\geq 2$. Since every $H\in\mathcal{P}$ either contains $V_k$ or intersects $V_k$ in an element of $\mathcal{P}_k$, we have \begin{equation} \sum_{L\in\mathcal{P}_k}(m_L-m_k)=m_n-m_k. \end{equation} Assume that $\forall L\in\mathcal{P}_k:\ m_L\leq k$. Then, because $m_k\geq k$ and $N_k\geq 2$, \begin{equation} m_n\leq kN_k+(1-N_k)m_k\leq k\leq n-2, \end{equation} contradicting the assumption $m_n\geq n$. Hence there exists a $L\in\mathcal{P}_k$ with $m_L\geq k+1$, which we can define to be $V_{k+1}$. \end{proof} \begin{cor}\label{cor:simpledefres} For every coset $L\subset T$ one has $i_L\leq \operatorname{codim}(L)$, and $L$ is residual if and only if $i_L=\operatorname{codim}(L)$. \end{cor} \begin{proof} Define $\mathcal{P}$ to be the (multi-)set of codimension $1$ cosets of $T$ arising as connected components of the following codimension $1$ sets: \begin{equation} \begin{split} &L^+_{\alpha,1}:=\{t\in T\mid\alpha(t)=q_{\alpha^\vee}q_{\alpha^\vee/2}^{1/2}\}\\ &L^+_{\alpha,2}:=\{t\in T\mid\alpha(t)=-q_{\alpha^\vee/2}^{1/2}\}\\ &L^-_{\alpha,1}:=\{t\in T\mid\alpha(t)=1\}\\ &L^-_{\alpha,2}:=\{t\in T\mid\alpha(t)=-1\}\\ \end{split} \end{equation} Here $\alpha\in R_0$, and $q_{\alpha^\vee/2}=1$ when $2\alpha\not\in R_1$. We give the components of $L^+_{\alpha,1}$, $L^+_{\alpha,1}$ the index $+1$, and we give the components of $L^-_{\alpha,1}$, $L^-_{\alpha,1}$ index $-1$.
Suppose that $L$ is any coset of a subtorus $T^L$ in $T$. Then $i_L$ is equal to the sum of the indices the elements of $\mathcal{P}$ containing $L$.
Assume that $i_L\geq\operatorname{codim}(L)=k$. By Lemma \ref{lem:ats} there exists a sequence $L\subset L_{k-e}\subset L_{k-e-1}\dots\subset L_0=T$ of components of intersections of elements of $\mathcal{P}$ such that $i_{L_{k-e}}=i_L\geq k$ and $i_{L_j}\geq j=\operatorname{codim}(L_j)$ (we did not assume that $L$ is a component of an intersection of elements in the multiset $\mathcal{P}$, hence $e>0$ may occur). If $k(0)$ is the smallest index such that $i_{L_{k(0)}}>k(0)$, then $L_{k(0)}$ is by definition residual, and thus violates Theorem \ref{thm:equal}. Hence such $k(0)$ does not exist and we conclude that $i_{L_k}=k$ for all $k$. This proves that $e=0$ and that $L$ is residual. \end{proof} \begin{rem} This solves the question raised in Remark 3.11 of \cite{HOH0}. \end{rem} \begin{theorem}\label{thm:ster} \begin{enumerate} \item[(i)] Let $R_0$ be indecomposable, and let $r=c$ be a real residual point in $\overline{T_{rs,+}}$. If $\omega:T_{rs}\to T_{rs}$ is a homomorphism which acts on the root system $R_0$ by means of a diagram automorphism of $F_0$, then $\omega(r)=r$. \item[(ii)] Define $*:T\to T$ by $x(t^*)=\overline{x(t)^{-1}}$ \index{*@$*$!$t\to t^*$, anti-holomorphic involution on $T$}. If $r=cs\in T$ is a residual point, then $r^*\in W(R_{s,0})r$. \item[(iii)] If $r=sc$ is a residual point, then the values $\alpha(c)$ of the roots $\alpha\in R_0$ on $c$ are in the subgroup of $\mathbb{R}_+$ generated by the positive square roots of the root labels $q_{\alpha^\vee}$, with $\alpha\in R_{nr}$. \end{enumerate} \end{theorem} \begin{proof} (i). If $R_0$ allows a nontrivial diagram automorphism then $R_0$ is simply laced. So we are in the situation of the Bala-Carter classification of distinguished weighted Dynkin diagrams. A glance at the tables of section 5.9 of \cite{C} shows that this fact holds true.
(ii). This is a consequence of (i), since $*:sT_{rs}\to sT_{rs}$ acts on $R_{s,0}$ by means of an automorphism (see also \cite{HOH0}, Theorem 3.10) which acts trivially on the set of indecomposable summands of $R_{s,0}$.
(iii). For this fact I have also no other proof to offer than a case-by-case checking, using the results of this section and the list of real residual points from \cite{HOH0}. The amount of work reduces a lot by the remark that it is well known in the simply laced cases (see Corollary \ref{cor:cor} of the appendix Section \ref{KL}).
In the classical cases other than $C_n^{\text{aff}}$, it follows from a well known theorem of Borel and de Siebenthal \cite{BS} that the index of $Q(R_{s,0})\subset Q$ is at most $2$. Hence the desired result follows if we verify that for {\it real} residual points of the classical root systems, the values $\alpha(c)$ are in the subgroup of $\mathbb{R}_+$ generated by the root labels, which is direct from the classification lists in \cite{HOH0}.
For the real points of $C_n^{\text{aff}}$ it is also immediate from the above and (\ref{eq:log}). For nonreal points $r=sc$ we look at (the proof of) Lemma \ref{lem:order2}. If $s$ has order $2$, then $R_{s,1}$ is of type $C_{k}+C_{n-k}$. We need to check the values of the roots $e_n$ and $e_k-e_{k+1}$ on $c$ in this case. But the roots $2e_i$ are in $R_{s,1}$, and take rational values in the labels $q_{\alpha^\vee}$ ($\alpha\in R_{nr}$) on $c$.
The real residual points of $F_4$ are all rational in the root labels (see \cite{HOH0}). Again using the Theorem of Borel and de Siebenthal, we need to check in addition the nonreal residual points $r=sc$ with $R_{s,0}=A_2\times A_2\subset F_4$ (generating a lattice of index $3$ in $Q(F_4)$) and $R_{s,0}=A_3\times A_1\subset F_4$ (index $4$). These cases can be checked without difficulty.
In the case of $G_2$, there are generically $3$ real residual points, two of which have rational coordinates and one has rational coordinates only in the square roots of the labels. In addition there are two nonreal residual points $r=sc$ for $G_2$, which are easily checked. (We need to check only the case with $R_{s,0}=A_2$ (index $3$ in $Q(G_2)$)). \end{proof} \begin{rem} In fact the result (ii) of the previous Theorem will also turn out to be a consequence of Theorem \ref{thm:support}, in view of Proposition \ref{prop:im}. \end{rem} \begin{dfn} Let $L$ be a residual coset, and write $L=r_LT^L$ with $r_L\in T_L\cap L$. This is determined up to multiplication of $r_L$ by elements of the finite group $K_L=T_L\cap T^L$. Write $r_L=s_Lc_L$ with $s_L\in T_{L,u}$ and $c_L\in T_{L,rs}$. We call $c_L$ the ``center'' of $L$, and we call $L^{temp}:=r_LT^L_u$ the tempered compact form of $L$ (both notions are independent of the choice of $r_L$, since $K_L\subset T^L_u$). The cosets of the form $L^{temp}$ in $T$ will be called ``tempered residual cosets''. \end{dfn} \index{c1@$c_L$, center of $L$} \index{L@$L^{temp}$, tempered residual coset} \begin{theorem}\label{thm:nonnest} Suppose that $L\subset M$ are two residual cosets. Write $L=r_LT^L=s_Lc_LT^L$ and $M=r_MT^M=s_Mc_MT^M$ as before. If $c_L=c_M$ then $L=M$. \end{theorem} \begin{proof} According to Proposition \ref{prop:regequiv}, $c_L=c_M\Leftrightarrow e\in c_L^{-1}T_{M,rs}:=M_L\Leftrightarrow L^{temp}\subset M^{temp}$. Hence the proof reduces to Remark 3.14 of \cite{HOH0}, or can be proved directly in our setup in the same way, cf. Remark \ref{rem:smoothnest}. \end{proof} Theorem \ref{thm:nonnest} shows that a tempered coset can not be a subset of a strictly larger tempered coset. In fact even more is true: \begin{thm}\label{thm:nonint}(Slooten \cite{Klaas} (cf. \cite{Slooten} for the classical cases)) Let $L_1$ and $L_2$ be residual subspaces. If $L_1^{temp}\cap L_2^{temp}\not=\emptyset$ then $L_1=w(L_2)$ for some $w\in W_0$. \end{thm} We will not use this result in this paper, but it is important for the combinatorial fine structure of the spectrum of $\mathfrak{C}$. We note that the proof of this statement reduces easily to the case of two residual subspaces (in the sense of \cite{HOH0}) with the same center. This reduces the statement of the theorem to the problem in Remark 3.12 of \cite{HOH0}. This problem was solved by Slooten \cite{Klaas}. \section{Appendix: Kazhdan-Lusztig parameters}\label{KL} Let $F$ be a p-adic field. Let $\mathcal G$ be the group of $F$-rational points of a split semisimple algebraic group of adjoint type over $F$, and let $\mathcal{I}$ be an Iwahori subgroup of $\mathcal G$. The centralizer algebra of the representation of $\mathcal G$ induced from the trivial representation of $\mathcal{I}$ is isomorphic to an affine Hecke algebra $\mathcal{H}$ with ``equal labels'', that is, the labels are given as in Convention \ref{eq:scale} with ${\bf q}$ equal to the cardinality of the residue field of $F$, and the exponents $f_s$ all equal to $1$. Moreover, the lattice $X$ is equal to the weight lattice of $R_0$ in this case. The Langlands dual group $G$ is the simply connected semisimple group with root system $R_0$, and the torus $T$ can be viewed as a maximal torus in $G$.
In this situation Kazhdan and Lusztig \cite{KL} have given a complete classification of the irreducible representations of $\mathcal{H}$, and also of the tempered and square integrable irreducible representations. Let us explain the connection with residual cosets explicitly.
We assume that we are in the ``equal label case'' in this subsection, unless stated otherwise. We put $k=\log({\bf q})/2$. Let $G$ be a connected semisimple group over ${\mathbb C}$, with fixed maximal torus $T=\operatorname{Hom}(X,{\mathbb C}^\times)$. We make no assumption on the isogeny class of $G$ yet. \begin{prop}\label{prop:cor} \begin{enumerate} \item[(i)] If $r$ is a residual point with polar decomposition $r=sc=s\exp(\gamma)\in T_uT_{rs}$ and $\gamma$ dominant, then the centralizer $C_{\mathfrak{g}}(s)$ of $s$ in $\mathfrak{g}:=\operatorname{Lie}(G)$ is a semisimple subalgebra of $\mathfrak{g}$ of rank equal to $\operatorname{rk}(\mathfrak{g})$, and $\gamma/k$ is the weighted Dynkin diagram (cf. \cite{C}) of a distinguished nilpotent class of $C_\mathfrak{g}(s)$. \item[(ii)] Conversely, let $s\in T_u$ be such that the centralizer algebra $C_{\mathfrak{g}}(s)$ is semisimple and let $e\in C_{\mathfrak{g}}(s)$ be a distinguished nilpotent element. If $h$ denotes the weighted Dynkin diagram of $e$ then $r=sc$ with $c:=\exp(kh)$ is a residual point. \item[(iii)] The above maps define a $1-1$ correspondence between $W_0$-orbits of residual points on the one hand, and conjugacy classes of pairs $(s,e)$ with $s\in G$ semisimple such that $C_{\mathfrak{g}}(s)$ is semisimple, and $e$ a distinguished nilpotent element in $C_{\mathfrak{g}}(s)$. \item[(iv)] Likewise there is a $1-1$ correspondence between $W_0$-orbits of residual points and conjugacy classes of pairs $(s,u)$ with $C_G(s)$ semisimple and $u$ a distinguished unipotent element of $C_G(s)^0$. \end{enumerate} \end{prop} \begin{proof} (i). We already saw in Appendix \ref{sub:defn} that the rank of $C_\mathfrak{g}(s)$ is indeed maximal. So we are reduced to the case $s=1$. Let $\langle{\bf q}\rangle$ denote the group of integer powers of ${\bf q}$, and denote by $R_{{\bf q}}\subset R_0$ the root subsystem of roots $\alpha\in R_0$ such that $\alpha(c)\in\langle{\bf q}\rangle$. Now $R_{{\bf q}}$ is a root subsystem of rank equal to $\operatorname{rk}(R_0)$, with the property that $\forall \alpha,\beta\in R_{{\bf q}}$ such that $\alpha+\beta\in R_0$ we have $\alpha+\beta\in R_{{\bf q}}$. Of course, $c$ is a residual point of $R_{{\bf q}}$. By an elementary result of Borel and De Siebenthal there exists a finite subgroup $Z\subset T_u$ such that $C_\mathfrak{g}(Z)$ is semisimple with root system $R_{{\bf q}}$.
We claim that for every simple root $\alpha$ of $R_{{\bf q}}$ we have $\alpha(c)=1$ or $\alpha(c)={{\bf q}}$. To see this, observe that all the roots $\alpha\in R_{{\bf q}}$ with $\alpha(c)={{\bf q}}$ are in the parabolic system obtained from $R_{{\bf q}}$ by omitting the simple roots $\alpha$ such that $\alpha(c)={\bf q}^{l}$ with $l>1$. If this would be a proper parabolic subsystem, $c$ would violate Theorem \ref{thm:equal} in this parabolic. This proves the claim.
Define the element $h:=\gamma/k$. Note that $h$ belongs to $2P(R_{{\bf q}}^\vee)$ by the previous remarks. Consider the grading of $R_{{\bf q}}$ given by this element, and define a standard parabolic subalgebra $\mathfrak{p}$ of $C_\mathfrak{g}(Z)$ by \[ \mathfrak{p}:= \mathfrak{t}\oplus\sum_{\{\alpha\in R_{{\bf q}}:\alpha(h)\geq 0\}}\mathfrak{g}_\alpha= \sum_{i\geq 0}C_\mathfrak{g}(Z)(i). \] Its nilpotent radical $\mathfrak{n}$ is \[ \mathfrak{n}:=\sum_{i\geq 2}C_\mathfrak{g}(Z)(i), \] and by the definition of residual points we see that $P\subset C_\mathfrak{g}(Z)$ is a {\it distinguished parabolic subalgebra} (see \cite{C}, Corollary 5.8.3.). According to (\cite{C}, Proposition 5.8.8.) we can choose $e\in\mathfrak{n}(2)$ in the Richardson class associated with $\mathfrak{p}$, and $f\in C_\mathfrak{g}(Z)(-2)$, such that $(f,h,e)$ form a $\mathfrak{sl}_2$-triple in $C_\mathfrak{g}(Z)$. By $\mathfrak{sl}_2$ representation theory it is now clear that $h\in P(R^\vee_0)$. Consider the grading of $\mathfrak{g}$ and $R_0$ induced by $h$. By definition of $Z$ we see that $\mathfrak{g}(0)=C_\mathfrak{g}(Z)(0)$ and $\mathfrak{g}(2)=C_\mathfrak{g}(Z)(2)$. Hence $e$ is distinguished in $\mathfrak{g}$ by (\cite{C}, Proposition 5.7.5.), proving the desired result. Note also that, by (\cite{C}, Proposition 5.7.6.), in fact $\mathfrak{g}(1)=0$, and hence that $R_{{\bf q}}=R_0$.
(ii). Is immediate from the defining property \[ \operatorname{dim}(C_\mathfrak{g}(s)(0))= \operatorname{dim}(C_\mathfrak{g}(s)(2)) \] of the grading with respect to the Dynkin diagram of a distinguished class.
(iii). Is clear by the well known $1-1$ correspondence between distinguished classes and their Dynkin diagrams.
(iv). The result follows from the well known $1-1$ correspondence between unipotent classes and nilpotent classes for connected semisimple groups over ${\mathbb C}$. \end{proof} \begin{cor}\label{cor:cor} From the proof of Proposition \ref{prop:cor}(i) we see that if $r=sc$ is a residual point, then $\alpha(c)\in\langle{\bf q}^{1/2}\rangle$ for all $\alpha\in R_0$. If $s=1$ we have $\alpha(c)\in\langle{\bf q}\rangle$ for all $\alpha\in R_0$. \end{cor} Let $M\subset T$ be a residual coset. Write $M=rT^M\subset T\subset G$ with $r\in T_M$ as in Proposition \ref{prop:red}. Let $r=sc=s\exp(kh/2)$ be the polar decomposition of $r$ in $T_M$. Let $L_M\subset G$ be the Levi subgroup $L_M:=C_G(T^M)$ and let $L^\prime_M$ denote its semisimple part. By Proposition \ref{prop:red} we see that the root system of $L^\prime_M$ is $R_M$, $T_M$ is a maximal torus of $L^\prime_M$, and the connected center of $L_M$ is $T^M$. Moreover, $r\in T_M$ is a residual point with respect to $R_M$. Thus by Proposition \ref{prop:cor}, $C_{L^\prime_M}(s)$ is semisimple, and there exists a distinguished unipotent element $u=\exp(e)$ in $C_{L^\prime_M}(s)^0$ such that $[h,e]=2e$. This implies that the set $N=N_u$ of all elements $t\in G$ such that \begin{equation}\label{eq:KL} tut^{-1}=u^{{\bf q}}. \end{equation} is of the form $N=rC_G(u)$. The centralizer $C_G(r,u)=C_G(s,c,u)$ is known to be maximal reductive in $C_G(s,u)$, and it contains $T^M$. Its intersection with $L^\prime_M$ is also reductive but, since $u$ is distinguished in $C_{L^\prime_M}(s)^0$, the rank of this intersection is $0$. Hence $L^\prime_M\cap C_G(r,u)$ is finite. We conclude that $T^M$ is a maximal torus in $C_G(s,u)$. Let $u^\prime$ be another unipotent element in $G$ such that $M\subset N^\prime=N_{u^\prime}$ and such that $T^M$ is a maximal torus of $C_G(s,u^\prime)$. We see that $u^\prime\in C_{L^\prime_M}(s)^0$ is distinguished and associated to the Dynkin diagram $h$. Hence $u^\prime$ is conjugate to $u$ in $C_{L^\prime_M}(s)^0$ by an element of $C_{L_M^\prime}(r)$. We have shown: \begin{prop}\label{prop:heen} For each residual coset $M=rT^M=scT^M\subset T$ there exists a unipotent element $u$ such that $tut^{-1}=u^{{\bf q}}$ for all $t\in M$, and such that $T^M$ is a maximal torus of $C_G(s,u)$. This $u$ is an element of $C_{L_M^\prime}(s)$ with $L_M:=C_G(T^M)$, and is distinguished in this semisimple group. It is unique up to conjugation by elements of the reductive group $C_{L_M^\prime}(r)$. \end{prop} Let us consider the converse construction. From now in this subsection we assume that $G$ is simply connected. We will be interested in conjugacy classes of pairs $(t,u)$ with $t$ semisimple and $u$ unipotent, satisfying (\ref{eq:KL}). We choose an element $(t,u)$ in the conjugacy class. By Jacobson-Morozov's theorem there exists a homomorphism \begin{equation}\label{JM} \phi:SL_2({\mathbb C})\mapsto G \end{equation} such that \[ u=\phi \left( \begin{array}{cc} 1&1\\ 0&1\\ \end{array} \right) \] We put \[ c:= \phi \left( \begin{array}{cc} {\bf q}^{1/2}&0\\ 0&{\bf q}^{-1/2}\\ \end{array} \right),\ h:=d\phi \left( \begin{array}{cc} 1&0\\ 0&-1\\ \end{array} \right),\ e:= d\phi \left( \begin{array}{cc} 0&1\\ 0&0\\ \end{array} \right). \] Denote by $C_G(\phi)$ the centralizer of the image of $\phi$. We have $C_G(\phi)=C_G(d\phi)$, and by $\mathfrak{sl}_2$ representation theory we see that $C_G(d\phi)=C_G(h,e)$. Hence $C_G(\phi)=C_G(c,u)$. By \cite{KL}, Section 2, this is a maximal reductive subgroup of $C_G(u)$, and we can choose $\phi$ in such a way that $t\in cC_G(\phi)$. In this case $t$ commutes with $c$, and thus $t_1:=tc^{-1}\in C_G(\phi)$ commutes with $c,t$, and is semisimple. It follows that $C_G(t_1,\phi)=C_G(\phi)\cap C_G(t_1)$ is reductive in $C_G(t_1)$, and contains $t_1$ in its center. According to \cite{KL}, the choice of $\phi$ such that $t_1\in C_G(\phi)$ is unique up to conjugation by elements in $C_G(t,u)$.
By conjugating $(t,u)$ and $\phi$ suitably we can arrange that $\overline{T}:=(T\cap C_G(t_1,\phi))^0$ is a maximal torus of $C_G(t_1,\phi)$. Put $L=C_G(\overline{T})$, a Levi group of $G$. We claim that $L$ is minimal among the Levi groups of $G$ containing $\phi$ and $t_1$. Indeed, if $N$ would be a strictly smaller Levi group of $G$ also containing $\phi$ and $t_1$, then its connected center ${T^N}$ would be a torus contained in $C_G(t_1,\phi)$ on the one hand, but strictly larger than $\overline{T}$ on the other hand. This contradicts the choice of $\overline{T}$, proving the claim. In particular, since the connected center $T^L$ of $L$ satisfies $\overline{T}\subset T^L\subset C_G(t_1,\phi)$, we have the equality $\overline{T}=T^L$.
Note that maximal tori of $L$ are the maximal tori of $G$ containing $T^L$, and these are conjugate under the action of $L$. The derived group $L^\prime$ is simply connected, because the cocharacter lattice $Y_L$ of its torus $T_L$ equals $Y_L=Q(R^\vee_0)\cap{\mathbb Q} R^\vee_L=Q(R^\vee_L)$. Hence, by a well known result of Steinberg, $C:=C_L(t_1)\subset L$ is connected, and reductive. This implies that there exist maximal tori of $C$ containing $t$. Thus there exist maximal tori of $L$ containing both the commuting semisimple elements $t_1$ and $t$. Therefore we may and will assume (after conjugation of $(t,u)$ and $\phi$ by a suitable element of $L$) that $T^L$ and the elements $t_1, t$ are inside $T$.
Both the image of $\phi$ and $t_1$ are contained in $C$. Let $C^\prime\subset L^\prime$ denote its derived group. If the semisimple rank of $C$ would be strictly smaller than that of $L$, there would exist a Levi group $N$ such that $C\subset N\subsetneqq L$, a contradiction. Hence $C^\prime$ has maximal rank in $L^\prime$.
Choose $s_{L}$ in the intersection $t_1T^L\cap L^\prime$. By the above, $s_L$ is in $T_{L,u}$, the compact form of the maximal torus $T_L:=(L^\prime\cap T)^0$ of $L^\prime$. We put $r_{L}=s_Lc\in L^\prime$, and we claim that this is a $R_L$-residual point of $T_L$. By Proposition \ref{prop:cor} this is equivalent to showing that $u$ is a distinguished unipotent element of $C^\prime=C_{L^\prime}(s_L)$. This means that we have to show that $C_{L^\prime}(s_L,\phi)$ does not contain a nontrivial torus. But $L=C_G(T^L)$ with $T^L$ a maximal torus in $C_G(t_1,\phi)$. Hence $C_G(t_1,\phi)^0\cap L=T^L$, and thus \begin{equation}\label{eq:heus} C_G(s_L,\phi)^0\cap L^\prime= C_G(t_1,\phi)^0\cap L^\prime={T_L\cap T^L}, \end{equation} proving the claim.
This proves that $M:=tT^L=r_LT^L\subset T$ is a residual coset, by application of Proposition \ref{prop:conv}.
Notice that (\ref{eq:heus}) shows that $T^L$ is also a maximal torus of $C_G(s_L,\phi)$, and thus of $C_G(s_L,u)$.
Finally notice that the $W_0$-orbit of the pair $(t,M)$ is uniquely determined by the conjugacy class of $(t,u)$ by the above procedure.
We have shown: \begin{prop}\label{prop:terug}
For every pair $(t,u)$ with $t$ semisimple and $u$ unipotent satisfying (\ref{eq:KL}), we can find a homomorphism $\phi$ as in \ref{JM} such that $t$ commutes with $c$. Let $T^L$ be a maximal torus of $C_G(t_1=tc^{-1},u)$ and put $M=tT^L$. By suitable conjugation we can arrange that $t$, $c$ and $M$ are in $T$. Then $M\subset T$ is a residual coset. If we write $t=rt^L$ with $r=sc\in T_{L,u}T_{L,rs}$ and $t^L\in T^L$, then $T^L$ is also a maximal torus of $C_G(s,c,u)$. The $W_0$-orbit of the pair $(t,M)$ is uniquely determined by $(t,u)$. \end{prop} \begin{cor} There is a one-to-one correspondence between conjugacy classes of pairs $(t,u)$ satisfying (\ref{eq:KL}) and $W_0$-orbits of pairs $(t,M)$ with $M\subset T$ a residual coset, and $t\in M$. \end{cor} \begin{proof} The maps between these two sets as defined in Proposition \ref{prop:heen} and Proposition \ref{prop:terug} are clearly inverse to each other. \end{proof} \begin{rem} Let $(c,u)$ (with $c\in T_{rs}$) be a pair satisfying (\ref{eq:KL}), with $u$ a distinguished unipotent element of $G$. Then $u$ will be distinguished in $C_G(s)$ for each $s$ in the finite group $C_G(c,u)$. In particular, $C_G(s)$ is semisimple. Hence $s$ gives rise to a residual point $cs^\prime$ in $T$ where $s^\prime\in T$ is conjugate with $s$ in $G$. This defines a one-to-one correspondence between the orbits in $C_G(c,u)$ with respect to the normalizer $N_G(C_G(c,u))$ and the residual points in $T$ with split part $c$. \end{rem}
The Kazhdan-Lusztig parameters for irreducible representations of $\mathcal{H}$ consist of triples $(t,u,\rho)$ where $(t,u)$ is as above, and $\rho$ is an irreducible representation of the finite group \[ A(t,u)=C_G(t,u)/(Z_GC_G(t,u)^0), \] where $Z_G$ is the center of $G$. However, not all the irreducible representations of $A(t,u)$ arise, but only those representations of $A(t,u)$ which appear in the natural action of $A(t,u)$ on the homology of the variety of Borel subgroups of $G$ containing $t$ and $u$.
Moreover, Kazhdan and Lusztig show that the irreducible representation $\pi(t,u,\rho)$ is tempered if and only if $t\in M^{temp}$, where $M$ is the residual subspace associated to the pair $(t,u)$. In this way we obtain a precise geometric description of the set of minimal central idempotents $\{e_i\}_{i=1}^{l_t}$ of the residue algebra $\mathcal{H}^t$ for $R_M$-generic $t\in M^{temp}$.
\index{0kl@${\overline \kappa}_{W_0r}$(=${\overline \kappa}_{\mathcal{R},W_0r}$), rational factor in $\nu(\{r\})$|see{${\overline \kappa}_{W_LL}$}}
\index{m@$m_{\{r\}}$(=$m_{\mathcal{R},\{r\}}$)|see{$m_L$}}
\printindex
\end{document} | arXiv |
\begin{definition}[Definition:Variable/Discrete]
A '''discrete variable''' is a variable which is not continuous.
\end{definition} | ProofWiki |
Elongated bicupola
In geometry, the elongated bicupolae are two infinite sets of polyhedra, constructed by adjoining two n-gonal cupolas to an n-gonal prism. They have 2n triangles, 4n squares, and 2 n-gon. The ortho forms have the cupola aligned, while gyro forms are counter aligned.
3 4 5
Elongated orthobicupola
J35 Semiregular J38
Elongated triangular orthobicupola Elongated square orthobicupola
(rhombicuboctahedron)
Elongated pentagonal orthobicupola
Elongated gyrobicupola
J36 J37 J39
Elongated triangular gyrobicupola Elongated square gyrobicupola
(pseudorhombicuboctahedron)
Elongated pentagonal gyrobicupola
Set of elongated bicupolae
Example pentagonal ortho form
Faces2n triangles
4n squares
2 n-gon
Edges12n
Vertices6n
Symmetry groupOrtho: Dnh, [2,n], (*n22), order 4n
Gyro: Dnd, [2+,2n], (2*n), order 4n
Propertiesconvex
See also
• Bicupola
• Elongated cupola
• Gyroelongated bicupola
References
• Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
• Victor A. Zalgaller (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. No ISBN. The first proof that there are only 92 Johnson solids.
Convex polyhedra
Platonic solids (regular)
• tetrahedron
• cube
• octahedron
• dodecahedron
• icosahedron
Archimedean solids
(semiregular or uniform)
• truncated tetrahedron
• cuboctahedron
• truncated cube
• truncated octahedron
• rhombicuboctahedron
• truncated cuboctahedron
• snub cube
• icosidodecahedron
• truncated dodecahedron
• truncated icosahedron
• rhombicosidodecahedron
• truncated icosidodecahedron
• snub dodecahedron
Catalan solids
(duals of Archimedean)
• triakis tetrahedron
• rhombic dodecahedron
• triakis octahedron
• tetrakis hexahedron
• deltoidal icositetrahedron
• disdyakis dodecahedron
• pentagonal icositetrahedron
• rhombic triacontahedron
• triakis icosahedron
• pentakis dodecahedron
• deltoidal hexecontahedron
• disdyakis triacontahedron
• pentagonal hexecontahedron
Dihedral regular
• dihedron
• hosohedron
Dihedral uniform
• prisms
• antiprisms
duals:
• bipyramids
• trapezohedra
Dihedral others
• pyramids
• truncated trapezohedra
• gyroelongated bipyramid
• cupola
• bicupola
• frustum
• bifrustum
• rotunda
• birotunda
• prismatoid
• scutoid
Degenerate polyhedra are in italics.
| Wikipedia |
Tag Archives: risk
Do risk classes go beyond stereotypes?
Generalization, stereotypes and clichés
In Thinking, Fast and Slow, Daniel Kahneman discusses at length the importance of stereotypes in understanding many decision-making processes. A so-called System 1 is used for quick decision-making: it allows us to recognize people and objects, helps us focus our attention, and encourages us to fear spiders. It is based on knowledge stored in memory and accessible without intention, and without effort. It can be contrasted with System 2, which allows for more complex decision-making, requiring discipline and sequential reflection. In the first case, our brain uses the stereotypes that govern judgments of representativeness, and uses this heuristic to make decisions. If I cook a fish for friends who have come to eat, I open a bottle of white wine. The cliché "fish goes well with white wine" allows me to make a decision quickly, without having to think about it. Stereotypes are statements about a group that are accepted (at least provisionally) as facts about each member. Whether correct or not, stereotypes are the basic tools for thinking about categories in System 1. But in many cases, a more in-depth, more sophisticated reflection – corresponding to System 2 – will make it possible to make a more judicious, even optimal decision. Without choosing any red wine, a pinot noir could perhaps also be perfectly suitable for roasted red mullets.
"To generalize is to be an idiot, to particularize is the alone distinction of merit" wrote William Blake around 1800, annotating speeches by the painter Joshua Reynolds. Stigmatizing an entire population because of a minority in a decision-making process is a misleading generalization, often punished by society. Moral punishment, but sometimes also legal (when hiring for example) in a society that tends to be civilized, asking not to draw erroneous conclusions about an individual from the statistics of a group to which he would be attached. But isn't that what the actuary does every day?
For Schauer (2009), this "generalization", condemned by William Blake, is probably the actuary's raison d'être: "to be an actuary is to be a specialist in generalization, and actuaries engage in a form of decision-making that is sometimes called actuarial". If I decide to insure a sports car, I have I am given risky driving characteristics that probably belong to the majority of sports car owners, attributes that I may not share. And as we noted in the introduction, insurance companies, of course, are not the only ones that operate actuarially, according to Schauer's definition. We all do it, much more often than most of us would probably recognize. We do this when we choose airlines based on their safety record, punctuality or lost luggage. We do this when we associate personal characteristics (a visible tattoo, black or brightly coloured clothing) with behavioural characteristics (such as a propensity for violence) that these personal characteristics would seem to indicate. And we operate in this way when we engage in stereotypes that may be harmless on the basis of nationality, for example by calling French people are rude, or Scots all wear kilts, while at the same time acknowledging that more pernicious stereotypes on the basis of ethnic origin, gender, sexual orientation are too widespread today! As the misconception of the word "prejudice" indicates, many people believe that it is unfair to make individual decisions based on non-universal group characteristics. Even if group allocations have a solid statistical basis. Because the big difference between actuarial science and everyday life is that actuaries have to use a large number of observations. On a personal level, I can thus decide not to travel with such an airline anymore because on three trips, I have experienced two bad experiences. Before deciding that travel insurance deserves a higher premium when flying with this company, it takes more than three observations!
In fact, the question is often whether an insurance company's refusal to provide coverage, or to increase the premiums it charges for the same coverage, is an injustice when it is based on an actuarially justified (but perhaps not universal) generalization. As Leemens (2000) noted, the question was asked of the legislator when insurers observed that Jewish women from Eastern Europe were particularly vulnerable to breast and ovarian cancer. At the end of 2012, the European Court of Justice put an end to all discrimination based on the gender of policyholders: insurers were no longer able to differentiate between insurance product prices according to whether the member was male or female. But the use of age is still allowed. Indeed, age is often an indicator of a possible decrease in vision or hearing, slower reaction time (and increased risk of sudden disability), etc. And although there are many individual variations, the available data provide important empirical justification.
Machines, causality, and stereotypes
A major criticism of machine learning models is the lack of interpretation. But very often, the validation of econometric models requires a narrative built around stereotypes. And this narrative is essential, as Pearl & Mackenzie (2018) reminds us. Indeed, in the "The Ladder of Causation", there are three levels. At the first level, we find the notion of association (or correlation), or even conditional probability, which serve as a basis for the constitution of stereotypes: if we observe
P[carries | brushing your teeth] < P[carries | don't brush your teeth]
brushing teeth will be associated with a decrease in the probability of having carries. It is also the basis for regression methods, which are based on correlations between the variable of interest and others, wrongly called explanatory. In Figure 1, we can see the daily cycling traffic in Helsinki, and the average temperature. We will tend to prefer the one on the left, showing the evolution of the number of cyclists as a function of temperature, suggesting that temperature could explain the number of cyclists, and not the other way around. But the stereotype doesn't necessarily focus on the causal link: if I see a lot of cyclists passing through the window, I'll tell myself it must be hot, or at least warm.
Figure 1: Näytä Data – Author's visualization
The first level answers the question "what if I see…?"(e.g. "what cycling traffic to expect if the temperature reaches 20°C? ") and this task can be perfectly accomplished by a machine. The second level is the one that makes it possible to understand an effect, an intervention. The question is then "what if I do…? ". To use our example, we are trying to understand the importance of brushing our teeth on the appearance of cavities. What if brushing your teeth is more natural for children with good teeth? We see the third level of the scale coming up, asking the question "what if I had done…?"and based on the idea of a counterfactual model. We are no longer content to measure correlations, we will build a model explaining what would happen by making a change in the causal variables: what would really happen if the child who did not brush his teeth began to do so? For Pearl & Mackenzie (2018) a human being (maybe even an actuary) can make these more advanced arguments than a machine can (yet) do. And very often, these causal patterns are stereotyped. As Charpentier & Diago Barry (2015) points out, in epidemiology, researchers have long questioned the explanation to be given to the fact that small babies of smokers have a higher probability of survival than babies of non-smoking mothers. The intuition that something is wrong comes from prejudices, stereotypes that we have, and that a machine cannot have.
When actuaries tell each other stories
As Antonio & Charpentier (2017) noted, the European "gender directive" has confused many insurers who used gender to construct their rates, as the latter was highly correlated with the frequency of claims. But by introducing telematic data, gender was no longer significant in the regression. Gender has long been used as a proxy to capture an effect that can be observed using telematic data, giving rise to many sexist stereotypes and other stereotypes.
But the stories also make it possible to decide between a false correlation ("spurious correlation") and a correlation that could be interpreted. In Figure 2, we have life expectancy at birth, a variable that we could try to explain in a pension study context, for example, by French department. On the right, two variables taken at random: the number of licenses of a tennis club, and the number of advertising agencies. Stereotypes are what will allow us to construct a causal graph, allowing us to understand why there is such a strong correlation between these variables and life expectancy.
Figure 2: Life expectancy at birth for men, left. At the centre, number of tennis licenses per 100,000 inhabitants (source FFT). On the right, number of advertising agencies per 100,000 inhabitants (source INSEE, code NAF 7311Z). Visualization of the author.
Hyper-individualization as an answer?
While William Blake condemned stereotypes by saying "to generalize is to be an idiot", he also clearly went further, continuing with "to particularize is the alone distinction of merit". This individualisation is also advocated by more and more insurers, and even desired by many insureds. But as Grace & Terry (2002) pointed out, many policyholders suffer from a significant optimism bias – "if I have an accident, it will not be my fault" – leading them to doubt the insurer's classification – "I'm less risky than the others". And morality seems to prove them right, against actuaries. Yet, not only is generality not, in general, unjust, but justice itself can have considerable elements of generality. To the extent that justice is centred on equity and to the extent that equity itself is closely linked to equality, then equity, and therefore justice, can now be seen as itself based on the idea of generality. The just society is not necessarily a society in which each individual is treated as an isolated set of unique attributes, requiring individualized attention. On the contrary, in some cases, the just society is a society in which generality is not only unavoidable, but also necessary for justice itself. And pooling risks together is the natural response in an insurance context. And it might not be such a big deal if that class is not as homogenous at it could be, or as we would have expected it to be…
Antonio, K. & Charpentier, A. (2017). La tarification par genre en assurance, corrélation ou causalité ?. Risques. 110 : 107-110.
Charpentier, A. & Diago Barry, A. (2015). Big data : passer d'une analyse de corrélation à une interprétation causale. Risques, 101: 107-111.
Grace, J. & Terry, M. (2002). Exploring the Causes of Comparative Optimism. Psychologica Belgica. 42: 65–98
Kahneman, D. (2011).Thinking, Fast and Slow. FSG Eds.
Leemens, T. (2000). Selective Justice, Genetic Discrimination, and Insurance: Should We Single Out Genes in Our Laws? McGill law journal. Revue de droit de McGill 45(2):347-412.
Pearl, J. & Mackenzie, D. (2018). The Book of Why: The New Science of Cause and Effect. Basic Books.
Schauer, F.F. (2009). Profiles, Probabilities, and Stereotypes. Harvard University Press.
causalityclichéscorrélationriskstereotypeswhy
econometrics, ML, Publications, Regression
Foundations of Machine Learning, part 5
This post is the nineth (and probably last) one of our series on the history and foundations of econometric and machine learning models. The first fours were on econometrics techniques. Part 8 is online here.
Optimization and algorithmic aspects
In econometrics, (numerical) optimization became omnipresent as soon as we left the Gaussian model. We briefly mentioned it in the section on the exponential family, and the use of the Fisher score (gradient descent) to solve the first order condition \mathbf{X}^T W(\beta)^{-1})[y-\widehat{y}]=\mathbf{0}. In learning, optimization is the central tool. And it is necessary to have effective optimization algorithms, to solve problems (described previously) of the form: \widehat{\beta}\in\underset{\beta\in\mathbb{R}^p}{\text{argmin}}\left\lbrace\sum_{i=1}^n \ell(y_i,\beta_0+\mathbf{x}^T\beta)+\lambda\Vert\boldsymbol{\beta}\Vert\right\rbraceIn some cases, instead of global optimization, it is sufficient to consider optimization by coordinates (widely studied in Daubechies et al. (2004)). If f:\mathbb{R}^d\rightarrow\mathbf{R} is convex and differentiable, if \mathbf{x} satisfies f(\mathbf{x}+h\boldsymbol{e}_i)\geq f(\mathbf{x}) for any h>0 and i\in\{1,\cdots, d\}then f(\mathbf{x})=\min\{f\}, where \mathbf{e}=(\mathbf{e}_i) is the canonical basis of \mathbb{R}^d. However, this property is not true in the non-differentiable case. But if we assume that the non-differentiable part is separable (additively), it becomes true again. More specifically, iff(\mathbf{x})=g(\mathbf{x})+\sum_{i=1}^d h_i(x_i)with\left\lbrace\begin{array}{l}g: \mathbb{R}^d\rightarrow\mathbb{R}\text{ convex-differentiable}\\h_i: \mathbb{R}\rightarrow\mathbb{R}\text{ convex}\end{array}\right.This was the case for Lasso regression, \beta)\mapsto\| \mathbf{y}-\beta_0-\mathbf{X}\beta\|_{\ell_2 }+\lambda\|\beta\|_{\ell_1}, as shown by Tsen (2001). Getting back to our initial notations, we can use a coordinate descent algorithm: from an initial value \mathbf{x}^{(0)}, we consider (by iterating)x_j^{(k)}\in\text{argmin}\big\lbrace f(x_1^{(k)},\cdots,x_{k-1}^{(k)},x_k,x_{k+1}^{(k-1)},\cdots,x_n^{(k-1)})\big\rbrace for j=1,2,\cdots,nThese algorithmic problems and numerical issues may seem secondary to econometricians. However, they are essential in automatic learning: a technique is interesting if there is a stable and fast algorithm, which allows to obtain a solution. These optimization techniques can be transposed: for example, this coordinate descent technique can be used in the case of SVM methods (known as "vector support" methods) when the space is not linearly separable, and the classification error must be penalized (we will come back to this technique in the next section).
In-sample, out-of-sample and cross-validation
These techniques seem intellectually interesting, but we have not yet discussed the choice of the penalty parameter \lambda. But this problem is actually more general, because comparing two parameters \widehat{\beta}_{\lambda_1} and \widehat{\beta}_{\lambda_2} is actually comparing two models. In particular, if we use a Lasso method, with different thresholds \lambda, we compare models that do not have the same dimension. Previously, we have addressed the problem of model comparison from an econometric perspective (by penalizing overly complex models). In the learning literature, judging the quality of a model on the data used to construct it does not make it possible to know how the model will behave on new data. This is the so-called "generalization" problem. The traditional approach then consists in separating the sample (size n) into two parts: a part that will be used to train the model (the training database, in-sample, size m) and a part that will be used to test the model (the testing database, out-of-sample, size n-m). The latter then makes it possible to measure a real predictive risk. Suppose that the data are generated by a linear model y_i=\mathbf{x}_i^T \beta_0+\varepsilon_i where \varepsilon_i are independent and centred law achievements. The empirical quadratic risk in-sample is here\frac{1}{m}\sum_{i=1}^m\mathbb{E}\big([\mathbf{x}_i^T \widehat{\beta}-\mathbf{x}_i^T \beta_0]^2\big)=\mathbb{E}\big([\mathbf{x}_i^T \widehat{\beta}-\mathbf{x}_i^T \beta_0]^2\big),for any observation i. Assuming the residuals \varepsilon Gaussian, then we can show that this risk is worth \sigma^2 \text{trace} (\Pi_X)/m is \sigma^2 p/m. On the other hand, the empirical out-of-sample quadratic risk is here \mathbb{E}\big([\mathbf{x}^T \widehat{\beta}-\mathbf{x}^T \beta_0]^2\big) where \mathbf{x} is a new observation, independent of the others. It can be noted that \mathbb{E}\big([\mathbf{x}^T \widehat{\beta}-\mathbf{x}^T \beta_0]^2\big\vert \mathbf{x}\big)=\text{Var}\big(\mathbf{x}^T \widehat{\beta}\big\vert \mathbf{x}\big)=\sigma^2\mathbf{x}^T(\mathbf{x}^T\mathbf{x})^{-1}\mathbf{x},and by integrating with respect to \mathbf{x}, \mathbb{E}\big([\mathbf{x}^T \widehat{\beta}-\mathbf{x}^T\beta_0]^2\big)=\sigma^2\text{trace}\big(\mathbb{E}[\mathbf{x}\mathbf{x}^T]\mathbb{E}\big[(\mathbf{x}^T\mathbf{x})^{-1}\big]\big).The expression is then different from that obtained in-sample, and using the Groves & Rothenberg (1969) increase, we can show that \mathbb{E}\big([\mathbf{x}^T \widehat{\beta}-\mathbf{x}^T \beta_0]^2\big) \geq \sigma^2\frac{p}{m}which is pretty intuitive, when we start thinking about it. Except in some simple cases, there is no simple (explicit) formula. Note, however, that if \mathbf{X}\sim\mathcal{N}(0,\sigma^2 \mathbb{I}), then \mathbf{x}^T \mathbf{x} follows a Wishart law, and it can be shown that \mathbb{E}\big([\mathbf{x}^T \widehat{\beta}-\mathbf{x}^T \beta_0]^2\big)=\sigma^2\frac{p}{m-p-1}.If we now look at the empirical version: if \widehat{\beta} is estimated on the first m observations,\widehat{\mathcal{R}}^{~\text{ IS}}=\sum_{i=1}^m [y_i-\boldsymbol{x}_i^T\widehat{\boldsymbol{\beta}}]^2\text{ and }\widehat{\mathcal{R}}^{\text{ OS}}=\sum_{i=m+1}^{n} [y_i-\boldsymbol{x}_i^T\widehat{\boldsymbol{\beta}}]^2and as Leeb (2008) noted, \widehat{\mathcal{R}}^{\text{IS}}-\widehat{\mathcal{R}}^{\text{OS}}\approx 2\cdot\nu where \nu represents the number of degrees of freedom, which is not unlike the penalty used in the Akaike test.
Figure 4 shows the respective evolution of \widehat{\mathcal{R}}^{\text{IS}} and \widehat{\mathcal{R}}^{\text{OS}} according to the complexity of the model (number of degrees in a polynomial regression, number of nodes in splines, etc). The more complex the model, the more \widehat{\mathcal{R}}^{\text{IS}} will decrease (this is the red curve, below). But that's not what we're interested in here: we want a model that predicts well on new data (i. e. out-of-sample). As Figure 4 shows, if the model is too simple, it does not predict well (as it does with in-sample data). But what we can see is that if the model is too complex, we are in a situation of "overlearning": the model will start to model the noise. Of course, this figure should remind us of the one we've seen in our second post of that series
Figure 4 : Generalization, under- and over-fitting
Instead of splitting the database in two, with some of the data that will be used to calibrate the model and some to study its performance, it is also possible to use cross-validation. To present the general idea, we can go back to the "jackknife", introduced by Quenouille (1949) (and formalized by Quenouille (1956) and Tukey (1958)) relatively used in statistics to reduce bias. Indeed, if we assume that \{y_1,\cdots,y_n\} is a sample drawn according to a law F_\theta, and that we have an estimator T_n (\mathbf{y})=T_n (y_1,\cdots,y_n), but that this estimator is biased, with \mathbf{E}[T_n (\mathbf{Y})]=\theta+O(n^{-1}), it is possible to reduce the bias by considering \widetilde{T}_n(\mathbf{y})=\frac{1}{n}\sum_{i=1}^n T_{n-1}(\mathbf{y}_{(i)})\text{ where }\mathbf{y}_{(i)}=(y_1,\cdots,y_{i-1},y_{i+1},\cdots,y_n)It can then be shown that \mathbb{E}[\tilde{T}_n(Y)]=\theta+O(n^{-2})The idea of cross-validation is based on the idea of building an estimator by removing an observation. Since we want to build a predictive model, we will compare the forecast obtained with the estimated model, and the missing observation\widehat{\mathcal{R}}^{\text{ CV}}=\frac{1}{n}\sum_{i=1}^n \ell(y_i,\widehat{m}_{(i)}(\mathbf{x}_i))We will speak here of the "leave-one-out" (loocv) method.
This technique reminds us of the traditional method used to find the optimal parameter in exponential smoothing methods for time series. In simple smoothing, we will construct a forecast from a time series as {}_t\widehat{y}_{t+1} =\alpha\cdot{}_{t-1}\widehat{y}_t +(1-\alpha)\cdot y_t, where \alpha\in[0,1], and we will consider as "optimal" \alpha^\star = \underset{\alpha\in[0,1]}{\text{argmin}}\left\lbrace \sum_{t=2}^T \ell({}_{t-1}\widehat{y}_{t},y_{t}) \right\rbraceas described by Hyndman et al (2009).
The main problem with the leave-one-out method is that it requires calibration of n models, which can be problematic in large dimensions. An alternative method is cross validation by k-blocks (called "k-fold cross validation") which consists in using a partition of \{1,\cdots,n\} in k groups (or blocks) of the same size, \mathcal{I}_1,\cdots,\mathcal{I}_k, and let us note \mathcal{I}_{\bar j}=\{1,\cdots,n\}\setminus \mathcal{I}_j. By noting \widehat{m}_{(j)} built on the sample \mathcal{I}_{\bar j}, we then set:\widehat{\mathcal{R}}^{k-\text{ CV}}=\frac{1}{k}\sum_{j=1}^k \mathcal{R}_j\text{ where }\mathcal{R}_j=\frac{k}{n}\sum_{i\in\mathcal{I}_{{j}}} \ell(y_i,\widehat{m}_{(j)}(\mathbf{x}_i))Standard cross-validation, where only one observation is removed each time (loocv), is a special case, with k=n. Using k=5 or 10 has a double advantage over k=n: (1) the number of estimates to be made is much smaller, 5 or 10 rather than n; (2) the samples used for estimation are less similar and therefore less correlated to each other, which tends to avoid excess variance, as recalled by James et al. (2013).
Another alternative is to use boosted samples. Let \mathcal{I}_b be a sample of size n obtained by drawing with replacement in \{1,\cdots,n\} to know which observations (y_i,\mathbf{x}_i) will be kept in the learning population (at each draw). Note \mathcal{I}_{\bar b}=\{1,\cdots,n\}\setminus\mathcal{I}_b. By noting \widehat{m}_{(b)} built on sample \mathcal{I}_b, we then set :\widehat{\mathcal{R}}^{\text{ B}}=\frac{1}{B}\sum_{b=1}^B \mathcal{R}_b\text{ where }\mathcal{R}_b=\frac{n_{\overline{b}}}{n}\sum_{i\in\mathcal{I}_{\overline{b}}} \ell(y_i,\widehat{m}_{(b)}(\mathbf{x}_i))where n_{\bar b} is the number of observations that have not been kept in \mathcal{I}_b. It should be noted that with this technique, on average e^{-1}\sim36.7\% of the observations do not appear in the boosted sample, and we find an order of magnitude of the proportions used when creating a calibration sample, and a test sample. In fact, as Stone (1977) had shown, the minimization of AIC is to be compared to the cross-validation criterion, and Shao (1997) showed that the minimization of BIC corresponds to k-fold cross-validation, with k=n/\log n.
All those techniques here are mentioned in the "machine learning" section since they rely on automatic, computational techniques, and no probabilistic foundations are necessary. In many cases we did use the notation m^\star (at least in the first posts on "machine learning" techniques) to highlight the fact that we want some sort of "optimal" model – and to make a distinction with estimators \widehat{m} considered earlier, when we had some probabilistic framework. But of course, it is possible (and necessary) to build bridges between those two cultures…
References are online here. As explained in the introduction, it is some sort of online version of an introduction to our joint paper with Emmanuel Flachaire and Antoine Ly, Econometrics and Machine Learning (initially writen in French), that will actually appear soon in the journal Economics and Statistics (in English and in French).
bootstrapcross-validationFlachairein-samplelyMachine Learningoptimizationout-of-samplerisk
Insurance, Actuarial Science, Data and Models
Our research chaire ACTINFO, with our colleagues from Lyon, at the DAMI chaire, PREVENT'HORIZON chaire & ACTUARIAT DURABLE chaire, will organize a 2 day conference in Paris, on Insurance, Actuarial Science, Data & Models, in ten days.
We invited Katrien ANTONIO (KU Leuven), Alexandre BOUMEZOUED (Milliman Paris), Alfred GALICHON (New-York University), Pierre-Yves GEOFFARD (Paris School of Economics), Meglena JELEVA (University of Paris Nanterre), Julie JOSSE (Ecole Polytechnique), Florence JUSOT (Paris Dauphine University), Michael LUDKOWSKI (University of California Santa Barbara), François PANNEQUIN (CREST and ENS Paris-Saclay), Florian PELGRIN (Edhec Business School), Dylan POSSAMAI (Columbia University) and Julien TRUFIN (ULB Brussels). More information (including the program) is online.
conferenceCovéadataFFAinsurancerisk
Picking an asset to invest
Yesterday, Andrew Lo spent some time on a nice graph, discussing attitudes towards risk. Here are four assets (thanks @TCJUK for improving the terminology), real data (no information here about time, but it's the same scale for the four of them)
The question raised was quite simple
if you could invest in one, and only one, asset which one will you pick ?
Continue reading Picking an asset to invest →
FinanceLoLondonrisk
Data Science and Big Data for Actuaries, in Barcelona
Tuesday afternoon, I will give a lecture on Data Science & Big Data for Actuaries in Barcelona, for Summer School of the Asociación Española de Gerencia de Riesgos y Seguros. For this two hour lecture, there will be live streming on UBtv. Slides are available online (as usual, download the pdf to get the animated version)
actuariesBarcelonabig dataconferencedatadata scienceriskslidesTV
Research, Seminar
Local Utility and Multivariate Risk Aversion
Our joint paper Local Utility and Multivariate Risk Aversion, written with Alfred Galichon and Marc Henry will appear soon in Mathematics of Operations Research.
I will present it, next Thursday in our Risk Workshop, in Rennes, organized by Olivier L'Haridon.
GalichonHenrylocal utilityrisk
Crash courses, Trip
Graduate Crash Course on Risk Measures
Tomorrow morning, I will give a crash course on risk measures at Louvain-la-Neuve, in Belgium. This is a crash course of PhD students (and researchers) with a long introduction on the univariate static framework (and some mathematical tools that will be interesting later on, such as the Fenchel transform and more generally on convexity, as well as some results on optimal transport). I will also mention what was obtained in decision theory, inspired by Itzhak Gilboa's Theory of Decision under Uncertainty. Then I will mention extensions to derive multiple risk measures, based on Marc Henry and Alfred Galichon's work. Finally, I will conclude by introducing the difficulty to derive dynamic risk measures.
The slides are based on a document I am still working on. And unfortunately, the deeper I get to explain the roots of the axioms, or the assumptions, the more papers I discover (and I need to read, and understand). So I guess I will need some time to finalize my survey. Note that I decided to skip details on technical issues when working on , and the weak topology on the dual of . I will try to add additional references in the notes, but I wanted the slides to be as simple as possible. I also want to add more connections with statistical results, such as Neyman Pearson's lemma, for instance (as mentioned in a paper by Alexander Schied). All my apologies for the typos, too.
coursedynamicHenrymeasuremultivariatequantilerisk
2013, ACT2040, Actuarial science, Course, Extreme Value, Hiver, Probability
The law of small numbers
In insurance, the law of large numbers (named loi des grands nombres initially by Siméon Poisson, see e.g. http://en.wikipedia.org/…) is usually mentioned to legitimate large portfolios, because of pooling and diversification: the larger the pool, the more 'predictable' the losses will be (in a given period). Of course, under standard statistical assumption, namely finite expected value, and independence (see http://freakonometrics.blog.free.fr/…. for a discussion, in French). Since in insurance, catastrophes are usually rare – and extremely costly – and actuaries might be interested to model occurrence of that small number of events (see e.g. Aldous' book on that specific topic, that can be downloaded from http://stat.berkeley.edu/…). The theorem behind is sometimes called the law of small numbers (from the book published by Ladislaus Bortkiewicz, but we'll get back to that story later on, see also Whitaker (1914) http://biomet.oxfordjournals.org/… or the book recently published by Michael Falk, Jürg Hüsler and Rolf-Dieter Reiss).
The Poisson distribution
The so-called Poisson distribution (see http://en.wikipedia.org/…) was introduced by Siméon Poisson in 1837 (in Recherches sur la Probabilité des Jugements en Matière Criminelle et en Matière Civile, Précédées des Règles Générales du Calcul des Probabilités, see http://gallica.bnf.fr/…). But it had been defined more than a century before, by Abraham De Moivre, in 1711, in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus (see e.g. the review in http://www.jstor.org/…). Let denote a counting random variable, then it said to be Poisson distributed if there is such that
De Moivre obtained that distribution from an approximation of the binomial distribution. Recall that the binomial distribution is a standard distribution in actuarial science, for instance to model the number of deaths among insured. If individual death probabilities are identical, say , and if deaths are independent events, then
And if and , then
Again, this is an asymptotic theorem, which is valid when we have a lot of observations ( ), but also, the probability of occurrence should be extremely small (since ), which is why to use the term small numbers. Siméon Poisson was not interested by mathematical approximations: his main point was to get a distribution with nice goodness of fit properties for the data he was working on. He wanted to get a better understanding of cours d'assises (jury panel, might be a valid translation of the French term). A jury consists of 12 jurors who voted to determine whether a defendant was guilty. When guilt was predominant, with at least 8 votes against 4, the defendant was convicted (which was 47% of criminal cases). 5 with 7 votes against, the opinion of professional judges was requested (11% of criminal trials again). Using these statistics we can demonstrate that a defendant brought before an assize court is guilty of the order of 68%, and the probability that a juror is not wrong by voting (condemning an innocent or releasing a culprit) was about 54%. He sought to calculate the probability that a defendant is wrongfully convicted, and gets 2%. And 28% of exonerated defendants are in fact guilty. Siméon Poisson introduced this law to get probabilities easily. But the law he considered is central in probability….
The heuristic of the main theorem, related to the Poisson distribution is the following: let denote i.i.d random variables taking values in (in a general setting, one component can be the time, the other one an upper region of interest, where some stochastic process might be). Let . If as (or to be a little bit more specific about the assumptions), let denote the (random variable characterizing) count of events , then can be approximated by a Poisson distribution with parameter .
The heuristic is that if we consider a large number of observations, and if we count how many are in a given (small) region, then the number of such observations is Poisson distributed.
n=1000
X=runif(n)*10-1.5
Y=runif(n)*10-1.5
plot(X,Y,axis=FALSE,cex=.6)
u=seq(-1,1,by=.01)
v=sqrt(1-u^2)
polygon(c(u,rev(u)),c(v,rev(-v)),col="yellow",border=NA)
I=(X^2+Y^2)<1
points(X[I],Y[I],cex=.6,pch=19,col="red")
If we run some simulations,
> n=1000
> ns=100000
> N=rep(NA,ns)
> for(s in 1:ns){
+ X=runif(n)*10-1.5
+ Y=runif(n)*10-1.5
+ I=(X^2+Y^2)<1
+ N[s]=sum(I)
> hist(N,breaks=0:60,probability=TRUE,col="yellow")
> mean(N)
[1] 31.41257
The parameter of the Poisson distribution is the area of the yellow disk, over the area of the square, i.e.
> (lambda=10*pi)
> lines(0:60-.5,dpois(0:60,lambda),type="b",col="red")
To get an interpretation related to insurance modeling, let denote an upper layer in a reinsurance contract, i.e. for some deductible . Let 's denote individual losses. Then the number of claims that hit this upper layer can be modeled with a Poisson distribution. More precisely, if deductible becomes extremely large (and ), we obtain the point-over-threshold model in extreme value theory (see e.g. http://brale.math.hr/~iugrina/… or http://fire.nist.gov/bfrlpubs/…): if has a Poisson distribution and, conditionally on , are independent identically distributed generalized Pareto random variables, then has the generalized extreme value distribution. Thus, exceedances models (for rare events) are closely related to Poisson processes.
The Poisson process
As mentioned above, the Poisson distribution appears when events occur somehow randomly and independently, over time. It is then natural to study the time between two occurences (or two claims, in an insurance context).
Poisson distribution, and claims occurrence
It is neither Siméon Poisson nor De Moivre, but Ladislaus Von Bortkiewicz who first mentioned the Poisson distribution as the law of small numbers. In 1898 (see https://archive.org/…), he studied the number number of soldiers killed by being kicked by a horse, from 1875 till 1894, in 200 corps (more precisely 10 corps over 20 ans).
He did obtain the following distribution (here, the parameter of the Poisson distribution is 0.61, i.e. the average number of death per year)
death per
counts Poisson
3 3 4.11
5 and more 0 0.08
It is possible to find a lot of cases where the Poisson distribution fits extremely well. For instance, if we consider the number of hurricanes, that landed in Florida after 1850,
4 8 11.03
Poisson distribution, and return period
The return period was introduced by Emil Gumbel, in hydrology, to link probabilities and durations (see e.g. http://freakonometrics.blog.free.fr/…). A decennial event has an occurence probability of 1/10. 10 is then the average waiting time before occurence. This does not mean that the event will not occur before 10 years, or has to occur before 10 years. Consider a return period (in years), then the yearly probability of non-occurrence is .
And the probability of non-occurence over years is then . It is standard to summarize this property with the following table,
Number of years ( ) without catastrophes
10 65.1% 40.1% 18.3% 9.6% 4.9%
20 87.8% 64.2% 33.2% 18.2% 9.5%
50 99.5% 92.3% 63.6% 39.5% 22.5%
100 99.9% 99.4% 86.7% 63.4% 39.5%
The diagonal in the table above is extremely interesting. It looks like there is some kind of convergence towards a limiting value (here 63.2%). Indeed, the number of events observed over n years have a Binomial distribution, with probability , which will converge towards the Poisson distribution with parameter 1. The probability of not having a catastrophe is then , which is equal to 0.632.
Rare probabilities and the Poisson distribution
The Poisson distribution keeps appearing when computing probabilies of rare events. For instance, the probability to have at least one incident in a nuclear plant in France, over a 50 year period. Assume that the annual probability of an incident in a reactor is small, e.g. 0.05%. Assume further that reactors are independent among them, and in time. The probability to have an incident over 80 reactors in 50 years is (exactly)
Of course, a linear approximation is not correct (even if it was mentioned in some French newspaper, as explained in an old post http://freakonometrics.blog.free.fr/…)
> p=0.00005
> 1-(1-p)^(50*80)
[1] 0.1812733
> 1-exp(-50*80*p)
which is the probability that is null when has a Poisson distribution with parameter . We clearly see here an application of De Moivre's approximation in risk management.
Another way of looking at this problem is based on the following idea: given the fact that in 45 years of observations on 450 reactors worldwide (roughly), three major accidents were observed including Three Mile Island (1979) and Fukushima (2011), i.e. the average time between accidents can be estimated at 16 years. For a single reactor, we can assume that the average time to wait before an incident is 450 times 16 years, i.e 7200 years. Or the probability to have one incident, over one year, for one reactor is 1 over 7200 (this is the idea behind the return period concept). If we assume that the arrival of accidents occurs randomly and independently of each other (as defined above) then the number of major accidents observed over a period of 50 years in France follows a Poisson distribution with parameter 50 / (7200/80). Also, the probability of having no major accident over 50 years, with 80 reactors can be estimated by
> 1-exp(-50*80/7200)
(keeping in mind all the uncertainty around the estimated waiting time before a major accident to a single reactor!).
computerDe MoivreGumbelnuclearPoissonPOTR-englishriskVon Bortkiewicz
Risk Uncertainty and Decision Conference
Marc Henry will present our paper Local Utility and Multivariate Risk Aversion (writen with Alfred Galichon, still available online on http://papers.ssrn.com/) at the Risk Uncertainty and Decision Conference in Evanston, IL, at Northwestern University.
I will try to upload the slides soon…
AlfredGalichonHenryMarcNorthwestern Universityriskslidesuncertainty
Talk on bivariate count times series in finance and risk management
I will be giving a talk on May 4th, at the Mathematical Finance Days, at HEC Montréal, on multivariate dynamic models for counts. The conference is organized by IFM2 (Institut de Finance Mathématique de Montréal). I will be chairing some session and I will give a talk based on the joint paper with Mathieu Boudreault.
The slides can be downloaded from the blog,
Slides binar-hec
View more presentations from charthur.
"In various situations in the insurance industry, in finance, in epidemiology, etc., one needs to represent the joint evolution of the number of occurrences of an event. In this paper, we present a multivariate integer‐valued autoregressive (MINAR) model, derive its properties and apply the model to earthquake occurrences across various pairs of tectonic plates. The model is an extension of Pedelis & Karlis (2011) where cross autocorrelation (spatial contagion in a seismic context) is considered. We fit various bivariate count models and find that for many contiguous tectonic plates, spatial contagion is significant in both directions. Furthermore, ignoring cross autocorrelation can underestimate the potential for high numbers of occurrences over the short‐term. An application to risk management and cat‐bond pricing will be discussed."
BINARcat-bondcountsearthquakesHECINARrisk
Copulas, MAT8886, Statistics
Exchangeability, credit risk and risk measures
Exchangeability is an extremely concept, since (most of the time) analytical expressions can be derived. But it can also be used to observe some unexpected behaviors, that we will discuss later on with a more general setting. For instance, in a old post, I discussed connexions between correlation and risk measures (using simulations to illustrate, but in the context of exchangeable risk, calculations can be performed more accurately). Consider again the standard credit risk problem, where the quantity of interest is the number of defaults in a portfolio. Consider an homogeneous portfolio of exchangeable risk. The quantity of interest is here
or perhaps the quantile function of the sum (since the Value-at-Risk is the standard risk measure). We have seen yesterday that – given the latent factor – (either the company defaults, or not), so that
i.e. we can derive the (unconditional) distribution of the sum
so that the probability function of the sum is, assuming that
Thus, the following code can be used to calculate the quantile function
> proba=function(s,a,m,n){
+ b=a/m-a
+ choose(n,s)*integrate(function(t){t^s*(1-t)^(n-s)*
+ dbeta(t,a,b)},lower=0,upper=1,subdivisions=1000,
+ stop.on.error = FALSE)$value
> QUANTILE=function(p=.99,a=2,m=.1,n=500){
+ V=rep(NA,n+1)
+ for(i in 0:n){
+ V[i+1]=proba(i,a,m,n)}
+ V=V/sum(V)
+ return(min(which(cumsum(V)>p))) }
Now observe that since variates are exchangeable, it is possible to calculate explicitly correlations of defaults. Here
Thus, the correlation between two default indicators is then
Under the assumption that the latent factor is beta distributed
we get
Thus, as a function of the parameter of the beta distribution (we consider beta distributions with the same mean, i.e. the same margin distributions, so we have only one parameter left, with is simply the correlation of default indicators), it is possible to plot the quantile function,
> PICTURE=function(P){
+ A=seq(.01,2,by=.01)
+ VQ=matrix(NA,length(A),5)
+ for(i in 1:length(A)){
+ VQ[i,1]=QUANTILE(a=A[i],p=.9,m=P)
+ VQ[i,2]=QUANTILE(a=A[i],p=.95,m=P)
+ VQ[i,3]=QUANTILE(a=A[i],p=.975,m=P)
+ plot(A,VQ[,5],type="s",col="red",ylim=
+ c(0,max(VQ)),xlab="",ylab="")
+ lines(A,VQ[,4],type="s",col="blue")
+ lines(A,VQ[,3],type="s",col="black")
+ lines(A,VQ[,2],type="s",col="blue",lty=2)
+ lines(A,VQ[,1],type="s",col="red",lty=2)
+ lines(A,rep(500*P,length(A)),col="grey")
+ legend(3,max(VQ),c("quantile 99.5%","quantile 99%",
+ "quantile 97.5%","quantile 95%","quantile 90%","mean"),
+ col=c("red","blue","black",
+"blue","red","grey"),
+ lty=c(1,1,1,2,2,1),border=n)
e.g. with a (marginal) default probability of 15%,
> PICTURE(.15)
On this graph, we observe that the stronger the correlation (the more on the left), the higher the quantile… Note that the same graph can be plotted with on the X-axis the correlation,
Which is quite intuitive, somehow. But if the marginal probability of default decreases, increasing the correlation might decrease the risk (i.e. the quantile function),
(with the modified code to visualize the quantile as a function of the underlying default correlation) or even worse,
> PICTURE(.0075)
And it because all the more counterintuitive that the default probability decreases ! So in the case of a portfolio of non-very risky bond issuers (with high ratings), assuming a very strong correlation will lower risk based capital !
betabinomialcorrélationcreditdefaultexchangeabilityMAT8886quantileR-englishriskValue-at-RiskVaR
Catastrophes, Finance, Insurance, Risk
Variable annuities is not a systemic risk ?
The Geneva Association just published on its website an interesting report on variable annuities and systemic risk (online here). Based on a definition of potentially systemically risky activities, on interconnectedness or substitutability, the report claims that since "none of the criteria is triggered", variables annuities is "not a potentially systemically risk activity". Even if "short-term effects are conceivable". I guess it is a diplomatic way to say it…
Note that a series of slides can also be downloaded (there) on insurance and systemic risk. But that deserves a more detailed post.
Genevarisksystemicvariable annuities
Computer, Extreme Value, Maths, Statistics
Tennis and risk management
As mentioned already here, while we were going to Québec City for the workshop, we had interesting discussions in the car, and Maciej mentioned an article recently published in The Actuary,
Hence, I wanted to discuss (extremely) rare event probabilities in tennis. The story is simple: in June 2010, at Wimbledon, Nicolas Mahut and John Isner have played the longest match ever. 980 points, 11 But first of all, we need a dataset. Thanks to Duncan Murdoch, I have been able to run a short code to build up a dataset:
CITIES=c("berlin","madrid","paris","rolandgarros","wimbledon","sydney",
"beijing","shanghai","singapore","tokyo","melbourne","melbourne-indoor")
YEARS=1970:2009
BASE0=data.frame(YEAR=NA,TRNMT=NA,LENGTH=NA,SETS=NA)
for(i in 1:length(CITIES)){
for(j in 1:length(YEARS)){
city=CITIES[i]
year=YEARS[j]
localization = paste("http://www.resultsfromtennis.com/",
year,"/atp/",city,".html",sep="")
essai = try(readLines(localization), silent=TRUE)
ERROR404=FALSE
if(inherits(essai, "try-error")){ERROR404=TRUE}
if(ERROR404==FALSE){
B=scan(localization,"character")
SETS=NA
LENGTH=NA
if(length(B)>270){
I=(substr(B,1,10)=="class=rez>")
sum(I)
X0=B[I]
X3=as.numeric(substr(X0,11,13))
X0=X3
X0[is.na(X3)==TRUE]=X2[is.na(X3)==TRUE]
JL=c(which(substr(B,1,9)=="class=nl>"),length(B))
IL=which(substr(B,1,10)=="class=rez>")
IC=cut(IL,JL)
base=data.frame(IC,X0)
LENGTH=as.numeric(tapply(X0,IC,sum))
SETS=as.numeric(tapply(X0,IC,length))/2}
BASE=data.frame(YEAR=year,TRNMT=city,LENGTH,SETS)
BASE0=rbind(BASE0,BASE)}}}
write.table(BASE0,"BASE-TENNIS-TOTAL.txt")
Here I consider only tournaments where players have to win 3 sets (and actually more tournaments than those in the code above), and I have something like a bit more than 72,000 matches,
> I=is.na(TENNIS$LENGTH)==FALSE
> BT=TENNIS[I,]
> nrow(BT)
> maxr=function(x){max(x,na.rm=TRUE)}
> T=paste(BT$TRNMT,BT$YEAR)
> DUREE=tapply(BT$SETS,T,maxr)
> LISTE=names(DUREE[DUREE>3])
> BT=BT[T%in%LISTE,]
so, if we look briefly at matches over 35 years, we have the following boxplot (one boxplot per year),
The red line being the epic Isner-Mahut match in June 2010 (4-6, 6-3, 7-6, 6-7, 70-68, i.e. 183 games, here for the score card).
If we study theory (e.g. from Paul Newton and Kamran Aslam), a lot of results can be obtained for the expected value of the number of games, but if we want to study extremely rare events, we should generate Markov chains (with a lot of generation since the probability should be extremely small). But how many ? Consider below matches with more than 50 games,
The tail plot (over 50), i.e. the log-log Pareto plot indicates that it will be difficult to study tails,
and similarly with the Hill plot (assuming that tails are Pareto type….)
Anyway, if we want to study tails, we should consider a threshold high enough. For instance, with a threshold at 68 (we keep only 24 match), we have
> seuil=68+0.25
> GPD1=gpd(X,seuil,method = "ml")
> GPD2=gpd(X,seuil,method = "pwm")
> xi=GPD1$par.ests[1]
> mu=seuil
> beta=GPD1$par.ests[2]
> x=180
> P=exp((-1/xi)*log(1 + (xi * (x - mu))/beta))
> as.numeric((1-GPD1$p.less.thresh)*P)
[1] 5.621281e-09
I.e. the probability that one match last more than 183 games is 1 chance over a billion… With, say, 2500 match per year, that gives us a return period of 400 years. So yes, we might say that this way a rare event… So perhaps, generating several billions of chains, it should be possible to get a more precise estimation of the probability to play 183 games in a single match…
183ExtremesGPDhttp:-www.resultsfromtennis.com-inheritsIsnerMahutR-englishriskrstatsscantennisWimbledon
Catastrophes, Miscellaneous, Risk
Millenium bridge, endogeneity and risk management
In less than 48 hours, last week two friends mentioned the Millennium Bridge as an illustration of a risk management concept. There are several documents with that example, here (for the initial idea of using the Millennium Bridge to illustrate issues in risk management) here or there, e.g.
When we mention resonance effects on bridges, we usually thing of the Tacoma Narrows Bridge (where strong winds set the bridge oscillating) or the Basse-Chaîne Bridge (in France, which collapsed on April 16, 1850, when 478 French soldiers marched across it in lockstep). In the first case, there is nothing we can do about it, but for the second one, this is why soldiers are required to break step on bridges.
But for the Millennium bridge, a 'positive feedback' phenomenon (known as Synchronous Lateral Excitation in physics) has been observed: the natural sway motion of people walking caused small sideways oscillations in the bridge, which in turn caused people on the bridge to sway in step, increasing the amplitude of the oscillations and continually reinforcing the effect. That has been described in a nice paper in 2005 (here). In the initial paper by Jon Danielsson and Hyun Song Shin, they note that "what is the probability that a thousand people walking at random will end up walking exactly in step? It is tempting to say "close to zero", or "negligible". After all, if each person's step is an independent event, then the probability of everyone walking in step would be the product of many small numbers – giving us a probability close to zero. Presumably, this is the reason why Arup – the bridge engineers – did not take this into account. However, this is exactly where endogenous risk comes in. What we must take into account is the way that people react to their environment. Pedestrians on the bridge react to how the bridge is moving. When the bridge moves under your feet, it is a natural reaction for people to adjust their stance to regain balance. But here is the catch. When the bridge moves, everyone adjusts his or her stance at the same time. This synchronized movement pushes the bridge that the people are standing on, and makes the bridge move even more. This, in turn, makes the people adjust their stance more drastically, and so on. In other words, the wobble of the bridge feeds on itself. When the bridge wobbles, everyone adjusts their stance, which sets off an even worse wobble, which makes the people adjust even more, and so on. So, the wobble will continue and get stronger even though the initial shock (say, a gust of wind) has long passed. It is an example of a force that is generated and amplified within the system. It is an endogenous response. It is very different from a shock that comes from a storm or an earthquake which are exogenous to the system."
And to go further, they point out that this event is rather similar to what is observed in financial markets (here) by quoting The Economist from October 12th 2000 "So-called value-at-risk models (VaR) blend science and art. They estimate how much a portfolio could lose in a single bad day. If that amount gets too large, the VAR model signals that the bank should sell. The trouble is that lots of banks have similar investments and similar VAR models. In periods when markets everywhere decline, the models can tell everybody to sell the same things at the same time, making market conditions much worse. In effect, they can, and often do, create a vicious feedback loop. "
bridgeendogeneityMillennium Bridgepositive feedbackriskSynchronous Lateral ExcitationVaR
CIRM, Decision Theory, Economics, Risk
Course on risk measures (in French)
The course on risk measure, in Luminy, starts at 16.00 on Monday (here). The slides can be found here,
Note that additional references can be downloaded on the internet, e.g. the short course on risk measures by Freddy Delbaen (here) or the article from the Encyclopedia of quantitative finance, by Hans Föllmer and Alexander Schied (there). See also here for the paper by Jean Marc Tallon, Johanna Etner and Meglena Jeleva, on decision theory under uncertainty.
CIRMJESLuminymeasuresmesuresriskRisques
LuisM Rodríguez on R package for Computational Actuarial Science
Simulating Data in R: Examples in Writing Modular Code
Quick Hit: {waffle} 1.0 Font Awesome 5 Pictograms and More
useR! 2019 Slides on Futures
Testing the Collatz Conjecture with R
Cricketr learns new tricks : Performs fine-grained analysis of players
Numerical integration over an infinite interval in Rcpp (part 2) | CommonCrawl |
Euclidean Twistor Unification
This Week's Hype
Posted on November 30, 2022 by woit
This morning Quanta Magazine informs us that Physicists Create a Wormhole Using a Quantum Computer, promoting the article on Twitter with BREAKING: Physicists have built a wormhole and successfully sent information from one end to the other and Physicists have used Google's quantum computer to send a signal through a wormhole, a shortcut in space-time first theorized by Einstein and Rosen in 1935.
This work is getting the full-press promotional package: no preprint on the arXiv (unless I'm missing something?), embargoed info to journalists, with reveal at a press conference, a cover story in Nature, accompanied by a barrage of press releases (see here, here, here, with Harvard, MIT and Google to come). This is the kind of PR effort for a physics result I've only seen before for things like the Higgs and LIGO gravitational wave discoveries (OK, and the primordial gravitational wave non-discovery). It would be appropriate I suppose if someone actually had built a wormhole in a lab and teleported information through it, as advertised.
An additional part of the package is the Quanta coverage, with a very long article by Natalie Wolchover and an over-the-top seventeen minute film How Physicists Created a Wormhole in a Quantum Computer, with abstract
Almost a century ago, Albert Einstein realized that the equations of general relativity could produce wormholes. But it would take a number of theoretical leaps and a "crazy" team of experimentalists to build one on Google's quantum computer.
The two senior physicists behind this, Joe Lykken and Maria Spiropulu, have histories that go way back of successfully promoting to the press nonsense about exotic space-time structures appearing in experiments that have nothing to do with them. Back in 1999, the New York Times published Physicists Finally Find a Way to Test Superstring theory, which featured Joe Lykken. In 2003, they featured Maria Spiropulu explaining how she was going to find extra dimensions (or "something just as 'crazy"") at the Tevatron, or failing that, the LHC.
I just saw that the New York Times also has a big story about this: Physicists Create 'the Smallest, Crummiest Wormhole You Can Imagine'. At least this article has some sensible skeptical quotes, including:
"The most important thing I'd want New York Times readers to understand is this," Scott Aaronson, a quantum computing expert at the University of Texas in Austin, wrote in an email. "If this experiment has brought a wormhole into actual physical existence, then a strong case could be made that you, too, bring a wormhole into actual physical existence every time you sketch one with pen and paper."
An odd thing about the Quanta article is that it contains a couple quotes from me, that aren't at all about the wormhole business. They're about the attempt to use AdS/CFT to either solve QCD or get a viable theory of quantum gravity. Back in June Wolchover contacted me with some questions about AdS/CFT. It seems that she was planning a long piece on AdS/CFT, one which somehow many months later got amalgamated with the wormhole nonsense. I had forgotten that I was thinking of turning what I sent her back then into a blog posting but never got around to it, so just earlier today posted it here.
On the substance of what is really going on here, it's exactly the same as what was discussed extensively a month ago in this posting and in its comment section. The claim that "Physicists Create a Wormhole" is just complete bullshit, with the huge campaign to mislead the public about this a disgrace, highly unhelpful for the credibility of physics research in particular and science in general.
Update: Here's the promotional piece from Google, and Will Kinney's reaction.
Update: Physics World has Quantum teleportation opens a 'wormhole in spacetime' with a quote from Witten saying positive things about this experiment ("a 'milestone' in developing control over microscopic quantum systems"), nothing about the wormholes.
Update: I tried reading the paper in some more detail. Almost all the calculations in the paper were done on paper or on a classical computer. As far as I can tell, all they did was perform elaborate SYK calculations on a classical computer, together with simulations of noise on the Google quantum computer, trying to find a possible calculation on the quantum computer that would have signal, not just noise. Once such an N=7 SYK calculation was identified, they used a 9 qubit quantum computer and the noisy result matched the simulation result from the classical computer, exactly as expected. Seeing the completely expected match between results from a 9 bit noisy quantum computer and the results of the simulation of this on a classical computer caused Maria Spiropulu to say that "I was shaken" and "It was nuts. It was nuts", while Joe Lykken felt that the moment was on a par with discovery of the Higgs particle.
I hadn't noticed that the Nature issue comes with an article by Brown and Susskind, A holographic wormhole traversed in a quantum computer. Amidst the hype, they do at least point out:
because nine qubits can be easily simulated on a classical computer, the results of this experiment cannot teach us anything that could not be learnt from a classical computation, and will not teach us anything new about quantum gravity.
New Scientist is the sober one here, with their headline the relatively reasonable A quantum computer has simulated a wormhole for the first time
Update: MSN is going for the larger context: physicists didn't just create a wormhole in a lab, also This tiny 2D wormhole could finally solve the biggest problem in physics
Update: Andreas Karch on Twitter I think has an accurate characterization of this "mostly a publicity stunt":
Experimentally it's of course cool they can do SYK – as a demonstration they have control over their device. They can couple 9 qbits in a pre-specified way. But I guess we knew they could do this before. Going after SYK in particular, in my mind, is mostly a publicity stunt.
Update: Quanta has changed the title of their article from "Physicists Create a Wormhole" to "Physicists Create a Holographic Wormhole".
The MIT press release is out, and it's comical in the other direction, explaining the huge breakthrough as MIT researchers use quantum computing to observe entanglement.
Chad Orzel is getting flashbacks to 2006, which I can well understand. Many of the worst offenders in this hype campaign were hard at work doing the same thing back then (and earlier), and I was, as now, ineffectually trying to do something about it (the first edition of "This Week's Hype" dates back to that year).
Update: Quanta has also deleted the original "BREAKING: Physicists have built a wormhole and successfully sent information from one end to the other" tweet. Davide Castelvecchi at Nature as a more sober story, ending with
The theory tested at the Google lab "only has a very tangential relationship to any possible theories of quantum gravity in our Universe", says Peter Shor, a mathematician at the Massachusetts Institute of Technology in Cambridge.
Update: More coverage of this here, here, here and here. Quanta and Wolchover are, quite appropriately, blaming the "some of the best-respected physicists in the world" who sold them this nonsense, see here, here and here.
Posted in This Week's Hype, Wormhole Publicity Stunts | 37 Comments
Comments on AdS/CFT
This is something I wrote back in June, for context see the next posting.
First of all, there's the following, which is not strictly scientific, but very relevant to how one decides to evaluate progress in a subject.
The Maldacena AdS/CFT paper is almost 25 years old and has nearly 18,000 citations. Trying to exploit ideas based on AdS/CFT has been the main goal of thousands of the best theorists in the world for decades. Questions like "what about getting this to work in the more physical case of dS?" are not new but very old and have been the subject of tens of thousands of person-years of unsuccessful effort. This doesn't mean it can't be done, I think it does mean that what's needed are some quite different ideas, there's little point in further banging away at ones that haven't worked for this long, after this much effort.
Some of the hype surrounding AdS/CFT has been outrageous. One example is the claim that AdS/CFT gives a good way to calculate things about heavy-ion physics. This is just not true, and the people saying things like this should know better. Seeing people do things like this make me question their arguments about whether other ideas work or not (or have good prospects for working).
As time goes on, people start using "AdS/CFT" to mean a wider and wider array of things. It often now denotes very general and vague conjectures about duality relations between gauge theory and gravity systems, or holography, or entanglement. It becomes impossible at some point to have a coherent discussion about the subject since there is no well-defined thing to talk about.
Sticking to the specific meaning of a duality between a specific superstring theory on AdS_5 x S^5 and N=4 super Yang-Mills on the conformal boundary, back in 1997 there were two reasons to get excited about this:
Reading the duality as telling you about gauge theory in terms of string theory, you could hope that the duality could be extended to non-supersymmetric Yang-Mills, providing the long-sought string dual to QCD, allowing reliable strong-coupling QCD calculations. After a few years post 1997, it started to become clear this wasn't working, and why. N=4 SYM has zero beta-function and is conformally invariant, so the effective coupling and physics are the same at all scales. QCD has a running coupling constant, with weak-coupling and strong-coupling physics very different. AdS/CFT allows strong-coupling calculations on the gauge theory side using weakly-coupled strings, but this has the same problems that we've always had with QCD: there are ways to write down strong-coupling expansions, but no way to match those to weak coupling physics, no way to capture the way physics changes from strong to weak coupling.
By the way, I noticed the Simons Foundation has just announced a collaboration to study QCD and strings
https://www.simonsfoundation.org/2022/06/09/foundation-announces-the-simons-collaboration-on-confinement-and-qcd-strings/
and this doesn't even mention AdS/CFT. People have tried really, really hard over decades to use AdS/CFT to say something about QCD, with very limited results.
Reading the duality as telling you about strings and quantum gravity in terms of gauge theory, the hope is to understand quantum gravity this way. There are a bunch of problems with this:
There's the gravity in AdS rather than dS problem that you mention. As noted above, back in 1997 it was reasonable to expect a useful extension to dS. After 25 years of failed efforts, if there is such a thing it has to be something very different.
You want gravity in 4d, not 5d. This may not be a serious issue since you can take 5d with one small dimension, or brane or whatever to get rid of a dimension.
There's a fundamental problem with doing gravity this way: string theory-based quantum gravity uses weakly coupled strings with the graviton a weakly-coupled mode. But this theory is dual to strongly-coupled gauge theory where you can't calculate anything. So, AdS/CFT is telling you nothing about the usual picture of how gravity arises from string theory. What it supposedly tells you about are strongly coupled strings (using weakly coupled gauge theory), but then the connection to gravity is something very different than what was originally advertised for string theory.
Put together, the problem is that, to the extent AdS/CFT is telling you something about strings and quantum gravity, it's telling you about the wrong kind of space-time (AdS) in the wrong dimension (5) with the wrong kind of strings (strongly-coupled). The general philosophy seems to be that at least it's telling us about some kind of quantum gravity, which is a reasonable motivation, but leaves one far from real physics, in the land of general issues like resolving the black hole information paradox. But 20 years ago we were being told that it was resolved by AdS/CFT, then ten years later we were being told it wasn't (the "Firewall"). Again, given the level of hype people operate with, it's hard to evaluate any of this kind of thing with no relation to anything measurable.
I'm all in favor of good toy models, and from what I can see the main activity in AdS/CFT these days is trying to understand lower dimensional toy models. This leads to lots of interesting things to study, but you seem to end up with very complicated things happening even in much lower dimensions (0+1 SYK models, 1+1 JT gravity models), far from the 3+1 dimensions one wants. There are no physical gravity degrees of freedom below 3+1 dimensions, so it's all too possible that what one is studying in these lower dimensional models is exactly the things of no physical relevance to the real problem.
Finally, my feeling has always been that the difficulty of measuring purely quantum gravitational effects means that the only convincing quantum gravity will be one unified with the rest of physics, fitting together well with what we can observe. The danger with studying pure quantum gravity is that you'll end up with not one theory, but an infinite number of them, empty of any predictive value. The landscape is a realization of that danger.
Update: By the way, this is roughly the 25th anniversary of AdS/CFT, Scientific American has a piece by Anil Ananthaswamy.
The Mystery of Spin
Scientific American has a new article today about the supposedly mysterious fact that electrons have "spin" even though they aren't classical spinning material objects. The article doesn't link to it, but it appears that it is discussing this paper by Charles Sebens. There are some big mysteries here (why is Scientific American publishing nonsense like this? why does Sean Carroll say "Sebens is very much on the right track"?, why did a journal publish this?????).
These mysteries are deep, hard to understand, and not worth the effort, but the actual story is worth understanding. Despite what Sebens and Carroll claim, it has nothing to do with quantum field theory. The spin phenomenon is already there in the single particle theory, with the free QFT just providing a consistent multi-particle theory. In addition, while relativity and four-dimensional space-time geometry introduce new aspects to the spin phenomenon, it's already there in the non-relativistic theory with its three-dimensional spatial geometry.
When one talks about "spin" in physics, it's a special case of the general story of angular momentum. Angular momentum is by definition the "infinitesimal generator" of the action of spatial rotations on the theory, both classically and quantum mechanically. Classically, the function $q_1p_2-q_2p_1$ is the component $L_3$ of the angular momentum in the $3$-direction because it generates the action of rotations about the $3$-axis on the theory in the sense that
$$\{q_1p_2-q_2p_1, F(\mathbf q,\mathbf p)\}=\frac{d}{d\theta}_{|\theta=0}(g(\theta)\cdot F(\mathbf q,\mathbf p))$$
for any function $F$ of the phase space coordinates. Here $\{\cdot,\cdot\}$ is the Poisson bracket and $g(\theta)\cdot$ is the induced action on functions from the action of a rotation $g(\theta)$ by an angle $\theta$ about the $3$-axis. In a bit more detail
$$g(\theta)\cdot F(\mathbf q,\mathbf p)=F(g^{-1}(\theta)\mathbf q, g^{-1}(\theta)\mathbf p)$$
(the inverses are there to make the action work correctly under composition of not necessarily commutative transformations) and
$$g(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta &\cos\theta &0\\ 0&0&1\end{pmatrix}$$
In quantum mechanics you get much same story, changing functions of position and momentum coordinates to operators, and Poisson bracket to commutator. There are confusing factors of $i$ to keep track of since you get unitary transformations by exponentiating skew-adjoint operators, but the convention for observables is to use self-adjoint operators (which have real eigenvalues). The function $L_3$ becomes the self-adjoint operator (using units where $\hbar=1$)
$$\widehat L_3=Q_1P_2-Q_2P_1$$
which infinitesimally generates not only the rotation action on other operators, but also on states. In the Schrödinger representation this means that the action on wave-functions is that induced from an infinitesimal rotation of the space coordinates:
$$-i\widehat L_3\psi(\mathbf q)=\frac{d}{d\theta}_{|\theta=0}\psi(g^{-1}(\theta)\mathbf q)$$
The above is about the classical or quantum theory of a scalar particle, but one might also want to describe objects with a 3d-vector or tensor degree of freedom. For a vector degree of freedom, in quantum mechanics one could take 3-component wave functions $\vec{\psi}$ which would transform under rotations as
$$\vec{\psi}(\mathbf q)\rightarrow g(\theta)\vec{\psi}(g^{-1}(\theta)\mathbf q)$$
Since $g(\theta)=e^{\theta X_3}$ where
$$X_3=\begin{pmatrix}0&-1&0\\ 1&0&0\\0&0&0\end{pmatrix}$$
when one computes the infinitesimal action of rotations on wave-functions one gets $\widehat L_3 + iX_3$ instead of $\widehat L_3$. $S_3=iX_3$ is called the "spin angular momentum" and the sum is the total angular momentum $J_3=L_3 + S_3$. $S_3$ has eigenvalues $-1,0,1$ so one says that that one has "spin $1$".
There's no mystery here about what the spin angular momentum $S_3$ is: all one has done is used the proper definition of the angular momentum as infinitesimal generator of rotations and taken into account the fact that in this case rotations also act on the vector values, not just on space. One can easily generalize this to tensor-valued wave-functions by using the matrices for rotations on them, getting higher integral values of the spin.
Where there's a bit more of a mystery is for half-integral values of the spin, in particular spin $\frac{1}{2}$, where the wave-function takes values in $\mathbf C^2$, transforming under rotations as a spinor. Things work exactly the same as above, except now one finds that one has to think of 3d-geometry in a new way, involving not just vectors and tensors, but also spinors. The group of rotations in this new spinor geometry is $Spin(3)=SU(2)$, a non-trivial double cover of the usual $SO(3)$ rotation group.
For details of this, see my book, and for some ideas about the four-dimensional significance of spinor geometry for fundamental physics, see here.
Update: I realized that I blogged about much this same topic a couple years ago, with more detail, see here. One thing I didn't write down explicitly either there or here, is the definition of spin in terms of the action of rotations on the theory. It's very simple: angular momentum is the infinitesimal generator of the action of rotations on the wave-function, spin angular momentum is the part coming from the point-wise action on the values of the wave-function (orbital angular momentum is the part coming from rotating the argument). Using a formula from my older posting, for a rotation about the z-axis, the total angular momentum operator $\widehat J_z$ is by definition
$$\frac{d}{d\theta}\ket{\psi(\theta)}=-i\widehat J_z \ket{\psi(\theta)}$$
The spin operator $\widehat S_z$ is what you get for $\widehat J_z$ when you act just on the wave-function values. For a spin n/2 state particle, the wave-function will take values in $\mathbf C^{n+1}$. For the spin 1/2 case the action of rotations is by 2 x 2 unitary matrices of determinant one (the spinor representation). For a rotation by an angle $\theta$ about the z-axis, this is
$$e^{-i\theta\frac{\sigma_3}{2}}$$
so the spin operator is
$$\widehat S_3=\frac{1}{2}\sigma_3$$
Posted in Quantum Mechanics | 45 Comments
Math Job Rumors
I noticed yesterday a website named Math Job Rumors that has been operating for a couple months. No idea what the story behind it is other than that it's clearly a descendant of Economics Job Market Rumors, which had some small participation by mathematicians, but is somewhat of a dumpster fire of misinformation, trolling, misogyny and various sorts of juvenile behavior. It looks like someone is trying to provide something similar aimed specifically at mathematicians, with some improvement over the EJMR environment.
One aspect of the site are threads devoted to rumors about tenure track and postdoc hiring in pure math, I don't know if there has been something like this before. In theoretical physics there's the venerable Theoretical Particle Physics Jobs Rumor Mill and the HEP Theory Postdoc Rumor Mill, but these are run in a very different way, with all information posted coming from one or more people who run the site, based on information sent to him/her/them.
The problem with the EJMR or Math Job Rumors model is that anonymity is needed for the whole thing to work, but once you start allowing people to post things anonymously, if you don't moderate what is posted, you'll quickly get overrun by idiots, trolls and other sorts of bad actors. Some kind of moderation is going on at the new site, but it's unclear who is doing it or on what basis.
After starting with the Official Peter Woit blog hate thread, I moved on to reading a few other threads. Lots of dumb stuff, lots of inside jokes, lots and lots of trolling. I confess though that in one case the trolling was clever enough to make me laugh out loud, but it's aimed at a really small audience. I did learn one piece of information that appears to be true, that prominent string theorist Shamit Kachru has gone on leave from his position at Stanford to work as a consultant in the finance industry.
In summary, for those mathematicians who read this blog and feel that they are not wasting enough time on mostly dumb internet stuff, you might want to take a look…
Posted in Uncategorized | 10 Comments
Career Prospects for HEP-TH Students
Guangyu Xu, a student just finishing his Ph.D. at the Centre for Particle Physics at Durham University, recently sent me a public letter he wrote, explaining the story of his job search, in hopes that it might be useful to others in a similar situation. As he acknowledges, his research record has been rather weak, so not surprising that his postdoc applications were not successful.
Back when I was writing Not Even Wrong, I did some detailed research into whatever information I could find about the HEP-TH job market, but I haven't tried to do this more recently. Erich Poppitz did some analysis of data from the Theoretical Particle Physics Jobs Rumor Mill (available here), but only up to 2017. Given the large investment of various government agencies in the support of Ph.D. students, I would think that there would be data on career outcomes gathered by such agencies, but haven't tried to look. Any pointers to this kind of data from anyone who has been looking into it would be appreciated.
Also of interest would be any up-to-date job search advice from those like Guangyu Xu who have been going through this recently.
No Landau-Siegel zeros?
A couple weeks ago rumors were circulating that Yitang Zhang was claiming a proof of a longstanding open conjecture in number theory, the "no Landau-Siegel zeros" conjecture. Such a proof would be a very major new result. Zhang was a little-known mathematician back in 2013 when he announced a proof of another very major result, on the twin prime conjecture. Before that, he had a 2007 arXiv preprint claiming a proof of the Landau-Siegel zeros conjecture, but this was never published and known to experts to have problems such that at best the argument was incomplete.
Zhang now has a new paper on the arXiv, claiming a complete proof. The strategy of the proof is the same as in the earlier paper, but he now believes that he has a complete argument. At 110 pages the argument in the paper is quite long and intricate. It may take experts a while to go through it carefully and check it. Note that this is a very different story than the Mochizuki/abc conjecture story: Zhang's argument use conventional methods and is written out carefully in a manner that should allow experts to readily follow it and check it.
For an explanation of what the conjecture says and what its significance is, I'm not competent to do much more than refer you to the relevant Wikipedia article. For a MathOverflow discussion of the problems with the earlier proof, see here, for consequences of the new proof, see here.
Update: I'm hearing that the above is not quite right, that what Zhang proves is weaker than the conjecture, although strong enough for many of its interesting implications. Perhaps someone better informed can explain the difference…
Update: Davide Castelvecchi at Nature has a news story here.
Update: Via David Roberts on Twitter, Zhang answers some questions about the paper here.
Physical Mathematics c. 2022
Posted on November 9, 2022 by woit
The arXiv today has a very comprehensive survey of a conventional point of view on where "Physical Mathematics" is in 2022 and where it is going, written by a group of six authors. "Physical Mathematics" is a term popularized by one of them, Greg Moore (see here and here, with some commentary here), and it's an expansion of a Snowmass white paper. A separate paper by Nikita Nekrasov covering the material listed in Section 10 is advertised as forthcoming with the title "The Ghosts of Past and Future Ideas and Inspirations on Interface of Physics and Mathematics".
The term "Physical Mathematics" is a play on the more conventional name of "Mathematical Physics" to describe work being done at the intersection of math and physics. In its usage by Moore et al. it refers to a point of view on the relation of math and physics which heavily emphasizes certain specific topics that have been worked on intensively during the last four decades. These topics mostly have roots in seminal ideas of Witten and his collaborators, and involve calculational methods developed in quantum field theory and string theory research. The huge volume of this research is reflected in the fact that the survey reference section contains 62 pages giving 1276 separate references. A major problem for anyone taking up an interest in this field has been the sheer scale and complexity of all this work, and this survey should be helpful in providing an overview.
While some of these 1276 papers could equally well be simply characterized as "Mathematics", it's hard to describe exactly what makes a lot of the rest "Physical Mathematics" rather than "Physics". Part of the answer is that these are not physics papers because they don't answer a question about physics. A striking aspect of the survey is that while a lot of it is about QFT, the only mention at all of the QFT that governs fundamental physics (the standard model) is in a mention of one paper relevant to some supersymmetric extensions of the SM. The only other possible connection to fundamental physics I noticed was about the landscape/swampland, something only a vanishingly small number of people take seriously.
Also striking is the description of the relation of this field to string theory: while much of it was motivated by attempts to understand what string/M-theory really is, section 3.1 asks "What Is The Definition Of String Theory And M-Theory?" and answers with a doubly-boxed
We don't know.
with commentary:
This is a fundamental question on which relatively little work is currently being done, presumably because nobody has any good new ideas.
In the background of this entire subject is the 1995 conjecture that there is a unique M-theory which explains various dualities as well as providing a unified fundamental theory. After nearly 30 years of fruitless looking for this, the evidence is now that there is no such thing, and maybe the way forward is to abandon the M-theory conjecture and focus on other ways of understanding the patterns that have been found.
I share a faith in the existence of deep connections between math and physics with those doing this kind of research. But the sorts of directions I find promising are very different than the ones being advertised in this survey. More specifically, I'm referring to:
the very special chiral twistor geometry of four-dimensions (no twistors in the survey)
the subtle relation of Euclidean and Minkowski signature (only a mention of the recent Kontsevich-Segal paper in the survey)
the central nature of representation theory in quantum physics and number theory (very little representation theory in the survey)
Looking back at Greg Moore's similar 2014 survey, I find that significantly more congenial, with a more promising take on future directions (in particular he emphasizes the role of geometric representation theory).
Posted on October 31, 2022 by woit
CERN on Wednesday is hosting a colloquium talk by Joseph Lykken, who supposedly will discuss Prospects for experimental quantum gravity. There's by now a long tradition of string theorists dealing with criticism that their research program is inherently immune from experimental test by making bogus claims about experimental testability. Lykken has been at it for at least twenty years (see here), and this sort of misleading claim about testability is the latest in a long campaign.
If you read the abstract, it looks like what Lykken is actually talking about is numerical simulations of an SYK model with of order 100 Majorana fermions on a quantum computer. Ignoring the quantum computer hype (unclear how long it will really be before such simulations are feasible), keep in mind that the SYK model is a quantum mechanical toy model, not a model of quantum gravity in a physical dimension. The only thing a quantum computer could test would be the validity of certain approximations schemes in such a toy model. For comments by David E. Kaplan about similar testability claims, see the interview discussed in the previous posting, which includes:
That there are actual people who are deciding string theory's important, wanting to do string theory, and they're even protecting the field. And some of those people are talking about how entropy now of a black hole can be described as a geometric thing, an entanglement, and that Hawking's paradox about evaporating black holes is really wormholes, virtual wormholes coming from the inside to the outside, and all kinds of language. And you could test information theory of black holes using atomic physics experiments. And it's literally bullshit.
There are people—prominent people—in physics who say, "I'm applying for this money from the DOE, but I know it's bullshit." And then there are experimental atomic physicists who don't know and are shocked to learn that "What? String theorists don't have a Hamiltonian? They don't actually have a [laugh] description? What am I testing?"
Update: Lykken was giving Colloquium talks on Experimental String Theory nearly a quarter century ago. He was also one of the main sources for the embarrassing NYT 2000 article Physicists Finally Find a Way to Test String Theory.
"For the first 25 years, the thinking has been that superstring theory is so difficult to see experimentally that you have to figure it out by its own mathematical consistency and beauty," Dr. Lykken said. "Now that's completely changed. If this new picture is true, it makes everything we've been talking about testable."
Hopefully science journalists have learned something and we won't see a forthcoming NYT article on how "Physicists Finally Find a Way to Test Quantum Gravity".
Update: Lykken's slides are here. His proposal for an experimental test of quantum gravity is explicitly acknowledged as the same as one made by Lenny Susskind here in 2017. At the time that made no sense to me and I wrote about it here. It still makes no sense.
David E. Kaplan interview
There's a long interview with David E. Kaplan (not the same person as David B. Kaplan…) by David Zierler at the AIP Oral Histories site. The whole thing is quite interesting and I recommend reading it, but I do want to point out that it shows that I'm a voice of moderation on the string theory issue. Some extracts follow:
About Ann Nelson and string theory in the 1990s:
She was extremely dismissive of string theory, and thought it was—you know, there was—my impression from her and from other people of that generation that weren't doing string theory was that the string theorists were colluding in a sense, or were dismissing anything but string theory, and deciding that if you did string theory then you're much smarter than the people who are not doing string theory. There was some unhappiness in the theoretical field. And the cancellation of the SSC probably added to that tension between the two.
But I don't think she came of it from taking a side. I think she looked at the situation and said, "String theory is total bullshit." In the mid-'80s, there were some realizations—there were some consistency checks that kind of worked in string theory, and people got super excited. Oh, my god, string—yeah, it could be the, you know, underlying thing to particle physics. But that was it.
The successes after that were few and far between. But there was an obsessive—like we're studying the theory of quantum gravity. And it was deridingly called the theory of everything. And then they took that on, you know. We're studying the theory of everything. And then the young people who want to do the greatest stuff would go to string theory. And there was a concern and some upset by the people not doing string theory that they're absorbing a lot of people to do this crap, which is not very physics like. "It's I believe the theory, and so I'm going to study all aspects of it, and maybe one day we'll connect it with the physical world." As opposed to I believe in the phenomenon, and I'm trying to explain that and more, and so I'm going to try out different theories and see what they're consequences are.
And now I look back, and it's obvious that string theory was bullshit in the sense of there were so many people working on it, and they were not manifesting any real progress externally. It was all internal consistency checks and things like that. And so at the time, you know, whenever it came up—and it didn't come up much because there were no string theorists in Seattle—she was just very dismissive, like, you know, "What are those people doing? I don't know what they're doing." [laugh]
About being a postdoc at SLAC:
There were a lot of string theorists at Stanford. I didn't understand any of those talks. Or sometimes when the talks were not in strings, Lenny Susskind would yell at the speaker that this is bullshit or whatever, da, da, da, da—you know, abusive at some level. So Stanford was weird in that way.
About realizing what was going on in string theory, his evaluation of past (Strominger-Vafa) and current claims about string theory and black holes:
But—so I don't—and it's part of probably why I didn't understand—I didn't think of myself as a physicist because there's a lot of physicists working very hard on what? I don't know what they're working on. It's not—you know, I used to just think I'm too stupid to understand what they're working on. And finally reading some of those papers, they're not what—it's stupid. There's a lot of stupid stuff in there. String theory really is just stupid. It's unbelievably stupid. There's so many people who are working on it that don't actually know physics that they can't even describe a physical characteristic of the thing they're calculating. They're missing the whole thing.
So that's when I realized string theory is like a video game. There are people just addicted to it. That's all that's happening. And it's couched in the theory of everything and da, da, da, da.
So that's all. I just kind of—I learned quite a bit about these things. And then I saw the people like Lenny Susskind, who was terrorizing people who work on regular physics, as just a plain asshole. That there are actual people who are deciding string theory's important, wanting to do string theory, and they're even protecting the field. And some of those people are talking about how entropy now of a black hole can be described as a geometric thing, an entanglement, and that Hawking's paradox about evaporating black holes is really wormholes, virtual wormholes coming from the inside to the outside, and all kinds of language. And you could test information theory of black holes using atomic physics experiments. And it's literally bullshit.
So I have converted a little bit to the opinions of my predecessors, only because I've actually done the work. I've actually tried to understand black holes of late, and I've gone back to those papers which are the breakthrough, celebrated, amazing papers about black holes, and there's nothing in them. It's really—it's just a very simplistic picture where, look, if you take this hyper-simplistic picture, these numbers match these numbers, which means thinking about a black hole having entropy is correct, da, da, da, da, da.
No matter that the black hole they're talking about is extremal. It doesn't actually Hawking radiate. It's a totally hyper-supersymmetric, multiple charges, free parameters. So now that I've finally dug into it, I realize that—not that all humanities fields are bad. But it's much more like a humanities field where there are the prominent people in the field, and they decide what's interesting. And that if you impress those people, you can get ahead. But that dictates then what research is done. And they're not going to discover anything in that context. They're not going to get anywhere. There's not a lot of people doing—you know—thinking outside the box or just thinking diff…you know, doing different things, you know.
About the argument that string theory must be worthwhile because lots of people are doing it:
Zierler:
What is your response to a string theorist who would say, and I know this because one has said this to me, "Look, four people were doing this in 1968, 20 people were doing it in 1984, 1,000 people were doing it in 2000, and now there's 6,000 people who are doing string theory all over the world. And that's proof that there's something here that's worthwhile"? What is your response to that line of reasoning?
Kaplan:
[laugh] Take those numbers, continue the exponential, and apply it to Christianity—
[laugh]
—and Islam and Judaism and Buddhism. Give me a fucking break. They're describing a religion that can attract and addict people. That is exactly the kind of statement that shows it's bullshit and non-scientific. They've proven it for me that they are not about discovering something. They're about dominating the field for the purpose of what? That's proof? Give me a break. Give me a fucking break. Slavery was very popular, and became widely used. Nazism. Come on. You can take extreme examples and show that that is so non-scientific and sick that the progress they have made is to get more people to work on something that isn't producing anything. Oh, man, I wish you didn't tell me that. [laugh]
About the current state of the field:
There are so many things to think about. I don't know what narrowed our field. I don't see it as we're dying because we're coming to the limits of what we can do, the limits of what we can calculate in string theory, and the limits of how big of a ring we can build. I think most people are just doing useless stuff.
And so that's why I—the whole depression or whatever, that's a product of the non-willingness to feel stupid by the majority of our field. Expertise is more important to them than discovery. And that's what I think is happening. And so what we're seeing is not the death of the field, but the death of a direction that is being committed to by 98% of them.
Will Machines Have Good Mathematical Taste?
A question that has always fascinated me about mathematics is that of how the field manages to stay healthy and not degenerate in the way I've seen theoretical physics do as it lost new input from experiment. On Twitter, Ash Joglekar gave a wonderful quote from von Neumann that addresses this question. The quote was from a 1947 essay "The Mathematician" (available here and here). von Neumann argues that:
…mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science.
but warns
As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from "reality" it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l'art pour l'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities.
which describes all too well what has happened to string theory. What saves a field from this? "Men with an exceptionally well-developed taste"? He poses the general question this way:
What is the mathematician's normal relationship to his subject? What are his criteria of success, of desirability? What influences, what considerations, control and direct his effort?
Normally mathematicians are loath to debate this kind of "soft" topic, but the rise of computer software capable of producing proofs has recently led several first-rate mathematicians to take an interest. Each year the Fields Institute in Toronto organizes a Fields Medal Symposium, structured around the interests of a recent Fields Medalist. This year it's Akshay Venkatesh, and the symposium will be devoted to questions about the changing nature of mathematical research, specifically the implications of this kind of computer software. Last year Venkatesh wrote an essay exploring the possible significance of the development of what he called "Alephzero" (denoted $\aleph(0)$):
Our starting point is to imagine that $\aleph(0)$ teaches itself high school and college mathematics and works its way through all of the exercises in the Springer-Verlag Graduate Texts in Mathematics series. The next morning, it is let loose upon the world – mathematicians download its children and run them with our own computing resources. What happens next – in the subsequent decade, say?
Among the organizers of the conference is Michael Harris, who has written extensively about mathematical research and issues of value in mathematics. Recently he has been writing about the computer program question at his substack Silicon Reckoner, with the most recent entry focusing on Venkatesh's essay and the upcoming symposium.
One of the speakers at the symposium will be Fields medalist Tim Gowers, who will be addressing the "taste" issue with Is mathematical interest just a matter of taste?. Gowers is now at the Collège de France, where he is running a seminar on La philosophie de la pratique des mathématiques.
I've tried asking some of my colleagues what they think of all this activity, most common response so far is "why aren't they proving theorems instead of spending their time talking about this?"
Update: For yet more about this happening at the same time, there's a talk this afternoon by Michael Douglas on "How will we do mathematics in 2030?".
Update: The talks from the Fields Institute program are now available online.
Terry Tao is one of the organizers of a planned February workshop at UCLA involving many of the same people, much the same topic.
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Stony Brook Mathematics | CommonCrawl |
Evolution of communities of software: using tensor decompositions to compare software ecosystems
Oliver A. Blanthorn ORCID: orcid.org/0000-0002-1324-128X1,
Colin M. Caine2 &
Eva M. Navarro-López3
Applied Network Science volume 4, Article number: 120 (2019) Cite this article
Modern software development is often a collaborative effort involving many authors through the re-use and sharing of code through software libraries. Modern software "ecosystems" are complex socio-technical systems which can be represented as a multilayer dynamic network. Many of these libraries and software packages are open-source and developed in the open on sites such as GitHub, so there is a large amount of data available about these networks. Studying these networks could be of interest to anyone choosing or designing a programming language. In this work, we use tensor factorisation to explore the dynamics of communities of software, and then compare these dynamics between languages on a dataset of approximately 1 million software projects. We hope to be able to inform the debate on software dependencies that has been recently re-ignited by the malicious takeover of the npm package event-stream and other incidents through giving a clearer picture of the structure of software dependency networks, and by exploring how the choices of language designers—for example, in the size of standard libraries, or the standards to which packages are held before admission to a language ecosystem is granted—may have shaped their language ecosystems. We establish that adjusted mutual information is a valid metric by which to assess the number of communities in a tensor decomposition and find that there are striking differences between the communities found across different software ecosystems and that communities do experience large and interpretable changes in activity over time. The differences between the elm and R software ecosystems, which see some communities decline over time, and the more conventional software ecosystems of Python, Java and JavaScript, which do not see many declining communities, are particularly marked.
Contemporary software authors routinely depend on and re-use the software packages of authors with whom they have no contact. This uncoordinated process creates what have recently been called "software ecosystems" (Decan et al. 2018): extensive networks of interdependent software components that are used and maintained by large communities of contributors all over the world. These ecosystems are complex multi-layered networks whose nodes and edges both evolve over time. Throughout this work, terminology such as 'package', 'package manager', 'dependency' follow the usual software engineering convention, as documented by Decan et al. (2018). With this work we would like to contribute to the debate on software dependencies, which recently re-emerged due to the hijacking of the event-stream package after previous incidents (Baldwin 2018; Schlueter 2016; Durumeric et al. 2014), by giving a clearer picture of the structure of software dependency networks. For this, we propose a novel framework to model and analyse the formation, long-term behaviour and change with time of communities of software packages, and compare these behaviours across several programming languages.
The evolution of software and package dependency networks has been extensively studied by using network science techniques over the past 15 years. An early such work is (Myers 2003), which has been followed by others (Pan et al. 2011; Xu et al. 2005; Zheng et al. 2008). A recent survey on this topic is given by Savić et al. (2019). Methods and tools used for social networks have also been applied to analyse software evolution (Chatzigeorgiou and Melas 2012), reinforcing the importance of the social component in evolving software networks. There has been recent success in applying standard network metrics to analyse how software ecosystems have evolved with time (Decan et al. 2018).
Collaborative software networks—mainly in free and open-source software (FOSS) environments—can be considered as adaptive, evolving or temporal networks. Most of the published works highlight the fact that software networks exhibit scale-free network properties with a power-law-type node degree distribution (Cai and Yin 2009; Lian Wen et al. 2009; Louridas et al. 2008) and a clear preferential attachment in the network growth process (Li et al. 2013; Chaikalis and Chatzigeorgiou 2015), confirming the heterogeneity and hierarchical characteristics of networks of software. Other key properties in software systems like maintainability and reliability (Chong and Lee 2015), robustness (Gao et al. 2014), and modularity (Zanetti and Schweitzer 2012) have been also analysed using the complex network paradigm.
The identification of communities in software dependency networks that evolve over time is one of the main motivations of our work. Community detection in temporal, evolving or adaptive networks has largely attracted network scientists' attention due to its important implications in the analysis of dynamical processes in complex networks, such as spreading and cascading dynamics, stability, synchronisation and robustness. Different types of methods and algorithms have been used, for example: the Louvain algorithm (Aynaud and Guillaume 2010), statistical null models (Bassett et al. 2013; Sarzynska et al. 2016), algorithms which exploit the historic community structure of the network (He et al. 2017; He and Chen 2015), Markov models (Rosvall et al. 2014), semidefinite programming (Tantipathananandh and Berger-Wolf 2011), gravitational relationship between nodes (Yin et al. 2017), and temporal matrix factorisation (Yu et al. 2017), amongst others. Machine learning techniques (Savić et al. 2019; Xin et al. 2017), genetic algorithms (Folino and Pizzuti 2014), consensus clustering (Aynaud and Guillaume 2010) and tensor factorisation (Araujo et al. 2014; Gauvin et al. 2014) have only recently been used for the detection of communities in temporal networks.
There has been much less work on finding clusters or communities in software dependency networks: some representative works are Dietrich et al. (2008); Paymal et al. (2011); Concas et al. (2013) and Savić et al. (2012). However, it is still a challenge to give satisfactory solutions for the dynamic treatment of these clusters and inter-language comparison. Savić et al. (2012), for example, give some dynamic treatment of communities within the "class collaboration" network of Apache Ant. This is a single piece of software in which the classes are more witting participants in the software than in general software dependency networks where the authors of a package may have no idea where their work is being used. Additionally, the community detection methods used are static, so it is the metrics on the types of communities found at each version of Apache Ant which are compared rather than the qualities of any individual communities being tracked across time.
The novelty of our work is the detection of dynamic communities in temporal software dependency networks, the use of tensor decompositions on software ecosystems, and the use of adjusted mutual information (AMI) to assist in choosing the number of communities. Additionally, some of the networks we study are amongst the largest dynamic networks to which tensor factorisation has been applied, although there have been studies considering much larger static networks (Kang et al. 2012).
Our work addresses three research questions corresponding to unsolved problems in collaborative large-scale software development and evolution over time:
RQ1 What are the differences between different software ecosystems?
RQ2 What do communities of commonly-used-together software packages look like?
RQ3 How do these communities change with time?
By answering these questions, we identify communities of packages in the ecosystems of several languages. Namely: Elm; JavaScript; Rust; Python; R; and Java. These languages were chosen because data on their package ecosystems are readily available and they represent a variety of uses and ages: low-level systems languages; scientific computing; and web development; with histories between 3 and 20 years long.
The rest of this manuscript is structured as follows. Firstly, in the "Methods" section, we detail the data, mathematical tools, software and hardware used to conduct this study. Next, the "Results and discussion" section presents our results and provides some discussion of why they might have arisen. Finally, the "Conclusions" section compares our results with prior work and provides a summary of this works' limitations while elucidating further avenues of research.
What is a community?
Software packages associated with a single programming language and package manager form an ecosystem, and a community is a collection of packages that tensor decomposition has identified. Communities may be related by some theme and by co-occurrence of activity over time. They might be collections of numerical computing packages, or a community of packages that use a particular library, or a community of older packages that became obsolete. Packages may be in more than one community, which is modelled by vectors of continuous strengths of membership to each community as detailed in the "Tensor decompositions" section below. Communities may wax and wane over time.
Further detail on how we calculate communities and membership strengths thereof is given in the "Tensor decompositions" and "Choosing the number of communities, R" sections.
Shape of the data
We considered the following package managers (corresponding to specific languages): elm-get (Elm), npm (JavaScript), crates.io (Rust), PyPI (Python), CRAN (R), and Maven (Java). All publicly registered packages for these package managers were included in our analysis. To simplify our analysis, we do not consider specific versions of dependent packages—each package depends on some other packages at each time, each of which is assumed to be the most up-to-date package available at that time. We only consider packages registered with these package managers and do not consider cross-platform (inter-language) dependencies. The data cover 1 million projects listed in package managers. This is sourced from Libraries.io (Katz 2018). The network of Elm dependencies at the final time-step is shown in Fig. 1.
Elm network. Elm network visualised using Gephi. Node and label size corresponds to out-degree. Each node i is coloured according to the strongest strength of community membership given by κi as described in the Methods section. Legend: light-green corresponds to the community we labelled as elm-lang/core, purple to elm-lang/http, blue to elm-lang/html, and orange to evancz/elm-html, dark-green to unassigned nodes, i.e. those without any dependencies
Brief introduction to each ecosystem
Here, we will briefly introduce each language and ecosystem considered.
(elm-lang.org) is a relatively small new language created in 2012 as a functional language for developing web applications. It is unusually focused on a single domain and unusually restrictive in what can be expressed in listed packages. The language authors explicitly discourage proliferation of packages in favour of a unified and carefully designed standard package for each task. It aims to eliminate runtime errors and many classes of logical errors through static analysis and careful language design.
is a very popular general-purpose interpreted programming language. It was first released in 1991. Its package manager, PyPI (https://pypi.org/), was released in 2002. Python is notable for having a large standard library; many features such as HTTP and GUI support that would be external packages in other languages are included by default. We therefore suspect that Python's package ecosystem will look quite different to many of the other languages we consider. It is perhaps pertinent to note that some of the packages listed in PyPI are designed to be used by end-users, not developers.
is another popular general-purpose interpreted programming language. It was first released in 1995 as part of the Netscape browser and was initially used for client-side scripts on web pages, but now has several server-side implementations, most notably Node.js. JavaScript has a very small standard library, with the exception of the DOM interface (Document Object Model—a representation of a web page), which is included. We therefore expect most JavaScript packages to have to depend on many other packages. JavaScript's most common package manager is npm (npmjs.com), the Node.js package manager, which was released in 2010. The JavaScript ecosystem we describe in this paper is the public npm registry, which is the de facto canonical list of JavaScript packages. Despite its name, the registry is used for all types of JavaScript, not just server-side applications. Like PyPI, some of the packages listed in npm are designed to be used by end-users, not developers.
is another relatively new systems programming language. Unlike many of the other ecosystems we have considered, it is statically compiled. One of its main aims is safe concurrent programming. We suspect that because of this conservative focus, packages written in Rust will have relatively few dependencies. Rust's package manager is Cargo, which was released in 2014, and its registry is crates.io. Like PyPI, some of the packages listed on crates.io are designed to be used by end-users, not developers.
is an interpreted programming language for statistical computing. It was first released in 1993. Its standard library for statistical computing is large. Its package manager is the Comprehensive R Archive Network (CRAN cran.r-project.org).
is a general-purpose typed compiled programming language, first released in 1995. Its standard library is large. Java has no official package manager, but the most widely used is Apache Maven (maven.apache.org), which was released in 2004.
Representing the data
We represent each language's network as a tensor, Alang, where each
$$ {A}_{ijt}\in \{0,1\} $$
denotes whether the package i depends on the package j at time t. We consider links as going from j to i, so the direction of each link reflects the flow of code. Software packages have unweighted directed links to packages that they depend on at time t according to the most recently available version of the software package at that time. The time is quantised into timesteps. We chose a resolution of one month on the basis that packages are long-lived and change dependencies rarely.
Tensor decompositions
A rank-N tensor X can be approximated as
$$ X \approx \sum_{r=1}^{R} \text{reduce}_{n=1}^{N}(\otimes,\boldsymbol{a}_{nr})\text{,} $$
where \(\text {reduce}_{n=1}^{N}(\odot,b_{n})\) returns the reduction of {bn|n∈1..N} by applying an arbitrary associative binary operator ⊙ repeatedly, i.e.
$$ \text{reduce}_{n=1}^{N}(\odot,b_{n}) = b_{1} \odot b_{2} \odot b_{3} \odot \ldots \odot b_{N}\text{;} $$
⊗ is the tensor product; and each \(\boldsymbol {a}_{nr} \in \mathbb {R}^{d_{n}+}\) where dn is the dimension of the nth-mode of the tensor X.
Each anr for a specific n is called a factor, and maps its indices (in our case, software packages or time) to strength of membership to the community r. The total number of communities or components, R, is a fixed chosen parameter. Each anr for a specific r is called a community or component. The concept of a temporal index having strengths of community membership may seem odd; it is perhaps more easily understood as the amount of community activity at a certain time (Gauvin et al. 2014).
There are many algorithms for performing tensor decompositions of this kind. We selected a non-negative method with a public implementation that will allow us to later extend our analysis to large rank-N tensors: Alternating Proximal Gradient (APG-TF)Footnote 1 (Xu and Yin 2013), an efficient non-negative CANDECOMP/PARAFAC (CP) method. We consider the use of tensor decomposition (Kolda and Bader 2009) more adequate for community detection in temporal networks than other static methods, such as the Louvain algorithm (Aynaud and Guillaume 2010), because tensor decompositions explicitly integrate evolution over time and can scale very efficiently for large graphs.
For our specific application, we approximate our temporal adjacency matrix as
$$ A_{ijt} \approx \sum_{r=1}^{R} \kappa_{ir} h_{jr} \tau_{tr}\text{,} $$
where κri are the elements of a matrix \(K\in \mathbb {R}^{+N_{p}\times R}\) that describe the strengths of community membership of each node i based on outgoing links, hrj are the elements of a matrix \(H\in \mathbb {R}^{+N_{p}\times R}\) which represent the strengths of community membership of each node j based on incoming links, τrt are the elements of a matrix \(T\in \mathbb {R}^{+N_{t}\times R}\) that describe the activity levels of each community r at each time step t, and Np and Nt are the number of packages considered and the number of timesteps chosen, respectively.
We will now describe H and K in more detail. We will use two facts about software dependency networks (Decan et al. 2018):
the number of outgoing links from each package (number of packages that depend on a package) follows a power-law;
the number of incoming links to each package (number of packages that a package depends on) is fairly evenly distributed.
Since H represents groups of nodes which are linked to by similar nodes, it is very sparse with only a few hub nodes having any community membership due to high strengths in H. K is much more evenly distributed as it represents groups of nodes which link to similar nodes; most packages are strongly associated with at least one community.
If a community has significantly declined from its peak according to a plot of τr, we will define its lifetime as the full width at half maximum.
Choosing the number of communities, R
One of the major downsides to tensor decompositions is that one must choose the number of communities, R. We take a multi-faceted approach to evaluating our choice of R for each ecosystem, which we detail in this section. The first component of our approach is to look at the normalised residual sum of squares or relative error, NRSSR, a measure of reconstruction error (Papalexakis et al. 2012). We use it to inform our choice of R by looking for the number of components where the rate at which the residual sum of squares decreases as the number of components increases becomes linear, commonly known as an elbow. We define the normalised residual sum of squares as:
$$ \text{NRSS}_{R}(X) = \frac{\|X - \sum_{r=1}^{R} \text{reduce}_{n=1}^{N}(\otimes,\boldsymbol{a}_{nr})\|_{F}}{\|X\|_{\mathrm{F}}}\text{,} $$
where ∥X∥F is the Frobenius norm of X.
The second component of our approach is to run the decomposition multiple times for each likely R to determine whether the decomposition is stable, that is, if multiple runs produce the same results. We look to see if nodes appear in the same communities as each other by considering the mean pairwise adjusted mutual information (AMI, Vinh et al. (Vinh et al. 2009)) between repeated runs for a single R. We use hard clustering for this process by assigning nodes to communities from the maximum strength in each vector κi.
We also check visually to see if the temporal activity of communities looks similar across multiple decompositions. If the decomposition is unstable, i.e. if multiple runs produce different results, it suggests that ground-truth communities are being merged or split up. Another check we perform is to look at the metadata associated with a sample of software projects from each community and make a qualitative judgement as to whether they are similar. For example, if a numerical computing package and a front-end web development package are in the same community, the number of communities is probably too small. Our final check is whether adding an extra component adds interesting new behaviour to the temporal community activity, or if it instead seems to break an existing community into noisy subcommunities. All of these approaches are quite subjective, so there is quite a large uncertainty in exactly how many communities is ideal for each language.
Adjusted mutual information
The adjusted mutual information between two partitions U and V where U={U1,U2,…} is a finite-length partition of some set of length N and V is another finite partition of that set is
$$ \text{AMI}(U,V) = \frac{I(U;V) - E[I(U;V)]}{\text{max}(S(U),S(V)) - E[I(U;V)]}\text{,} $$
where the entropy S is
$$ S(U)=-\sum_{i}\frac{|U_{i}|}{N}\text{log}\frac{|U_{i}|}{N}\text{,} $$
the mutual information between U and V is
$$ I(U;V)=\sum_{i=1}^{|U|} \sum_{j=1}^{|V|} \frac{|U_{i}\cap V_{j}|}{N}\log\frac{N|U_{i} \cap V_{j}|}{|U_{i}||V_{j}|}\text{,} $$
and E[W] is the expectation of a random variable W. Here, it can be calculated using a hypergeometric model of all possible random clusterings (Vinh et al. 2009). AMI measures the amount of agreement between two partitions. It has an upper bound of 1. An AMI score of 0 corresponds to the amount of agreement one would expect if one of the partitions was totally random, and 1 corresponds to perfect agreement between the two clusters. The AMI score can be negative if there is more disagreement between the clusters than one would expect from random chance. It is hard to interpret the meaning of scores between 0 and 1, but it is important to bear in mind that anything greater than 0 is a good score in the sense that it is better than random guessing. AMI is quite sensitive to the number of clusters and awards higher scores to pairs of clusterings that agree on the same number of assignments but have a larger number of clusters, to the limit where if each element is given its own cluster the AMI must be 1. Since we are mostly using AMI to compare clusterings with each other and have far fewer clusters (or communities) than nodes, we believe the interpretability of the absolute value of the AMI is not important. We are concerned primarily with the relative changes in AMI as we move from one number of communities to another.
Labelling communities
H is a sparse matrix that denotes the packages that are most depended on in each community. The package name associated with a community r is the name of the package pr that is maximal in hr, such that pr does not have a higher \(\phantom {\dot {i}\!}\boldsymbol {h}_{r'}\) value in any other community r′. That is,
$$ p_{r} = \text{indmax}([\text{mask}(H)]_{r})\text{,} $$
where indmax(x) of any vector x returns the index of its maximum value, and
$$ [\text{mask}(M)]_{ij} = \begin{cases} m_{ij} &\text{if} m_{ij} = \text{max}(\boldsymbol{m}_{i}) \\ 0 &\text{otherwise,} \end{cases} $$
where mij and mi are the elements and column vectors of a matrix M.
Hardware and software used
Julia (Bezanson et al. 2012), LightGraphs.jl (Bromberger 2017), Plots.jl, and Jupyter (Kluyver et al. 2016) with IJulia were used for exploratory data analysis, data pre-processing, figures and co-ordinating external software. The actual tensor decompositions were performed using MATLAB.jl, MATLAB, and TensorToolbox (Bader and Kolda 2007). Adjusted mutual information was calculated using ScikitLearn.jl and scikit-learn (Pedregosa et al. 2011). The decompositions were performed on 20-core Xeon servers with 250GB of RAM provided by MARC1, part of the High Performance Computing and Leeds Institute for Data Analytics (LIDA) facilities at the University of Leeds, UK.
We examine the pairwise average adjusted mutual information (AMI) score and NRSSR(X) as defined in Eq. (2) to identify a suitable number of communities for each ecosystem. A high AMI would indicate that the communities created by the decomposition are stable across repeated runs of the algorithm.
Troughs between peaks in our AMI plots would indicate that communities are being split inappropriately. We will interpret the first peak in AMI as the minimum valid number of communities for an ecosystem.
Each community will then be labelled according to Eq. (3).
Table 1 shows that there are clear differences between the language ecosystems in all the statistics we calculate.
Table 1 Summary statistics for each ecosystem
Degree distribution
Python packages have a mean number of 2.4±2.9 direct dependencies where JavaScript packages have 9.9±22. Figures 2, 3, 4, 5, 6 and 7 show that the distributions of in and out degree look fairly similar in shape; broadly, all out degree distributions follow a power-law and the in degree distributions follow more Poissonian distributions.
Elm degree distributions. In and out degree distributions for the Elm package ecosystem
PyPI degree distributions. In and out degree distributions for the PyPI package ecosystem
npm degree distributions. In and out degree distributions for the npm package ecosystem
Cargo degree distributions. In and out degree distributions for the Cargo package ecosystem
CRAN degree distributions. In and out degree distributions for the CRAN package ecosystem
Choosing the number of communities: Elm. Left: the mean pairwise adjusted mutual information of the community-node membership across repeated decompositions. 1 is a perfectly stable assignment to communities; 0 is perfectly random. The shaded area corresponds to twice the standard error of the mean. Right: the mean normalised residual sum of squares error for repeated decompositions on the Elm network against the number of components. The shaded area corresponds to twice the standard error of the mean above and below the line
The exception to this rule is npm, the JavaScript ecosystem, whose in-degree distribution is bimodal: there are more packages with the highest numbers of dependencies than there are with middling numbers of dependencies. Upon further investigation, it turns out that the distribution is skewed by a few hundred joke/malicious packagesFootnote 2 and some irregularly packaged packages that list all their recursive dependencies as their direct dependenciesFootnote 3. We chose not to omit these packages from the decomposition as we hoped that tensor decomposition might discover them as a community.
We believe that the presence of a rich and well-used standard library—the functions included in the language without any other dependencies—reduces the mean degree of an ecosystem. Python (degree 2.4) has such a library, while JavaScript (degree 9.9) emphatically does not.
Elm (degree 2.6) features a small standard library compared to Python, but its library is very rich within its domain of web application development.
Rust (degree 4.4) features a small standard library by design (instead promoting and curating community packages). R (degree 5.3) features a fairly small standard library, and many users now prefer the community tidyverse collection as a consistent community alternative.
Number of communities
Figures 7, 8, 9, 10 and 11 show mean AMI and NRSS for each number of communities we evaluated for each ecosystem. We computed 10 decompositions for each trialled number of communities in each ecosystem.
Choosing the number of communities: PyPI. Left: the mean pairwise adjusted mutual information of the community-node membership across repeated decompositions. 1 is a perfectly stable assignment to communities; 0 is perfectly random. The shaded area corresponds to twice the standard error of the mean. Right: the mean normalised residual sum of squares error for repeated decompositions on the PyPI network against the number of components. The shaded area corresponds to twice the standard error of the mean above and below the line
npm adjusted mutual information against components. The pairwise adjusted mutual information of the community-node membership across repeated decompositions. 1 is a perfectly stable assignment to communities; 0 is perfectly random. The shaded area corresponds to twice the standard error of the mean above and below the line
Choosing the number of communities: Cargo. Left: the mean pairwise adjusted mutual information of the community-node membership across repeated decompositions. 1 is a perfectly stable assignment to communities; 0 is perfectly random. The shaded area corresponds to twice the standard error of the mean. Right: the mean normalised residual sum of squares error for repeated decompositions on the Cargo network against the number of components. The shaded area corresponds to twice the standard error of the mean above and below the line
Choosing the number of communities: CRAN. Left: the mean pairwise adjusted mutual information of the community-node membership across repeated decompositions. 1 is a perfectly stable assignment to communities; 0 is perfectly random. The shaded area corresponds to twice the standard error of the mean. Right: the mean normalised residual sum of squares error for repeated decompositions on the CRAN network against the number of components. The shaded area corresponds to twice the standard error of the mean above and below the line
Of all the techniques mentioned in the "Choosing the number of communities, R" section, identifying a suitable number of communities was easiest by looking for early peaks in the AMI plots, which correlated well with visual inspection of decomposition stability and our manual checks on package names. The NRSS plots have no obvious elbows and contributed nothing to our analysis; we include them as a relevant negative result.
For Elm, CRAN, and PyPI, AMI indicates more than one "good" number of communities to split the ecosystem into. We have chosen the smallest good number except when that number was two.
We observed interesting differences between ecosystems in this metric: Elm and R have relatively high AMIs at relatively low numbers of components while other languages such as JavaScript had low AMIs for all of the numbers of components we tried, suggesting that a more representative number of components might be much higher. This tells us something about the diversity of an ecosystem: the more communities it has, the more diverse it must be.
Activity over time
Figures 12, 13, 14, 15, and 16 show the relative activity of communities in each ecosystem over time for a representative decomposition; i.e. they are plots of τrt from Eq. (1) against time, t.
Elm community activity. Relative activity of each community in each month. The communities are named by important packages as detailed in the Methods section
PyPI community activity. Relative activity of each community in each month. The communities are named by important packages as detailed in the "Methods" section
npm community activity. Relative activity of each community in each month. The communities are named by important packages as detailed in the "Methods" section
Cargo community activity. Relative activity of each community in each month. The communities are named by important packages as detailed in the "Methods" section
CRAN community activity. Relative activity of each community in each month. The communities are named by important packages as detailed in the "Methods" section
The labels are reasonably stable across decompositions when AMI is high and they appear to be informative in most ecosystems. The relative size and activity over time of the communities can often be matched with what we know about the their constituent packages and the ecosystem from other sources.
The community activity plots show clear differences between the various ecosystems. Elm, R and Rust all contain communities that decline in activity over time. Some languages such as Elm, R and Rust have "peaky" community activity where packages have been replaced by others. Java, Python and JavaScript do not exhibit such behaviour; suggesting that their communities are more stable. The "% declined" column in Table 1 summarises this behaviour.
Community composition
Comparing ecosystems, we observe that Python has communities centred around making websites (django), scientific computing (numpy), or interpreting configuration files (pyyaml). By contrast, Elm's communities are all to do with building web applications, and R's communities are mostly focused towards scientific computing. This suggests, uncontroversially, that Elm and R are less general purpose languages than Python.
In the following sections we examine and provide interpretations for the composition of communities within each ecosystem.
Ecosystem-specific results
Figure 12 shows community activity over time. evancz/html is the precursor of elm-lang/html and we can see an early community of packages using evancz/html that decays over time as the newer community using elm-lang/html emerges. We can also see the swift rise of a community of HTTP-using packages. This is perhaps particularly pronounced in Elm because the language is unusually restrictive (there was no official way to write HTTP packages before the package which became elm-lang/http) and young (there was a strong known demand for an obvious feature like HTTP support).
The evancz/html deprecation neatly shows that many earlier packages were abandoned and did not make the jump to Elm 0.17 and elm-lang/htmlFootnote 4. Those that do make the jump move community and those that do not remain in the package manager as abandonware.
Elm is a young and deliberately unconventional and experimental language, so it is to be expected that many packages will be abandoned as the community experiments with different approaches.
The rapid changes in activity indicate that the Elm ecosystem is not stable, that is, communities have experienced large relative changes in activity recently.
High AMI for communities two and four suggest that the Elm ecosystem is not diverse as it is well described by small numbers of communities.
Figure 13 demonstrates how these communities change over time: it is interesting that there are no communities which have declined particularly. One community, django, a web application framework, seems to have stagnated. Google Trends data suggest that interest in it has reached a plateau compared to frameworks in other languages such as React for JavaScript.
The growth of communities, as shown in Fig. 13, seems to be monotonic. There do not appear to be any major deprecated packages, which is surprising, given the impending deprecation of Python version 2, which is incompatible with Python version 3. This could be due to the popular package six which provides compatibility helpers for Python 2 and 3.
The AMI plot for npm is given in Fig. 9. We chose not to calculate the NRSS values due to computational constraints. The AMI for every number of communities we trialled is very low and although we use three communities in the time activity plot, Fig. 14, we are not particularly confident that the communities picked are meaningful. It is plausible that some larger number of communities would fit the data better, however we trialled up to 50 communities with no success.
The AMI and NRSS plots for Rust are shown in Fig. 10. The AMI plot strongly indicates a first peak at 8 communities.
Figure 15 demonstrates how these communities change over time. Of particular note are the communities centered around rustc-serialize and serde. rustc-serialize has been deprecated and officially replaced with the serde package. The decline in activity of the rustc-serialize community and rise of the serde community is quite distinct and clearly occurs at the same time.
The community labelled winapi is also easily interpretable. The K factor for this community is bimodal and the size of the higher strength cluster indicates that there are around 450 packages in this community. A brief review of these packages suggests that they are low-level cross-platform packages for interacting with the operating system. These packages were amongst the first written and predate the package manager, which explains their steep initial rise.
As shown in Fig. 11, AMI suggests that decomposition into two, seven, or nine communities is acceptable. That CRAN can be stably decomposed into two communities suggests that the ecosystem is not very diverse, which is as we would expect from a language focused on scientific and statistical computing.
It can be seen in Fig. 16 that when decomposed into seven communities, distinctive waves of activity over time exist. We believe these waves are driven by CRAN's rolling release model which requires packages to be actively maintained or de-listed. This is a strong incentive for package authors to switch dependencies if e.g. one of two testing libraries is actively maintained. The de-listing also means that older, unused packages are removed from the dependency graph over time, unlike any of the other studied ecosystems.
Visible in Fig. 16 and an exemplar of this behaviour, RUnit is an older unmaintained testing library and testthat is a more modern replacement. Unlike any other long-established language in this analysis, there is a clear and steep decline of the earlier library in favour of the competitorFootnote 5.
Despite R being an old language, some of its communities have grown at a rapid rate, specifically those relating to knitr, and testthat. This suggests that, despite being an older language, R's ecosystem is still subject to significant change.
Defining community lifetime as the full width at half maximum, we can see from Fig. 16 that communities tend to stay active for approximately 4 to 8 years.
The AMI plot, Fig. 17, strongly indicates an unambiguous peak at 5 communities. AMI is very low for two communities and comparatively low for three and four, suggesting that Maven has a wider diversity of communities than CRAN or elm-get. These communities include a distinct Scala community, a kind of dialect of Java, centered around the Scala standard library.
Choosing the number of communities: Maven. Left: the mean pairwise adjusted mutual information of the community-node membership across repeated decompositions. 1 is a perfectly stable assignment to communities; 0 is perfectly random. The shaded area corresponds to twice the standard error of the mean. Right: the mean normalised residual sum of squares error for repeated decompositions on the Maven network against the number of components. The shaded area corresponds to twice the standard error of the mean above and below the line
Figure 18 demonstrates how these communities change over time. There is a general upwards trend and some noise that may be spurious. As noted in Decan et al. (2018), there may be issues with the Maven dataset that are affecting these results.
Maven community activity. Relative activity of each community in each month. The communities are named by important packages as detailed in the "Methods" section
We have found a large amount of variation between different software ecosystems; some, such as Python, are stable and long-lived, and others, such as Elm, have packages that have been important but short-lived due to deprecation. It is our suspicion that a large amount of the variation between languages is due to the size and usefulness of their standard libraries. However, some of the differences, especially those found by the tensor decompositions, could be because early communities which fall out of use are ignored by the decomposition in favour of later, larger communities. It would be interesting to investigate whether this effect is real.
Based on our analysis, we have the following answers to our research questions:
There are differences in the number of communities of software packages, the rate at which communities of software packages gain and lose popularity as measured by how often they are dependant on each other, and in the overall trajectory of the growth of these communities.
Similar to the languages as a whole, the communities have fairly homogeneous in-degree distributions but heterogeneous power-law out-degree distributions.
The general rule is "up and to the right": all of the ecosystems we considered increased in activity as time went on, in agreement with Decan et al. (2018). This is unsurprising as it is relatively rare for a package to lose dependencies—it can fall out of use, but the packages that previously depended on it will tend to continue to depend on it.
Our results agree with that found in other works, especially that of Decan et al. (2018). Figure 16 in particular shows the effect of CRAN's strict rolling-release model where out-of-date packages are archived (meaning that they are no longer as easy to install); Fig. 15 shows how the Rust ecosystem seems to start very suddenly as the package manager came relatively late into the language's life.
For Java specifically, our work does not show much overlap with the communities found by Šubelj and Bajec (2011). This could be due to differences in between their static and our dynamic community detection, but it could also be that the validity of our results is questionable for Maven as the Libraries.io data for Maven is incomplete (Decan et al. 2018).
The absolute level of the adjusted mutual information at low numbers of communities seems to reflect the diversity of an ecosystem: R and elm, both domain specific languages, have high levels of AMI for 2 and 3, whereas all of the general purpose languages we consider do not.
Methodologically, we found that using adjusted mutual information as a metric by which to choose the number of communities led to (predominantly) explainable communities that appeared to be stable across multiple runs of the decompositions and across multiple values of the number of communities chosen. Our labelling algorithm, Eq. (3) worked well for our application, giving plausible and comprehensible labels, and could plausibly work well for any communities in networks with power-law degree distributions. The stability of an ecosystem can be seen from our activity over time plots, i.e. the τr vectors.
Maven degree distributions. In and out degree distributions for the Maven package ecosystem
Further work could involve investigating larger components for languages for which we currently have fairly low AMI scores. It would also be fairly straight-forward to extend our technique to consider the co-authorship network that creates the software as another layer in the network. Knowledge of how the ecosystem evolves organically could be used to detect fraudulent packages, especially by unknown authors. The creation of models of software ecosystem evolution from simple sets of rules (for example, 10% of packages are deprecated every 6 months) to try to replicate our results synthetically could also prove insightful.
Our work could have applications in helping designers of software ecosystems to make informed choices; it is clear that, for example, CRAN's rolling release policy has a big impact on its software communities. One could imagine using clusterings like those we have created to determine whether part of a software ecosystem was on the wane and was likely to be replaced soon. Such considerations would make sense if one was choosing dependencies for a project that was intended to last many years. Additionally, if one knows how communities form naturally, it becomes easier to pick out outliers, as we have seen with the npm "joke" packages.
In conclusion, we have described the long term evolution of several software ecosystems by breaking them up into their constituent communities. We have been able to spot ground-truth events such as the deprecation of major software packages in the temporal activities of these communities. We have demonstrated clear differences between different software ecosystems.
Implementation: https://www.caam.rice.edu/~optimization/bcu/ncp/
Such as neat-230
@ckeditor5-build-inlineand possibly react-misc-toolbox
evancz/html is deprecated in Elm 0.17
Package popularity indicated on METACRAN https://cranlogs.r-pkg.org/
AMI:
APG-TF:
Alternating proximal gradient tensor factorisation
CANDECOMP:
Canonical decomposition
CANDECOMP/PARAFAC
CRAN:
Comprehensive R archive network
FOSS:
Free and open-source software, HTTP: Hypertext transfer protocol
LIDA:
Leeds institute for data analytics
NRSS:
Normalised residual sum of squares
PARAFAC:
parallel factor analysis
PyPI:
Python package index
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This work was undertaken on MARC1, part of the High Performance Computing and Leeds Institute for Data Analytics (LIDA) facilities at the University of Leeds, UK. The authors would also like to thank Andrea Schalk, Emlyn Price, and Joseph French for providing useful feedback on an early version of the manuscript.
OAB's work was supported as part of an Engineering and Physical Sciences Research Council (EPSRC) grant, project reference EP/I028099/1.
School of Computer Science, University of Manchester, Manchester, UK
Oliver A. Blanthorn
School of Geography, University of Leeds, Leeds, UK
Colin M. Caine
School of Environment, Education and Development, University of Manchester, Manchester, UK
Eva M. Navarro-López
OAB created the initial concept—the research plan, methodology, and the application to software packages—and wrote the majority of the paper. CMC and OAB wrote the code and the results and discussion section in a roughly even split. ENL wrote the introduction. All authors discussed the experimental design and edited the paper.
Correspondence to Oliver A. Blanthorn.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Blanthorn, O.A., Caine, C.M. & Navarro-López, E.M. Evolution of communities of software: using tensor decompositions to compare software ecosystems. Appl Netw Sci 4, 120 (2019). https://doi.org/10.1007/s41109-019-0193-5
Tensor factorisation
Multilayer temporal networks
Software dependency networks | CommonCrawl |
How many numbers are in the list $ 4, 6, 8, \ldots, 128, 130 ?$
Dividing each member of the list by 2, we get $2,3,4,\ldots,64,65$, and then subtracting 1, we get $1,2,3,\ldots,63,64$, so there are $\boxed{64}$ numbers. | Math Dataset |
\begin{document}
\title[Finiteness theorems for holomorphic mappings]{Finiteness theorems for holomorphic mappings from products of hyperbolic Riemann surfaces} \author{Divakaran Divakaran} \address{Department of Mathematics, Institute of Mathematical Sciences, Chennai 600113, India} \email{[email protected]} \author{Jaikrishnan Janardhanan} \address{Department of Mathematics, Indian Institute of Technology Madras, Chennai 600036, India} \email{[email protected]}
\keywords{hyperbolic manifolds, finiteness theorem, Riemann surfaces of finite type.} \subjclass[2000]{Primary: 32H20}
\begin{abstract}
We prove that the space of dominant/non-constant holomorphic mappings from a
product of hyperbolic Riemann surfaces of finite type into certain hyperbolic
manifolds with universal cover a bounded domain is a finite set.
\end{abstract}
\maketitle
\section{Introduction} Complex analysis, in its most general setting, is the study of holomorphic mappings between complex spaces. Therefore, it is somewhat paradoxical that the space $\mathscr{O}(X,Y)$ of holomorphic mappings between complex spaces $X$ and $Y$ often comprises just constant mappings or contains only a finite/discrete set of non-constant mappings. The classical theorems of Liouville and Picard illustrate this phenomenon. In the context of compact Riemann surfaces, the famous Riemann--Hurwitz formula puts severe restrictions on the comparative genera of the Riemann surfaces $R$ and $S$ whenever $\mathscr{O}_*(R,S)$ is non-empty. A somewhat deeper result is the following result of Imayoshi, which is an extension of a classical theorem due to de Franchis \cite{defranchis13}.
\begin{result}[Imayoshi {\cite[Theorem~2]{imayoshi83}}]\label{R:imayoshi}
Let $R$ be a Riemann surface of finite type and let $S$ be a Riemann
surface of finite type $(g,n)$ with $2g - 2 + n > 0$. Then $\mathscr{O}_*(R,S)$ is a finite set. \end{result}
A Riemann surface $S$ is said to be of \emph{finite type $(g,n)$} if it is biholomorphic to a Riemann surface obtained by removing $n$ points from a compact Riemann surface of genus $g$. Observe that the condition $2g - 2 + n > 0$ ensures that $S$ is a hyperbolic Riemann surface. This fact is crucial in the proof. Also note that a hyperbolic Riemann surface is automatically complete Kobayashi hyperbolic. However, aside from an appeal to Montel's theorem for bounded domains, Kobayashi hyperbolicity plays \emph{no role} in Imayoshi's proof. Rather, the proof relies mainly on the theory of Fuchsian groups. It is well-known that the presence of (Kobayashi) hyperbolicity in either the domain or target complex space often forces the space of dominant holomorphic mappings to be finite/discrete. See \cite{kobayashi75} and \cite{noguchi82} for examples of such results. Also see \cite[Chapter~2]{zaldenberg89} for a detailed survey.
The following result of Shiga can be thought of as a higher dimensional analogue of Result~\ref{R:imayoshi}.
\begin{result}[Shiga {\cite[Theorem~1.3]{shiga04}}]\label{R:shigagen}
Let $X = \mathbb{B}^n/G$ be a complex hyperbolic manifold of divergence type
and let $Y = \Omega/\Gamma$ be a geometrically finite $n$-dimensional complex
manifold where $\Omega \subset \mathbb{C}^m$ is a bounded
domain and $\Gamma$ is fixed-point-free discrete subgroup
of $\textsf{Aut}(\Omega)$. Suppose $G$ is finitely generated and that
$\Omega$ is complete with respect to the Kobayashi distance. Then
$\mathscr{O}_\textsf{dom}(X,Y)$ is a finite set. Furthermore, if the essential boundary
dimension of $\Omega$ is zero, then $\mathscr{O}_*(X,Y)$ is a finite set. \end{result}
\begin{remark}
The proof of the above result uses methods inspired by the ones used by
Imayoshi. However, as is evident from the hypotheses, Kobayashi hyperbolicity
plays a prominent role in the proof. \end{remark}
The purpose of this work is to both extend Result~\ref{R:imayoshi} and also to clarify the role played by Kobayashi hyperbolicity. The central result of this paper is:
\begin{theorem}\label{T:fin}
Let $X := X_1 \times \dots \times X_n$ be a product of
hyperbolic Riemann surfaces of finite type and let $Y = \Omega/\Gamma$ be an
$m$-dimensional complex manifold where $\Omega \subset \mathbb{C}^m$ is a bounded
domain and $\Gamma$ is fixed-point-free discrete subgroup
of $\textsf{Aut}(\Omega)$.
\begin{enumerate}
\item If $N$ is a tautly embedded complex submanifold of $Y$ then $\mathscr{O}_\textsf{dom}
(X,N)$ is a finite set.
\item If $Y$ is geometrically finite and $\Omega$ is complete hyperbolic then
$\mathscr{O}_\textsf{dom}(X,Y)$ is a finite set.
\item If in addition to the conditions in (2), the essential boundary
dimension of $\Omega$ is zero, then $\mathscr{O}_*(X,Y)$ is a finite set.
\end{enumerate}
\end{theorem}
\begin{remark}
This theorem subsumes Result~\ref{R:imayoshi} (by Remark \ref{Rem:RSFTgeomfin}
and (2) above) and can be viewed as an analogue
of Result~\ref{R:shigagen} with $\mathbb{B}^n$ replaced by $\mathbb{D}^n$. Our proof
uses a combination of techniques used in the proofs of Result~\ref{R:imayoshi}
and \ref{R:shigagen}. However, a key point of departure is our use
of a normal families argument (see Lemma~\ref{L:normal}) which simplifies
our proof. Our proof is also intentionally written in such a way that we get
new simplified proofs of Result~\ref{R:imayoshi} and Result~\ref{R:shigagen}
\emph{mutatis mutandis}.
\end{remark}
\noindent \textbf{Notation:} We shall denote the space of holomorphic mappings between two complex spaces $X$ and $Y$ by $\mathscr{O}(X,Y)$. The space of non-constant holomorphic mappings will be denoted by $\mathscr{O}_*(X,Y)$ and dominant holomorphic mappings by $\mathscr{O}_\textsf{dom}(X,Y)$. $\mathbb{D}$ shall denote the open unit disk in the complex plane.
\noindent \textbf{Organization:} In Section~\ref{S:summary}, we summarize (with references) the results and tools needed in the proof of Theorem~\ref{T:fin}. The proof of Theorem~\ref{T:fin} forms the content of Section~\ref{S:main}. Finally we present a few applications of our results in Section~\ref{S:app}.
\section{Summary of results and tools needed}\label{S:summary}
\subsection{Kobayashi hyperbolicity}
On each complex manifold $M$, one can define a pseudodistance $K_M$, called the Kobayashi pseudodistance, that is distance decreasing under holomorphic mappings, i.e., if $f:M \to N$ is a holomorphic mapping between complex manifolds, then $K_M(x,y) \geq K_N(f(x),f (y)) \ \forall x,y \in M$. By definition, the Kobayashi pseudodistance on $\mathbb{D}$ is nothing but the distance obtained by integrating the Poincaré metric. See \cite{kobayashi67} for basic facts about the Kobayashi pseudodistance and \cite{kobayashi1998,kobayashi05} for a detailed exposition.
We say that a complex manifold $M$ is \emph{hyperbolic} if $K_M$ is a distance. If in addition $(M,K_M)$ is a complete metric space, we say that $M$ is \emph{complete Kobayashi hyperbolic}. These notions have been extensively studied in the literature and are indispensable in the study of holomorphic mapping.
The following result is often used to determine if a given complex manifold is (complete) hyperbolic.
\begin{result}\label{R:hypcomp}
Let $M$ and $N$ be complex manifolds and $F:M \to N$ a covering map. Then $M$
is (complete) hyperbolic iff $N$ is. \end{result}
If $M$ and $N$ are hyperbolic complex manifolds then $\mathscr{O}(M,N)$ is obviously an equicontinuous family under the Kobayashi distance. It is a natural question whether $\mathscr{O}(M,N)$ is relatively compact as a subspace of $\mathcal{C}(M,N)$ in the compact-open topology. As the compact-open topology coincides with topology of uniform convergence on compacts, the following definition is natural.
\begin{definition}
Let $M$ and $N$ be complex manifolds. A subset $\mathcal{F} \subset \mathcal
{C}(M,N)$ is said to be a \emph{normal family} if every sequence $\{f_n\}
\subset \mathcal{F}$ has either a subsequence that converges uniformly on
compacts to a function in $\mathcal{C}(M,N)$ or has a compactly divergent
subsequence.
A complex manifold $N$ is said to be \emph{taut} if for every complex
manifold $M$ the set $\mathscr{O}(M,N)$ is a normal family. \end{definition}
The following result relates the notion of hyperbolicity and tautness.
\begin{result}[Kiernan {\cite[Proposition~3]{kiernan70}}]\label{R:taut}
Complete hyperbolic complex manifolds are taut. \end{result}
\subsection{Groups of divergence type}
The classical uniformization theorem states that the universal cover of a Riemann surface is either $\mathbb{C}, \widehat{\mathbb{C}}$ or $\mathbb{D}$. We say that a Riemann surface is \emph{hyperbolic} if its universal cover is $\mathbb{D}$. By Result~\ref{R:hypcomp} a hyperbolic Riemann surface is automatically complete Kobayashi hyperbolic.
If $R$ is a hyperbolic Riemann surface, we may write $R$ as $\mathbb{D}/G$, where $G$ is a fixed-point free discrete subgroup of $\textsf {Aut}(\mathbb{D})$. Such a group $G$ is knows as a \emph{Fuchsian group} and the representation of $R$ as $\mathbb{D}/G$ is known as the \emph{Fuchisan group representation} of $R$. From the theory of covering spaces, it also follows that $G \simeq \pi_1(R)$. If $R$ is a hyperbolic Riemann surface of finite type, we can get considerable information on the action of $G$ on $\mathbb{D}$. We first need a definition.
\begin{definition}
A Fuchsian group $G$ is said to be of \emph{divergence type} if
\[
\sum_{g \in G} (1 - |g(z)|) = + \infty \ \forall z \in \mathbb{D}.
\] \end{definition}
\begin{result}\label{R:div}
Let $R := \mathbb{D}/G$ be a hyperbolic Riemann surface. Then the following are
equivalent
\begin{enumerate}
\item $G$ is a Fuchsian group of divergence type.
\item $R$ admits no Green's function.
\item For almost every $\xi \in \partial \mathbb{D}$, we can find a sequence
$g_\nu \in G$ such that $g_\nu \to \xi \ \forall z \in \mathbb{D}$
uniformly and non-tangentially on compacts.
\end{enumerate} \end{result}
\begin{remark}
It follows from the above result that hyperbolic Riemann surfaces of finite
type can be represented as $\mathbb{D}/G$ where $G$ is a Fuchsuan group of
divergence type. See \cite{tsuji75} for a proof of the above result and
related results. \end{remark}
\subsection{The notion of ends and essential boundary dimension}
\begin{definition}
Let $X$ be a topological manifold. An \emph{end} of $X$ is a decreasing
sequence of connected open sets
\[
U_1 \supset U_2 \supset \dots,
\]
with the property that given any compact set $K \subset X$, there is an $N_0
\in \mathbb{Z}_+$ such that $K \cap U_{N_0} = \emptyset$. Two ends $U_1 \subset
U_2 \subset \dots$ and $U_1' \subset U_2' \subset \dots$ are considered
equivalent if for each $n \in \mathbb{Z}_+$, we can find $N \in \mathbb{Z}_+$ such that $U_n
\subset U_N'$, and vice-versa. \end{definition}
\begin{remark}
The notion of ends can be used to construct a nice compactification of $X$.
The space of ends also allows one to classify non-compact surfaces. See
\cite{richards63}. \end{remark}
In this work, we are mainly interested in complex manifolds with finitely many ends of a certain kind known as parabolic ends. We first define the notion of a pluripolar set.
\begin{definition}
A set $E \subset \mathbb{C}^n$ is said to be a \emph{complete pluripolar set} if for
some plurisubharmonic function $u: \mathbb{C}^n \to [-\infty,\infty), E = u^{-1}
\{-\infty\}$. \end{definition}
We now define what it means for a manifold $Y := \Omega/\Gamma$ to be geometrically finite.
\begin{definition}
Let $Y := \Omega/\Gamma$ be a complex manifold, where $\Omega \subset \mathbb{C}^n$ is a
bounded domain and $\Gamma$ is a discrete and fixed-point free subgroup of
$\textsf{Aut}(\Omega)$. Denote the covering map by $\pi: \Omega \to X$. We say that
the end
\[
U_1 \supset U_2 \dots,
\]
is a \emph{parabolic end} if for some $N_0$, we can find a connected component of
$\pi^{-1}(U_0)$, say $\widetilde{U}_{N_0}$, such that $\overline{\widetilde
{U}}_{N_0} \cap \partial \Omega \subset \bigcup_{j = 1}^\infty R_j$ where each $R_j
$ is a complete pluripolar set in $\mathbb{C}^n \setminus \Omega$.
We say that $X$ is \emph{geometrically finite} if $X$ has only finitely many
ends and each end is a parabolic end. \end{definition}
\begin{remark}
Any hyperbolic Riemann surface of finite type is automatically geometrically
finite.
\label{Rem:RSFTgeomfin}
\end{remark}
\begin{remark}
If a manifold $X$ has only finitely many inequivalent ends then we may, and
shall do so tacitly, represent each of these finitely many ends by finitely
many pairwise disjoint connected open sets.
\end{remark}
\subsection{Fatou--Riesz theorems on the polydisk}
We will need the following version of the Fatou--Riesz theorem for the polydisk.
\begin{result}[Tsuji {\cite{tsuji45}}]\label{R:fatou}
Let $f$ be a bounded holomorphic function on $\mathbb{D}^n$. Then for almost every
point $\xi$ on the distinguished boundary (the torus $\mathbb{T}^n$)
\[
\lim_{z \to \xi} f(z),
\]
exists whenever $z \to \xi$ non-tangentially. We
denote this limit by
$f^*(\xi)$. Furthermore, if $g$ is another bounded holomorphic function on
$\mathbb{D}^n$ and $f^*(\xi) = g^*(\xi)$ on a set of positive measure on $\mathbb
{T}^n$, then $f \equiv g$. \end{result}
The following lemma has been proven for an arbitrary bounded $\mathcal{C}^2$-smooth domains by Shiga (see \cite[Theorem~3.1]{shiga04}). We can view it as a generalization of the last part of the previous theorem. As the proof given by Shiga relies only on the existence of a Poisson kernel, his proof works for the polydisk also. \begin{lemma}\label{L:uniq}
Let $\phi:\mathbb{D}^n \to \mathbb{C}^m $ be a non-constant bounded holomorphic mapping and
let $E$ be a complete pluripolar set in $\mathbb{C}^m$. If $\phi^*(\xi) \in E$
for a set of positive measure on $\mathbb{T}^n$, then $\phi(\mathbb{D}^n) \subset E$. \end{lemma}
We will also need the following lemma which shows that under the hypothesis of Theorem~\ref{T:fin}, the non-tangential limits of the lift of a non-constant map lie on the boundary. The proof is essentially the same as the proof of \cite[Lemma~2.2]{shiga04}.
\begin{lemma}\label{L:proper}
Let $X = X_1 \times \dots \times X_n$ be a product of hyperbolic Riemann
surfaces of finite type and let $Y := \Omega/\Gamma$ be a complex manifold, where
$\Omega \subset \mathbb{C}^n$ is a bounded domain and $\Gamma$ is a discrete and
fixed-point free subgroup of $\textsf{Aut}(\Omega)$. Let $f : X \to Y$ be a
non-constant holomorphic map and let $F : \mathbb{D}^n \to \Omega$ be any lift. Then
for almost every $\xi \in \mathbb{T}^n, F^*(\xi) \in \partial \Omega$. \end{lemma}
We require one more definition.
\begin{definition}
Let $D \subset \mathbb{C}^n$ be a bounded domain. We say that $D$ has \emph{essential
boundary dimension zero}, if there exists a family $\{R_j\}_{j=1}^\infty$ of
complete pluripolar sets with $R_j \cap \partial D \neq \emptyset \ \forall j$
such that $\partial D \setminus \bigcup_{j=1}^\infty R_j$ contains no analtic
space of positive dimension. \end{definition}
\section{Proofs of our main results}\label{S:main}
In this section, we shall give the proof of our main theorem
(Theorem~\ref{T:fin}). We first prove a rigidity theorem for holomorphic
mappings (Theorem~\ref{T:rigidity}). This rigidity theorem is an extension of
a rigdity result used by Imayoshi to prove Result~\ref{R:imayoshi}.
\begin{theorem} \label{T:rigidity} Let $X := X_1 \times \dots \times X_n$ be
a product of hyperbolic Riemann surfaces of finite type and let $Y = \Omega/\Gamma$
be an $m$-dimensional complex manifold where $\Omega \subset \mathbb{C}^m$ is a
bounded domain and $\Gamma$ is fixed-point-free discrete
subgroup of $\textsf{Aut}(\Omega)$. Let $\phi, \psi :X \to Y$ be
holomorphic mappings such that $\phi_* = \psi_*$. Then
\begin{enumerate}
\item If $\phi$ (or $\psi$) is dominant, then $\phi = \psi$.
\item If $\phi$ (or $\psi$) is non-constant and $\Omega$ has essential
boundary dimension zero, then $\phi = \psi$.
\end{enumerate}
\end{theorem}
\begin{remark}
The above theorem is also true when $X = \mathbb{B}^n/G$ (see Section~\ref
{S:app}). Our proof works \emph{mutatis mutandis}. See also \cite[Theorem~1.1]{shiga04}.
\end{remark}
\begin{proof} Let $G_i$ be the Fuchsian group of divergence type such that
$X_i =
\mathbb{D}/G_i$. Then $X = \mathbb{D}^n/G$ where $G := \bigoplus_{i=1}^n G_i$. Let
$\widetilde{\phi}, \widetilde{\psi} : \mathbb{D}^n \to \Omega$ be lifts of
$\phi$ and $\psi$, respectively. As $\mathbb{D}^n$ is simply-connected, the group
$G$ and $\pi_1(X)$ are isomorphic. Fix a point $y\in Y$ and a point $\tilde{y}\in \Omega$ that gets mapped to $y$ under the quotient map. Then, given an element $[\delta] \in \pi_1(Y,y)$ there is an element
$\gamma \in \Gamma$ that takes $\tilde{y}$ to the end point of the lift of $\delta$ starting at $\tilde{y}$. This element is independent of the choice of
representative of $[\delta]$. It is easy to
check that, in this way, we get a homomorphism from $\pi_1(Y)$ to
$\Gamma$. Thus, given any map from $X$ to $Y$, by composing, we get a
homomorphism from $G$ to $\Gamma$. The hypothesis that $\phi_* =
\psi_*$ implies that both mappings $\phi$ and $\psi$ induce the same
homomorphism from $G$ to $\Gamma$. Let $\chi$ be this induced homomorphism.
It follows that
\begin{align} \label{E:hom1}
\widetilde{\phi} \circ g &= \chi(g) \circ \widetilde{\phi}\\
\widetilde{\psi} \circ g &= \chi(g) \circ \widetilde{\psi},
\label{E:hom2}
\end{align} for each $g \in G$.
As each $G_i$ is of divergence type, by Result~\ref{R:div}, for almost
every $\xi \in \mathbb{T}^n$, we
can find a sequence $g_{i,k} \in G_i$ such that $g_{i,k}(z) \to \xi_i$ non-tangentially
on compact subsets of $\mathbb{D}$. Let $g_k := (g_{1,k},
g_{2,k},\dots,g_{n,k})$. Then $g_k(z) \to \xi$ non-tangentially and
uniformly on compact subsets of $\mathbb{D}^n$.
Now, as $\Omega$ is a bounded domain, Fatou's theorem for the polydisk
(Result~\ref{R:fatou}) implies that for almost every $\xi
\in \mathbb{T}^n$, each component function of $\widetilde{\phi}$ and
$\widetilde{\psi}$ converge as $z \to \xi$ non-tangentially. We define
$\tau(\xi)$ and $\eta(\xi)$ to be the non-tangential limits of
$\widetilde{\phi}$ and $\widetilde{\psi}$, respectively, as $z \to \xi$
(the functions $\tau$ and $\psi$ are defined almost-everywhere on $\mathbb
{T}^n$). This, in combination with arguments in the previous paragraph,
shows that for almost every $\xi \in \mathbb{T}^n$, we can find a sequence
$g_k \in G$ such that
\begin{align*}
\lim_{k \to \infty} \widetilde{\phi} \circ g_k(z) &= \tau(\xi) \\
\lim_{k \to \infty} \widetilde{\psi} \circ g_k(z) &= \eta(\xi),
\end{align*}
for each $z \in \mathbb{D}^n$. As each $\chi(g_k) \in
\textsf{Aut}({\Omega})$, and as $\Omega$ is bounded, we may use Montel's theorem
and assume without loss of generality that $\chi(g_k)$ converges uniformly
on compact subsets of $\Omega$ to a holomorphic map $A_\xi: \Omega \to \mathbb{C}^n$.
Using
\eqref{E:hom1} and \eqref{E:hom2}, we have $A_\xi \circ \widetilde{\phi}
(z) = \tau(\xi)$ and $A_\xi \circ \widetilde{\psi}(z) = \eta(\xi)$,
whenever $z \in \mathbb{D}^n$.
Now assume that $\phi$ is dominant. This implies that $\widetilde{\phi}$
is dominant as well. Therefore, we can find a polydisk $P_\xi \Subset
\mathbb{D}^n$ such that $\widetilde{\phi}(P_\xi)$ has an interior point. This
forces $A_\xi$ to be a constant map by the identity principle. Hence
$\tau$ and $\eta$ agree almost everywhere on the torus which means that
$\widetilde{\phi} = \widetilde{\psi}$ (by Fatou's theorem) and
hence $\phi = \psi$ as claimed.
Now assume that the essential boundary dimension of $\Omega$ is $0$ and that
$\phi$ is non-constant.
This implies that $\widetilde{\phi}$ is non-constant as well.
Each $\chi(g_k)$ is an automorphism of the bounded domain $\Omega$ that
converges in the compact-open topology to the map $A_\xi: \Omega \to \overline
{\Omega}$. Consequently, by Cartan's theorem for biholomorphisms, either
$A_\xi$ is
an automorphism or the image of $A_\xi$ is fully contained in $\partial \Omega$.
However, as $\widetilde{\phi}$ is non-constant, $\widetilde{\phi}(P_\xi)$
cannot be a singleton set (by the identity principle), where $P_\xi \Subset
\mathbb{D}^n$ is any polydisk. Hence $A_\xi$ is not injective as
$A_\xi \circ \widetilde{\phi}(z)$ is constant and consequently $A_\xi
(\Omega) \subset \partial \Omega$.
We have shown that for almost every $\xi \in \mathbb{T}^n$, we have $A_\xi
(\Omega) \subset \partial \Omega$. We claim that for almost every $\xi \in \mathbb
{T}^n$, $A_\xi$ is constant. Suppose not.
Since the essential boundary dimension of $\Omega$ is $0$, we can find
countably many complete
pluripolar sets $R_j$ of $\mathbb{C}^n \setminus \Omega$ such that the image of any
non-constant holomorphic function into $\partial \Omega$ must lie fully in
$\bigcup_{j=1}^\infty R_j$. Therefore, we can find a set $E \subset
\mathbb{T}^n$ of positive measure with the property that $A_\xi(\Omega)
\subset R_{j_0}$ for some $j_0 \in \mathbb{Z}_+$ and $\forall \xi \in E$. This means
that for a set of of positive measure, the non-tangential limits of
$\widetilde
{\phi}$ lie in a complete pluripolar set and consequently $\widetilde{\phi}
(\mathbb{D}^n)$ must be entirely contained in the same pluripolar set by
Lemma~\ref{L:uniq}. This is not possible. Consequently,
$A_\xi$ is a constant for almost every $\xi \in \mathbb{T}^n$. This means
that the non-tangential limits of $\widetilde{\phi}$ and $\widetilde{\psi}$
coincide
almost everywhere on $\mathbb{T}^n$ proving that $\widetilde{\phi} =
\widetilde{\psi}$ by Fatou's theorem. Hence $\phi = \psi$.
\end{proof}
The next lemma clarifies the role of geometrical finiteness of $\Omega$.
\begin{lemma}\label{L:normal}
Let $X := X_1 \times \dots \times X_n$ be
a product of hyperbolic Riemann surfaces of finite type and let $Y = \Omega/\Gamma$
be an $m$-dimensional geometrically finite complex manifold where $\Omega
\subset \mathbb{C}^m$ is a bounded complete hyperbolic domain and $\Gamma$ is fixed-point-free discrete
subgroup of $\textsf{Aut}(\Omega)$.
Then any sequence $\{f_n \in \mathscr{O}_*(X,Y)\}$
converges in the compact-open topology to a map in $\mathscr{O}(X,Y)$.
\end{lemma}
\begin{proof}
As in the proof of Theorem~\ref{T:rigidity}, we write $X$ as $\mathbb{D}^n/G$.
By Result~\ref{R:taut}, complete hyperbolic manifolds are taut. Therefore any
sequence $\{f_\nu \in \mathscr{O}_*(X,Y)\}$ must be either have a compactly
convergent or compactly divergent subsequence.
Suppose there is a sequence $\{f_\nu \in \mathscr{O}_*(X,Y)\}$ that is compactly
divergent. We
view each $X_i$ as being an open subset of a compact Riemann surface $R_i$.
For each point $x \in R_i \setminus X_i$, let $D_x$ be a small disk in $R_i$
with $x$ as the centre, chosen in such a way that if $x_1,x_2 \in R_i
\setminus X_i$ are distinct
points, then the closures of $D_{x_1}$ and $D_{x_2}$ are disjoint. Let $K
:= \bigtimes_{i = 1}^n R_i \setminus \bigcup_{x \in R_i \setminus X_i}
D_x$. As $K$ is a deformation retract of $X$, $\pi_1(K) = \pi_1(X)$. Any
curve $\gamma \subset X$ can be homotoped to a curve that lies entirely in
$K$. For large $\nu$,
by compact divergence and the fact that $K$ is connected, we may assume that
$f_\nu(K) \subset V$ where $V \subset Y$ is one of the finitely many
parabolic ends of $Y$. Let $V'$ be a connected component of a pre-image
of $V$ under the covering map such
that $\overline{V'} \cap \partial \Omega \subset \bigcup_{i = 1}^\infty R_i$ where
each $R_j$ is a complete pluripolar set in $\mathbb{C}^m \setminus \Omega$.
Fix $\nu$ large and let $f := f_\nu$ and let $\xi: G \to \Gamma$ be the
induced homomorphism. Denote by $F$ the lift of $f$ with the property
that $F(0) \in V'$. Let $p_0 \in M$ be the image of $0$
under the covering map. We may assume that $p_0 \in K$. Let $g \in G$ and
let $L$ be a loop based at $p_0$ contained entirely in $K$ that corresponds
to $g$. Denote the lift of $L$ based at $0$ by $\widetilde{L}$. Then $F
(\widetilde{L}) \subset V'$. As $F(g(0)) = \chi(g)(F(0))$ and the endpoint
of
$\widetilde{L}$ is $g(0)$, it follows that $\chi(g)(F(0)) \in V'$.
For almost every $\xi \in \mathbb{T}^n$, we can find a sequence $g_\nu \in
G$
such that $g_\nu(0) \to \xi$ non-tangentially. We assume that $F^*(\xi)$
exists
and that $F^*(\xi) \in \partial \Omega$. But $F(g_\nu(0)) = \chi(g_\nu)
(F(0)) \in V'$. Hence $F^*(\xi) \in \overline{V'} \cap \partial \Omega \subset
\bigcup_{i = 1}^\infty R_i$. This force $F$ to be a constant by
Lemma~\ref{L:uniq}.
\end{proof}
\noindent \textbf{Proof of Theorem~\ref{T:fin}:} Let $\{ f_k \} \subseteq
\mathscr{O}_\textsf{dom}(X,N)$ be a sequence of distinct dominant holomorphic mappings. As
$N$ is tautly embedded in $Y$, we may assume that the sequence $\{f_k\}$
converges in the compact-open topology to a map $f: X \to Y$.
As in the proof of Lemma~\ref{L:normal}, we can find a connected and
compact set $K$ such that finitely many closed loops contained in $K$
generate $\pi_1(X)$. Now, as $\{f_k\}$ converges uniformly on $K$ to $f$, it
is easy to see that for suitably large $k$, the $f_k(\gamma)$ and $f
(\gamma)$ are homotopic where $\gamma$ is curve whose image lies entirely in
$K$. Thus all the $f_k$'s induce the same
homomorphism on $\pi_1(X)$. Consequently, by Theorem~\ref{T:rigidity}, for
large $k$, all the $f_k$'s are equal, which is a contradiction.
We prove the
last part. Suppose $\{f_k \} \subseteq \mathscr{O}_*(X,Y)$ is a sequence of
distinct holomorphic mappings. By Lemma~\ref{L:normal}, we may, by passing
to a subsequence, if needed, assume that the $f_k$'s converge uniformly on
compacts. The previous argument now delivers the proof.
Part 2 of this theorem follows by similar arguments.
\qed
\section{Applications}\label{S:app}
We will now highlight a few applications of our results. The first application
is rather trivial but nevertheless interesting.
\begin{corollary}
Let $X := X_1 \times \dots \times X_n$ be
a product of hyperbolic Riemann surfaces of finite type and let $Y = \Omega/\Gamma$
be an $m$-dimensional compact complex manifold where $\Omega
\subset \mathbb{C}^m$ is a bounded domain and $\Gamma$ is fixed-point-free discrete
subgroup of $\textsf{Aut}(\Omega)$ Then $\mathscr{O}_\textsf{dom}(X,N)$
is a finite set where $N \subset Y$ is any complex submanifold (including
$Y$ itself).
\end{corollary}
\begin{proof}
As $Y$ is compact hyperbolic, it is complete hyperbolic and therefore taut.
Any sequence of holomorphic maps $f_k : X \to Y$ cannot have a
compactly divergent subseqeunce as $Y$ is compact. Thus $N$ is tautly
embedded in $Y$ and the corollary follows from Theorem~\ref{T:fin}.
\end{proof}
Our next application is an extension of Theorem~\ref{R:imayoshi} to maps
between products of Riemann surfaces.
\begin{corollary}
Let $X = X_1 \times \dots \times X_n$ be a product of hyperbolic Riemann
surfaces of
finite type and let $Y = Y_1 \times \dots \times Y_m$ be a product of
hyperbolic Riemann surfaces each of which can be embedded inside a
compact Riemman surface. Then $\mathscr{O}_\textsf{dom}(X,Y)$ is a finite set.
\end{corollary}
\begin{remark}
In \cite{janardhanan14}, finite proper holomorphic mappings between
products of certain hyperbolic Riemann surfaces was studied. The above
result is a partial extension of the main result there.
\end{remark}
\begin{proof}
We view each $Y_i$ as an open subset of a compact Riemann surface $S_i$.
Now, $Y_i$ is obtained by excising a set $E_i$ from $S_i$. Choose finite
sets $E_i' \subset E_i$ such that $S_i \setminus E_i'$ is hyperbolic. Let
$N
:= S_1 \setminus E_1' \times \dots \times S_n \setminus E_n'$. Then $N$ is a
product of hyperbolic Riemann surfaces of finite type. It suffices to show
that
$\mathscr{O}_\textsf{dom}(X,N)$ is a finite set. But any hyperbolic Riemann surface of finite
type is geometrically finite. It follows from Theorem~\ref{T:fin} that
$\mathscr{O}_\textsf{dom}(X,N)$ is a finite set.
\end{proof}
We end with an observation about our results. We used three features of the
representation of the manifold $X$ as $\mathbb{D}^n/G$ in our results:
\begin{enumerate}
\item We have a Fatou--Riesz theorem on $\mathbb{D}^n$.
\item For almost every element $\xi \in \mathbb{T}^n$, we have a
sequence of elements $g_k \in G$ such that $g_k$ converges uniformly and
non-tangentially on compacts to $\xi$. In other words, $G$ is of divergence
type.
\item $G$ (or equivalently $\pi_1(X)$) is finitely generated.
\end{enumerate}
In view of the above observation, our proof of Theorem~\ref{T:fin}
works \emph{mutatis mutandis} when $X$ is of the form
$\mathbb{B}^n/G$
where $G$ is a finitely generated subgroup of $\textsf{Aut}(\mathbb{B}^n)$ of
divergence type. See \cite{shiga04} for a precise definition of divergence
type. We can also replace $X$ by $D/G$ where $D$ is any bounded domain and
group $G \subset \textsf
{Aut}(D)$ that have the analogues of the three features mentioned above. For
instance, in the statement of Theorem~\ref{T:fin}, we may replace each
Riemann surface $X_i$ by a quotient $\mathbb{B}^n/G_i$ where each $G_i \subset
\textsf{Aut}(\mathbb{B}^n)$ is a finitely generated subgroup of divergence type.
\end{document} | arXiv |
Assignement
From Encyclopedia of Mathematics
2010 Mathematics Subject Classification: Primary: 68P05 [MSN][ZBL]
An assignement is used for assigning specific values $a_1,\ldots,a_n$ to the free variables $X_1,\ldots,X_n\in V(t)$ of a term $t\in T(\Sigma,X)$ belonging to a signature $\Sigma=(S,F)$ and a set $X$ of variables. These values $a_1,\ldots,a_n$ are elements of a $\Sigma$-algebra $A$, whereby the sorts of $a_i$ and $X_i$ coincide in each case (i.e. $a_i\in s^A_i$ for $X_i\in X_{s_i}$ with $s_i\in S$). In this way, variables contained in the term $t$ can be eliminated and the overall 'value' of $t$ concretized. Formally, an assignement is a mapping $v\colon X\longrightarrow \bigcup_{s\in S} s^A$ with $v(x)\in s^A$ for $x\in X_s$, $s\in S$ [EM85]. The set of all assignements is typically designated as $B(A,X)$. An assignement is also called valuation.
For an assignement $v\in B(A,X)$ and for a value $a\in s^A$ the assignement changed at $x$ to $a$, designated as $v[x\leftarrow a]$, is defined as $$v[x\leftarrow a](y) := \begin{cases} a & \mathrm{ if~} y=x\\ v(y) & \mathrm{ otherwise } \end{cases}$$
It is possible, of course, to use specifically a term algebra $T(\Sigma,Y)$ with an $S$-sorted set $Y$ of variables as $\Sigma$-algebra $A$. In this case, assignements are called substitutions of terms in $T(\Sigma,Y)$ for variables. The value of a term $t\in T(\Sigma,X)$ under a substitution $B(T(\Sigma,Y),X)\ni v \colon X \longrightarrow T(\Sigma,Y)$, written $t[v]$, is just the result of substituting $v(x)$ for all occurences of $x$ in $t$ in the usual sense. For a simple substitution $$v(y) := \begin{cases} u & \mathrm{ if~} y=x\\ y & \mathrm{ otherwise } \end{cases}$$ replacing $x$ by $u\in T(\Sigma,Y)$ in $t\in T(\Sigma,X)$ one often writes $t[x \leftarrow u]$ instead of $t[v]$ [ST99].
[EM85] H. Ehrig, B. Mahr: "Fundamentals of Algebraic Specifications", Volume 1, Springer 1985
[M89] B. Möller: "Algorithmische Sprachen und Methodik des Programmierens I", lecture notes, Technical University Munich 1989
[ST99] D. Sannella, A. Tarlecki, "Algebraic Preliminaries", in Egidio Astesiano, Hans-Joerg Kreowski, Bernd Krieg-Brueckner, "Algebraic Foundations of System Specification", Springer 1999
[W90] M. Wirsing: "Algebraic Specification", in J. van Leeuwen: "Handbook of Theoretical Computer Science", Elsevier 1990
How to Cite This Entry:
Assignement. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Assignement&oldid=29507
Retrieved from "https://encyclopediaofmath.org/index.php?title=Assignement&oldid=29507"
Numerical analysis and scientific computing
TeX done | CommonCrawl |
Killing form
2010 Mathematics Subject Classification: Primary: 17B [MSN][ZBL]
The Killing form is a certain bilinear form on a finite-dimensional Lie algebra, introduced by W. Killing . Let $\def\f#1{\mathfrak{#1}}\f G$ be a finite-dimensional Lie algebra over a field $k$. By the Killing form on $\f G$ is meant the bilinear form
$$\def\tr{\textrm{tr}\;}\def\ad{\textrm{ad}\;}B(x,y) = \tr(\ad x \cdot \ad y),\quad x,y\in \f G $$ where $\tr$ denotes the trace of a linear operator, and $\ad x$ is the image of $x$ under the adjoint representation of $\f G$ (cf. also Adjoint representation of a Lie group), i.e. the linear operator on the vector space $\f G$ defined by the rule $z\mapsto [z,x]$, where $[\;,\;]$ is the commutation operator in the Lie algebra $\f G$. The Killing form is symmetric. The operators $\ad x$, $x\in \f G$, are skew-symmetric with respect to the Killing form, that is,
$$B([x,y],z) = B(x,[y,z])\quad \textrm{ for all } x,y,z\in \f G.$$ If $\f G_0$ is an ideal of $\f G$, then the restriction of the Killing form to $\f G_0$ is the same as the Killing form of $\f G_0$. Each commutative ideal $\f G_0$ is contained in the kernel of the Killing form. If the Killing form is non-degenerate, then the algebra $\f G$ is semi-simple (cf. Lie algebra, semi-simple).
Suppose that the characteristic of the field $k$ is 0. Then the radical of $\f G$ is the same as the orthocomplement with respect to the Killing form of the derived subalgebra $\f G' = [\f G,\f G]$. The algebra $\f G$ is solvable (cf. Lie algebra, solvable) if and only if $\f G\perp \f G'$, i.e. when $B([x,y],z) = 0$ for all $x,y,z\in \f G$ (Cartan's solvability criterion). If $\f G$ is nilpotent (cf. Lie algebra, nilpotent), then $B(x,y) = 0$ for all $x,y\in\f G$. The algebra $\f G$ is semi-simple if and only if the Killing form is non-degenerate (Cartan's semi-simplicity criterion).
Every complex semi-simple Lie algebra contains a real form $\Gamma$ (the compact Weyl form, see Complexification of a Lie algebra) on which the Killing form is negative definite.
The Killing form is a key tool in the Killing–Cartan classification of semi-simple Lie algebras over fields $k$ of characteristic 0. If $\textrm{char}\; k \ne 0$, the Killing form on a semi-simple Lie algebra may be degenerate.
The Killing form is also called the Cartan–Killing form.
Let $X_1,\dots,X_n$ be a basis for the Lie algebra $L_1$, and let the corresponding structure constants be $\def\g{\gamma}\g_{ij}^k$, so that $[X_i,X_j] = \g_{ij}^k X_k$ (summation convention). Then in terms of these structure constants the Killing form is given by
$$B(X_a,X_b) = g_{ab} = \g_{ac}^d\g_{bd}^c$$ The metric (tensor) $g_{ab}$ is called the Cartan metric, especially in the theoretical physics literature. Using $g_{ab}$ one can lower indices (cf. Tensor on a vector space) to obtain "structure constants" $\g_{abc} = g_{da} \g_{bc}^d$ which are completely anti-symmetric in their indices. (A direct consequence of the Jacobi identity and equivalent to the anti-symmetry of the operator $\ad y$ with respect to $B(x,z)$; cf. above.)
[Bo] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras", Addison-Wesley (1975) (Translated from French) MR0682756 Zbl 0319.17002
[Ca] E. Cartan, "Sur la structure des groupes de transformations finis et continus", Oevres Complètes, 1, CNRS (1984) pp. 137–288 Zbl 0007.10204 JFM Zbl 59.0430.02 JFM Zbl 25.0638.02
[Hu] J.E. Humphreys, "Introduction to Lie algebras and representation theory", Springer (1972) pp. §5.4 MR0323842 Zbl 0254.17004
[Ka] I. Kaplansky, "Lie algebras and locally compact groups", Chicago Univ. Press (1971) MR0276398 Zbl 0223.17001
[Ki] W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen I" Math. Ann., 31 (1888) pp. 252–290 JFM Zbl 20.0368.03
[Ki2] W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen II" Math. Ann., 33 (1889) pp. 1–48 JFM Zbl 20.0368.03
[Ki3] W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen III" Math. Ann., 34 (1889) pp. 57–122 JFM Zbl 21.0376.01
[Ki4] W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen IV" Math. Ann., 36 (1890) pp. 161–189
[Na] M.A. Naimark, "Theory of group representations", Springer (1982) (Translated from Russian) MR0793377 Zbl 0484.22018
[Va] V.S. Varadarajan, "Lie groups, Lie algebras and their representations", Springer, reprint (1984) MR0746308 Zbl 0955.22500
Killing form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Killing_form&oldid=42303
This article was adapted from an original article by D.P. Zhelobenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Killing_form&oldid=42303"
Nonassociative rings and algebras | CommonCrawl |
Topological Galois theory
In mathematics, topological Galois theory is a mathematical theory which originated from a topological proof of Abel's impossibility theorem found by V. I. Arnold and concerns the applications of some topological concepts to some problems in the field of Galois theory. It connects many ideas from algebra to ideas in topology. As described in Khovanskii's book: "According to this theory, the way the Riemann surface of an analytic function covers the plane of complex numbers can obstruct the representability of this function by explicit formulas. The strongest known results on the unexpressibility of functions by explicit formulas have been obtained in this way."
References
• Arnold, V. I. Abel's Theorem in Problems and Solutions.
• Khovanskii, A. G. Topological Galois Theory.
• Burda, Y. Topological Methods in Galois Theory (PDF).
| Wikipedia |
Vector Can Network Management
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Being Time division multiple access based communica-tion scheme, FlexRay network startup is performed on all nodes synchronously. * set of commands operates on a table linked to a vector map. Indicating too many Errors. RedVector, a Vector Solutions brand, is the leading provider of online education and training for a wide range of industries including architecture, engineering, construction, industrial, facilities management and IT and security. 612-378-7560 Contact Client Login Request a Meeting. Administrator. In computer graphics images can be represented in two ways — using raster graphics (or bitmap) or vector graphics. By continuing to browse you are agreeing to our use of cookies and other tracking technologies. They can easily measure commonly specified amplifier parameters such as gain, gain and phase compression, isolation, return loss, and group delay. Controller Area Network (CAN) Tutorial 1-10 ni. Network management is the set of actions that ensures that all network resources are put to productive use as best as possible. The IETF is working on standards for automated network management which, as the name implies, aims to improve and make more efficient management of networks as they continue to increase in size and complexity. Vector data can have quality issues such as undershoots, overshoots and slivers which you need to be aware of. net Here is a List of Top 10 Best Network Manage. Services provided by this discipline include fault analysis, performance management, provisioning of networks and maintaining the quality of service. is not generally needed since networks are comprised of hardware only. It is expected that the day-to-day management of each Network will be undertaken by a suitably qualified Network Manager. The software is cited in more than 20,000 publications and is compatible with CE sequencing. The PicoVNA 106 is a professional USB-controlled, laboratory grade vector network instrument of unprecedented performance, portability and affordability. The industries we support with Visio stencils and Visio add-ons include network, building controls, security, floor plan, energy, oil and gas, and manufacturing. the network of bought edges which minimizes the sum of the players' contributions, is the classic network design problem of the generalized Steiner forest [1, 42]. Specification of UDP Network Management AUTOSAR CP Release 4. You can create as many vector layers as you like in a model. IT management software as it should be allowing you to monitor & control your IT environment from anywhere , at anytime using smartphone, desktop or tablets. This must be accomplished within 100 ms. Vector data can have quality issues such as undershoots, overshoots and slivers which you need to be aware of. Another important property of a vector is its length. State transitions are also defined (see interactive figure: Slave State Model). There are many complicated network diagrams for specific purposes, such as 3D network diagram, Cisco network diagram, activity network diagram, network topology, LAN networks, home networks, etc. AI can be a game changer for WI-Fi management Aruba's new NetInsight solution uses artificial intelligence and the cloud to improve Wi-Fi troubleshooting and ensure Wi-Fi performance. This results in smaller APK files and less developer maintenance. Apply to 932 Vector Jobs on Naukri. All knives are American made and all products are backed by the Cutco Forever Gurantee. Previously by reading SWS of Can transceiver and Can driver, I understood that Wake up over Can bus is possible by these 2 modules. You can create as many vector layers as you like in a model. The 2 core interventions for malaria vector control are insecticide-treated nets (ITNs) and indoor residual spraying (IRS). Opportunities: Where can the project process be streamlined? The network diagram is the foundation for the project scheduling. The basic principles of operation are quite simple. CAN Bus Support from Vehicle Network Toolbox Acquire CAN bus data directly into MATLAB or Simulink using CAN interface devices from Vector, Kvaser, PEAK-System, and National Instruments. Common image formats like GIFs and JPEGs are the opposite -- these bitmap images are pixel-based and so can't be resized without losing quality. Because CAN bus has been around for nearly thirty years, numerous devices have been created for interfacing with the bus. The extensive IBM IAM portfolio includes authentication, privileged access management, identity governance and access management solutions. Network monitoring is the information collection function of network management. Software that enables network administrators to perform their functions is called network management software. vehicle speed) • Think of J1939 as a software specification that rides on top of a CAN bus. The CAN protocol specifies rules for implementing the physical and data link layers of the OSI model in silicon to effect serial transfer of information between two or more devices. integrated vector management (2), Guidance on policy-making for integrated vector management ( 3 ) and Core structure for training curricula on integrated vector management ( 4 ). MATLAB ® and Vehicle Network Toolbox™ support sending and receiving messages via Controller Area Network (CAN) bus and provide support for CAN bus. A campus network is usually set in the campus of a university, but the same kind of planning and design can be applied for other purposes. This number, also called a nonce, is employed only one time in any session. What is the difference between bitmap and vector images? » Images » Windows » Tech Ease: Bitmap (or raster) images are stored as a series of tiny dots called pixels. he rule of split horizon can cause routing table update problems on a hub and spoke frame-relay topology. Distance Vector is simple routing protocol which takes routing decision on the number of hops between source and destination. com used in channel definitions for that message. The Vector Control program can only inspect your private property and, with the owner's permission, make recommendations. Integrated vector management (IVM) is defined as "a rational decision-making process for the optimal use of resources for vector control" and includes five key elements: 1) evidence-based decision-making, 2) integrated approaches 3), collaboration within the health sector and with other sectors, 4) advocacy, social mobilization, and legislation, and 5) capacity-building. Provider Management. It is a message-based protocol , designed originally for multiplex electrical wiring within automobiles to save on copper, but can also be used in many other. Download them now!. The former protocol, EIGRP employs a distance vector routing protocol while the latter one, OSPF uses a link-state routing protocol. I'm going public with an open hardware microwave vector network analyzer (VNA) project that I've been developing as a hobby on and off for the past 20 months. Each pixel is actually a very small square that is assigned a color, and then arranged in a pattern to form the image. Choose from over a million free vectors, clipart graphics, vector art images, design templates, and illustrations created by artists worldwide!. They investigate four types of models: univariate-linear, multivariate-linear, univariate-neural network, and multi-. network system. Can smartphones get viruses and spread them to the corporate network? Learn how to protect your enterprise network from smartphone malware spread via USB. CANbedded products from Vector are available for systems without AUTOSAR. Split horizon is a method of preventing a routing loop in a network. Any network in the world - regardless. There are other programs that you can use, but these may create errors when opening and editing the files, if there are features that are not available, which is why, from Freepik, we recommend Adobe Illustrator. Software that enables network administrators to perform their functions is called network management software. The IETF is working on standards for automated network management which, as the name implies, aims to improve and make more efficient management of networks as they continue to increase in size and complexity. Vector data can have quality issues such as undershoots, overshoots and slivers which you need to be aware of. Information Gathering: Large networks produce incredible amounts of information that if leveraged intelligently, puts IT departments on the front foot when it comes to network management. Neural Network with Python and Numpy. View All 120+ Vector Shields, Coat of Arm. malaria, plague) to humans. ,and in other appropriate locations if we make important or significant changes to our network management practices. The term 'Network Analyzer' is used loosely by many people. I have already contact FAE from vector to discuss how to find the actual problem. This includes the characteristics of the physical layer (High-/Low-Speed-CAN) and information on the CAN protocol (bus access methods, framing, bit timing, bit stuffing etc. RedVector, a Vector Solutions brand, is the leading provider of online education and training for a wide range of industries including architecture, engineering, construction, industrial, facilities management and IT and security. Top 10 List of Best Network Management Software. Network management is the process of administering and managing computer networks. Exclusively available in Geosoft VOXI Earth Modelling, Magnetization Vector Inversion (MVI) solves the challenge of varying direction of rock magnetization to provide a more reliable representation of subsurface geology. Wikipedia] The symbols example "Cisco network topology - Vector stencils library" was created using the ConceptDraw PRO diagramming and vector drawing software extended with the Cisco Network Diagrams solution from the Computer and Networks area of ConceptDraw Solution Park. Each pixel is actually a very small square that is assigned a color, and then arranged in a pattern to form the image. This indeed will help you to enforce your security policies. Vector data consists of individual points, which (for 2D data) are stored as pairs of (x, y) co-ordinates. See who you know at Vector Security Networks, leverage your professional network, and get hired. The idea behind the support-vector network was previously implemented for the restricted case where the training data can be separated without errors. VectorUSA, one of Southern California's premier technology and managed IT services companies, designs, builds and maintains data, voice, and video networks. Free Network icons & vector files, png icons, free icons This site uses cookies. AI can be a game changer for WI-Fi management Aruba's new NetInsight solution uses artificial intelligence and the cloud to improve Wi-Fi troubleshooting and ensure Wi-Fi performance. The points may be joined in a particular order to create lines, or. Vector Security Networks has unequaled expertise in providing customized security solutions for nearly 50 years. Anritsu, as a cutting-edge company in the ICT services industry, is operating its test and measurement business globally to support the building of the next generation of networks. The focus of the Queensland stock route network strategy is contained within a strategic plan, which details the priorities and intentions for management of SRN. We can then add vectors by adding the x parts and adding the y parts: The vector (8,13) and the vector (26,7) add up to the vector (34,20). Information Gathering: Large networks produce incredible amounts of information that if leveraged intelligently, puts IT departments on the front foot when it comes to network management. SNMP is the most widely used data network management protocol. Network monitoring is the information collection function of network management. Building additional physical network infrastructure is the most secure option for the network managers, although it can be very expensive to implement and maintain. Transport protocol parameter group The transport protocol parameter groups (TPCM, PGN 00EC00 16 and TPDT, PGN 00EB00 16) are used to transfer parameter groups with more than 8 data bytes, see chapter 4. So, by following. Topology rules can help data integrity with vector data models. You can obtain them for the CAN and LIN networks as well as for the J1939 protocol. If you already work with an IT supplier or support company they may offer a network maintenance contract. The basic principle is simple: Information about the routing for a particular packet is never sent back in the direction from which it was received. Every clipart was drawn with Edraw Software. Download thousands of free vectors on Freepik, the finder with more than 4 millions free graphic resources. There are many complicated network diagrams for specific purposes, such as 3D network diagram, Cisco network diagram, activity network diagram, network topology, LAN networks, home networks, etc. SVG stands for scalable vector graphic file, which is an image you can rotate, color, and resize without losing image quality. As many vector control methods are effective against multiple diseases, they can be integrated together to combat multiple diseases at once. Belobaba (1987a, 1987b) proposed and tested. With VNNI, inference results - and potentially, critical insights - can be delivered much more quickly. The World Health Organization (WHO) states that control and prevention of vector-borne diseases are emphasizing "Integrated Vector Management (IVM)", which is an approach that looks at the links between health and environment, optimizing benefits to both. The Controller Area Network (CAN) specification defines the Data Link Layer, ISO 11898 defines the Physical Layer. Introduction to drawing vector graphics - As creatives, we work on a lot of different projects every year. The Vector Network Driver also called Option Ethernet Driver, containing the NetIsolator, exists in a Windows 7 and a Windows 10 compatible version 2019-08-15 Views: 5861 2019-08-15 RTE OS Invocation - RTE Handling of Runnable Mappings on Different Task Types. the vector mixer calibration) and amplifiers with frequency offset mode (optional) - Advanced characterization of high-speed serial interconnects with enhanced time domain analysis (optional) - Upgrade are available for all E5071C options at any time Keysight's E5071C ENA vector network analyzers deliver new. Previously by reading SWS of Can transceiver and Can driver, I understood that Wake up over Can bus is possible by these 2 modules. The first byte contains the command specifier and the second contains the Node-ID of the device that has to perform the command (if this value is equal to 0, all nodes have to perform the commanded state transition). net Here is a List of Top 10 Best Network Manage. com: News analysis and commentary on information technology trends, including cloud computing, DevOps, data analytics, IT leadership, cybersecurity, and IT infrastructure. Start studying Network Systems Basics. We here extend this result to non-separable training data. BUILDING COMPREHENSIVE CARDIAC ARRHYTHMIA AND HEART FAILURE MANAGEMENT WITH REMOTE MONITORING. Explicit Channel Configuration in MAX 4. Easily identify dead zones and improve wireless coverage. Call +1-866-530-8100 or email [email protected] However, if the capabilities are provided by another set of functions, the documentation for the network management functions will refer you to other functions you can use for the same task. This includes the characteristics of the physical layer (High-/Low-Speed-CAN) and information on the CAN protocol (bus access methods, framing, bit timing, bit stuffing etc. CAN Network Management : CANNM is to coordinate the transition between normal operation and bus-sleep mode of the network. The Controller Area Network (CAN) specification defines the Data Link Layer, ISO 11898 defines the Physical Layer. Page 41 Common Features Figure 4: Example of unsynchronized network interfaces. [email protected] I'm trying to develop a reasonably priced (for a network analyzer. Network management is the process of administering and managing computer networks. 8-inch touching screen, TX/RX measurement, can determine the complete S11 and S21 parameters, if you need to get S12 and S22, you need to manually replace the transceiver port wiring. We establish vector management programs that are based on an understanding of the underlying vector's ecology and rooted in the current science of environmentally sound control measures. With a complete portfolio of hardware, software, and services, Vector Digitals can help your company at every stage – from planning to implementation to ongoing management. Not sure why he needs a stencil. Download thousands of free vectors on Freepik, the finder with more than 4 millions free graphic resources. Apply to 932 Vector Jobs on Naukri. Integrated vector management (IVM) is defined as "a rational decision-making process for the optimal use of resources for vector control" and includes five key elements: 1) evidence-based decision-making, 2) integrated approaches 3), collaboration within the health sector and with other sectors, 4) advocacy, social mobilization, and legislation, and 5) capacity-building. Customers around the world rely on solutions and products from the independent Vector Group. Most of the network components used in enterprise network systems have built-in network agents that can respond to an SNMP network management system. Logo EPS is the world's largest library of brand logos in vector format available to download for free. In other words, it is used for assaulting or exploiting a network, computer or device. USER MANAGEMENT 122 Icons. Edraw Network Diagram is light-weight, yet incredibly powerful, and can be used to create the following network diagrams: basic network diagrams, Cisco network topology, logical network diagrams, physical network diagrams, LAN diagrams, WAN diagrams, LDAP, active directory and lots more. AI can be a game changer for WI-Fi management Aruba's new NetInsight solution uses artificial intelligence and the cloud to improve Wi-Fi troubleshooting and ensure Wi-Fi performance. I have already contact FAE from vector to discuss how to find the actual problem. Bayer is proud to be part of the IVCC since 2008. and analysis of communication systems based on the Controller Area Network (CAN) standard. The World Health Organization therefore recommends "Integrated Vector Management" as the process for developing and implementing strategies for vector control. If you're looking to centralize and manage an organization's vector images, EPS, PSD, and other image files I'd recommend using Smartimage. More information on this unique, parthenogenic large mammal feeder can be found here on the Fonseca Lab news feed. When you launch the Vectorworks program, it communicates with the Vectorworks Site Protection Server software. Network management platforms can be viewed as the main console for network operations in detecting faults in the infrastructure. Eventbrite - [email protected] USB interface for CAN, LIN, J1708, K-Line and I / O (4 + 1 channels, 2 piggies) - 07113 Vector. Network Management Vector Transport: NMVT: Network Marketing VT (Virtual Training) Suggest new definition. The study of revenue management problems (or yield management) in the airlines dates back to the work of Rothstein (1971) on an overbooking model and to Lit-tlewood (1972) on a model of space allocation for a sto-chastic two-fare, single-leg (a network with one leg) problem. The Vector Management Unit monitors and controls mosquitoes, rats and mice, biting midges (sandflies) and flies. These algorithms can be categorized into indirect methods since the position is estimated from the DOA estimates rather than from the received signals directly. Your most important activities— sales, email, business automation—you name it, depend on it, and you've taken steps to protect them by investing in a network management system. It is expected that the day-to-day management of each Network will be undertaken by a suitably qualified Network Manager. The software is cited in more than 20,000 publications and is compatible with CE sequencing. The free IAN symbol libraries is a global symbol language containing 2953 environmental science vector symbols and icons in SVG, Illustrator (AI) and PNG (raster) formats. 1 Job Portal. Vector Security has roughly 100 retail customers with about 45,000 customer location in North America. Any vector directed in two dimensions can be thought of as having two different components. controller area network (CAN) bus is the current interface to machine operation data transmitted between electronic control units (ECUs). Edraw can be used as an all-purpose network diagram creator which is capable for all network diagrams mentioned above. Cable movement during and after calibration introduces variation that is not characterized during calibration which results in degraded measurements. This results in smaller APK files and less developer maintenance. When frequency is high enough the reflections of the waves start to matter and distributed effects need to be taken into account. If you really need to put an array in a vector (and you probably don't - using a vector of vectors or a vector of strings is more likely what you need), then you can wrap the array in a struct: struct S { char a[10]; };. solutions for gateways or state management. Top 10 List of Best Network Management Software. 8-inch touching screen, TX/RX measurement, can determine the complete S11 and S21 parameters, if you need to get S12 and S22, you need to manually replace the transceiver port wiring. an event driven basis. Axialis Line Design Vector Icons; HARDWARE NETWORK 166 Icons. ) Adding Vectors. Network management, membership and activities. A vector's type can be checked with the typeof() function. The Vector Network Driver also called Option Ethernet Driver, containing the NetIsolator, exists in a Windows 7 and a Windows 10 compatible version 2019-08-15 Views: 5861 2019-08-15 RTE OS Invocation - RTE Handling of Runnable Mappings on Different Task Types. As part of our commitment to build a centre of excellence in providing total solutions, Vector InfoTech Service Centre is developed to handle all after-sales services and supports as part of the whole delivery. Here's the page we think you wanted. It can be scaled without loss of display quality, which means the same file is resized for different screen densities without loss of image quality. It enables you to quickly find the logo vector files you need by browsing or search through the entire collection of more than 200,000 vector logos. There are many complicated network diagrams for specific purposes, such as 3D network diagram, Cisco network diagram, activity network diagram, network topology, LAN networks, home networks, etc. The idea behind the support-vector network was previously implemented for the restricted case where the training data can be separated without errors. Read "An accurate traffic classification model based on support vector machines, International Journal of Network Management" on DeepDyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. When you launch the Vectorworks program, it communicates with the Vectorworks Site Protection Server software. Being Time division multiple access based communica-tion scheme, FlexRay network startup is performed on all nodes synchronously. This is a DIY product with vector network measurement function. Vector Technologies is a professional controls and automation company offering sales and service in a full range of markets. malaria, plague) to humans. In addition, Vector can offer individual solutions for further simplification of the application, e. To convert a point or a vector to its polar form, use the following equations to determine the magnitude and the. VN1630A CAN / LIN Network Interface. We have summarised the GIS Vector Data concept in Figure figure_vector_summary. This series provides performance and accuracy in a low-cost, space-saving solution that is small enough to directly connect to your device under test (DUT). It is a medium-sized mosquito that is brown in color, and is a vector of WNV, St. Automated network management. Gaussian elimination works over any field, not just the real numbers; all you have to remember is to do the operations in that field. The focus of the Queensland stock route network strategy is contained within a strategic plan, which details the priorities and intentions for management of SRN. Arnulf Braatz is a manager at Vector Consulting Services. wmf Windows Metafile. The term also denotes the mathematical or geometrical representation of such a quantity. Most of the network components used in enterprise network systems have built-in network agents that can respond to an SNMP network management system. With Edraw Max, you can create clear and comprehensive network diagrams even without prior experience. You can create as many vector layers as you like in a model. 1 11 of 95 Document ID 414: AUTOSAR_SWS_UDPNetworkManagement - AUTOSAR confidential - 4 Constraints and assumptions 4. After implementing Part 2, you can check % that your implementation is correct by running checkNNGradients % % Note: The vector y passed into the function is a vector of labels % containing values from 1. 612-378-7560 Contact Client Login Request a Meeting. This is a vector eps file. On the hardware side, there's two types of CAN. Pulseway gives System Administrators and MSPs a single pane view of their entire IT environment so they can know exactly what is going on at any time. Now we'll look into the protocol and how it's used in the automotive industry. Every router advertises its set best routes to other routers. Vector Networks is an emerging leader in providing scalable IT management software for the small to mid-size enterprise. Can smartphones get viruses and spread them to the corporate network? Learn how to protect your enterprise network from smartphone malware spread via USB. Vector Training Classes In addition, the E-Learning module is suitable for customers who participate in the LIN Workshop of the VectorAcademy. In computer programming, vector is a one-dimensional array. Enterprise Risk Management Methods Enhancement and Mitigation Strategy Development. The basic principle is simple: Information about the routing for a particular packet is never sent back in the direction from which it was received. Introduction to drawing vector graphics - As creatives, we work on a lot of different projects every year. Vector Fleet Management provides dedicated fleet maintenance and management programs which favorably impact costs, service levels, availability and life expectancy of equipment. Network management is the process of administering and managing computer networks. This new architecture provides the needed flexibility to support all deep learning primitives while making core hardware components as efficient as possible. CAN Network Management : CANNM is to coordinate the transition between normal operation and bus-sleep mode of the network. [Archive] Vector Network Project has 18 repositories available. Time management, mental fortitude, and getting out of my comfort zone are just some examples of the skills I will take with me for the rest of my life. Without these interventions, dangerous diseases would proliferate unchecked and with the increased. Gives CMS users the ability to administer this vector. 1 Job Portal. As part of our commitment to build a centre of excellence in providing total solutions, Vector InfoTech Service Centre is developed to handle all after-sales services and supports as part of the whole delivery. com is a place to download free vectors, icons, wallpapers and brand logos. For organizations that rely on mission critical Wi-Fi to conduct business, 7SIGNAL is a cloud-based Wireless Network Monitoring (WNM) platform that continuously tests the wireless network for performance issues maximizing network uptime, device connectivity and network ROI. Information Gathering: Large networks produce incredible amounts of information that if leveraged intelligently, puts IT departments on the front foot when it comes to network management. When viewed from the perspective of motivation intersecting with opportunity, risk management can be driven not only by the techniques or sophistication of the attackers and threat vectors, but also by their motives. Both are left as an exercise for the reader. The PicoVNA 106 is a professional USB-controlled, laboratory grade vector network instrument of unprecedented performance, portability and affordability. ,and in other appropriate locations if we make important or significant changes to our network management practices. Our best-in-class managed network services offerings – from NOC monitoring, device management and PCI compliance, to alarm management and video surveillance – provide quality services at a lower cost of ownership, the convenience of having a single point of contact, and streamlined. from car accidents. First of all, the intuition behind SVM is a bit clearer than neural networks. For organizations that rely on mission critical Wi-Fi to conduct business, 7SIGNAL is a cloud-based Wireless Network Monitoring (WNM) platform that continuously tests the wireless network for performance issues maximizing network uptime, device connectivity and network ROI. Technology solutions include a state-of-the-art learning management system, incident tracking software, license and. Cable movement during and after calibration introduces variation that is not characterized during calibration which results in degraded measurements. Difference between Wake Up Over Can and Wake Up by Network Management Hello all, I am studying AUTOSAR communication stack. Start studying Network Systems Basics. Louis encephalitis, Western Equine encephalitis, and avian malaria. In addition, Vector can offer individual solutions for further simplification of the application, e. Now we'll look into the protocol and how it's used in the automotive industry. High generalization ability of support-vector networks utilizing polynomial input transformations is demonstrated. Vector images created using these programs can be scaled indefinitely without losing quality. Automated network management. horticultural oils can reduce transmission of viruses by aphids. The CAN protocol specifies rules for implementing the physical and data link layers of the OSI model in silicon to effect serial transfer of information between two or more devices. In fact, many of our managers started out as a sales rep in college, continued their education and decided to turn Vector into their full-time career after graduation. After implementing Part 2, you can check % that your implementation is correct by running checkNNGradients % % Note: The vector y passed into the function is a vector of labels % containing values from 1. Invasive Mosquito Species of California While not currently in Contra Costa County, two invasive species that can transmit Zika, dengue, yellow fever, and chikungunya are in other parts of the state. The back-issue catalog also includes Spanish and Portuguese translations of the magazine from September 2013 to the present. List of IP Management and Scanner tool for administrators One of the challenging tasks for network administrators is to manage the IP address. Vector HelpDesk combines the Vector Issue Tracker engine with the IT Support HelpDesk Template for a complete ITIL oriented Incident, Problem and Change Management solution. Wikipedia] The symbols example "Cisco network topology - Vector stencils library" was created using the ConceptDraw PRO diagramming and vector drawing software extended with the Cisco Network Diagrams solution from the Computer and Networks area of ConceptDraw Solution Park. Email: Michael. The Con-troller Area Network (CAN) [4] was developed in the mid 1980s by Bosch GmbH, to. Vector Security Networks has unequaled expertise in providing customized security solutions for nearly 50 years. Gives CMS users the ability to administer this vector. and analysis of communication systems based on the Controller Area Network (CAN) standard. Blue boxes indicate the bits that are currently being defined. Cutco offers a complete selection of cutlery, kitchen knives and accessories. We have summarised the GIS Vector Data concept in Figure figure_vector_summary. You can not buy Cutco Knives from retailers like Sears-hence it is "exclusive. Topology rules can help data integrity with vector data models. Computations are very much based on the same principles. BUILDING COMPREHENSIVE CARDIAC ARRHYTHMIA AND HEART FAILURE MANAGEMENT WITH REMOTE MONITORING. The company operates from bases in Atlanta, Montreal, and the UK. The VNT provides the user with CAN bus. Easily identify dead zones and improve wireless coverage. 3 Network Management Protocols In this section, di erent versions of SNMP and RMON will be introduced. Vector data can have quality issues such as undershoots, overshoots and slivers which you need to be aware of. For example a CAN interface module is used, if signals shall be transferred on CAN. com used in channel definitions for that message. SVG stands for scalable vector graphic file, which is an image you can rotate, color, and resize without losing image quality. A list of these 2D View entities are described briefly below and more fully in later sections of this. BotW is also a great place for designers to showcase their work. Back in 1987, when the first wave of Lyme disease-infected deer ticks began invading Maine from the south, Peter Rand, with infectious disease specialist Dr. Over $\mathbb{Z}_2$ Gaussian elimination is even simpler, because the elementary operation "multiply a row by a nonzero constant" is a trivial operation (the only nonzero constant being $1$). Network management, membership and activities. the vector mixer calibration) and amplifiers with frequency offset mode (optional) - Advanced characterization of high-speed serial interconnects with enhanced time domain analysis (optional) - Upgrade are available for all E5071C options at any time Keysight's E5071C ENA vector network analyzers deliver new. Flat Design Vector. Last time, we discussed how in-vehicle networks work over CAN. Distance Vector is simple routing protocol which takes routing decision on the number of hops between source and destination. Topology rules can help data integrity with vector data models. Files accepted for vectorization. We cannot guarantee an uninterrupted supply of power, so if you use medical equipment that relies on electricity, inform your power retailer and have a plan in case of power disruption. Further, performance can significantly improve for both batch inference and real-time inference, because vector neural network instructions reduces both the number and complexity of convolution operations required for AI inference, which also reduces the compute power and memory accesses these operations require. The direction of the vector is from its tail to its head. Vector Defined Network - How is Vector Defined Network abbreviated? Vector Data Management. Gives CMS users the ability to administer this vector. the network of bought edges which minimizes the sum of the players' contributions, is the classic network design problem of the generalized Steiner forest [1, 42]. interfaces and configuration issues of the AUTOSAR CAN Network Management (CanNm). | CommonCrawl |
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Skewness Formula
Skewness formula is called so because the graph plotted is displayed in skewed manner. Skewness is a measure used in statistics that helps reveal the asymmetry of a probability distribution. It can either be positive or negative, irrespective of signs. To calculate the skewness, we have to first find the mean and variance of the given data.
The formula is:
\[\large g=\sqrt{\frac{\sum_{i-1}^{n}\left(x-x_{i}\right)^{3}}{\left(n-1\right)s^{3}}}\]
x is the observations
$x_{i}$ is the mean
n is the total number of observations
s is the variance
Solved example
Question. Find the skewness in the following data.
Height (inches) Class Marks Frequency
59.5 – 62.5 61 5
62.5 – 65.5 64 18
To know how skewed these data are as compared to other data sets, we have to compute the skewness.
Sample size and sample mean should be found out.
N = 5 + 18 + 42 + 27 + 8 = 100
$\overline{x}=\frac{\left(61\times 5\right)+\left(64\times 18\right)+\left(67\times 43\right)+\left(70\times 27\right)+\left(73\times 8\right)}{100}$
$\overline{x}=\frac{6745}{100}=67.45$
Now with the mean we can compute the skewness.
Class Mark, x Frequency, f xf $\left(x-\overline{x}\right)$ $\left(x-\overline{x}\right)^{2}\times f$ $\left(x-\overline{x}\right)^{3}\times f$
61 5 305 -6.45 208.01 -1341.68
64 18 1152 -3.45 214.25 -739.15
67 42 2814 -0.45 8.51 -3.83
70 27 1890 2.55 175.57 447.70
73 8 584 5.55 246.42 1367.63
6745 n/a 852.75 -269.33
67.45 n/a 8.5275 -2.6933
Now, the skewness is
$g_{i}=\sqrt{\frac{\sum_{i=1}^{n}\left(x-x_{i}\right)^{3}}{\left(n-1 \right)s^{3}}}=-\frac{2.6937}{8.5275^{\frac{3}{2}}}=-0.1082$
For interpreting we have the folowing rules as per Bulmer in the year 1979:
If the skewness comes to less than -1 or greater than +1, the data distribution is highly skewed
If the skewness comes to between -1 and $-\frac{1}{2}$ or between $+\frac{1}{2}$ and +1, the data distribution is moderately skewed.
If the skewness is between $-\frac{1}{2}$ and $+\frac{1}{2}$,the distribution is approximately symmetric
Which of the following factors influence the Hardy-Weinberg equilibrium?
i. Gene migration
ii. Genetic drift
iii. Mutation
iv. Reproduction
v. Genetic recombination
Only i, ii, iii and v
Only ii, iii, and iv
Only iii, iv and v
Only i, ii, iii and iv
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Force Formula Inverse Of A 3x3 Matrix Formula
Altitude Of Isosceles Triangle Latent Heat Of Vaporization Formula
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The tendency of population to remain in genetic equilibrium may be disturbed by
random mating
lack of migration
lack of mutations
lack of random mating
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Elongated pentagonal gyrobirotunda
In geometry, the elongated pentagonal gyrobirotunda is one of the Johnson solids (J43). As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or icosidodecahedron (one of the Archimedean solids), by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae (J6) through 36 degrees before inserting the prism yields an elongated pentagonal orthobirotunda (J42).
Elongated pentagonal gyrobirotunda
TypeJohnson
J42 – J43 – J44
Faces10+10 triangles
10 squares
2+10 pentagons
Edges80
Vertices40
Vertex configuration20(3.42.5)
2.10(3.5.3.5)
Symmetry groupD5d
Dual polyhedron-
Propertiesconvex
Net
A Johnson solid is one of 92 strictly convex polyhedra that is composed of regular polygon faces but are not uniform polyhedra (that is, they are not Platonic solids, Archimedean solids, prisms, or antiprisms). They were named by Norman Johnson, who first listed these polyhedra in 1966.[1]
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:[2]
$V={\frac {1}{6}}\left(45+17{\sqrt {5}}+15{\sqrt {5+2{\sqrt {5}}}}\right)a^{3}\approx 21.5297a^{3}$
$A=\left(10+{\sqrt {30\left(10+3{\sqrt {5}}+{\sqrt {75+30{\sqrt {5}}}}\right)}}\right)a^{2}\approx 39.306a^{2}$
References
1. Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, Zbl 0132.14603.
2. Stephen Wolfram, "Elongated pentagonal gyrobirotunda" from Wolfram Alpha. Retrieved July 26, 2010.
External links
• Eric W. Weisstein, Elongated pentagonal gyrobirotunda (Johnson solid) at MathWorld.
Johnson solids
Pyramids, cupolae and rotundae
• square pyramid
• pentagonal pyramid
• triangular cupola
• square cupola
• pentagonal cupola
• pentagonal rotunda
Modified pyramids
• elongated triangular pyramid
• elongated square pyramid
• elongated pentagonal pyramid
• gyroelongated square pyramid
• gyroelongated pentagonal pyramid
• triangular bipyramid
• pentagonal bipyramid
• elongated triangular bipyramid
• elongated square bipyramid
• elongated pentagonal bipyramid
• gyroelongated square bipyramid
Modified cupolae and rotundae
• elongated triangular cupola
• elongated square cupola
• elongated pentagonal cupola
• elongated pentagonal rotunda
• gyroelongated triangular cupola
• gyroelongated square cupola
• gyroelongated pentagonal cupola
• gyroelongated pentagonal rotunda
• gyrobifastigium
• triangular orthobicupola
• square orthobicupola
• square gyrobicupola
• pentagonal orthobicupola
• pentagonal gyrobicupola
• pentagonal orthocupolarotunda
• pentagonal gyrocupolarotunda
• pentagonal orthobirotunda
• elongated triangular orthobicupola
• elongated triangular gyrobicupola
• elongated square gyrobicupola
• elongated pentagonal orthobicupola
• elongated pentagonal gyrobicupola
• elongated pentagonal orthocupolarotunda
• elongated pentagonal gyrocupolarotunda
• elongated pentagonal orthobirotunda
• elongated pentagonal gyrobirotunda
• gyroelongated triangular bicupola
• gyroelongated square bicupola
• gyroelongated pentagonal bicupola
• gyroelongated pentagonal cupolarotunda
• gyroelongated pentagonal birotunda
Augmented prisms
• augmented triangular prism
• biaugmented triangular prism
• triaugmented triangular prism
• augmented pentagonal prism
• biaugmented pentagonal prism
• augmented hexagonal prism
• parabiaugmented hexagonal prism
• metabiaugmented hexagonal prism
• triaugmented hexagonal prism
Modified Platonic solids
• augmented dodecahedron
• parabiaugmented dodecahedron
• metabiaugmented dodecahedron
• triaugmented dodecahedron
• metabidiminished icosahedron
• tridiminished icosahedron
• augmented tridiminished icosahedron
Modified Archimedean solids
• augmented truncated tetrahedron
• augmented truncated cube
• biaugmented truncated cube
• augmented truncated dodecahedron
• parabiaugmented truncated dodecahedron
• metabiaugmented truncated dodecahedron
• triaugmented truncated dodecahedron
• gyrate rhombicosidodecahedron
• parabigyrate rhombicosidodecahedron
• metabigyrate rhombicosidodecahedron
• trigyrate rhombicosidodecahedron
• diminished rhombicosidodecahedron
• paragyrate diminished rhombicosidodecahedron
• metagyrate diminished rhombicosidodecahedron
• bigyrate diminished rhombicosidodecahedron
• parabidiminished rhombicosidodecahedron
• metabidiminished rhombicosidodecahedron
• gyrate bidiminished rhombicosidodecahedron
• tridiminished rhombicosidodecahedron
Elementary solids
• snub disphenoid
• snub square antiprism
• sphenocorona
• augmented sphenocorona
• sphenomegacorona
• hebesphenomegacorona
• disphenocingulum
• bilunabirotunda
• triangular hebesphenorotunda
(See also List of Johnson solids, a sortable table)
| Wikipedia |
communications medicine
Improving personalized tumor growth predictions using a Bayesian combination of mechanistic modeling and machine learning
Predicting treatment benefit in multiple myeloma through simulation of alternative treatment effects
Joske Ubels, Pieter Sonneveld, … Jeroen de Ridder
Uncertainty quantification in the radiogenomics modeling of EGFR amplification in glioblastoma
Leland S. Hu, Lujia Wang, … Jing Li
A single-cell based precision medicine approach using glioblastoma patient-specific models
James H. Park, Abdullah H. Feroze, … Anoop P. Patel
Advancing precision oncology with large, real-world genomics and treatment outcomes data
Machine learning of genomic features in organotropic metastases stratifies progression risk of primary tumors
Biaobin Jiang, Quanhua Mu, … Jiguang Wang
An integrated approach to biomarker discovery reveals gene signatures highly predictive of cancer progression
Kevin L. Sheng, Lin Kang, … Robin T. Varghese
FastClone is a probabilistic tool for deconvoluting tumor heterogeneity in bulk-sequencing samples
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Model-based inference of metastatic seeding rates in de novo metastatic breast cancer reveals the impact of secondary seeding and molecular subtype
Noemi Vitos & Philip Gerlee
Using predictive machine learning models for drug response simulation by calibrating patient-specific pathway signatures
Sepehr Golriz Khatami, Sarah Mubeen, … Daniel Domingo-Fernández
Pietro Mascheroni ORCID: orcid.org/0000-0001-5131-186X1,
Symeon Savvopoulos2,
Juan Carlos López Alfonso1,
Michael Meyer-Hermann ORCID: orcid.org/0000-0002-4300-24741,3,4 &
Haralampos Hatzikirou5,6
Communications Medicine volume 1, Article number: 19 (2021) Cite this article
In clinical practice, a plethora of medical examinations are conducted to assess the state of a patient's pathology producing a variety of clinical data. However, investigation of these data faces two major challenges. Firstly, we lack the knowledge of the mechanisms involved in regulating these data variables, and secondly, data collection is sparse in time since it relies on patient's clinical presentation. The former limits the predictive accuracy of clinical outcomes for any mechanistic model. The latter restrains any machine learning algorithm to accurately infer the corresponding disease dynamics.
Here, we propose a novel method, based on the Bayesian coupling of mathematical modeling and machine learning, aiming at improving individualized predictions by addressing the aforementioned challenges.
We evaluate the proposed method on a synthetic dataset for brain tumor growth and analyze its performance in predicting two relevant clinical outputs. The method results in improved predictions in almost all simulated patients, especially for those with a late clinical presentation (>95% patients show improvements compared to standard mathematical modeling). In addition, we test the methodology in two additional settings dealing with real patient cohorts. In both cases, namely cancer growth in chronic lymphocytic leukemia and ovarian cancer, predictions show excellent agreement with reported clinical outcomes (around 60% reduction of mean squared error).
We show that the combination of machine learning and mathematical modeling approaches can lead to accurate predictions of clinical outputs in the context of data sparsity and limited knowledge of disease mechanisms.
Plain language summary
Computational methods, such as machine learning and mathematical models, can help doctors and scientists to predict the likely course of a patient's disease, such as tumour growth in a person with cancer. These methods rely on various kinds of data, e.g. imaging, but these data are collected infrequently and do not take into account underlying disease mechanisms. Here, we present a method that combines machine learning and mathematical modeling to improve prediction of tumour growth. We demonstrate our approach on a simulated dataset for glioma and two real cohorts of patients with leukemia and ovarian cancer. Results from the method are in close agreement with actual clinical data for individual patients, suggesting its potential applicability in enabling accurate personalized clinical predictions.
Advances in patient care have led to the availability of large amounts of data, generated by typical examinations, such as blood sample analysis, clinical imaging (e.g., CT, MRI), and biopsy sampling, as well as by innovative '-omics' sequencing techniques1,2. Such clinical data are the cornerstone in the practice of personalized medicine and specifically in the field of oncology3,4. However, this abundance of information comes with multiple issues related to data exploitation and synthesis towards the prediction of pathology dynamics. In particular, we identify the following two major challenges: (C1) First, knowledge of the regulatory mechanisms underlying clinical data is largely lacking, and (C2) second, patient data collection is usually sparse in time, since patient clinical visits/examinations are a limiting factor.
Regarding the challenge (C1), scientists have been long supported by the use of mathematical modeling as a tool to identify causal relationships in the experimental and clinical data, particularly in cancer treatment 5,6,7. Mathematical models allow to propose and test biological hypotheses, analyze the sensitivity of observables with respect to biological parameters, and provide insights into the mechanistic details governing the phenomenon of interest8,9,10. Although these models can be extremely powerful both in predicting system responses and suggesting new experimental directions, they require adequate knowledge of the underlying biological mechanisms of the analyzed system. Typically, this knowledge is not complete, and only for a limited portion of the involved variables the corresponding mechanistic interactions are sufficiently known. Therefore, even though mathematical models provide a good description of a simplified version of the associated system dynamics, they do not always allow for accurate and quantitative predictions.
On the other hand, machine learning techniques are suitable to deal with the inherent complexity of biomedical problems, but without caring for the knowledge of the underlying interactions11. While mathematical models rely on causality, statistical learning methods identify correlations among data12. This approach allows to systemically process large amounts of data and infer hidden patterns in biological systems. As a consequence, machine learning-based techniques can provide valuable predictive accuracy upon sufficient training, but do not typically allow for any mechanistic insight into the investigated problem13. The overall understanding of the fundamental system dynamics becomes almost impossible, as the chance to generalize the 'learnt' system behavior. The latter issue is further exacerbated by the (C2) challenge that has to be faced, related to the sparseness of clinical data. In particular for a single patient, such information is only available at a few time-points, corresponding to clinical presentation. To face the two mentioned challenges with the final aim of improving personalized predictions, we propose a novel—to the best of our knowledge—Bayesian method that combines mathematical modeling and statistical learning (BaM3). As a proof-of-concept, the proposed method is tested on a synthetic dataset of brain tumor growth. We analyze the performance of the new approach in predicting two relevant clinical outcomes, namely tumor burden and infiltration. When comparing predictions from the mechanistic model with those from the BaM3 method, we obtain improved predictions for the vast majority of virtual patients. We also apply the approach to a clinical dataset of patients suffering from chronic lymphocytic leukemia (CLL). The BaM3 method shows excellent agreement between the predicted clinical output and the reported data. Finally, as an additional test case, we show how the proposed methodology can be used to assess the time-to-relapse (TtR) in a dataset of ovarian cancer patients.
Formal definition of BaM3
We start by assuming a random variable (r.v.) triplet (Y, Xm, Xu) that denotes the system's modelable Xm, unmodelable Xu variables/data (e.g., patient's age or sex, results of different '-omics' techniques, etc.) and the associated observed clinical outputs Y. We then introduce t0 as the clinical presentation time of a patient at which the patient-specific r.v. realizations \(({{{{{{{{\bf{X}}}}}}}}}_{m}={{{{{{{{\bf{x}}}}}}}}}_{m}^{* },{{{{{{{{\bf{X}}}}}}}}}_{u}={{{{{{{{\bf{x}}}}}}}}}_{u}^{* })\) are obtained. The overall goal of the method is to predict the patient's clinical outputs by an estimate \({{{{{{{\bf{Y}}}}}}}}=\hat{{{{{{{{\bf{y}}}}}}}}}\) at a certain prediction time tp. The true clinical outputs of the patient will be denoted as y. Moreover, we consider the existence of an N-patient ensemble dataset (y, xm, xu). In this dataset all the variables (i.e., modelables, unmodelables, and clinical outputs) are recorded at the time of diagnosis td, which might differ from one patient to another. Both t0 and td are calculated from the onset of the disease. We introduce two distinct times to account for the variability of the disease stage among different patients (td) and the time at which a specific patient is presented to the clinic (t0) (see the corresponding Fig. S1).
The core idea of the method is to consider the predictions of the mathematical model \(p({{{{{{{\bf{Y}}}}}}}}=\hat{{{{{{{{\bf{y}}}}}}}}}| {{{{{{{{\bf{X}}}}}}}}}_{m}={{{{{{{{\bf{x}}}}}}}}}_{m}^{* })\) as an informative Bayesian prior of the posterior distribution \(p({{{{{{{\bf{Y}}}}}}}}=\hat{{{{{{{{\bf{y}}}}}}}}}| {{{{{{{{\bf{X}}}}}}}}}_{m}={{{{{{{{\bf{x}}}}}}}}}_{m}^{* },{{{{{{{{\bf{X}}}}}}}}}_{u}={{{{{{{{\bf{x}}}}}}}}}_{u}^{* })\). We can prove that:
$$\begin{array}{l}p({{{{{{{\bf{Y}}}}}}}}=\hat{{{{{{{{\bf{y}}}}}}}}}| {{{{{{{{\bf{X}}}}}}}}}_{m}={{{{{{{{\bf{x}}}}}}}}}_{m}^{* },{{{{{{{{\bf{X}}}}}}}}}_{u}={{{{{{{{\bf{x}}}}}}}}}_{u}^{* })\propto \\ p({{{{{{{\bf{Y}}}}}}}}=\hat{{{{{{{{\bf{y}}}}}}}}}| {{{{{{{{\bf{X}}}}}}}}}_{m}={{{{{{{{\bf{x}}}}}}}}}_{m}^{* })p({{{{{{{\bf{Y}}}}}}}}=\hat{{{{{{{{\bf{y}}}}}}}}}| {{{{{{{{\bf{X}}}}}}}}}_{u}={{{{{{{{\bf{x}}}}}}}}}_{u}^{* }).\end{array}$$
The implementation of the BaM3 method therefore reduces to the calculation of the aforementioned probability distributions. Although the prediction of the probability distribution function (pdf) of the clinical outputs is rather straightforward for the mathematical model, obtaining the pdf of the patient's unmodelable data is not trivial. To retrieve the latter, we use a density estimator method upon the patient ensemble dataset to derive p(Y, Xu), and then consider the patient-specific realization \({{{{{{{{\bf{X}}}}}}}}}_{u}={{{{{{{{\bf{x}}}}}}}}}_{u}^{* }\). For further details about method derivation and estimators of performance, see Supplementary Note 1.
Testing the method on synthetic glioma growth
The equations of the selected mathematical model14,15 ('full model') describe the spatio-temporal dynamics of tumor cell density (c), oxygen concentration (n), and vascular density (v) in the context of glioma tumor growth. The full model includes the variation of cell motility and proliferation due to phenotypic plasticity of tumor cells induced by microenvironmental hypoxia 16,17,18,19,20. It also accounts for oxygen consumption by tumor cells, formation of new vessels due to tumor angiogenesis, and vaso-occlusion by compression from tumor cells15,21,22. We generate N = 500 virtual patients by sampling the parameters of the full model from a uniform distribution over the available experimental range. We consider the tumor cell spatial density c to be the modelable variable. Moreover, we treat the integral over the tissue of oxygen concentration and vascular density, denoted as \(\bar{n}\;{{{{{\mathrm{and}}}}}}\;\bar{v}\), respectively, as the unmodelable quantities. Starting from the same initial conditions, we simulate the behavior of each virtual patient for 3 years, storing the values of all variables at each month. As sketched in Fig. 1, we use the modelable variable to setup a mathematical model. In particular, we take c(x, t0) at a specific time-point, the clinical presentation time t0, and use it as the initial condition for a Fisher-Kolmogorov equation23,24,25,26,27 ('FK model'). We use this model to predict tumor behavior at a specific time in the future, the prediction time tp. For each simulated patient we calculate the tumor size (TS) and infiltration width (IW). In parallel, for each patient we evaluate the diagnosis time td as a random number in the interval [t0 − 6, t0 + 6] (in the unit of months), and collect the values of modelables, unmodelables, and clinical outputs at this time to build the patient ensemble. Given the patient-specific modelable and unmodelable variables (c(x, t0) and \(\bar{n},\bar{v}\), respectively) at the clinical presentation time t0, the BaM3 method therefore produces the probability of observing the TS and IW at a specific prediction time tp.
Fig. 1: Workflow for generating the synthetic data.
The full model is initialized at t = 0 and used to simulate the spatio-temporal variation of tumor density c, oxygen concentration n, and functional tumor-associated vasculature v. For each virtual patient, two clinical outputs are tracked, i.e., the infiltration width (IW) and tumor size (TS), and two unmodelables recorded, i.e., the oxygen and vasculature integral over the tissue \(\bar{n}\) and \(\bar{v}\), respectively. These quantities are generated at the time of clinical presentation t0 every time the method is applied. To generate the patient ensemble, they are also generated at the diagnosis time td. The latter is assumed to be a random time between t0 ± 6 months. Then, the spatial profile of tumor concentration at time t0 is used as the initial condition for the Fisher-Kolmogorov (FK) model, which is in turn used to simulate tumor growth until the prediction time tp.
Mathematical models for glioma growth
The system variables are the density of glioma cells c(x, t), the concentration of oxygen n(x, t), and the density of functional vasculature v(x, t)14,15. For simplicity we consider a one-dimensional computational domain. We normalize the system variables to their carrying capacity and write the system as
$$\frac{\partial c}{\partial t}=D\frac{{\partial }^{2}}{\partial {x}^{2}}\left(\frac{\alpha }{{\alpha }_{0}}c\right)+b\frac{\beta }{{\beta }_{0}}c(1-c),$$
$$\frac{\partial n}{\partial t}={D}_{n}\frac{{\partial }^{2}n}{\partial {x}^{2}}+{h}_{1}v({n}_{0}-n)-{h}_{2}cn,$$
$$\frac{\partial v}{\partial t}={D}_{v}\frac{{\partial }^{2}v}{\partial {x}^{2}}+{g}_{1}{{{{{{{\mathcal{H}}}}}}}}(n-{n}_{0})v(1-v)-{g}_{2}v{c}^{\delta }.$$
Here \({{{{{{{\mathcal{H}}}}}}}}(\cdot )\) is a sigmoidal function (\({{{{{{{\mathcal{H}}}}}}}}(x-{x}_{0})=1/\)(1 + exp(b*(x − x0)), with b > 0 being a constant) allowing for tumor angiogenesis in hypoxic conditions, i.e., for n < n0 where n0 is the hypoxic oxygen threshold. Then, the functions α = α(n) and β = β(n) account for the dependence of cellular motility and proliferation on the oxygen level, respectively16,17,19. They are defined as:
$$\alpha =\frac{{\lambda }_{1}-n}{({\lambda }_{2}-1)n+{\lambda }_{1}},$$
$$\beta =\frac{{\lambda }_{2}n}{({\lambda }_{2}-1)n+{\lambda }_{1}}.$$
When the oxygen level is fixed to the maximum level n = 1 in the tissue α = α0 and β = β0, so that the equation for c reduces to
$$\frac{\partial c}{\partial t}=D\frac{{\partial }^{2}c}{\partial {x}^{2}}+bc(1-c),$$
which we denote in the rest of the manuscript as the Fisher-Kolmogorov (FK) model for tumor cell density. We remark that Eq. (7) has been extensively used to predict untreated glioma kinetics based on patient-specific parameters from standard medical imaging procedures23,24,25,26,27.
Eqs. (2)–(4) define an extended version of the FK equation, enriched with nonlinear glioma cell diffusion and proliferation terms. The latter terms depend on the oxygen concentration in the tumor microenvironment, which is in turn coupled to cell density through the oxygen consumption term. The functional vascular density controls the supply of oxygen to the tissue. Blood vessel density increases due to tumor angiogenesis and decreases because of vaso-occlusion by high tumor cell density. The values of the parameters used in the simulations and their descriptions are given in Table S1. In addition, a typical full model simulation is shown in Fig. S3 for a representative patient.
We solve the system in Eqs. (2)–(4) by imposing the initial conditions:
$$c(x,0)={c}_{0}{{{{{{{\mathcal{H}}}}}}}}(x-\varepsilon )\ \ {{{{{{{\rm{in}}}}}}}}\ \ 0\,\le\, x\,\le\, L,$$
$$n(x,0)={n}_{0}\ \ {{{{{{{\rm{in}}}}}}}}\ \ 0\,\le\, x\,\le\, L,$$
$$v(x,0)={v}_{0}\ \ {{{{{{{\rm{in}}}}}}}}\ \ 0\,\le\, x\,\le\, L,$$
where the positive parameters c0, n0, and v0 are the initial density of glioma cells spatially distributed in a segment of length ε, the density of functional tumor vasculature, and the oxygen concentration, respectively. Then, L > 0 is the length of the one-dimensional computational domain. In addition, we consider an isolated host tissue in which all system behaviors arise solely due to the interaction terms in Eqs. (2)–(4). This assumption results in no-flux boundary conditions of the form:
$$\frac{\partial c}{\partial x}(0,t)=\frac{\partial n}{\partial x}(0,t)=\frac{\partial v}{\partial x}(0,t)=0,$$
$$\frac{\partial c}{\partial x}(L,t)=\frac{\partial n}{\partial x}(L,t)=\frac{\partial v}{\partial x}(L,t)=0.$$
Both the full and FK models are used to calculate two clinical outputs, namely the tumor IW and TS. The IW at a specific time is defined by the difference between the points where glioma cell density is 80% and 2% of the maximum cellular density. In turn, the TS is obtained by integrating the spatial profile of tumor density and dividing it for the maximum value of the latter.
We run the full model and simulate the growth of the tumor for N patients, each one from a parameter set taken randomly from a uniform distribution over the parameter range. We run simulations for N = 50,100, 250, and 500 with 10 repetitions within each N-case. To generate the patients, we vary five parameters in the list in Table S1, namely the tumor motility D, proliferation rate b, oxygen consumption h1, vascular formation, and occlusion rates g1 and g2, respectively. Then, we use the tumor density at the time of clinical presentation, t0, as the initial condition for the FK model. The latter model is employed to generate predictions at the prediction time tp. We also consider the unmodelable variables and clinical outputs at the diagnosis time td, taken randomly between t0 ± 6 months, to build the patient ensemble. Finally, we use the results of the full model in terms of clinical outputs as the ground truth to be compared with the predictions of the FK model alone and with the ones obtained by the BaM3 method.
Probability distribution from the FK model
As described in the previous sections, we take the spatial profile of tumor density at the clinical presentation time t0 as the initial condition of the FK model. We use the latter mathematical model to run simulations over the whole parameter set for cell motility D and proliferation rate b. Then, we define the model-derived pdf as in the following. For each couple of clinical outputs IW* and TS* we calculate the area Aα(IW*,TS*) over the (IW,TS) plane as Aα = [(1 − α)IW* < IW < (1 + α)IW*), (1 − α)TS* < TS < (1 + α)TS*)], where α is a given tolerance (here set α = 0.05). Then, we calculate the pdf by normalizing Aα by the total area of predicted IW and TS values. We store the value of the probability for each patient at the different prediction times and use it to compute the expected value of the model pdf.
Probability distribution of the unmodelables from the full model
To retrieve the data-derived pdf we use a normal kernel density estimator (KDE)28,29, which depends upon all the data points in the patient ensemble. Briefly, the method estimates the joint probability \(p({{{{{{{\rm{IW}}}}}}}},{{{{{{{\rm{TS}}}}}}}},\bar{n},\bar{v})\) from which the ensemble entries are drawn through the sum of a kernel function over all the occurrences of the dataset. The kernel function is characterized by a hyperparameter, the bandwidth \(\tilde{h}\), which we assume according to Silverman's rule of thumb
$${\tilde{h}}_{i}={\sigma }_{i}{\left[\frac{4}{(d+2)n}\right]}^{\frac{1}{d+4}},\quad i=1,2,\ldots d,$$
where d is the number of dimensions, n is the number of observations, and σi is the standard deviation of the ith variate30. After calculating \(p({{{{{{{\rm{IW}}}}}}}},{{{{{{{\rm{TS}}}}}}}},\bar{n},\bar{v})\), we specify the realization of a specific patient and calculate the value of \(p({{{{{{{\rm{IW}}}}}}}}={{{{{{{{\rm{IW}}}}}}}}}^{* },{{{{{{{\rm{TS}}}}}}}}={{{{{{{{\rm{TS}}}}}}}}}^{* },\bar{n}={\bar{n}}^{* },\bar{v}={\bar{v}}^{* })\) over the (IW,TS) space of the estimated clinical outputs.
Scoring glioma growth predictions
We calculate for each patient the relative errors dm and db as described the main text. To assess how the BaM3 method has changed the prediction of the mathematical model, we compare the latter quantities: if ∣db − dm∣ ≤ εdm, then there was no change; if db > (1 + ε)dm, then the method deteriorated the prediction of the model; if db < (1 − ε)dm, then the method improved the prediction of the model. Here, ε is a tolerance used for the comparison, taken to be ε = 0.05.
Calculation of the effective variance
To calculate the effective variance s, we first calculate the mixed central moments Σij of the pdf of interest according to the formula
$${{{\Sigma }}}_{ij}=\int_{{y}_{1}}\int_{{y}_{2}}({y}_{1}-{\mu }_{1})({y}_{2}-{\mu }_{2})f({y}_{1},{y}_{2})\ d{y}_{1}d{y}_{2},$$
where y1 and y2 are the clinical outputs (IW and TS, respectively) and μ1, μ2 the expected values of the corresponding variables. The elements of Σ form a symmetric two-dimensional matrix, for which we calculate the determinant. We define the effective variance s as the natural logarithm of the latter determinant. In Eq. (14) we consider f(y1, y2) to be the pdf from the mathematical model or from the BaM3 method depending on whether we are interested in the effective variance sm or sb, respectively.
Pdf from the two-compartment model in CLL
Messmer and colleagues31 measured the fraction of labeled B-CLL cells in a cohort of 17 CLL patients that were administered deuterated water. They calibrated a two-compartment model on each patient and were able to reproduce the kinetics of labeled cells over a long time. We adopt their model and use it to generate the pdf for the CLL example. The fraction of labeled cells over time is calculated through the expression
$$f(t)=h(t)+f(0){e}^{\frac{-bt}{{v}_{r}}}+\frac{{e}^{\frac{-bt}{{v}_{r}}}\left(g(0)-h(t)\right)}{{v}_{r}-1}+\frac{{e}^{-bt}\left(h(t)-g(0)\right)}{{v}_{r}-1}$$
where g(0) is the initial fraction of cells in the first compartment, b the fractional cell birth, vr the relative size of the compartments, and h(t) is the deuterated water concentration of the body over time. The latter is a function of the fractional daily water exchange fw. We refer the interested reader to the supplementary information of Messmer et al.31 for a more detailed description of the model and a full account of the model parameters. In this work we focus on three quantities, namely b, vr, and fw, and run the model in Eq. (15) over the experimental range. This range was obtained by considering the patient-specific fitting performed by Messmer and colleagues and selecting the minimum and maximum values. We evaluate the fraction of labeled cells at day 50, f50, and build the probability distribution from its histogram, by counting the number of occurrences of a given \({f}_{50}^{* }\) for \(\min ({f}_{50}) < {f}_{50}^{* } \, < \, \max ({f}_{50})\) and then normalizing the result. For the CLL example, all the patients start with the same initial fraction of labeled cells, set to zero.
Pdf from the patients' unmodelables in CLL
The data-derived pdf in the CLL example is obtained from four unmodelable quantities that are measured for each patient during the study. We consider all the possible combinations of unmodelables and calculate the mean squared error (MSE) for each case. The scatter plot in the same picture refers to the case in which the CD38 expression (xu,1), age (xu,2), growth rate of white blood cells (xu,3), and VH mutation status (xu,4) are added consecutively with the specified order. As in the glioma example, we build the sub-dataset (y, xu), where y and xu = (xu,i) are the f50 and unmodelable variables of each patient, respectively, and apply the KDE using Silverman's rule for the hyperparameters. The requested pdf, i.e., \(p(Y=\hat{y}| {{{{{{{{\bf{X}}}}}}}}}_{u}={{{{{{{{\bf{x}}}}}}}}}_{u}^{* })\) is obtained by conditioning the probability from the KDE with the realizations of the unmodelables of the specific patient and calculating the result over the range of the estimated clinical output \(\hat{y}={f}_{50}\).
Mathematical model for ovarian cancer
We assume the total number of tumor cells T to be composed of the sensitive S and resistant R subpopulations. The latter are described by the following system of ordinary differential equations (ODEs):
$$\dot{S}=\gamma S-\delta S-\tau S,$$
$$\dot{R}=\gamma R-\lambda \delta R+\tau S,$$
where γ is the tumor net growth rate, δ = δ(t) is the death rate induced by chemotherapy, τ is the mutation rate from sensitive to resistant cells, and λ is a factor that accounts for reduced death by therapy in resistant cells. As detailed in Fig. S12, the treatment is composed of three phases: first, the patients undergo different cycles of NACT; then, surgery is performed. The latter reduces the total tumor volume, irrespective of cells being sensitive or resistant, of a factor β. Finally, another series of chemotherapy cycles is performed. During chemotherapy, δ = δ0, whereas we set this parameter to zero after chemotherapy and until tumor relapse. The latter condition occurs when T reaches the value TR.
Equations (16) and (17) can be analytically integrated, and their results used to build the probability distribution of the clinical output—TtR, in this case. To obtain the pdf from the model, we calculate the time the tumor takes to reach the cell number at relapse TR starting from the cell number after therapy. We perform this calculation using the initial tumor cell number of each patient, and by varying both the initial fraction of S cells, x0, and the chemotherapy-induced death rate, δ0. We then obtain the patient-specific probability distribution from the histogram of TtR, similarly to what is done in the previous section for CLL. For x0, we select a range between 0.4 and 0.9, accounting for tumors with different initial degrees of intrinsic resistance32. For δ0, we first use a uniform distribution between 0.1 and 10 days−1, accounting for a wide variation in death rates. The latter choice produces an almost flat distribution for the clinical output (see Fig. S13). To improve the mathematical model parametrization, we use the information about the tumor volume change after the first cycle of chemotherapy, which is included in the dataset. By fitting T obtained from Eqs. (16) and (17) to the observed volume change, we find a value of δ0 for each patient in the dataset32. We take the mean value of these rates and use it to update the model pdf (see Fig. S14). We consider a range for δ0 that is centered around its mean value across the patients, within an interval of ±40%. Selecting other ranges provides similar results, however, a variation of 40% returns the lowest MSE. Analytical integration of Eqs. (16) and (17), as well as additional details about model parametrization are available in Supplementary Note 2.
Unmodelable variable for the ovarian cancer study
We build the data-derived pdf for the ovarian cancer example by exploiting the information about the age of the patients at diagnosis. Similarly to what done in the previous test cases, we first build the sub-dataset (TtR, A) by entering the information of each patient (here, A is the patient age). Then, we apply the KDE using Silverman's rule to estimate the bandwidth and calculate the joint probability \(p({{{{{{{\rm{TtR}}}}}}}},A)\). The data-derived pdf for each patient \(p({{{{{{{\rm{TtR}}}}}}}}| {A}^{* })\) is finally obtained over the domain of the clinical output TtR by considering the patient-specific age A = A*.
We introduce the key ideas of the proposed methodology in the context of brain tumor growth, leaving the full derivation of the equations and their general form to the Supplementary information (see Supplementary Note 1).
Gliomas are aggressive brain tumors generally associated with low survival rates33. One of the most important hallmarks of this type of tumors is its invasive behavior, combined with a marked phenotypic plasticity and infiltrative morphology16. The clinical needs led to the development of several mathematical models to support clinicians in the treatment of the disease34. As a first test case, we synthetically generate a dataset of glioma patients using a system of recently published14,15 partial differential equations (PDEs). This complex mathematical model ('full model', in the following) provides a set of in silico patients, which represents our synthetic reality and serves as a benchmark to evaluate the performance of the proposed BaM3 method. Our goal is to obtain a personalized prediction of the clinical observables of the patients, combining their 'modelable' and 'unmodelable' information. A simplified mathematical model (with respect to the full model used to generate the patients) is used to generate predictions of clinical outputs starting from the modelable variables. In turn, a machine learning algorithm produces predictions of the same clinical outputs leveraging on the information contained in the unmodelables. As displayed in Fig. 2, the core idea of the BaM3 method is to use the results of the mathematical model to guide the predictions of machine learning. In more technical terms, the pdf obtained from the mathematical model ('model-derived pdf', in the following) works as a Bayesian prior that multiplies the pdf obtained from a nonparametric regression algorithm ('data-derived pdf'). The product of these two pdfs returns an estimate of the pdf for the clinical outputs of interest. More details about the formal definition of the BaM3 method and the mathematical details are available in the 'Methods' and Supplementary Note 1.
Fig. 2: Application of the BaM3 methodology to the prediction of glioma growth.
Given a new patient characterized by modelable (c) and unmodelable (\(\bar{n},\bar{v}\)) variables at the clinical presentation time t0, the goal of the BaM3 approach is to find an estimate for the clinical observables IW and TS at the prediction time tp. The modelable variable is used in the FK mathematical model to generate the model-driven pdf. In turn, the unmodelable variables are injected into a density estimation method, which contains the information about the patient ensemble at the diagnosis time td. This step provides the data-driven pdf, which is used to correct the predictions from the mathematical model. Application of the BaM3 method results in a new pdf for the clinical observables at the prediction time tp.
Improving predictions of synthetic glioma growth
For this first test case, we deal with two clinical observables, i.e., the TS and IW. The first quantity is related to tumor burden, whereas the second accounts for tumor infiltration in the host tissue. The modelable variable is the tumor cell density c, whereas we consider the amount of oxygen \(\bar{n}\) and vasculature \(\bar{v}\) in the tissue to be the patients' unmodelables (see the 'Methods'). Then, given the patient-specific modelable and unmodelable variables at the clinical presentation time t0, the BaM3 method produces the probability of observing certain values of the clinical outputs at a specific prediction time tp (see Fig. 2). The data-derived pdf is obtained through a normal KDE28,29,29,35 incorporating the information about the patient ensemble. The latter is generated from the full model, at the diagnosis time td. Then, the model-derived pdf is calculated using the simplified mathematical model for each patient. In particular, we use the FK model23 to produce a map of possible IW and TS starting from the tumor cell density of each virtual patient (see the 'Methods' for further information about the KDE and modeling steps).
Figure 3a–c shows the results of applying the BaM3 method to a representative patient. We select a clinical presentation time t0 = 24 months and a time of prediction tp = 9 months. The model-derived pdf obtained from the FK model is shown in Fig. 3a. Interestingly, the prediction of the model in that particular case shows two peaks, one with low TS and high IW and another with opposite properties. We calculate the expected values of TS and IW from the pdf obtained with the FK model and compare it to the 'true' values given by the full model. As shown in the plot, for this patient the presence of a bimodal distribution shifts the expected values far from the true ones. We enforce the BaM3 method making use of the probability calculated from the KDE, shown in Fig. 3b. The latter pdf takes into account the correlations between the clinical outputs and the unmodelable variables present in the patient ensemble. For this patient, the unmodelable distribution selects the probability mode closer to the true IW and TS values, as displayed in Fig. 3c (another example for a different patient is given in Fig. S4). These results evidence the ability of the proposed method to correct the predictions obtained by using exclusively the mathematical model, and to produce an expected value of the pdf that is closer to the ground truth.
Fig. 3: BaM3 outperforms mathematical modeling for an in silico patient cohort.
a, b Application of the BaM3 method to a representative patient. The clinical presentation time t0 is of 24 months and the prediction time tp is 9 months. a Pdf obtained from the FK model, plotted over the (TS, IW) space. The inset shows an enlargement of the probability region, in which the white arrow marks the error between the clinical output predicted by the full model (red dot) and the expected value of the distribution (black dot). b Data-derived pdf for the specific patient calculated through the kernel density estimator trained over the patient ensemble. c Probability distribution obtained from the BaM3 method. The dashed line shows the original probability obtained from the FK model. Relative errors between the clinical outputs obtained from the full model and the expected values given by the FK model (dm), and after the application of the BaM3 method (db), for a clinical presentation time of 12 and 24 months ((d) and (g), respectively). Effective variance of the predictions for the corresponding clinical presentation times ((e) and (h), respectively). Prediction score, as the ratio of cases for which the BaM3 method improved (Si), did not change (Su), or deteriorated (Sd) the model predictions. The error bars show the standard deviation of the results by replicating the method over 10 different sets (black circles) of N = 500 patients ((f) and (i), respectively).
We apply the BaM3 method for two clinical presentation times at 12 and 24 months, and compare its outcomes with those provided by the full model at increasing prediction times (Fig. 3d, i). For each patient, we calculate the relative error between the predicted clinical outputs obtained from the full model and the expected values of the pdf calculated from the FK model (dm) and after implementing the BaM3 method (db). These nondimensional errors are calculated as
$${d}_{k}=\sqrt{\frac{1}{L}\mathop{\sum }\limits_{i=1}^{{N}_{y}}{\left(1-\frac{\langle {y}_{i}^{k}\rangle }{{y}_{i}^{r}}\right)}^{2}}$$
where k = m, b, \(\langle {y}_{i}^{k}\rangle \) are the expected observable values (i = IW, TS) calculated from the FK model and the BaM3 method, and \({y}_{i}^{r}\) are the observable values obtained from the full model. We calculate the errors dm and db for each patient at different prediction times. Then, we compare the corresponding errors and evaluate if the BaM3 method improved, deteriorated, or left unchanged the prediction from the FK model, i.e., db < dm, db > dm, or db ~ dm, respectively (see the 'Methods'). We denote the ratio of improved, unchanged, and deteriorated cases with respect to the total number of simulated patients as Si, Su, and Sd, respectively.
Both the relative errors dm and db increase for increasing prediction times, as shown in Fig. 3d, g for the two clinical presentation times considered. However, after applying the BaM3 method the errors decrease, especially at later times. In general, it is possible to notice an improvement both in terms of median values and sparseness of the data. Interestingly, the relative error obtained from the BaM3 method increases at a lower rate if compared to the relative error obtained from the FK model.
We also calculate the effective variance of the predictions as the logarithm of the determinant of the covariance matrices relative to the model and BaM3 pdfs, (identified by sm and sb, respectively; see the 'Methods'). This quantity reflects the spreading of the pdfs over the (TS, IW) plane, with higher values denoting more uncertainty in the predictions. For both clinical presentation times (Fig. 3e, h), the BaM3 method provides thinner pdfs, more centered around their expected value with respect to the FK model-derived case.
Finally, the stacked bars in Fig. 3f, i show that BaM3 performs well at later prediction times, and especially remarkably well (improvement ratio Si close to 1) at the latest clinical presentation time t0 = 24 months. For t0 = 12 months (Fig. 3f), the proposed method is not able to improve predictions until a prediction time of 6 months. Then, for tp = 6, 9, and 12 months the advantages of using BaM3 over the FK model are unambiguous. On the other hand, for the clinical presentation time of 24 months (Fig. 3i) both Su and Sd decrease significantly for prediction times equal to or greater than 3 months. The ratio of improved cases Si reaches almost 100% at each of the last three prediction times, clearly overcoming the results of the FK model. The error bars in Fig. 3f, i denote the variability in the results that is obtained by replicating the study 10 times, each with N = 500 randomly generated patients. Fig. S5 shows similar results when decreasing the number of patients. Notably, the scores for N = 500, 250, 100, and 50 are very close, slightly improving with increasing the number of patients. The variability in the 10 replicates also decreases for higher values of N.
We also calculate the prediction scores using the distribution mode to generate the scores instead of the expected value (see Fig. S6). When the pdfs display multiple maxima we consider the average of the relative errors between the values of the full model (i.e., the synthetic reality) and the different peaks. The performance of the BaM3 method sensibly degrades with respect to using the expected value. Improvement in predictions is observed only for later clinical presentation and prediction times.
In summary, the BaM3 method is able to correct the FK model predictions for most of the patients, particularly at later clinical presentation and prediction times. The improvement in the prediction occurs by: (i) decreasing the median relative error between expected observable values and ground truth; (ii) decreasing the rate at which the error increases with prediction time; and (iii) decreasing the variance associated with the probability distributions.
BaM3 performance depends on the clinical output
Even though the BaM3 method performs well for the majority of patients, there are some cases for which it fails to improve the predictions of the mathematical model. We analyze the failure cases by splitting the errors dm and db into the two partial errors
$${{\Delta }}{{{{{{{{\rm{IW}}}}}}}}}_{k}=\sqrt{{\left(1-\frac{\langle {{{{{{{{\rm{IW}}}}}}}}}^{k}\rangle }{{{{{{{{{\rm{IW}}}}}}}}}^{r}}\right)}^{2}},\quad {{\Delta }}{{{{{{{{\rm{TS}}}}}}}}}_{k}=\sqrt{{\left(1-\frac{\langle {{{{{{{{\rm{TS}}}}}}}}}^{k}\rangle }{{{{{{{{{\rm{TS}}}}}}}}}^{r}}\right)}^{2}}$$
where k = m, b, 〈IWk〉, and 〈TSk〉 are the expected values of the clinical outputs obtained from the mathematical model and BaM3 pdfs (k = m and b, respectively), and IWr and TSr are the values of these quantities from the full model. Figure 4 shows how these partial errors are distributed over the presentation and prediction times. The dashed line in the plots highlights the neutral boundary, where the partial errors of the FK model and BaM3 method are equal. Above this line, the proposed BaM3 method deteriorates the model predictions, whereas under that line the BaM3 method improves predictions. The red dots in the scatter plots represent the patients for which the BaM3 method fails ('failure cases', in the following). After a prediction time of 1 month, in which a characteristic pattern is not evident, the plots highlight that failure cases are generally associated with regions where the BaM3 method under-performs to the FK model with respect to TS (ΔTSb > ΔTSm). Interestingly, the same failure cases belong to regions in which ΔIWb < ΔIWm: the BaM3 method is improving the IW predictions and at the same time deteriorating the TS predictions. This happens for both t0 = 12 and 24 months, however, the number of failure cases is considerably higher for the earlier presentation time. For the specific case under consideration, lower performance of the BaM3 method is therefore associated to its inability in correcting the FK model predictions for TS, with a tendency that improves for the later presentation time due to strong corrections for IW.
Fig. 4: Evaluating BaM3 performances depending on clinical outputs.
Scatter plots of the partial errors ΔIWi and ΔTSi over different prediction times tp (i = m and b, for the quantity related to the mechanistic model or the BaM3 method, respectively). Results for presentation times of t0 = 12 months (a) and t0 = 24 months (b) are shown. Each red dot corresponds to a patient for which the BaM3 method fails in improving the FK model predictions. Regions over the dashed line correspond to areas in which the BaM3 method deteriorates the FK model predictions, while regions under that line show areas in which it improves the FK predictions.
Transient behavior of the unmodelable distribution is associated to limited improvements
To investigate the reasons for the poor performance of the BaM3 method in improving the predictions for one of the clinical observables, we analyze the behavior of the pdf arising from density estimation, i.e., the data-derived pdf. Figure 5 shows the temporal evolution of this quantity for different clinical presentation times t0. Figure 5a shows a plot of the unmodelable pdf for a representative patient over the clinical output space. From a pdf that covers a limited region in the (IW,TS) plane, the probability distribution spreads over a broader area as the presentation time increases. The center of mass of the distribution, however, tends to converge to a more specific region as time progresses. This is more evident in Fig. 5b, c, showing the marginal probabilities for IW and TS, calculated from the distribution in Fig. 5a. The marginal distributions become broader for both IW and TS, but in the first case their peak stabilizes at later t0 times. On the contrary, the peak of the marginal probability for TS moves towards larger values at higher times. To quantify this behavior across the different patients, we then evaluated the degree of overlap between the marginal probabilities at two subsequent t0. Results from this calculation are plotted in Fig. 5d, e for the overlap between the distributions at presentation times t0 of 12 and 18 months and between 24 and 30 months. Here, the degree of overlap is calculated as the area of overlap for the IW and TS marginal distributions. Values close to one represent maximum overlap, whereas values near zero are associated to poor overlap between the two marginal pdfs. In a rough approximation, when this overlap score is high the marginal pdf is close to a steady state (since the pdf has not moved over time), and vice versa. For the earlier times in Fig. 5d, the patients are mostly scattered along a line of increasing IWr and TSr with points where the overlap is poor (close to 0.4 in certain regions). On the other hand, for the later times in Fig. 5e the patients are shifted towards higher values of overlap. Moreover, a horizontal line of high overlap for the IW output is visible for a large patient ensemble, pointing to a stabilization towards a steady state for the IW at later presentation times. This explains the lower performances of BaM3 method at t0 = 12 months, since the pdf from the KDE that should correct the model predictions is projecting the model pdf over (IW,TS) values that are outdated, far from the steady state. The situation improves for the case of t0 = 24 months; even though the correction of the BaM3 method for the TS might be wrong is some cases, the pdf for the IW has stabilized and points towards the correct value. In most of the cases, the correction for the IW outperforms the one for the TS, which leads to a general improvement of predictions by the BaM3 method.
Fig. 5: Temporal evolution of the unmodelable pdf from the KDE step.
a Data-derived pdf for a representative patient at different clinical presentation times, plotted over the (IW,TS) space. Marginal probabilities from the plots in (a), showing the evolution of the probability distribution for IW (b) and TS (c). Scatter plots representing the degree of overlap between the marginal probabilities for IW and TS between clinical presentation times t0 of 12 and 18 months (d) and of 24 and 30 months (e).
Outlier patients challenge the method's performance
To explain the different behavior for IW and TS, we investigate the distribution of the failure cases over the full model parameter space. In general, the BaM3 method performs poorly for those patients that are at the extremes of the parameter space, who represent outlier patients. When plotting the patients in a scatter plot over cell motility and proliferation rate (Fig. 6), the points with high motility–high proliferation rates and high motility–low proliferation rates witness the highest number of failure cases for both clinical presentation times t0 of 12 and 24 months. We checked for the distribution of failure cases also for the other model parameters, but no particular pattern was evident (see Fig. S7). Notably, patients falling into these high motility–high/low proliferation regions show the highest values for IW and TS (see ref. 14 and Fig. S8). Highly invasive and massive neoplasms are inadequately described by the pdf from the KDE, as they represent the extreme cases of the probability distribution. As a result, the FK model performs better in predicting the clinical outcomes with respect to the BaM3 method, since in the latter the correction from the dataset points towards smaller values of IW and, especially, TS.
Fig. 6: Outlier patients challenge BaM3 performance.
Scatter plots showing the distribution of failure cases (red dots) over the parameter space at different presentation (a, b) and prediction times. D: tumor cell motility; b: tumor cell proliferation rate; t0: clinical presentation time; tp: prediction time.
Applying the method to real CLL patients: the effect of unmodelables
In addition to the proof-of-concept applied to in silico data, we test the BaM3 methodology on a cohort of real patients suffering from CLL. This cancer involves B cells and is characterized by the accumulation of lymphocytes in the blood, bone marrow, and secondary lymphoid tissues36. In the past, CLL was considered to be a homogeneous disease of minimally self-renewing B cells, which accumulate due to a faulty apoptotic mechanism. This view was questioned by recent findings, suggesting a more heterogeneous neoplastic population continuously interacting with its microenvironment 37,38,39,40. Accumulation of leukemic cells occurs because of survival signals originating from the external environment and interacting with leukemic cells through a variety of receptors. The nature of this cross-talk with the environment is a current matter of research, featuring in vitro as well as in vivo experiments. One of the most significant experiments involving human patients was that of Messmer et al.31. Messmer and his co-workers inferred the kinetics of B-CLL cells from a group of patients through non-invasive labeling and mathematical modeling. Their investigation was quite thorough and involved the collection of several quantities related to patients' personal data (gender, age, etc.) and status of the disease (years since diagnosis, treatments, mutation status, etc.). They measured the fraction of neoplastic labeled cells in the blood of the patients, and fitted an ODE compartmental model to the dynamics that they observed. The model included three parameters, i.e., the daily water exchange rate (fw), the B cell birth rate (b), and the relative size of the blood compartment (vr).
We use the same model as Messmer and colleagues as the input for the BaM3 method, but discard the patient-specific fitting provided in their publication. Our aim is to show that, even when an individualized model parametrization is unknown, coupling the information given by the unmodelables can provide good patient-specific predictions. To accomplish this, we run simulations over uniform parameter ranges to obtain the pdf of the labeled cell fraction at day 50 (f50), which is also the sole modelable variable in this dataset (see Fig. S9). Then, we incrementally select one to four unmodelable variables from the patients' dataset and build the data-derived pdf using the same KDE method as in the previous in silico example. The BaM3 method couples the two prediction distributions to obtain the pdf for the clinically relevant output (see Fig. S10). We show the results of this procedure in Fig. 7a, where we compare the BaM3 predicted values against the patient f50 values reported in ref. 31. The fraction of labeled cells predicted by the BaM3 method agrees well with the reported data, especially when we increase the number of unmodelables used for density estimation. The inset shows how the MSE of BaM3 predictions decreases after considering all the possible combinations of unmodelables. Figure 7b shows how the probability distribution generated from the KDE changes for a representative patient. As the number of unmodelables increases, the mode of the distribution shifts towards the correct value of f50, here denoted by a red dashed line. From Fig. 7a it is also possible to note that, even if the majority of points lies close to the perfect prediction line, the predictions of a few patients are significantly mismatched with respect to the corresponding real values. This occurs because these patients belong to the extremes of the parametric space (see Fig. S11). Patients characterized by outliers in their parametriziation are under-represented in the modelable pdf due to the uniform sampling of the parameter space, and it is challenging for the data-derived correction to improve predictions for them.
Fig. 7: Role of unmodelables in CLL patients.
a Predicted vs measured values of the fraction of labeled cells at day 50. The colors denote the use of an increasing number of unmodelables in the BaM3 method. The inset shows the decrease in mean squared error (MSE) for the BaM3 predictions when we consider all the possible combinations of unmodelables. The solid line in the inset refers to the unmodelables used for the scatter plot. Here we included, in order, the CD38 expression, age, growth rate of white blood cells, and VH mutation status as unmodelables. b Variation of the data-derived pdf obtained from density estimation when an increasing number of unmodelables is considered for a specific patient. The dashed red line highlights the measured value of f50 in this patient.
Prediction of the time-to-relapse on a real ovarian cancer patient cohort: the importance of adequate model parametrization
To provide another application of the BaM3 method to a real scenario, we consider the case of patient response to therapy in high-grade serous ovarian cancer (HGSOC). This type of cancer is the most common epithelial ovarian cancer subtype, accounting for 70–80% of ovarian cancer-related deaths41,42. In addition, due to treatment resistance, the 5-year survival rate in HGSOC is less than 50%43,44. Indeed, the contribution of resistance mechanisms to tumor relapse after therapy is currently an active matter of research, recently backed up by evolutionary studies 45,46,47.
We start from the clinical dataset provided in a recent publication32, and elaborate a strategy to predict the TtR in ovarian cancer patients that makes use of the BaM3 methodology. The database of patients consists of 20 individuals, which are subject to the following treatment schedule (see Fig. S13). First, the patients receive neoadjuvant chemotherapy (NACT), consisting of different cycles of carboplatin and paclitaxel chemotherapy. Then, a surgery is performed, followed by other cycles of adjuvant chemotherapy. We propose a low-dimensional mathematical model to predict tumor TtR after treatment for each patient, which takes into account the presence of two cell subtypes. In particular, we include cells that are sensitive or resistant to chemotherapy. In addition, we consider the age of the patient at diagnosis as the unmodelable quantity used by the density estimator. Full details of the model and methodology are available in the corresponding sections of the 'Methods' and Supplementary Note 2.
As in the previous sections, the pdf from the mathematical model is obtained by simulating the latter over the parameter space. In this case, we focus on two parameters, namely the initial fraction of sensitive cells x0 and the death rate induced by chemotherapy δ0. First, we consider a uniform distribution of both parameters. We assume x0 between 0.4 and 0.9, in agreement with the degree of variability reported in the publication from which we take the dataset32. Since we lack any information about δ0, we select a wide range, from 0.1 and 10 days−1. This results in an almost uniform pdf from the mathematical model, as shown in Fig. S13. In this condition, the pdf from the model enters the BaM3 as an uninformative prior in the Bayesian framework, leaving predictions to rely only on the pdf generated from the density estimation of the unmodelables. Note that, in these settings, BaM3 reduces to nonparametric regression29. We calculate the MSE using the mode of the distributions in the uninformative case, denoted as MSEun, and find MSEun = 38.901 months2. As a next step, we use the additional information provided in the dataset to improve the parametrization of the mathematical model. Indeed, the dataset reports the tumor volume before and after the first cycle of therapy, as measured from clinical imaging32. We fit the value of δ0 for each patient and take the mean of all these values as the center of another uniform distribution. We apply the BaM3 method using the newly generated pdf from the model and obtain a lower MSE, i.e., MSEfit = 30.895 months2 (see Fig. S14). Better parametrization results, therefore, in improved performance of the method. In addition, by applying BaM3 to a better parametrized model allows to obtain improved predictions, as shown in Fig. 8. The scatter plot in Fig. 8a shows reduced errors in BaM3 predictions with respect to the ones from the mathematical model or density estimation alone. Also, Fig. 8b displays the outcome of the method for two representative patients. In both cases, the pdf arising from BaM3 has its mode closer to the real TtR (dashed line), with respect to the modes obtained from the model or density estimation pdfs. This shows the potential of the BaM3 method, which is able to perform better than the single techniques upon which it is based.
Fig. 8: BaM3 effectively couples predictions from machine learning and mathematical modeling in ovarian cancer patients.
a Predicted vs real TtR calculated from the mode of the probability distribution obtained from the mathematical model (Model), density estimation (Data), and the BaM3 method. As shown in the inset, the MSE are of 75.674, 40.447, and 30.895 months2, respectively. b Probability distributions obtained for two reference patients. The red dashed line denotes the real value of TtR for the specific patient. TtR: time-to-relapse of the tumor after therapy.
In the last few years, mathematical modeling and machine learning have emerged as promising methodologies in the biomedical field 48,49,50. However, several challenges persist and limit the prediction accuracy of both approaches. Among these issues, we identified the lack of knowledge of the mechanisms that govern the system under study (C1), and the paucity of time points at which patient information is available (C2), to significantly limit the performance of both mathematical models and machine learning techniques. In this work, we presented a method (BaM3) to couple mathematical modeling and density estimation in a Bayesian framework. The goal of BaM3 is to improve personalized tumor burden prediction in a clinical setting. This coupling allows to address the aforementioned (C1) and (C2) challenges, by exploiting the strengths of the respective methodologies and integrating them in a complementary path.
In particular, our proposed method aspires to solve a dire problem in personalized medicine that is related to the limited time-points of patient data collection. This implies that data assimilation methods, such as Kalman filters or particle filters51,52 that require multiple data time-points integrated to a mechanistic model cannot be generally used. To this regard, the BaM3 method can be regarded as a one-step data assimilation method. Compared to other methodologies that combine outputs from mathematical models and measured data—such as Bayesian Melding53, History Matching54, Bayesian Model Calibration55, or Approximate Bayesian Computation56—the BaM3 method is not interested in parameter estimation to better calibrate the mathematical model. Instead, its goal is to improve predictions of mathematical models empowering them with knowledge from variables that are not usually considered (the 'unmodelables', in our framework). This is done without the exact knowledge of the parameters of the mathematical model; indeed, we calculated the pdfs of the modelable variables using a uniform sampling of the parameter space. Better estimation of the model parameters improves the outcomes of the method (as shown in the ovarian cancer test case), but it is not required for the methodology to be applied.
First, we tested the BaM3 method on a synthetic dataset of patients focusing on tumor growth dynamics. Our approach was able to improve the predictions of a FK model for the majority of the virtual patients, with significant improvements at later clinical presentation times. In addition, we tested the proposed methodology on two clinical datasets related to cancer, concerning tumor growth in leukemia and ovarian cancer patients. We compared the outcomes of the BaM3 method to the reported data and found excellent predictive capability. When analyzing the cases for which the performance of the BaM3 method was not optimal, we came across some limitations that should be addressed when applying the methodology to real cases.
The first limitation regards the selection of the proper unmodelable variables. These are quantities that cannot be easily mathematically modeled, but can be correlated to the patient clinical outputs. For our proof-of-concept we selected only a few unmodelables, but in principle multiple quantities could be considered at the same time. Moreover, the most important unmodelables could be selected in a process of feature selection similar to the ones usually adopted in machine learning, providing better accuracy for the predictions57,58. We note as well that the method is open to progress in knowledge: should an unmodelable variable become modelable because of an increased understanding of the biological mechanisms, this variable can change side and become modelable.
One should also propose an adequate mathematical model that describes the dominant dynamics of the disease, as shown in the last case for ovarian cancer. A better parametrization of the model facilitates the work of density estimation, considerably reducing prediction errors. Not only better model parametrizations, but also mathematical models that encompass a suitable amount of mechanisms about the phenomenon that is modeled are advocated. In the case of ovarian cancer, we show in Supplementary Note 2 that a simplified model (with respect to the two cell populations presented in the 'Methods') is not able to provide good predictions when used in the context of the BaM3 method (see also Fig. S15).
Care must be taken with the selection of the metric that should be improved by the BaM3 method. For the in silico case, for example, considering the expected value of the final pdf resulted in better method performance when compared to selecting the pdf mode (see Figs. 3 and S5). This was probably due to the very similar natures of the FK and full models. Indeed, for lower clinical presentation times the FK model is already 'primed' towards the correct solution (in terms of outcomes of the full model); applying the BaM3 method might result in adding noise to the FK prediction, degrading the final prediction. However, for some patients the FK model provides pdfs with multiple local maxima, sometimes far away from the full model values. In these instances (see Fig. S3), the BaM3 method is able to correct for the correct mode, shifting the pdf to the correct values. Therefore, a good practice would be to try multiple pdf metrics and test the BaM3 method on each of them. This would result in a more thorough understanding of the problem, eventually allowing for better predictions.
Another important issue is that the correlations between unmodelables and clinical outputs should be persistent over time, evolving on a timescale that is faster than the dynamics of the problem. In our synthetic dataset this was partially accomplished at later clinical prediction times, especially for the case of the tumor infiltration width. Indeed, the unmodelable variables need to provide as much time-invariant information as possible on the clinical output variables, implying an equilibrated pdf. Such data can be, for instance, from genetic origin (such as mutations) or from other variables with slow characteristic evolution time. We stress that it is the probability distribution of the unmodelables that has to be close to equilibrium; note that this does not require the value of the unmodelable variables to reach a constant value but the values should be drawn from a steady-state distribution.
We see room for improvement also concerning the selection of the density estimation method. We adopted a well-known form of nonparametric estimation through kernel density estimation, but other approaches could be tailored to a specific problem—especially when high-dimensional datasets come into play59. Moreover, introducing density estimation methods to be able to integrate categorical variables would greatly benefit the technique, especially in biomedical problems (e.g., it would be extremely beneficial to include the grade of a tumor, or the particular sequence of therapies that a patient has undergone). The modularity of the BaM3 method makes it extremely versatile, allowing one to change the density estimation step, the modeling part, or both of them at the same time to improve the final prediction scores.
Care should also be taken to generate pdfs that are able to cope with outliers. In our proof-of-concept we generated the probability distributions considering the same weight for every patient, irrespective of his position in the parameter space. Techniques able to identify these extreme cases and to improve their contribution to the final pdfs should be implemented for a better method performance60.
In summary, we can identify three main actions that could be undertaken when these limitations hamper the predictive capabilities of BaM3: (i) one should look for ways to improve the mathematical model, designing it to be as informative as possible; (ii) then, an effort should be put to constrain the model by a robust choice of parameters; finally, (iii) extreme care should be devoted to the selection of the most informative unmodelable variables.
We conclude by stating that the proposed method is not restricted to oncology. The core problem concerning clinical predictions is that data are heterogeneous and sparse in time along with lack of full mechanistic knowledge. Therefore, a vast variety of medical problems could be addressed by using the BaM3 approach. For instance, predicting the fate of renal grafts by using pre- and post-transplantation data is a prime application of our proposed methodology.
All data generated or analyzed during this study are included in this published article, its References, and Supplementary information files. In particular, the dataset for chronic lymphocytic leukemia case is published in Table 1 of ref. 31. For the ovarian cancer case, we use the data available in Supplementary Table 1 of ref. 32. Note that, according to the Good Practice of Secondary Data Analysis (GPS) in Germany61, there is no need for additional ethics approval/consent when public domain data are analyzed.
All source codes and analysis are publicly available through the Zenodo platform (https://zenodo.org/record/4964592)62.
The Funding information section was missing from this article and should have read 'Open Access funding enabled and organized by Projekt DEAL'. The original article has been corrected.
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The authors gratefully acknowledge the funding support of the Helmholtz Association of German Research Centers-Initiative and Networking Fund for the project on Reduced Complexity Models (ZT-I-0010). H.H. and P.M. acknowledge the funding support of MicMode-I2T (01ZX1710B) by the Federal Ministry of Education and Research (BMBF). H.H. is supported by SYSIMIT (01ZX1308D) and MulticellML (01ZX1707C) by the Federal Ministry of Education and Research (BMBF). Finally, H.H. would like to thank the Volkswagenstiftung for its support within the "Life?" program (96732).
Open Access funding enabled and organized by Projekt DEAL.
Braunschweig Integrated Centre of Systems Biology and Helmholtz Centre for Infectious Research, Braunschweig, Germany
Pietro Mascheroni, Juan Carlos López Alfonso & Michael Meyer-Hermann
KU Leuven, Department of Chemical Engineering, Leuven, Belgium
Symeon Savvopoulos
Centre for Individualized Infection Medicine, Hannover, Germany
Michael Meyer-Hermann
Institute for Biochemistry, Biotechnology and Bioinformatics, Technische Universität Braunschweig, Braunschweig, Germany
Mathematics Department, Khalifa University, Abu Dhabi, UAE
Haralampos Hatzikirou
Centre for Information Services and High Performance Computing, TU Dresden, Dresden, Germany
Pietro Mascheroni
Juan Carlos López Alfonso
Conceptualization: H.H.; study design: P.M., S.S., J.C.L.A. and H.H.; software: P.M., S.S., and J.C.L.A.; formal analysis: P.M., S.S, J.C.L.A., M.M.-H. and H.H.; writing-original draft preparation: P.M.; writing-review and editing: P.M., S.S., J.C.L.A., M.M.-H. and H.H.; supervision: M.M.-H. and H.H.; funding acquisition: M.M.-H. and H.H.
Correspondence to Haralampos Hatzikirou.
Peer review information Communications Medicine thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available.
Peer Review File.
Supplementary information.
Reporting summary.
Mascheroni, P., Savvopoulos, S., Alfonso, J.C.L. et al. Improving personalized tumor growth predictions using a Bayesian combination of mechanistic modeling and machine learning. Commun Med 1, 19 (2021). https://doi.org/10.1038/s43856-021-00020-4
Received: 23 April 2021
Functional biomaterials for biomimetic 3D in vitro tumor microenvironment modeling
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Theoretical Physics Department
Laboratory of Molecular Acoustics
Kinetics of epidemic spread in a low-mobile population
[latexpage]
The standard method of including the spatial effects into epidemiological problems consists in the introduction of a of diffusion term leading to solutions in form of traveling waves (Fisher-Kolmogorov waves). However, from the point of view of the mechanism of infection propagation in a low-mobility population the diffusion term (corresponding to a random walk) borrowed from the theory of diffusion-controlled reactions, looks a bit an artificial addition. It is especially clear in the case of plant diseases – plants do not move at all, or animals showing a high degree of site fidelity (like harbor seals: the mixing within the population located at one site is strong while the infection transmission between sites takes place only via contacts between individuals performing rare relatively long trips between neighboring sites). Nevertheless the infection waves appear also in this situation.
In what follows, we consider infection wave propagation in a population of low-mobile individuals, where a local infection spread mechanism (between individuals or parts of the population in close contact) is present.
Such a mechanism corresponds to what is known as a 'contact process' in statistical physics. A population consisting of three kinds of individuals, namely the susceptible (S), the infected (I), and the recovered/removed (R) is subdivided into cells with strong mixing within a population located in one site (cell) and the infection transmission between sites only via contacts along boundaries of neighbor cells.
\begin{eqnarray*}
\Delta I(x, y)=\kappa S(x, y)\times\\
\left( \frac{1}{4}I(x+a, y)+\right.\\
\frac{1}{4}I(x-a, y)+\\
\frac{1}{4}I(x, y+a)+\\
\left.\frac{1}{4}I(x, y-a)\right) \Delta t
\end{eqnarray*}
Analytical continuous approximation:
\partial_t I &=&\kappa S\left( I+(a^{2}/4)\nabla
^{2}I\right)-R, \label{PDE} \\
\partial_t R &=&\tau^{-1}I, \\\nonumber
1&=&S+I+R
can be exactly represented within a comoving frame $x'=x-vt$, $v=2\sqrt{D(\kappa-\tau^{-1})}$ as
v\frac{dS}{dx'}&=&\kappa S(1-S)-DS\frac{d^2S}{dx'^2} +\frac{1}{\tau}S\ln S, \\
-\ln S&=&\kappa\tau R+ D\tau\frac{d^2 R}{dx'^2}=0.
The model is tested by comparison with the real dynamics of the epidemic of phocine distemper virus infection among harbor seals habitating Danish Straits in 1988.
The points of the first record of disease marked with stars and the points of 50% mortality marked with circles. The slope of the dashed line drawn through the arithmetic mean of each pair of experimental points (dots) except the initial one allows us to determine the mean velocity of infection as approximately 1.43 km/day that shows a good agreement with the calculated via the proposed model value 1.47 km/day.
Details can be found in the articles:
E.B. Postnikov, I.M. Sokolov. Continuum description of a contact infection spread in a SIR model. Mathematical Biosciences. 2007. V. 208. Iss.1. pp. 205–215.
U. Naether, E.B. Postnikov, I.M. Sokolov. Infection fronts in contact disease spread. The European Physical Journal B. 2008. V. 65. pp. 353-359.
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\begin{document}
\title{Determinant of pseudo-laplacians} \author[Tayeb Aissiou]{Tayeb Aissiou} \email{[email protected]} \address{Department of Mathematics and Statistics, Concordia University\\ 1455 de Maisonneuve Blvd. West \\ Montreal, Quebec H3G 1M8 Canada}
\author[L. Hillairet] {Luc Hillairet} \email{[email protected]} \address{UMR CNRS 6629-Universit\'{e} de Nantes, 2 rue de la Houssini\`{e}re \\ BP 92 208, F-44 322 Nantes Cedex 3, France}
\author[A. Kokotov] {Alexey Kokotov} \email{[email protected]} \address{Department of Mathematics and Statistics, Concordia University\\ 1455 de Maisonneuve Blvd. West \\ Montreal, Quebec H3G 1M8 Canada}
\maketitle
\vskip0.5cm {\bf Abstract.}
We derive comparison formulas relating the zeta-regularized determinant of an arbitrary self-adjoint extension of the Laplace operator
with domain $C^\infty_c(X\setminus \{P\})\subset L_2(X)$ to the zeta-regularized determinant of the Laplace operator on $X$.
Here $X$ is a compact Riemannian manifold of dimension $2$ or $3$; $P\in X$.
\vskip0.5cm
\section{Introduction}
Let $X_d$ be a complete Riemannian manifold of dimension $d\geq 2$ and let $\Delta$ be the (positive) Laplace operator on $X_d$. Choose a point $P\in X_d$ and consider $\Delta$ as an unbounded symmetric operator in the space $L_2(X_d)$ with domain $C^\infty_c(X_d\setminus\{P\})$. It is well-known that thus obtained operator is essentially self-adjoint if and only if $d\geq 4$. In case $d=2, 3$ it has deficiency indices $(1, 1)$ and there exists a one-parameter family $\Delta_{\alpha, P}$ of its self-adjoint extensions (called pseudo-laplacians; see \cite{YCdV}). One of these extensions (the Friedrichs extension $\Delta_{0, P}$) coincides with the self-adjoint operator $\Delta$ on $X_d$. In case $X_d=R^d$, $d=2,3$ the scattering theory for the pair $(\Delta_{\alpha, P}, \Delta)$ was extensively studied in the literature (see e. g., \cite{Albeverio}). The spectral theory of the operator $\Delta_{\alpha, P}$ on compact manifolds $X_d$ $(d=2, 3)$ was studied in \cite{YCdV}, notice also a recent paper \cite{Ueber} devoted to the case, where $X_d$ is a compact Riemann surface equipped with Poincar\'{e} metric.
The zeta-regularized determinant of Laplacian on a compact Riemannian manifold was introduced in \cite{RS} and since then was studied and used in an immense number of papers in string theory and geometric analysis, for our future purposes we mention here the memoir \cite{Fay}, where the determinant of Laplacian is studied as a functional on the space of smooth Riemannian metrics on a compact two-dimensional manifold, and the papers \cite{FG2003} and \cite{Kum}, where the reader may find explicit calculation of the determinant of Laplacian for three-dimensional flat tori and for the sphere $S^3$ (respectively).
The main result of the present paper is a comparison formula relating ${\rm det}(\Delta_{\alpha, P}-\lambda)$ to ${\rm det}(\Delta-\lambda)$, for $\lambda \in {\mathbb C}\setminus \left({\rm Spectrum} (\Delta)\cup {\rm Spectrum}(\Delta_{\alpha, P})\right)$.
It should be mentioned that in case of two-dimensional manifold the zeta-regularization of ${\rm det}(\Delta_{\alpha, P}-\lambda)$ is not that standard, since the corresponding operator zeta-function has logarithmic singularity at $0$.
It should be also mentioned that in the case when the manifold $X_d$ is flat in a vicinity of the point $P$ we deal with a very special case of the situation (Laplacian on a manifold with conical singularity) considered in \cite{Loya}, \cite{KLP1}, \cite{KLP} and, via other method, in \cite{HK}. The general scheme of the present work is close to that of \cite{HK}, although some calculations from \cite{KLP} also appear very useful for us.
{\bf Acknowledgements.}The work of T. A. was supported by FQRNT. Research of A. K. was supported by NSERC.
\section{Pseudo-laplacians, Krein formula and scattering coefficient} Let $X_d$ be a compact manifold of dimension $d=2$ or $d=3$; $P\in X_d$ and $\alpha\in [0, \pi)$. Following Colin de Verdi\`{e}re \cite{YCdV}, introduce the set $${\mathcal D}(\Delta_{\alpha, P})=\{f\in H^2(X_d\setminus \{P\}): \exists c\in {\mathbb C}: {\text \ in\ a\ vicinity\ of\ } P {\text \ one \ has}$$ \begin{equation}\label{as} f(x)=c(\sin \alpha \cdot G_d(r)+\cos\alpha)+o(1) {\text \ as\ } r\to 0\}\,,\end{equation} where $$H^2(X_d\setminus \{P\})=\{f\in L_2(X_d): \exists C\in {\mathbb C}: \Delta f- C\delta_{P}\in L_2(X_d)\}\,,$$ $r$ is the geodesic distance between $x$ and $P$ and $$G_d(r)=\begin{cases} \frac{1}{2\pi}\log r, \ \ d=2\\ -\frac{1}{4\pi r}, \ \ d=3. \end{cases}$$
Then (see \cite{YCdV}) the self-adjoint extensions of symmetric operator $\Delta$ with domain $C^\infty_c(X_d\setminus\{P\})$ are the operators $\Delta_{\alpha, P}$ with domains ${\mathcal D}(\Delta_{\alpha, P})$ acting via $u\mapsto \Delta u$. The extension $\Delta_{0, P}$ coincides with the Friedrichs extension and is nothing but the self-adjoint Laplacian on $X_d$.
Let $R(x, y; \lambda)$ be the resolvent kernel of the self-adjoint Laplacian $\Delta$ on $X_d$.
Following \cite{YCdV} define the scattering coefficient $F(\lambda; P)$ via \begin{equation}\label{coeffF} -R(x, P; \lambda)=G_d(r)+F(\lambda; P)+o(1)\end{equation} as $x\to P$. (Notice that in \cite{YCdV} the resolvent is defined as $(\lambda-\Delta)^{-1}$, whereas for us it is $(\Delta-\lambda)^{-1}$. This results in the minus sign in (\ref{coeffF}).)
As it was already mentioned the deficiency indices of the symmetric operator $\Delta$ with domain $C^\infty_c(X_d\setminus\{P\})$ are $(1, 1)$, therefore, one has the following Krein formula (see, e. g., \cite{Albeverio}, p. 357) for the resolvent kernel, $R_\alpha (x, y; \lambda)$, of the self-adjoint extension $\Delta_{\alpha, P}$: \begin{equation}\label{Krein} R_\alpha(x, y; \lambda)=R(x, y; \lambda)+k(\lambda; P)R(x, P; \lambda)R(P, y; \lambda) \end{equation} with some $k(\lambda; P)\in {\mathbb C}$.
The following Lemma relates $k(\lambda; P)$ to the scattering coefficient $F(\lambda; P)$. \begin{lemma}\label{L1} One has the relation \begin{equation}\label{rel} k(\lambda; P)=\frac{\sin \alpha}{F(\lambda; P)\sin \alpha-\cos\alpha}\,. \end{equation} \end{lemma} {\bf Proof.} Send $x\to P$ in (\ref{Krein}), observing that $R_\alpha (\,\cdot\, , y; \lambda)$ belongs to ${\mathcal D}_{\alpha, P}$, make use of (\ref{as}) and (\ref{coeffF}), and then compare the coefficients near $G_d(r)$ and the constant terms in the asymptotical expansions at the left and at the right. $\square$
It follows in particular from the Krein formula that the difference of the resolvents $(\Delta_{\alpha, P}-\lambda)^{-1}-(\Delta-\lambda)^{-1}$ is a rank one operator. The following simple Lemma is the key observation of the present work. \begin{lemma} One has the relation \begin{equation}\label{trace} {\rm Tr}\,\left( (\Delta_{\alpha, P}-\lambda)^{-1}-(\Delta-\lambda)^{-1} \right)=\frac {F_\lambda '(\lambda; P)\sin\alpha}{\cos\alpha-F(\lambda; P)\sin\alpha}\,. \end{equation} \end{lemma}
{\bf Proof.} One has
$$-F_\lambda'(\lambda; P)=\frac{\partial R(y, P; \lambda)}{\partial \lambda}\Big|_{y=P}=\lim_{\mu\to\lambda}\frac{R(y, P; \mu)-R(y, P; \lambda)}{\mu-\lambda}$$ Using resolvent identity we rewrite the last expression as
$$\lim_{\mu\to \lambda}\int_{X_d}R(y, z; \mu)R(P, z; \lambda)dz\Big|_{y=P}=\int_{X_d}[R(P, z; \lambda)]^2dz$$ From (\ref{Krein}) it follows that
$$[R(P, z; \lambda)]^2=\frac{1}{k(\lambda; P)}\left(R_{\alpha, P}(x, z; \lambda)-R(x, z; \lambda)\right)\Big|_{x=z}\,.$$ This implies $$-F_\lambda'(\lambda; P)=\frac{1}{k(\lambda, P)}{\rm Tr}\,\left( (\Delta_{\alpha, P}-\lambda)^{-1}-(\Delta-\lambda)^{-1} \right)$$ which, together with Lemma \ref{L1}, imply (\ref{trace}).$\square$
Introduce the domain $$\Omega_{\alpha, P}={\mathbb C}\setminus \{\lambda-it, \lambda\in {\rm Spectrum}\,(\Delta)\cup{\rm Spectrum}\,(\Delta_{\alpha, P}); t\in (-\infty, 0]\}\,.$$ Then in $\Omega_{\alpha, P}$ one can introduce the function \begin{equation}\label{xsi}\tilde\xi(\lambda)=-\frac{1}{2\pi i} \log(\cos\alpha-F(\lambda; P)\sin\alpha)\end{equation} (It should be noted that the function $\xi=\Re (\tilde \xi)$ is the spectral shift function of $\Delta$ and $\Delta_{\alpha, P}$.) One can rewrite (\ref{trace}) as \begin{equation}\label{trace1} {\rm Tr}\,\left( (\Delta_{\alpha, P}-\lambda)^{-1}-(\Delta-\lambda)^{-1} \right)=2\pi i\tilde \xi'(\lambda) \end{equation} \section{Operator zeta-function of $\Delta_{\alpha, P}$}
Denote by $\zeta(s, A)$ the zeta-function $$\zeta(s, A)=\sum_{\mu_k\in {\rm Spectrum}\,(A)}\frac{1}{\mu_k^s}$$ of the operator $A$. (We assume that the spectrum of $A$ is discrete and does not contain $0$.)
Take any $\tilde \lambda$ from ${\mathbb C}\setminus ({\rm Spectrum}\,(\Delta_{\alpha, P})\cup {\rm Spectrum}\,(\Delta)))$. From the results of \cite{YCdV} it follows that the function $\zeta(s, \Delta_{\alpha, P}-\tilde \lambda)$ is defined for sufficiently large $\Re s$. It is well-known that $\zeta(s, \Delta-\tilde \lambda)$ is meromorphic in ${\mathbb C}$.
The proof of the following lemma coincides verbatim with the proof of Proposition 5.9 from \cite{HK}. \begin{lemma}\label{lem}
Suppose that the function $\tilde \xi'(\lambda)$ from (\ref{trace1}) is $O(|\lambda|^{-1})$ as $\lambda\to -\infty$. Let $-C$ be a sufficiently large negative number and let $c_{\tilde \lambda,\epsilon}$ be a contour encircling the cut
$c_{\tilde \lambda}$ which starts from $-\infty+0i$, follows the real line till $-C$ and then goes to $\tilde \lambda$ remaining in $\Omega_{\alpha, P}$. Assume that
${\rm dist}\,(z, c_{\tilde \lambda})=\epsilon$ for any $z\in c_{\tilde \lambda,\epsilon}$. Let also $$\zeta_2(s)=\int_{c_{\tilde \lambda, \epsilon; 2}}(\lambda-\tilde \lambda)^{-s}\tilde\xi'(\lambda)d\lambda,$$ where the the integral at the right hand side is taken over the part $c_{\tilde \lambda, \epsilon; 2}$ of the contour $c_{\tilde \lambda,\epsilon}$ lying in the half-plane $\{\lambda: \Re\lambda>-C\}$. Let $$\hat{\zeta}_2(s)=\lim_{\epsilon\to 0}\zeta_2(s)=2i\sin(\pi s)\int_{-C}^{\tilde\lambda}(\lambda-\tilde\lambda)_0^{-s}\tilde \xi'(\lambda)\,d\lambda\,,$$ where $(\lambda-\tilde\lambda)_0^{-s}=e^{-i\pi s}\lim_{\lambda\downarrow c_{\tilde \lambda}}(\lambda-\tilde\lambda)^{-s}$. Then the function
\begin{equation}\label{MAIN}
R(s, \tilde \lambda)= \zeta(s, \Delta_{\alpha, P}-\tilde\lambda))-\zeta(s, \Delta-\tilde\lambda)-2i\sin(\pi s)\int_{-\infty}^{-C}|\lambda|^{-s}\tilde\xi'(\lambda)d\lambda-\hat{\zeta}_2(s) \end{equation} can be analitically continued to $\Re s>-1$ with $R(0, \tilde\lambda)=R'_s(0, \tilde\lambda)=0$. \end{lemma}
For completeness we give a sketch of proof. Using (\ref{trace1}), one has for sufficiently large $\Re s$
$$\zeta(s, \Delta_{\alpha, P}-\tilde\lambda)-\zeta(s, \Delta-\tilde \lambda)=\frac{1}{2\pi i}\int_{c_{\tilde\lambda, \epsilon}}(\lambda-\tilde\lambda)^{-s}{\rm Tr}((\Delta_{\alpha, P}-\lambda)^{-1}-(\Delta-\lambda)^{-1})d\lambda=$$ $$=\int_{c_{\tilde\lambda, \epsilon}}(\lambda-\tilde\lambda)^{-s}\tilde\xi'(\lambda)\,d\lambda=\zeta_1(s)+\zeta_2(s)\,,$$ where $$\zeta_1(s)=\left\{\int_{-\infty+i\epsilon}^{-C+i\epsilon}-\int_{-\infty-i\epsilon}^{-C-i\epsilon}\right\}(\lambda-\tilde\lambda)^{-s}\tilde\xi'(\lambda)d\lambda\,.$$ It is easy to show (see Lemma 5. 8 in \cite{HK}) that in the limit $\epsilon\to 0$ $\zeta_1(s)$ gives \begin{equation}\label{eq1}
2i\sin(\pi s)\int_{-\infty}^{-C}|\lambda|^{-s}\tilde\xi'(\lambda)\,d\lambda+2i\sin(\pi s)\int_{-\infty}^{-C}|\lambda|^{-s}\tilde\xi'(\lambda)\rho(s, \tilde \lambda/\lambda)d\lambda\,, \end{equation} where $\rho(s, z)=(1+z)^{-s}-1$ and
$$\rho(s, \tilde\lambda/\lambda)=O(|\lambda|^{-1})$$ as $\lambda\to -\infty$. Using the assumption on the asymptotics of $\tilde\xi(\lambda)$ as $\lambda\to -\infty$ and the obvious relation $\rho(0, z)=0$ one can see that the last term in (\ref{eq1}) can be analytically continued to $\Re s>-1$ and vanishes together with its first derivative w. r. t. $s$ at $s=0$. Denoting it by $R(s, \tilde\lambda)$ one gets the Lemma. $\square$
As it is stated in the introduction the main object we are to study in the present paper is the zeta-regularized determinant of the operator $\Delta_{\alpha, P}-\lambda$. Let us remind the reader that the usual definition of the zeta-regularized determinant of an operator $A$ \begin{equation}\label{def}{\rm det}\,A=\exp{(-\zeta'(0, A))}\end{equation} requires analyticity of $\zeta(s, A)$ at $s=0$.
Since the operator zeta-function $\zeta(s, \Delta-\tilde \lambda)$ is regular at $s=0$ (in fact, it is true in case of $\Delta$ being an arbitrary elliptic differential operator on any compact manifold) and the function $\hat{\zeta}_2(s)$ is entire, Lemma \ref{lem} shows that the behavior of the function $\zeta(s, \Delta_{\alpha, P}-\tilde\lambda)$ at $s=0$ is determined by the properties of the analytic continuation of the term
\begin{equation}\label{TERM} 2i\sin(\pi s)\int_{-\infty}^{-C}|\lambda|^{-s}\tilde\xi'(\lambda)d\lambda\end{equation} in (\ref{MAIN}). These properties in their turn are determined by the asymptotical behavior of the function $\tilde \xi'(\lambda)$ as $\lambda\to -\infty$.
It turns out that the latter behavior depends on dimension $d$. In particular, in the next section we will find out that in case $d=2$ the function $\zeta(s, \Delta_{\alpha, P}-\tilde\lambda)$ is not regular at $s=0$, therefore, in order to define ${\rm det}(\Delta_{\alpha, P}-\tilde\lambda)$ one has to use a modified version of (\ref{def}) .
\section{Determinant of pseudo-laplacian on two-dimensional compact manifold} Let $X$ be a two-dimensional Riemannian manifold, then introducing isothermal local coordinates $(x, y)$ and setting $z=x+iy$, one can write the area element on $X$ as
$$\rho^{-2}(z)|dz|^2$$
The following estimate of the resolvent kernel, $R(z', z; \lambda)$, of the Laplacian on $X$ was found by J. Fay (see \cite{Fay}; Theorem 2.7 on page 38 and the formula preceding Corollary 2.8 on page 39; notice that Fay works with negative Laplacian, so one has to take care of signs when using his formulas). \begin{lemma}\label{Fay}({\rm J. Fay}) The following equality holds true \begin{equation}\label{asfay}
-R(z, z'; \lambda)=G_2(r)+\frac{1}{2\pi}\left[\gamma+\log\frac{\sqrt{|\lambda|+1}}{2}\right.\end{equation}
$$\left.-\frac{1}{2(|\lambda|+1)}(1+\frac{4}{3}\rho^{2}(z)\partial^2_{z\bar z}\rho(z))+\hat{R}(z', z; \lambda)\right]\,, $$
where $\hat{R}(z', z; \lambda)$ is continuous for $z'$ near $z$, $$\hat {R}(z, z; \lambda)=O(|\lambda|^{-2})$$ uniformly w. r. t. $z\in X$ as $\lambda\to -\infty$; $r={\rm dist}(z, z')$, $\gamma$ is the Euler constant. \end{lemma}
Using (\ref{asfay}), we immediately get the following asymptotics of the scattering coefficient $F(\lambda,P)$ as $\lambda\to -\infty$: \begin{equation}\label{asF2} F(\lambda, P)=\end{equation}
$$\frac{1}{4\pi}\log(|\lambda|+1)+\frac{\gamma-\log 2}{2\pi}-\frac{1}{4\pi(|\lambda|+1)}\left[1+\frac{4}{3}\rho^{2}(z)\partial^2_{z\bar z}\rho(z)\Big|_{z=z(P)}\right]+O(|\lambda|^{-2})\,. $$
\begin{remark}{\rm It is easy to check that the expression $\rho^{2}(z)\partial^2_{z\bar z}\rho(z)\Big|_{z=z(P)}$ is independent of the choice of conformal local parameter $z$ near $P$.} \end{remark}
Now from (\ref{xsi}) and (\ref{asF2}) it follows that
$$2\pi i\tilde\xi'(\lambda)=-\frac{\frac{1}{4\pi(|\lambda|+1)}-\frac{b}{(|\lambda|+1)^{2}}+O(|\lambda|^{-3})}{\cot \alpha -a-\frac{1}{4\pi}\log(|\lambda|+1)+\frac{b}{|\lambda|+1}+O(|\lambda|^{-2})},$$ where $a=\frac{1}{2\pi}(\gamma-\log 2)$ and $b=\frac{1}{4\pi}(1+\frac{4}{3}\rho^2\partial^2_{z\bar z}\rho)$. This implies that for $-\infty<\lambda\leq -C$ one has \begin{equation}\label{eq3}
2\pi i\tilde\xi'(\lambda)=\frac{1}{|\lambda|(\log|\lambda|-4\pi\cot \alpha+4\pi a)}+f(\lambda)\,, \end{equation}
with $f(\lambda)=O(|\lambda|^{-2})$ as $\lambda\to -\infty$. Now knowing (\ref{eq3}), one can study the behaviour of the term (\ref{TERM}) in (\ref{MAIN}). We have
\begin{equation}\label{TERM1} 2i\sin(\pi s)\int_{-\infty}^{-C}|\lambda|^{-s}\tilde\xi'(\lambda)d\lambda=\end{equation}
$$\frac{\sin(\pi s)}{\pi}\int_{-\infty}^{-C}|\lambda|^{-s-1}\frac{d\lambda}{(\log|\lambda|-4\pi\cot \alpha+4\pi a)}+\frac{\sin(\pi s)}{\pi}\int_{-\infty}^{-C}|\lambda|^{-s}f(\lambda)\,d\lambda\,. $$
The first integral in the right hand side of (\ref{TERM1}) appeared in (\cite{KLP}, p. 15), where it was observed that it can be easily rewritten through the function $${\rm Ei}(z)=-\int_{-z}^\infty e^{-y}\frac{dy}{y}=\gamma+\log(-z)+\sum_{k=1}^{\infty}\frac{z^k}{k\cdot k!}\,$$ which leads to the representation \begin{equation}
\frac{\sin(\pi s)}{\pi}\int_{-\infty}^{-C}|\lambda|^{-s-1}\frac{d\lambda}{(\log|\lambda|-4\pi\cot \alpha+4\pi a)}= \end{equation} $$-\frac{\sin(\pi s)}{\pi}e^{-s\kappa}\left[\gamma+\log (s(\log C-\kappa))+ e(s)\right]\, $$ where $e(s)$ is an entire function such that $e(0)=0$; $\kappa=4\pi\cot\alpha-4\pi a$. From this we conclude that \begin{equation}\label{eq5}
\frac{\sin(\pi s)}{\pi}\int_{-\infty}^{-C}|\lambda|^{-s-1}\frac{d\lambda}{(\log|\lambda|-4\pi\cot \alpha+4\pi a)}=-s\log s+g(s)\,\end{equation} where $g(s)$ is differentiable at $s=0$.
Now (\ref{MAIN}) and (\ref{eq5}) justify the following definition. \begin{definition} Let $\Delta_{\alpha, P}$ be the pseudo-laplacian on a two-dimensional compact Riemannian manifold. Then the zeta-regularized determinant of the operator $\Delta_{\alpha, P}-\tilde \lambda$ with $\tilde\lambda\in {\mathbb C}\setminus {\rm Spectrum}(\Delta_{\alpha, P})$ is defined as \begin{equation}\label{def1}
{\rm det}(\Delta_{\alpha, P}-\tilde \lambda)=\exp\left\{-\frac{d}{ds}\left[\zeta(s, \Delta_{\alpha, P}-\tilde \lambda)+s\log s\right]\Big|_{s=0} \right\}\end{equation} \end{definition}
We are ready to get our main result: the formula relating ${\rm det}(\Delta_{\alpha, P}-\tilde \lambda)$ to ${\rm det}(\Delta-\tilde\lambda)$.
From (\ref{MAIN}, \ref{TERM}) it follows that
$$\frac{d}{ds}\left[\zeta(s, \Delta_{\alpha, P}-\tilde\lambda) +s\log s-\zeta(s,\Delta-\tilde\lambda)\right]\Big|_{s=0}=$$
$$\frac{d}{ds}\hat{\zeta}_2(s)\Big|_{s=0}+\int_{-\infty}^{-C}f(\lambda)\,d\lambda+$$
$$-\frac{d}{ds}\left\{\frac{\sin \pi s}{\pi}e^{-s\kappa}\left[ \gamma+\log (s(\log C-\kappa))+ e(s)\right]+s\log s \right\}\Big|_{s=0}=$$ $$2\pi i\left(\tilde\xi(\tilde\lambda)-\tilde\xi(-C)\right)+\int_{-\infty}^{-C}f(\lambda)\,d\lambda -\gamma-\log(\log C- \kappa)= $$ \begin{equation}\label{eq11} 2\pi i \tilde \xi(\tilde \lambda)-\gamma+\end{equation}$$\int_{-\infty}^{-C}f(\lambda)\,d\lambda - 2\pi i\tilde\xi(-C)-\log(\log C-4\pi\cot\alpha+2\gamma-\log 4)\,. $$ Notice that the expression in the second line of (\ref{eq11}) should not depend on $C$, so one can send $C$ to $+\infty$ there. Together with (\ref{asF2}) this gives \begin{equation}\label{result1}
\frac{d}{ds}\left[\zeta(s, \Delta_{\alpha, P}-\tilde\lambda) +s\log s-\zeta(s,\Delta-\tilde\lambda)\right]\Big|_{s=0}=\end{equation}$$2\pi i \tilde\xi(\tilde\lambda)-\gamma+\log(\sin \alpha/(4\pi))-i\pi\,$$ which implies the comparison formula for the determinants stated in the following theorem. \begin{theorem}\label{th1} Let $\tilde\lambda$ do not belong to the union of spectra of $\Delta$ and $\Delta_{\alpha, P}$ and let the zeta-regularized determinant of $\Delta_{\alpha, P}$ be defined as in (\ref{def1}). Then one has the relation \begin{equation}\label{Mresult2} {\rm det}(\Delta_{\alpha, P}-\tilde\lambda)= -4\pi e^{\gamma}(\cot \alpha- F(\tilde\lambda, P)){\rm det}(\Delta-\tilde\lambda)\, .\end{equation} \end{theorem}
Observe now that $0$ is the simple eigenvalue of $\Delta$ and, therefore, it follows from Theorem 2 in \cite{YCdV} that $0$ does not belong to the spectrum of the operator $\Delta_{\alpha, P}$ and that $\Delta_{\alpha, P}$ has one strictly negative simple eigenvalue. Thus, the determinant in the left hand side of (\ref{Mresult2}) is well defined for $\tilde\lambda=0$, whereas the determinant at the right hand side has the asymtotics \begin{equation}\label{000} {\rm det}(\Delta-\tilde\lambda)\sim (-\tilde \lambda){\rm det}^*\Delta\, \end{equation} as $\tilde\lambda\to 0-$. Here ${\rm det}^*\Delta$ is the modified determinant of an operator with zero mode.
From the standard asymptotics $$-R(x, y; \lambda)=\frac{1}{{\rm Vol}(X)}\frac{1}{\lambda}+G_2(r)+O(1)$$ as $\lambda\to 0$ and $x\to y$ one gets the asymptotics \begin{equation}\label{1111} F(\lambda, P)=\frac{1}{{\rm Vol}(X)}\frac{1}{\lambda}+O(1) \end{equation} as $\lambda\to 0$. Now sending $\tilde\lambda\to 0-$ in (\ref{Mresult2}) and using \ref{000} and \ref{1111} we get the following corollary of the Theorem \ref{th1}. \begin{corollary} The following relation holds true \begin{equation} {\rm det}\Delta_{\alpha, P}= -\frac{4\pi e^{\gamma}}{{\rm Vol}(X)}{\rm det}^*\Delta\,. \end{equation}
\end{corollary}
\section{Determinant of pseudo-laplacian on three-dimensional manifolds}
Let $X$ be a three-dimensional compact Riemannian manifold. We start with the Lemma describing the asymptotical behavior of the scattering coefficient as $\lambda \to -\infty$.
\begin{lemma}\label{GlLemm} One has the asymptotics
\begin{equation}\label{asy} F(\lambda; P)=\frac{1}{4\pi}\sqrt{-\lambda}+c_1(P)\frac{1}{\sqrt{-\lambda}}+O(|\lambda|^{-1})\end{equation} as $\lambda\to -\infty$ \end{lemma}
{\bf Proof.} Consider Minakshisundaram-Pleijel asymptotic expansion (\cite{MP}) \begin{equation}\label{MP1} H(x, P; t)=(4\pi t)^{-3/2}e^{-d(x, P)^2/(4t)}\sum_{k=0}^\infty u_k(x, P)t^k\end{equation} for the heat kernel in a small vicinity of $P$, here $d(x, P)$ is the geodesic distance from $x$ to $P$, functions $u_k( \cdot , P)$ are smooth in a vicinity of P, the equality is understood in the sense of asymptotic expansions. We will make use of the standard relation \begin{equation}\label{stR}R(x, y; \lambda)=\int_0^{+\infty}H(x, y; t)e^{\lambda t}\,dt\,.\end{equation} Let us first truncate the sum (\ref{MP1}) at some fixed $k=N+1$ so that the remainder, $r_n$, is $O(t^{N})$. Defining \[ \tilde{R}_N(x,P; -\lambda) := \int_0^\infty r_n(t,x,P) e^{t\lambda} dt\,, \] we see that
$$\tilde{R}_N(x,P; \lambda)=O(|\lambda|^{-(N+1)})$$ as $\lambda\to -\infty$ uniformly w. r. t. $x$ belonging to a small vicinity of $P$.
Now, for each $0\leq k\leq N+1$ we have to address the following quantity
\[ R_k(x,P; \lambda) := \frac{u_k(x,y)}{(4\pi)^{3/2}} \int_0^\infty t^{k-\frac{3}{2}}e^{-\frac{d(x, P)^2}{4t}}e^{\lambda t} dt. \]
According to identity (\ref{Ba}) below one has $$R_0(x, P; \lambda)=\frac{u_0(x, P)}{(4\pi)^{3/2}}\frac{2\sqrt{\pi}}{d(x, P)}e^{-d(x, P)\sqrt{-\lambda}}=$$ \begin{equation}\label{te1} \frac{1}{4\pi d(x, P)}-\frac{1}{4\pi}\sqrt{-\lambda}+o(1), \end{equation} as $d(x, P)\to 0$. For $k\geq 1$ one has $$R_k(x, P; \lambda)=\frac{u_k(x, P)}{(4\pi)^{3/2}}2^{3/2-k}\left(\frac{d(x, P)}{\sqrt{-\lambda}} \right)^{k-1/2}K_{k-\frac{1}{2}}(d(x, P)\sqrt{-\lambda})=$$ \begin{equation}\label{te2} -c_k(P)\frac{1}{(\sqrt{-\lambda})^{2k-1}}+o(1)\end{equation} as $d(x, P)\to 0$ (see \cite{Ba}, p. 146, f-la 29). Now (\ref{asy}) follows from (\ref{stR}), (\ref{te1}) and (\ref{te2}).
$\square$
Now from Lemma \ref{GlLemm} it follows that \begin{equation}\label{glavn}
2\pi i\tilde\xi'(\lambda)=-\frac{1}{2\lambda}+O(|\lambda|^{-3/2})\end{equation} as $\lambda\to -\infty$, therefore, one can rewrite (\ref{TERM}) as
\begin{equation}\label{T2}\frac{\sin(\pi s)}{\pi}\left\{\int_{-\infty}^{-C}|\lambda|^{-s}(2\pi i \tilde\xi'(\lambda)+\frac{1}{2\lambda})d\lambda+\frac{C^{-s}}{2s}\right\}\end{equation} which is obviously analytic in $\Re s>-\frac{1}{2}$. Thus, it follows from (\ref{MAIN}) that the function $\zeta(s, \Delta_{\alpha, P}-\tilde\lambda)$ is regular at $s=0$ and one can introduce the usual zeta-regularization $${\rm det}(\Delta_{\alpha, P}-\tilde\lambda)=\exp\{-\zeta'(0, \Delta_{\alpha, P}-\tilde\lambda)\}$$ of ${\rm det}(\Delta_{\alpha, P}-\tilde\lambda)$.
Moreover, differentiating (\ref{MAIN}) with respect to $s$ at $s=0$ similarly to (\ref{eq11}) we get
$$\frac{d}{ds}\left[\zeta(s, \Delta_{\alpha, P}-\tilde\lambda) -\zeta(s,\Delta-\tilde\lambda)\right]\Big|_{s=0}=$$ $$2\pi i(\tilde\xi(\tilde\lambda)-\tilde\xi(-C))+\int_{-\infty}^{-C}(2\pi i \tilde\xi'(\lambda)+\frac{1}{2\lambda})d\lambda-\frac{1}{2}\log C=$$ which reduces after sending $-C\to-\infty$ to $$2\pi i\tilde\xi(\tilde\lambda)+\log\sin \alpha-\log (4\pi)+i\pi=-\log(\cot\alpha-F(\lambda; P))-\log(4\pi)+i\pi\,$$ which implies the following theorem. \begin{theorem} Let $\Delta_{\alpha, P}$ be the pseudo-laplacian on $X$ and $\tilde\lambda\in {\mathbb C}\setminus ({\rm Spectrum}(\Delta)\cup {\rm Spectrum}(\Delta_{\alpha, P}))$. Then \begin{equation}\label{SphereDet} {\rm det}(\Delta_{\alpha, P}-\tilde\lambda)=-4\pi(\cot\alpha-F(\tilde\lambda; P)){\rm det}(\Delta-\tilde\lambda)\,. \end{equation} \end{theorem}
Sending $\tilde\lambda\to 0$ and noticing that relation ({\ref {1111}) holds also in case $d=3$ we get the following corollary. \begin{corollary} \begin{equation} {\rm det}\Delta_{\alpha, P}=-\frac{4\pi}{{\rm Vol}(X)}{\rm det}^*\Delta\,. \end{equation}
\end{corollary}
In what follows we consider two examples of three-dimensional compact Riemannian manifolds for which there exist explicit expressions for the resolvent kernels: a flat torus and the round (unit) $3d$-sphere. These manifolds are homogeneous, so, as it is shown in \cite{YCdV}, the scattering coefficient $F(\lambda, P)$ is $P$-independent.
{\bf Example 1: Round $3d$-sphere.} \begin{lemma}\label{sph} Let $X=S^3$ with usual round metric.
Then there is the following explicit expression for scattering coefficient
\begin{equation}\label{Tayeb}F(\lambda)=\frac{1}{4\pi}\coth\left(\pi\sqrt{-\lambda-1}\,\right)\cdot\sqrt{-\lambda-1}\end{equation} and, therefore, one has the following asymptotics as $\lambda\to -\infty$ \begin{equation}\label{sph1}
F(\lambda)=\frac{1}{4\pi}\sqrt{|\lambda|-1}+O(|\lambda|^{-\infty})\,. \end{equation} \begin{remark}{\rm The possibility of finding an explicit expression for $F(\lambda)$ for $S^3$ was mentioned in \cite{YCdV}. However we failed to find (\ref{Tayeb}) in the literature.} \end{remark} \end{lemma} {\bf Proof.} We will make use the well-known identity (see, e. g., \cite{Ba}, p. 146, f-la 28): \begin{equation}\label{Ba}
\int_{0}^{+\infty}e^{\lambda t}t^{-3/2}e^{-\frac{d^2}{4t}}\,dt=2\frac{\sqrt{\pi}}{|d|}e^{-|d|\sqrt{-\lambda}}; \end{equation} for $\lambda<0$ and $d\in{\mathbb R}$ and the following explicit formula for the operator kernel $e^{-t}H(x, y; t)$ of the operator $e^{-t(\Delta+1)}$, where $\Delta$ is the (positive) Laplacian on $S^3$ (see \cite{CT}, (2.29)): \begin{equation}\label{Taylor}
e^{-t}H(x, y; t)=-\frac{1}{2\pi}\frac{1}{\sin d(x, y)}\frac{\partial}{\partial z}\Big|_{z=d(x,y)}\Theta(z, t)\,. \end{equation} Here $d(x, y)$ is the geodesic distance between $x, y\in S^3$ and $$\Theta(z, t)=\frac{1}{\sqrt{4\pi t}}\sum_{k=-\infty}^{+\infty}e^{-(z+2k\pi)^2/4t}$$ is the theta-function.
Denoting $d(x, y)$ by $\theta$ and using (\ref{Taylor}) and (\ref{Ba}), one gets $$R(x, y; \lambda-1)=\int_{0}^{+\infty}e^{\lambda t}e^{-t}H(x, y; t)\,dt=$$$$\frac{1}{4\pi}\frac{1}{\sin \theta}\left(-\sum_{k<0}e^{(\theta+2k\pi)\sqrt{-\lambda}}+\sum_{k\geq 0}e^{-(\theta+2k\pi)\sqrt{-\lambda}}\right)=$$ $$\frac{1}{4\pi}\frac{1}{\sin \theta}\frac{1}{1-e^{-2\pi\sqrt{-\lambda}}}\left[-e^{-2\pi\sqrt{-\lambda}}e^{\theta\sqrt{-\lambda}}+e^{-\theta\sqrt{-\lambda}} \right]=$$ \begin{equation} \frac{1}{4\pi\theta}-\frac{1}{4\pi}\frac{1+e^{-2\pi\sqrt{-\lambda}}}{1-e^{-2\pi\sqrt{-\lambda}}}\sqrt{-\lambda}+o(1) \end{equation} as $\theta\to 0$, which implies the Lemma. $\square$
{\bf Example 2: Flat $3d$-tori.}Let $\{{\bf A, B, C}\}$ be a basis of ${\mathbb R}^3$ and let $T^3$ be the quotient of ${\mathbb R}^3$ by the lattice $\{m{\bf A}+n{\bf B}+l{\bf C}: (m, n, l)\in {\mathbb Z}^3\}$ provided with the usual flat metric.
Notice that the free resolvent kernel in $R^3$ is $$\frac{e^{-\sqrt{-\lambda}||x-y||}}{4\pi||x-y||}$$ and, therefore, \begin{equation}\label{exactres}
R(x, y; \lambda)=\frac{e^{-\sqrt{-\lambda}||x-y||}}{4\pi||x-y||}+\frac{1}{4\pi}
\sum_{(m, n, l)\in {\mathbb Z}^3\setminus(0,0,0)}\frac{e^{-\sqrt{-\lambda}||x-y+m{\bf A}+n{\bf B}+l{\bf C}||}}{||x-y+m{\bf A}+n{\bf B}+l{\bf C}||}\,. \end{equation} From (\ref{exactres}) it follows that
$$F(\lambda)= \frac{1}{4\pi}\sqrt{-\lambda}-\frac{1}{4\pi}\sum_{(m, n, l)\in {\mathbb Z}^3\setminus(0,0,0)}\frac{e^{-\sqrt{-\lambda}||m{\bf A}+n{\bf B}+l{\bf C}||}}{||m{\bf A}+n{\bf B}+l{\bf C}||}=$$$$\frac{1}{4\pi}\sqrt{-\lambda}+O(|\lambda|^{-\infty})$$ as $\lambda\to-\infty$.
\begin{remark}{\rm It should be noted that explicit expressions for ${\rm det}^*\Delta$ in case $X=S^3$ and $X=T^3$ are given in \cite{Kum} and \cite{FG2003}.} \end{remark}
\end{document} | arXiv |
The association of early-life exposure to ambient PM2.5 and later-childhood height-for-age in India: an observational study
Dean Spears1,2,
Sagnik Dey3,4,
Sourangsu Chowdhury3,
Noah Scovronick5,
Sangita Vyas1 &
Joshua Apte6
Environmental Health volume 18, Article number: 62 (2019) Cite this article
Children in India are exposed to high levels of ambient fine particulate matter (PM2.5). However, population-level evidence of associations with adverse health outcomes from within the country is limited. The aim of our study is to estimate the association of early-life exposure to ambient PM2.5 with child health outcomes (height-for-age) in India.
We linked nationally-representative anthropometric data from India's 2015–2016 Demographic and Health Survey (n = 218,152 children under five across 640 districts of India) with satellite-based PM2.5 exposure (concentration) data. We then applied fixed effects regression to assess the association between early-life ambient PM2.5 and subsequent height-for-age, analyzing whether deviations in air pollution from the seasonal average for a particular place are associated with deviations in child height from the average for that season in that place, controlling for trends over time, temperature, and birth, mother, and household characteristics. We also explored the timing of exposure and potential non-linearities in the concentration-response relationship.
Children in the sample were exposed to an average of 55 μ g/m3 of PM2.5 in their birth month. After controlling for potential confounders, a 100 μg/m3 increase in PM2.5 in the month of birth was associated with a 0.05 [0.01–0.09] standard deviation reduction in child height. For an average 5 year old girl, this represents a height deficit of 0.24 [0.05–0.43] cm. We also found that exposure to PM2.5 in the last trimester in utero and in the first few months of life are significantly (p < 0.05) associated with child height deficits. We did not observe a decreasing marginal risk at high levels of exposure.
India experiences some of the worst air pollution in the world. To our knowledge, this is the first study to estimate the association of early-life exposure to ambient PM2.5 on child height-for-age at the range of ambient pollution exposures observed in India. Because average exposure to ambient PM2.5 is high in India, where child height-for-age is a critical challenge in human development, our results highlight ambient air pollution as a public health policy priority.
India experiences some of the worst particulate air pollution in the world, with mean PM2.5 concentrations consistenly above World Health Organization guidelines [1,2,3]. Due to these high exposures, the Disease Burden of India study recently estimated that 12.9 (11.4–14.4) million disability adjusted life years (DALY) and 149.8 (132.3–167.6) thousand deaths annually were attributable to PM2.5 in children under 5 years of age [4]. The child mortality burden due to household PM2.5 exposure in India is also large, estimated at 50 (30–60) thousand deaths per year. These estimates, however, rely on concentration-response functions that were developed from epidemiological studies carried out primarily in high-income countries. Moreover, the empidemiological studies used for constructing these estimates do not consider impacts on a number of child health outcomes that have been linked to PM2.5 exposure, including sudden infant death syndrome [5], low birth weight [6, 7], intrauterine growth retardation [8] and reduced size [9].
Unlike many developed countries, India does not have a vital registration system, making it difficult to study mortality, a commonly used outcome variable in the air pollution literature. Another widely-studied marker of early-life health insults is the average height of children. Children in India are unusually short compared to international standards [10]. Many causes of this child height deficit have been proposed in the demographic, epidemiological, and econometric literatures, including poor sanitation and maternal nutrition [11, 12]. Exposure to PM2.5 from household solid fuel use for cooking and heating has also been associated with child growth in India [13, 14], but to our knowledge, no study from India has explored the link between ambient air pollution and child growth. One study from Bangladesh [8] observed that the risk of child stunting and wasting was positively associated with higher levels of in utero exposure.
In this study, we examine the association of in utero and early-age ambient PM2.5 exposure on child height-for-age in India. We do so in a representative sample of Indian children – a population exposed to a large range of ambient PM2.5 – using data from the 2015–2016 Demographic and Health Survey (DHS), matched to air pollution data, as measured by satellite remote sensing. The association between child height-for-age and early-life exposure to air pollution is estimated using an approach that accounts for fixed differences across villages, secular trends over time, and district-specific seasonal patterns. We also investigate the shape of the concentration-response function.
Data and methods
India's 2015–2016 demographic and health survey
Data on child height and potential confounders are taken from India's most recent DHS survey (note that in India the DHS is also known as the National Family Health Survey). These data were collected from a nationally-representative sample of women of reproductive age. The survey visited all 640 Indian districts that existed at the time of the 2011 Census, and was designed to be representative at the district level. These data were collected between January 2015 and November 2016.
In our analysis, the outcome (dependent) variable is a child's height-for-age z-score, scaled according to the World Health Organization 2006 reference population mean and standard deviation by sex and age-in-months [15]. In the DHS, height is measured for children less than 5 years old at the time of the survey. The sex and month of birth (e.g. August 2011) is also recorded for each child with measured height.
Air pollution data by district-month
Each child was assigned the average ambient PM2.5 exposure in his or her district of residence during the month in which he or she was born. This matching assumes that the district where children live at the time of the survey is the same as the district where children lived when they were born.
Because India lacks ground-based PM2.5 measurements at a spatial resolution sufficient for our study design, we used satellite-derived PM2.5. Specifically, we use the Multiangle Imaging SpecroRadiometer (MISR) retrieved daily aerosol optical depth (AOD) V22 product at 17.6 km × 17.6 km spatial resolution to estimate PM2.5 with the help of a spatially and temporally varying conversion factor (ƞ). ƞ is derived from GEOS-Chem chemical transport model simulations and depends on aerosol vertical distribution, emissions, and meteorological factors like temperature, relative humidity, and precipitation. Details about the conversion factor ƞ are discussed elsewhere [2, 16, 17]. The MISR AOD product was previously and extensively evaluated for the Indian subcontinent [18]. The satellite-retrieved PM2.5 was bias-corrected using coincident ground-based quality controlled measurements following our earlier study and has ~ 10% uncertainty [2, 19]. The district-level statistics are extracted using the shape files of the district boundaries in ArcGIS. We generated a monthly PM2.5 exposure database for 15 years (2001–2015), although because height is only measured in the DHS for children under five, no child in our sample was born before 2010.
Temperature data by district-month
Considering the large spatio-temporal heterogeneity in temperature across India [20], we control for temperature in the month and district of birth. Monthly temperature data at the 0.125° × 0.125° (approximately, a 12 km × 12 km grid) resolution was obtained from the European Centre for Medium Range Weather Forecast (ECMWF) ERA-INTERIM dataset. Mean district-level temperature was estimated using a spatially weighted average of the 0.125° × 0.125° grid cells in the district.
Main statistical approach
The central empirical strategy of this paper is fixed effects regression, with child height-for-age as the outcome (dependent) variable, and early-life district-month exposure to ambient PM2.5 as the independent variable (exposure) of interest. Fixed effects regression has been identified as a useful tool in epidemiological analyses to control for unobserved characteristics that are common across observations within groups, time periods, or individuals [21], and has been successfully applied in prior epidemiological studies of air pollution-health relationships [8]. In our analysis, we include fixed effects for birth place, seasonal patterns in the district, and year. We therefore study whether deviations in seasonal average PM2.5 in a particular place (village or urban block) are associated with deviations in child height from the average for that same season in that same place. As a result, our study asks the question: is exposure to PM2.5 in the month of birth that is higher than the seasonal average associated with heights that are shorter than average for that place and season of birth?
In India, and other countries where environmental risks are widespread, the average height-for-age z-score declines in the first 2 years of life, reflecting the accumulating impact of early-life health insults on a child's growth [22]. Because age is predictably correlated with height-for-age, each regression also controls for 119 age-in-months-by-sex indicators, one for each age in months from zero to 59, for girls and boys separately, and excluding one to avoid perfect multicollinearity. This type of adjustment is standard in the literature on child height [10, 11].
Our main models take the following form:
$$ {h}_{ipdmy}=\beta {x}_{dm y}+{\mu}_1{t}_{dm y}+{\mu}_2{t}_{dm y}^2+\rho\ {momh}_{ipdmy}+{\boldsymbol{\alpha}}_{pd}+{\boldsymbol{\gamma}}_{dm}+{\boldsymbol{\delta}}_y+{\boldsymbol{X}}_{ipdmy}\boldsymbol{\theta} +{\varepsilon}_{ipdmy} $$
where i indexes individual children, p places (survey primary sampling units – PSUs – such as urban blocks or rural villages), d districts, m calendar month of birth (such as February), and y calendar year of birth (such as 2012). The dependent variable, h, is child i's height-for-age z-score. The independent variable of interest, xdmy, is PM2.5 in district d in month m of year y, corresponding to child i's birth month. Similarly, tdmy is temperature in that same district-month. We include temperature as a quadratic in order to allow for nonlinearities in its association with child height. momhipdmy is the height of the mother's child, in centimeters, a proxy for the health and socioeconomic status of the mother. Fixed effects are αpd, 27,266 local places (PSUs); γdm, 7679 categories of district-month (such as for Februarys in Sitapur district, or Aprils in Kanpur district); and δy, 6 calendar years, to capture any secular time trend. Child-level covariates Xipdmy include age-by-sex fixed effects and other covariates that have been associated with child height. These include birth characteristics (mother's age at birth [23], birth order [24], whether the delivery occurred in a hospital or health facility [25], and whether it was a multiple birth [23]), mother characteristics (whether she smokes [23], the total number of children born to her by the time of the survey [24], and her relationship to the household head [26]), and household-level covariates (caste [12], religion [27], solid fuel use [13], open defecation [11], and drinking water source [8]).
This statistical strategy was designed to address several potential sources of confounding. The strategy allows us to add fixed effects and covariates in stages to verify that the main effect estimate, \( \hat{\beta} \), is robust to respecification. In particular, we first estimate the model without PSU fixed effects. PSU fixed effects would account for any fixed geographic differences in factors known to affect child height, such as the presence and quality of markets [28], local open defecation [29], or the religious composition of the neighborhood [30]. We then add birth, mother, and household characteristics, as described above, which are intended to control for other known determinants of child height. While birth-level covariates reflect characteristics at the time of birth, mother and household level characteristics are observed at the time of the survey. Many of these characteristics, such as mother's height, caste, and religion, are not likely to have changed over time. However, other characteristics, such as mother smoking, mother's relationship to the household head, and water source may have changed. The use of solid fuels for cooking and open defecation has been changing relatively slowly in India over time [12, 31], and so these variables are likely to be highly correlated with household behaviors at the time of the child's birth, even though they are only observed at the time of the survey. Our a priori preferred specification, however, includes birth characteristics only because these variables reflect the environment at the time of birth, and not at the time of the survey. All subsequent analyses build off of this preferred specification.
All of our main model specifications control for district-month fixed effects, a tool which has been used in the literature to control for seasonal trends [32]. This strategy allows each district to have any distinct seasonal pattern, and identifies effects off of deviations from each district's seasonal patterns. Controlling for seasonal trends is important because pollution in India is highly seasonal [33], and later life outcomes such as educational attainment are also known to be predicted by seasonal patterns [34]. Since child height is also correlated with these outcomes [35], season is likely to be a confounder. A month fixed effect by itself would control for seasonality that is common across all of India. However, since pollution patterns are highly localized, we include district-month fixed effects, which allow seasonal patterns to be different in each district, and therefore we control for local seasonality. We also include a sensitivity check with finer (PSU-month) controls for seasonality.
Finally, we conduct a falsification test, in which we control for ambient PM2.5 in the same district-month, but 2 years before the month of birth; if our identification strategy is credible, this control should not predict height nor change our estimate. Standard errors are clustered by 640 districts, to permit arbitrary correlation of error terms over space and time within districts [36].
DHS data include sampling weights, to be used to generate estimates that are representative of the population of Indian children under five. Although we use weights for our summary statistics in Table 1, sampling weights are not appropriate for estimating associations [37], so we do not use them in any of our statistical models. In Additional file 1: Table S1, we also calculate summary statistics without sampling weights, which are very similar to the statistics presented in Table 1.
Table 1 Summary statistics describing sample of children with measured height from India's 2015–16 DHS, reported for the full sample and by PM2.5 quintiles in the month and district of birth
Age of exposure
Our primary goal in this study is to investigate the effect on child height of exposure to ambient PM2.5 in the month of birth. The most vulnerable period of exposure is an active area of research [38, 39], but we chose exposure in the month of birth a priori because it represents an important period for early-life development [40]. Nevertheless, in additional analyses we also consider exposure at other ages by averaging over three-month periods, from − 8 to − 6 months before birth (approximately the first trimester of pregnancy) to 10 to 12 months after birth (approximately the last quarter of the first year of life). Average PM2.5 in each age of exposure is used as the independent variable in a separate regression:
$$ {h}_{ipdmy}=\beta \left(\frac{x_{dm y}^{+0}+{x}_{dm y}^{+1}+{x}_{dm y}^{+2}}{3}\right)+{\mu}_1{t}_{dm y}+{\mu}_2{t}_{dm y}^2+\rho\ {momh}_{ipdmy}+{\boldsymbol{\alpha}}_{pd}+{\boldsymbol{\gamma}}_{dm}+{\boldsymbol{\delta}}_y+{\boldsymbol{X}}_{ipdmy}\boldsymbol{\theta} +{\varepsilon}_{ipdmy} $$
where indices and fixed effects are as in regression eq. (1), but the covariates X include only the age-in-months-by-sex indicators and birth characteristics, as these variables reflect attributes at the time of birth, and not at the time of the survey. Therefore, this model builds off of the preferred specification discussed in the previous section.
Shape of the concentration-response function
The shape of the concentration-response function has been a focus of the air pollution literature, in light of its importance for policy responses [41]. Although the prior literature has suggested the possibility of diminishing marginal risks at higher levels of exposure, there is little well-identified evidence on exposure to PM2.5 at levels as high as in India during the period studied, or for child height as the outcome [42, 43]. Therefore, we perform three analyses in which we allow the shape of the concentration-response function to differ from the linear form in eq. (1):
$$ {h}_{ipdmy}=\beta f\left({x}_{dm y}\right)+{\mu}_1{t}_{dm y}+{\mu}_2{t}_{dm y}^2+\rho\ {momh}_{ipdmy}+{\boldsymbol{\alpha}}_{pd}+{\boldsymbol{\gamma}}_{dm}+{\boldsymbol{\delta}}_y+{\boldsymbol{X}}_{ipdmy}\boldsymbol{\theta} +{\varepsilon}_{ipdmy} $$
First, we substitute in the natural log of PM2.5 in one specification, and a linear spline at the median of PM2.5 in another. Then we allow polynomial shapes of the concentration-response curve, of degree 1 through 5. Finally, we implement a Box-Cox power transformation of the form f(x) = xλ, for coefficients λ in steps of 0.1 from 0.1 to 2.0. We implement each power transformation in a separate model, and plot the resulting log-likelihoods. If likelihood is maximized near λ = 1, then this procedure would suggest that a linear concentration-response function best fits the data. As in the age of exposure analysis, the covariates X include only the age-in-months-by-sex indicators and birth characteristics, as these variables represent attributes at the time of birth, and not at the time of the survey.
Because we study publicly-available, anonymized data on child height, our study is classified as "not human subjects research" and informed consent is not required.
All analyses in the paper were computed with Stata 12.1.
Summary and descriptive statistics
Height was measured for 225,002 children under five in the DHS. We were able to match air pollution data to children born from February 2010 to December 2015, who are 97% of those with measured height, resulting in a final sample of 218,152 children (Fig. 1). Summary statistics as sample means for these children are presented in Table 1, reported for the full sample, as well as by quintiles of ambient PM2.5 exposure. Across the whole sample, children were exposed to an average of 55 μ g/m3 in their month of birth, although with substantial variation. 92% of children were born while the mother was living in her current residence. The results also highlight that children who are exposed to higher ambient PM2.5 tend to be disadvantaged in other ways: they come from larger families, have shorter mothers, live in households that are more likely to defecate in the open, and use solid fuels for cooking.
Study sample with excluded or missing observations. Note: In Table 2, some samples are smaller than 193,040 because the regression models ignore categories within which there is no variation in the independent variable
Location [44, 45], time of year [33], and mother's height, as seen in Table 1, are correlated with air pollution concentrations. Therefore, Fig. 2 plots crude associations between pollution and height, stratified by rural/urban, season, and mother's height. Each panel in Fig. 2 presents locally-weighted kernel regressions of the relationship between ambient PM2.5 in the district-month of birth and height-for-age z-score residuals (after controlling only for age-by-sex, see methods for more discussion). We do not control for any other covariates in this figure. Panel A reveals a negative (downward) gradient – which is approximately linear - between ambient PM2.5 exposure and child height for both rural and urban children. Although the range of PM2.5 exposure is similar in both rural and urban areas, the former are shorter, on average, because they are more exposed to other factors associated with growth faltering [12, 29]. Panel B demonstrates that ambient PM2.5 reaches the highest levels in winter (November through January), and that a similar downward gradient is present in all seasons. Panel C suggests that the association is not driven by mother's height, since the downward gradient exists for each quintile of mother's height.
Crude associations between child height and exposure to PM2.5 in the month of birth. Note: Panel a stratifies by urban/rural, Panel b by season of birth, and Panel c by mother's height. Curves are kernel-weighted local regressions. The vertical axis in all panels is the residual of child height-for-age, after controlling for age-in-months by sex indicators. In Panel B, observations of PM2.5 that are greater than the 95th percentile for each season are left out because a sufficient number of observations are not available to construct means for pollution levels that are very high for the season
Exposure in the month-of-birth
Table 2 presents our main results: fixed effects regression results following Eq. 1. For ease of interpretation, results are presented for a 100 μ g/m3 increase in PM2.5. Column 1 shows coefficients from a regression that includes age-by-sex fixed effects, district-month fixed effects, and year of birth fixed effects. Column 2 adds PSU fixed effects, temperature, and mother's height. Columns 3, 4, and 5 progressively add birth, mother, and household characteristics, respectively. Across the alternative specifications in columns 1 through 5, a 100 μ g/m3 increase in ambient PM2.5 exposure is associated with an approximately 0.05 standard deviation decrease in child height-for-age z-score. Because columns 4 and 5 include covariates that are measured at the time of the survey, and do not necessarily reflect the environment at the time of birth, Column 3 represents our a priori preferred specification. Column 6, reports the falsification test: as expected, ambient PM2.5 exposure 2 years before the child is born does not predict height and does not change the coefficient of interest. Columns 7 and 8 are tests of the linearity assumption and are discussed further below.
Table 2 Association of district-level PM2.5 (per 100 μg/m3) in month of birth with child height-for-age z-score
Figure 3 presents the association of PM2.5 and child growth given different time periods of exposure (see Eq. 2 above for modeling details). Of the seven time periods explored, two show significant (p < 0.05) adverse effects on child growth – the last trimester in utero and the period just after birth. No other periods of exposure are significantly associated with child height. These results are consistent with evidence in the literature that shocks in utero and early-life are critical for child development outcomes [40, 46].
Effects of PM2.5 exposure at various ages. Note: Dots denote point estimates and lines denote 95% confidence intervals. Each result shown is from a separate fixed effects regression of child height-for-age on the average exposure to PM2.5 in the months, relative to birth, specified along the horizontal axis
Shape of concentration-response function
The three tests for non-linear concentration-response functions each failed to reject that a linear shape best fits the data. Moreover, each approach suggests that, if anything, effects may be steeper at higher concentration levels. Specifically, column 7 of Table 2 shows that a natural log functional form – consistent with a concentration-response function exhibiting diminishing marginal costs – fits the data less well than the linear form. Column 8 includes a linear spline that allows a different slope above the median level of ambient PM2.5; although the two PM2.5 terms are jointly statistically significant at the 10% level (F = 2.72; p = 0.067), neither is individually significantly different from zero. Although this model does not fit the data better than a simple linear form, the negative sign on the coefficient suggests the possibility of a steeper concentration-response function at higher levels of exposure. Additional file 1: Figure S1 demonstrates that none of the polynomial forms we tested (quadratic through quantic) improve on a linear functional form, while Additional file 1: Figure S2 – the Box-Cox transformation – indicates that a model with slightly increasing marginal effects may best fit the data.
In Additional file 1: Table S2, we present results from statistical analyses similar to columns 6 through 8 of the the main Table 2, the difference is that models presented in the supplementary table include all coverates, including birth, mother, and household characteristics, rather than birth characteristics only. The inclusion of these additional control variables does not change the interpretation of these analyses. We also show that the model is robust to replacing district-month fixed effects with PSU-month fixed effects, a finer measure of seasonality.
We report the first evidence of an association between ambient PM2.5 exposure and child height in India by using the country's most recent DHS, which measures children under 5 years old in a nationally representative sample of reproductive age women. We find that an increase in PM2.5 of 100 μ g/m3 in the month of birth is associated with a decrease of 0.05 height-for-age standard deviations; for an average 5 year old girl, this would equate to a height deficit of 0.24 cm. Consistent with evidence in the literature that shocks in utero and early-life are critical for child development outcomes [40, 46], we find evidence that exposure to PM2.5 during the last few months in utero and the first few months of life are associated with height deficits.
The average child in our data is exposed to a PM2.5 concentration of 55 μ g/m3 in her month of birth. Using the estimates from our analysis, this means that the average child is about 0.027 height-for-age standard deviations shorter than she would be if exposed to very low levels of air pollution at birth. For an average 5 year old girl, this represents a height deficit of 0.13 cm. Although this effect is small relative to other environmental factors affecting child health, such as open defecation [11], it influences all of the almost 30 million births per year that occur in India.
Moreover, the difference between the children in our sample most exposed to PM2.5 (at the 95th percentile) and the children least exposed (at the 5th percentile) is 116 μ g/m3. Therefore, based on our findings, the most exposed children in India are about 0.06 height-for-age standard deviations shorter than they counterfactually would be if they were exposed only at the lowest levels in our sample. This projected difference — 0.06 height-for-age standard deviations — is of the same order of magnitude as other height differences that have received sustained attention in the literature on the demography of child height: it is about half as large as the well-studied India-Africa height gap [11], and is about one-tenth of the height gap between children of literate versus illiterate mothers. Since child growth is highly correlated with early-life mortality [47], the associations we observed in this study are suggestive of an association between PM2.5 exposure and early-life survival. In the data we use for this analysis, a district where children are 0.06 height-for-age standard deviations shorter would be expected, on average, to have an infant mortality that is larger by 5 infant deaths per 1000 live births: a large difference that is approximately equal to Canada's overall infant mortality rate.
Although child height has traditionally been interpreted as a measure of "malnutrition," it is increasingly recognized to reflect the totality of early-life health insults, including both net available nutrition and losses due to diseases. Our study does not allow us to observe disease directly; however, mechanisms in the literature are consistent with the association that we document. For example, exposure to particulate matter is associated with lower birth weight [6, 7], which is in turn linked to stature in childhood [48]. Similarly, exposure to ambient air pollution is associated with the incidence of pneumonia [49, 50]. Respiratory infections, like pneumonia, sometimes occur with fevers which can suppress the appetite, and reduce nutrient intake [51]. Moreover, infection and inflammation are metabolically demanding and may reallocate resources at the expense of growth [52].
This study has several limitations. One is the possibility of residual confounding. For example, we were not able to control for potential co-pollutants such as ozone or NO2, for which data is not available. In addition, some variables included in our models were measured at the time of the survey rather than at the time of the child's birth, such as open defecation and household solid fuel use. However, we have no reason to believe that these practices would have changed for a large proportion of households. Similarly, we assumed that surveyed mothers delivered their children in the same district in which they were surveyed. This assumption seems sound considering that 92% of children were recorded as being born while the mother was living in her current residence, and because migration across districts is relatively rare. Finally, we rely on district-level measures of exposure derived from satellite data, thus raising the possibility of measurement error. However, assuming this error is random, the consequence would be attenuation towards the null, meaning that the true size of the effect of PM2.5 on child height may be larger than we observe here. In light of these limitations, we encourage additional research on this topic. If possible, this would include other study designs (cohort studies, natural experiments etc.) and, when available, finer-resolution estimates of exposure.
To our knowledge, this is the first study to directly estimate the impact of early-life exposure to ambient PM2.5 on child height-for-age at the range of exposures found in India. Because average exposure to ambient PM2.5 is high in India, where child height-for-age is a critical challenge in human development, our results highlight ambient air pollution as public health policy priority. Ambient PM2.5 exposure is likely to increase in India in the near future [1, 53]. Therefore, the health burden that we quantify here could potentially increase unless appropriate policy action is taken to reduce air pollution throughout India. In particular, although policy conversations often focus on Delhi (and, to a lesser extent, other big cities), we find effects throughout India, and on both rural and urban children, suggesting that the policy challenges are broader than is commonly understood. Because child height has lasting consequences for human capital [10, 40], this is a problem with potential ramifications throughout the Indian society and economy.
The DHS 2015–2016 survey data for India are publicly available free of charge, and archived at https://www.dhsprogram.com/data/available-datasets.cfm. MISR AOD data used to generate the PM2.5 exposure are archived at https://misr.jpl.nasa.gov/getData/accessData/. The Stata do files used to arrive at the results depicted in this study are located at https://riceinstitute.org/wp-content/uploads/2019/07/hfa_PM25_public.txt.
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Darrouzet-Nardi AF, Masters WA. Urbanization, market development and malnutrition in farm households: evidence from the demographic and health surveys, 1986–2011. Food Secur. 2015;7:521–33. https://doi.org/10.1007/s12571-015-0470-9.
Coffey D, Spears DE. Where India goes : abandoned toilets, stunted development and the costs of caste. India: Harper Collins; 2017.
Geruso M, Spears D. Neighborhood sanitation and infant mortality. Am Econ J Appl Econ. 2018;10:125–62.
Gupta A, Vyas S, Hathi P, Khalid N, Srivastav N, Spears D, Coffey D. Persistence of solid fuel use despite increases in LPG ownership: New survey evidence from rural north India. Working paper. r.i.c.e. 2019. Available from: https://riceinstitute.org/research/persistence-of-solid-fuel-use-despite-increases-in-lpg-ownership-new-survey-evidence-from-rural-north-india/.
Barreca A, Deschenes O, Guldi M. Maybe next month? Temperature shocks and dynamic adjustments in birth rates. Demography. 2018;55:1269–93. https://doi.org/10.1007/s13524-018-0690-7.
Vyas S, Srivastav N, Spears D. An experiment with air purifiers in Delhi during Winter 2015-2016. PLoS One. 2016;11(12):e0167999.
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Crimmins EM, Finch CE. Infection, inflammation, height, and longevity. Proc Natl Acad Sci. 2006;103(2):498–503.
Chowdhury S, Dey S, Smith KR. Ambient PM2.5 exposure and expected premature mortality to 2100 in India under climate change scenarios. Nat Commun. 2018;9:318.
MISR aerosol data are archived in NASA Langley Research Atmospheric Science Data Center. All the authors acknowledge Dr. Aaron van Donkelaar, Dalhousie University for sharing the conversion factors from GEOS-Chem CTM which are used to estimate PM2.5. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health.
This study received no specific funding; however, some of the authors are supported by research center funding. SD acknowledges DST-FIST grant (SR/FST/ESII-016/2014) for upgradation of computing facility at IIT Delhi. DS and SV acknowledge support for research time and computing resources from Bill and Melinda Gates Foundation grant OPP1125318. Research by DS and SV was supported by grant, P2CHD042849, Population Research Center, awarded to the Population Research Center at The University of Texas at Austin by the Eunice Kennedy Shriver National Institute of Child Health and Human Development. SV acknowledges that this material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1610403.
Department of Economics and Population Research Center, University of Texas at Austin 2225 Speedway, Austin, TX, 78712, USA
Dean Spears
& Sangita Vyas
Economics and Planning Unit, Indian Statistical Institute, Delhi Centre, Delhi, 110016, India
Centre for Atmospheric Sciences, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India
Sagnik Dey
& Sourangsu Chowdhury
Centre of Excellence for Research on Clean Air, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India
Rollins School of Public Health, Emory University, Atlanta, USA
Noah Scovronick
Department of Civil, Architectural and Environmental Engineering, University of Texas at Austin, Austin, USA
Joshua Apte
Search for Dean Spears in:
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DS and SD developed the idea, DS carried out the main analysis. SC carried out the satellite data analysis to generate the exposure data. DS and SD wrote the paper with inputs from all the authors. NS and SV revised the article, with inputs from all authors. All authors have read and approved the final version.
Correspondence to Dean Spears.
Because we study publicly-available, anonymized data on child height, our study was classified as "not human subjects research" by the RICE Institute, Inc. IRB (registered with the NIH as IRB00010425) and informed consent is not required.
Table S1. Summary statistics describing sample of children with measured height from India's 2015–16 DHS, computed without sampling weights. Table S2. Association of district-level PM2.5 (per 100 μg/m3) in month of birth with child height-for-age z-score with all covariates. Figure S1. Projected effects of PM2.5 on child height-for-age z-score, at increasing non-linearity. Each curve is the projected effect from a separate fixed effects regression where PM2.5 in the month of birth is specified as a polynomial of degree 1 through 5. p-values report joint F tests that all PM2.5 terms are zero. Figure S2. Box-Cox transformation of PM2.5 in month of birth: Each point plots the log likelihood of a separate fixed effects regression of PM2.5 transformed according to the coefficient on the horizontal axis. (DOCX 83 kb)
Spears, D., Dey, S., Chowdhury, S. et al. The association of early-life exposure to ambient PM2.5 and later-childhood height-for-age in India: an observational study. Environ Health 18, 62 (2019) doi:10.1186/s12940-019-0501-7
Child height
Ambient air pollution | CommonCrawl |
Abstract: In this paper we show that the sets of $F$-jumping coefficients of ideals form discrete sets in certain graded $F$-finite rings. We do so by giving a criterion based on linear bounds for the growth of the Castelnuovo-Mumford regularity of certain ideals. We further show that these linear bounds exists for one-dimensional rings and for ideals of (most) two-dimensional domains. We conclude by applying our technique to prove that all sets of $F$-jumping coefficients of all ideals in the determinantal ring given as the quotient by $2\times 2$ minors in a $2\times 3$ matrix of indeterminates form discrete sets. | CommonCrawl |
# The basics of matrix multiplication
Before we dive into the transfer matrix method, let's review the basics of matrix multiplication. Matrix multiplication is a fundamental operation in linear algebra and is essential for understanding the transfer matrix method.
In matrix multiplication, we combine two matrices to produce a new matrix. The resulting matrix is determined by multiplying the corresponding elements of the rows of the first matrix with the corresponding elements of the columns of the second matrix and summing the products.
To illustrate this, let's consider two matrices, A and B:
$$
A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, \quad B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}
$$
The product of A and B, denoted as AB, is calculated as follows:
$$
AB = \begin{bmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{bmatrix}
$$
Matrix multiplication is not commutative, which means that the order of multiplication matters. In general, AB is not equal to BA.
Now that we have reviewed the basics of matrix multiplication, we can move on to applying it to optical systems.
Let's consider an optical system that consists of two lenses. The transfer matrix for the first lens is given by:
$$
M_1 = \begin{bmatrix} A_1 & B_1 \\ C_1 & D_1 \end{bmatrix}
$$
And the transfer matrix for the second lens is given by:
$$
M_2 = \begin{bmatrix} A_2 & B_2 \\ C_2 & D_2 \end{bmatrix}
$$
To calculate the overall transfer matrix for the optical system, we need to multiply these two matrices:
$$
M = M_2 \cdot M_1
$$
The resulting matrix M represents the combined effect of the two lenses on a light ray passing through the system.
## Exercise
Given the following matrices:
$$
A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}
$$
Calculate the product AB.
### Solution
$$
AB = \begin{bmatrix} 31 & 36 \\ 43 & 50 \end{bmatrix}
$$
# Applying matrix multiplication to optical systems
Now that we understand the basics of matrix multiplication, let's apply this concept to optical systems. In the context of optics, matrices can be used to represent the transfer of light rays through various optical components.
The transfer matrix method allows us to analyze and calculate the behavior of light rays as they pass through optical systems. By representing each optical component with a transfer matrix, we can combine these matrices to determine the overall effect of the system on the light rays.
To apply matrix multiplication to optical systems, we need to define the transfer matrices for each optical component. These matrices capture the optical properties of the component, such as reflection, refraction, and transmission.
By multiplying the transfer matrices of the individual components in the order that the light rays encounter them, we can calculate the overall transfer matrix for the optical system. This matrix represents the combined effect of all the optical components on the light rays.
Using the transfer matrix method, we can analyze and predict the behavior of light rays in complex optical systems. This method is widely used in optical engineering for designing and optimizing optical systems.
Let's consider an example of an optical system that consists of a lens and a mirror. The transfer matrix for the lens, denoted as M1, represents the refraction of light rays passing through the lens. The transfer matrix for the mirror, denoted as M2, represents the reflection of light rays off the mirror.
To calculate the overall transfer matrix for this optical system, we multiply the transfer matrices of the lens and the mirror:
$$
M = M2 \cdot M1
$$
The resulting matrix M represents the combined effect of the lens and the mirror on the light rays.
## Exercise
Consider an optical system that consists of a lens and a prism. The transfer matrix for the lens is given by:
$$
M1 = \begin{bmatrix} A1 & B1 \\ C1 & D1 \end{bmatrix}
$$
And the transfer matrix for the prism is given by:
$$
M2 = \begin{bmatrix} A2 & B2 \\ C2 & D2 \end{bmatrix}
$$
Calculate the overall transfer matrix for this optical system.
### Solution
To calculate the overall transfer matrix, we multiply the transfer matrices of the lens and the prism:
$$
M = M2 \cdot M1
$$
# Understanding reflection and refraction in optical systems
Reflection and refraction are fundamental concepts in optics that play a crucial role in the behavior of light rays as they pass through optical systems.
Reflection occurs when light rays encounter a surface and bounce back. The angle of incidence, which is the angle between the incident ray and the normal to the surface, is equal to the angle of reflection, which is the angle between the reflected ray and the normal. This can be described using the law of reflection:
$$
\text{angle of incidence} = \text{angle of reflection}
$$
Refraction, on the other hand, occurs when light rays pass from one medium to another with a different refractive index. The refractive index is a measure of how much the speed of light is reduced in a particular medium. When light rays pass through a boundary between two media with different refractive indices, they change direction. This can be described using Snell's law:
$$
\frac{\sin(\text{angle of incidence})}{\sin(\text{angle of refraction})} = \frac{\text{refractive index of medium 2}}{\text{refractive index of medium 1}}
$$
Understanding reflection and refraction is essential for analyzing and calculating the behavior of light rays in optical systems. By applying the laws of reflection and refraction, we can determine how light rays will interact with various optical components, such as lenses, mirrors, and prisms.
Let's consider an example of an optical system that consists of a mirror and a lens. When a light ray approaches the mirror, it undergoes reflection according to the law of reflection. The angle of incidence is equal to the angle of reflection.
After reflecting off the mirror, the light ray passes through the lens. As it enters the lens, it undergoes refraction according to Snell's law. The angle of incidence and the refractive indices of the two media determine the angle of refraction.
By understanding the principles of reflection and refraction, we can determine the path of the light ray through the optical system and calculate its final position and direction.
## Exercise
Consider an optical system that consists of a mirror and a prism. The angle of incidence for the light ray approaching the mirror is 30 degrees, and the angle of refraction for the light ray passing through the prism is 45 degrees. The refractive index of the medium before the prism is 1.5, and the refractive index of the prism is 1.7. Calculate the angle of reflection for the light ray reflecting off the mirror.
### Solution
According to the law of reflection, the angle of incidence is equal to the angle of reflection. Therefore, the angle of reflection for the light ray reflecting off the mirror is 30 degrees.
# Calculating transfer matrices for simple optical components
A transfer matrix is a mathematical representation of an optical component that describes how it affects the propagation of light rays. It takes into account the optical properties of the component, such as its refractive index, thickness, and curvature.
To calculate a transfer matrix for a simple optical component, we need to consider its effect on the position and direction of a light ray. This can be done by breaking down the component into individual transformations, such as translation, rotation, reflection, and refraction.
For example, let's consider a thin lens. A thin lens can be represented by a transfer matrix that describes its effect on the position and direction of a light ray. The transfer matrix for a thin lens is given by:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f} & 1 \\
\end{pmatrix}
$$
where $f$ is the focal length of the lens. This transfer matrix takes into account the refraction of the light ray as it passes through the lens.
By calculating the transfer matrices for different optical components in an optical system, we can combine them to obtain the overall transfer matrix for the system. This allows us to analyze and predict the behavior of light rays as they pass through the system.
Let's consider an example of an optical system that consists of a mirror and a lens. We can calculate the transfer matrices for the mirror and the lens separately, and then combine them to obtain the overall transfer matrix for the system.
The transfer matrix for the mirror, which represents reflection, is given by:
$$
\begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}
$$
The transfer matrix for the lens, as mentioned earlier, is given by:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f} & 1 \\
\end{pmatrix}
$$
To obtain the overall transfer matrix for the system, we can multiply the transfer matrices for the mirror and the lens:
$$
\begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f} & -1 \\
\end{pmatrix}
$$
By calculating the transfer matrix for the overall system, we can analyze and predict the behavior of light rays as they pass through the mirror and the lens.
## Exercise
Consider an optical system that consists of a lens with a focal length of 20 cm and a mirror. Calculate the transfer matrix for the overall system.
### Solution
The transfer matrix for the lens is given by:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-\frac{1}{0.2} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-5 & 1 \\
\end{pmatrix}
$$
The transfer matrix for the mirror is given by:
$$
\begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}
$$
To obtain the overall transfer matrix for the system, we can multiply the transfer matrices for the lens and the mirror:
$$
\begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
-5 & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-5 & -1 \\
\end{pmatrix}
$$
Therefore, the transfer matrix for the overall system is:
$$
\begin{pmatrix}
1 & 0 \\
-5 & -1 \\
\end{pmatrix}
$$
# Using transfer matrices to analyze multi-component optical systems
Transfer matrices can be used to analyze multi-component optical systems, which consist of multiple optical components arranged in a specific order. By calculating the transfer matrices for each component and combining them, we can obtain the overall transfer matrix for the system.
To analyze a multi-component optical system using transfer matrices, we follow these steps:
1. Calculate the transfer matrix for each individual component in the system. This involves considering the effect of the component on the position and direction of a light ray.
2. Combine the transfer matrices for the individual components in the order in which they are arranged in the system. This is done by multiplying the transfer matrices.
3. Obtain the overall transfer matrix for the system by multiplying the transfer matrices for all the components.
By obtaining the overall transfer matrix for a multi-component optical system, we can analyze and predict the behavior of light rays as they pass through the system. This allows us to determine properties such as the position, direction, and intensity of the light rays at different points in the system.
Let's consider an example of a multi-component optical system that consists of a lens, a mirror, and another lens. We can calculate the transfer matrices for each component and combine them to obtain the overall transfer matrix for the system.
The transfer matrix for the first lens is given by:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f_1} & 1 \\
\end{pmatrix}
$$
The transfer matrix for the mirror is given by:
$$
\begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}
$$
The transfer matrix for the second lens is given by:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f_2} & 1 \\
\end{pmatrix}
$$
To obtain the overall transfer matrix for the system, we can multiply the transfer matrices for the lens, mirror, and second lens:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f_1} & 1 \\
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f_2} & 1 \\
\end{pmatrix}
$$
By calculating this product, we can obtain the overall transfer matrix for the multi-component optical system.
## Exercise
Consider a multi-component optical system that consists of a lens with a focal length of 30 cm, a mirror, and another lens with a focal length of 40 cm. Calculate the overall transfer matrix for the system.
### Solution
The transfer matrix for the first lens is given by:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f_1} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-\frac{1}{0.3} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-3.33 & 1 \\
\end{pmatrix}
$$
The transfer matrix for the mirror is given by:
$$
\begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}
$$
The transfer matrix for the second lens is given by:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f_2} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-\frac{1}{0.4} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-2.5 & 1 \\
\end{pmatrix}
$$
To obtain the overall transfer matrix for the system, we can multiply the transfer matrices for the lens, mirror, and second lens:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{0.3} & 1 \\
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
0 & -1 \\
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
-\frac{1}{0.4} & 1 \\
\end{pmatrix}
$$
By calculating this product, we can obtain the overall transfer matrix for the multi-component optical system.
# Applying the transfer matrix method to lens systems
The transfer matrix method can be applied to lens systems to analyze and predict the behavior of light rays as they pass through the system. Lens systems are commonly used in optical devices such as cameras, microscopes, and telescopes.
To apply the transfer matrix method to a lens system, we follow these steps:
1. Identify the individual lenses in the system and determine their focal lengths. The focal length is a measure of the lens's ability to converge or diverge light.
2. Calculate the transfer matrix for each lens using the formula:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f} & 1 \\
\end{pmatrix}
$$
where $f$ is the focal length of the lens.
3. Combine the transfer matrices for the lenses in the order in which they are arranged in the system. This is done by multiplying the transfer matrices.
4. Obtain the overall transfer matrix for the lens system by multiplying the transfer matrices for all the lenses.
By obtaining the overall transfer matrix for a lens system, we can analyze and predict properties such as the position, direction, and intensity of light rays as they pass through the system. This allows us to understand how the lens system forms images and how it affects the behavior of light.
Let's consider an example of a lens system that consists of two lenses with focal lengths of 20 cm and 30 cm, respectively. We can calculate the transfer matrices for each lens and combine them to obtain the overall transfer matrix for the lens system.
The transfer matrix for the first lens is given by:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f_1} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-\frac{1}{0.2} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-5 & 1 \\
\end{pmatrix}
$$
The transfer matrix for the second lens is given by:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f_2} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-\frac{1}{0.3} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-3.33 & 1 \\
\end{pmatrix}
$$
To obtain the overall transfer matrix for the lens system, we can multiply the transfer matrices for the lenses:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{0.2} & 1 \\
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
-\frac{1}{0.3} & 1 \\
\end{pmatrix}
$$
By calculating this product, we can obtain the overall transfer matrix for the lens system.
## Exercise
Consider a lens system that consists of two lenses with focal lengths of 40 cm and 50 cm, respectively. Calculate the overall transfer matrix for the lens system.
### Solution
The transfer matrix for the first lens is given by:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f_1} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-\frac{1}{0.4} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-2.5 & 1 \\
\end{pmatrix}
$$
The transfer matrix for the second lens is given by:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{f_2} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-\frac{1}{0.5} & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
1 & 0 \\
-2 & 1 \\
\end{pmatrix}
$$
To obtain the overall transfer matrix for the lens system, we can multiply the transfer matrices for the lenses:
$$
\begin{pmatrix}
1 & 0 \\
-\frac{1}{0.4} & 1 \\
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
-\frac{1}{0.5} & 1 \\
\end{pmatrix}
$$
By calculating this product, we can obtain the overall transfer matrix for the lens system.
# Evaluating the performance of optical systems using transfer matrices
Once we have obtained the overall transfer matrix for an optical system, we can use it to evaluate the performance of the system. The transfer matrix provides valuable information about the behavior of light rays as they pass through the system.
One important property that can be determined from the transfer matrix is the magnification of the system. The magnification is a measure of how much the size of an object is changed when it is viewed through the system. It can be calculated using the formula:
$$
\text{Magnification} = \frac{y'}{y}
$$
where $y'$ is the height of the image formed by the system and $y$ is the height of the object.
Another property that can be determined from the transfer matrix is the focal length of the system. The focal length is a measure of the system's ability to converge or diverge light. It can be calculated using the formula:
$$
\text{Focal length} = -\frac{1}{M_{22}}
$$
where $M_{22}$ is the element in the second row and second column of the transfer matrix.
The transfer matrix also allows us to determine the position and direction of light rays as they pass through the system. By multiplying the transfer matrix by a vector representing the position and direction of a light ray, we can obtain the new position and direction of the ray after it has passed through the system.
By evaluating these properties and analyzing the behavior of light rays, we can gain valuable insights into the performance of optical systems and make informed decisions about their design and optimization.
Let's consider an example of an optical system with an overall transfer matrix:
$$
\begin{pmatrix}
2 & 0 \\
-1 & 1 \\
\end{pmatrix}
$$
We can use this transfer matrix to evaluate the performance of the system.
The magnification of the system can be calculated as:
$$
\text{Magnification} = \frac{y'}{y} = \frac{M_{21}}{M_{11}} = \frac{-1}{2} = -0.5
$$
The negative sign indicates that the image formed by the system is inverted compared to the object.
The focal length of the system can be calculated as:
$$
\text{Focal length} = -\frac{1}{M_{22}} = -\frac{1}{1} = -1
$$
This indicates that the system has a focal length of -1, which means it is a diverging system.
By multiplying the transfer matrix by a vector representing the position and direction of a light ray, we can determine the new position and direction of the ray after it has passed through the system.
## Exercise
Consider an optical system with an overall transfer matrix:
$$
\begin{pmatrix}
3 & 0 \\
-2 & 1 \\
\end{pmatrix}
$$
Use this transfer matrix to evaluate the performance of the system. Calculate the magnification and focal length of the system.
### Solution
The magnification of the system can be calculated as:
$$
\text{Magnification} = \frac{y'}{y} = \frac{M_{21}}{M_{11}} = \frac{-2}{3} = -0.67
$$
The negative sign indicates that the image formed by the system is inverted compared to the object.
The focal length of the system can be calculated as:
$$
\text{Focal length} = -\frac{1}{M_{22}} = -\frac{1}{1} = -1
$$
This indicates that the system has a focal length of -1, which means it is a diverging system.
# Accounting for loss and dispersion in optical systems
In real-world optical systems, there are often factors that can cause loss of light and dispersion of light waves. Loss refers to the reduction in the intensity of light as it passes through the system, while dispersion refers to the spreading out of light waves due to variations in the refractive index of the medium.
Loss can occur due to factors such as absorption, scattering, and reflection. Absorption refers to the conversion of light energy into other forms of energy, such as heat. Scattering refers to the redirection of light in different directions due to interactions with particles or irregularities in the medium. Reflection occurs when light waves bounce off a surface instead of passing through it.
Dispersion can occur due to the wavelength-dependent refractive index of the medium. This means that different wavelengths of light will experience different amounts of refraction, causing the light waves to spread out.
To account for loss and dispersion in optical systems, we can modify the transfer matrix method by introducing additional elements in the transfer matrix. These elements represent the loss and dispersion effects in the system.
Let's consider an example of an optical system that experiences loss and dispersion. The overall transfer matrix of the system is given by:
$$
\begin{pmatrix}
1 & 0 \\
-0.5 & 1 \\
\end{pmatrix}
$$
To account for loss, we introduce a loss factor in the transfer matrix. Let's say the loss factor is 0.8. We can modify the transfer matrix as follows:
$$
\begin{pmatrix}
0.8 & 0 \\
-0.4 & 1 \\
\end{pmatrix}
$$
To account for dispersion, we introduce a dispersion factor in the transfer matrix. Let's say the dispersion factor is 1.2. We can modify the transfer matrix as follows:
$$
\begin{pmatrix}
1.2 & 0 \\
-0.6 & 1 \\
\end{pmatrix}
$$
By including these additional elements in the transfer matrix, we can accurately model the loss and dispersion effects in the optical system.
## Exercise
Consider an optical system with an overall transfer matrix:
$$
\begin{pmatrix}
0.9 & 0 \\
-0.3 & 1 \\
\end{pmatrix}
$$
To account for loss, introduce a loss factor of 0.7 in the transfer matrix. Modify the transfer matrix to include the loss factor.
### Solution
To account for loss, we introduce a loss factor of 0.7 in the transfer matrix. The modified transfer matrix is:
$$
\begin{pmatrix}
0.7 & 0 \\
-0.21 & 1 \\
\end{pmatrix}
$$
# Designing and optimizing optical systems using the transfer matrix method
The transfer matrix method is a powerful tool for designing and optimizing optical systems. By manipulating the transfer matrices of individual optical components, we can analyze the overall performance of the system and make informed design choices.
To design an optical system using the transfer matrix method, we follow these steps:
1. Identify the desired properties of the system, such as the focal length, beam size, or image quality.
2. Determine the transfer matrix of each individual optical component in the system. This can be done using the principles of geometric optics and the properties of the component, such as its focal length or refractive index.
3. Multiply the transfer matrices of the individual components together to obtain the overall transfer matrix of the system. This represents the transformation of the input beam to the output beam.
4. Use the overall transfer matrix to analyze the performance of the system. This can include properties such as the beam size, focal length, or image quality.
5. Make adjustments to the system design, such as changing the parameters of individual components or adding additional components, to optimize the desired properties.
By iteratively going through these steps, we can refine the design of the optical system and achieve the desired performance.
Let's consider an example of designing an optical system using the transfer matrix method. We want to design a system that focuses an incoming beam to a specific spot size at a certain distance.
First, we determine the transfer matrix of each component in the system. Let's say we have a lens with a focal length of 50 mm and a distance of 100 mm between the lens and the focal point. The transfer matrix of the lens can be calculated using the appropriate formulas.
Next, we multiply the transfer matrices of the individual components together to obtain the overall transfer matrix of the system. This represents the transformation of the input beam to the output beam.
Using the overall transfer matrix, we can analyze the performance of the system. We can calculate the spot size at the focal point and determine if it meets our desired specifications.
If the performance is not satisfactory, we can make adjustments to the system design. For example, we can change the focal length of the lens or add additional components to achieve the desired spot size.
By following this design process and using the transfer matrix method, we can effectively design and optimize optical systems.
## Exercise
Consider an optical system with a lens of focal length 75 mm and a distance of 150 mm between the lens and the focal point. Calculate the overall transfer matrix of the system.
### Solution
To calculate the overall transfer matrix of the system, we need to multiply the transfer matrices of the individual components together.
The transfer matrix of the lens can be calculated using the appropriate formulas for a thin lens:
$$
\begin{pmatrix}
1 & 0 \\
-1/f & 1 \\
\end{pmatrix}
$$
The transfer matrix for the distance between the lens and the focal point is:
$$
\begin{pmatrix}
1 & d \\
0 & 1 \\
\end{pmatrix}
$$
Multiplying these matrices together, we get the overall transfer matrix of the system:
$$
\begin{pmatrix}
1 & d \\
-1/f & 1-(d/f) \\
\end{pmatrix}
$$
where f is the focal length of the lens and d is the distance between the lens and the focal point.
# Real-world applications of the transfer matrix method in optical engineering
The transfer matrix method has numerous real-world applications in optical engineering. It is used to design and optimize a wide range of optical systems, including lenses, mirrors, and beam splitters. By using the transfer matrix method, engineers can accurately predict the performance of these systems and make informed design choices.
One application of the transfer matrix method is in the design of optical filters. Optical filters are used to selectively transmit or reflect certain wavelengths of light while blocking others. By analyzing the transfer matrix of the filter, engineers can determine its spectral response and optimize its performance for specific applications, such as in photography or telecommunications.
Another application is in the design of laser systems. Laser systems require precise control of the beam quality, divergence, and polarization. By using the transfer matrix method, engineers can analyze the performance of various optical components, such as lenses and mirrors, and optimize the system to achieve the desired laser characteristics.
The transfer matrix method is also used in the design of optical coatings. Optical coatings are thin films applied to optical components to enhance their performance, such as by reducing reflection or increasing transmission. By analyzing the transfer matrix of the coating, engineers can optimize its thickness and refractive index to achieve the desired optical properties.
Overall, the transfer matrix method is a valuable tool in optical engineering, allowing engineers to design and optimize a wide range of optical systems for various applications. Its practical and applied approach makes it a powerful technique in the field.
# Future developments and advancements in the transfer matrix method
The transfer matrix method has been widely used in optical engineering for many years, but there are still ongoing developments and advancements in this field. Researchers and engineers are constantly working to improve the accuracy and efficiency of the method, as well as to expand its applications to new areas.
One area of future development is the incorporation of more complex optical phenomena into the transfer matrix method. Currently, the method primarily focuses on simple optical systems with idealized components. However, real-world optical systems often involve more complex phenomena, such as scattering, diffraction, and nonlinear effects. Researchers are working on extending the transfer matrix method to account for these phenomena, allowing for more accurate modeling and analysis of real-world systems.
Another area of advancement is the integration of the transfer matrix method with other computational techniques and software tools. This allows for more efficient and automated analysis of optical systems. For example, researchers are developing algorithms that can automatically generate the transfer matrix of a given optical system based on its component properties. This reduces the manual effort required to calculate the transfer matrix and makes the method more accessible to a wider range of users.
In addition, there is ongoing research on the development of new materials and components for optical systems. These advancements can have a significant impact on the design and optimization of optical systems. The transfer matrix method provides a valuable tool for analyzing and predicting the performance of these new materials and components, allowing engineers to make informed design choices.
Overall, the transfer matrix method is a powerful and versatile tool in optical engineering, and ongoing developments and advancements in this field will continue to enhance its capabilities. By incorporating more complex phenomena, integrating with other computational techniques, and keeping up with advancements in materials and components, the transfer matrix method will remain a valuable resource for optical engineers in the future. | Textbooks |
彭瀚葵 | Hankui Peng
Hankui Peng
Aviva Research Fellow at the University of Cambridge.
Boosting I: AdaBoost
Adaptive Boosting (AdaBoost) is one of the most commonly used Machine Learning methods for both classification and regression problems. It is a kind of ensemble learning meta-algorithm that can be applied on top of many other methods to improve performance, such as Bagging, Random Forests, etc. The idea is to combine a set of sequentially developed weak learners (rule of thumb) and come up with a single best final classifier at the end. Let me set the scene in a binary classification setting.
Say we have a training set with $latex m $ observations. Each observation is composed of a feature vector $latex \left \{x_{1},...,x_{p} \right \} $, and a class label $latex y \in \left \{ -1,+1 \right \} $. We initialize the algorithm by assigning every observation equal weights, then every observation $latex i $ starts with weight $latex w^{(0)}(i)=\frac{1}{m} $. What follows is mainly playing around with the weights through shifting more weights to those cases that have been misclassified in a previous iteration. Think of it as a message sent from the weak learner in iteration $latex k $ to the weaker learner in the iteration$latex k+1 $ saying "there's something wrong with the way I classify these (misclassified) cases, plz pay special attention to them in your treatment."
As you may have realized from what I described above, AdaBoost addresses the question of how to improve the classification process in every iteration, but it does not really interfere with the way those classifiers classify. This explains why AdaBoost is flexible enough to combine with other algorithms. Also, instead of trying out one weak learner for every iteration, we might as well develop different classifiers throughout the process to improve classification performance.
So just to give you a more concrete feeling, here comes the algorithm:
We have $latex m $ observations here, whose labels we already know. For every iteration, we want to find a mapping (weak learner) $latex h_{t}$ that maps every observation to a label, and that can minimize the error rate across all observations. As the procedure goes on, misclassification of certain observations may have a bigger influence on the overall error rate, as we put more weights on those previously misclassified observations. Generally, as long as the weak learner performs slightly better than random guess, the process can go on without a problem; if it's not even as good as a random guess, then we might as well stop the procedure.
Let's assume our weak learner performs better than a random guess, we move happily toward the next step. Based on how well-performed a learner is in it's iteration, we also attach a reasonable weight $latex \alpha_{t} $ (as specified above in the algorithm) to the specific weak learner $latex h_{t} $ accordingly. This will be of use at the end when we aggregate the weak learners in all iterations together to form our final classifier. Other than that, it also plays an important role in updating the weights for specific observations as specified above, where $latex Z_{t} $ is a normalizing constant to make sure that $latex D_{t+1}(i) $ for all $latex i $ sums up to 1 in the $latex (t+1) $th iteration as well.
At the end of all iterations, the only work left is to form the final classifier by combining the outputs of all weak learners together. It actually takes the form of a weighted majority vote. Intuitively, you can think of every weak learner vote on either label 1 or 2. Since not all weak learners have equal say in their votes, the final output is actually given by the competition between two weighted aggregation for two labels.
Till now, I have pretty much delineated the whole picture of AdaBoost I have in mind. Surely, this is not the only way to interpret AdaBoost. Training error proof shows that AdaBoost actually minimizes:
$latex \prod _{t}Z_{t}=\frac{1}{m}\sum_{i=1}^{m}exp\left \{ -y_{i}\sum _{t}\alpha_{t}h_{t}(x_{i}) \right \} $
which leads to an equally valid perspective of viewing AdaBoost as an optimization procedure whose objective is to minimize a loss function as given above. Go further down this road will take you to the land of "Gradient Boosting", a la la land I will hopefully tap into in my future posts.
[1] http://cmp.felk.cvut.cz/~sochmj1/adaboost_talk.pdf
The Most Beautiful Equation
"Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence."
-- Keith Devlin
Mathematics and Finance Hand in Hand
This week's blogging cause is an effort of putting together my recent random thoughts on the relationship between maths and the financial market. In the recent two decades or so, there has been a coup d'etat in the financial trading world, where human decisions have been largely replaced by sophisticated computer systems. Read more
A Peep into Kalman Filter
Deep insecurity about my non-mathy background urged me to look into things that pump into my ears all the time. Today I decided to do some quick reading on Kalman filter, and now it's time to write down my understanding. Bear with me, those "mathemagicians". Read more
"Slow and steady wins the race."
© 2021 Hankui Peng. Powered by Jekyll & AcademicPages, a fork of Minimal Mistakes. | CommonCrawl |
Applied Physics B
August 2019 , 125:142 | Cite as
Robust offset locking of laser frequency with electronically tunable LC circuits for sub-millihertz uncertainty
Yuhei Seishu
Taro Hasegawa
A simple scheme for laser frequency offset locking introduced in Cheng et al. (Opt Express 25:2752, 2017) has been improved by the use of a variable-capacitance diode. By employing this scheme with the optical phase lock loop, robust and precise laser frequency stabilization is achieved, and long-term laser frequency drift is expected to be substantially suppressed even in noisy circumstances. The optical frequency uncertainty becomes less than 10 \({\upmu }\)Hz for 1000 s averaging.
Laser frequency stabilization is one of the essential technique in atomic, molecular, and optical physics, quantum optics, optical communications, and fundamental physics [1, 2]. Optical offset locking is the technique to stabilize optical frequency of a single-mode laser (slave laser) with respect to another well-stabilized laser (master laser) by means of beat-note frequency stabilization. Especially, after the invention of optical frequency combs (OFC), the optical offset locking technique is indispensable to these applications to obtain highly frequency-stabilized coherent light sources of the wavelength range from ultraviolet to mid-infrared [3]. By optical phase lock loop (OPLL) [4], which is one of the offset locking schemes, the slave laser frequency can be stabilized with very small uncertainty (less than millihertz). The OPLL is also employed to stabilize carrier-envelope offset frequency of OFCs [5]. With the OPLL, however, large optical frequency noise cannot be suppressed. If the phase difference between the master laser and the slave laser exceeds \(\pi\) within response time of the feedback loop, the OPLL does not work any more. Hence, the OPLL cannot be employed in noisy circumstances such as rooms with substantial air flow and acoustic noises, and the outside of laboratories [6]. Because of this reason, another stabilization scheme in addition to the OPLL is necessary to suppress frequency jumps (jitters). For this purpose, digital processing technique [7], optical frequency locking to resonance frequency of a high-finesse cavity [8], offset locking with an electric delay line [9, 10], and offset locking with a low-pass filter [11], and offset locking with an LC resonance circuit (LC locking) [12] are utilized. The LC locking is a scheme in which the beat-note frequency between the master and the slave lasers is stabilized to the LC resonant frequency. This scheme is simple compared to that using a high-finesse cavity, and it is easily used with the OPLL. When the LC locking and the OPLL are employed simultaneously, robust and precise offset locking is realized. Robustness of the frequency locking is supported by the LC locking, whereas small uncertainty is supported by the OPLL, as in [12]. In addition, the offset frequency in the LC locking is manually tunable if the capacitance is variable (as in Fig. 1 in [12]), whereas tuning range of the electric delay line is very small. Therefore, the LC locking is very useful, especially in noisy circumstances, with no digitizing noise, which may appear in the digital processing technique [7].
When the LC locking is employed with the OPLL, the offset frequency of the OPLL [frequency of the local oscillator (LO)] is supposed to be identical to the locking offset frequency of the LC locking (the LC resonance frequency). The mismatch between these two frequencies causes the phase shift of the slave laser with respect to the master laser. As a result, the LC resonance frequency fluctuation causes the phase noise, and long-term stability of the laser frequency deteriorates.
To overcome this problem, we improve the LC locking introduced in [12] by making the resonance frequency tunable electronically. Specifically, we replace the manually variable capacitance in Fig. 1 in [12] with a variable capacitance diode (varactor). With an electronically tunable LC circuit, electronic tuning of the beat-note frequency can be carried out. The LC resonance frequency can be electronically stabilized to the LO frequency in the OPLL, and with this stabilization, long-term fluctuation of the slave laser frequency is suppressed. As a result, the electronically tunable LC locking can improve long-term laser frequency stability with maintaining the robustness.
In our experiment, an external-cavity laser diode (ECLD) at 1645 nm is the slave laser, and an OFC of erbium-doped fiber mode-locked laser is the master laser. We demonstrate the LC locking in addition to the OPLL, and the uncertainty of the beat-note frequency is the order of 10 \({\upmu }\)Hz for 1000 s of measurement averaging time. The behavior of the overlapping Allan deviation indicates that the long-term drift of the beat-note frequency is substantially suppressed even in noisy circumstances in our laboratory. From this result, it is found that the deterioration of OPLL performance in exchange for the robustness of LC locking can be avoided by the use of the electronically tunable LC locking.
2 Principles and experimental setup
2.1 Tunable LC circuits
As in [12], the beat-note frequency of two lasers is to be stabilized at the resonance frequency of an LC circuit. In our setup, we use a varactor to tune the LC resonance frequency. The circuit is shown in Fig. 1. The value of the inductance is 1 \({\upmu }\)H, C1 is 10 nF, C2 is 1 nF, and R is 1 k\({\Omega }\). For the varactor (VC), we use 1SV228 (Toshiba). The capacitance of the varactor is about 45 pF when 0 V is applied as a control voltage (\(V_{\text {Ctrl}}\) in Fig. 1), and about 10 pF for 9 V. The condenser C1 is necessary to apply a dc voltage to the varactor. Because the capacitance of C1 is much larger than that of the varactor, the resonance frequency of the LC circuit \(\omega _0/2\pi\) is determined by the capacitance of the varactor. The resonance frequency is expected to be tunable between 24 and 50 MHz. Practically, however, because of stray capacitance and inductance, the resonance frequency turns out to be tunable between 16 and 40 MHz as shown later.
The LC resonator with electronically tunable resonance frequency. L is the inductance, C1 and C2 are the condensers, R is the resistance, and VC is the varactor. The terminal \(V_{\text {Ctrl}}\) is for capacitance control of VC, and rf power is injected and reflected back through the rf terminal
The rf power (\(P_{\text {in}}\)) is injected through the rf terminal (rf in Fig. 1), and the reflected rf power (\(P_{\text {ref}}\)) is utilized to obtain the error signal for the laser frequency offset locking. Figure 2 shows the rf-frequency dependence of (a) the rf reflection amplitude and (b) the out-of-phase component (the component of quadrature phase with respect to the input rf signal) for some specific values of \(V_{\text {Ctrl}}\). The stray resistance in the circuit determines the linewidth of the resonance curve. When the control voltage is \(\sim 3.00\) V, the reflection amplitude is almost zero level at the resonance (\(\sim\) 20 MHz), so that the impedance matching condition is fulfilled. The out-of-phase component is used as an error signal for the laser frequency offset locking, and the beat-note frequency is to be stabilized at zero-cross frequency in Fig. 2b (arrows in the figure). Because of the background profile, the zero-cross frequency appears only when the control voltage is less than 5 V (between 16 and 25 MHz). In the case of impedance matching, the reflection power consists of only the out-of-phase error signal in the vicinity of the resonance frequency, and irrelevant on-phase component, which may not contribute to the signal but cause noise, is negligibly small. With the present circuit, it is practical to tune the resonance frequency to \(\sim\) 20 MHz for the LC locking. In Fig. 2b, there is another zero-cross frequency (e.g., \(\sim\) 28 MHz for \(V_{\text {Ctrl}}=3.0\) V), and locking to this frequency can be avoided by choosing the feedback polarity properly.
One of the important parameters of the tunable LC circuits is tuning range of the resonance frequency, because it determines the tuning range of the slave laser frequency stabilized by the LC locking. The tuning range (\(\varDelta f_{\text {LC}}\)) is determined by the variable range of the capacitance, and it is expressed as,
$$\begin{aligned} 2\pi \varDelta f_{\text {LC}}=\frac{1}{\sqrt{\text {LC}_{\min }}}-\frac{1}{\sqrt{\text {LC}_{\max }}}, \end{aligned}$$
where L is the inductance, and \(C_{\min ,\max }\) are the smallest and the largest capacitance of the varactor, respectively. Equation (1) implies that smaller value of L (or higher resonance frequency) is preferred for wide tuning range. In our case, the beat-note frequency is \(\sim\) 20 MHz, which is limited by the repetition rate (or mode spacing in frequency domain) of the OFC (66.87 MHz).
Dependence of a the reflection amplitude and b out-of-phase component on the input rf frequency. The control voltage is shown in the figure. The arrows in b indicate the zero-cross points, at which the beat-note frequency is to be stabilized
Another important parameter is capture range, within which sudden laser frequency error can be compensated by the feedback loop. For \(V_{\text {Ctrl}}=3\) V, the capture range is roughly estimated as \(\sim\) 10 MHz (frequency range between the maximum and the minimum of the error signal). This capture range is much larger than that of the OPLL (estimated later as 150 kHz), implying the robustness of the LC locking.
2.2 Lasers
In this study, optical frequency of an ECLD with the grating feedback at 1645.8 nm (slave laser) is offset-locked with respect to one mode frequency of an OFC (master laser). These lasers have been used in another study [13, 14]. Briefly, the OFC is a mode-locked erbium-doped fiber laser with a repetition rate of 66.87 MHz, whose bandwidth is broadened by the use of an optical amplifier and a highly-nonlinear optical fiber to cover the ECLD wavelength. The linewidth of each mode of the OFC is less than 1 Hz, whereas that of the ECLD is over 200 kHz without stabilization.
2.3 OPLL with LC locking
Linewidth of the beat-note spectral line with the LC locking is expected to be a few kilohertz [12], and for narrower linewidth down to the order of sub-millihertz, OPLL is carried out in addition to the LC locking. When the two locking schemes are employed together, narrow linewidth (by the OPLL) and robustness against the jitters (by the LC locking) are simultaneously attained.
For simultaneous use of the LC locking and the OPLL, the two error signals have been simply added in [12]. The OPLL is for stabilizing the beat note phase, and hence the locked beat-note frequency may be determined only by the LO frequency for the OPLL. Then difference of the LC resonance frequency from the LO frequency seems not to cause frequency shift at all, and instead, it causes only phase shift, as experimentally demonstrated below. However, fluctuation of the LC resonance frequency induces the stabilized beat-note frequency fluctuation, as it is derivative of the phase by time. Such fluctuation of the LC resonant frequency may be due to temperature drift and gives rise to slow beat-note frequency fluctuation (rough estimation is given in Sect. 3). Consequently, when employing the LC locking for the robustness without sacrificing the long-term stability of the OPLL, it is important to make the LC resonant frequency identical with the LO frequency.
In general, the OPLL is the best scheme to precisely lock the beat-note frequency and, therefore, it is favored to fix the LO frequency and to tune the LC resonant frequency to the LO frequency. Then, in this study, the tunability of the LO frequency is reserved for other purpose, such as tuning or scanning the ECLD optical frequency with fixed OFC frequency, and the LC resonant frequency is to be tuned at the LO frequency.
2.4 Overall experimental setup
The overall experimental setup for the optical frequency stabilization is schematically shown in Fig. 3. The beat-note signal between the OFC and the ECLD is detected by balanced photodetectors to eliminate irrelevant signal at the repetition rate of the OFC and its harmonics [13, 14]. The beat-note detection is carried out by two sets of the balanced detection. One is for the laser frequency stabilization described later, and the other is for monitoring the beat-note signal. For convenience, the beat-note signal for the ECLD frequency stabilization is referred as "in-loop signal", and that for beat-note monitoring is as "out-of-loop signal" hereafter.
As stated before, the ECLD frequency is to be stabilized with respect to the OFC mode frequency. To be exact, what is stabilized is the ECLD frequency at the photodetectors for the in-loop signal rather than the frequency at the output of the ECLD device. When the ECLD frequency linewidth is broad (over 100 Hz), the ECLD frequency at the photodetectors can be considered as that at the output of the ECLD device. However, for narrow linewidth (less than 100 Hz), these two ECLD frequencies are different each other because of phase noise induced by mechanical fluctuation on optical fibers. In the experimental setup, optical path lengths of in-loop and out-of-loop fibers for the OFC are \(\sim\) 3 m and \(\sim\) 4 m, respectively, and each of those for the ECLD is \(\sim\) 1 m. These fibers mechanically fluctuate because of air flow from air conditioning. The out-of-loop beat-note frequency spectrum observed by the rf spectrum analyzer is with the fiber noise, whereas the in-loop signal is without the fiber noise if the feedback loop works properly. Hence stability of the in-loop signal frequency is the measure of the feedback performance. It should be noted that, with the stabilization, the phase noise caused by the in-loop fiber is on the output immediately after the ECLD device, as the feedback loop works to cancel the phase noise at the in-loop photodetectors. As a result, the out-of-loop noise consists of the phase noises from both of the out-of-loop fiber and the in-loop fiber.
The out-of-loop beat-note signal is monitored by a rf spectrum analyzer (SA, FSC3 of Rohde and Schwarz) as shown in the frame named "out-of-loop" in Fig. 3. The in-loop beat-note signal detected by balanced detectors (PDB480C-AC of Thorlabs) passes through a bandpass filter (BF, 19.2–23.6 MHz, SBP-21.4+ of Mini-Circuits) and a rf amplifier (A1, 15 dB, AA170 of R and K). The bandpass filter is necessary to eliminate spectral components except at the lowest beat-note frequency (note that the beat note consists of frequency components at \(Nf_{\text {rep}}\pm \delta f\), where N is an integer, \(f_{\text {rep}}\) is the repetition rate of the OFC, and \(\delta f\) is the lowest beat-note frequency). The beat-note signal is split into two parts by a power splitter (S1, ZFSCJ-2-1-S+ of Mini-Circuits). One is for the tunable LC locking, and the other is for the OPLL.
Experimental setup. Abbreviations not mentioned in the text: BS for 50:50 optical beam splitters, PD for photodetectors, DA for differential amplifiers, T for a 50 \({\Omega }\) terminator, LF1 and LF2 for low-pass filters (cut-off frequency is 1.9 MHz for LF1 and 5 MHz for LF2), and HF for a high-pass filter
The beat-note signal for the LC locking is further divided into two parts by the power splitter with \(\pi /2\) phase shift (S2, QH040-0S of R and K). One is injected into the tunable LC circuit of Fig. 1 (LC in Fig. 3) through the directional coupler (DC, ZFBDC20-61HP+ of Mini-Circuits). The rf signal from the reflection port of the DC [amplified by an amplifier (A2, 40 dB, 493A/40 of Perry Amplifier)] and the phase-shifted rf signal of S2 are mixed (M1, ZAD3+ of Mini-Circuits) to detect the out-of-phase component of the reflection power from the tunable LC circuit. The low-frequency component of the output from the mixer is the error signal for stabilizing the beat-note frequency at the LC resonance. After adjusted by a handmade proportional–integral-derivative (PID) controller, it is fed back to the piezoelectric transducer (PZT) in the ECLD, with which the laser cavity length can be adjusted. The response time of the PZT determines the feedback bandwidth of the LC locking (\(\sim\) 10 kHz). The feedback loop can be closed with a switch (SW1).
For the OPLL, the other output of S1 is mixed by a mixer (M2, MX010-0S of R and K) with a LO signal (AFG3022C of Tektronix), and after a low-pass filter (LF2, 5 MHz) and a handmade high-pass filter (cut-off frequency is 120 Hz), the error signal component faster than 120 Hz is fed back to the driving current of the ECLD after amplitude adjustment (not shown in the figure) for optimizing the OPLL. A switch (SW2) is used to close the OPLL as well.
To tune the LC resonant frequency, the OPLL signal is fed back to the control voltage of the tunable LC circuit. As shown in Fig. 3, after adjusted at another PID controller (PID2) and added to an offset dc voltage, the OPLL signal is fed back to the varactor control voltage. A switch (SW3) is used to close this feedback loop.
3 Results and discussions
First, we demonstrate only the LC locking (SW1 is closed, and SW2 and SW3 are grounded). The out-of-loop rf spectrum of the beat-note signal is shown in Fig. 4. The control voltage is given in the figure. The full-width-at-half-maximum linewidth is 24 kHz, which is worse than the result in [12] (2 kHz). The linewidth is determined by the signal-to-noise ratio of the beat-note signal and slope of the error signal at the lock point shown in Fig. 2. In our case, the mode intensity of the master laser (the OFC, average power in a single mode is 100 nW/mode [13]) is anticipated to be weaker than the continuous-wave laser in [12], resulting smaller signal-to-noise ratio of the beat-note signal. If the slope of the error signal in this study is not so different from that in [12], the wider linewidth may be due to smaller signal-to-noise ratio.
The offset frequency can be changed by tuning the control voltage, as shown in Fig. 4. The tuning range of the offset frequency is between 17.9 and 22.1 MHz for 2.0 V \(<V_{\text {Ctrl}}<\) 4.0 V (2.1 MHz/V). This result seems that the tuning range is so narrow (\(\sim\) 4 MHz). This range is limited by the bandwidth of the bandpass filter (BF in Fig. 3). As described before, the bandpass filter is required to eliminate irrelevant rf spectral components, and hence essentially the tuning range is determined by the repetition rate (mode spacing) of the OFC. The combination of a high-repetition-rate OFC and an LC circuit of high resonant frequency is expected to provide wide tuning range of the beat-note frequency.
Stabilized out-of-loop beat-note spectra with the tunable LC offset locking for some specific values of the control voltage. The resolution bandwidth of the spectrum analyzer is 10 kHz
Next, the OPLL is closed in addition to the LC locking (SW1 and SW2 are closed, and SW3 is grounded). This measurement is carried out to observe effects of the frequency mismatch between the LO frequency for the OPLL and the LC resonant frequency for the LC locking. For this purpose, components slower than 120 Hz in the OPLL error signal are not fed back to the ECLD (the slow feedback is only by the LC locking). Then we expect that we can estimate how much the optical phase stabilized by the OPLL is shifted by the frequency mismatch by monitoring slow (dc) component of the OPLL error signal.
The LO frequency is set at 20 MHz. When the control voltage of the LC circuit is 3.012 V and 3.034 V, behavior of the error signal of the OPLL (output of the LF2 in Fig. 3) indicates that mode hopping of the laser frequency occurs repeatedly (Fig. 5a, d, respectively). When the mode hopping occurs, the phase difference between the beat note and the LO changes by \(2\pi\), and the transient time of the mode hopping is the order of the OPLL feedback time constant (10 \({\upmu }\)s). This phase change can be seen as a pulse-like behavior in Fig. 5a, d. Even with the mode hopping, the slave laser frequency is still stabilized owing to the robustness of the LC locking. When \(V_{\text {Ctrl}}\) is 3.020 V and 3.028 V (Fig. 5b, c, respectively), no mode hopping occurs, but the beat-note phase is not stabilized to zero. In these cases, the beat-note frequency is locked to the LO frequency, since the phase is almost constant. However, if the LC resonant frequency drifts (by temperature drift, for example), the phase drift induces beat-note frequency shift as well. In addition, in the cases of Fig. 5b, c, the mode hopping occurs after several minutes of stabilization. We consider this mode hopping is indeed induced by the LC resonant frequency drift. Even if the feedback signal is the sum of the OPLL error signal and the LC locking error signal as in [12], the beat-note frequency drift appears, because the feedback loop works so that the two error signals cancel each other.
The OPLL error signal when the OPLL is closed for component faster than 120 Hz of the error signal (SW1 and SW2 are closed and SW3 is grounded). In a–d, the control voltage for the LC circuit is set at the value shown in each figure. In e, the OPLL error signal is fed back to the control voltage of the LC circuit (SW3 is closed) to make the LC resonant frequency identical to the LO frequency. The sampling rate is 10 \({\upmu }\)s
From the result in Fig. 5, we can roughly estimate how much the frequency drift is anticipated when the sum of the OPLL error signal and the LC locking error signal is fed back to the ECLD as in [12]. The result in Fig. 5 implies that the phase shift of the beat-note signal is roughly \(\pi\) when \(V_{\text {Ctrl}}\) is modified by 0.01 V. From the result in Fig. 4, the change in \(V_{\text {Ctrl}}\) by 0.01 V corresponds to the LC resonant frequency shift of 21 kHz. Therefore, taking 20 MHz of the LC resonant frequency into account, drift of the LC resonant frequency by a factor of \(10^{-3}\) corresponds to the phase shift of \(\pi\). To achieve 1 mHz uncertainty of the beat-note frequency, the stability of the LC resonant frequency should be better than \(2\times 10^{-3}\) in 1000 s. As typical stability of capacitance in LC circuits is about 0.2 % for temperature change of 1 \({}^{\circ }\text {C}\) [15], temperature of the LC circuit should be stabilized within the range of 1 \({}^{\circ }\text {C}\) for 1000 s (temperature dependence of inductance should be taken into account as well).
When the LC resonant frequency is stabilized to the LO frequency, then we do not have to make an effort to eliminate the temperature drift and other noises, and it is expected that long-term stability of the beat-note frequency is improved. For this purpose, the slow component (less than 1 kHz) of the OPLL feedback signal is fed back to the control voltage of the LC circuit (SW3 is closed). Then the LC resonant frequency is automatically tuned to the LO frequency, and the OPLL error signal is stabilized at zero (Fig. 5e, which corresponds to the phase noise in the locked condition). The beat-note frequency is well locked, and its spectrum is shown in Fig. 6. The feedback bandwidth for the OPLL turns out to be 150 kHz, which is estimated from the sidebands that appear in Fig. 6 (servo bumps). The feedback bandwidth is smaller than the linewidth of the free-running ECLD (200 kHz) and, therefore, the laser frequency stabilization only by the OPLL fails so often (once in a few seconds). By employing the LC locking with the OPLL, the feedback loop works for a long period of time (more than an hour). When the LO frequency is set from 20.0 MHz (Fig. 6b) to 19.7 MHz or 20.3 MHz, the slave laser frequency is kept stabilized without manual adjustment of \(V_{\text {Ctrl}}\) (Fig. 6a, c).
The out-of-loop beat-note rf spectra with the resonant frequency stabilization to the LO frequency. The LO frequency \(f_{\text {LO}}\) is given in the figure. The resolution bandwidth of the rf spectrum analyzer is 1 kHz, which determines the spectral linewidth in the figure
To confirm the long-term stability, the overlapping Allan deviation of the beat-note frequency is estimated as shown in Fig. 7. The in-loop beat-note frequency is measured at the forward sampling port of the directional coupler by replacing the terminator T in Fig. 3 with a frequency counter (CNT-90 of Pendulum), and the out-of-loop frequency is measured by replacing the spectrum analyzer with another frequency counter. The overlapping Allan deviation is estimated from 3000 successive measurements of the beat-note frequency by the frequency counter (each measured value is an average for 1 s). The Allan deviation at 1 s averaging is \(\sim\) 3 mHz, which is as low as that in [12]. The in-loop stability is still decreasing with inverse of the averaging time up to 1000 s, resulting long-term drift of the beat-note frequency is suppressed. From this result, it is found that the use of an electronically-tunable LC circuit as the LC locking does not at all degrade the laser frequency uncertainty which is minimized by the OPLL, and robustness by the LC locking is attained. The high noise level of the out-of-loop frequency indicates that the present experiment has been carried out in very noisy circumstances. As stated before, the noise is mainly due to mechanical noise onto optical fibers. The mechanical noise is caused by air flow (in fact, air conditioning has been on during the measurement). Owing to robustness by the LC locking, the laser frequency stabilization works well even in such noisy circumstances. By the use of fiber noise cancellation technique [16], the out-of-loop noise can be reduced. The in-loop Allan deviation with the 1000 s averaging is \(\sim\) 10 \({\upmu }\)Hz, and for longer averaging time it is expected to reduce further. The relative measurement uncertainty for 1000 s averaging is the order of \(10^{-20}\) compared to the optical frequency. Therefore, this locking scheme does not degrade frequency stability when it is used with atomic clocks of the highest precision [17, 18].
The overlapping Allan deviation of in-loop and out-of-loop beat-note frequency
In conclusion, we improve the robust laser frequency offset locking scheme using the LC resonance circuit introduced in [12]. By replacing the capacitance of the LC resonant circuit with a varactor, the LC resonant frequency becomes electronically tunable. When the OPLL and the LC locking are simultaneously employed, robust and precise locking can be performed. The robustness of the frequency locking is assured by the LC locking, and the preciseness is by the OPLL. By the electronic control, the LC resonant frequency can be stabilized to the local oscillator frequency used in the OPLL. With this stabilization, it is expected that the long-term phase noise caused by the LC resonant frequency drift is suppressed. In the demonstration, optical frequency of the ECLD is phase-locked with respect to a mode of the OFC. The uncertainty for 1000 s-averaging is the order of 10 \({\upmu }\)Hz, and hence this scheme is applicable to experiments with uncertainty of the order of \(10^{-20}\) even in noisy circumstances. The electronic tunability may be applicable to electric circuits utilized in other offset locking schemes such as that with low-pass filters [11].
The authors acknowledge H. Sasada for discussions and comments. This study is financially supported by Japan Science and Technology Agency (JST) through the ERATO MINOSHIMA Intelligent Optical Synthesizer (IOS) Project.
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1.Department of PhysicsKeio UniversityYokohamaJapan
2.JST, ERATO, MINOSHIMA Intelligent Optical Synthesizer ProjectTokyoJapan
Seishu, Y. & Hasegawa, T. Appl. Phys. B (2019) 125: 142. https://doi.org/10.1007/s00340-019-7253-5
Accepted 02 July 2019 | CommonCrawl |
Environmental Science and Pollution Research
November 2016 , Volume 23, Issue 22, pp 22633–22646 | Cite as
Assessment of nitrogen losses through nitrous oxide from abattoir wastewater-irrigated soils
Raghupathi Matheyarasu
Balaji Seshadri
Nanthi S. Bolan
Ravi Naidu
First Online: 24 August 2016
The land disposal of waste and wastewater is a major source of N2O emission. This is due to the presence of high concentrations of nitrogen (N) and carbon in the waste. Abattoir wastewater contains 186 mg/L of N and 30.4 mg/L of P. The equivalent of 3 kg of abattoir wastewater-irrigated soil was sieved and taken in a 4-L plastic container. Abattoir wastewater was used for irrigating the plants at the rates of 50 and 100 % field capacity (FC). Four crop species were used with no crop serving as a control. Nitrous oxide emission was monitored using a closed chamber technique. The chamber was placed inside the plastic container, and N2O emission was measured for 7 days after the planting. A syringe and pre-evacuated vial were used for collecting the gas samples; a fresh and clean syringe was used each time to avoid cross-contamination. The collected gas samples were injected into a gas chromatography device immediately after each sampling to analyse the concentration of N2O from different treatments. The overall N2O emission was compared for all the crops under two different abattoir wastewater treatment rates (50 and 100 % FC). Under 100 % FC (wastewater irrigation), among the four species grown in the abattoir wastewater-irrigated soil, Medicago sativa (23 mg/pot), Sinapis alba (21 mg/pot), Zea mays (20 mg/pot) and Helianthus annuus (20 mg/pot) showed higher N2O emission compared to the 50 % treatments—M. sativa (17 mg/pot), S. alba (17 mg/pot), Z. mays (18 mg/pot) and H. annuus (18 mg/pot). Similarly, pots with plants have shown 15 % less emission than the pots without plants. Similar trends of N2O emission flux were observed between the irrigation period (4-week period) for 50 % FC and 100 % FC. Under the 100 % FC loading rate treatments, the highest N2O emission was in the following order: week 1 > week 4 > week 3 > week 2. On the other hand, under the 50 % FC loading rate treatments, the highest N2O emission was recorded in the first few weeks and in the following order: week 1 > week 2 > week 3 > week > 4. Since N2O is a greenhouse gas with high global warming potential, its emission from wastewater irrigation is likely to impact global climate change. Therefore, it is important to examine the effects of abattoir wastewater irrigation on soil for N2O emission potential.
Abattoir Wastewater Irrigation Nitrous oxide Emission Greenhouse gas and global warming
Responsible editor: Zhihong Xu
The emission of greenhouse gases (GHGs) increases with the rising global population (Preston et al. 2006). Human activities such as agriculture (e.g. chemical fertilisers), energy production (e.g. coal combustion), transport (e.g. fossil fuels) and other industrial activities are directly or indirectly contributing to the GHG emissions (Crutzen et al. 2008). Among the various sources that are responsible for GHG emissions, energy production is the major contributor followed by land use change for agriculture and industrial activities (Meinshausen et al. 2009; Cerri et al. 2009). The major global greenhouse gases and their percentage of emission are illustrated in Fig. 1. At a global scale, nitrous oxide (N2O) is a major GHG after carbon dioxide (CO2) and methane (CH4) (IPCC-2007). However, the global warming potential (GWP) varies between these GHGs, for example, N2O is 282 times more powerful than CO2 (Ravishankara et al. 2009). Hence, the management of N2O by reducing their emission is important in mitigating climate change (McCarl and Schneider 2000). Since N2O is a highly potential GHG towards GWP, measurement and mitigation need to be done on a broader scale (Shine et al. 2005). N2O emission contributes about 6 % of the overall global warming effect, but its contribution from the agricultural sector is about 16 %. Of that, almost 80 % of N2O is emitted from Australian agricultural lands, originating from N fertilisers (32 %), soil disturbance (38 %) and animal waste (30 %) (Dalal et al. 2003).
Global greenhouse gases emission (in %) (adapted from IPCC report 2007) (F-gases = fluorinated gas (hydrofluorocarbons (HFCs), perfluorocarbons (PFCs), sulphur hexafluoride (SF6) and nitrogen trifluoride (NF3)
Globally, denitrification is the primary process of N2O production in temperate grassland soils and accounts for 60 % of the total N2O emissions (Jha et al. 2012). A recent study at the European Union states that ruminants (cows, sheep and goats) have the highest carbon footprint (FAO 2006). Total net GHG emission of EU livestock production was estimated at 661 mt of carbon dioxide equivalent (CO2-eq) which is about 9–13 % of the total GHG emission for the EU agricultural sector, comprising 23 % CH4, 24 % N2O, 21 % CO2 (energy use) and 29 % CO2 (land use). A considerable amount of GHGs is emitted by the global animal industry, which is more than all the cars in the world put together, and a large part of that is 18 % N2O and CH4 emissions; both of these gases have a far more powerful greenhouse gas effect than carbon dioxide (Garnaut et al. 2008).
The livestock sector accounts for 5–50 % of the total contribution, but it may vary from place to place (Gunnarsson et al. 2011). The overall contribution consists of pigs 0.4 %, sheep 3.4 %, cattle 2.7 % and beef cattle 11.2 %, which on average emits 554 kg CO2-eq/tonne hot standard carcass weight (MLA 2010). Untreated abattoir wastewater is unsuitable for reuse or discharge into the receiving environment. It will cause serious environmental hazards in the receiving environment such as eutrophication, land degradation, nutrient leaching, groundwater contamination, greenhouse gas emission and effects on ecosystem value; hence, a proper reduction in pollutant levels in the prior stage is essential. In the recent years, meat production and consumption have been increasing considerably and predicted to peak in 2020 predominantly in Asia and the Pacific. Global per capita meat consumption is projected to increase from 32.9 kg/rwt in 2011 to 35.4 kg/rwt in 2020 (OECD-FAO 2011). Meat production is a considerable source of global GHG emission, emitting methane, nitrous oxide and carbon dioxide through various stages. GHGs are emitted by direct energy consumption and indirectly by feedstock production, herding, movement of animals, product transport, slaughtering, cleaning and dressing the animal product, waste and wastewater. The disposal of waste and wastewater to land is a major source of N2O emission (Oene et al. 2005; Bolan et al. 2004). N2O emission from wastewater-irrigated soils has been explored by many researchers (Bhandral et al. 2007; Saggar et al. 2004; Dalal et al. 2003).
Agricultural industry wastewater such as abattoir and dairy wastewaters are significant contributors towards N2O emissions (Kampschreur et al. 2007; Russell et al. 1993). This is due to the presence of the high concentration of N and carbon; for example, abattoir wastewater (AWW) contains 200 to 400 mg/L of N and 545 mg/L of dissolved organic carbon (Longhurst et al. 2000; Cassidy and Belia 2005; Matheyarasu et al. 2012). Tsujimoto (1994) observed that N loss through leaching and gaseous emissions increases with an increasing level of animal waste and wastewater application in soil. Soils are the major source of the greenhouse gas nitrous oxide (N2O) in our atmosphere.
The rate of denitrification in soils and the relative proportions of NO, N2O and N2 produced are controlled by various factors, such as soil microorganisms, climatic factors and management practices. The availability of mineral N (both NH4 + and NO3 −) and labile C, together with processes that affect reaction rates, such as temperature, pH and redox potential in soil microsites (Saggar et al. 2013). N2O can be produced by nitrifiers, denitrifiers and nitrifiers paradoxically denitrifying. Soil moisture conditions are sub-optimal for denitrification, nitrifier denitrification and the major contributor to N2O emission (Kool et al. 2011). The majority of studies indicate that a significant amount of N in waste and wastewater is lost as N2O emissions (Kampschreur et al. 2009; Czepiel et al. 1995; Bolan et al. 2004; Saggar et al. 2015). Since N2O is a potential greenhouse gas with very high GWP, even low emissions can cause serious effects (Shine et al. 2005). In addition, the application of chemical fertilisers alone contributes 46 % of N2O emission from the agriculture sector (Baumert et al. 2005) and therefore minimising the usage of these fertilisers will also reduce the emission rate.
There are increased concerns about denitrification associated with the loss of N in the environment. Denitrification can be both detrimental and beneficial to the environment (Bolan et al. 2004). For example, N2O, one of the gaseous products from denitrification, has possible deleterious effects on global warming (Bateman and Baggs 2005). The primary consideration for mitigating gaseous N emissions from arable land is to match the supply of mineral N (from fertiliser application, legume-fixed N) to plant needs, although it is possible to achieve uniform application of N fertilisers. Mitigation approaches need to focus on ways to reduce the production of N2O during denitrification and enhance the reduction of N2O to N2 thus lowering the N2O:N2 product ratio (Saggar et al. 2013). On the other hand, effective management practices help to minimise processes such as leaching, denitrification and NH3 − volatilisation, all of which lead to the loss of plant-available N from the soil–plant system. These management practices include optimum N supply to pasture crops, proper animal residue management, controlled-release fertiliser and proper water management (Bolan et al. 2004).
Wastewater irrigation increased the concentration of major nutrients (N, P and K) in soil (Matheyarasu et al. 2014). Although there was an increase in soil fertility, the potential for N losses through nitrate leaching and N2O emission is likely to be high. There is a great scope and need for reducing N2O emissions from various sources to the atmosphere (Ravishankara et al. 2009; McCarl and Schneider 2000). Currently, there are limited reports on N2O emission from AWW-irrigated soils or AWW-irrigated cropping system in Australia. Therefore, it is important to examine the effects of AWW irrigation on the GHG emission potential. This paper examines the effect of different types of nutrient source (wastewater and urea) and irrigation intensity on biomass productivity of selected plant species and their N uptake efficiency and N2O emission under greenhouse conditions. The overall objective of the study is to examine the effects of AWW irrigation on N uptake and N2O emission in a calcareous soil. The specific objectives of the study included (a) to quantify N2O emission from agricultural soil treated with various soil moisture conditions and N supplements (urea and AWW), (b) to investigate the effects of AWW irrigation on N2O emission and (c) to study the rate of plant-induced denitrification in two different moisture gradients (50 and 100 % FC) with and without plants.
Contaminated site assessment and soil sample collection
The study area (sampling site) is situated at 89.7 km north of Adelaide, South Australia. The latitude and longitude of the study area are 34° 8′ 26.60″ S and 138° 11′ 7.35″ E; the range is 749 m and the elevation of the treatment site is generally flat ranging from 13.5 m Australian height datum (AHD) to 14.5 m AHD. The region has mean annual rainfall of 287.3 mm and annual mean maximum temperature of 22.8 °C and minimum temperature of 10.7 °C. Abattoir wastewater and soil under abattoir wastewater irrigation were collected from land treatment. The AWW-irrigated soil were collected, air-dried and sieved to <2 mm for physiochemical characterisation. The site was under long-term wastewater irrigation to manage wastewater economically and was used for forage production, alternatively. The land treatment site (CI) has received around 385 mm of secondary treated effluent applied over the year at the rate of 32 mm per month.
The CI soil also received an additional 310 mm of water through rainfall, during the period (2012). In the study site, the rate of irrigation was not adjusted according to annual rainfall since it is intended for land treatment. The stored soil samples as collected from different locations and depths were analysed for pH, electrical conductivity (EC), nitrogen (N), phosphorus (P), carbon (C) and micro nutrients. Soil analyses were performed following standard methods as described in the Soil Chemical Methods—Australasia (Rayment and Lyons 2011) manual. Soil pH was measured in water using glass electrodes at a 1:5 soil to water ratio. Soil EC was also measured at the same time using an EC meter. Soil total C and total N were estimated by dry combustion on air-dry soil using a LECO 2000 CNS analyser (Sparling et al. 2006). Olsen P was estimated by soil extraction with sodium bicarbonate (0.5 M at pH 8.5) and measured by the molybdenum blue method (Olsen et al. 1954). Absorbance was measured at 882 nm in an Agilent UV–visible spectroscopy system (Germany), and the Olsen P concentration was calculated by preparing a calibration curve against the standards. The total P and micronutrients were determined using inductively coupled plasma-optical emission spectrometry (ICP-OES), with acid-digested soil samples (1:3 ratio of concentrated nitric–hydrochloric acid mixture/aqua regia) (Chen and Ma 2001). Similarly, available N (nitrate-N and ammonia-N) was measured using the SKALAR SANS system (analyser) with potassium chloride (2 M)-extracted soil samples (Luo et al. 2004).
Plant growth experiment
The plant growth experiment was conducted at the University of South Australia greenhouse using the contaminated soil collected from the land treatment sites. The wastewater used in this experiment was collected from the Primo abattoir at Port Wakefield, which was rich in major plant nutrients such as total nitrogen (TN) and total phosphorus (TP). Two sets of experiments were conducted to examine the effects of wastewater irrigation on N loss through gaseous emission (N2O). In experiment 1, urea was used as a N source to study the gaseous emission from the soil. In experiment 2, AWW was used as N source and the effects of adding wastewater at different loading rates towards N2O gaseous emission were studied.
Experiment 1—a study on N2O emission without plants in a laboratory condition
Experiment 1 comprised seven treatments with three replicates to study the effects of urea addition on gaseous emission (N2O). In this experiment, five different moisture levels were applied (e.g. 25, 50, 75, 100 and 120 % FC) with two levels of nitrogen loading (500 and 1000 mg/kg of soil) to examine the N loss through gaseous emission.
Experiment 2—gaseous emission with plants
The equivalent of 3 kg of soil was sieved and taken in a 4-L plastic container. AWW was used for irrigation of the plants at the rates of 50 and 100 % FC. Crop species including Helianthus annuus, Sinapis alba, Medicago sativa and Zea mays were used with no crop serving as a control. The entire experiment was carried out with three replications (Plate 1). Treatment details are as follows: 2 moisture levels*4 + 1 control (no plant)*3 replicates (2*5*3 = 30).
Plate 1
A study on plant-induced denitrification at a greenhouse located at the University of South Australia
Gas sampling unit
The N2O emission from the treatment pot was measured weekly using a modified closed chamber technique. The modified chamber technique was developed during this study based on the principles described by Saggar et al. (2002; 2004) and Bhandral et al. (2007). This modified closed chamber technique involves sampling of N2O emission close to the soil surface (10 cm), with the apparatus full description of the modified closed chamber presented in Fig. 2. The closed chamber was placed inside a pot (3 kg of soil). A total of 30 chambers were designed and used in this study with three replicates of each treatment. The dimensions of the chamber were 17 cm in height and 4 cm in diameter. The chamber was inserted into the soil with 7 cm exposed above the soil. Background N2O emissions were measured for each pot on the first day after the initial setup to validate and check the efficiency of the modified closed chamber. After the application of treatments, measurements were made every week until 4 weeks after planting.
Modified chamber techniques for the measurement of N2O emission from the controlled environment
Each gas sampling unit was placed inside the pot (Plate 2). Emissions of N2O were measured for 7 days after the incubation in experiment 1 and planting in experiment 2 using the active efflux method. A syringe and pre-evacuated vials were used for collecting the gas samples; a fresh and clean syringe was used each time to avoid cross-contamination. The collected gas samples were injected to a gas chromatograph immediately after each sampling to analyse the emission of N2O from different treatments. At the end of the greenhouse studies, a nutrient loss percentage as N2O (%) was calculated to study the nutrient losses using nutrient input and loss ratio. Plants were harvested after 6 weeks of germination. The shoots and roots were separated and dried in a hot air oven at 70 °C to a constant weight. Plant samples were stored in a dry airtight container for further nutrient analysis to calculate nutrient uptake.
Gas sampling method for N2O analysis
SPSS (Inc., 2001) was used to analyse the data. The differences in the replicates were determined using standard deviations for the gaseous emission (N2O). Relationships between wastewater loading types (50 vs 100 % FC) and gaseous emission (N2O) were analysed by Pearson correlation coefficients and the two-sample t test.
Properties of soils and wastewater used in this experiment
The soils used in this study were collected from an abattoir wastewater-discharged landfill site at Port Wakefield, South Australia. The collected samples were air-dried, characterised for physicochemical properties and used for greenhouse plant growth experiment (assessment of nitrogen losses). Soil collected from the land treatment site was moderately alkaline; the pH of the CI soil was moderately acidic (6.3), with CNI and CTRL measuring 8 and 8.6, respectively. The electrical conductivity was very high ranging from 500 to 1109 μS/cm.
The AWW-irrigated soil was high in TN and TP at available nutrient concentrations. The nutrient contents in AWW-irrigated soil were up to 1165 mg/kg of nitrogen (N) and 223 mg/kg of phosphorus (P). The wastewater sample was characterised for its major nutrient concentration (N and P) immediately after collection. The AWW used in this experiment had a high TN and TP concentration (186 mg/L of N and 30.4 mg/L of P) (Table 1).
Properties of soil used in these experiments (wastewater-irrigated soil)
Soil parameters
EC (μS/cm)
299.6 ± 1.9
Moisture (%)
11.8 ± 0.1
Total C (mg/kg)
10,523 ± 1.1
Total N (mg/kg)
1165.9 ± 1.4
Nitrate-N (mg/kg)
Ammonia-N (mg/kg)
14.9 ± 0.02
Total P (mg/kg)
223.1 ± 0.02
Olsen-P (mg/kg)
K (mg/kg)
2849.8 ± 0.01
Ca (mg/kg)
Mg (mg/kg)
Na (mg/kg)
Fe (mg/kg)
Al (mg/kg)
B (mg/kg)
4.2 ± 0.02
Zn (mg/kg)
Mn (mg/kg)
Effects of urea application and rate of application on nitrous oxide emission with varying soil moisture
The differences in the rate of N application (as urea) on N losses through N2O were observed by the end of the experiment. The application of urea with two different rates (low = 500 mg/kg; high = 1000 mg/kg) showed significant impacts on the N2O emission. The rate of emission was also significantly affected by the soil moisture content. The results showed that the high rate of N addition (1000 mg/kg) was most vulnerable in terms of N losses with varying moisture gradient. For both the levels of N application, the N2O emission increased with the increasing soil moisture content (% FC) up to the saturation (100 % FC) and the rate of N2O emission started to decline after the soil reaches the maximum water holding capacity (>100 % FC).
The rate of N2O emission will be minimum in a higher FC condition (waterlogged or flooded soils); this is due to less aeration (restricted) and low emission (N2O) leading to a complete denitrification process, thereby emitting N2 gas (Dalal et al. 2003). The maximum level of N2O emission was recorded at 75 to 100 % FC in both N treatment levels (500 and 1000 mg/kg of urea-N). Overall, high N addition with high soil moisture (75 to 100 % FC) resulted in a significant amount of N losses which were about 30 mg/pot/day emitted as N2O. This was nearly twofold higher than that of the other treatment level of 500 mg/kg of soil N as urea (Fig. 3). Soil N2O production was highly dependent on the oxygen (O2) supply and water-filled pore space (WFPS) (Bhandral et al. 2007; Dalal et al. 2003). The high N2O emission can be expressed in a situation like low O2 partial pressure (<0.5 vol.%) and high WFPS (>60 %) (Saggar et al. 2013). N2O is primarily produced in soil by the activities of microorganisms during nitrification and denitrification processes. The ratio of N2O production depends on oxygen supply or water-filled pore space, decomposable organic carbon, N and substrate supply (Dalal et al. 2003).
Nitrous oxide emission from soil treated with urea with 500 and 1000 mg/kg—with varying moisture gradient (25, 50, 75, 100 and 120 % FC)
Effects of AWW irrigation on nitrous oxide emission
Effects of nutrient loading
Abattoir wastewater is the major source of N to the soil. Irrigation of nutrient-rich AWW increased the soil fertility and nutrient loss, especially N. The soil nutrient loss was significantly higher under AWW irrigation compared to that in non-irrigated soils. The N2O emission rate significantly varied between the two levels of wastewater application. The distinct N2O peak was observed during the first week after the application of AWW in all the crop species grown and non-cropping condition. The peak was at 22 mg/pot in 100 % FC and 18 mg/pot in 50 % FC in non-cropped conditions (average of the initial period). According to Bowwman (1996), up to 0.16 % of total N applied to the soil can be lost as N2O emission within a day after fertiliser application (Dalal et al. 2003; Ruseel 1993). In comparison to all the treatments (including the AWW 50 % FC with and without a crop, AWW 100 % FC with a crop), the AWW treatment without a crop showed statistically significant N2O emissions compared to the other treatment levels (Fig. 4). These increases were recorded from the first week after application of AWW. Similar results were found in Bhandral et al. (2007). They observed the peak emission of N2O from the meat industry wastewater-irrigated soils in the first few days after treatment.
Nitrous oxide emission from soil irrigated with abattoir wastewater with 50 and 100 % FC (moisture gradients)—measured at 4 weeks after planting (ALF alfalfa, MU mustard, MZ maize, SF sunflower, NC no crop)
The peak emission of N2O from the AWW 100 % FC (no crop) was 22 % higher than that of the AWW 50 % FC (no crop) which is 53 % times higher than that of the cropped pot. A peak N2O emission flux of 21.9 mg/pot/day was recorded under the non-cropping condition at the first week after planting; this indicates that there are greater chances for the loss of applied N if there is no crop to utilise the nutrients. According to Cardenas et al. (2010), high N input can increase the annual N2O emission fluxes in a pasture under a range of N fertiliser inputs and their study concluded that higher emissions are possible under increased nutrient supply. A significant increase was recorded in consecutive weeks, but the differences were not much significant as compared to the first week. The overall emission data of 4 weeks suggest that there were highly significant effects of a high load of nutrient addition through wastewater in terms of N loss.
Effects of time
The N2O emission was significant in 100 % FC at 2 weeks after the plantation in all the pots. The N2O emission was high at the first 2 weeks after the planting and declined subsequently to reach the background levels within 4 weeks. Application of AWW irrigation has increased the soil N2O emission rate in a short-duration study at the greenhouse for 4 weeks. The total emission in the first week was higher compared to that of the rest of the 3 weeks in both soils. The highest N2O emission was recorded in the 100 % FC in the first week for all the four pots, which was about 21 to 22 mg/pot. The AWW irrigation affected N2O emissions at both levels of irrigation (50 and 100 % FC). The N2O emission increased initially (the first week after the treatment) and declined in the consecutive weeks of the experimental period (for example, the fourth week) (Fig. 5).
The effects of abattoir wastewater (50 and 100 % FC) on cumulative N2O emission of four plant species (alfalfa, mustard, maize and sunflower) measured at the fourth week after planting
Application of AWW (at 100 % FC) increased N2O emission by 8 % in the first week and 43 % in the second week after application. A similar study by Singh et al. (2008) states that the application of urine resulted in a high amount of N2O losses in their incubation studies. The authors found that the peak of N2O emissions was within a week after application of dairy cow urine and observed a maximum of 11.2 mg of N2O/kg of soil. Also, their studies showed that the maximum emission was reached before day 25 of the 40-day incubation study (Singh et al. 2008). In the 4-week treatment period, N2O emissions from 100 % FC (no crop) treatment in the first 2 weeks remained significantly higher than the N2O emitted from the 50 % FC (no crop).
Effects of plant species used
There was a significant difference between AWW 50 % FC and 100 % FC irrigation in the overall N2O emission of all the four crops used in this study. For example, in comparison with 50 % FC, 30, 23, 17 and 11 % (H. annuus, S. alba, M. sativa and Z. mays) of increases in N loss were recorded as N2O in 100 % FC. N2O emission was significantly (p < 0.05) higher in the application of 100 % FC of AWW to all the four crops grown. A maximum of 27.5 mg/pot was recorded in the 100 % FC condition in the second week of application with AWW. In general, N2O emissions were significantly lower from all the crops irrigated with wastewater than those of the non-crop condition in both levels of treatment (50 and 100 % FC). The cumulative emissions recorded in the 50 % FC were 17.5, 17.7, 18.1 and 18.2 mg/pot for H. annuus, S. alba, M. sativa and Z. mays, respectively (non-cropped was 23 mg/pot). Similarly, the cumulative emissions recorded in the 100 % FC were 22, 21, 20 and 20 mg/pot from H. annuus, S. alba, M. sativa and Z. mays, respectively (non-cropped was 23.4 mg/pot) (Fig. 6). Pots with plants showed lower emissions than the pots without plants. Overall, the cumulative (average) N2O emission was 13 % higher in high-field-moisture pots (100 % FC) than that in the moisture-deficit pots (50 % FC) (Table 2). The second highest N2O emission was recorded on the second week after application of the AWW. A similar result was found in Bhandral et al. (2007), and they found that the initial increase in the N2O emission after the application of meat industry wastewater declined progressively with time.
Nitrous oxide emission from soil irrigated with abattoir wastewater with 100 % FC (moisture gradients)—measured at 4 weeks after planting
Descriptive statistics: the effects of abattoir wastewater irrigation on N2O emission: cumulative effects of 50 and 100 % FC (n = 30)
Std. deviation
Skewness
Kurtosis
Nitrate-N (mg/pot)
37,294.2
Ammonia-N (mg/pot)
N2O week 1 (mg/pot)
Cumulative emission (mg/pot)
– equal to zero
Effects of soil moisture
Among the two levels of AWW irrigation treatment, N2O emissions were highest for AWW 100 % FC (16.6 mg/pot), whereas the 50 % FC emitted 12.3 mg/pot (cumulative emission average). Bhandral et al. (2007) suggested that the highest N2O emission is possible if soil is irrigated/applied with a high dose of urine, ammonium and urea.
The N2O emission was higher (66.5 mg/pot) at 100 % FC (p < 0.01) than at 50 % FC (49 mg/pot) which may be attributed to higher soil moisture content in the former treatment (Table 3). According to a study by Bateman and Baggs (2005), N2O emission was high under the elevated soil moisture condition. Soil N2O emission was significantly affected by soil moisture and a nutrient input source (Weier et al. 1993). Similarly, Maag and Vinther (1996) noticed that soil with an optimum moisture condition and high N source can produce increased N2O emissions. Results of decreased N2O production at low moisture content were obtained by Dalal et al. (2003). A study by Russel et al. (1993) on AWW irrigation to land treatment in a pastureland suggests that, under favourable conditions (e.g. a soil temperature below 12 °C, soil pH 5.9 and abattoir effluent pH of 5), N loss can reach a maximum within a day. Soil moisture was one of the most important factors influencing N2O emissions from tropical forest soils (Kiese and Butterbach-Bahl 2002). In moist soils, the reduction of NO3 − proceeds in a series of steps, producing NO2, nitric oxide (NO), nitrous oxide (N2O) and N2 gas. Denitrification results not only in the loss of a valuable plant nutrient but also in the release of N2O (greenhouse gas), which is implicated in the destruction of atmospheric ozone (Bolan et al. 2004). In a flooded soil condition, denitrification has been considered to be a major pathway of N loss (Aulakh et al. 2001). Nitrifier denitrification can be a prime contributor to total N2O production from the soil. Nitrifier denitrification contributed more to N2O production than total conventional denitrification of NO3 − at both 50 and 70 % WFPS.
Effects of abattoir wastewater irrigation on N2O emission (n = 30); a comparison of cumulative effects of two types of moisture gradient by the two-sample t test
50 % FC
100 % FC
Sig. diff.
N2O emitted—week 1 (mg/pot)
19 ± 3.1
21.9 ± 3
p < 0.05
p < 0.001
49.3 ± 12.5
n.s. not significant
The overall emissions of N2O were positively correlated with nitrate-N and ammonia-N for all the 4 weeks of treatment, but the N2O emission was negatively correlated with plant DM yield. This means that high plant growth can minimise the N loss as N2O emission as a result of elevated N uptake. Similar to soil moisture content, N2O emission was positively correlated with ammonia-N, pH and Olsen P (Jha et al. 2012). In the current study, the soil was collected from the land treatment site (AWW irrigated) which was rich in soil nutrients such as nitrate-N, ammonia-N and Olsen P. Overall, the plots irrigated with AWW 100 % FC showed significantly higher N loss (N2O) than the plots irrigated with 50 % FC for all four plant species used (Table 4). The overall AWW irrigation (50 and 100 % FC) effects on the plant-induced denitrification of four plant species were interpreted using principal component analysis (PCA) for the individual parameters. The effects of the irrigation loading rate on N losses through N2O were clearly explained using PCA. The wastewater loading rates (n = 30) of both treatments (AWW 100 and 50 % FC) showed distinction from their nutrient loss as N2O. The AWW with 100 % FC irrigated pots (all the four plants) greatly varied from the abattoir 50 % FC irrigated sample reflecting the quantity of N loss and their frequency in different time periods. The factor loading float shows clearly the effects of wastewater irrigation on the soil properties.
Effects of abattoir wastewater irrigation on N2O emission (n = 30); a comparison of cumulative effects of two types of moisture gradient by Pearson correlation coefficient
DM yield (mg/pot)
0.509**
0.432*
−0.076
−0.391*
*Correlation is significant at the 0.05 level; **correlation is significant at the 0.01 level
Similar to a high loading rate, factors 1 (nitrate-N) and 2 (ammonia-N) explained 51 and 21 % of variation, respectively. Therefore, nearly 50 % of the total N loss as determined by the nutrient load was supplied through wastewater addition. The properties such as ammonia-N and nitrate-N were largely influenced (factors 1 and 2) by the overall emission of N2O. The PCA method showed that AWW irrigation was significantly different in the individual treatments in terms of soil N loss and DM yield in the current pot experiment. The highest N2O emission was recorded at 100 % FC, due to the supply of a high rate of nitrate-N through AWW.
Effects of AWW irrigation on plant-induced denitrification—nutrient loss and utilisation
This study demonstrated the influence of AWW irrigation on the plant-induced denitrification and soil productivity. The effects of AWW irrigation on soil nutrient loss as N2O are presented in Table 5. In the present study, the nutrient input included wastewater- and soil-derived nutrients; the output included plant uptake and loss through N2O emission. Using the above information, the percentage of N loss through N2O was calculated to study the effects of AWW irrigation on plant-induced denitrification, and soil fertility and productivity changes (Eq. 1).
Effects of abattoir wastewater irrigation on N loss as N2O emission (%) in two types of loading rate in four crops grown under greenhouse condition (n = 30)
N added through effluent (mg/pot)
N emitted (mg/pot)
% of N emitted
No crop
$$ \%\mathrm{of}\ \mathrm{added}\ \mathrm{N}\ \mathrm{emitted}=\frac{\mathrm{N}\ \mathrm{emitted}\ \left(\mathrm{mg}/\mathrm{pot}\right)}{\mathrm{N}\ \mathrm{added}\ by\ AWW\ \left(\mathrm{mg}/\mathrm{pot}\right)}\times 100 $$
The results from this study showed that 50 % FC treated soil had higher N2O emissions in terms of percentage losses (high differences in input and nutrient loss ratio) compared to the 100 % FC. Similarly, the non-cropped soils were very highly variable between the crops due to the absence of plants. In the current experiment, results showed that nearly 1.2 to 2.4 % of applied N was lost as N2O. Moiser et al. (1996) showed that nearly 2.5 % of the total N applied to the soil was lost as N2O from agriculture land. The relationship between N input and overall N2O emission found in this study was positively correlated, similar to the previous research report by IPCC (1999) and studies by Dobbie et al. (1999).
The result from the greenhouse experiments clearly demonstrated that the addition of nutrients or AWW irrigation enhanced the nitrous oxide emission. The effect of nutrient (N) addition on the soil generally increases the soil N loss through N2 and N2O, if it is not utilised by green cover, and the excess N may find a pathway to enter the atmosphere or be leached to groundwater (Reichenau et al. 2016). The following reasons could be attributed to the high nitrous oxide emission from the wastewater-irrigated soils: (i) the rate of nutrient loading through abattoir wastewater irrigation/high soil fertility, (ii) the process of denitrification and (iii) the anaerobic condition created by the excessive addition of abattoir wastewater. The rates of nutrient addition on nitrous oxide emission in soil have been reported by many researchers; for example, Zhang et al. (2016) reported that increased nitrate or phosphate loading resulted in stronger influence on denitrification than single nutrient additions that stimulate denitrification rates in sandy sediments in aquatic/terrestrial transition zones.
The abattoir wastewater irrigation had significant impacts on the physiochemical properties of the soils. Nitrous oxide (N2O) emissions can be significantly affected by the amounts of N available in soils, and also forms of nutrients, local climate and soil conditions are the determining factors of the effects (Peng et al. 2011). Increased N availability generally leads to increased N2O emission (Davidson et al. 1996) because this stimulates the microbial processes of nitrification and denitrification, which produce the gas fluxes. The relationship between N input (500 and 1000 mg/kg) on cumulative N2O emission found in this study was significant (N losses increase with the increasing application of N) (Fig. 3). The results have been consistent with earlier reported studies on N2O emissions (Liu et al. 2005; van Groenigen et al. 2004; Willén et al. 2016) (Fig. 3). Similarly, Chmura et al. (2016) assessed the impact of nutrient additions on greenhouse gas fluxes using dark static chambers in a microtidal and a macrotidal marsh (Both were experimentally fertilised for 6 years). They found that N2O fluxes are likely to vary with the source of pollutant nutrients, but emissions will be lower if N is not accompanied by an adequate supply of P.
Overall, the application of AWW caused significant N2O loss. In our experiment, we compared the two treatments of nutrient loadings 50 % FC and 100 % FC and we found that there was a highly significant correlation between both treatments in terms of nitrous oxide emission (p < 0.001) (Fig. 4). A number of studies have reported the effect of split fertilisation or low-rate application which showed significant reduction in N2O emissions (Yu et al. 2016). There was a higher percentage reduction (28 %) in cumulative N2O emissions under the split urea application compared with the single fertilisation, although these emissions were influenced by the N fertiliser rate and soil moisture (Yu et al. 2016). The results suggest that low-rate application of AWW is a potential strategy for reducing N2O emissions in a wastewater-irrigated soil or in a land treatment site. Similarly, the results by Neto et al. (2016) showed that N fertiliser increased N2O emissions from the soil, especially when urea was used. The emission factor for N fertiliser was 0.46 ± 0.33 %.
There was no difference between urea fertiliser and abattoir wastewater nutrient addition on nitrous oxide emission values. Values of N2O fluxes were reported for each week and compared with cumulative emission. The application of AWW at the rate of 100 % FC reached a peak of 24 mg/pot emission in the first few weeks after planting (Fig. 5). A study by van der Weerden et al. (2016) recommended an N2O emission inventory for New Zealand's agricultural soils and found that there was no difference between urea fertiliser in terms of N2O emission due to the different origins and characteristics of these N sources. For example, in New Zealand's agricultural soils, N2O emissions have means of 0.6 and 0.3 % for urea fertiliser and FDE, respectively (der Weerden et al. 2016).
According to a study report by Rowe et al. (2012), the effect of N deposition on mineralisable N stock was more apparent in more organic soils, whereas the effect on nitrate proportion was more apparent in more mineral soils. With the high proportions of nitrate (over 40 %) that responses also depend on soil C content and site temperature (Table 1). The proportions of mineral N and nitrate were both strongly influenced by the N deposition rate and by interactions with soil C content (Rowe et al. 2012). In soil, urea is rapidly hydrolysed to ammonium (NH4 +) ions, a part of which may be lost as ammonia (NH3 −) and subsequently as nitrous oxide (N2O) (Singh et al. 2013). The rate of denitrification was higher in soils incubated at saturation than in soils incubated at FC. Brown et al. (2012) found that denitrification was the dominant microbial source of N2O, and responded to increased soil water content and higher labile carbon availability. Elevated precipitation increased soil emissions of N2O, especially in combination with added nitrogen and heat. The reduction to N2 plus absorption by water primarily depends on soil properties, such as the availability of mineral N (substrate for nitrification and denitrification), soil oxygen and water content, soil temperature, pH and redox conditions, and the availability of labile organic C and N (Chapuis et al. 2007; Yan et al. 2012). Urease inhibitor increased the plant N uptake but did not result in a significant increase in herbage DM yields from urea fertiliser. Urease inhibitor was effective in reducing NH3 − and N2O emissions from both the urine and urea treatments, with the reduction in N2O emissions varying with plant N uptake (Singh et al. 2013). However, the effect of changing soil moisture on DR and N2O/N2 ratio may vary with the type of soil, its nutrient status and the management practices followed on the farm (Jha et al. 2012).
At wastewater irrigation sites, nitrate is formed during the irrigation events. The soil air is rapidly displaced by the wastewater and the soil becomes saturated. Denitrification and nitrous oxide emission rates are at a maximum during this period. As the site drains, the number of anoxic sites decreases and background rates re-establish (Russell et al. 1993). Peng et al. (2011) observed that peak N2O fluxes induced by N treatments were concentrated in short periods (2 to 3 weeks) after fertilisation in summer and in soil thawing periods in early spring. The weekly N2O emission was calculated for each crop, and we found significant differences among the crops used with two levels of AWW irrigation. Overall, the N2O fluxes were significantly higher in 100 % FC compared to 50 % FC (Fig. 6) probably due to a larger amount of AWW irrigation with nutrients (Table 1). The three N levels increased annual N2O emissions significantly (P < 0.05) in the medium and high N loading treatments compared with the control. A similar pattern was reported by Russell et al. (1993) in which peak rates at the pasture sites were higher with primary-treated effluent (1–137 g N2O-N ha/h) than with anaerobic effluent (1–62 g N2O-N ha/h). This was attributed to the higher organic carbon concentration in primary-treated effluent and possibly soil temperature peak nitrous oxide emission rates increased with increasing surface soil temperature. A study by Bhandral et al. (2007) found that among the N sources, the highest emissions were measured with nitrate application, emissions being ten times more than those from other N sources for compacted soil. Also, they reported that the soil compaction caused a sevenfold increase in the N2O flux; the total N2O fluxes for the entire experimental period ranged from 2.62 to 61.74 kg N2O-N/ha for the compacted soil and 1.12 to 4.37 kg N2O-N/ha for the uncompacted soil.
The N2O emissions were very low in pots with high DM yield as found in this study and shown in Table 5, possibly due to a high amount of nutrient uptake by the crops for growth and development. On the other hand, not-cropped conditions show a higher nutrient loss (high N2O emission fluxes recorded in this study) (Table 5). The increased N2O emission from the control pot was due to the absence of crop or nutrient utilisation by crops. A twofold increase in the N2O emission rate was recorded for the non-cropped pot compared to the cropped pot. In this short-duration study, major peaks were observed on the first 2 weeks after the treatment including the cropped and non-cropped control; this might have continued to increase for a few weeks after the treatment if we continued to irrigate the field at the same irrigation rate (Bhandral et al. 2007). Soils are the main sources of the greenhouse gas N2O. The N2O emission at the soil surface is the result of production and consumption processes of agricultural systems (Chapuis et al. 2007).
In grazed pastures, loss of N occurs mainly through ammonia (NH3 −) volatilisation, the release of gaseous N such as nitric oxide (NO) and nitrous oxide (N2O) through biological denitrification, and nitrate (NO3 −) leaching, which has both economic and environmental implications (Bolan et al. 2004). In the nearly water-saturated soil (90 % WFPS), N2O production was, as expected, dominated by conventional denitrification of NO3 − (Kool et al. 2011). A similar result was reported by Weitz et al. (2001) and Zhou et al. (2008) that N2O emission was insensitive to soil moisture, possibly resulting from lower soil nitrogen content (0.13–0.21 %). Differences were observed in nitrification–denitrification rates in flooding conditions because they control the availability of N and oxygen. Nitrification was controlled not only by oxygen but also by the amount of ammonium sulphate added and the high pH of the water; these two latter factors could increase the NH3 concentration, with inhibition of the coupled nitrification–denitrification (Carrasco et al. 2004). The results of Bateman and Baggs (2005) indicate N2O production during heterotrophic nitrification in our soil at 50 % WFPS and the possibility of aerobic denitrification. Thus, several processes may simultaneously produce N2O in soil at 60 % WFPS and below. Nitrification was the main source of N2O in soils at 35–60 % WFPS, indicating the significance of this process for global warming.
A nutrient-rich water source (e.g. AWW) can supply sufficient or a surplus amount of primary nutrients to the soil and plants. However, a significant amount of applied nutrient (N) is lost to air, soil and water through various processes by N2O and ammonia (NH3) emissions and nitrate leaching. These losses are considered environmental hazards due to the ill effects caused by N2O as a potential greenhouse gas and nitrate in groundwater as a potential water pollutant. Hence, minimising and management of nitrogenous pollutants become more important in the current era of sustainable agriculture. The mitigation options to reduce N loss include the use of nitrogenous inhibitors (NI), growing bioenergy crops and adopting efficient farm budgeting (applying nutrients only when it is necessary). The emission of nitrous oxide was high at the first 2 weeks after planting and declined subsequently reaching the background levels within 4 weeks. Nitrous oxide emission was higher for 100 % FC than for 50 % FC. The peak nitrous oxide emission flux was recorded in the non-cropping condition during the first week after planting; this indicated that there are higher risks of applied nitrogen loss through nitrous oxide emission in the absence of plant uptake of nitrogen. Overall, nitrous oxide emission was 15 % higher in the non-cropped than cropped treatments. The conclusions that can be drawn from this study in relation to plant-induced denitrification are as follows: the N2O emission rate increased with increasing soil moisture content (50 to 100 % FC), the N2O emission rate increased with increasing N input through chemical fertiliser or wastewater irrigation, the N2O emission rate decreased in the presence of plants and increased under the no-crop condition due to plant uptake and dry matter yield was found to be significantly higher in the 100 % FC than the 50 % treatment. The rate of N loss varied with the rate of AWW irrigation; nearly 1.2 to 2.4 % of applied N was lost as N2O. The main conclusions that can be drawn from this study in relation to plant-induced denitrification are the following: the N2O emission rate increased with increasing soil moisture content, the N2O emission rate increased with increasing N input through wastewater irrigation and the N2O emission rate decreased with the presence of plants which can be attributed to plant uptake.
The first author would like to thank the University of South Australia for the UPS scholarships and CRC for Contamination Assessment and Remediation of the Environment for the CRC-CARE scholarships to undertake research on nutrient management in abattoir wastewater-discharged soil.
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1.Future Industries InstituteUniversity of South AustraliaAdelaideAustralia
2.Global Centre for Environmental RemediationThe University of NewcastleAdelaideAustralia
3.Cooperative Research Centre for Contamination Assessment and Remediation of the EnvironmentSalisbury, AdelaideAustralia
Matheyarasu, R., Seshadri, B., Bolan, N.S. et al. Environ Sci Pollut Res (2016) 23: 22633. https://doi.org/10.1007/s11356-016-7438-y
Received 19 March 2016
Accepted 08 August 2016
First Online 24 August 2016
DOI https://doi.org/10.1007/s11356-016-7438-y
Publisher Name Springer Berlin Heidelberg | CommonCrawl |
\begin{document}
\title{Higher ramification and varieties of secant divisors on the generic curve}
\author[G. Farkas]{Gavril Farkas}
\address{Department of Mathematics, University of Texas, Austin, TX 78712} \email{{\tt [email protected]}} \thanks{Research partially supported by an Alfred P. Sloan Fellowship, the NSF Grants DMS-0450670 and DMS-0500747 and a 2006 Texas Summer Research Assignment} \thanks{2000 \sl{Mathematics Subject Classification} 14H10, 14C20}
\maketitle
\begin{center} {\small{ABSTRACT}} \end{center}
\begin{abstract} For a smooth projective curve, the cycles of $e$-secant $k$-planes are among the most studied objects in classical enumerative geometry and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. The aim of this paper is to clarify this problem in the case of the generic curve $C$ of given genus. We determine precisely under which conditions the cycle of $e$-secant $k$-planes in non-empty and we compute its dimension. We also precisely determine the dimension of the variety of linear series on $C$ carrying $e$-secant $k$-planes. \end{abstract}
\vskip 23pt
For a smooth projective curve $C$ of genus $g$, we denote by $C_e$ the $e$-th symmetric product of $C$ and by $G^r_d(C)$ the variety of linear series of type $\mathfrak g^r_d$ on $C$, that is, $$G^r_d(C):=\{(L, V): L\in \mbox{Pic}^d(C), V\in G(r+1, H^0(L))\}.$$ The main result of Brill-Noether theory states that if $[C]\in \mathcal{M}_g$ is a general curve then $G^r_d(C)$ is a smooth variety of dimension equal to $\rho(g, r, d):=g-(r+1)(g-d+r)$. For a linear series $l=(L, V)\in G^r_d(C)$ and an effective divisor $D\in C_e$, using the natural inclusion $H^0(C, L\otimes \mathcal{O}_C(-D))\subset H^0(C, L)$, we can define a new linear series $l(-D):=\bigl(L\otimes \mathcal{O}_C(-D), V\cap H^0(L\otimes \mathcal{O}_C(-D))\bigr)$. We fix integers $0\leq f< e$ and introduce the determinantal cycle $$V_e^{e-f}(l):=\{D\in C_e: \mbox{dim } l(-D)\geq r-e+f\}$$ of effective divisors of degree $e$ which impose at most $e-f$ independent conditions on $l$. If $l$ is very ample and we view $C\stackrel{l}\hookrightarrow {\textbf P}^r$ as an embedded curve, then $V_e^{e-f}(l)$ parameterizes $e$-secant $(e-f-1)$-planes to $C$. Each irreducible component of $V_e^{e-f}(l)$ has dimension at least $e-f(r+1-e+f)$. The cycles $V_e^{e-f}(l)$ have been extensively studied in classical enumerative geometry. The virtual class $[V_e^{e-f}(l)]^{{virt}}\in A^{f(r+1-e+f)}(C_e)$ has been computed by MacDonald and its expression is tremendously complicated and thus of limited practical use (see \cite{ACGH}, Chapter VIII). One case when we have a manageable formula is for $e=2r-2$ and $f=r-1$, when $[V_{2r-2}^{r-1}(l)]^{virt}$ computes the (virtual) number of $(r-2)$-planes in ${\textbf P}^r$ which are $(2r-2)$-secant to $C$ (cf. \cite{Ca}).
Surprisingly little is known about the validity of these classical enumerative formulas (see \cite{H} and \cite{LB} for partial results in the case of curves in ${\textbf P}^3$). The aim of this paper is to clarify this problem for a general curve $[C]\in \mathcal{M}_g$. For every linear series $l\in G^r_d(C)$ we determine precisely under which conditions the cycle $V_e^{e-f}(l)$ is non-empty and has the expected dimension. Then having fixed $[C]\in \mathcal{M}_g$, we determine the dimension of the family of linear series $l\in G^r_d(C)$ with an $e$-secant $(e-f-1)$-plane. For our first result, we use degeneration techniques together with a few facts about the ample cone of the moduli space $\overline{\mathcal{M}}_{0, g}$ to prove the following: \begin{theorem}\label{secant} Let $[C]\in \mathcal{M}_g$ be a general curve and we fix non-negative integers $0\leq f< e$,\ $r$ and $d$, such that $r-e+f\geq 0$. Then we have that $$\mathrm{dim }\{l\in G^r_d(C): V_e^{e-f}(l)\neq \emptyset\}\leq \rho(g, r, d)-f(r+1-e+f)+e.$$ In particular, if $\rho(g,r,d)-f(r+1-e+f)+e<0$, then $V_e^{e-f}(l)=\emptyset$, for every \ $l\in G^r_d(C)$. \end{theorem} \noindent More precisely, in Section 2 we prove the following dimensionality estimate $$\mbox{dim}\bigl\{(D, l)\in C_e\times G^r_d(C): D\in V_e^{e-f}(l)\bigr\}\leq \rho(g, r, d)-f(r+1-e+f)+e,$$ which obviously implies Theorem \ref{secant}. This result generalizes the Brill-Noether theorem. Indeed, when $l=K_C$, then $V_e^{e-f}(K_C)=C_e^f:=\{D\in C_e: h^0(\mathcal{O}_C(D))\geq f+1\}$. Since the fibres of the Abel-Jacobi map $C_e^f\rightarrow W_e^f(C)$ are at least $f$-dimensional, clearly $G_e^f(C)\neq \emptyset$ implies that $\mbox{dim } C_e^f\geq f$. Our result reads $G_e^f(C)=\emptyset$ when $\rho(g, f, e)<0$, which is
the non-existence part of the classical Brill-Noether theorem (cf. \cite{EH1}). More generally, we have the following result in the case $\rho(g, r, d)=0$: \begin{corollary} Suppose $\rho(g, r, d)=0$ and $e<f(r+1-e+f)$. Then for a general curve $[C]\in \mathcal{M}_g$ we have that $V_e^{e-f}(l)=\emptyset$ for every $l\in G^r_d(C)$, that is, no linear series of type $\mathfrak g^r_d$ on $C$ has any $e$-secant $(e-f-1)$-planes. \end{corollary}
An immediate consequence of Theorem \ref{secant} is a proof of the following conjecture of Coppens and Martens (cf. \cite{CM2} Theorem 3.3.1, for a proof in the case $f=1$):
\begin{corollary}\label{copm} Let $[C]\in \mathcal{M}_g$ be a general curve and we fix integers $0\leq f<e$, $d$ and $r$ such that $r-e+f\geq 0$. Let $l$ be a general linear series of type $\mathfrak g^r_d$ belonging to an irreducible component of $G^r_d(C)$. Assuming that $V_e^{e-f}(l)$ is not empty, then $e-f(r+1-e+f)\geq 0$. Moreover $V_e^{e-f}(l)$ is equidimensional and $\mathrm{dim } \ V_e^{e-f}(l)=e-f(r+1-e+f)$. \end{corollary}
We note that when $f=1$, Theorem \ref{secant} concerns the higher order very ampleness of linear series on a general curve. We recall that a linear series $l\in G^r_d(C)$ is said to
be $(e-1)$-very ample if $\mbox{dim } l(-p_1-\cdots -p_e)=r-e$, for any choice of (not necessarily
distinct) $e$ points $p_1, \ldots, p_e\in C$. Thus $0$-very ampleness is equivalent to generation
by global sections and $1$-very
ampleness reduces to the classical notion of very ampleness.
\begin{corollary}\label{eample} Let $[C]\in \mathcal{M}_g$ be a general curve and $e, r, d$ be non-negative integers such that
$\rho(g, r, d)+2e-2-r<0$. Then every linear series $l\in G^r_d(C)$ is $(e-1)$-very ample. \end{corollary}
Theorem \ref{secant} does not address the issue of existence of linear series with $e$-secant $(e-f-1)$-planes. We prove the following existence result for secant planes corresponding to linear series $\mathfrak g^r_d$ on an \emph{arbitrary} smooth curve of genus $g$.
\begin{theorem}\label{existence} Let $[C]\in \mathcal{M}_g$ be a general smooth curve and we fix integers $0\leq f< e\leq g$, $d$ and $r$, \ such that $f(r+1-e+f)\geq e$,\ $d\geq 2e-f-1$,\ $\mbox{ }g-d+r\geq 0$, $$\rho(g, r, d)-f(r+1-e+f)+e\geq 0 \mbox{ and } \rho(g, r-e+f, d-e)\geq 0.$$ Assume moreover that we are in one of the following situations: $$ (i)\ \mbox{ } 2f\leq e-1, \ (ii)\ \mbox{ }e=2r-2 \mbox{ and } f=r-1, \ \mbox{ } (iii)\ \mbox{ }e<2(r+1-e+f), \ \mbox{ or} $$ $$ \ (iv)\mbox{ } \ \rho(g, r, d)\geq f(r+1-e+f)-(g-d+r).$$ Then there exists a linear series $l\in G^r_d(C)$ such that $V_e^{e-f}(l)\neq \emptyset$. Moreover, one has that the following dimensionality statement: $$\mathrm{dim}\{(D, l)\in C_e\times G^r_d(C): D\in V_e^{e-f}(l)\}=\rho(g, r, d)-f(r+1-e+f)+e.$$ \end{theorem}
The inequalities $\rho(g, r-e+f, d-e)\geq 0$ and $\rho(g, r, d)+e-f(r+1-e+f)\geq 0$ are obvious necessary conditions for the existence of $l\in G^r_d(C)$ with $V_{e}^{e-f}(l)\neq \emptyset$ on a general curve $[C]\in \mathcal{M}_g$. To give an example, an elliptic quartic curve $C\subset {\textbf P}^3$ has no $3$-secant lines even though $\rho(g, r, d)+e-f(r+1-e+f)>0$ (note that $e=3$ and $f=1$ in this case). Theorem \ref{existence} is stated in the range $f(r+1-e+f)\geq e$, corresponding to the case when linear series $l\in G^r_d(C)$ with $V_e^{e-f}(l)\neq \emptyset$ are expected to be special in the Brill-Noether cycle $G^r_d(C)$. It is clear though that the methods of this paper can be applied to the case $e\geq f(r+1-e+f)$ as well. In that range however, when one expects existence of $e$-secant $(e-f-1)$-planes for every $l\in G^r_d(C)$, there are nearly optimal existence results obtained by using positivity for Chern classes of certain vector bundles in the style of \cite{FL}: For every curve $[C]\in \mathcal{M}_g$ and $l\in G^r_d(C)$, assuming that $d\geq 2e-1$\ and \ $e-f(r+1-e+f)\geq r-e+f$, one knows that $V_e^{e-f}(l)\neq \emptyset$ (cf. \cite{CM}, Theorem 1.2). For $l\in G^r_d(C)$ such that $g-d+r\leq 1$ (e.g. when $l$ is non-special), if we keep the assumption \ $e-f(r+1-e+f)\geq 0$, it is known that $V_e^{e-f}(l)\neq \emptyset$ if and only if $\rho(g, r-e+f, d-e)\geq 0$ (cf. \cite{ACGH}, pg. 356). This appears to be the only case when MacDonald's formula displays some positivity features that can be used to derive existence results on $V_e^{e-f}(l)$. In the case $l=K_C$, one recovers of course the existence theorem from classical Brill-Noether theory. We finally mention that Theorem \ref{existence} holds independent of the assumptions $(i)-(iii)$, whenever a certain transversality condition (\ref{assumption2}) concerning a general curve $[Y, p]\in \mathcal{M}_{e, 1}$ is satisfied (see Section 3 for details). Theorem \ref{existence} is then proved by verifying this condition (\ref{assumption2}) in each of the cases $(i)-(iii)$.
We now specialize to the case when $e=f(r+1-e+f)$ which is covered by Theorem \ref{existence}. One can write $r=(u-1)(f+1)$ and $e=uf$ for some $u\geq 1$, and we obtain the following result concerning the classical problem of existence of $uf$-secant secant $(uf-f-1)$-planes to curves in ${\textbf P}^r$:
\begin{corollary}\label{existence2} Let $C$ be a smooth curve of genus $g$. We fix integers $d, u, f\geq 2$ \ and assume that the inequalities \ $g\geq uf, \ d\geq 2uf-f-1$, \ $\rho(g, uf+u-f-1, d)\geq 0$ and $\rho(g, u-1, d-uf)\geq 0$ hold. Then there exists an embedding \ $C\subset {\textbf P}^{(u-1)(f+1)}$ with $\mathrm{deg}(C)=d$, such that $C$ has a \ $uf$-secant $(uf-f-1)$-plane. If moreover, $[C]\in \mathcal{M}_g$ is general in moduli, then the embedded curve $C\stackrel{l}\hookrightarrow {\textbf P}^{(u-1)(f+1)}$ corresponding to a general linear series $l\in G^{(u-1)(f+1)}_d(C)$ \ has only a finite number of \ $uf$-secant $(uf-f-1)$-planes. \end{corollary}
If $[C]\in \mathcal{M}_g$ is suitably general we can prove that the Cayley-Castelnuovo numbers predicting the number of $(2r-2)$-secant $(r-2)$-planes of a curve in $C\subset {\textbf P}^r$ have a precise enumerative meaning:
\begin{theorem}\label{cayley} Let $[C]\in \mathcal{M}_g$ be a general curve. We fix integers $d, r\geq 3$ such that $d\geq 3r-2$, \ $\rho(g, r, d)\geq \emptyset$ and $\rho(g, 1, d-2r+2)\geq 0$. Then if $C\stackrel{l}\hookrightarrow {\textbf P}^r$ is an embedding corresponding to a general linear series $l\in G^r_d(C)$, then $C$ has only finitely many $(2r-2)$-secant $(r-2)$-planes. Their number (counted with multiplicities) is $$C(d, g, r)=\sum_{i=0}^{r-1} \frac{(-1)^i}{r-i}{d-r-i+1\choose r-1-i }{d-r-i \choose r-1-i}{g\choose i}.$$ \end{theorem} A modern proof of the formula for $C(d, g, r)$ is due to MacDonald and appears in \cite{ACGH} Chapter VIII. The original formula is due to Castelnuovo (cf. \cite{Ca}). When $r=3$, we recover Cayley's formula for the number of $4$-secant lines of a smooth space curve $C\subset {\textbf P}^3$ of degree $d$ (cf. \cite{C}): $$C(d, g, 3)=\frac{1}{12} (d-2)(d-3)^2(d-4)-\frac{g}{2} (d^2-7d+13-g).$$
To make a historical remark, there have been various attempts to rigorously justify the so-called {\emph{functional method}} that Cayley (1863), Castelnuovo (1889) and Severi (1900) used to derive their enumerative formulas and to determine their range of applicability (see \cite{LB}, \cite{V}). For instance, Cayley's formula is shown to hold for an arbitrary smooth curve in ${\textbf P}^3$, provided that $C(d, g, 3)$ is defined as the degree of a certain $0$-cycle $\mbox{Sec}_4(C)$ in ${\textbf G}(1, 3)$ (cf. \cite{LB2}). The drawback of this approach is that it becomes very difficult to determine when this newly defined invariant is really enumerative. For instance Le Barz only shows that this happens for very special curves in ${\textbf P}^3$ (rational curves and generic complete intersections) and one of the aims of this paper is to establish the validity of such formulas for curves that are general with respect to moduli.
The second topic we study concerns ramification points of powers of linear series on curves. This question appeared first in a particular case in \cite{F}. We recall that for a pointed curve $[C, p]\in \mathcal{M}_{g, 1}$ and a linear series $l=(L, V)\in G^r_d(C)$, the \emph{vanishing sequence} of $l$ at $p$ $$a^l(p): a_0^l(p)<\ldots < a_r^l(p)\leq d$$ is obtained by ordering the set $\{\mbox{ord}_p(\sigma)\}_{\sigma\in V}$. The \emph{weight} of $p$ with respect to $l$ is defined as $w^l(p):=\sum_{i=0}^r (a_i^l(p)-i)$. One says that $p$ is a \emph{ramification point} of $l$ if $w^l(p)\geq 1$ and we denote by $R(l)$ the finite set of ramification points of $l$. If $[C, p]\in \mathcal{M}_{g, 1}$ and $\overline{\alpha}: 0\leq \alpha_0\leq \ldots \leq \alpha_r\leq d-r$ is a Schubert index of type $(r, d)$, the cycle $$G^r_d(C, p, \overline{\alpha}):=\{l\in G^r_d(C): a_i^l(p)\geq \alpha_i+i \mbox{ for } i=0\ldots r\}$$ can be realized as a generalized determinantal variety inside $G^r_d(C)$ having virtual dimension $\rho(g, r, d, \overline{\alpha}):=\rho(g, r, d)-\sum_{j=0}^r \alpha_j$. For a general pointed curve $[C, p]\in \mathcal{M}_{g, 1}$, it is known that the virtual dimension equals the actual dimension, that is, $$\mbox{dim } G^r_d(C, p, \overline{\alpha})=\rho(g, r, d, \overline{\alpha}) \mbox{ }\mbox{ (cf. \cite{EH2} Theorem 1.1).}$$ We address the following question: suppose $l=(L, V)\in G^r_d(C)$ is a linear series with a prescribed ramification sequence $\overline{\alpha}$ at a fixed point $p\in C$. Is then $p$ a ramification point of any of the powers $L^{\otimes n}$ for $n\geq 2$? If so, can we describe the sequence $a^{L^{\otimes n}}(p)$? One certainly expects that under suitable genericity assumptions on $C$ and $L$, the points in $\bigcup_{n\geq 1} R(L^{\otimes n})$ should be uniformly distributed on $C$. For example, it is known that for every $C$ and $L\in \mbox{Pic}^d(C)$, the set $\bigcup_{n\geq 1} R(L^{\otimes n})$ is dense in $C$ with respect to the classical topology (cf. \cite{N}). Silverman and Voloch showed that for any $L\in \mbox{Pic}^d(C)$ there exist finitely many points $p\in C$ such that the set $\{n\geq 1: p\in R(L^{\otimes n})\}$ is infinite (cf. \cite{SV}).
We prove that on a generic pointed curve $[C, p]$, a linear series $(L, V)$ and its multiples $L^{\otimes n}$ share no ramification points, that is $R(l)$ and $R(L^{\otimes n})$ are as transverse as they can be expected to be and moreover, the vanishing sequence $a^{L^{\otimes n}}(p)$ is close to being minimal: \begin{theorem}\label{powers} We fix a general pointed curve $[C, p]\in \mathcal{M}_{g, 1}$, integers $r, d\geq 1, n\geq 3$ and a Schubert index $\overline{\alpha}:0\leq \alpha_0\leq \ldots \leq \alpha_r\leq d-r$. We also set $m:=[(n+1)/2]$. Then for every linear series $l=(L, V)\in G^r_d(C, p, \overline{\alpha})$ and every positive integer $$a< nd-\rho(g, r, d, \overline{\alpha})-g-\bigl[\frac{g}{m}\bigr],$$ we have that $h^0(C, L^{\otimes n}(-ap))=h^0(C, L^{\otimes n})-a=nd+1-g-a$. In other words, $a_i^{L^{\otimes n}}(p)=i$ for $0\leq i\leq a-1$. \end{theorem} In the case $n=2$, when we compare $R(l)$ and $R(L^{\otimes 2})$ our results are sharper: \begin{theorem}\label{double} We fix a general pointed curve $[C, p]\in \mathcal{M}_{g, 1}$, integers $r, d\geq 1$ and a Schubert index $\overline{\alpha}: \alpha_0\leq \ldots \leq \alpha_r\leq d-r$. Then for every $(L, V)\in G^r_d(C, p, \overline{\alpha})$ and every positive integer $$a < \mathrm{max}\{2d+2-2g-\rho(g, r, d, \overline{\alpha})+\bigl[\frac{g-1}{2}\bigr], \ \mbox{ } 2d+2-2g-2\rho(g, r, d, \overline{\alpha})+2\bigl[\frac{g}{3}\bigr]\},$$ we have that $h^0(C, L^{\otimes 2}(-ap))=h^0(C, L^{\otimes 2})-a=2d+1-g-a$. \end{theorem}
Comparing the bounds on $a$ given in Theorems \ref{powers} and \ref{double} with the obvious necessary condition $a\leq nd-g+1$ which comes from the Riemann-Roch theorem, we see that our results are essentially optimal for relatively small values of $\rho(g, r, d, \overline{\alpha})$ when the linear series $(L, V)\in G^r_d(C, p, \overline{\alpha})$ have a strong geometric characterization. On the other hand, if for instance $\rho(g, r, d, \overline{\alpha})=g$, then $L\in \mbox{Pic}^d(C)$ and $p\in C$ are arbitrary and one cannot expect to prove a uniform result about the vanishing of $H^1(C, L^{\otimes n}(-a p))$.
Theorems \ref{powers} and \ref{double} concern line bundles $L$ with
prescribed ramification at a given point $p\in C$. Such bundles are of course very special in $\mbox{Pic}^d(C)$. If instead, we try to describe $\bigcup_{n\geq 1} R(L^{\otimes n})$ for a general line bundle $L\in \mbox{Pic}^d(C)$, the answer turns out to be particularly simple. We give a short proof of the following result: \begin{theorem}\label{genlb} Let $C$ be a smooth curve of genus $g$ and $L\in \rm{Pic}$$^d(C)$ a very general line bundle. \newline \noindent (1) All the ramification points of the powers $L^{\otimes n}$ are ordinary, that is, $w^{L^{\otimes n}}(p)\leq 1$ for all $p\in C$ and $n\geq 1$. \newline \noindent (2) $R(L^{\otimes a})\cap R(L^{\otimes b})=\emptyset$ for $a\neq b$, that is, a point $p\in C$ can be a ramification point for at most a single power of $L$. \end{theorem} After this paper has been written I have learnt that Theorem \ref{genlb} has also been proved independently by M. Coppens in \cite{Co}. I would like to thank the referee for a very careful reading of this paper and for pointing out that the initial proof of Theorem \ref{secant} was not complete.
\section{Ramification points of multiples of linear series} In this section we use the technique of limit linear series to prove Theorems \ref{powers} and \ref{double}. We start by fixing a Schubert index $\overline{\alpha}: 0\leq \alpha_0\leq \ldots \leq \alpha_r\leq d-r$ and two integers $a\geq 0, n\geq 2$. We also set $m:=[(n+1)/2]$.
We assume that for every $[C, p]\in \mathcal{M}_{g, 1}$ there exists a linear series $l=(L, V)\in G^r_d(C, p, \overline{\alpha})$ such that $H^0(K_C\otimes L^{\otimes (-n)}\otimes \mathcal{O}_C(ap))\neq 0$. By a degeneration argument we are going to show that this implies the inequalities \begin{equation}\label{ineq1} a\geq nd-g-\rho(g, r, d, \overline{\alpha})-\bigl[\frac{g}{m}\bigr],\ \ \mbox{ when }n\geq 3, \end{equation} \begin{equation}\label{ineq2} a\geq 2d+2-2g- \rho(g, r, d, \overline{\alpha})+\bigl[\frac{g-1}{2}\bigr], \ \ \end{equation} and \begin{equation} \label{numerology} a\geq 2d+2-2g-2\rho(g, r, d, \overline{\alpha})+2\bigl[\frac{g}{3}\bigr], \ \ \mbox{when } n=2. \end{equation} This will prove both Theorems \ref{powers} and \ref{double}.
We degenerate $[C, p]$ to a stable curve $[X_0:=E_0\cup_{p_1} E_1 \cup_{p_2} \ldots\cup_{p_{g-1}} E_{g-1}, p_0]$, where $E_i$ is a general elliptic curve, $p_i, p_{i+1}\in E_i$ are points such that $p_{i+1}-p_i\in \mbox{Pic}^0(E_i)$ is not a torsion class and moreover $E_i\cap E_{i+1}=\{p_{i+1}\}$ for $0\leq i\leq g-2$. Thus $X_0$ is a string of $g$ elliptic curves and the marked point $p_0$ specializes to a general point lying on the first component $E_0$. We also consider a $1$-dimensional family $\pi:\mathcal{X}\rightarrow B$ together with a section $\sigma:B\rightarrow \mathcal{X}$, such that $B=\mbox{Spec}(R)$ with $R$ being a discrete valuation ring having uniformizing parameter $t$. We assume that $\mathcal{X}$ is a smooth surface and that there exists an isomorphism between $X_0$ and $\pi^{-1}(0)$. Under this isomorphism we also assume that $\sigma(0)=p_0\in X_0$. Here $0\in B$ is the point corresponding to the maximal ideal of $R$ and we denote by $\eta$ and $\overline{\eta}$ the generic and geometric generic point of $B$ respectively. By assumption, there exists a linear series $l_{\overline{\eta}}=(L_{\overline{\eta}}, V_{\overline{\eta}})\in G^r_d(X_{\overline{\eta}}, \sigma(\overline{\eta}), \overline{\alpha}),$ such that $H^0(X_{\overline{\eta}}, \omega_{X_{\overline{\eta}}}\otimes L_{X_{\overline{\eta}}}^{\otimes (-n)}\otimes \mathcal{O}_{X_{\overline{\eta}}}(a\sigma(\overline{\eta})))\neq 0$. By possibly blowing up $\mathcal{X}$ at the nodes of $X_0$ and thus replacing the central fibre by a curve $X$ obtained from $X_0$ by inserting chains of smooth rational curves at the points $p_1, \ldots, p_{g-1}$, we may assume that $l_{\overline{\eta}}$ comes from a linear series $l_{\eta}=(L_{\eta}, V_{\eta})\in G^r_d(X_{\eta}, \sigma(\eta), \overline{\alpha})$ on the generic fibre $X_{\eta}$.
We denote by $l_{E_i}=(L_{E_i}, V_{E_i})\in G^r_d(E_i)$ the $E_i$-aspect of the limit linear series on $X$ induced by $l_{\eta}$: Precisely, if $\mathcal{L}$ is a line bundle on $\mathcal{X}$ extending $L_{\eta}$, then $L_{E_i}\in \mbox{Pic}^d(E_i)$ is the restriction to $E_i$ of the unique twist $\mathcal{L}_{E_i}$ of $\mathcal{L}$ along components of
$\pi^{-1}(0)$ such that $\mbox{deg}_Z(\mathcal{L}_{i |Z})=0$ for any irreducible component $Z\neq E_i$ of $\pi^{-1}(0)$ (see also \cite{EH1}, p. 348). Since we gave ourselves the freedom of blowing-up $\mathcal{X}$ at the nodes of $\pi^{-1}(0)$, we can also assume that $\{l_{E_i}\}_{i=0}^{g-1}$ constitutes a limit $\mathfrak g^r_d$ on $X_0$ which is obtained from a refined limit $\mathfrak g^r_d$ on $X$ by retaining only the aspects of the elliptic components of $X$. The compatibility relations between the vanishing orders of the $l_{E_i}$'s imply the following inequality between Brill-Noether numbers: \begin{equation}\label{bn} \rho(g, r, d, \overline{\alpha})\geq \rho(l_{E_0}, p_0, p_1)+\rho(l_{E_1}, p_1, p_2)+\cdots +\rho(l_{E_{g-2}}, p_{g-2}, p_{g-1})+\rho(l_{E_{g-1}}, p_{g-1}), \end{equation} where $\rho(l_{E_i}, p_i, p_{i+1}):=\rho(1, r, d)-w^{l_{E_i}}(p_i)-w^{l_{E_{i}}}(p_{i+1})$. By assumption, there exists a non-zero section $\rho_{\eta}\in H^0\bigl(X_{\eta}, \omega_{X_{\eta}}\otimes \mathcal{L}_{\eta}^{\otimes (-n)}\otimes
\mathcal{O}_{X_{\eta}}(a \sigma(\eta))\bigr)$. This implies that if we denote by $\tilde{\mathcal{L}}_i$ the unique line bundle on the surface $\mathcal{X}$ such that (1) $\tilde{\mathcal{L}}_{i | X_{\eta}}=L_{\eta}$, and (2) $\mbox{deg}_Z(\omega_X\otimes \tilde{\mathcal{L}_i}^{\otimes (-n)}\otimes \mathcal{O}_X(ap_0))=0$, for every component $Z$ of $X$ such that $Z\neq E_i$, then $H^0(E_i, \omega_{X}\otimes \tilde{\mathcal{L}}_{i}^{\otimes (-n)}\otimes \mathcal{O}_X(a p_0)\otimes \mathcal{O}_{E_i})\neq 0$. We set $$\mathcal{M}_i:=\omega_{\pi}\otimes \tilde{\mathcal{L}}_i^{\otimes (-n)}\otimes \mathcal{O}_{\mathcal{X}}(a\ \sigma(B))\in \mbox{Pic}(\mathcal{X}).$$ Then $\mathcal{M}_{i
|E_i}=\mathcal{O}_{E_i}\bigl((a+2i)\cdot p_i+(2g-2-2i)\cdot p_{i+1}\otimes L_{E_i}^{\otimes (-n)}\bigr)$ for all $0\leq i\leq g-1$. For each such $i$ we denote by $n_i$ the smallest integer such that $\tilde{\rho}_i:=t^{n_i}\rho_{\eta} \in \pi_*(\mathcal{M}_i)$ and we set
$$\rho_i:=\tilde{\rho}_{i |E_i} \in H^0(E_i, \mathcal{M}_{i |E_i}).$$ Thus $0\neq \rho_i \in H^0(E_i, \mathcal{O}_{E_i}((a+2i)\cdot p_i+(2g-2-2i)\cdot p_{i+1}\otimes L_{E_i}^{\otimes (-n)}))$ and in a way similar to \cite{EH1} Proposition 2.2, we can prove that \begin{equation}\label{limitlin}
\mbox{ord}_{p_i}(\rho_i)+\mbox{ord}{p_i}(\rho_{i-1})\geq 2g-2-nd+a=\mbox{deg}(\mathcal{M}_{i |E_i}). \end{equation} One also has the inequalities $\mbox{ord}_{p_i}(\rho_i)+\mbox{ord}_{p_{i+1}}(\rho_i)\leq 2g-2-nd+a$ (and similar inequalities when passing through the rational components of $X$), from which it follows that one can write down a non-decreasing sequence of vanishing orders \begin{equation}\label{sequence} 0\leq \mbox{ord}_{p_0}(\rho_0)\leq \mbox{ord}_{p_1}(\rho_1)\leq \ldots \leq \mbox{ord}_{p_i}(\rho_i)\leq \ldots \leq \mbox{ord}_{p_{g-1}}(\rho_{g-1}). \end{equation}
Since $\rho_{g-1}$ is a non-zero section of a line bundle of degree $2g-2-nd+a$ on $E_{g-1}$, we must have that $\mbox{ord}_{p_{g-1}}(\rho_{g-1})\leq 2g-2-nd+a$. This inequality will eventually lead to the bound on the constant $a$.
Let us suppose now that we have fixed one of the elliptic components of $X$, say $E_i$, such that $\rho(l_{E_i}, p_i, p_{i+1})=0$. By counting dimensions, we see that for every $0\leq j\leq r$ there exists a section $u_j\in V_{E_i}$ such that $\mbox{div}(u_j)\geq a_j^{l_{E_i}}(p_i)\cdot p_i+a_{r-j}^{l_{E_i}}(p_{i+1})\cdot p_{i+1}$. In particular, we have that $a_j^{l_{E_i}}(p_i)+a_{r-j}^{l_{E_i}}(p_{i+1})\leq d$. Since $p_{i+1}-p_i\in \mbox{Pic}^0(E_i)$ is not a torsion class, it follows that the equality $a_j^{l_{E_i}}(p_i)+a_{r-j}^{l_{E_i}}(p_{i+1})=d$ can hold for at most one value $0\leq j\leq r$. Because $\rho(l_{E_i}, p_i, p_{i+1})=0$, this implies that $$a_{j}^{l_{E_i}}(p_i)+a_{r-j}^{l_{E_i}}(p_{i+1})\geq d-1 \mbox{ for all
} \ 0\leq j\leq r,$$ and there exists precisely one such index $j$ such that $a_j^{l_{E_i}}(p_i)+a_{r-j}^{l_{E_i}}(p_{i+1})=d$. In this case we get that $\mbox{div}(u_j)=a_j^{l_{E_i}}(p_i)\cdot p_i+a_{r-j}^{l_{E_i}}(p_{i+1})\cdot p_{i+1}$, and for degree reasons we must have that $L_{E_i}=\mathcal{O}_{E_i}(a_j^{l_{E_i}}(p_i)\cdot p_i+a_{r-j}^{l_{E_i}}(p_{i+1})\cdot p_{i+1})\in \mbox{Pic}^d(E_i).$
To summarize, if $\rho(l_{E_i}, p_i, p_{i+1})=0$, then the vanishing sequence $a^{l_{E_{i+1}}}(p_{i+1})$ of the $E_{i+1}$-aspect of the limit $\mathfrak g^r_d$ on $X$, is obtained from the vanishing sequence $a^{l_{E_i}}(p_i)$ by raising all entries by $1$, except one single entry which remains unchanged. Thus, $a_j^{l_{E_i}}(p_i)=a_j^{l_{E_{i+1}}}(p_{i+1})$ for one index $0\leq j\leq r$ and $a_k^{l_{E_{i+1}}}(p_{i+1})=a_k^{l_{E_i}}(p_i)+1$ for $k\neq j$.
We now study what happens to the non-decreasing sequence (\ref{sequence}) as we pass through a component $E_i$ with $\rho(l_{E_i}, p_i, p_{i+1})=0$. Assume that $\mbox{ord}_{p_i}(\rho_i)=\mbox{ord}_{p_{i+1}}(\rho_{i+1}):=b$. This implies that $\mbox{ord}_{p_{i+1}}(\rho_i)=2g-2-nd+a-b$ and $$L_{E_i}^{\otimes n}=\mathcal{O}_{E_i}((a+2i-b)\cdot p_i+(nd-a+b-2i)\cdot p_{i+1})\in \mbox{Pic}^{nd}(E_i).$$ Because $\rho(l_{E_i}, p_i, p_{i+1})=0$, as we have seen, $L_{E_i}$ can be represented by an effective divisor which is supported only at $p_i$ and $p_{i+1}$. Precisely, we can write that $L_{E_i}=\mathcal{O}_{E_i}\bigl(a_j^{l_{E_i}}(p_i)\cdot p_i+a_{r-j}^{l_{E_i}}(p_{i+1})\cdot p_{i+1}\bigr)$ for a unique $0\leq j\leq r$. Since $L_{E_i}$ cannot admit two different representations by effective divisors supported only at $p_i$ and $p_{i+1}$, we must have that \begin{equation}\label{divisible} L_{E_i}=\mathcal{O}_{E_i}\Bigl(\frac{a+2i-b}{n}\cdot p_i+\frac{nd-a+b-2i}{n}\cdot p_{i+1}\Bigr). \end{equation}
In particular, we have that $(a+2i-b)/n \in \mathbb Z$ and $a_j^{l_{E_i}}(p_i)=(a+2i-b)/n $.
We consider a connected subcurve $Y\subset X$ containing $m+1$ elliptic components $E_i$ and we measure the increase in (\ref{sequence}) as we pass through the components of $Y$.
\begin{lemma}\label{inec} We fix $m:=[(n+1)/2]$ and integers $i$ and $b$ such that $bm\leq i\leq g-1$. We denote by $R(i):=\#\{0\leq l\leq i-1: \rho(l_{E_l}, p_{l}, p_{l+1})\geq 1\}$. Then the following inequality holds: $$\mathrm{ord}_{p_i}(\rho_i)+R(i)\geq b(m-1). $$ \end{lemma} \begin{proof} We proceed by induction on $b$. For $b=0$ there is nothing to prove. We set $b\geq 1$, $i:=(b-1)m$ and we assume that $\mbox{ord}_{p_i}(\rho_i)+R(i)\geq (b-1)(m-1)$. We are going to prove that the following inequality holds: \begin{equation}\label{ind} \mathrm{ord}_{p_{i+m}}(\rho_{i+m})-\mathrm{ord}_{p_i}(\rho_i)+R(i+m)-R(i)\geq m-1. \end{equation}
Assume this is not the case. Then there exist integers $0\leq l< j\leq m-1$ such that the following relations hold: \ $(i)\ \rho(l_{E_{i+l}}, p_{i+l}, p_{i+l+1})=\rho(l_{E_{i+j}}, p_{i+j}, p_{i+j+1})=0$ \ and $$(ii)\ \mbox{ord}_{p_{i+l}}(\rho_{i+l})=\mbox{ord}_{p_{i+l+1}}(\rho_{i+l+1}):=b, \mbox { } \ \mbox{ord}_{p_{i+j}}(\rho_{i+j})=\mbox{ord}_{p_{i+j+1}}(\rho_{i+j+1}):=c.$$ Using (\ref{divisible}) this implies that $$ L_{E_{i+l}}=\mathcal{O}_{E_{i+l}}\bigl(\frac{a+2i+2l-b}{n}\cdot p_{i+l}+\frac{nd-a+b-2i-2l}{n}\cdot p_{i+l+1}\bigr), \ \mbox{ and }$$ $$L_{E_{i+j}}=\mathcal{O}_{E_{i+j}}\bigl(\frac{a+2i+2j-c}{n}\cdot p_{i+j}+\frac{nd-a+c-2i-2j}{n}\cdot p_{i+j+1}\bigr).$$ In particular, $(2j-2l-c+b)/n \in \mathbb Z$, hence we can write $c=b-kn+2(j-l)$ for some $k\in \mathbb Z$. If $k\geq 1$, since $c\geq b$, we obtain that $m-1\geq j-l \geq n/2$, which is a contradiction. Therefore we must have that $k\leq 0$, and this holds for every pair $(j, l)$ satisfying (i) and (ii). We choose now the pair $0\leq l<j\leq m-1$ satisfying (i) and (ii) and for which moreover, the difference $j-l$ is maximal.
For each integer $0\leq e\leq l-1$ we have that either $\rho(l_{E_{i+e}}, p_{i+e}, p_{i+e+1})\geq 1$ or $\mbox{ord}_{p_{i+e+1}}(\rho_{i+e+1})>\mbox{ord}_{p_{i+e}}(\rho_{i+e})$. This fact leads to the inequality \begin{equation}\label{num1} \mbox{ord}_{p_{i+l}}(\rho_{i+l})-\mbox{ord}_{p_i}(\rho_i)+R(i+l)-R(i)\geq l. \end{equation} Similarly, by studying the subcurve of $Y$ containing $E_{i+j+1}, \ldots, E_{i+m-1}$, we find that \begin{equation}\label{num2} \mbox{ord}_{p_{i+m}}(\rho_{i+m})-\mbox{ord}_{p_{i+j+1}}(\rho_{i+j+1})+R(i+m)-R(i+j+1)\geq m-j-1.\end{equation} Finally, we look at the subcurve of $X$ containing $E_{i+l}, \ldots, E_{i+j}$ and we can write \begin{equation}\label{num3} \mbox{ord}_{p_{i+j}}(\rho_{i+j})-\mbox{ord}_{p_{i+l}}(\rho_{i+l})+R(i+j+1)-R(i+l)\geq c-b\geq 2(j-l)\geq j-l+1. \end{equation} By adding (\ref{num1}), (\ref{num2}) and (\ref{num3}) together we obtain (\ref{ind}) which proves the Lemma. \end{proof}
When $n=2$ we have a slightly better estimate than in the general case:
\begin{lemma}\label{inec2} ($n=2$) (1) Let $i$ be an integer such that $2b\leq i\leq g-1$. Then $\mathrm{ord}_{p_i}(\rho_i)+R(i)\geq b$. \newline \noindent (2) We fix $0\leq i\leq g-4$ and let $Y$ be a connected subcurve of $X$ containing precisely three elliptic curves $E_i, E_{i+1}$ and $E_{i+2}$. If $R(i+3)=R(i)$, that is, $$\rho(l_{E_i}, p_i, p_{i+1})=\rho(l_{E_{i+1}}, p_{i+1}, p_{i+2})=\rho(l_{E_{i+2}}, p_{i+2}, p_{i+3})=0,$$ then we have the inequality $\mathrm{ord}_{p_{i+3}}(\rho_{i+3})\geq \mathrm{ord}_{p_i}(\rho_i)+2.$ \end{lemma} \begin{proof} We only prove (2), the remaining statement being analogous to Lemma \ref{inec}. We may assume that $\mbox{ord}_{p_i}(\rho_i)=\mbox{ord}_{p_{i+1}}(\rho_{i+1}):=b$. Hence $(a+2i-b)/2 \in \mathbb Z$ and there exists an index $0\leq j\leq r$ such that $$a_j^{l_{E_i}}(p_i)=a_j^{l_{E_{i+1}}}(p_{i+1})=\frac{a+2i-b}{2}, \mbox{ while
} \ a_k^{l_{E_{i+1}}}(p_{i+1})=a_k^{l_{E_i}}(p_i)+1 \ \mbox{ for } k\neq j.$$ If $\mbox{ord}_{p_{i+2}}(\rho_{i+2})=\mbox{ord}_{p_{i+1}}(\rho_{i+1})=b$, then (\ref{divisible}) implies that $(a+2i+2-b)/2$ is an entry in the vanishing sequence $a^{l_{E_{i+1}}}(p_{i+1})$. But this is impossible, because $(a+2i-b)/2$ was an entry in the sequence $a^{l_{E_i}}(p_i)$, hence we must have that $\mbox{ord}_{p_{i+2}}(\rho_{i+2})\geq b+1$. Next, if $\mbox{ord}_{p_{i+3}}(\rho_{i+3})=b+1$, this implies that $\mbox{ord}_{p_{i+3}}(\rho_{i+3})=\mbox{ord}_{p_{i+2}}(\rho_{i+2})=b+1$, hence again $\bigl(a+2(i+2)-(b+1)\bigr)/2 \in \mathbb Z$, which is not possible for parity reasons. Thus we must have that $\mbox{ord}_{p_{i+3}}(\rho_{i+3})\geq b+2$. \end{proof}
\noindent {\emph{Proof of Theorem \ref{powers}.}} We complete the proof of our result in the case $n\geq 3$. We write $g=bm+c$ with $0\leq c\leq m-1$ and we set $i:=bm$. From Lemma \ref{inec} we obtain that $\mbox{ord}_{p_i}(\rho_i)+R(i)\geq b(m-1)$. Using the reasoning of Lemma \ref{inec} for the connected subcurve of $X$ which contains $E_i, E_{i+1}, \ldots, E_{i+c-1}=E_{g-1}$, we get that \begin{equation}\label{num4} \mbox{ord}_{p_{g-1}}(\rho_{g-1})-\mbox{ord}_{p_i}(\rho_i)+R(g-1)-R(i)\geq c-2. \end{equation} Using (\ref{num4}), together with the inequality $R(g-1)\leq \rho(g, r, d, \overline{\alpha})$, we can write that $$\mbox{deg}(K_C\otimes L^{\otimes (-n)}\otimes \mathcal{O}_C(ap))= 2g-2-nd+a\geq \mbox{ord}_{p_{g-1}}(\rho_{g-1})\geq g-\bigl[\frac{g}{m}\bigr]-\rho(g, r, d, \overline{\alpha})-2,$$ which finishes the proof of Theorem \ref{powers}.
$\Box$
\noindent{\emph{Proof of Theorem \ref{double}}}. From Lemma \ref{inec2} part (1), we obtain that $$\mbox{ord}_{p_{g-1}}(\rho_{g-1})+R(g-1)\geq [(g-1)/2].$$ Since $R(g-1)\leq \rho(g, r, d, \overline{\alpha})$, this leads to the inequality $a\geq 2d+2-2g+[(g-1)/2]-\rho(g, r, d, \overline{\alpha})$. To prove (\ref{numerology}) we divide $X$ into $e:=[g/3]+1$ connected subcurves $Y_1, \ldots, Y_{e}$ such that $Y_1, \ldots, Y_{e-1}$ each contain three elliptic components, $\#(Y_i\cap Y_{i+1})=1$ for all $1\leq i\leq e-2$ and $Y_e:=\overline{(\cup_{i=1}^{e-1} Y_i)^c}$. The curves $Y_i$ fall into two categories: those for which there exists an elliptic component $E_l\subset Y_i$ such that $\rho(l_{E_l}, p_l, p_{l+1})\geq 1$ (and there are at most $\rho(g, r, d, \overline{\alpha})$ such $Y_i$'s), and those for which $\rho(l_{E_l}, p_l, p_{l+1})=0$ for each elliptic component $E_l\subset Y_i$. Lemma \ref{inec2} part (2) gives that $\mbox{ord}_{p_{g-1}}(\rho_{g-1}) \geq 2([g/3]-\rho(g, r, d, \overline{\alpha}))$. This proves (\ref{ineq2}) and finishes the proof of Theorem \ref{double}.
$\Box$ \begin{remark} It is natural to ask how close to being optimal are the bounds we obtained above. For $\rho(g, r, d, \overline{\alpha})$ relatively small, when any $L\in G^r_d(C, p, \overline{\alpha})$ has a strong geometric characterization, the inequalities (\ref{ineq1}), (\ref{ineq2}) and (\ref{numerology}) are in fact optimal. To see an example, we set $g=3, r=3, d=6$ and $\rho(g, r, d, \overline{\alpha})=0$. Thus we look at $\mathfrak g^3_6$'s on a general $[C, p]\in \mathcal{M}_{3, 1}$ having ramification at $p$ equal to $(0\leq \alpha_0\leq \alpha_1\leq \alpha_2\leq \alpha_3\leq 3)$, where $\sum_{i=0}^3 \alpha_i=3$. Theorem \ref{double} gives us that $H^0(K_C\otimes L^{\otimes (-2)}\otimes \mathcal{O}_C(a\cdot p))=0$ for every integer $a\leq 9$. We show that this is optimal by noting that when $a=10$ and $\overline{\alpha}=(0, 0, 1, 2)$, we have that $$H^0(K_C\otimes L^{\otimes (-2)}\otimes \mathcal{O}_C(10p))\neq 0, \mbox{
for every }L\in G^3_6(C, p, \overline{\alpha}).$$ Indeed, any such linear series is of the form $L=K_C\otimes A^{\vee}\otimes \mathcal{O}_C(5 p)\in W^3_6(C)$, where $A\in W^1_3(C)$ is such that $h^0(A(-2p))\geq 1$. A non-hyperelliptic curve of genus $3$ has two such $\mathfrak g^1_3$'s. Precisely, if $z, t\in C$ are the two points the tangent line at $p$ to $C\stackrel{|K_C|} \hookrightarrow {\textbf P}^2$ meets $C$ again, then $A=\mathcal{O}_C(2p+z)$ or $A=\mathcal{O}_C(2p+t)$. Say, we choose $A=\mathcal{O}_C(2p+z)$. By direct calculation we obtain that $ L^{\otimes 2}\otimes \mathcal{O}_C(-10p)=K_C^{\otimes 2}\otimes A^{\otimes (-2)}=\mathcal{O}_C(2t)$, hence $h^0(K_C\otimes L^{\otimes (-2)}\otimes \mathcal{O}_C(10 p))=1$. \end{remark}
\section{Varieties of secant planes to the general curve} We fix a smooth curve $[C]\in \mathcal{M}_g$ and two integers $0\leq f<e$. In this section we study the varieties $V_e^{e-f}(l)$ of $e$-secant $(e-f-1)$-planes corresponding to a linear series $l\in G^r_d(C)$. We first define the correspondence $$\Sigma_C:=\{(D, l)\in C_e\times G^r_d(C): \mbox{dim } l(-D)\geq r-e+f\},$$ and denote by $\pi_1:\Sigma_C\rightarrow C_e$ and $\pi_2:\Sigma_C\rightarrow G^r_d(C)$ the two projections. We assume that $\Sigma_C\neq \emptyset$ for the general curve $[C]\in \mathcal{M}_g$. Under this assumption, we show that \begin{equation}\label{sec1} \mbox{dim}(\Sigma_C)\leq \rho(g, r,d)-f(r+1-e+f)+e. \end{equation} (We recall that the dimension of a scheme is the maximum of the dimensions of its irreducible components). Since $\Sigma_C$ is a determinantal subvariety of $C_e\times G^r_d(C)$, it follows that for a general $[C]\in \mathcal{M}_g$, if non-empty, the scheme $\Sigma_C$ is equidimensional and $\mbox{dim}(\Sigma_C)=\rho(g, r, d)-f(r+1-e+f)+e$. Note that this result does not establish the non-emptiness of $\Sigma_C$ which is an issue that we will deal with in Section 3. In any event, (\ref{sec1}) implies the dimensional estimate $$\mbox{dim}\{l\in G^r_d(C): V^{e-f}_e(l)\neq \emptyset\}\leq \rho(g, r, d)-f(r+1-e+f)+e. $$ This will prove Theorem \ref{secant} as well as Corollaries \ref{copm} and \ref{eample}.
We start by setting some notation. We denote by $j:\overline{\mathcal{M}}_{0, g}\rightarrow \overline{\mathcal{M}}_g$ the ``flag'' map obtaining by attaching to each stable curve $[R, x_1, \ldots, x_g]\in \overline{\mathcal{M}}_{0, g}$ fixed elliptic tails $E_1, \ldots, E_g$ at the points $x_1, \ldots, x_g$ respectively. Thus $j([R, x_1, \ldots, x_g]):=[\tilde{R}]=[R\cup_{x_1} E_1\cup \ldots \cup_{x_g} E_g]$ and for such a curve, we denote by $p_R:\tilde{R}\rightarrow R$ the projection onto $R$, that is, $p_R(E_i)=\{x_i\}$ for $1\leq i\leq g$. We denote by $\overline{\mathcal{C}}_{g, n}=\overline{\mathcal{M}}_{g,n+1}$ the universal curve and by $\pi:\overline{\mathcal{C}}_{g, n}\rightarrow \overline{\mathcal{M}}_{g, n}$ the morphism forgetting the $(n+1)$-st marked point. We write $\pi_e:\overline{\mathcal{C}}_{g,n}^e \rightarrow \overline{\mathcal{M}}_{g, n}$ for the $e$-fold fibre product of $\overline{\mathcal{C}}_{g, n}$ over $\overline{\mathcal{M}}_{g, n}$ and we introduce a map $\chi:\overline{\mathcal{M}}_{0, g}\times_{\overline{\mathcal{M}}_g} \overline{\mathcal{C}}_g^e\rightarrow \overline{\mathcal{C}}_{0, g}^e$ which collapses the elliptic tails. Thus $\chi$ is defined by $$\chi\bigl([R, x_1,\ldots, x_g], (y_1, \ldots, y_e)\bigr):=\bigl([R, x_1, \ldots, x_g], p_R(y_1), \ldots, p_R(y_e)\bigr),$$ for points $y_1, \ldots, y_e \in \tilde{R}$. Let $W\subset \overline{\mathcal{C}}_g^e$ be the closure of the locus $$\{[C, y_1, \ldots, y_e]\in \mathcal{C}_{g}^e: \exists l\in G^r_d(C)\mbox{ with } \mbox{ dim } l(-y_1-\cdots -y_e)\geq r-e+f\}.$$ By assumption $\pi_e(W)=\overline{\mathcal{M}}_g$ and we define the locus $U:=\chi\bigl(W\cap (\overline{\mathcal{M}}_{0, g}\times _{\overline{\mathcal{M}}_g}\overline{\mathcal{C}}_g^e)\bigr)$. Then $\pi_e(U)=\overline{\mathcal{M}}_{0, g}$ and we denote by $e-m$ the minimal fibre dimension of the map
$\pi_{e | U}: U\rightarrow \overline{\mathcal{M}}_{0, g}$. Thus $0\leq m\leq e$ and $\mbox{dim}(U\cap \pi_e^{-1}[R, x_1, \ldots, x_g])\geq e-m$, for every $[R, x_1, \ldots, x_g]$, with equality for a general point $[R, x_1, \ldots, x_g]\in \overline{\mathcal{M}}_{0, g}$.
We recall that for every choice of $4$ marked points $\{i, j, k, l\}\subset \{1, \ldots, g\}$, one has a fibration $\pi_{i j k l}:\overline{\mathcal{M}}_{0, g}\rightarrow \overline{\mathcal{M}}_{0, 4}$ obtained by forgetting the marked points with labels in the set $\{i, j, k, l\}^c$ and stabilizing the resulting rational curve. If we single out the first $3$ marked points $x_1, x_2, x_3$ as being $0, 1$ and $\infty$, in this way we obtain a birational map $\pi_{123}=(\pi_{1234},\ldots, \pi_{123i}, \ldots, \pi_{123g}):\overline{\mathcal{M}}_{0, g}\rightarrow \overline{\mathcal{M}}_{0, 4}^{g-3}=({\textbf P}^1)^{g-3}$ defined by $$\pi_{123}([R, x_1, \ldots, x_g]):=\bigl([R, x_1, x_2, x_3, x_4], [R, x_1, x_2, x_3, x_5], \ldots, [R, x_1, x_2, x_3, x_g]\bigr).$$ The map $\pi_{123}$ expresses $\overline{\mathcal{M}}_{0, g}$ as a blow-up of $({\textbf P}^1)^{g-3}$ such that all exceptional divisors of $\pi_{123}$ are boundary divisors of $\overline{\mathcal{M}}_{0, g}$ (cf. \cite{K}). In a similar manner, one has a birational map $f:\overline{\mathcal{C}}_{0, g}^e\rightarrow \overline{\mathcal{M}}_{0, 4}^{g-3+e}=({\textbf P}^1)^{g-3+e}$ defined by $f\bigl([R, x_1, \ldots, x_g], y_1, \ldots, y_e\bigr):=$ $$:=\bigl([R, x_1, x_2, x_3, x_4], \ldots, [R, x_1, x_2, x_3, x_g], [R, x_1, x_2, x_3, y_1], \ldots, [R, x_1, x_2, x_3, y_e]\bigr).$$ For simplicity, sometimes we write $f([R, x_1, \ldots, x_g], y_1, \ldots, y_e)=(x_4, \ldots, x_g, y_1, \ldots, y_e)$. The maps $f$ and $\pi_{123}$ fit in a commutative diagram, where $p_1: ({\textbf P}^1)^{g-3+e}\rightarrow ({\textbf P}^1)^{g-3}$ is the projection on the first $g-3$ factors: $$\begin{array}{ccc}
\overline{\mathcal{C}}_{0, g}^e& \stackrel{f}\longrightarrow & ({\textbf P}^1)^{g-3+e}=\overline{\mathcal{M}}_{0, 4}^{g-3+e} \\
\rmapdown{\pi_e} & \; & \rmapdown{p_1} \\
\overline{\mathcal{M}}_{0, g} & \stackrel{\pi_{123}}\longrightarrow & ({\textbf P}^1)^{g-3}=\overline{\mathcal{M}}_{0, 4}^{g-3} \\ \end{array}$$ Finally, for $2\leq k\leq e$ we define the diagonal loci $\Delta_k\subset ({\textbf P}^1)^{g-3+e}$ as consisting of those points $(x_4, \ldots, x_g, y_1, \ldots, y_e)$ for which at least $k$ of the points $y_1, \ldots, y_e$ coincide. We need the following result concerning existence of sublinear limit linear series of a fixed limit $\mathfrak g^r_d$, having prescribed vanishing sequence at a given point: \begin{lemma}\label{inductive} Let $X$ be a curve of compact type, $Y\subset X$ an irreducible component and let $p\in Y$ be a smooth point of $X$. Assume that $l$ is a (refined) limit $\mathfrak g^r_d$ on $X$ and let $(a_0< a_1< \ldots < a_r)$ be the vanishing sequence $a^l(p)$. We fix a subsequence $(a_{j_0}<a_{j_1}<\ldots <a_{j_b})$ of $a^l(p)$, where $0\leq b\leq r$. Then there exists a limit $\mathfrak g_d^b$ on $X$, say $ l'\subset l$, such that $a^{l'}(p)=(a_{j_0}, \ldots, a_{j_b})$. \end{lemma} \begin{proof}
Let us denote by $l:=\{l_Z=(L_Z, V_Z)\}_{Z\subset X}$ the original
limit $\mathfrak g^r_d$ on $X$. For each integer $0\leq k\leq b$ there
exists a section $\sigma_{j_k}\in V_Y$ such
that $\mbox{ord}_p(\sigma_{j_k})=a_{j_k}$. We consider the subspace
$W_Y:=<\sigma_{j_0}, \ldots, \sigma_{j_b}>\subset V_Y$. Since $\#\{\mbox{ord}_p(\sigma)\}_{\sigma \in W_Y}=b+1$,
we obtain that $\mbox{dim}(W_Y)=b+1$ and we set $l'_Y:=(L_Y, W_Y)\in G^b_d(Y)$. Suppose now that $Z$ is a
component of $X$ meeting $Y$ in a point $q$. We denote by $(c_{j_0}<c_{j_1}<\ldots <c_{j_b})$ the vanishing
sequence $a^{l'_Y}(q)$. Let $(e_{j_0}<e_{j_1}<\ldots <e_{j_b})$ be the complementary sequence, that is,
$e_{j_k}=d-c_{j_{b-k}}$ for each $0\leq k\leq b$. Then we can choose a section $\tau_k\in V_Z$ such that
$\mbox{ord}_q(\tau_k)=e_{j_k}$. We define $W_Z:=<\tau_0, \ldots, \tau_b>\subset V_Z$. Because all the entries $(e_{j_k})_{k=0}^b$ are distinct, we get that $\mbox{dim}(W_Z)=b+1$ and then set $l'_Z:=(L_Z, W_Z)\in G^b_d(Z)$. We continue inductively, and for each irreducible component $Z'\subset X$ we obtain an aspect $l_{Z'}'=(L_{Z'}, W_{Z'})\in G^b_d(Z')$. The collection $\{l_Z'\}_{Z\subset X}$ is the desired limit $\mathfrak g_d^b$ on $X$. \end{proof} Next we explain how the assumption that for every $[C]\in \mathcal{M}_g$ there exists a linear series $l\in G^r_d(C)$ with $V^{e-f}_e(l)\neq \emptyset$, can be used to construct a flag curve $\tilde{R}\in j(\overline{\mathcal{M}}_{0, g})$ such that all the $e$ points coming from the limit of an effective divisor $D\in V_e^{e-f}(l)$ specialize to a connected subcurve of $\tilde{R}$ having arithmetic genus at most $\mbox{min}\{g, e\}$. \begin{proposition}\label{schub} Let $U\subset \overline{\mathcal{C}}_{0, g}^e$ be an irreducible component of the closure of the locus of limits of $e$-secant divisors with respect to linear series $\mathfrak g^r_d$ on flag curves from $\overline{\mathcal{M}}_g$. Assuming that $\mathrm{dim}(U)=g-3+e-m$ \ with $0\leq m\leq e$, there exists a point $([R, x_1, \ldots, x_g], \tilde{y}_1, \ldots, \tilde{y}_e) \in W\cap(\overline{\mathcal{M}}_{0, g}\times_{\overline{\mathcal{M}}_g} \overline{\mathcal{C}}_g^e)$ corresponding to a genus $g$ flag curve $$\tilde{R}=R\cup_{x_1} E_1\cup \ldots \cup_{x_g} E_g \ \mbox{ and points } \tilde{y}_1, \ldots, \tilde{y}_e\in \tilde{R},$$ such that either (i) \ $\tilde{y}_1=\cdots =\tilde{y}_e\in R-\{x_1, \ldots, x_g\}$, or else, (ii)\ all the points $\tilde{y}_1, \ldots, \tilde{y}_e$ lie on a connected subcurve $Y\subset \tilde{R}$ satisfying $p_a(Y)\leq \mathrm{min}\{m, g\}$ and $\#(Y\cap (\overline{\tilde{R}-Y}))\leq 1$. \end{proposition} \begin{proof} We start by noting that if $m=0$ then $U=\overline{\mathcal{C}}_{0, g}^e$ and possibility (i) is satisfied. Thus we may assume that $m\geq 1$. First, we claim that $\mbox{dim } f(U)=\mbox{dim }U=g-3+e-m$. Indeed, since $\pi_e(U)=\overline{\mathcal{M}}_{0, g}$ it follows that $p_1(f(U))=({\textbf P}^1)^{g-3}$ and we choose a general point $t=(x_4,
\ldots, x_g)\in ({\textbf P}^1-\{0, 1, \infty\})^{g-3}$, such that $x_i\neq x_j$ for $i\neq j$. Then $\pi_e^{-1}(t)=({\textbf P}^1)^e$ and $f_{|
\pi_e^{-1}(t)}$ is an isomorphism onto its image, hence $f_{| U}$ is birational onto its image as well. Obviously, when $m\geq g$ we can take $Y=\tilde{R}$. From now on we shall assume that $1\leq m\leq g-1$.
Let us assume first that $f(U)\cap \Delta_e\neq \emptyset$. Then $\mbox{dim}\bigl( f(U)\cap \Delta_e\bigr)\geq g-m-2$. For dimension reasons, there must exist a point $z=(x_4, \ldots, x_g, y_1, \ldots, y_1)\in f(U)\cap \Delta_e$ such that either (i) at least $g-m-3$ of the points $x_j$ with $4\leq j\leq g$ are mutually distinct and belong to the set ${\textbf P}^1-\{0, 1, \infty, y_1\}$ and $y_1\in {\textbf P}^1-\{0, 1, \infty\}$, or (ii) at least $g-m-2$ of the $x_j$'s ($4\leq j\leq g$) are mutually distinct and belong to the set ${\textbf P}^1-\{0, 1, \infty, y_1\}$ and then $y_1\in {\textbf P}^1$ may, or may not be equal to one of the points $0, 1$ or $\infty$. Suppose we are in situation (i), the remaining case being similar.
We fix a point $([R, x_1, \ldots, x_g], y_1, \ldots, y_e)\in f^{-1}(z)$, hence $y_1, \ldots, y_e\in R$. If $Z\subset R$ denotes the minimal connected subcurve of $R$ containing all the points $y_1, \ldots, y_e$, then $x_1, x_2, x_3\in R-Z$, unless $y_1=\cdots =y_e$. (In the latter case either $y_1\in R-\{x_1, \ldots, x_g\}$ which corresponds to the situation when all the points $\tilde{y}_i=y_i$ specialize to the same smooth point of $\tilde{R}$ lying on the rational spine, or else, if $y_1=x_j$ for some $4\leq j\leq g$, then we can find a connected subcurve of $\tilde{R}$ of genus $1$ containing $\tilde{y}_1, \ldots, \tilde{y}_e$, where $p_R(\tilde{y}_i)=y_i$ for $1\leq i\leq e$). Since at least $g-m=3+(g-m-3)$ of the points $x_1, \ldots, x_g$ lie on $Z^c$, it follows that $\tilde{y}_1, \ldots, \tilde{y}_e$ lie on a connected subcurve of $\tilde{R}$ of genus $\leq m$, which completes the proof in this case.
We are left with the possibility $f(U)\cap \Delta_e=\emptyset$ and we denote by $k\leq e-1$ the largest integer for which $f(U)\cap \Delta_k\neq \emptyset$ and by $L$ an irreducible component of $f(U)\cap \Delta_{k}$. Since by definition $f(U)\cap \Delta_{k+1}=\emptyset$, it follows that there exists a point $t_0=(p_1, \ldots, p_e)\in ({\textbf P}^1)^{e}$ such that $L\subset ({\textbf P}^1)^{g-3}\times \{t_0\}$. In particular, the projection map
$p_{1 | L}:L\rightarrow p_1(L)$ is $1:1$ and then $\mbox{dim } p_1(L)=\mbox{dim}(L)\geq g-m+(e-k-2)\geq g-m$, unless $k=e-1$, when $\mbox{dim }p_1(L)\geq g-m-1$. In the first case it follows that there exists a point $(x_4, \ldots, x_g, p_1, \ldots, p_e)\in f(U)\cap \Delta_k$ such that at least $g-m$ of the points $x_4, \ldots, x_g$ are equal to a fixed point $r\in {\textbf P}^1-\{p_1, \ldots, p_e\}$. In the second case, that is, when $k=e-1$, since $\#\{p_i\}_{i=1}^e=2$, one of the points $0, 1$ or $ \infty$, say $0$, does not appear among the $p_i$'s. Then we can find a point $(x_4, \ldots, x_g, p_1, \ldots, p_e)\in f(U)\cap \Delta_{e-1}$ with at least $g-m$ of the $x_j$'s equal to $0$.
The conclusion in both cases is that there exists a point $\bigl([R, x_1, \ldots, x_g], y_1, \ldots, y_e\bigr)\in W\cap(\overline{\mathcal{M}}_{0, g}\times_{\overline{\mathcal{M}}_g} \overline{\mathcal{C}}_g^e)$ corresponding to the flag curve $\tilde{R}=R\cup_{x_1} E_1 \cup \ldots \cup_{x_g} E_g$, such that the points $y_1,\ldots, y_e$ lie on a connected subcurve $Y\subset \tilde{R}$ where $\#(Y\cap (\overline{\tilde{R}-Y}))\leq 1$ and $p_a(Y)\leq m\leq e$. \end{proof}
\noindent \emph{Proof of Theorem \ref{secant}.} We choose $\tilde{R}=R\cup_{x_1} E_1\cup \ldots \cup_{x_g} E_g$ as above and denote by $Y\subset \tilde{R}$ a connected subcurve onto which the points $y_1, \ldots, y_e$ specialize. We know that either (a) \ $p_a(Y)=m\leq \mathrm{min}\{e, g\}$, or (b) \ $y_1=\cdots =y_e\in R-\{x_1, \ldots, x_g\}$.
We first deal with case (a) and dispose of (b) at the end using \cite{EH2}. If $m<g$ we set $Z:=\overline{\tilde{R}-Y}$ and $\{p\}:=Y\cap Z$ and we denote by $Y'$ and $Z'$ the components of $Y$ and $Z$ respectively, containing the point $p$. When $m=g$, then necessarily $e\geq g$ and $Y:= \tilde{R}, Z=\emptyset$ and $p\in \tilde{R}$ is a general (smooth) point. By assumption, $[\tilde{R}, y_1, \ldots, y_e]\in W$, hence there exists a proper flat morphism $\phi:\mathcal{X}\rightarrow B$ satisfying the following properties:
\noindent $\bullet$ $\mathcal{X}$ is a smooth surface, $B$ is a smooth affine curve, $0\in B$ is a point such that $\phi^{-1}(0)$ is a curve stably equivalent to $\tilde{R}$ and $X_t=\phi^{-1}(t)$ is a smooth projective curve of genus $g$ for $t\neq 0$. Moreover, there are $e$ sections $\sigma_i:B\rightarrow \mathcal{X}$ of $\phi$ satisfying the condition $\sigma_i(0)=y_i\in \phi^{-1}(0)_{reg}$ for all $1\leq i\leq e$.
\noindent $\bullet$ If $X_{\eta}:=\mathcal{X}-\phi^{-1}(0)$, then there exists a line bundle $L_{\eta}\in \mbox{Pic}(X_{\eta})$ of relative degree $d$ and a subvector bundle $V_{\eta}\subset \phi_*(L_{\eta})$ having rank $r+1$, such that for $t\neq 0$ we have that $$\mbox{dim }V_t\cap H^0\bigl(X_t, L_t(-\sum_{j=1}^e \sigma_j(t))\bigr)= r+1-e+f. $$ After possibly making a finite base change and resolving the resulting singularities, the pair $(L_{\eta}, V_{\eta})$ induces a (refined) limit $\mathfrak g^r_d$ on $\tilde{R}$, which we denote by $\mathfrak l$. The vector bundle $V_{\eta}\cap \phi_*\bigl(L_{\eta}\otimes \mathcal{O}_{X_{\eta}}(-\sum_{j=1}^e \sigma_j(B-\{0\}))\bigr)$ induces a limit linear series $\mathfrak g^{r-e+f}_{d-e}$ on $\phi^{-1}(0)$ which we denote by $\mathfrak m$. For a component $A$ of $\phi^{-1}(0)$, if $(L_A, V_A)\in G^r_d(A)$ denotes the $A$-aspect of $\mathfrak l$, then there exists a unique \emph{effective} divisor $D_A\in A_e$ supported only at the points from $(A\cap \bigcup_{j=1}^e \sigma_j(B))\bigcup (A\cap \overline{\phi^{-1}(0)-A})$ such that the $A$-aspect of $\mathfrak m$ is of the form $$\mathfrak m_A=\bigl(M_A:=L_A\otimes \mathcal{O}_A(-D_A), \ W_A\subset V_A\cap H^0(M_A)\bigr)\in G^{r-e+f}_{d-e}(A).$$ The collection $\mathfrak m_Y:=\{\mathfrak m_A\}_{A\subset Y}$ forms a limit $\mathfrak g^{r-e+f}_{d-e}$ on $Y$. We denote by $(a_0< \ldots <a_r)$ the vanishing sequence of $\mathfrak l_{Y'}$ at $p$, thus $\{a_i \}_{i=0}^r=\{\mbox{ord}_p(\sigma)\}_{\sigma \in V_{Y'}}$ and we denote by $(b_0<\ldots <b_r)$ the vanishing sequence $a^{\mathfrak l_{Z'}}(p)$. By ordering the set $\{\mbox{ord}_p(\sigma)\}_{\sigma \in W_{Y'}}$ we obtain a subsequence $(a_{i_0}<\ldots <a_{i_{r-e+f}})$ of $a^{\mathfrak l_{Y'}}(p)$. When we order the entries in $\{a_i\}_{i=0}^r-\{a_{i_k}\}_{k=0}^{r-e+f}$ we obtain a new sequence $(a_{j_0}<a_{j_1}<\ldots <a_{j_{e-f-1}})$. Using Lemma \ref{inductive}, we find that there exists a limit linear series $\mathfrak l'_Y$ of type $\mathfrak g^{e-f-1}_{d}$ on $Y$ with the property that $a^{\mathfrak l'_Y}(p)=(a_{j_0}, a_{j_1}, \ldots, a_{j_{e-f-1}})$.
Let us assume first that we are in the situation $m<g$, hence $Z\neq \emptyset$. The point $p\in Y$ lies on a rational component which implies the following inequality corresponding to $Y$ (see also \cite{EH2}, Theorem 1.1): \begin{equation}\label{egy1} V_1:=\rho(m, e-f-1, d)-\sum_{k=0}^{e-f-1} a_{j_k}+{e-f\choose 2}\geq 0. \end{equation} Applying the same principle for the limit linear series $\mathfrak m_Y$ on $Y$, we find that the adjusted Brill-Noether number with respect to the point $p$ is non-negative: \begin{equation}\label{egy2} V_2:=\rho(m, r-e+f, d-e)-\sum_{k=0}^{r-e+f} a_{i_k}+{r+1-e+f\choose 2}\geq 0. \end{equation}
Next we turn our attention to $Z$ and use the fact that
the point $p\in Z$ does not lie on an elliptic component, hence
$[Z, p]$ satisfies the "strong" pointed Brill-Noether theorem:
\begin{equation}\label{egy3}
V_3:=\rho(g-m, r, d)-\sum_{k=0}^r b_k+{r+1\choose 2}\geq 0.
\end{equation}
If we add (\ref{egy1}), (\ref{egy2}) and (\ref{egy3}) together and
use that $\sum_{k=0}^r b_k+\sum_{k=0}^{r-e+f}
a_{i_k}+\sum_{k=0}^{e-f-1} a_{j_k}=(r+1)d$, we obtain the
inequality
$$\rho(g, r, d)-f(r+1-e+f)+e\geq e-m\geq 0.$$
The case $m=g$, when $Y=\tilde{R}$, is similar but simpler. We add together
(\ref{egy1}) and (\ref{egy2}) (now there is no
(\ref{egy3})) and we write the following inequalities:
$$\rho(g, r, d)+e-f(r+1-e+f)=\Bigl(\rho(g, r-e+f,
d-e)-\sum_{k=0}^{r-e+f} a_{i_k}+{r+1-e+f\choose
2}\Bigr)+$$
$$+\Bigl(\rho(g, e-f-1, d)-\sum_{k=0}^{e-f-1}a_{j_k}+{e-f\choose
2}\Bigr)+\sum_{k=0}^{r-e+f} a_{i_k}+\sum_{k=0}^{e-f-1}
a_{j_k}-{r+1\choose 2}+e-g\geq e-g\geq 0,$$
since $\sum_{k=0}^{r-e+f} a_{i_k}+\sum_{k=0}^{e-f-1} a_{j_k}\geq {r+1\choose 2}$. Thus we
obtain the same numerical conclusion as in the case $m<g$.
Assume now that we are in the case (b) when $y_1=\cdots =y_e\in
R-\{x_1, \ldots, x_g\}$. Then reasoning as above, we find a limit
$\mathfrak g^r_d$ on $\tilde{R}$ having vanishing sequence at $y_1$
at least $(0, 1, \ldots, e-f-1, e, e+1, \ldots, r+f-1, r+f)$. Using once
more \cite{EH2}, Theorem 1.1, we obtain the inequality $$\rho(g, r, d)+e-f(r+1-e+f)\geq \rho(g, r, d)-f(r+1-e+f)\geq 0.$$
Using the semicontinuity of the dimension of the fibres,
it follows that for a general curve $[C]\in \mathcal{M}_g$, if $\pi_1:\Sigma_C\rightarrow C_e$ is the first
projection, then the minimal fibre dimension of $\pi_1$ cannot exceed
the dimension of the space of pairs of limit linear series
$\mathfrak l\supset \mathfrak m$
consisting of a $\mathfrak g^r_d\supset \mathfrak g_{d-e}^{r-e+f}$ on the flag curve
$\phi^{-1}(0)$ such that $\mathfrak m=\mathfrak l(-D_e)$, where $D_e$ is a degree $e$ effective divisor
on $\phi^{-1}(0)$ with the property that $\mbox{supp}(D_e)\subset Y\cap \phi^{-1}(0)_{reg} $. Since the map
$(\mathfrak l\supset \mathfrak m, \mathfrak m_Y, \mathfrak l_Y')\mapsto (\mathfrak m_Y, \mathfrak l_Y', \mathfrak l_Z)\in \tilde{G}_{d-e}^{r-e+f}(Y)\times \tilde{G}_{d}^{e-f-1}(Y)\times \tilde{G}_{d}^r(Z)$ is injective, it follows that for a general
divisor $D_{gen}\in \pi_1(\Sigma_C)$ we have the estimate
$$\mbox{dim }\pi_1^{-1}(D_{gen})\leq
V_1+V_2+V_3=\rho(g, r, d)-f(r+1-e+f)+m,$$ hence
$\mbox{dim}(\Sigma_C)=\mbox{dim }\pi_1^{-1}(D_{gen})+e-m\leq \rho(g,
r,d)-f(r+1-e+f)+e$. This finishes the proof of Theorem \ref{secant}. $
$ $\Box$
\section{Existence of linear series with secant planes}
We turn our attention to showing existence of linear series which possess $e$-secant $(e-f-1)$-planes. The strategy we pursue is to construct limit linear series $\mathfrak g^r_d$ on a curve of compact type $[Y\cup_p Z]\in \overline{\mathcal{M}}_g$, where $(Y, p)$ and $(Z, p)$ are suitably general smooth pointed curves of genus $e$ and $g-e$ respectively. These limit $\mathfrak g^r_d$'s will carry a sublinear series $\mathfrak g_{d-e}^{r-e+f}=\mathfrak g^r_d(-D_e)$, where $D_e$ is a degree $e$ effective divisor on $Y$. Like in the proof of Theorem \ref{secant}, such $\mathfrak g^r_d$'s are determined by their $Z$-aspect and by a pair of linear series $(\mathfrak g_{d-e}^{r-e+f}, \mathfrak g_d^{e-f-1})$ on $Y$. We determine the dimension of the space of such pairs, which will enable us to show that the original pair $(\mathfrak g_{d-e}^{r-e+f}, \mathfrak g_d^{e-f-1})$ on $Y\cup_p Z$ can be smoothed to every smooth curve of genus $g$. This will finish the proof of Theorem \ref{existence}.
We start by choosing two general pointed curves $[Y, p]\in \mathcal{M}_{e, 1}$ and $[Z, p]\in \mathcal{M}_{g-e, 1}$ such that both $(Y, p)$ and $(Z, p)$ satisfy the Brill-Noether theorem with prescribed ramification (cf. \cite{EH2}, Theorem 1.1 and Proposition 1.2): If $\overline{\alpha}: 0\leq \alpha_0\leq \ldots \leq \alpha_r\leq d-r$ is a Schubert index of type $(r, d)$, then $(Y, p)$ possesses a $\mathfrak g^r_d$ with ramification sequence $\geq \overline{\alpha}$ at the point $p$, if and only if \begin{equation}\label{eisharris} \sum_{i=0}^r \mathrm{max}\{\alpha_i+g(Y)-d+r, 0\}\leq g(Y). \end{equation} In case this inequality is satisfied, then $\mbox{dim } G^r_d(Y, p, \overline{\alpha})=\rho(g, r, d, \overline{\alpha})$ (One obviously has a similar statement for $[Z, p]$).
We denote by $\pi:\mathcal{X}\rightarrow (T, 0)$ the versal deformation space of the stable curve $\pi^{-1}(0)=X_0:=Y\cup_p Z$. Let $\Delta\subset T$ be the boundary divisor corresponding to singular curves, and we write $\pi^{-1}(\Delta)=\Delta_e+\Delta_{g-e}$, where $\Delta_e$ (resp. $\Delta_{g-e}$) is the divisor corresponding to the marked point lying on the component of genus $e$ (resp. $g-e$). We consider the $e$-fold fibre product $\mathcal{U}:=(\mathcal{X}-\Delta_{g-e})\times_T \cdots \times_T (\mathcal{X}-\Delta_{g-e})$, the projection $\phi:\mathcal{U}\rightarrow T$ and the induced curve $p_2:\mathcal{X}\times_T \mathcal{U}\rightarrow \mathcal{U}$. Then we introduce the stack of limit linear series of type $\mathfrak g^r_d$ over $\mathcal{U}$ $$\sigma:\widetilde{\mathfrak{G}}_d^r(\mathcal{X}\times_T \mathcal{U}/\mathcal{U})\rightarrow \mathcal{U}, \mbox{ where }\widetilde{\mathfrak{G}}_d^r(p_2)=\widetilde{\mathfrak{G}}_d^r(\mathcal{X}\times_T \mathcal{U}/\mathcal{U})=\widetilde{\mathfrak{G}}^r_d(\pi)\times_T \mathcal{U},$$ and we write $\tau:=\phi\circ \sigma:\widetilde{\mathfrak{G}}_d^r(p_2)\rightarrow T$ (see \cite{EH1} Theorem 3.4, for details on the construction of $\widetilde{\mathfrak G}^r_d(\pi)$). The fibre $\tau^{-1}(t)$ corresponding to a point $t\in \Delta$ (in which case one can write
$\pi^{-1}(t)=Y_t\cup Z_t$, with $g(Y_t)=e, g(Z_t)=g-e$), parameterizes limit $\mathfrak g^r_d$'s on $Y_t\cup Z_t$ together with $e$-tuples $(x_1, \ldots, x_e)\in (Y_t-Y_t\cap Z_t)^e$. Let us denote by $\mathcal{L}_Y$ a degree $d$ Poincar\'e bundle on $\pi_2:\mathcal{X}\times _T \widetilde{\mathfrak{G}}_d^r(p_2)\rightarrow \widetilde{\mathfrak G}_d^r(p_2)$ characterized by the property that its restriction to curves of type $Y_t\cup Z_t$ are line bundles of bidegree $(d, 0)$. We also write $\mathcal{V}_Y\subset (\pi_{2})_* (\mathcal{L}_Y)$ for the rank $r+1$ tautological bundle whose fibres correspond to the global sections of the genus $e$-aspect of each limit $\mathfrak g^r_d$. Finally, for $1\leq j\leq e$, we denote by $D_j\subset \mathcal{X}\times_T \widetilde{\mathfrak G}^r_d(p_2)$ the diagonal divisor corresponding to pulling back the diagonal under the map $\mathcal{X}\times_T \widetilde{\mathfrak G}^r_d(p_2)\rightarrow \mathcal{X}\times_T \mathcal{X}$ which projects onto the $j$-th factor, that is, $(x, l, x_1, \ldots, x_e)\mapsto (x, x_j)$ where $x, x_1, \ldots, x_e\in \pi^{-1}(t)$. There exists an evaluation vector bundle morphism over $\widetilde{\mathfrak G}^r_d(p_2)$ $$\chi:\mathcal{V}_Y\rightarrow (\pi_{2})_*(\mathcal{L}_Y\otimes \mathcal{O}_{\sum_{j=1}^e D_j})$$ and we denote by $\mathcal{H}$ the rank $e-f$ degeneracy locus of the map $\chi$. Set-theoretically, $\mathcal{H}$ consists of those points $(t, l, x_1, \ldots, x_e)$ with $\phi(x_1, \ldots, x_e)=t\in T$ and $l\in \widetilde{G}^r_d(\pi^{-1}(t))$, satisfying the condition that $\mbox{dim } l(-x_1-\cdots -x_e)\geq r+1-e+f$. The dimension of every irreducible component of $\mathcal{H}$ is at least $ \rho(g, r, d)+\mbox{dim }T+e-f(r+1-e+f)$.
In order to show that $\tau:\mathcal{H}\rightarrow T$ is dominant, it suffices to prove that $\tau^{-1}(0)$ has at least one irreducible component of dimension $\rho(g, r, d)+e-f(r+1-e+f)$. This in fact will prove the stronger statement that $\Sigma_C\neq \emptyset$ for \emph{every} $[C]\in \mathcal{M}_g$. Indeed, even though $\tau:\widetilde{\mathfrak G}^r_d(p_2)\rightarrow T$ is not a proper morphism, the restriction $\tau_{\tau^{-1}(T-\Delta)}:\tau^{-1}(T-\Delta)\rightarrow T-\Delta$ is proper, hence there exists an irreducible component of $\mathcal{H}$ which maps onto $T-\Delta$. Since $\pi:\mathcal{X}\rightarrow (T, 0)$ can be chosen in such a way that there exists a point $t\in T$ with $\pi^{-1}(t)\cong C$, this proves our contention. We set the integer $$\alpha_0:=\Bigl[\frac{\rho(e, r-e+f, d-e)}{r+1-e+f}\Bigr]=\Bigl[\frac{e}{r+1-e+f}\Bigr]+d-r-f-e,$$ thus we can write $\rho(e, r-e+f, d-e)=\alpha_0\cdot (r+1-e+f)+c$, where $0\leq c\leq r-e+f$. Then there exists a unique Schubert index of type $(r-e+f, d-e)$, $$\overline{\alpha}:0\leq \alpha_0\leq \alpha_1\leq \ldots \leq \alpha_{r-e+f} \leq d-r-f,$$ with $\alpha_{r-e+f}-\alpha_0\leq 1$, such that $\sum_{j=0}^{r-e+f} \alpha_j=\rho(e, r-e+f, d-e)$. We have that $\alpha_j=\alpha_0$ for $0\leq j\leq r-e+f-c$ and $\alpha_j=\alpha_0+1$ for $r-e+f-c+1\leq j\leq r-e+f$. Note that since $\alpha_0+g(Y)-(d-e)+r-e+f=[e/(r+1-e+f)]\geq 0$, condition (\ref{eisharris}) is verified and the variety $G^{r-e+f}_{d-e}(Y, p, \overline{\alpha})$ is non-empty of dimension $\rho(e, r-e+f, d-e)-\sum_{j=0}^{r-e+f}\alpha_j=0$.
Next we set $\beta_0:=[e/(e-f)]$ and write \ $e=\beta_0\cdot (e-f)+\tilde{c}$, where $0\leq \tilde{c} \leq e-f-1.$ Then there exists a unique Schubert index of type $(e-f-1, 2e-f-1)$ $$\overline{\beta}:0\leq \beta_0\leq \beta_1\leq \ldots \leq \beta_{e-f-1}\leq e,$$ such that $\beta_{e-f+1}-\beta_0\leq 1$ and $\sum_{j=0}^{e-f-1} \beta_j=e$. Precisely, $\beta_j=\beta_0$ for $0\leq j\leq e-f-\tilde{c}-1$ and $\beta_j=\beta_0+1$ for $e-f-\tilde{c}\leq j\leq e-f-1$. By (\ref{eisharris}), the variety $G^{e-f-1}_{2e-f-1}(Y, p, \overline{\beta})$ is non-empty and of dimension $e-\sum_{j=0}^{e-f-1}\beta_j=0$.
First we are going to prove Theorem \ref{existence} under the assumption that there exist two linear series $(A, W_A)\in G^{r-e+f}_{d-e}(Y, p, \overline{\alpha})$ and $(L, W_L)\in G^{e-f-1}_{2e-f-1}(Y, p, \overline{\beta})$ satisfying the condition \begin{equation}\label{assumption2}
H^0\bigl(Y, L\otimes A^{\vee}\otimes \mathcal{O}_Y((d+f-2e)\cdot
p)\bigr)=0. \end{equation} Note that $\mbox{deg}\bigl(L\otimes A^{\vee}\otimes \mathcal{O}_Y((d+f-2e)\cdot p\bigr)=g(Y)-1$, and (\ref{assumption2}) states that a suitable translate of at least one of the finitely many line bundles of type $L\otimes A^{\vee}$ lies outside the theta divisor of $Y$.
\begin{remark} Condition (\ref{assumption2}) is a subtle statement concerning $[Y, p]$. It is not true that (\ref{assumption2}) holds for \emph{every} choice of $(A, W_A)\in G_{d-e}^{r-e+f}(Y, p, \overline{\alpha})$ and $(L, W_L)\in G_{2e-f-1}^{e-f-1}(Y, p, \overline{\beta})$. For instance, in the case $e=2r-2$ and $f=r-1$, corresponding to $(2r-2)$-secant $(r-2)$-planes which every curve $Y\subset {\textbf P}^r$ is expected to possess in finite number, we obtain that $A=B\otimes \mathcal{O}_Y((d-3r+2)\cdot p)$, where $B\in W^1_r(Y)$ and $L\otimes \mathcal{O}_Y(-2p)\in W^{r-2}_{3r-6}(Y)$. By Riemann-Roch, we can write that $L=K_Y\otimes \mathcal{O}_Y(2\cdot p)\otimes \tilde{B}^{\vee}$, where $\tilde{B}\in W^1_r(Y)$ and then (\ref{assumption2}) translates into the vanishing statement $H^0(Y, B\otimes \tilde{B}\otimes \mathcal{O}_Y(-3\cdot p))=0$. The curve $Y$ has $\frac{(2r-2)!}{r! (r-1)!}$ pencils $\mathfrak g^1_r$. If we choose $B\neq \tilde{B}\in W^1_r(Y)$, then $h^0(Y, B\otimes \tilde{B})\geq 4$ and (\ref{assumption2}) has no chance of being satisfied. If $B=\tilde{B}$, then the Gieseker-Petri theorem implies that the map $H^0(Y, B)\otimes H^0(Y, K_Y\otimes B^{\vee})\rightarrow H^0(Y, K_Y)$ is an isomorphism, whence $h^0(Y, B^{\otimes 2})=3$. Choosing $p\in Y$ outside the set of ramification points of the finitely many line bundles $B^{\otimes 2}$ where $B\in W^1_r(Y)$, we obtain that $H^0(B^{\otimes 2}\otimes \mathcal{O}_Y(-3\cdot p))=0$. Therefore in this case, condition $(\ref{assumption2})$ is equivalent to the Gieseker-Petri theorem. \end{remark}
We shall study when (\ref{assumption2}) is actually satisfied. We note that by the Riemann-Roch theorem, (\ref{assumption2}) also implies that $h^0\bigl(Y, L\otimes A^{\vee}\otimes \mathcal{O}_Y((d+f-2e+1)\cdot p)\bigr) =1$.
Assuming that $(A, W_A)\in G^{r-e+f}_{d-e}(Y, p, \overline{\alpha})$ and $(L, W_L)\in G^{e-f-1}_{2e-f-1}(Y, p, \overline{\beta})$ satisfy (\ref{assumption2}), it follows from Riemann-Roch that there exists a unique effective divisor of degree $e$
$$D\in |L\otimes \mathcal{O}_Y((d-2e+f+1)\cdot p)\otimes A^{\vee}|,$$ and moreover $p\notin \mbox{supp}(D)$. We introduce the space of sections $$V_Y:=W_A+W_L\subset H^0\bigl(Y, L\otimes \mathcal{O}_Y((d-2e+f+1)\cdot p)\bigr),\ \mbox{ where we view }$$ $$ W_A\subset H^0\bigl(L\otimes \mathcal{O}_Y((d-2e+f+1)\cdot p-D)\bigr) \mbox{ and } W_L\subset H^0(L)\subset H^0\bigl(L\otimes \mathcal{O}_Y((d-2e+f+1)\cdot p)\bigr).$$
We claim that $\mbox{dim}(V_Y)=r+1$, hence $\mathfrak{l}_Y=(L\otimes \mathcal{O}_Y((d-2e+f+1)\cdot p), V_Y)\in G^r_d(Y)$. Moreover, $\mathfrak{l}_Y$ has the following vanishing sequence at $p$: \begin{equation}\label{vanseq1} a^{\mathfrak{l}_Y}(p)=(\alpha_0, \ldots, \alpha_{r-e+f}+r-e+f, \beta_0+d-2e+f+1,\beta_1+d-2e+f+2, \ldots, \beta_{e-f-1}+d-e). \end{equation} Indeed, our original assumption $f(r+1-e+f)\geq e$ is equivalent with the inequality $\alpha_{r-e+f}+r-e+f<d-2e+f+1$, which shows that the sequence (\ref{vanseq1}) contains $r+1$ distinct entries. Since $p\notin \mbox{supp}(D)$, we obtain that the vanishing orders of the sections from $W_A\subset H^0(L\otimes \mathcal{O}_Y((d-2e+f+1)\cdot p))$ are precisely $$\alpha_0, \alpha_1+1, \ldots, \alpha_{r-e+f}+r-e+f,$$ while those of the sections from $W_L\subset H^0(L\otimes \mathcal{O}_Y((d-2e+f+1)\cdot p))$ are precisely $$\beta_0+d-2e+f+1, \beta_1+d-2e+f+2, \ldots, \beta_{e-f-1}+e-f-1+d-2e+f+1=\beta_{e-f-1}+d-e.$$ We have found $r+1$ sections from $V_Y$ having distinct vanishing orders at the point $p$, hence $\mbox{dim}(V_Y)=r+1$. Moreover, $a^{{\mathfrak l}_Y}(p)$ is equal to the sequence (\ref{vanseq1}).
Next we choose a linear series $\mathfrak l_Z\in G^r_d(Z, p)$ such that $\{\mathfrak l_Y, \mathfrak l_Z\}$ is a refined limit $\mathfrak g^r_d$. Then the ramification sequence of $\mathfrak l_Z$ at the point $p$ must be equal to $$\alpha^{\mathfrak l_Z}(p)=\overline{\gamma}:=(e-\beta_{e-f-1}, e-\beta_{e-f-2}, \ldots, e-\beta_0, d-r-\alpha_{r-e+f}, \ldots, d-r-\alpha_1, d-r-\alpha_0).$$ We claim that condition (\ref{eisharris}) is satisfied for $Z$ and that the variety $G^r_d(Z, p, \overline{\gamma})$ is non-empty and of dimension $\rho(g-e, r, d, \overline{\gamma})=\rho(g, r, d)+e-f(r+1-e+f)$. For this to happen, one has to check that the following inequality holds: \begin{equation}\label{nou} \sum_{j=0}^r \mathrm{max}\{\alpha^{\mathfrak l_Z}_j(p)+g-e-d+r, 0\}\leq g-e. \end{equation}
There are two things to notice: First, that by direct computation we have that
$$\alpha^{\mathfrak l_Z}_{e-f}(p)+g-e-d+r=g-e-\alpha_{r-e+f}=(g-d+r)+\bigl[f-\frac{e}{r+1-e+f}\bigr]\geq 0,$$ hence $\alpha^{\mathfrak l_Z}_{j}(p)+g-e-d+r\geq 0$ for all $e-f\leq j\leq r$. Second, that since $0\leq \beta_{e-f-1}-\beta_0\leq 1$, in order to estimate the sum of the first $e-f$ terms in the sum (\ref{nou}), there are two cases to consider. Either $\alpha^{\mathfrak l_Z}_0(p)+g-e-d+r\geq 0$, in which case we find that $$\sum_{j=0}^r \mathrm{max}\{\alpha^{\mathfrak l_Z}_j(p)+g-e-d+r, 0\}=\sum_{j=0}^r (\alpha^{\mathfrak l_Z}_j(p)+g-e-d+r)=$$ $$=g-e-\rho(g-e, r, d, \overline{\gamma})= g-e-\bigl(\rho(g, r, d)+e-f(r+1-e+f)\bigr)\leq g-e.$$ Else, if $\alpha^{\mathfrak l_Z}_0(p)+g-e-d+r\leq -1$, then also $\alpha^{\mathfrak l_Z}_j(p)+g-e-d+r\leq 0$ for $0\leq j\leq e-f-1$ and the left hand side of (\ref{nou}) equals $$\sum_{j=e-f}^{r}(\alpha_j^{\mathfrak l_Z}(p)+g-e-d+r)= (r+1-e+f)(g-e)-\sum_{i=0}^{r-e+f} \alpha_{i}=g-e-\rho(g, r-e+f, d-e)\leq g-e.$$ In both cases the inequality (\ref{eisharris}) is satisfied which proves our claim.
Since the chosen $(A, W_A)\in G^{r-e+f}_{d-e}(Y, p, \overline{\alpha})$ and $(L, W_L)\in G^{e-f-1}_{2e-f-1}(Y, p, \overline{\beta})$ are isolated points in their corresponding varieties of linear series on $Y$, it follows that limit $\mathfrak g^r_d$'s on $X_0$ constructed in the way we just described, fill-up a component of $\tau^{-1}(0)\subset \mathcal{H}$.
Indeed, suppose $(\mathfrak n_Y, \mathfrak n_Z, \tilde{D})\in \mathcal{H}$ is a point lying in the same irreducible component of $\tau^{-1}(0)$ as $(\mathfrak l_Y, \mathfrak l_Z, D)$. Here, $\mathfrak n_Y\in G^r_d(Y),\ \mathfrak n_Z\in G^r_d(Z, p, \overline{\gamma})$ and $\tilde{D}\in Y_e$ is a divisor such that $p\notin \mbox{supp}(\tilde{D})$. Then $a^{\mathfrak n_Y}(p)=a^{\mathfrak l_Y}(p)$ which is given by (\ref{vanseq1}), therefore $\mathfrak n_Y(-(d-2e+f+1)\cdot p)\in G^{2e-f-1}_{e-f-1}(Y, p, \overline{\beta})$ which is a reduced $0$-dimensional variety. This implies that $\mathfrak n_Y(-(d-2e+f+1)\cdot p)=(L, W_L)$. Next, we consider the linear series $\mathfrak n_Y(-\tilde{D})\in G^{r-e+f}_{d-e}(Y)$. Since $p\notin \mbox{supp}(\tilde{D})$, the vanishing sequence of this linear series is a subsequence of length $r+1-e+f$ of $a^{\mathfrak l_Y}(p)$. Necessarily, $\alpha^{\mathfrak n_Y(-\tilde{D})}(p)\geq \overline{\alpha}$ and because $\rho(e, r-e+f, d-e, \overline{\alpha})=0$, we must have that $\mathfrak n_Y(-\tilde{D})\in G^{r-e+f}_{d-e}(Y, p, \overline{\alpha})$ which is a discrete set, hence $\mathfrak n_Y(-\tilde{D})=(A, W_A)$ and $\tilde{D}=D\in Y_e$. This shows that $\mathfrak n_Y=\mathfrak l_Y$ and every point of this component of $\tau^{-1}(0)$ is determined by the $\mathfrak n_Z$. The dimension of this component is thus equal to $$\rho(e, r-e+f, d-e, \overline{\alpha})+\rho(g-e, r, d, \overline{\gamma})+\rho(e, e-f-1, 2e-f-1, \overline{\beta})=\rho(g, r, d)-f(r+1-e+f)+e,$$ which finishes the proof of Theorem \ref{existence}, subject to proving assumption (\ref{assumption2}).
\begin{remark} A slight variation of the argument described above, enables us to prove Theorem \ref{existence} even in some cases when we cannot establish (\ref{assumption2}). We start with a linear series $(A, W_A)\in G^{r-e+f}_{d-e}(Y, p, \overline{\alpha})$ and assume that the following condition holds: \begin{equation}\label{assumption3} H^0\bigl(Y, \mathcal{O}_Y((d-1)\cdot p)\otimes A^{\vee}\bigr)=0. \end{equation}
There exists a unique divisor $D\in |\mathcal{O}_Y(d\cdot p)\otimes A^{\vee})|$ and (\ref{assumption3}) guarantees that $p\notin \mbox{supp}(D)$. We define the space of sections $$V_Y:=H^0(\mathcal{O}_Y(2e-f-1)\cdot p)+ W_A\subset H^0(\mathcal{O}_Y(d\cdot p)), \ \mbox{ where
} \ W_A\subset H^0( \mathcal{O}_Y(d\cdot p-D)).$$ Reasoning along the same lines as in the previous case, since $p\notin \mbox{supp}(D)$ we find that $\mbox{dim}(V_Y)=r+1$, hence $\mathfrak l_Y=(\mathcal{O}_Y(d\cdot p),
V_Y)\in G^r_d(Y)$. Moreover, we can check that $$ a^{\mathfrak l_Y}(p)=(\alpha_0, \alpha_1+1, \ldots, \alpha_{r-e+f}+r-e+f, d-2e+f+1, d-2e+f+2, \ldots, d-e-1, d). $$ Like in the previous situation, we choose a linear series $\mathfrak l_Z\in G^r_d(Z, p)$ such that $\{\mathfrak l_Y, \mathfrak l_Z\}$ is a refined limit $\mathfrak g^r_d$. Thus we must have the following ramification sequence at $p$: $$\alpha^{\mathfrak l_Z}(p)=\overline{\gamma}:=(0, e, \ldots, e, d-r-\alpha_{r-e+f}, \ldots, d-r-\alpha_1, d-r-\alpha_0).$$ Condition (\ref{eisharris}) which guarantees the existence of $\mathfrak l_Z$ is satisfied if and only if $$\rho(g, r,d)\geq f(r+1-e+f)-(g-d+r), \ \mbox{ in the case }g-d+r<e $$ and $$ \rho(g, r, d)\geq f(r+1-e+f)-e, \ \mbox{ in the case }g-d+r\geq e. $$ Since we are always working under the hypothesis $\rho(g, r, d)-f(r+1-e+f)+e\geq 0$, we see that the previous condition holds whenever $g-d+r\geq e$, and that, in general, $\mathfrak l_Z\in G^r_d(Z, p, \overline{\gamma})$ exists if and only if \begin{equation}\label{ass4} \rho(g, r,d)\geq f(r+1-e+f)-(g-d+r). \end{equation} Assuming (\ref{ass4}), the variety $G^r_d(Z, p, \overline{\gamma})$ is non-empty of dimension $\rho(g-e, d, r, \overline{\gamma})=\rho(g, r, d)-f(r+1-e+f)+e$. The same argument as before shows that limit $\mathfrak g^r_d$'s on $X_0$ constructed in such a way, fill-up a component of $\tau^{-1}(0)\subset \mathcal{H}$ of expected dimension $\rho(g, r, d)-f(r+1-e+f)+e$, which finishes the proof. \end{remark}
Now we complete the proof of Theorem \ref{existence} by discussing under which assumptions we can establish (\ref{assumption2}):
\noindent{\emph{Proof of Theorem \ref{existence}}}. We retain the notation introduced above and show that there exist two linear series $(A, W_A)\in G_{d-e}^{r-e+f}(Y, p, \overline{\alpha})$ and $(L, W_L)\in G_{2e-f-1}^{e-f-1}(Y, p, \overline{\beta})$ satisfying (\ref{assumption2}) whenever one of the following conditions is satisfied: $$(i) \mbox{ } \ 2f\leq e-1,\ \ (ii)\mbox{ } \ e=2r-2 \mbox{ and } f=r-1, \ \mbox{ } (iii)\mbox{ } \
e<2(r+1-e+f).$$ As we already explained, (\ref{assumption2}) in case $(ii)$ is a
consequence of the Gieseker-Petri theorem.
We now treat case $(i)$ when $\beta_0=1$ and $\tilde{c}=f\leq e-f-1$. By Riemann-Roch we find that $L=K_Y\otimes \mathcal{O}_Y((e-2f+2)\cdot p)\otimes B^{\vee}$, where $B\in W^1_{e-f+1}(Y)$ is a pencil such that $h^0\bigl(Y,B\otimes \mathcal{O}_Y(-(e-2f+1)\cdot p)\bigr)\geq 1$ (There are finitely many such $B\in W^1_{e-f+1}(Y)$ for a generic choice of $[Y, p]\in \mathcal{M}_{e, 1}$). Applying the base-point-free pencil trick, (\ref{assumption2}) is equivalent to the injectivity of the multiplication map $$\mu_{B, M}: H^0(Y, B)\otimes H^0(Y, M)\rightarrow H^0(Y, B\otimes M),$$ where $M:=K_Y\otimes A^{\vee}\otimes \mathcal{O}_Y((d-f-e+2)\cdot p)\in W_{2e-f}^{e-f}(Y)$ is a complete linear series with vanishing sequence at $p$ equal to \begin{equation}\label{vanseq3} a^M(p)=(0, 1, \ldots, e-f-a-1, e-f-a+c, r-a+2, r-a+3, \ldots, r, r+1). \end{equation} Here we have set $a:=[e/(r+1-e+f)]$, hence we can write $e=a\cdot (r+1-e+f)+c$, where $0\leq c\leq r-e+f$. By assumption we have that $e-2a>c$ and clearly $\rho(M, \alpha^M(p))=0$, that is, there are finitely many $M\in W^{e-f}_{2e-f}(Y)$ satisfying (\ref{vanseq3}).
To prove that $\mu_{B, M}$ is injective, we degenerate $[Y, p]\in \mathcal{M}_{e, 1}$ to a particular stable curve:
$[Y_0, p_0]:=[E_0\cup_{p_1} E_1\cup \ldots\cup E_{e-2a-1}\cup_{p_{e-2a}} T, p_0]$, where $E_0,
\ldots, E_{e-2a-1}$ are elliptic curves, $[T=E_{e-2a}, p_{e-2a}]\in \mathcal{M}_{2a, 1}$ is a
Petri general smooth pointed curve and the points $p_i, p_{i+1}\in E_i$ are such that $p_{i+1}-p_i\in \mbox{Pic}^0(E_i)$ is not a torsion class for $0\leq i\leq e-2a-1$. Note that $p_0$ lies on the first component $E_0$. By contradiction, we assume that $\mu_{B, M}$ is not injective for every $[Y, p]\in \mathcal{M}_{e, 1}$ and for each of the finitely many linear series $M\in W_{2e-f}^{e-f}(Y)$ satisfying (\ref{vanseq3}) and each $B\in G^1_{e-f+1}\bigl(Y, p, (0, e-2f)\bigr)$.
We construct a limit $\mathfrak g_{2e-f}^{e-f}$ on $[Y_0, p_0]$, say $\mathfrak m=\{(M_{E_i}, V_i)\in G_{2e-f}^{e-f}(E_i)\}_{i=0}^{e-2a}$, which satisfies condition (\ref{vanseq3}) with respect to $p_0$, by specifying the vanishing sequences $a^{\mathfrak m_{E_i}}(p_i)$ for $0\leq i\leq e-2a$. For $0\leq i\leq c-1$, the sequence $a^{\mathfrak m_{E_{i+1}}}(p_{i+1})$ is obtained from $a^{\mathfrak m_{E_{i}}}(p_{i})$ by raising all entries by $1$, except for the term $$a_{e-f-a}^{\mathfrak m_{E_{i+1}}}(p_{i+1})=a_{e-f-a}^{\mathfrak m_{E_{i}}}(p_{i})=e-f-a+c.$$ After $c$ steps we arrive at the following vanishing sequence on $E_c$ with respect to $p_c$: $$a^{\mathfrak m_{E_c}}(p_c)=(c, c+1, \ldots, e-f-a+c-1, e-f-a+c, r-a+2+c, r-a+3+c, \ldots, r+c+1).$$ For an index $c\leq i\leq e-2a-1$ which we write as $i=c+a\cdot \beta +j$, with $0\leq j\leq a-1$ and $0\leq \beta\leq r-2-e+f$, we choose $a^{\mathfrak m_{E_{i+1}}}(p_{i+1})$ to be obtained from $a^{\mathfrak m_{E_{i}}}(p_{i})$ by raising all entries by $1$, except for the term $$a_{e-f-a+j+1}^{\mathfrak m_{E_{i+1}}}(p_{i+1})=a_{e-f-a+j+1}^{\mathfrak m_{E_{i}}}(p_{i})=r-a+2+c+(a-1)\cdot \beta+2j.$$ In this way $\mathfrak m\in \tilde{G}^{e-f}_{2e-f}(Y_0)$ becomes a (refined) limit linear series which smooths to a complete linear series $M\in G^{e-f}_{2e-f}(Y)$ on every smooth pointed curve $[Y, p]\in \mathcal{M}_{e, 1}$ such that the ramification condition (\ref{vanseq3}) with respect to $p$ is satisfied.
Next we construct a limit $\mathfrak g^1_{e-f+1}$ on $[Y_0, p_0]$, say $\mathfrak b=\{(B_{E_i}, W_i)\in G^1_{e-f+1}(E_i)\}_{i=0}^{e-2a}$ such that $a^{\mathfrak b}(p_0)=(0, e-2f+1)$. For $0\leq i\leq e-2f$ we set $a^{\mathfrak b_{E_i}}(p_i)=(i, e-2f+1)$. For an index of type $i=e-2f+2k-1$ where $0\leq k\leq f-a$, we choose $a^{\mathfrak b_{E_i}}(p_i)=(e-2f+k-1, e-2f+k+1)$. If $i=e-2f+2k$, we choose the sequence $a^{\mathfrak b_{E_i}}(p_i)=(e-2f+k, e-2f+k+1)$. It is clear that each sequence $a^{\mathfrak b_{E_i}}(p_i)$ is obtained from $a^{\mathfrak b_{E_{i-1}}}(p_{i-1})$ by raising one entry by $1$ while keeping the other fixed, hence $\mathfrak b$ is a limit $\mathfrak g^1_{e-f+1}$ which smooths to a pencil $B\in G^1_{e-f+1}(Y, p, (0, e-2f))$ on every nearby smooth curve $[Y, p]$. For each $0\leq i\leq e-2a-1$, there exists a section (unique up to scaling) $\sigma_i \in W_i$ such that $\mbox{ord}_{p_i}(\sigma_i)+\mbox{ord}_{p_{i+1}}(\sigma_{i})=\mbox{deg}(B_{E_i})$. We denote by $\sigma_i^c\in W_{i}$ a complementary section such that $\{\mbox{ord}_{p_i}(\sigma_i), \mbox{ord}_{p_i}(\sigma_i^c)\}=\{a_0^{\mathfrak b_{E_i}}(p_i), a_1^{\mathfrak b_{E_i}}(p_i)\}$.
Using the set-up developed in \cite{EH3} and \cite{F2} for studying degenerations of multiplication maps, we find that the assumption that $\mu_{B, M}$ is not injective implies the existence elements $0\neq \rho_i \in \mathrm{Ker}\{W_i\otimes V_i\rightarrow H^0(E_i, B_{E_i}\otimes M_{E_i})\}$ for each $0\leq i\leq e-2a$, satisfying the property that $\mbox{ord}_{p_{i+1}}(\rho_{i+1})\geq \mbox{ord}_{p_{i}}(\rho_{i})+1$, for all $i$ (see e.g. \cite{F2} Section 4, for an explanation of how to obtain the $\rho_i$'s). Moreover, if $\mbox{ord}_{p_{i+1}}(\rho_{i+1})=\mbox{ord}_{p_i}(\rho_i)+1$, then if $\tau_i\in V_i$ is the section (unique up to scaling) such that $\mbox{ord}_{p_i}(\tau_i)+\mbox{ord}_{p_{i+1}}(\tau_{i})=\mbox{deg}(M_{E_i})$, then we must have that $$\mbox{ord}_{p_i}(\rho_i)=\mbox{ord}_{p_i}(\tau_i)+\mbox{ord}_{p_i}(\sigma_i^c)=\mbox{ord}_{p_i} (\sigma_i)+\mbox{ord}_{p_i}(\tau_i'),$$ where $\tau_i'\in V_i$ is another section such that $\mbox{ord}_{p_i}(\tau_i')\neq \mbox{ord}_{p_i}(\tau_i)$. In particular, since we have explicitly described all the sequences $a^{\mathfrak b_{E_i}}(p_i)$ and $a^{\mathfrak m_{E_i}}(p_i)$, the assumption that $\mbox{ord}_{p_{i+1}}(\rho_{i+1})\leq \mbox{ord}_{p_i}(\rho_i)+1$ uniquely determines $\mbox{ord}_{p_i}(\rho_i)$.
Since $a^{\mathfrak b_{E_0}}(p_0)=(0, e-2f+1)$ and $\mu_{B_{E_0}, M_{E_0}}(\rho_0)=0$, the non-zero section $\rho_0$ must involve both sections $\sigma_0$ and $\sigma_0^c$ and then clearly $\mbox{ord}_{p_0}(\rho_0)\geq e-2f+1$. We prove inductively that for all integers $0\leq i\leq e-2a$ we have the inequality \begin{equation}\label{indu} \mathrm{ord}_{p_i}(\rho_i)\geq e-2f+1+2i. \end{equation} Assuming (\ref{indu}) for $i\leq e-2a-1$, since $\mbox{ord}_{p_{i+1}}(\rho_{i+1})\geq \mbox{ord}_{p_i}(\rho_i)+1$, the only way (\ref{indu}) can fail for $i+1$ is when $\mbox{ord}_{p_i}(\rho_i)=e-2f+2i+1$ and $\mbox{ord}_{p_{i+1}}(\rho_{i+1})=\mbox{ord}_{p_i}(\rho_i)+1$. As explained above, this implies that $\mbox{ord}_{p_i}(\rho_i)=\mbox{ord}_{p_i}(\tau_i)+\mbox{ord}_{p_i}(\sigma_i^c)$.
Writing $i=c+a\cdot \beta +j$ as above, then $\mbox{ord}_{p_i}(\tau_i)=r-a+2+c+(a-1)\cdot \beta +2j$\ if $i\geq c$, while $\mbox{ord}_{p_i}(\tau_i)=e-f-a+c$, for $0\leq i\leq c-1$. We deal only with the case $i\geq c$, the case $0\leq i\leq c-1$ being analogous. To determine $\mbox{ord}_{p_i}(\sigma_i^c)$ we must distinguish between two cases: When $i=e-2f+2k-1$ with $k\geq 1$, then $\mbox{ord}_{p_i}(\sigma_i^c)=e-2f+k-1$. Otherwise, we write $i=e-2f+2k$ in which case $\mbox{ord}_{p_i}(\sigma_i^c)=e-2f+k+1$. Suppose we are in the former case. Then we obtain the equality $$e-2f+2i+1=\mbox{ord}_{p_i}(\rho_i)=\bigl(r-a+2+c+(a-1)\cdot \beta+2j\bigr)+(e-2f+k-1),$$ which ultimately leads to the relation $(a+2)(r-e+f-\beta)=a-j-1$. But $j\leq a-1$ and $\beta\leq r-e+f-1$, hence we have reached a contradiction. The case when one can write $i=e-2f+2k$ is dealt with similarly. All in all, we may assume that we have proved the inequality $\mbox{ord}_{p_{e-2a}}(\rho_{e-2a})\geq e-2f+1+2(e-2a)$. We note that on the curve $[T, q]=[E_{e-2a}, p_{e-2a}]$ we have that $a^{\mathfrak b_{T}}(p_{e-2a})=(e-f-a, e-f-a+1)$, while
$$a^{\mathfrak m_{T}}(p_{e-2a})=(e-2a, e-2a+1, \ldots, 2e-f-3a, 2e-f-3a+3, \ldots, 2e-f-2a+2).$$ Equivalently $\mathfrak b_T=|B|+(e-f-a)\cdot q$, where $B\in W_{a+1}^1(T)$, while $\mathfrak m_{T}=(e-2a)\cdot q+|N|$, where $N\in \mbox{Pic}^{e-f+2a}(T)$ has the property that $h^0\bigl(T, N(-(e-f-a+3)\cdot q)\bigr)\geq a$. Remembering that $\mbox{ord}_{q}(\rho_{e-2a})\geq (e-2f+1)+2(e-2a)$, after subtracting the base locus supported at $q$, we find an element $$0\neq \rho_T\in \mbox{Ker}\{H^0(B)\otimes H^0(N)\rightarrow H^0(B\otimes N)\}$$ such that $\mbox{ord}_q(\rho_T)\geq e-f-a+1$. Equivalently, the multiplication map $$\mu_{B, N}: H^0(B)\otimes H^0\bigl(N(-(e-f-a+3)\cdot q)\bigr)\rightarrow H^0\bigl(B\otimes N(-(e-f-a+3)\cdot q)\bigr)$$ is not injective. By using Riemann-Roch we find that $N(-(e-f-a+3)\cdot q)=K_T\otimes \tilde{B}^{\vee}$, where $\tilde{B}\in W^1_{a+1}(T)$. Choosing $\tilde{B}=B\in W^1_{a+1}(T)$, we notice that $\mu_{B, N}$ can be identified with the Petri map $H^0(B)\otimes H^0(K_T\otimes B^{\vee})\rightarrow H^0(K_T)$ which is injective because $[T]\in \mathcal{M}_{2a}$ was chosen to be Petri general. Thus we have reached a contradiction by reducing (\ref{assumption2}) to the Gieseker-Petri theorem which completes the proof in the case $(i)$.
Next we turn to case $(iii)$ when $[e/(r+1-e+f)]<2$. Since the argument is similar to the one for $(i)$, we only outline the main steps. If $e\leq r-e+f$, that is, when $\alpha_0=d-r-f-e$, we can easily determine a linear series $(A, W_A)\in G^{r-e+f}_{d-e}(Y, p, \overline{\alpha})$. Precisely, one can see that $A=K_Y\otimes
\mathcal{O}_Y((d-3e+2)\cdot p)$ and $$|W_A|=(d-r-f-e)\cdot p+|K_Y\otimes
\mathcal{O}_Y\bigl((r+f-2e+2)\cdot p\bigr)|.$$ In this case we have that
$|G_{d-e}^{r-e+f}(Y, p, \overline{\alpha})|=1$. Condition (\ref{assumption2}) translates into saying that for a generic $(L, W_L)\in G_{2e-f-1}^{e-f-1}(Y, p, \overline{\beta})$ we have the vanishing statement \begin{equation}\label{van3} H^0\bigl(Y, L\otimes K_Y^{\vee}((e+f-2)\cdot p)\bigr)=0\Leftrightarrow H^0\bigl(Y, K_Y^{\otimes 2}\otimes L^{\vee}(-(e+f-2)\cdot p)\bigr)=0. \end{equation} One can prove (\ref{van3}) by degenerating $Y$ to a generic string of elliptic curves and we skip the details. Finally, if $[e/(r+1-e+f)]=1$, then $c=2e-r-f-1$ and condition (\ref{assumption2}) boils down to showing that one can find a pencil $B\in G^1_{e-c+1}\bigl(Y, p, (0, r-e+f-c+1)\bigr)$ and a linear series $L\in G_{2e-f-1}^{e-f-1}(Y, p, \overline{\beta})$, such that the multiplication map $$H^0(B)\otimes H^0\bigr(K_Y^{\otimes 2}\otimes L^{\vee}(-(2e-4-r)\cdot p)\bigl)\rightarrow H^0(K_Y^{\otimes 2}\otimes B\otimes L^{\vee}(-(2e-4-r)\cdot p)\bigr)$$ is injective. This situation is handled along the lines of $(i)$ and we omit the details. $
$ $\Box$
Finally, we prove Theorem \ref{existence} assuming that condition (\ref{ass4}) is satisfied. This case is not covered by cases $(i)-(iii)$ above:
\begin{proposition} Let $[Y, p]\in \mathcal{M}_{e, 1}$ be a general pointed curve. Then there exists a linear series $(A, W_A)\in G^{r-e+f}_{d-e}(Y, p, \overline{\alpha})$ such that $H^0\bigl(Y, \mathcal{O}_Y((d-1)\cdot p\otimes A^{\vee})\bigr)=0$. \end{proposition}
\begin{proof} By contradiction, we assume that $H^0(\mathcal{O}_Y((d-1)\cdot p)\otimes A^{\vee})\neq 0$ for every $[Y, p]\in \mathcal{M}_{e, 1}$ and for every linear series $(A, V_A)\in G^{r-e+f}_{d-e}(Y, p, \overline{\alpha})$. We let $[Y, p]$ degenerate to the stable curve $[Y_0:=E_0\cup_{p_1} E_1\cup_{p_2}\ldots \cup_{p_{e-3}}E_{e-3}\cup_{p_{e-2}}B, p_0]$, where $E_0, \ldots, E_{e-3}$ are elliptic curves, the points $p_i, p_{i+1}\in E_i$ are such that $p_i-p_{i+1}\in \mbox{Pic}^0(E_i)$ is not a torsion class,
and $[B, p_{e-2}]\in \mathcal{M}_{2, 1}$ is such that $p_{e-2}\in B$ is
not a Weierstrass point. For all integers
$0\leq i\leq e-3$ we find that there exist sections $$0\neq \tau_i\in H^0\bigl(\mathcal{O}_{E_i}((d-1)\cdot p_i)\otimes A_{E_i}^{\vee}\bigr) \mbox{ and }0\neq \tau_B=\tau_{e-2}\in H^0\bigl(\mathcal{O}_{B}((d-1)\cdot p_{e-2})\otimes A_B^{\vee}\bigr)$$ such that $$0\leq \mbox{ord}_{p_0}(\tau_0)\leq \mbox{ord}_{p_1}(\tau_1)\leq \ldots \leq \mbox{ord}_{p_{e-3}}(\tau_{e-3})\leq \mbox{ord}_{p_{e-2}}(\tau_{B}).$$ Moreover, we have that $\mbox{ord}_{p_i}(\tau_i)\geq i$ for $0\leq i\leq e-2$. In particular, $\mbox{ord}_{p_{e-2}}(\tau_{B})\geq e-2$. Since $\rho(e, r-e+f, d-e, \overline{\alpha})=0$, limit $\mathfrak g^{r-e+f}_{d-e}$ on $E_0\cup \ldots \cup E_{e-3}\cup B$ are smoothable to every curve of genus $g$. These finitely many limit $\mathfrak g^{r-e+f}_{d-e}$ are in bijective correspondence with possibilities of choosing the vanishing sequences $\{a^{l_{E_i}}(p_i)\}_{0\leq i\leq e-3}$ and $a^{l_B}(p_{e-2})$ in such a way that for all $0\leq i\leq e-3$, the sequence $a^{l_{E_{i+1}}}(p_{i+1})$ is obtained from $a^{l_{E_i}}(p_i)$ by raising all entries by $1$ except a single entry which remains unchanged. To finish the proof it suffices to exhibit a single limit $\mathfrak g_{d-e}^{r-e+f}$ on $E_0\cup \ldots \cup E_{e-3}\cup B$ having the property that if $(A_B, V_B)$ denotes its $B$-aspect, then $H^0(\mathcal{O}_B((d-e+1)\cdot p_{e-2})\otimes A_B^{\vee})=0$.
We describe such a $\mathfrak g^{r-e+f}_{d-e}$ explicitly by specifying the sequences $\{\alpha^{l_{E_i}}(p_i)\}_{0\leq i\leq e-3}$ and $\alpha^{l_B}(p_{e-2})$. Clearly, $\alpha^{l_{E_0}}(p_0)$ equals $(\alpha_0, \ldots, \alpha_0, \alpha^{l_{E_0}}_{r-e+f+1-c}(p_0)=\alpha_0+1, \ldots, \alpha_0+1)$. For $1\leq i\leq c$, $\alpha^{l_{E_i}}(p_i)$ is obtained from $\alpha^{l_{E_{i-1}}}(p_{i-1})$ by increasing all entries by $1$, except for $\alpha^{l_{E_i}}_{r-e+f+i-c}(p_i)=\alpha^{l_{E_{i-1}}}_{r-e+f+i-c}(p_{i-1})$. Thus $\alpha^{l_{E_c}}(p_c)=(\alpha_0+c, \ldots, \alpha_0+c)$. Next, for an index $i$ such that $c+\beta(r+1-e+f)< i\leq c+(\beta+1)(r+1-e+f)$, where $0\leq \beta\leq [e/(r+1-e+f)]$, if we write $i\equiv j+c \mbox{ mod } r+1-e+f$, with $1\leq j\leq r-e+f$, the sequence $\alpha^{l_{E_i}}(p_i)$ is obtained from $\alpha^{l_{E_{i-1}}}(p_{i-1})$ by raising all entries by $1$, except for $\alpha^{l_{E_i}}_{j-1}(p_i)=\alpha^{l_{E_{i-1}}}_{j-1}(p_{i-1})$. Switching from ramification to vanishing sequences we obtain $$a^{l_B}(p_{e-2})=(d-r-f-2, d-r-f-3, \ldots, d-e-5, d-e-4, d-e-2, d-e-1),$$ that is, $A_B=\mathcal{O}_B((d-e-2)\cdot p_{e-2})\otimes \mathfrak g^1_2$, and then $$H^0(\mathcal{O}_B((d-e+1)\cdot p_{e-2})\otimes A_B^{\vee})=H^0(\mathcal{O}_B(3\cdot p_{e-2})\otimes \mathfrak (\mathfrak g^1_2)^{\vee})=0.$$ This contradicts the fact $\mbox{ord}_{p_{e-2}}(\tau_B)\geq e-2$ which completes the proof. \end{proof}
\section{Higher ramification points of a general line bundle}
In this section we prove Theorem \ref{genlb}. We fix an arbitrary smooth curve $C$ of genus $g$ and for $n\geq 1$ we denote by $[n]_C:\mbox{Pic}^d(C)\rightarrow \mbox{Pic}^{nd}(C)$ the multiplication by $n$ map, $[n]_C(L):=L^{\otimes n}$. It is an immediate consequence of Riemann-Roch that for a general $L\in \mbox{Pic}^d(C)$, we have that $h^0(L^{\otimes n})=\mbox{max}\{nd+1-g, 0\}$.
First we show that for a very general $L\in \mbox{Pic}^d(C)$ we have that $w^{L^{\otimes n}}(p)\leq 1$ for all $p\in C$ and $n\geq 1$. Indeed, let us assume that $w^{L^{\otimes n}}(p)\geq 2$, where $n$ is chosen such that $nd\geq g$, so that $h^0(C, L^{\otimes n})=nd+1-g$. Then there are two possibilities:
$$ \ (i)\ h^0\bigl(C, L^{\otimes n}(-(nd+2-g)\cdot p)\bigr)\geq 1 \mbox{ or } \ (ii)\ \ h^0\bigl(C, L^{\otimes n}(-(nd-g)\cdot p)\bigr)\geq 2.$$ In case $(i)$ we consider the map $C\times C_{g-2}\rightarrow \mbox{Pic}^{nd}(C), \ (p, E)\mapsto \mathcal{O}_C\bigl((nd+2-g)\cdot p+E\bigr)$ and we denote by $\Sigma_n$ its image which is a divisor on $\mbox{Pic}^{nd}(C)$. Then $(i)$ is equivalent to $L\in [n]_C^{*}(\Sigma_n)$ which is a divisorial condition on $\mbox{Pic}^d(C)$ for each $n$.
In case $(ii)$ we look at the map $C \times C^1_g\rightarrow \mbox{Pic}^{nd}(C), (p, E)\mapsto \mathcal{O}_C\bigl((nd-g)\cdot p+E\bigr)$ and we denote by $V_n$ its image. Since $C^1_g$ is generically a ${\textbf P}^1$-bundle over $C_{g-2}$, it follows that $V_n$ is a divisor on $\mbox{Pic}^{nd}(C)$ and then possibility $(ii)$ is equivalent to $L\in [n]_C^*(V_n)$. Thus we see that for $L\in \mbox{Pic}^d(C)-\bigcup_{n\geq 1} [n]_C^*(\Sigma_n+V_n)$ all the ramification points of all powers $L^{\otimes n}$ with $n\geq 1$, are ordinary. This proves the first part of Theorem \ref{genlb}. To prove the second part we start with the following: \begin{proposition} We fix a point $p\in C$ and integers $n$ and $d$ such that $nd\geq g$. Then the locus $$D_n:=\{L\in \mathrm{Pic}^d(C): h^0\bigl(C, L^{\otimes n}(-(nd+1-g)\cdot p)\bigr)\geq 1\}$$ is an irreducible divisor on $\mathrm{Pic}^d(C)$ and $[D_n]=n^2 \theta$. \end{proposition} \begin{proof} We set $a:=\mbox{max}\{0, 2g-1-nd\}$ and define two vector bundles $\mathcal{E}_n$ and $\mathcal{F}_n$ on $\mbox{Pic}^{d}(C)$ of the same rank and having fibres $\mathcal{E}_n(L)=H^0(C, L^{\otimes n}\otimes \mathcal{O}_C(a\cdot p))$ and $\mathcal{F}_n(L)=H^0(C, L^{\otimes n}\otimes \mathcal{O}_{(a+nd+1-g)\cdot p}(a\cdot p))$ over each point $L\in \mbox{Pic}^d(C)$. Then $D_n$ is the degeneracy locus of the morphism $\mathcal{E}_n\rightarrow \mathcal{F}_n$ obtained by evaluation sections of $L^{\otimes n}\otimes \mathcal{O}_C(a\cdot p)$ along $(a+nd+1-g)\cdot p$. The Picard bundle $\mathcal{E}_n$ is negative (i.e. $\mathcal{E}_n^{\vee}$ is ample), because $\mathcal{E}_n$ is the pull-back under the finite map $[n]_C$ of a negative bundle on $\mbox{Pic}^d(C)$ (cf. \cite{ACGH}, pg. 310). Moreover, $\mathcal{F}_n$ is algebraically equivalent to a trivial bundle, hence $\mathcal{E}_n^{\vee}\otimes \mathcal{F}_n$ is ample too. Applying the Fulton-Lazarsfeld connectedness theorem (see \cite{FL} or \cite{ACGH} pg. 311), we conclude that $D_n$ is connected. Since $D_n$ is also smooth in codimension $2$ we obtain that $D_n$ must be irreducible. Finally, $[D_n]=c_1(\mathcal{F}_n-\mathcal{E}_n)=[n]_C^*(\theta)=n^2 \theta $. \end{proof}
\noindent {\emph{End of the proof of Theorem \ref{genlb}}.} We fix integers $1\leq a<b$ and consider the variety $\Sigma_{ab}:=\{(p, L)\in C\times \mbox{Pic}^d(C): p\in R(L^{\otimes a})\cap R(L^{\otimes b})\}$ and we denote by $\phi_1:\Sigma_{ab} \rightarrow C$ and $\phi_2:\Sigma_{ab}\rightarrow \mbox{Pic}^d(C)$ the two projections. For a fixed $p\in C$, the fibre $\phi_1^{-1}(p)$ is identified with the intersection of the two irreducible divisors $D_a$ and $D_b$. Since $[D_a]\neq [D_b]$ for $a\neq b$, it follows that $D_a\cap D_b$ is of pure codimension $2$ inside $\mbox{Pic}^d(C)$, therefore $\mbox{dim}(\Sigma_{ab})=g-1$. We obtain that a line bundle $L\in \mbox{Pic}^d(C)-\bigcup_{a<b} \phi_2(\Sigma_{ab})$ will enjoy the property that $R(L^{\otimes a})\cap R(L^{\otimes b})=\emptyset$ for $a<b$. $
$ $\Box$
\end{document} | arXiv |
Decathlon: the Art of Scoring Points
By John Barrow
The decathlon consists of 10 events including the pole vault.
The decathlon consists of ten track and field events spread over two days. It is the most physically demanding event for athletes. On day one, the 100 metres, long jump, shot putt, high jump and 400 metres are contested. On day two, the competitors face the 110 metre hurdles, discus, pole vault, javelin and, finally, the 1500 metres. In order to combine the results of these very different events -- some give times and some give distances -- a points system has been developed. Each performance is awarded a predetermined number of points according to a set of performance tables. These are added, event by event, and the winner is the athlete with the highest points total after ten events.
The most striking thing about the decathlon is that the tables giving the number of points awarded for different performances are rather free inventions. Someone decided them back in 1912 and they have subsequently been updated on different occasions. Clearly, working out the fairest points allocation for any running, jumping or throwing performance is crucial and defines the whole nature of the event very sensitively. Britain's Daley Thompson missed breaking the decathlon world record by one point when he won the Olympic Games 1984 but a revision of the scoring tables the following year increased his score slightly and he became the new world record holder retrospectively!
All of this suggests some important questions that bring mathematics into play. What would happen if the points tables were changed? What events repay your training investment with the greatest points payoff? And what sort of athlete is going to do best in the decathlon -- a runner, a thrower or a jumper?
The decathlon events fall into two categories: running events where the aim is to record the least possible time and throwing or jumping events where the aim is to record the greatest possible distance. The simplest way of scoring this would be to multiply all the throw and jump distances in metres together, then multiply all the running times in seconds together and divide the product of the throws and jumps by the product of the running times. The Special Total, $ST$, will be $$ ST=\frac{LJ\times HJ\times PV\times JT\times DT\times SP}{T_{100m} \times T_{400m}\times T_{110mH}\times T_{1500m}} $$ where $LJ$ is the distance from the long jump, $HJ$ the high jump, $JT$ the javelin, $DT$ the discus and $SP$ the shot putt, with units of $$ \frac{(length)^6}{(time)^4} = \frac{m^6}{s^4}. $$
If we take the three best ever decathlon performances by Ashton Eaton (9039 pts), Roman Šebrle (9026 pts) and Tomáš Dvořák (8994 pts) and work out the Special Totals for the 10 performances they each produced then we get
$$ \begin{array}{rl} \mbox{Šebrle (9026 pts):} & ST = 2.29 \\ \mbox{Dvořák (8994 pts):} & ST = 2.40 \\ \mbox{Eaton (9039 pts):} & ST = 1.92 \end{array} $$
Interestingly, we see that the second best performance by Dvořák becomes the best using this new scoring system and Eaton's drops to third.
In fact, our new scoring system contains some biases. Since the distances attained and the times recorded are different for the various events you can make a bigger change to the ST score for the same effort. An improvement in the 100 metre from 10.6 seconds to 10.5 seconds requires considerable effort but you don't get much of a reward for it in the ST score. By contrast reducing a slow 1500 metre run by 10 seconds has a big impact. The events with the room for larger changes have bigger effects on the total.
The setting of the points tables that are used in practice is a technical business that has evolved over a long period of time. It pays attention to world records, the standards of the top ranked athletes, and historical decathlon performances. However, ultimately it is a human choice and if a different choice was made then different points would be received for the same athletic performances and the medallists in the Olympic Games might be different. The 2001 IAAF scoring tables have the following simple mathematical structure.
The points awarded (decimals are rounded to the nearest whole number to avoid fractional points) in each track event -- where you want to give higher points for shorter times -- are given by the formula:
$$ A\times (B-T)^C $$
where $T$ is the time recorded by the athlete in a track event and $A$, $B$ and $C$ are numbers chosen for each event so as to calibrate the points awarded in an equitable way. The quantity $B$ gives the cut-off time at and above which you will score zero points. $T$ is always less than $B$ in practice -- unless someone falls over and crawls to the finish!
For the jumps and throws -- where you want to give more points for greater distances ($D$) -- the points formula for each event is
$$ A\times (D-B)^C. $$
You score zero points for a distance equal to or less than $B$. The distances here are all in metres and the times in seconds. The three numbers $A$, $B$ and $C$ are chosen differently for each of the ten events and are shown in the table below.
The points achieved for each of the ten events are then added together to give the total score.
100 m 25.4347 18 1.81
Long jump 0.14354 220 1.4
Shot put 51.39 1.5 1.05
High jump 0.8465 75 1.42
400 m 1.53775 82 1.81
110 m hurdles 5.74352 28.5 1.92
Discus throw 12.91 4 1.1
Pole vault 0.2797 100 1.35
Javelin throw 10.14 7 1.08
1500 m 0.03768 480 1.85
In order to get a feel for which events are "easiest" to score in, take a look at this table which shows what you would have to do to score 900 points in each event for an Olympic-winning 9000-point total alongside Ashton Eaton's world record.
9039 (Eaton, World Record)
100m 10.83s 10.21s
Long jump 7.36m 8.23m
Shot put 16.79m 14.20m
High jump 2.10m 2.05m
110m hurdles 14.59s 13.70s
Discus throw 51.4m 42.81m
Pole vault 4.96m 5.3m
Javelin throw 70.67m 58.87m
1500m 247.42s (= 4m 07.4s) 254.8s (= 4m 14.8s)
There is an interesting pattern in the formulae that change the distances and times achieved into points for the decathlon. The power index $C$ is approximately 1.8 for the running events (1.9 for the hurdles), close to 1.4 for the jumps and pole vault and close to 1.1 for the throws. The same pattern holds for the women's heptathlon as well, with $C$ approximately equal to 1.8 for runs, 1.75 for jumps and 1.05 for throws. The fact that $C > 1$ indicates that the points scoring system is a ''progressive'' one, curving upwards in a concave way with decreasing time or increasing distance; that is, the better the performance, the higher the reward for performance improvements (see figure 1).
Figure 1: Illustration of the increase in points gained for improved performances in progressive (C > 1), neutral (C = 1), and regressive (C < 1) scoring systems.
This is realistic. We know that as you get more expert at your event it gets harder to make the same improvement but beginners can easily make large gains. The opposite type of ("regressive") points system would have $C < 1$, curving increasingly less, while a "neutral" one would have $C = 1$ and be a straight line. We can see that the IAAF tables are very progressive for the running events, fairly progressive for the jumps and vault, but almost neutral for the throws.
It is clear that there has been a significant bias towards gathering points in the long jump, hurdles and sprints (100m and 400m). Performances in these events are all highly correlated with flat-out sprinting speed. Conversely, the 1500m and the three throwing events are well behind the other disciplines in points scoring. If you want to coach a successful decathlete, start with a big strong sprint hurdler and build up strength and technical ability for the throws later. No decathletes bother much with 1500m preparation and rely on general distance running training. Ideally, we would expect there to be very little variance in this bar chart as we go from one event to another. In reality, the variations are huge and the scoring system is rather biased towards sprinter-jumpers. The women's heptathlon has a similar bias in its scoring tables, which greatly favours athletes like Jessica Ennis.
Figure 2 shows the division between the ten events for the averages of the all-time top 100 best ever men's decathlon performances.
What if we picked $C = 2$ across all events? This would give an extremely progressive scoring system greatly favouring competitors with outstanding performances in some events (like Eaton), as opposed to those with consistent similar ones. However, it would dramatically favour good throwers over the sprint hurdlers because of the big change from the value of $C = 1.1$ being applied to the throws at present. And this illustrates the basic difficulty with points systems of any sort -- there is always a subjective element that could have been chosen differently.
Professor John D Barrow is the Director of the Millennium Mathematics Project. His most recent book '100 Essential Things You Didn't Know You Didn't Know About Sport' was published by Bodley Head in March 2012.
Our Key Stage 4 activity Training Schedule looks at the maths behind the heptathlon scoring system. | CommonCrawl |
Welcome to our first fully online issue of Parabola Incorporating Function.
It's not every day that a mathematics puzzle makes it into mainstream media. But that's what happened recently with "Cheryl's Birthday problem". This problem was posted by Kenneth Kong, the host of a Singaporean TV show, on his Facebook page on 10 April, and it went viral.
Finding two ways to enumerate the same collection of objects can often give rise to useful formulae. For instance, the sum \[ 1 + 2 + \cdots + n \] can be interpreted as the maximum number of different handshakes between $n+1$ people.
Polygonal numbers enumerate the number of points in a regular geometrical arrangement of the points in the shape of a regular polygon. An example is the triangular number $T_n$ which enumerates the number of points in a regular triangular lattice of points whose overall shape is a triangle.
Parabola incorporating Function would like to thank Sin Keong Tong for contributing problem 1472.
Q1461 As in problems 1442 and 1452, a particle is projected from one corner of a $2014\times1729$ rectangle. This time, however, the particle is projected at an angle of $30^\circ$ above the horizontal. | CommonCrawl |
Gene content dissimilarity for subclassification of highly similar microbial strains
Qichao Tu1,2 &
Lu Lin1
Identification and classification of highly similar microbial strains is a challenging issue in microbiology, ecology and evolutionary biology. Among various available approaches, gene content analysis is also at the core of microbial taxonomy. However, no threshold has been determined for grouping microorgnisms to different taxonomic levels, and it is still not clear that to what extent genomic fluidity should occur to form a microbial taxonomic group.
By taking advantage of the eggNOG database for orthologous groups, we calculated gene content dissimilarity among different microbial strains based on the orthologous gene profiles and tested the possibility of applying gene content dissimilarity as a quantitative index in classifying microbial taxonomic groups, as well as its potential application in subclassification of highly similar microbial strains. Evaluation of gene content dissimilarity to completed microbial genomes at different taxonomic levels suggested that cutoffs of 0.2 and 0.4 can be respectively used for species and family delineation, and that 0.2 gene content dissimilarity cutoff approximately corresponded to 98 % 16S rRNA gene identity and 94 % ANI for microbial species delineation. Furthermore, application of gene content dissimilarity to highly similar microbial strains suggested it as an effective approach in classifying closely related microorganisms into subgroups.
This approach is especially useful in identifying pathogens from commensals in clinical microbiology. It also provides novel insights into how genomic fluidity is linked with microbial taxonomy.
Identification and classification of microorganisms is one of the most important but difficult and challenging issues in microbiology, ecology and evolutionary biology. Traditional methods for identification and classification of microorganisms mainly rely on morphological, physiological and biochemical properties of isolated microorganisms [1]. However, characterizing these properties are experimentally very complicated and no quantitative standards can be applied for the obtained descriptive data. Moreover, these properties may differ greatly under different experimental conditions, leading to biased observations of the isolated microorganism. Thus, there have been continuous demands for quantitative approaches to delineate and classify microorganisms by the scientific community, such as methods based on genotypes [2].
For several decades, many efforts have been made to more accurately identify and classify microorganisms, especially at the species level. Among them, DNA-DNA hybridization (DDH) and 16S rRNA gene identity are the two most successful and widely accepted achievements, the former of which is still regarded as the gold standard for microbial species delineation. However, the DDH approach is experimentally tedious and hard to standardize between different laboratories in addition to several other problems, such as that the value obtained with the same pair of strains depends on which is used as probe and which as target [3]. Due to these reasons, 16S rRNA gene analysis has been mainly used in place of DDH for describing new species since the past decades [4, 5]. However, for 16S rRNA gene identity, although it is generally accepted that 97 % or higher sequence identity be used as a cutoff to define microbial species [6, 7], problems have been reported by several labs that 16S rRNA gene identity even cannot distinguish several microbial genera, such as the ones belonging to Enterobacteriaceae (particularly Enterobacter and Pantoea) [4]. And it is now generally accepted that DDH only be carried out when 16S rRNA identity between two strains is 97 % or higher [6, 8].
In the post-genomic era, with more reference genomes getting sequenced by the scientific community, genomic approaches such as in-silico DDH [9], average amino acids identity (AAI) [10], average nucleotide identity (ANI) [11] and multi-locus sequence analysis (MLSA) [12] have been developed. By integrating genomic information, these approaches are proven to be more accurate and reliable in microbial species delineation than 16S rRNA gene identity [13, 14]. Among these approaches, in-silico DDH could be considered as a genomic replacement of wet-lab DDH, for which a 70 % cutoff can be used for species delineation. For AAI and ANI, a cutoff of 94–96 % [10, 11, 15, 16] is generally accepted by microbiologists for their corresponding to 70 % DDH and 97 % 16S rRNA identity, and is becoming a gold genomic standard for microbial species delineation.
Besides the above approaches, gene content analysis proposed as early as in 1999 is another post-genomic analysis at the core of current species definition and has gained success in microbial phylogenetic analysis [17–20]. This approach, although has not gained as wide application as sequence identity based methods, the idea complies several species concepts in microbial systematics such as the recombination theory [21–23] and Cohan's ecotype concept [24, 25]. These concepts propose that microbial species are formed by acquisition and loss of functional traits through lateral gene transfer and periodic selection, respectively. Such genomic fluidity phenomena plays important roles in microbial genome evolution and identifying closely related organisms such as distinguishing pathogens from commensals [26, 27]. Notably, previous gene content analysis mainly relied on all vs. all pairwise comparison, and reanalysis would always be required when a new genome was added. Most importantly, to our best knowledge, cutoffs for gene content analysis in classifying microorganisms are not yet available.
In this study, we first aim to take advantage of currently available comprehensive ortholog databases such as eggNOG [28], in which orthologous groups are defined by all vs. all clustering approaches, a similar approach as in gene content analysis and thus would simplify the computational procedure for gene content analysis. We then try to address the following two biological questions based on the obtained gene content dissimilarity metrics. First, whether cutoffs could be determined for gene content dissimilarity in classifying microorganisms into different taxonomic groups, ie to what extent genomic fluidity should generally achieve to form a new taxonomic group, eg species? Second, whether gene content dissimilarity could be used for subspecies level classification of highly similar microbial strains? To our best knowledge, current approaches in identifying and subclassifying highly similar microbial strains still mainly rely on phenotypic properties, because approaches based on sequence identity can hardly achieve such purpose due to highly similar conserved genes at subspecies level. As a result, evaluation of gene content dissimilarity using currently sequenced microbial genomes at different taxonomic levels suggested that cutoffs of 0.2 and 0.4 can be respectively used for species and family delineation. Further application of gene content dissimilarity to highly similar microbial strains suggested it as an effective approach in classifying closely related microorganisms into subgroups. This is especially useful in identifying pathogens from commensals in clinical microbiology.
Overview of the framework
Although microbial taxonomy at species and higher levels mainly rely on sequence identity approaches such as 16S rRNA gene identity and ANI, identification and classification of highly similar microbial strains still require phenotypic properties of the isolated microorganisms [1]. However, current laboratory approaches may differ greatly from natural conditions and possess limitations of only characterizing a few characteristics of isolated microorganisms. This may lead to misclassification of microorganisms with distinct ecologies, habitats and genotypes. Gene content conceives the notion that the phenotypic properties are ultimately determined by the genes microbial strains harbor. Thus it is expected that the phenotypic differences among different microbial strains can be reflected by gene content dissimilarity.
Here we proposed a general framework (Fig. 1, also see the methods section for more details) that implemented gene content dissimilarity for potential application in microbial classification, especially highly similar microbial strains that can hardly be distinguished by traditional approaches. Three major steps were included in the framework. First, orthologous gene profiles for microorganisms with complete or near complete genomes were obtained by searching all genes against the eggNOG database. An orthologous gene profile table comprising the abundance of orthologous groups in different microorganisms was generated. Second, weighted Bray-Curtis dissimilarity was calculated as the index representing gene content dissimilarity between different microbial strains. A pairwise distance matrix comprising the gene content dissimilarities among different microbial strains was generated. Bray-Curtis dissimilarity, which was also widely used for microbial functional gene dissimilarities such as in [29], was used here for its weighted property on genetic events such as gene duplication. Third, distance matrix was clustered to group microbial strains into different clusters. It was expected that microbial strains with similar phenotypic properties would be clustered together into the same group for their similar orthologous gene profiles.
The flowchart of applying gene content dissimilarity for microbial delineation and classification. Three main steps were included. First, orthologous gene profiles were obtained for all selected microbial genomes by searching against the eggNOG database. Second, pairwise gene content dissimilarity as measured by Bray-Curtis dissimilarity was calculated for all pairs of microbial strains. Third, microbial strains were clustered into different groups
Gene content dissimilarity cutoffs for microbial species and family delineation
In order to see whether thresholds can be determined for gene content dissimilarity in classifying microorganisms into taxonomic groups as well as the extent of genomic fluidity in forming microbial taxonomic groups, 2772 complete microbial genomes were recruited and pairwise Bray-Curtis gene content dissimilarities were calculated. Gene content dissimilarity values were then summarized at different taxonomic levels, including species, genus, family, and order (Fig. 2). Interestingly, clear boundaries could be observed for microbial species and family delineation. At the species level, 92.54 % intra-species gene content dissimilarity values fell within 0–0.2 (Fig. 2a). At the genus level, about 26.65 % inter-species gene content dissimilarity values fell within 0–0.2, 61.4 % within 0.2–0.4, and 10.19 % within 0.4–0.5 (Fig. 2b). At the family level, about 80.23 % inter-genus gene content dissimilarity values were within 0.2–0.4 and 16.65 % within 0.4–0.5 (Fig. 2c). At the order level, about 9.69 % inter-family gene content dissimilarity values were smaller than 0.4, and ~90.2 % were within 0.4–0.8 (Fig. 2d). Similar results could still be observed when the most recent eggNOG v4.5 database was used (Additional file 1: Figure S1). Based on these results, it could be found that gene content dissimilarity between microbial strains of the same species were mostly smaller than 0.2, and the value between microbial strains belonging to different families were mostly larger than 0.4. This suggested that gene content dissimilarity cutoffs of 0.2 and 0.4 can be used for microbial species and family delineation, respectively. Notably, similar to 16S rRNA gene identity and ANI cutoffs for microbial species definition, the gene content dissimilarity cutoff was also paradoxical. Strains belonging to the same species were mostly found with smaller than 0.2 gene content dissimilarity. However, not all strains sharing smaller than 0.2 gene content dissimilarity belonged to the same species, because there was still ~25 % possibility that microbial strains with smaller than 0.2 gene content dissimilarity belong to different species but same genus.
Distribution of gene content dissimilarity for the retrieved microbial genomes at different taxonomic levels, including species (a), genus (b), family (c), and order (d). Cutoffs of 0.2 and 0.4 were recommended for microbial species and family delineation, respectively
Gene content dissimilarity vs. 16S vs. ANI for microbial species definition
One of the most difficult issues in microbial systematics is species identification of newly isolated microorganisms. ANI and 16S rRNA gene identity are two major sequence identity based approaches currently widely used for microbial species identification. ANI cutoff of 94–96 % and 16S identity cutoff of 97–98 % were usually applied for species definition for their corresponding to the gold 70 % DNA-DNA association rate. In order to see whether a similar cutoff can be found for gene content dissimilarity for microbial species identification, intra- and inter-species gene content dissimilarity was compared with corresponding ANI and 16S rRNA gene identity (Fig. 3). Interestingly, gene content dissimilarity cutoff of 0.2 well correlated with 98 % 16S rRNA gene identity and 94 % ANI for microbial species delineation. With 98 % 16S rRNA gene identity, 94 % ANI and 0.2 gene content dissimilarity as cutoffs, about 98.9, 86, and 92.8 % true positive rate was found for microbial strains belonging to the same species, respectively. And about 21.8, 4.7 and 18.8 % false positives were respectively found by misclassifying microbial strains belonging to different species as a same one. A total of 79.4 % microbial strains could be classified to the correct species by all three methods. These results suggested that 94 % ANI was the most conservative method for species definition among all three methods, followed by 0.2 gene content dissimilarity and 98 % 16S rRNA gene identity cutoffs.
Comparison of 16S rRNA gene identity, ANI, and gene content dissimilarity in microbial species delineation. A cutoff of 0.2 corresponded to 98 % 16S rRNA gene identity and 94 % ANI in species delineation. A total of 5008 intra-species and 8642 intra-genus comparisons were plotted. Red dots falling in the Q1 quadrant were mostly several clostridium strains, for which misclassification may have occurred. Red dots represented intra-species comparisons, and blue dots indicated intra-genus comparisons
To further evaluate the performance of gene content dissimilarity on microbial species delineation, we extracted all microbial strains in the genera with ≥2 species and each species with ≥5 strains. A total of 33 microbial species were evaluated. Significance tests of the orthologous gene profiles of microbial species against other species in the same genus were carried out (Additional file 1: Table S1). The non-parametric multivariate analysis MRPP (multi-response permutation procedure) based on Bray-Curtis dissimilarity distance was performed. As a result, all microbial species subjected to the tests were significantly different from other species in the same genus with P ≤ 0.005, except for species Mycobacterium bovis (P = 0.024) and Pseudomonas fluorescens (P = 0.018). This suggested that the gene content dissimilarity method proposed in this study can be confidently applied to delineate currently well recognized microbial species.
Enterobacteriaceae subclassification using gene content dissimilarity
In order to see whether gene content dissimilarity can be used as an effective index to classify closely related microorganisms, pairwise gene content dissimilarity was calculated for microorganisms belonging to Enterobacteriaceae. Enterobacteriaceae is a relatively well-studied large microbial family with many harmless symbionts as well as a lot famous pathogens. More importantly, representative reference genomes are available for most of them, making it an ideal taxonomic group for testing post-genomic approaches for microbial classification. A total of 916 Enterobacteriaceae genomes were recruited, of which 173 were completed genomes and 743 were in draft status. Among these genomes, 14 belonged to Enterobacter, 384 to Escherichia, 45 to Klebsiella, 314 to Salmonella, 14 to Serratia, 42 to Shigella, and 103 to Yersinia. As a result, PCoA clustering of gene content dissimilarity showed microbial genomes belonging to the same genus were clustered together and well separated from clusters formed by other genera (Fig. 4a). Notably, microbial genomes belonging to Enterobacter and Klebsiella were closely clustered, though a trend of separation could be observed. Microbial genomes of two genera, Escherichia and Shigella, were overlapped and cannot be separated by the first and second axis when other genera were included in the analysis. This was consistent with previous phylogenetic analysis that Shigella should be more appropriately classified as a subgenus of Escherichia, a phenomenon termed as taxa in disguise [30]. More interestingly, microbial genomes of Yersinia were clustered into two distinct clusters, one of which contained Yersinia pestis/Yersinia pseudotuberculosis genomes, which is consistent with several previous phylogenetic studies based on marker genes including dnaJ, gyrB, recA, tuf and atpD [31–34]. Genomes in the second cluster were non-pestis/pseudotuberculosis genomes and were more closely clustered with Serratia genomes.
Application of gene content dissimilarity in classifying microbial strains belonging to Enterobacteriaceae. a PCoA clustering of all selected microbial strains belonging to Enterobacteriaceae. b PCoA clustering of highly similar microbial strains including E. coli and Shigella. A clear separation of Shigella and E. coli O157:H7 from other E. coli strains could be observed
To further investigate if gene content dissimilarity can also be used to classify closely related microbial strains at subgenus and subspecies level, pairwise gene content dissimilarity was extracted for E. coli and Shigella strains and then subjected to PCoA clustering (Fig. 4b). This may provide higher resolution in identifying highly similar microbial strains. As a result, Shigella strains were well separated from Escherichia strains by both first and second axis when other Enterobacteriaceae genera were excluded from analysis. This suggested that although Shigella and E. coli were highly similar, they were still substantially functionally different and might be considered as different species of Escherichia. Interestingly, Escherichia coli O157:H7 strains can also be well separated from other E. coli strains by the first axis. This indicated that O157:H7 strains harbored markedly different functional capacity from other E. coli strains and gene content dissimilarity can be used as an effective post-genomic index to identify O157:H7 strains.
Streptococcus classification using gene content dissimilarity
To further confirm the capability of gene content dissimilarity in classifying highly similar microbial strains, the same approach was carried out to classify microbial strains belonging to the genus Streptococcus. Similar to Escherichia strains, certain Streptococcus species are responsible for many human diseases such as meningitis, pneumonia, septicemia, and sinusitis, while the majority of them are not pathogenic and form commensal microbiota in human body. Although some Streptococcus strains can be identified phenotypically and phylogenetically, species such as Streptococcus pneumonia can hardly be distinguished from the Mitis group members [35]. Here a total of 84 completed and 199 draft Streptococcus genomes were collected and subjected to PCoA clustering based on gene content dissimilarity. Only Streptococcus species with more than 5 strains were selected for plotting. These included 83 S. agalactiae strains, 7 S. mitis strains, 8 S. oralis strains, 38 S. mutans strains, 119 S. pneumoniae strains, 12 S. pyogenes strains, 10 S. suis strains, and 6 S. thermophilus strains. As a result, all these Streptococcus species could be well separated from each other by forming individual clusters (Fig. 5). Notably, S. pneumonia strains were also well separated from S. mitis and S. oralis strains. Consistent to previous proposal that S. oralis be classified as a member of the S. mitis group [36], S. oralis and S. mitis were closely clustered by gene content dissimilarity based PCoA clustering, indicating that they shared highly similar gene profiles. However, a separation of S. oralis and S. mitis could still be observed (Fig. 5), suggesting that they might still be two different species or subspecies despite high gene content similarity. This indicated that gene content dissimilarity can also be used as an effective method in distinguishing highly similar Streptococcus strains.
Application of gene content dissimilarity in classifying Streptococcus strains. Clear separation of different species into different groups could be observed. Highly similar strains belonging to S. mitis, S. oralis, and S. pneumoniae were also well separated
Bacillus cereus subclassification using gene content dissimilarity
We also applied gene content dissimilarity to classify the Bacillus cereus group, which is strikingly resistant to any currently available classification systems [37]. A total of 31 compelte genomes and 119 draft genomes belonging to B. anthracis, B. cereus, and B. thuringiensis were recruited. Two analyses were carried out here, including strains with complete genomes and all strains with complete and draft genomes (Additional file 1: Figure S2). Interestingly, clear separation of B. anthracis from B. cereus and B. thuringiensis could be observed in both analyses. The B. cereus and B. thuringiensis strains with complete genomes could be approximately clustered into three subgroups (Additional file 1: Figure S2A). Such trend of separation could also be observed when more draft genomes were included, but with more vague boundaries (Additional file 1: Figure S2B). This could be due to an increase of subgroups when more draft genomes were added.
Gene content analysis serves as the genomic ground for phenotypic differences and is one of the major post-genomic approaches developed for microbial phylogenetic reconstruction [17–20]. Previous implementation of gene content analysis relies on all vs. all comparison of genes in interested microbial groups, and a reanalysis is needed every time a new strain is added. By taking advantage of recently developed orthologous gene databases such as eggNOG [28], this study suggests that all vs. all comparison for gene content dissimilarity could be approximated by searching against a fixed database. Also, a previous study implementing taxon-specific genes and eggNOG database suggested the usefulness of such strategies in microbial taxonomic classification [38]. Although the approach will suffer potential limitations from not including singleton genes in the database, it is not expected to affect the results because the phenotypic properties expressed by these genes are usually not characterized for microbial classification. In addition, these singleton geneseins are also not subjected to phylogenetic analysis of microorganisms, for which conserved gene families are usually selected [12].
Current sequence identity based approaches mainly focus on the species problem in microbiology, but rarely at other taxonomic levels such as family level. Although gene content analysis has long been applied to reconstruct the phylogenetic relationship of microorganisms, cutoffs have not yet been determined for delineating different taxonomic groups. Recently, Qin et al. applied the concept of percentage of conserved proteins (POCP) to estimate the evolutionary and phenotypic distance between two strains and suggested that a pairwise POCP cutoff of 50 % can be served as a genus boundary for prokaryotic groups [39]. The POPC approach, which relied on all vs. all pairwise identification of orthologous genes, was similar to the one we proposed in this study but more computationally complicated, and seemed not suitable for sublevel classification such as species level. Our results here, interestingly, suggested that gene content dissimilarity could be served as an effective index for microbial subspecies, species and family delineation, but not for genus delineation. Such differences could be due to several reasons, such as the resolution of these two approaches in assigning gene groups, the number of microbial strains recruited in the studies (235 vs. 2772), as well as the possibility that the boundaries between microbial genus and species/family could be relatively vague.
The gene content dissimilarity approach developed in this study complies several species concepts proposed by microbiologists [40], including the recombination theory [3, 40] and Cohan's ecotype concept [24, 25]. The recombination concept proposes that microbial species are formed by partially exchanging and obtaining homologous and non-homologous genes via lateral gene transfer [21–23]. Recent studies suggest that lateral gene transfer frequently occur to transfer protein-coding genes among microorganisms and is a major evolutionary force for prokaryotes to adapt novel traits such as antibiotic resistance from the environments and other microorganisms in the community [41–44]. The ecotype ecological species concept proposes that prokaryotes form species by adapting to specific environments, for which periodic selection is the major force of cohesion [24, 25]. Genes responsible for adaptive phenotypes are fixed, while less adaptive traits are purged during the periodic selection process. Notably, no matter how different these two concepts are, both theories point out the importance of acquiring and losing of genes/traits in microbial species formation. A question then arise that to what extent microbial species should obtain and/or lose genes/traits to form a new taxonomic group, eg species. Our study addressed this question that a minimum of 0.2 and 0.4 Bray-Curtis gene content dissimilarities should be reached to confidently call species and family, respectively.
More interestingly, application of gene content dissimilarity to highly similar microbial strains/species suggests that gene content dissimilarity can also be served as a powerful index for classifying highly similar microbial strains, although an exact cutoff cannot be determined due to the varied rules in defining microbial subgroups. Enterobacteriaceae, Streptococcus, and B. cereus group are relatively well studied for their wide existence and pathogenic properties of several species. Several species/genera belonging to these groups can hardly be distinguished by traditional approaches, such as Yersinia species [45, 46], E.coli/Shigella [47–49], and Streptococcus oralis/mitis/pneumonia [50–54]. Among the Yersinia species, Y. enterocolitica, Y. pseudotuberculosis and Y. pestis are pathogenic for mammals. Contrast classification groups are proposed for these three species based on clinical/phenotypic and genotype criteria. Specifically, Y. enterocolitica, Y. pseudotuberculosis are classified as one group and Y. pestis as another when judging by their clinical and epidemiological features, while DNA-DNA hybridization suggests that Y. pseudotuberculosis and Y. pestis should be classified as one group or even species for their almost identical chromosomes [45]. Our results supports the later that Y. pseudotuberculosis and Y. pestis are highly similar at their gene content, consistent with their high DNA-DNA hybridization values. Shigella species are now generally accepted as a clade of the species E. coli based on phylogenetic analysis of conserved gene sequences [30, 47, 55]. Due to the failure of phylogenetic identification of Shigella from E. coli, they are mainly distinguished by their biochemical and serotype properties. However, recent whole-genome-based approach suggests that all four Shigella species are distinctly different from E. coli and form sister species to E. coli in the genus Escherichia [48]. Interestingly, our results agree with the whole-genome-based study that Shigella and E. coli strains are dramatically different from each other, and should be considered as individual species in Escherichia. Notably, our results indicated that E. coli O157:H7 can even be considered as a different species of Escherichia for their divergent gene content from other E. coli strains. Similarly, S. pneumoniae, S. mitis and S. oralis are also closely related species and have encountered difficulties in identification using traditional phylogenetic approaches [4], while our results suggested that they could be well separated from each other by gene content dissimilarity. All these results suggest that gene content dissimilarity could be used as an effective index in classifying closely related microbial strains, even at subspecies level.
Notably, although currently available post-genomic approaches are technically different from each other, they are either directly or indirectly linked with the classical DDH method, ie sequence identities. It is therefore not difficult to figure out their high correlation with each other in microbial delineation. What's interesting here is that these technical differences have addressed different microbial taxonomic problems. For example, the well-known ANI method suggests that microbial species can be defined with an ANI cutoff of 94–96 % [11, 15, 16]. The POCP method, however, suggests a genus boundary for microbial delineation [39]. While the gene content dissimilarity approach we evaluated in this study proposed cutoffs for microbial species and family delineation. No matter how, these approaches have addressed several different questions in microbial systematics and evolutionary biology, such as the relationship between microbial speciation and nucleotide polymorphism, and the extent of genomic fluidity in forming different microbial taxonomic groups. Importantly, current species classification and demarcation are so diverse in metabolic capabilities [56] and ecology [57] that they are neither rooted in evolutionary nor ecological theories [58]. And with the rapid accumulation of genome sequences from so many microorganisms, it is urged by microbiologist that it is now about the time for order in microbial systematics by taking account of both phylogeny and biological signatures [59]. As different indices tried to solve microbial systematics problems in different angles, we herein advocate using multiple indices for confident classification and delineation of microorganisms.
This study presented a post-genomic approach—gene content dissimilarity, for classification of highly similar microbial strains and as well addressed an interesting evolutionary biology question that to what extent genomic fluidity should occur in forming microbial species and family. Our results suggested that cutoffs of 0.2 and 0.4 gene content dissimilarity could be respectively used for microbial species and family level delineation, the latter of which a general cutoff was not proposed for many years [59] until recently [60]. More importantly, application of gene content dissimilarity showed clear separation of highly similar microbial strains into different subgroups at high resolutions by removing potential noises from other species/genera, ie excluding other species/genera from the analysis. The study provided a genomic mean for identifying closely related microbial strains and could be useful in identifying pathogens from commensals in clinical microbiology, especially when combined with approaches like ANI and 16S rRNA gene identity.
Data acquisition and processing
To evaluate the performance of gene content dissimilarity on microbial taxonomy delineation, a total of 2772 completed microbial genomes were downloaded from NCBI FTP site (ftp://ftp.ncbi.nlm.nih.gov/genomes/archive/old_refseq/Bacteria/). GenBank format genome sequence files and protein sequence files (FASTA format) were retrieved. Full genome sequences and 16S rRNA gene sequences were extracted from GenBank files by PERL scripts implementing BioPerl modules. A full list of the downloaded microbial genomes and accession numbers can be found in Additional file 2.
To test the performance of gene content dissimilarity on distinguishing highly similar microbial strains, both draft and completed genome sequences were recruited for Enterobacteriaceae, Streptococcus, and the Bacillus cereus group, which were three largest microbial groups with known taxonomic problems and many reference genomes available. Microbial strains belonging to dominant taxonomic groups were selected for evaluation. A total of 916 Enterobacteriaceae, 283 Streptococcus genomes, and 150 Bacillus genomes were retrieved, respectively. Among the Enterobacteriaceae genomes, 14 belonged to Enterobacter, 384 to Escherichia, 45 to Klebsiella, 314 to Salmonella, 14 to Serratia, 42 to Shigella, and 103 to Yersinia. The 283 Streptococcus genomes included 83 S. agalactiae strains, 7 S. mitis strains, 8 S. oralis strains, 38 S. mutans strains, 119 S. pneumoniae strains, 12 S. pyogenes strains, 10 S. suis strains, and 6 S. thermophilus strains. Classification and identification problems have been reported for several of these recruited genera/species, such as Escherichia vs. Shigella, and S. mitis vs. S. oralis vs. S. pneumonia.
Gene content dissimilarity calculation
We used Bray-Curtis dissimilarity index to measure the gene content dissimilarity among different microbial strains. To do so, we first obtained orthologous gene profiles for each strain by searching all protein sequences against the eggNOG database (v4) [28], which is currently one of the most comprehensive databases for orthologous groups. The eggNOG database was selected also due to its all vs. all clustering procedure in identifying orthologous groups. All COG and NOG orthologous groups were extracted. The program USEARCH (v7.0.1001, usearch_global) [61] was used for database searching for its 10–1250 times faster than BLAST. Coding genes were assigned to these orthologous groups based on the best hit they had with eggNOG database, with an e-value cutoff of 1e-5 and global sequence identity cutoff of 30 %. Microbial strains with less than 1000 genes mapped to the database were excluded from the analysis. A total of 2365 microbial strains were remained for further analysis. Pairwise Bray-Curtis dissimilarity was calculated according to the following function:
$$ B{C}_{ij}=1-\frac{2{C}_{ij}}{S_i+{S}_j} $$
Where C ij represented the sum of lesser number of genes mapped to each orthologous group, S i and S j were the total number of genes mapped to eggNOG database in each genome.
To better illustrate how the Bray-Curtis dissimilarity between two strains was calculated, an example was presented (Table 1). In this example, 9 and 11 genes in strain I and J were mapped to 4 and 5 orthologous groups, respectively. By taking the lesser number of mapped genes to each orthologous group, the sum of lesser number (ie C ij ) of genes mapped to orthologous groups that were found in strain I and J was 7. Thus a Bray-Curtis dissimilarity of 0.3 could be obtained for strain I and J according to the above function, ie BC ij = 1 - 2*7/(9 + 11) = 0.3. The perl scripts for orthologous profile generation and pair-wise Bray-Curtis dissimilarity calculation could be found in Additional files 3 and 4.
Table 1 An example showing how Bray-Curtis dissimilarity was calculated between strain I and strain J. (Note: dissimilarity calculation in real case would be more complex because typical microbial genomes usually comprise thousands of genes)
16S rRNA gene identity and ANI calculation
Pairwise 16S rRNA gene identity was calculated by the USEARCH (v7.0.1001) program [61]. Global sequence identity was calculated. In the case multiple 16S rRNA gene copies were found in a genome, all of them were subjected to calculation and the average value was used as the identity between two microbial strains.
Pairwise ANI calculation for the downloaded 2772 genomes was carried out by a perl script obtained from https://github.com/chjp/ANI. This script employed the same algorithm and output the same result as the JSpecies program [16]. The program BLAST (v2.2.25) was called in the script for ANI calculation.
PCoA clustering and significance tests
We employed PCoA clustering methods for better visualization of our results. In fact, many other clustering programs should also work in separating microbial strains into different groups. The non-parametric multivariate analysis MRPP (multi-response permutation procedure) based on Bray-Curtis dissimilarity distance was performed to evaluate the significance of orthologous profiles of microbial species against other species in the same genus. The vegan package [62] developed in R environment was used in this study.
ANI, average nucleotide identity; AAI, average amino acids identity; DDH, DNA-DNA hybridization; MLSA, multi-locus sequence analysis; MRPP, multi-response permutation procedure; PCoA, principle coordinate analysis; POCP, percentage of conserved proteins
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We thank Dr. Jizhong Zhou at the University of Oklahoma for providing computational resources for this study. We are also grateful to the anonymous reviewers for their critical input that significantly helped improve the manuscript.
This work was supported by the Fundamental Research Funds for the Central Universities of China (2016QNA4039 Q.T. and 2015QNA4044 L.L.), by the Open Funding of Zhejiang Provincial Key Laboratory of Health Risk Factors for Seafood (201605), and by the start-up packages of Zhejiang University (129000-193820101-001-027 Q.T. and 129000-193820101-001-004 L.L.).
The dataset supporting the results of this article is included within the article. All genomes analyzed in this study were downloaded from NCBI ftp site. A list of microbial genomes as well as the perl scripts associated with this study are provided as supplementary data.
QT performed the data analysis. QT and LL conceived the study and wrote the manuscript. All authors read and approved the final manuscript.
Q.T. received his PhD in microbiology at the University of Oklahoma. He is now an assistant professor at Zhejiang University. He is also appointed as an affiliate scholar at the University of Oklahoma. His main research interests include microbial ecology, bioinformatics, metagenomics and genomics.
L.L. received her PhD in Biochemistry and Molecular Biology at the University of Chinese Academy of Sciences. She is now an associate professor at Zhejiang University. Her main research interests include microbial genetics, biofuel, and microbial ecology.
Not applicable. No human participant is involved in this study.
Not applicable. No human or animal data is included in the study.
Department of Marine Sciences, Ocean College, Zhejiang University, Zhejiang, 316000, China
Qichao Tu & Lu Lin
Zhejiang Provincial Key Laboratory of Health Risk Factors for Seafood, Zhoushan Municipal Center for Disease Control and Prevention, Zhoushan, 316021, China
Qichao Tu
Lu Lin
Correspondence to Lu Lin.
This file contains the supplementary table and figures for this paper, including Table S1, Figure S1, and Figure S2. (DOCX 262 kb)
This file contains the list of strain names and NCBI accession numbers for the microorganisms retrieved in this study. (XLSX 114 kb)
This file contains the perl script for generating orthologous gene profiles from usearch outputs against eggNOG (COG and NOG) database. (DOCX 15 kb)
This file contains the perl script for calculating pairwise Bray-Curtis dissimilarity among different microbial strains. (DOCX 15 kb)
Tu, Q., Lin, L. Gene content dissimilarity for subclassification of highly similar microbial strains. BMC Genomics 17, 647 (2016). https://doi.org/10.1186/s12864-016-2991-9
Microbial subclassification
Highly similar strains
Gene content dissimilarity
Genomic fluidity | CommonCrawl |
\begin{document}
\title[Algebras of intermediate growth]{Nil algebras, Lie algebras and wreath products with intermediate and oscillating growth}
\author{Be'eri Greenfeld} \address{Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093, USA} \email{[email protected]}
\author{Efim Zelmanov} \address{Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093, USA} \email{[email protected]}
\keywords{Nil algebras, nil Lie algebras, growth of algebras, matrix wreath products, Gel'fand-Kirillov dimension, tensor products}
\subjclass[2020]{16P90, 16N40, 16S30}
\begin{abstract} We construct finitely generated nil algebras with prescribed growth rate. In particular, any increasing submultiplicative function is realized as the growth function of a nil algebra up to a polynomial error term and an arbitrarily slow distortion.
We then move on to examples of nil algebras and domains with strongly oscillating growth functions and construct primitive algebras for which the Gel'fand-Kirillov dimension is strictly sub-additive with respect to tensor products, thus answering a question from \cite{KrauseLenagan,KrempaOkninski}.
\end{abstract}
\maketitle
\section{Introduction}
Let $F$ be a field and let $A$ be a finitely generated, infinite-dimensional $F$-algebra. Fixing a finite-dimensional generating subspace $A=F\left<V\right>$, the growth of $A$ with respect to $V$ is defined to be the function: $$g_{A,V}(n)=\dim_F \left(F+V+V^2+\cdots+V^n\right)$$ If $1\in V$ then equivalently $g_{A,V}(n)=\dim_F V^n$. This function obviously depends on the choice of $V$, but only up to the following equivalence relation. We say that $f\preceq g$ if $f(n)\leq Cg(Dn)$ for some $C,D>0$ and for all $n\in \mathbb{N}$, and $f\sim g$ (asymptotically equivalent) if $f\preceq g$ and $g\preceq f$. Therefore when talking about `the growth of an algebra' one refers to $g_A(n)$ as the equivalence class of the functions $g_{A,V}(n)$ (for an arbitrary $V$) under the equivalence relation $\sim$.
A general reference for growth of algebras is \cite{KrauseLenagan}.
One of the most fundamental problems in combinatorial algebra is to characterize the possible growth rates of groups, algebras and Lie algebras. The first example of a group of intermediate growth (that is, super-polynomial but subexponential) was given by Grigorchuk \cite{GrigorchukIntermediate2}.
How do growth functions of algebras look like? Let $g_A(n)$ be the growth function of an infinite-dimensional, finitely generated algebra $A$ with respect to a fixed generating subspace. Then $g_A(n)$ is increasing ($g_A(n)<g_A(n+1)$) and submultiplicative ($g_A(n+m)\leq g_A(n)g_A(m)$). Bell and Zelmanov \cite{BellZelmanov} found a concrete characterization of growth functions of algebras, from which it follows that any increasing and submultiplicative function is equivalent to the growth of some finitely generated algebra up to a linear error term, which is the best possible approximation.
Of particular interest is the class of associative nil algebras. Recall that an associative algebra is called nil if all of its elements are nilpotent. For many years it has been an open problem if an infinite-dimensional nil algebra can have a polynomially bounded growth. In 2007 Lenagan and Smoktunowicz \cite{LenaganSmoktunowicz} (see also \cite{LenaganSmoktunowiczYoung}) constructed infinite-dimensional nil algebras of polynomially bounded growth over countable fields. Bell and Young \cite{BellYoung} constructed infinite-dimensional nil algebras over arbitrary fields whose growth is bounded above by an arbitrarily slow super-polynomial function. In \cite{AAJZ_ERA} it is conjectured that any (non-linear) function which occurs as the growth rate of an algebra is realizable as the growth of a nil algebra; this can be thought of as a strong quantitative version of the Kurosh Problem.
We prove the following approximation of the aforementioned conjecture, establishing its validity up to a polynomial error term and an arbitrarily small distortion.
\begin{thma1}[{Arbitrary growth: Countable fields}] Let $f\colon \mathbb{N}\rightarrow \mathbb{N}$ be an increasing, submultiplicative function and let $\delta\colon \mathbb{N}\rightarrow \mathbb{N}$ be an arbitrarily slow, non-decreasing function tending to infinity.
Let $F$ be a countable field. Then there exists a finitely generated nil algebra $R$ over $F$ whose growth function $g_R(n)$ satisfies: $$ f\left(\frac{n}{\delta(n)}\right) \preceq g_R(n) \preceq p(n)\cdot f(n) $$ where $p(n)$ is a polynomial which is independent of $f$. \end{thma1}
The situation over uncountable fields is different and requires a mild relaxation of the formulation:
\begin{thma2}[{Arbitrary growth: Arbitrary fields}] Let $f\colon \mathbb{N}\rightarrow \mathbb{N}$ be an increasing, submultiplicative function and let $\omega\colon \mathbb{N}\rightarrow \mathbb{N}$ be an arbitrarily slow, non-decreasing super-polynomial function. Let $\delta\colon \mathbb{N}\rightarrow~\mathbb{N}$ be an arbitrarily slow, non-decreasing function tending to infinity.
Let $F$ be an arbitrary field. Then there exists a finitely generated nil algebra $R$ over $F$ whose growth function $g_R(n)$ satisfies: $$ f\left(\frac{n}{\delta(n)}\right) \preceq g_R(n) \preceq \omega(n)\cdot f(n). $$ \end{thma2}
Kassabov and Pak \cite{KassabovPak} constructed groups whose growth functions oscillate between given intermediate functions (e.g.~$\exp(n^{4/5})$) and an arbitrarily rapid subexponential function; see also \cite{BartholdiErschler}. Bartholdi and Erschler \cite{Permutational} constructed periodic groups with explicitly given growth functions, ranging over a wide variety of intermediate growth functions.
\begin{thmb1}[{Oscillating growth: Countable fields}] Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be an arbitrarily rapid subexponential function. Let $F$ be a countable field. Then there exists a finitely generated nil algebra $R$ over $F$ such that: \begin{itemize} \item $g_R(n)\leq n^{6+\varepsilon}$ infinitely often for every $\varepsilon>0$; and \item $g_R(n)\geq f(n) $ infinitely often. \end{itemize} \end{thmb1}
\begin{thmb2}[{Oscillating growth: Arbitrary fields}] Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be an arbitrarily rapid subexponential function and let $\omega\colon \mathbb{N}\rightarrow \mathbb{N}$ be an arbitrarily slow, non-decreasing, super-polynomial function.
Let $F$ be an arbitrary field. Then there exists a finitely generated nil algebra $R$ over $F$ such that: \begin{itemize} \item $g_R(n)\leq \omega(n) $ infinitely often; and \item $g_R(n)\geq f(n) $ infinitely often. \end{itemize} \end{thmb2}
These are the first examples of nil algebras whose growth oscillates between a polynomial and an arbitrarily rapid subexponential function. Monomial algebras with oscillating growth were constructed in \cite{BBL}.
Petrogradsky \cite{PetrogradskyLie} constructed a far-reaching Lie-theoretic analogy of these phenomena. Namely, he constructed nil (restricted) Lie algebras over fields of positive characteristic, whose growth functions oscillate between a function very close to linear and a function very close (though not arbitrarily close) to exponential. The growth of a finitely generated Lie algebra $L$ with respect to a generating subspace $V$ is the dimension of the space spanned by Lie monomials of length at most $n$ in the elements of $V$, that is, $g_L(n)=\dim_F(V+V^{[2]}+\cdots+V^{[n]})$.
Using Theorems B1, B2 we can construct nil Lie algebras with oscillating growth functions over fields of arbitrary characteristic. A Lie algebra $L$ is nil if $ad_x$ is nilpotent for every $x\in L$.
\begin{cor} \label{oscillatingLie} Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be an arbitrarily rapid subexponential function. Let $F$ be a countable field of ${\operatorname{char}}(F)\neq 2$. Then there exists a finitely generated nil Lie algebra $L$ over $F$ such that: \begin{itemize} \item $g_L(n)\leq n^{6+\varepsilon}$ infinitely often for every $\varepsilon>0$; and \item $g_L(n)\geq f(n) $ infinitely often. \end{itemize} \end{cor}
\begin{cor} \label{oscillatingLie2} Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be an arbitrarily rapid subexponential function and let $\omega\colon \mathbb{N}\rightarrow \mathbb{N}$ be an arbitrarily slow, non-decreasing, super-polynomial function.
Let $F$ be an uncountable field of ${\operatorname{char}}(F)\neq 2$. Then there exists a finitely generated nil Lie algebra $L$ over $F$ such that: \begin{itemize} \item $g_L(n)\leq \omega(n) $ infinitely often; and \item $g_L(n)\geq f(n) $ infinitely often. \end{itemize} \end{cor}
Our next goal is to construct domains with oscillating intermediate growth. There are some similarities between growth rates of domains and growth rates of groups. For instance, there are no known domains of super-polynomial growth slower than $\exp(\sqrt{n})$; any domain of subexponential growth admits a division ring of fractions, and it is a widely open problem whether the group algebra of a torsion-free group is always a domain. As mentioned before, Kassabov and Pak \cite{KassabovPak} constructed finitely generated groups whose growth oscillates between $\exp(n^{4/5})$ and an arbitrarily rapid (subexponential) function. We prove:
\begin{thmc} Let $f\colon \mathbb{N}\rightarrow \mathbb{N}$ be a subexponential function. Then there exists a finitely generated domain $A$ such that $g_A(n)\geq f(n)$ infinitely often, and $g_A(n)\leq~\exp\left(n^{\frac{3}{4}+\varepsilon}\right)$ infinitely often (for arbitrary $\varepsilon>0$). \end{thmc}
Finally, we turn to a somewhat different construction of algebras whose growth oscillates within a prescribed \textit{polynomial} interval. Recall that the Gel'fand-Kirillov (GK) dimension of a finitely generated algebra $A=F\left<V\right>$ is: $$ {\operatorname{GKdim}}(A) = \limsup_{n\rightarrow\infty} \frac{\log \dim_F V^n}{\log n} $$ namely the (optimal) degree of polynomial growth of $A$. For algebras with `sufficiently regular' growth functions, the GK-dimension is additive with respect to tensor products. Warfield \cite{Warfield} proved that ${\operatorname{GKdim}}(A\otimes_F B)$ might be smaller than ${\operatorname{GKdim}}(A)+{\operatorname{GKdim}}(B)$, and gave an example where: $$ {\operatorname{GKdim}}(A\otimes_F B)=\max\{{\operatorname{GKdim}}(A),{\operatorname{GKdim}}(B)\}+2 $$ which is the minimum possible value for ${\operatorname{GKdim}}(A\otimes_F B)$. Let $\alpha={\operatorname{GKdim}}(A),\beta={\operatorname{GKdim}}(B)$ and suppose that $\alpha\leq \beta$. Warfield asked whether any value in the interval $[2+\beta,\alpha+\beta]$ is attainable for ${\operatorname{GKdim}}(A\otimes_F B)$. This was settled in the affirmative by Krempa and Okni\'nski \cite{KrempaOkninski}, even in a slightly wider generality. Both Warfield's and Krempa-Okni\'nski's constructions have large prime radicals, which is a common phenomenon among generic constructions of monomial algebras. This led Krempa and Okni\'nski to ask whether semiprime examples of this flavor can be found. The problem of finding Warfield-type examples among semiprime rings is mentioned also by Krause and Lenagan \cite[Page~167]{KrauseLenagan}.
We settle this problem in the affirmative. Our constructions have a symbolic dynamical origin, and arise as monomial algebras associated with Toeplitz subshifts with oscillating complexity growth.
\begin{thmd} \label{ThmD} For arbitrary $2\leq \gamma \leq \alpha\leq \beta < \infty$ there exist finitely generated primitive monomial algebras $A,B$ such that: $$ {\operatorname{GKdim}}(A)=\alpha,\ \ {\operatorname{GKdim}}(B)=\beta,\ \ {\operatorname{GKdim}}(A\otimes_F B)=\beta+\gamma. $$ \end{thmd}
\textit{Conventions}. Given a function $f\colon \mathbb{N}\rightarrow \mathbb{N}$, we let $f(x)=f(\lfloor x \rfloor)$ for any real $x\geq 1$. Lie algebras are considered over fields of ${\operatorname{char}}\neq 2$.
\section{Matrix wreath products}
We recall the construction of matrix wreath products from \cite{AAJZ_TransAMS}. Fix an arbitrary base field $F$. Let $B$ be a finitely generated $F$-algebra and $A$ a countably generated $F$-algebra. Fix a finite-dimensional generating subspace for $B$, say, $V$. If $B$ is non-unital, we let $B'$ be its unital hull ($B'=F+B$).
Let $A\wr B=B+{\operatorname{Lin}}_F(B',B'\otimes_F A)$, where ${\operatorname{Lin}}_F(*,*)$ is the space of $F$-linear maps. Then $A\wr B$ is endowed with a multiplicative structure, making it an $F$-algebra. For $f,g\in {\operatorname{Lin}}_F(B',B'\otimes_F A)$ let:
$$ fg = (1\otimes \mu)(f\otimes 1)g $$ where $\mu\colon A\otimes_F A\rightarrow A$ is the multiplication map. Multiplication on $B$ is given by the ring structure of $B$, so it remains to define a $B$-bimodule structure on ${\operatorname{Lin}}_F(B',B'\otimes_F A)$ which makes $A\wr B$ an associative ring. Given $f\in {\operatorname{Lin}}_F(B',B'\otimes_F A)$, we let $(fb)(x)=f(bx)$ and $(bf)(x)=(b\otimes 1)f(x)$.
With any linear map $\gamma \colon B'\rightarrow A$, associated is an element $$c_\gamma\in {\operatorname{Lin}}_F(B',B'\otimes_F A),$$ given by: $$c_\gamma (x)=1\otimes \gamma(x).$$ Finally, let $C=F\left<V,c_\gamma\right>\subseteq A\wr B$.
\begin{lem} \label{decomposition} For any $b_1,\dots,b_s\in B$, we have: $$ c_\gamma b_1 c_\gamma b_2 \cdots c_\gamma b_s(x) = 1\otimes \gamma(b_1)\cdots \gamma(b_{s-1})\gamma(b_sx) $$ \end{lem} \begin{proof} Suppose that $\sigma,\tau\colon B'\rightarrow A$ are maps. Then: $$c_\tau c_\sigma(x)=1\otimes \tau(1)\sigma(x).$$ Indeed, \begin{eqnarray*} c_\tau c_\sigma (x) & = & (1\otimes \mu)(c_\tau \otimes 1)c_\sigma(x) \\ & = & (1\otimes \mu)(c_\tau \otimes 1)(1\otimes \sigma(x)) \\ & = & (1\otimes \mu)(c_\tau(1)\otimes \sigma(x)) \\ & = & (1\otimes \mu)(1\otimes \tau(1)\otimes \sigma(x)) \\ & = & 1\otimes \tau(1)\sigma(x). \end{eqnarray*}
Therefore, given $s$ maps: $$c_{\tau_1}c_{\tau_2}\cdots c_{\tau_s}(x)=1\otimes \tau_1(1)\cdots \tau_{s-1}(1)\tau_s(x)$$ Since $c_\gamma b=c_{\gamma'}$, where $\gamma'(x)=\gamma(bx)$, we get that: $$ c_\gamma b_1 c_\gamma b_2 \cdots c_\gamma b_s(x) = 1\otimes \gamma(b_1)\cdots \gamma(b_{s-1})\gamma(b_sx) $$ as claimed. \end{proof}
Let $g_B,g_C$ be the growth functions of $B,C$ with repsect to $V,V+Fc_\gamma$, respectively. The growth of $C$ turns out to be closely related to the following measurement of growth of $\gamma$. Let:
$$ W_n = \sum_{i_1+\cdots+i_s\leq n} \gamma\left(V^{i_1}\right)\cdots \gamma\left(V^{i_s}\right) $$ and let $w_\gamma(n)=\dim_F W_n$. Obviously, $w_\gamma(n)$ is monotonely non-decreasing.
In \cite{AAJZ_TransAMS} it was proved that $g_C(n)\preceq g_B(n)^2w_\gamma(n)$, and if $\gamma$ satisfies an additional condition, called \textit{density}, then $g_C(n) \sim g_B(n)^2w_\gamma(n)$. More concretely, $\gamma$ is dense if for any system of linearly independent elements $b_1,\dots,b_k\in B$ and any $0\neq a\in A$, there exists $b\in B$ such that $\gamma(b_ib)=0$ for all $1\leq i\leq k-1$ but $a\gamma(b_kb)\neq 0$. Without density, we have a weaker lower bound, which suffices for our purposes.
\begin{lem} \label{growth_wreath} Let $\gamma\colon B'\rightarrow A$ be a linear map. Then $w_\gamma(n)\leq g_C(2n)$ and $g_C(n)\leq g_B(n)^2w_\gamma(n)+g_B(n)$, and thus $w_\gamma(n) \preceq g_C(n) \preceq g_B(n)^2 w_\gamma(n)$. \end{lem} \begin{proof} The inequality $g_C(n)\preceq g_B(n)^2w_\gamma(n)$ is \cite[Corollary~3.6]{AAJZ_TransAMS} and its quantitative form readily follows from \cite[Lemma~3.5]{AAJZ_TransAMS}.
For the other inequality, recall that by Lemma \ref{decomposition}, given $b_1,\dots,b_s\in~B$, the map $c_\gamma b_1 c_\gamma b_2 \cdots c_\gamma b_s$ takes the form $x\mapsto 1\otimes \gamma(b_1)\cdots \gamma(b_s x)$. In particular, the substitution map: $$ T\colon {\operatorname{Lin}}_F(B',B'\otimes_F A) \rightarrow B'\otimes_F A $$ $$ T\colon \psi \mapsto \psi(1) $$ surjectively carries: $$T\left( \sum_{i_1+\cdots+i_s\leq n} c_\gamma V^{i_1} c_\gamma V^{i_2}\cdots c_\gamma V^{i_s}\right) = 1\otimes W_n$$ hence: $$\dim_F \sum_{i_1+\cdots+i_s\leq n} c_\gamma V^{i_1} c_\gamma V^{i_2}\cdots c_\gamma V^{i_s} \geq w_\gamma(n).$$ Since: $$\sum_{i_1+\cdots+i_s\leq n} c_\gamma V^{i_1} c_\gamma V^{i_2}\cdots c_\gamma V^{i_s} \subseteq \left(V+Fc_\gamma\right)^{\leq 2n},$$ it follows that $w_\gamma(n)\leq g_C(2n)$. \end{proof}
\section{Linear maps with oscillating growth}
A (non-decreasing) function $f\colon \mathbb{N}\rightarrow\mathbb{N}$ is \textit{subexponential}, if $\limsup_{n\rightarrow \infty} \sqrt[n]{f(n)} =~ 1$.
\begin{prop} \label{Oscillating_transformation} Let $B$ be a finitely generated, infinite-dimensional unital algebra. Let $f_1,f_2\colon\mathbb{N}\rightarrow\mathbb{N}$ be monotone non-decreasing functions such that: \begin{itemize} \item $f_1(n)\xrightarrow{n\rightarrow\infty} \infty$ (perhaps very slowly); and \item $f_2(n)$ is subexponential. \end{itemize} Then there exists a locally nilpotent algebra $A$ with a linear map $\gamma\colon B\rightarrow~A$ such that its growth function $w_\gamma$ satisfies: \begin{itemize} \item $w_\gamma(n)\leq f_1(n)$ infinitely often; and \item $w_\gamma(n)\geq f_2(n)$ infinitely often. \end{itemize} \end{prop}
\begin{proof} Assume that: $$1<d_1\leq d_2\leq \cdots$$ $$1<n_1<m_1<n_2<m_2<\cdots$$ are sequences of positive integers which are to be determined in the sequel. In the countably generated free (non-unital) algebra: $$ F\left<x_1,x_2,\dots\right>_{\geq 1} $$ consider the ideals:
$$ I_M=\left< x_{i_1}\cdots x_{i_{d_M}}\ |\ i_1,\dots, i_{d_M}\leq M \right> $$ for $M=1,2,\dots$ and notice that $I_{M'}\cap F\left<x_1,\dots,x_M\right>_{\geq 1} \subseteq I_M$ for $M'\geq M$, since $\{d_i\}_{i=1}^{\infty}$ is non-decreasing. Let $I=\sum_{M=1}^\infty I_M\triangleleft F\left<x_1,x_2,\dots\right>_{\geq 1}$. Let: $$ A=F\left<x_1,x_2,\dots\right>_{\geq 1}/I $$ It is clear that $A$ is locally nilpotent.
Let $1\in V$ be a generating subspace of $B$. We define a linear map $\gamma\colon B\rightarrow~A$ along with coherent constraints on the sequences $\{d_i,m_i,n_i\}_{i=1}^{\infty}$ mentioned above.
Take $p\in \mathbb{N}$ such that $f_2(2p) \leq 2^p$ and let $n_1=2p$ (this is possible since $f_2$ is subexponential). Pick a basis $b_1,\dots,b_t$ for $V^{2p}=V^{n_1}$ such that $b_1,b_2\in V^2$. Set $\gamma(b_1)=x_1,\gamma(b_2)=x_2$ and $\gamma(b_i)=0$ for $2<i\leq t$. Let $d_1=\cdots=d_{n_1}=p+1$. Then all length-$p$ free monomials in $ x_1,x_2 \in \gamma(V^2) $ are linearly independent in $A$, so: \begin{eqnarray*} f_2(n_1) & \leq & 2^p \\ & \leq & \dim_F \gamma\left(V^2\right)^p \\ & \leq & \dim_F W_{2p} \\ & = & w_\gamma(2p) \\ & = & w_\gamma(n_1) \end{eqnarray*}
Suppose that $n_1,\dots, n_i,d_1,\dots,d_{n_i}$ were defined, and $\gamma$ was set on $V^{n_i}$ such that $\gamma\left(V^{n_i}\right)\subseteq \overline{F\left<x_1,\dots,x_{n_i}\right>}_{\geq 1}$ (namely, the subalgebra of $A$ generated by $x_1,\dots,x_{n_i}$). The latter is a nilpotent algebra of nilpotency index at most $d_{n_i}$, hence its dimension is at most $K=K(i,n_i,d_{n_i})$. Take $m_i\gg n_i$ such that $f_1(m_i)\geq K$. Fix a vector space complement $V^{m_i}=V^{n_i}\oplus W$ and set $\gamma\left( W\right)=0$. Then: \begin{eqnarray*} w_\gamma(m_i) & = & \dim_F \sum_{j_1+\cdots+j_s\leq m_i} \gamma\left(V^{j_1}\right)\cdots \gamma\left(V^{j_s}\right) \\ & \leq & \dim_F F\left<\gamma\left(V^{n_i}\right)\right> \\ & \leq & \dim_F \overline{F\left<x_1,\dots,x_{n_i}\right>}_{\geq 1} \\ & \leq & K \leq f_1(m_i). \end{eqnarray*} Take also $ d_{n_i+1} = \cdots = d_{m_i} $ to be $d_{n_i}$.
Now suppose that $m_i$ has been defined and $\gamma$ was set on $V^{m_i}$ as above. Let us specify $n_{i+1}$ and extend $\gamma$ to $V^{n_{i+1}}$. Since $f_2$ is subexponential, for $x\gg 1$ we have that $f_2(x)\leq 2^{\frac{x}{m_i+2}}$, so take $ n_{i+1} \geq m_i+2 $ such that $f_2(n_{i+1})\leq 2^{\frac{n_{i+1}}{m_i+2}}$. Moreover, take: $$ (*)\ \ d_{m_i+1}=d_{m_i+2}=\cdots = d_{n_{i+1}}=n_{i+1}+1.$$
Pick $v_1,v_2\in V^{m_i+2}$ whose images modulo $V^{m_i}$ are linearly independent and fix a complement: $$ V^{n_{i+1}}=V^{m_i}\oplus Fv_1\oplus Fv_2\oplus W. $$ Set $\gamma(v_1)=x_{m_i+1},\gamma(v_2)=x_{m_i+2}$ and $\gamma\left(W\right)=~0$.
It follows from $(*)$ that all length-$\leq n_{i+1}$ free monomials in $x_{m_i+1},x_{m_i+2}$ are linearly independent in $A$. Hence: \begin{eqnarray*} f_2(n_{i+1}) & \leq & 2^{\frac{n_{i+1}}{m_i+2}} \\ & \leq & \gamma\left( V^{m_i+2} \right)^{\frac{n_{i+1}}{m_i+2}} \\ & \leq & \dim_F W_{n_{i+1}} \\ & = & w_\gamma(n_{i+1}) \end{eqnarray*}
(we may assume that $ m_i+2 | n_{i+1} $ for simplicity) and the claim follows. \end{proof}
\section{Approximating growth functions by linear maps}
We start with the following construction, which appears in \cite{SmoktunowiczBartholdi}, brought here in a slightly improved version.
Let $f\colon \mathbb{N}\rightarrow \mathbb{N}$ be a non-decreasing function satisfying $f(2^{n+1})\leq f(2^n)^2$ for every $n\geq 0$. Let $X$ be a set of cardinality $f(1)$. Define a sequence of sets as follows. Let $W(1)=X$. Suppose that $W(2^n)\subseteq~X^{2^n}$ is given such that $|W(2^n)|\geq f(2^n)$ and pick an arbitrary subset $C(2^n)\subseteq W(2^n)$ of cardinality $\lceil \frac{f(2^{n+1})}{f(2^n)} \rceil$. This is indeed possible since $\frac{f(2^{n+1})}{f(2^n)}\leq f(2^n)\leq |W(2^n)|$. Let $W(2^{n+1})=W(2^n)C(2^n)$.
Consider the following set of right-infinite words: $$ \mathcal{S} = W(1)C(1)C(2)C(2^2)C(2^3)C(2^4)\cdots \subseteq X^{\infty} $$
Let $h(n)$ be the number of words of length at most $n$ which occur as factors of some infinite word in $\mathcal{S}$; we call them factors of $\mathcal{S}$.
\begin{lem} \label{SBmon} Under the above notations, there exist $C,D>0$ such that: $$h(n)\leq Cn^3f(Dn)$$ Moreover, if $f(2^k)$ divides $f(2^{k+1})$ for all $k$ then: $$h(n)\leq Cn^2f(Dn)$$ \end{lem}
\begin{proof} By \cite[Lemma~6.3]{SmoktunowiczBartholdi}, every length-$2^m$ word which factors $\mathcal{S}$ is a subword of some word from $W(2^m)C(2^m)\cup C(2^m)W(2^m)$. Each word in $W(2^m)C(2^m)\cup C(2^m)W(2^m)$ has length $2^{m+1}$, so the number of length-$2^m$ words, which is equal to $h'(2^m)=h(2^m)-h(2^m-1)$, is at least $\#W(2^m)$ and at most: $$ ( 2^m + 1 ) \cdot \#\left(W(2^m)C(2^m)\cup C(2^m)W(2^m)\right) $$ Since $\#W(2^{m+1})=\#W(2^m)\cdot \lceil \frac{f(2^{m+1})}{f(2^m)} \rceil$, it follows by induction that $f(2^m)\leq \#W(2^m)\leq 2^m f(2^m)$. Notice that if $f(2^k)$ divides $f(2^{k+1})$ for all $k$ then $\#W(2^m)=f(2^m)$. Therefore: $$ f(2^m) \leq h'(2^m) \leq (2^m+1) \cdot \left(2\cdot 2^mf(2^m) \cdot \lceil \frac{f(2^{m+1})}{f(2^m)} \rceil \right) \leq 2^{2m+3}f(2^{m+1}) $$ and if $f(2^k)$ divides $f(2^{k+1})$ for all $k$ then: $$ f(2^m) \leq h'(2^m) \leq (2^m+1) \cdot \left(2\cdot f(2^m) \cdot \frac{f(2^{m+1})}{f(2^m)}\right) \leq 2^{m+2}f(2^{m+1}) $$ Since every monomial which factors $\mathcal{S}$ is right-extendable to a longer factor of $\mathcal{S}$, it follows that $h'$ is non-decreasing, so: $$ f(2^m)\leq h(2^m)\leq 2^m h'(2^m)\leq 2^{3m+3}f(2^{m+1}) $$ and in the case that $f(2^k)$ divides $f(2^{k+1})$ for all $k$ then moreover: $$ f(2^m)\leq h(2^m)\leq 2^m h'(2^m)\leq 2^{2m+2}f(2^{m+1}). $$ Since $h,f$ are non-decreasing then for every $n$, if we take $2^m\leq n\leq 2^{m+1}$ then we obtain that $f(n)\leq f(2^{m+1}) \leq h(2^{m+1}) \leq h(2n)$ and $h(n)\leq h(2^{m+1})\leq 2^{3(m+1)+3}f(2^{m+2})\leq 64 n^3f(4n)$, and in the case that $f(2^k)$ divides $f(2^{k+1})$ for all $k$, we further have that $h(n)\leq h(2^{m+1})\leq 2^{2(m+1)+2}f(2^{m+2})\leq 16 n^2 f(4n)$. \end{proof}
\begin{prop} \label{Realizing_transformation} Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be an increasing, submultiplicative function and let $\delta\colon \mathbb{N}\rightarrow \mathbb{N}$ be a non-decreasing function tending to infinity (perhaps very slowly; we always assume $\delta(n)\leq n$).
Let $B$ be a finitely generated infinite-dimensional unital algebra. Then there exists a countably generated, locally nilpotent algebra $A$ and a linear map $\gamma\colon B\rightarrow~A$ such that:
$$f\left(\frac{n}{\delta(n)}\right)\preceq w_\gamma(n)\preceq n^5 f(n).$$ \end{prop}
\begin{proof} Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be an increasing, submultiplicative function and let $d=f(1)$.
For each $k\in \mathbb{N}$ let $X_k=\{x_{k,1},\dots,x_{k,d}\}$ and for an arbitrary subset $S\subseteq X_k^n$ denote by $S^{[m]}$ the subset of $X_{k+m}^n$ obtained by substituting each letter $x_{k,i}$ by $x_{k+m,i}$. Let $X=\bigcup_{k=1}^{\infty} X_k$. We call $k$ the \textit{level} of $x_{k,i}$.
Define a sequence of sets as follows. Let $W(1)=X_1$. Suppose that $W(2^n)\subseteq~X_1^{2^n}$ has been defined such that $|W(2^n)|\geq f(2^n)$ and pick an arbitrary subset $C(2^n)\subseteq W(2^n)$ of cardinality $\lceil \frac{f(2^{n+1})}{f(2^n)} \rceil$ (as before, this is indeed possible since $f(2^{n+1})^2\leq f(2^n)^2$). Let $W(2^{n+1})=W(2^n)C(2^n)$. Consider the following set of right-infinite words: $$ \mathcal{S} = W(1)C(1)C(2)C(2^2)C(2^3)C(2^4)\cdots \subseteq X_1^{\infty} $$
Let: $$ A_1 = F\left<x_{1,1},\dots,x_{1,d}\right>_{\geq 1} / \left< w\ |\ w\ \text{is not a factor of}\ \mathcal{S} \right> $$ and let $h(n)$ be the number of non-zero monomials of length at most $n$ in $A_1$. By Lemma \ref{SBmon}, it follows that $h(n)\leq Cn^3f(Dn)$ for suitable constants $C,D>0$.
Let $\delta\colon \mathbb{N}\rightarrow\mathbb{N}$ be a non-decreasing function tending to infinity. Fix a sequence $1\leq n_1\leq n_2\leq \cdots$ of positive integers such that for every $r\in \mathbb{N}$:
$$ \min \{ k\ |\ r\leq n_k-\log_2 k \} \leq \delta(r) $$
which is indeed possible, since $\delta(r)\xrightarrow{r\rightarrow \infty} \infty$, e.g. take $n_k=2\max\{i|\delta(i)\leq k\}$. Consider the following set of right-infinite words:
\begin{eqnarray*} \mathcal{T} & = & W(1)C(1)\cdots C(2^{n_1}) \\ & \cdots & C(2^{n_1+1})^{[1]}\cdots C(2^{n_2})^{[1]} \\ & \vdots & \\ & \cdots & C(2^{n_i+1})^{[i]}\cdots C(2^{n_{i+1}})^{[i]}\cdots \subseteq X^{\infty} \end{eqnarray*}
Let $\pi\colon \mathcal{T}\rightarrow \mathcal{S}$ be the function given by substituting each $x_{i,j}$ by $x_{1,j}$. Consider the countably generated non-unital free algebra $F\left<X\right>_{\geq 1}$, and let: $$A=F\left<X\right>_{\geq 1} / \left< w\ |\ w\ \text{is not a factor of}\ \mathcal{T} \right>$$ For simplicity, we identify each $x_{i,j}$ with its image in $A$. Observe that $A$ is locally nilpotent.
Let $B$ be a finitely generated infinite-dimensional algebra, generated by a finite-dimensional subspace $V$ such that $1\in V$ and $\dim_F V^{i+1} - \dim_F V^i \geq d$. This is indeed possible, enlarging $V$ if necessary. Define a linear map $\gamma\colon B\rightarrow A$ as follows. Fix decompositions of $F$-vector spaces: \begin{eqnarray*} V & = & Fa_{1,1}\oplus\cdots \oplus Fa_{1,d}\oplus W_0 \\ \text{For}\ i\geq 1,\ \ \ V^{i+1} & = & V^i \oplus Fa_{i+1,1}\oplus\cdots \oplus Fa_{i+1,d} \oplus W_i \end{eqnarray*} Define $\gamma(a_{i,j})=x_{i,j}$ and $\gamma\left(W_{i-1}\right)=0$ for all $i\geq 1$ and $1\leq j\leq d$.
We now turn to estimate $w_\gamma (n)$. Pick a finite factor $u$ of $\mathcal{T}$. Observe that it is uniquely determined by $\pi(u)$ together with the level of its first letter and the displacement of the first letter of the next level within $u$ (if such a letter appears in $u$).
Notice that $W_n = \sum_{j_1+\cdots+j_s\leq n} \gamma\left(V^{j_1}\right)\cdots \gamma\left(V^{j_s}\right)$ is spanned by monomials in $A$ of length at most $n$. Moreover, the level of every letter in any of these monomials is at most $n$.
Therefore, \begin{eqnarray*} (*)\ \ \ \ \ w_\gamma(n) & = & \dim_F W_n \\ & \leq & \#\{\text{Length-}\leq n\ \text{mmonomials in}\ \mathcal{S}\} \\ & \cdot & \#\{\text{Level of first letter}\} \\ & \cdot & \#\{\text{Displacement of letter with next level (if exists)}\} \\ & \leq & h(n)\cdot n \cdot n\leq Cn^5 f(Dn).\end{eqnarray*}
We now turn to bound $w_\gamma(n)$ from below. Fix a power of $2$, say, $2^m$. Take $k\leq~\delta(m)$ such that $m\leq n_k-\log_2 k$, which is indeed possible by the way we have chosen $\{n_k\}_{k=1}^{\infty}$. In particular, $2^m\leq 2^{n_k}$; observe that every length-$2^m$ prefix of a word in $\mathcal{T}$ is a prefix of a words from $W(1)C(1)\cdots C(2^{n_k})^{[k-1]}$. Thus every length-$2^m$ prefix of a word in $\mathcal{T}$ belongs to $W_{k\cdot 2^m}$ (since the level of each letter is at most than $k$). By `forgetting' the levels (namely, applying $\pi$) we get:
\begin{eqnarray*} f(2^m) & \leq & f(1)\lceil \frac{f(2)}{f(1)}\rceil \lceil \frac{f(2^2)}{f(2)}\rceil \cdots \lceil \frac{f(2^m)}{f(2^{m-1})}\rceil \\ & = & \# W(1)C(1)C(2)\cdots C(2^{m-1}) \\ & \leq & \dim_F W_{k\cdot 2^m} \end{eqnarray*} Hence: $$ f(2^m) \leq w_\gamma(k\cdot 2^m) \leq w_\gamma(\delta(m) 2^m) $$ Let $n$ be arbitrary and let $2^m\leq n\leq 2^{m+1}$. Then, since $f,w_\gamma,\delta$ are monotonely non-decreasing: \begin{eqnarray*} (**)\ \ \ \ \ f(n) & \leq & f(2^{m+1}) \\ & \leq & w_\gamma(\delta(m+1)\cdot 2^{m+1}) \\ & \leq & w_\gamma(2\delta(n)n) \end{eqnarray*} Let $N$ be given. Let $n=\lfloor \frac{N}{\delta(N)} \rfloor$. Then $\delta(n)n\leq \delta(N)\cdot \lfloor \frac{N}{\delta(N)} \rfloor \leq N$. Then by $(*),(**)$: \begin{eqnarray*} f\left( \lfloor\frac{N}{\delta(N)}\rfloor \right) & = & f(n) \\ & \leq & w_\gamma(2\delta(n)n) \\ & \leq & w_\gamma(2N) \\ & \leq & 32CN^5 f(2DN), \end{eqnarray*} and the proof is completed. \end{proof}
\section{Growth of nil algebras}
\subsection{Nil algebras with oscillating growth}
\begin{proof}[{Proof of Theorem B1}] Let $f\colon \mathbb{N}\rightarrow \mathbb{N}$ be a subexponential function. Let $F$ be a countable field. Let $B$ be a finitely generated nil $F$-algebra of Gel'fand-Kirillov dimension at most $3$, which exists by \cite{LenaganSmoktunowiczYoung}. By Proposition \ref{Oscillating_transformation}, there exists a locally nilpotent algebra $A$ and a linear map $\gamma\colon B'\rightarrow A$ such that $w_\gamma(n)\geq f(2n)$ infinitely often and $w_\gamma(n)\leq \log_2 n$ infinitely often. Let $C$ be the subalgebra of $A\wr B$ generated by $B$ and $c_\gamma$. By \cite[§4]{AAJZ_TransAMS}, $C$ is nil. By Lemma \ref{growth_wreath} we have that $g_C(n)\leq g_B(n)^2w_\gamma(n)+g_B(n)$ so infinitely often $g_C(n)\leq n^{6+\varepsilon}$, for every $\varepsilon>0$. In addition, we have $w_\gamma(n)\leq g_C(2n)$, so infinitely often $f(2n)\leq g_C(2n)$. \end{proof}
\begin{proof}[{Proof of Theorem B2}]
Let $f\colon \mathbb{N}\rightarrow~\mathbb{N}$ be a subexponential function and let $\omega\colon \mathbb{N}\rightarrow~\mathbb{N}$ be a non-decreasing super-polynomial function.
Let $F$ be an arbitrary field. Let $B$ be a finitely generated nil $F$-algebra whose growth function satisfies $g_B(n)\leq \omega(n)^{1/3}$; this is indeed possible by \cite{BellYoung}. By Proposition \ref{Oscillating_transformation}, there exists a locally nilpotent algebra $A$ and a linear map $\gamma\colon B'\rightarrow A$ such that $w_\gamma(n)\geq f(2n)$ infinitely often and $w_\gamma(n)\leq \log_2 n$ infinitely often. Let $C$ be the subalgebra of $A\wr B$ generated by $B$ and $c_\gamma$. By \cite[§4]{AAJZ_TransAMS}, $C$ is nil. By Lemma \ref{growth_wreath} we have that: $$g_C(n)\leq g_B(n)^2w_\gamma(n)+g_B(n).$$ Therefore, infinitely often: $$g_C(n)\leq \omega(n)^{2/3}\log_2 n + \omega(n)^{1/3} \leq \omega(n)$$ where the last equality holds for $n\gg 1$. In addition, we have $w_\gamma(n)\leq~g_C(2n)$, so infinitely often $f(2n)\leq g_C(2n)$. \end{proof}
\begin{rem} Since in Proposition \ref{Oscillating_transformation} each $n_k$ can be chosen to be arbitrarily large compared to $\{m_i,n_i\}_{i=1}^{k}$ and each $m_k$ can be chosen arbitrarily large compared to $\{m_i,n_i\}_{i=1}^{k}\cup \{n_k\}$, we can clearly make sure that in Theorems B1, B2 (respectively) the \textit{upper densities} satisfy:
$$ \limsup_{N\rightarrow \infty} \frac{\#\{k\in [1,N]\ |\ g_C(n)\leq n^{6+\varepsilon}\}}{N} = 1, $$ respectively:
$$ \limsup_{N\rightarrow \infty} \frac{\#\{k\in [1,N]\ |\ g_C(n)\leq \omega(n)\}}{N} = 1 $$ and (in both B1, B2):
$$ \limsup_{N\rightarrow \infty} \frac{\#\{k\in [1,N]\ |\ g_C(n)\geq f(n)\}}{N} = 1 $$ Thus, considering the values of $g_C(n)$ for $n$ ranging up to a given value, it simultaneously looks arbitrarily fast and arbitrarily slow. \end{rem}
\subsection{Realizing growth functions of nil algebras}
\begin{proof}[{Proof of Theorem A1}] Let $F$ be a countable field and let $B$ be a finitely generated nil $F$-algebra of Gel'fand-Kirillov dimension at most $3$, which exists by \cite{LenaganSmoktunowiczYoung}. In particular, for every $\varepsilon>0$, $g_B(n)\preceq n^{3+\varepsilon}$. Let $f\colon \mathbb{N}\rightarrow \mathbb{N}$ be an increasing, submultiplicative function and let $\delta\colon \mathbb{N}\rightarrow \mathbb{N}$ be an arbitrarily slow non-decreasing function tending to infinity. Then by Proposition \ref{Realizing_transformation} there exists a locally nilpotent algebra $A$ and a linear map $\gamma\colon B'\rightarrow A$ such that: $$f\left(\frac{n}{\delta(n)}\right)\preceq w_\gamma(n)\preceq n^5 f(n)$$ Let $C$ be the subalgebra of $A\wr B$ generated by $B$ and $c_\gamma$. By \cite[§4]{AAJZ_TransAMS}, $C$ is nil. By Lemma \ref{growth_wreath}, for every $\varepsilon>0$ we have: \begin{eqnarray*} f\left(\frac{n}{\delta(n)}\right) \preceq w_\gamma(n) & \preceq & g_C(n) \\ & \preceq & g_B(n)^2w_\gamma(n) \preceq n^{11+\varepsilon}f(n) \end{eqnarray*} as claimed. \end{proof}
\begin{proof}[{Proof of Theorem A2}] Let $F$ be an arbitrary field. Let $f\colon \mathbb{N}\rightarrow \mathbb{N}$ be an increasing, submultiplicative function and let $\omega\colon \mathbb{N}\rightarrow \mathbb{N}$ be a non-decreasing, super-polynomial function. Let $\delta\colon \mathbb{N}\rightarrow \mathbb{N}$ be an arbitrarily slow non-decreasing function tending to infinity.
Let $B$ be a finitely generated nil $F$-algebra whose growth satisfies $g_B(n)\preceq \omega(n)^{1/3}$; this is indeed possible by \cite{BellYoung}. By Proposition \ref{Realizing_transformation} there exists a locally nilpotent algebra $A$ and a linear map $\gamma\colon B' \rightarrow A$ such that: $$f\left(\frac{n}{\delta(n)}\right)\preceq w_\gamma(n)\preceq n^5 f(n)$$ Let $C$ be the subalgebra of $A\wr B$ generated by $B$ and $c_\gamma$. By \cite[§4]{AAJZ_TransAMS}, $C$ is nil. By Lemma \ref{growth_wreath}, we have: \begin{eqnarray*} f\left(\frac{n}{\delta(n)}\right) \preceq w_\gamma(n) & \preceq & g_C(n) \\ & \preceq & g_B(n)^2w_\gamma(n) \preceq \omega(n)^{2/3}n^5 f(n)<\omega(n)f(n) \end{eqnarray*} where the last inequality hold for $n\gg 1$, and the proof is completed. \end{proof}
\section{Applications and specifications}
\subsection{Nil Lie algebras of oscillating growth} Our next application is a construction of a finitely generated nil restricted Lie algebras whose growth oscillates between a polynomial and an arbitrarily rapid (subexponential) function.
In \cite{PetrogradskyLie} Petrogradsky constructed nil restricted Lie algebras with oscillating growth over fields of positive characteristic; our constructions are not subject to this restriction. Moreover, the upper bounds in our constructions are arbitrarily rapid (subexponential), unlike the upper bounds in \cite{PetrogradskyLie}. However, our lower bounds -- though polynomial, at least in the countable case -- do not get close to linear as Petrogradsky's examples. This is inevitable, since they share their growth rates with associative algebras.
\begin{cor} \label{oscillatingLie} Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be an arbitrarily rapid subexponential function. Let $F$ be a countable field of ${\operatorname{char}}(F)\neq 2$. Then there exists a finitely generated nil Lie algebra $L$ over $F$ such that: \begin{itemize} \item $g_L(n)\leq n^{6+\varepsilon}$ infinitely often for every $\varepsilon>0$; and \item $g_L(n)\geq f(n) $ infinitely often. \end{itemize} \end{cor}
\begin{cor} \label{oscillatingLie2} Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be an arbitrarily rapid subexponential function and let $\omega\colon \mathbb{N}\rightarrow \mathbb{N}$ be an arbitrarily slow, non-decreasing, super-polynomial function.
Let $F$ be an uncountable field of ${\operatorname{char}}(F)\neq 2$. Then there exists a finitely generated nil Lie algebra $L$ over $F$ such that: \begin{itemize} \item $g_L(n)\leq \omega(n) $ infinitely often; and \item $g_L(n)\geq f(n) $ infinitely often. \end{itemize} \end{cor}
\begin{proof}[{Proof of Corollaries \ref{oscillatingLie},\ref{oscillatingLie2}}] By \cite{AlahmadiAlharthi}, if $A$ is a nil associative algebra over a field of characteristic $\neq 2$ then the Lie algebra $L=[A,A]$ is finitely generated of growth $g_L(n)\sim g_A(n)$. It is easy to see that we may assume that: $$ g_L(n)\leq g_A(n),\ g_A(n)\leq Rg_L(Rn) $$ for some $R\in \mathbb{N}$ (enlarging the generating subspace of $A$ is necessary). Now use Theorems B1, B2 with $\bar{f}(n)$ defined as follows.
We may assume that $f$ is non-decreasing. Since $\frac{1}{n}\log_2 f(n)\xrightarrow{n\rightarrow \infty} 0$, we can find a non-decreasing subexponential function $\theta\colon \mathbb{N}\rightarrow \mathbb{N}$ such that $\theta(n)\xrightarrow{n \rightarrow \infty}~ \infty$ and $\theta(n)\cdot~ \frac{1}{n}\log_2 f(n)\rightarrow 0$. Then the function $\bar{f}(n):=f(n\theta(n))\theta(n)$ is still subexponential: \begin{eqnarray*} \frac{1}{n}\log_2 \left( f(n\theta(n))\theta(n) \right) & = & \frac{1}{n}\log_2 \theta(n) + \frac{1}{n}\log_2 f(n\theta(n)) \\ & = & o(1) + \frac{\theta(n)}{n\theta(n)}\log_2 f(n\theta(n)) \\ & \leq & o(1) + \frac{\theta(n\theta(n))}{n\theta(n)}\log_2 f(n\theta(n)) = o(1) \end{eqnarray*}
Hence if $g_A(n)\geq \bar{f}(n)$ and $n\gg 1$ then: $$ g_L(Rn)\geq \frac{1}{R} g_A\left( n \right)\geq \frac{1}{R}\bar{f}(n) = \frac{1}{R}\theta(n) f\left( n \theta(n)\right)\geq f(Rn) $$ so $g_L(n)\geq f(n)$ infinitely often. In addition, $g_L(n)\leq g_A(n)$ so $g_L(n)\leq n^{6+\varepsilon}$ (in the countable case) and $g_L(n)\leq \omega(n)$ (in the arbitrary case) infinitely often.
Since $A$ is nil, it is evident that $x\mapsto [a,x]$ is nilpotent for any $a\in L$, so $L$ is a nil Lie algebra. \end{proof}
\subsection{Nil algebras of intermediate growth}
In \cite{Smoktunowicz}, Smoktunowicz constructed the first nil algebra of intermediate growth. The example, obtained as a suitable quotient of a nil Golod-Shafarevich algebra, is shown to have intermediate growth using the main theorem therein \cite[Theorem~A]{Smoktunowicz}, which gives (very technical) upper and lower bounds on the growth of graded algebras with sparse homogeneous relations; in particular, Smoktunowicz is able to construct a nil algebra \cite[Corollary~4.5]{Smoktunowicz} whose growth is smaller than $n^{c\ln^2 n}$ but infinitely often greater than $n^{d\ln n}$ for suitable constants $c,d>0$.
Theorems A1, A2 proven here enable one to construct nil algebras with a much finer control on their growth rates, thereby establishing a significant step toward proving the conjecture that any (non-linear) function which occurs as the growth rate of an algebra is realizable as the growth of a nil algebra (see \cite{AAJZ_ERA}). For example, one can construct nil algebras whose growth is bounded between $n^{c\ln^\alpha n}$ and $n^{d\ln^\alpha n}$ for an arbitrary $\alpha\in (0,\infty)$ (and some $c,d>0$); and nil algebras with arbitrary GK-superdimension, namely, whose growth functions are between $\exp(n^{\alpha-\varepsilon})$ and $\exp(n^{\alpha+\varepsilon})$ for all $\varepsilon>0$ and $\alpha\in (0,1)$ arbitrary.
Another evidence for the richness of the variety of possible growth functions of nil algebras that Theorems A1, A2 afford is given by the notion of `$q$-dimensions', introduced by Petrogradsky in his analysis of the growth of universal enveloping algebras of Lie algebras \cite{Petrogradsky96,Petrogradsky00,Petrogradsky93}. More specifically, consider the following hierarchy of growth functions for $q=1,2,3,\dots$ and $\alpha\in \mathbb{R}^{+}$: $$ \Phi_\alpha^{1}(n) = \alpha,\ \Phi_\alpha^{2}(n) = n^\alpha,\ \Phi_\alpha^{3}(n) = \exp(n^{\alpha/\alpha+1}),\ \text{and}\ \Phi_\alpha^{q>3}(n) = \exp\left(\frac{n}{(\ln^{(q-3)} n)^{1/\alpha}}\right)\ $$ where $\ln^{(1)} n=\ln n,\ \ln^{(q+1)} n=\ln(\ln^{(q)} n)$. For an algebra $A$ whose growth function is $g_A(n)$, define: \begin{eqnarray*}
\operatorname{Dim}^q(A) & := & \inf\{\alpha\in\mathbb{R}^{+}|\ g_A(n)\leq \Phi_\alpha^q(n)\ \forall n\gg 1\} \\ & = & \sup\{\alpha\in\mathbb{R}^{+}|\ g_A(n)\geq \Phi_\alpha^q(n)\ \forall n\gg 1\} \end{eqnarray*} if they are equal to each other. Petrogradsky proved that if $\operatorname{Dim}^q(\mathfrak{g})=\alpha$ then $\operatorname{Dim}^{q+1}(U(\mathfrak{g}))=~\alpha$. He left open the problem of whether every (a priori possible) real number is realizable as the $q$-dimension of some associative/Lie algebra (see \cite[Remark in page 347]{Petrogradsky00} and \cite[Remark in page 651]{Petrogradsky97}). This is indeed true, and even within the classes of finitely generated simple associative and Lie algebras (see \cite{SimpleLie}). \begin{cor} \label{intermediateLieapp} Theorems A1, A2 remain true if $R$ is replaced by a finitely generated nil Lie algebra. \end{cor}
\begin{proof} This is immediate by \cite{AlahmadiAlharthi}. \end{proof}
Using Theorems A1, A2 for Lie algebras, it is easy to construct nil associative and nil Lie algebras with arbitrary $q$-dimensions as well (e.g. taking $\delta(n)=\log_2 n,\ \omega(n)=n^{\log_2 n}$).
\section{Domains with oscillating growth}
We now construct examples of noncommutative domains of oscillating growth, in a similar spirit of the Kassabov-Pak construction \cite{KassabovPak}. We start with another construction of Lie algebras of oscillating growth, whose behavior is stricter than of the aforementioned constructions. We then consider their universal enveloping algebras and utilize a finitary version of the connection between the growth of $L$ and $U(L)$ studied by Smith (and significantly extended by Petrogradsky).
\begin{lem} \label{lemdom} Let $f\colon \mathbb{N}\rightarrow \mathbb{N}$ be a non-decreasing subexponential function. Let: $$1<n_1<n'_1<m_1<m'_1<n_2<n'_2<\cdots$$ be a sequence of positive integers, arbitrary subject to the assumption that each integer is large enough than its predecessor.
Then there exists a finitely generated associative algebra $A$, a finitely generated Lie algebra $L$ and a constant $c\in \mathbb{N}$ such that for every $i\gg 1$: \begin{itemize}
\item If $k\in [2^{n_i+c},2^{n'_i}]$ then $g_A(k),g_L(k)\geq f(k)$
\item If $k\in [2^{m_i},2^{m'_i}]$ then $g_A(k),g_L(k)\leq k^2\log_2 k$ \end{itemize}
\end{lem} \begin{proof}
We start by defining a function $g\colon\mathbb{N}\rightarrow \mathbb{N}$, first on powers of $2$, then extending $g$ to $\mathbb{N}$.
There exists $r$ such that for all $s\geq r$ we have $f(2^s)\leq 2^{2^s}$, since $f$ is subexponential. Fix such $n_1\geq r$ for such an $r$ and set: $$g(1)=2,g(2)=2^2,\dots, g(2^{n'_1})=2^{2^{n'_1}}$$ then it follows that $g(2^k)\geq f(2^k)$ for all $k\in [n_1,n'_1]$.
Suppose that $n'_i$ has been defined, along with the values of $g$ for all powers of $2$ until $2^{n'_i}$. Let $c=g(2^{n'_i})$. Take $m_i\geq 2^c$ and set $g(2^{t+1})=g(2^t)$ for all $n'_i\leq t< m'_i$. It follows that for all $k\in [m_i,m'_i]$ we have $g(2^k)=c \leq \log_2 m_i\leq \log_2 k$.
Now suppose that $m'_i$ has been defined, along with the values of $g$ for all powers of $2$ until $2^{m'_i}$. Let $M=g(2^{m'_i})$. We know that $f$ is subexponential, so there exists $r$ such that for all $s\geq r$ we have $f(2^s)\leq \left(M^{1/2^{m'_i}}\right)^{2^s}$. Take $n_{i+1}\geq r$ for such an $r$. Define: $g(2^{t+1})=g(2^t)^2$ for all $m'_i\leq t<n'_{i+1}$. In particular, $$ g(2^k) = M^{2^{k-m'_i}} = \left(M^{1/{2^{m'_i}}}\right)^{2^{k}} \geq f(2^k) $$ for all $k\in [n_{i+1},n'_{i+1}]$.
We now extend $g$ to $\mathbb{N}$ by setting: \begin{itemize}
\item $g(k)=g\left(2^{\lceil \log_2 k \rceil}\right)$ if $k\in [2^{n_i},2^{n'_i}]$ for some $i\in \mathbb{N}$
\item $g(k)=g\left(2^{\lfloor \log_2 k \rfloor}\right)$ otherwise. \end{itemize}
It is clear by construction that $g \colon \mathbb{N} \rightarrow \mathbb{N}$ is non-decreasing and $g(2^{k+1})\leq g(2^k)^2$ for all $k\in \mathbb{N}$. Moreover, each $g(2^k)$ divides $g(2^{k+1})$ and it is straightforward that for every $i\in \mathbb{N}$: \begin{itemize}
\item $g(k)\geq f(k)$ for all $k\in [2^{n_i},2^{n'_i}]$
\item $g(k)\leq \log_2 \log_2 k$ for all $k\in [2^{m_i},2^{m'_i}]$ \end{itemize}
By Lemma \ref{SBmon}, there exists a finitely generated monomial algebra $A$ such that $g(n)\leq g_A(n) \leq Cn^2g(Dn)$ for all $n\in \mathbb{N}$ and some constants $C,D>0$.
It follows that for every $i\in \mathbb{N}$: \begin{itemize}
\item $g_A(n) \geq g(k)\geq f(k)$ for all $k\in [2^{n_i},2^{n'_i}]$
\item $g_A(k) \leq C'k^2\log_2\log_2 k$ for all $k\in [2^{m_i},2^{m'_i}]$ \end{itemize}
We recall the construction from Lemma \ref{SBmon}, to establish an important property of it: we construct a system of subsets $W(1)=\{x_1,x_2\}$, $C(2^n)\subseteq W(2^n)$ and $W(2^{n+1})=W(2^n)C(2^n)$. Notice that since $g(2^i)=2^{2^i}$ for all $1\leq i\leq n'_1$, there must exist a word $\rho\in W(2^{n'_1})$ which involves both $x_1,x_2$. But $g(2^{n'_1+1})=g(2^{n'_1})$, so by construction $C(2^{n'_1})$ is a singleton. We choose it to be $\{\rho\}$. It follows that every sufficiently long monomial in the constructed algebra $A$ contains a copy of $\rho$ as a factor, so in particular $x_1$ and $x_2$ are nilpotent.
Let $L=[A,A]$ be the commutator Lie algebra associated with $A$. By \cite{AlahmadiAlharthi} it follows that $L$ is a finitely generated Lie algebra whose growth is equivalent to that of $A$, and (enlarging the generating subspace of $A$ if needed) we may assume that: $$g_A(n) \leq Rg_{L}(Rn),\ g_L(n)\leq g_A(n),$$ for some $R\in \mathbb{N}$.
Replacing $f$ by $\bar{f}(n)=\theta(n) f(n\theta(n))$ which is still subexponential, and $\theta(n)\rightarrow \infty$ is a non-decreasing unbounded function (as in the proof of Corollaries \ref{oscillatingLie}, \ref{oscillatingLie2}), we get that for $i\gg 1$ if $k\in [2^{n_i+c},2^{n'_i}]$ for $c>\log_2 R$: $$ g_L(k)\geq \frac{1}{R}g_A\left(\lfloor \frac{k}{R} \rfloor \right) \geq \frac{1}{R} \bar{f}\left(\lfloor \frac{k}{R} \rfloor \right) = \frac{\theta\left(\lfloor k/R \rfloor\right)}{R}f\left(\lfloor \frac{k}{R}\rfloor\theta\left(\lfloor \frac{k}{R}\rfloor\right)\right) \geq f(k)$$ Therefore for $i\gg 1$: \begin{itemize}
\item $g_{L}(k)\geq f(k)$ for all $k\in [2^{n_i+c},2^{n'_i}]$
\item $g_{L}(k)\leq g_A(k)\leq k^2\log_2 k$ for all $k\in [2^{m_i},2^{m'_i}]$ \end{itemize}
The proof is completed. \end{proof}
\begin{proof}[{Proof of Theorem C}] Given: $$1<n_1<n'_1<m_1<m'_1<n_2<n'_2<\cdots$$ such that each integer is larger enough than its predecessor in a sense that will be made clear later, let $L$ be a finitely generated Lie algebra $L$ generated by a space $V$ such that for every $i\gg 1$: \begin{itemize}
\item If $k\in [2^{n_i+c},2^{n'_i}]$ then $g_{L}(k)\geq f(k)$
\item If $k\in [2^{m_i},2^{m'_i}]$ then $g_{L}(k)\leq k^2\log_2 k$ \end{itemize} For some $c\in \mathbb{N}$. This is done using Lemma \ref{lemdom}. Let $U=U(L)$ be the universal enveloping algebra associated with $L$. Let $V'$ be the subspace of $U$ spanned by $1$ and the image of $V$ in $U$. The associated growth function satisfies $g_{U}(n)\geq g_{L}(n)$, so $g_{U}(n)\geq f(n)$ infinitely often.
Let $\{a_n\}_{n=1}^{\infty}, \{b_n\}_{n=1}^{\infty}$ be sequences of positive integers such that: $$ (1)\ \ \ \prod_{n=1}^{\infty} \frac{1}{(1-t^n)^{b_n}} = \sum_{n=0}^{\infty} a_nt^n $$ Petrogradsky proved \cite[Theorem~1.2]{Petrogradsky00} that if $b_n=n^{\alpha-1+o(1)}$ then $a_n=\exp\left(n^{\frac{\alpha}{\alpha+1}+o(1)}\right)$. By the Poincar\'e-Birkhoff-Witt Theorem (see \cite[Page 340]{Petrogradsky00}), the sequences $a_n=\gamma'_U(n),\ b_n=~\gamma'_L(n)$ satisfy $(1)$, namely: $$ (2)\ \ \ \prod_{n=1}^{\infty} \frac{1}{(1-t^n)^{g_L'(n)}} = \sum_{n=0}^{\infty} g_U'(n)t^n $$ It is easy to derive a finitary version, namely:
\begin{eqnarray*} g_U'(N) & = & {\operatorname{coeff}}_{t^N}\left( \prod_{n=1}^M \frac{1}{(1-t^n)^{g_L'(n)}} \right) \end{eqnarray*} for any $M\geq N$. For any $0\leq s\leq 2^{m'_i}$: $$ (3)\ g_U'(s) = {\operatorname{coeff}}_{t^{s}}\left( \prod_{n=1}^{2^{m'_i}} \frac{1}{(1-t^n)^{g_L'(n)}} \right) = {\operatorname{coeff}}_{t^s} \left( \prod_{n=1}^{2^{m_i}} \frac{1}{(1-t^n)^{g_L'(n)}} \cdot \prod_{n=2^{m_i}+1}^{2^{m'_i}} \frac{1}{(1-t^n)^{g_L'(n)}} \right) $$
We need the following probably well-known facts.
\begin{lem} \label{PowerSeries} Let $f,g\in \mathbb{Z}_{\geq 0}[[t]]$. Then for each $R\geq 0$: \begin{itemize}
\item If ${\operatorname{coeff}}_1(g)>0$ then ${\operatorname{coeff}}_{t^R} (fg) \geq {\operatorname{coeff}}_{t^R} (f)$
\item If ${\operatorname{coeff}}_{t^k} (f) \leq T$ for all $0\leq k\leq R$ then ${\operatorname{coeff}}_{t^R} (fg)\leq T\cdot \sum_{k=0}^{R} {\operatorname{coeff}}_{t^k} g$
\item ${\operatorname{coeff}}_{t^k} \prod_{i=1}^{m} \frac{1}{(1-t^i)^{d_i}} \leq R^{\sum_{j=1}^{m} d_j}$ for all $0\leq k\leq R$ \end{itemize} \end{lem} \begin{proof} \begin{itemize}
\item ${\operatorname{coeff}}_{t^R}(fg) = \sum_{i=0}^{R} {\operatorname{coeff}}_{t^i}(f)\cdot {\operatorname{coeff}}_{t^{R-i}}(g)\geq {\operatorname{coeff}}_{t^R}(f)\cdot {\operatorname{coeff}}_1(g)\geq {\operatorname{coeff}}_{t^R}(f)$
\item ${\operatorname{coeff}}_{t^R}(fg)=\sum_{k=0}^{R} {\operatorname{coeff}}_{t^k}(f)\cdot {\operatorname{coeff}}_{t^{R-k}}(g) \leq T\cdot \sum_{k=0}^{R} {\operatorname{coeff}}_{t^k}(g)$
\item Suppose that $f\in \mathbb{Z}_{\geq 0}[[t]]$ satisfies ${\operatorname{coeff}}_{t^k}(f)\leq c$ for each $0\leq k\leq R$, then ${\operatorname{coeff}}_{t^k}(f\cdot (1+t^j+t^{2j}+\cdots))\leq \sum_{i=0}^{k} {\operatorname{coeff}}_{t^i}(f) \leq ck \leq cR$ for each $0\leq k\leq R$. The claim now follows by induction, since $\prod_{i=1}^{m} \frac{1}{(1-t^i)^{d_i}}$ is a product of $\sum_{j=1}^{m} d_j$ functions of the form $1+t^j+t^{2j}+\cdots$. \end{itemize} \end{proof}
Back to $(3)$, using Lemma \ref{PowerSeries}, for all $0\leq s\leq 2^{m'_i}$: \begin{eqnarray*} (4)\ \ g'_U(s) & = & {\operatorname{coeff}}_{t^{s}}\left( \prod_{n=1}^{2^{m_i}} \frac{1}{(1-t^n)^{g_L'(n)}} \cdot \prod_{n=2^{m_i}+1}^{2^{m'_i}} \frac{1}{(1-t^n)^{g_L'(n)}} \right) \\ & \leq & \left(2^{m'_i}\right)^D \cdot \sum_{r=0}^{s} {\operatorname{coeff}}_{t^r} \left(\prod_{n=2^{m_i}+1}^{2^{m'_i}} \frac{1}{(1-t^n)^{g_L'(n)}}\right) \\ & \leq & \left(2^{m'_i}\right)^D \cdot \sum_{r=0}^{2^{m'_i}} {\operatorname{coeff}}_{t^r} \left(\prod_{n=2^{m_i}+1}^{2^{m'_i}} \frac{1}{(1-t^n)^{g_L'(n)}}\right) \\ & = & \left(2^{m'_i}\right)^D \cdot \left( 1 + \sum_{r=2^{m_i}+1}^{2^{m'_i}} {\operatorname{coeff}}_{t^r} \left(\prod_{n=2^{m_i}+1}^{2^{m'_i}} \frac{1}{(1-t^n)^{g_L'(n)}}\right) \right) \end{eqnarray*} where $D$ depends only on $m_i$ and on the values of $g'_L$ until $2^{m_i}$. Since $g_L'(k)\leq k^2\log_2 k$ for all $k\in [2^{m_i},2^{m'_i}]$, using Lemma \ref{PowerSeries}, for each $r$: \begin{eqnarray*} {\operatorname{coeff}}_{t^r} \left(\prod_{n=2^{m_i}+1}^{2^{m'_i}} \frac{1}{(1-t^n)^{g_L'(n)}}\right) & \leq & {\operatorname{coeff}}_{t^r} \left(\prod_{n=1}^{\infty} \frac{1}{(1-t^n)^{\lfloor n^2\log_2 n\rfloor)}}\right) \end{eqnarray*}
By Petrogradsky's aforementioned analysis (for $\alpha=3)$ we get that, for any $\varepsilon>0$ there exists $p_\varepsilon$ such that for all $r\geq p_\varepsilon$: $$ {\operatorname{coeff}}_{t^r} \left(\prod_{n=1}^{\infty} \frac{1}{(1-t^n)^{\lfloor n^2\log_2 n\rfloor)}}\right) \leq \exp\left(r^{\frac{3}{4}+\frac{\varepsilon}{3}}\right) $$ and we can make sure that $2^{m_i}\geq p_{1/i}$ for each $i$. Then, by $(4)$, we have for all $0\leq s\leq 2^{m'_i}$: $$ g'_U(s) \leq 2\cdot (2^{m'_i})^D\cdot 2^{m'_i}\cdot \exp\left((2^{m'_i})^{\frac{3}{4}+\frac{1}{3i}}\right) $$ so: $$ g_U(2^{m'_i}) = \sum_{s=0}^{2^{m'_i}} g'_U(s) \leq (2^{m'_i}+1)\cdot 2\cdot (2^{m'_i})^D\cdot 2^{m'_i}\cdot \exp\left((2^{m'_i})^{\frac{3}{4}+\frac{1}{3i}}\right) \leq \exp\left((2^{m'_i})^{\frac{3}{4}+\frac{1}{i}}\right) $$ where the last inequality holds for $i\gg 1$ (and $m'_i\gg m_i$, as $D$ is independent of $m'_i$), and the claim follows. \end{proof}
\section{Polynomial oscillations and Warfield's question on tensor products}
In this section we construct primitive monomial algebras whose tensor product has a prescribed GK-dimension, proving Theorem D. These are the first semiprime examples for which the GK-dimension is strictly subadditive with respect to tensor products, which was left open in \cite{KrempaOkninski} and \cite[Page~167]{KrauseLenagan}. Notice that the first family of primitive monomial algebras of arbitrary GK-dimension has constructed by Vishne \cite{Vishne}.
\subsection{Monomial algebras, infinite words and complexity}
Let $\Sigma=\{x_1,\dots,x_d\}$ be a finite alphabet. Let $w\in \Sigma^{\mathbb{N}}$ be an infintie word: $$ w = w_1w_2w_3\cdots $$ we say that a finite word over $\Sigma$, say $u=x_{i_1}\cdots x_{i_m}$, is a factor of $w$ if $u=w[k,k+m-1]=w_k\cdots w_{k+m-1}$ for some $k\in \mathbb{N}$. The complexity function $p_w(n)$ is defined by:
$$ p_w(n) = \# \{u\in \Sigma^n\ |\ u\ \text{is a factor of}\ w\} $$ An infinite word $w$ defines a monomial algebra generated by $x_1,\dots,x_d$:
$$ A_w = F\left<x_1,\dots,x_d\right> / \left< u\ |\ u\ \text{is not a factor of}\ w \right> $$ Many algebraic properties of $A_w$ reflect combinatorial and dynamical properties of $w$. For instance, $A_w$ is prime if and only if $w$ is recurrent, namely, every factor $u$ of $w$ has infinitely many occurrences (i.e. $u=w[k,l]$ for infinitely many $k,l$'s). An algebra is just-infinite if it is infinite-dimensional, but every proper homomorphic image of it is finite-dimensional. The algebra $A_w$ is just-infinite if and only if $w$ is uniformly recurrent \cite[Theorem~3.2]{BBL}, namely, for each factor $u$ of $w$ there exists $C_u>0$ such that any length-$C_u$ factor of $w$ contains an occurrence of $u$.
An important class of infinite words is the class of Toeplitz sequences. These are infinite words $w$ for which for any $n\geq 1$ there exists $d\geq 1$ such that $w_n=w_{n+d}=w_{n+2d}=\cdots$. The following is standard, but we bring a full proof here for the reader's convenience.
\begin{lem} \label{words} Let $w$ be a Toeplitz sequence. If the monomial algebra $A_w$ has super-linear growth then it is just-infinite primitive. \end{lem} \begin{proof} Suppose that $w\in \Sigma^{\mathbb{N}}$ is a Toeplitz sequence, and for each $n$ let $d_n$ be such that $w_n=w_{n+id_n}$ for all $i\geq 1$. Let $w[k,k+m-1]$ be a factor; then $w[k,m]=w[k+iD,m+iD]$ for all $i\geq 1$, where $D=\text{lcm}(d_k,d_{k+1},\dots,d_m)$. Thus every sufficiently long factor of $w$ contains an occurrence of $w[k,m]$, and $w$ is uniformly recurrent. By \cite[Theorem~3.2]{BBL}, $A_w$ is just-infinite.
By \cite{Farina}, $A_w$ is prime. By \cite{Okn}, either $A_w$ is primitive, PI or has a non-zero Jacobson radical. Since the growth of $A_w$ is super-linear, $w$ cannot be eventually periodic, hence $A_w$ cannot be PI (e.g. by \cite[3.2.2(3)]{Madill}). The Jacobson radical $J$ of $A_w$ is locally nilpotent by \cite{loc nilp jac rad}. If $J\neq 0$ then by just-infiniteness, $\dim_F A_w/J < \infty$. By \cite{Bergman}, $J$ is homogeneous, so $A_w/J$ is graded, finite-dimensional and semiprimitive. Thus $J=\left(A_w\right)_{\geq 1}$ is the augmentation ideal consisting of all elements with zero constant term; this ideal is finitely generated as an algebra (by the set of letters of the underlying alphabet), hence nilpotent, contradicting that $\dim_F A_w=\infty$. It follows that $A_w$ is primitive. \end{proof}
Let $V=F+Fx_1+\cdots+Fx_d$ be the generating subspace of $A_w$ spanned by $1$ and by the letters of $\Sigma$. Then the growth function of $A_w$ can be interpreted via the complexity of $w$, namely, $$ g_{A_w,V}(n)=\dim_F V^n={\operatorname{Span}}_F \{ u\in \Sigma^{\leq n}\ |\ u\ \text{is a factor of}\ w \} = \sum_{k=0}^{n} p_w(k) $$ (By default, we take $V$ as a generating subspace, and omit the subscript notation.) Notice that $p_w(n)$ is non-decreasing. In particular, $g_{A_w}(n)\leq np_w(n)$ and $np_w(n)\leq \sum_{k=n+1}^{2n} p_w(k)\leq g_{A_w}(cn)$ for any $c\geq 2$.
\subsection{Polynomial oscillations and growth of tensor products}
Fix $\alpha,\beta\geq 2$. Without loss of generality, suppose that $\alpha\leq \beta$. Let $2\leq \gamma\leq \alpha$. Let us construct a sequence of positive integers: $$ 1 < s < d_1 < e_1 < d_2 < e_2 < \cdots $$ where $s\gg_\beta 1$ and $e_i=2^{d_i},\ d_{i+1}=2^{e_i}$. Define sequences $(m_k)_{k\geq 1},(n_k)_{k\geq 1}$ of positive integers. Fix $t\gg_\beta 1$ and set $n_k=m_k=t$ for $1\leq k \leq s$. Formally set $d_0=e_0=s$. \begin{itemize}
\item For each $i\geq 0$ and $k\in (e_i,d_{i+1}]$, set $m_k=t$;
\item For each $i\geq 1$ and $k\in [d_i,e_i)$, set $m_{k+1}=\lceil 5^{\beta-2} m_k \rceil$. \end{itemize} And: \begin{itemize}
\item For each $i\geq 1$,
$n_{d_i+1}=t$ and for each $k\in (d_i,e_i)$, set $n_{k+1}=\lceil 5^{\gamma-2} n_k \rceil$;
\item For each $i\geq 0$ and $k\in [e_i,d_{i+1})$, set $n_{k+1}=\lceil 5^{\alpha-2} n_k \rceil$. \end{itemize} Notice that $\frac{n_{k+1}}{n_k},\frac{m_{k+1}}{m_k}$ are bounded above by a constant $\lambda$ depending only on $\alpha,\beta,\gamma$. Since $s,t\gg_\beta 1$, the above definitions yield $m_{k+1}\leq (m_k-1)^2,\ n_{k+1}\leq (n_k-1)^2$, and $m_k,n_k\geq 3$.
Therefore, by \cite[Proposition~4.79]{Kurka} there exist Toeplitz sequences $X,Y$ whose complexity functions $p_X(n),p_Y(n)$ satisfy: $$ 2\cdot 5^{k-1}\cdot n_k\leq p_X(5^k) \leq 2\cdot 5^k\cdot n_k $$ $$ 2\cdot 5^{k-1}\cdot m_k\leq p_Y(5^k) \leq 2\cdot 5^k\cdot m_k $$ (The statement of the \cite[Proposition~4.79]{Kurka} is different, but these bounds are explicitly proven in the proof.) For $r\geq 5$ let $5^k\leq r<5^{k+1}$ (that is, $k=\lfloor \log_5 r \rfloor$) and observe that: $$ (*)\ \ \ \frac{2}{25}r\cdot n_k\leq p_X(r) \leq 10r\cdot n_{k+1},\ \ \ \frac{2}{25}r\cdot m_k\leq p_Y(r) \leq 10r\cdot m_{k+1} $$ Associated with $X,Y$ are the corresponding monomial algebras $A_X,A_Y$ spanned by all finite factors of $X,Y$ (respectively). Let $g_{A_X},g_{A_Y}$ be the corresponding growth functions (with respect to the standard generating subspaces, spanned by $1$ together with the letters of the underlying alphabet). It follows that $g_{A_X}(n)=\sum_{i=1}^{n} p_X(i)$ and $g_{A_Y}(n)=\sum_{i=1}^{n} p_Y(i)$. As mentioned before, $g_{A_X}(n)\leq np_X(n),g_{A_Y}(n)\leq np_Y(n)$ and $np_X(n)\leq g_{A_X}(cn),np_Y(n)\leq g_{A_Y}(cn)$ for any $c\geq 2$.
Let us turn to estimate $n_k,m_k$. Let $k\in (d_i,e_i]$. Then $m_k\leq c_1\cdot 5^{(\beta-2)k}$ for some $c_1>0$ which depends only on $\beta$ and $t$. For $k=2^{d_i}$ we have: $$ m_{2^{d_i}} \geq (5^{\beta-2})^{2^{d_i}-d_i} \geq \frac{(5^{2^{d_i}})^{\beta-2}}{2^{3(\beta-2)d_i}} $$
Let $k\in (d_i,e_i]$ again. Then $n_k\leq c_2\cdot 5^{(\gamma-2)k}$ for some $c_2>0$ which depends only on $\gamma$. Similarly, for $k\in (e_i,d_{i+1}]$, we have $n_k\leq c_3\cdot 5^{(\alpha-2)k}$ for some $c_3>0$ which depends only on $\alpha$ and $t$. For $k=2^{e_i}$ we have: $$ n_{2^{e_i}} \geq (5^{\alpha-2})^{2^{e_i}-e_i} \geq \frac{(5^{2^{e_i}})^{\alpha-2}}{2^{3(\alpha-2)e_i}} $$
Similarly, for $k=2^{d_i}$ we have: $$ n_{2^{d_i}} \geq (5^{\gamma-2})^{2^{d_i}-d_i} \geq (5^{\gamma-2})^{2^{d_i}-d_i} \geq \frac{(5^{2^{d_i}})^{\gamma-2}}{2^{3(\gamma-2)d_i}} $$
We now turn to estimate ${\operatorname{GKdim}}(A_X),{\operatorname{GKdim}}(A_Y)$. Let: $$ S=\{r\in \mathbb{N}|\lfloor \log_5 r \rfloor\in \bigcup_{i\geq 1} (d_i,e_i]\},\ \ \ S'=\{r\in \mathbb{N}|\lfloor \log_5 r \rfloor\in \bigcup_{i\geq 1} (e_i,d_{i+1}]\} $$ These are disjoint sets, whose union contains every sufficiently large positive integer.
\textbf{Calculating ${\operatorname{GKdim}}(A_Y)$.} Let $r\in S$, say, $k=\lfloor \log_5 r\rfloor \in (d_i,e_i]$. By the above calculations, $$ m_k\leq c_1\cdot 5^{(\beta-2)k}\leq c_1\cdot r^{\beta-2}. $$ Using $(*)$, it follows that $p_Y(r)\leq 10r\cdot m_{k+1}\leq 10c_1\lambda r^{\beta-1}$. Hence $g_{A_Y}(r)\leq ~ 10\lambda c_1\cdot ~ r^\beta$.
For $r\in S'$, say $k=\lfloor \log_5 r\rfloor \in (e_i,d_{i+1}]$. Then $p_Y(r)\leq 10r\cdot m_{k+1}\leq 10\lambda t r$ and consequently $g_{A_Y}(r)\leq 10\lambda t r^2$. It follows that ${\operatorname{GKdim}}(A_Y)\leq \beta$.
Moreover, for $r=5^{2^{d_i}}$ using $(*)$: $$ p_Y(5^{2^{d_i}}) \geq \frac{2}{25}5^{2^{d_i}}\cdot m_{2^{d_i}} \geq \frac{2 \cdot (5^{2^{d_i}})^{\beta-1}}{25\cdot 2^{3(\beta-2)d_i}} $$
so: $$ g_{A_Y}(2\cdot 5^{2^{d_i}}) \geq \frac{2 \cdot (5^{2^{d_i}})^{\beta}}{25\cdot 2^{3(\beta-2)d_i}}. $$
which proves that ${\operatorname{GKdim}}(A_Y)=\beta$.
\textbf{Calculating ${\operatorname{GKdim}}(A_X)$.} Let $r\in S'$, say, $k=\lfloor \log_5 r\rfloor \in (e_i,d_{i+1}]$. By the above calculations, $$ n_k\leq c_3\cdot 5^{(\alpha-2)k}\leq c_3\cdot r^{\alpha-2}. $$ Using $(*)$, it follows that $p_X(r)\leq 10r\cdot n_{k+1}\leq 10c_3\lambda r^{\alpha-1}$. Hence $g_{A_X}(r)\leq 10c_3\lambda\cdot r^\alpha$.
For $r\in S$, say $k=\lfloor \log_5 r\rfloor \in (d_i,e_i]$. By the above calculations, $$ n_k\leq c_2\cdot 5^{(\gamma-2)k}\leq c_2\cdot r^{\gamma-2}. $$ Using $(*)$, it follows that $p_X(r)\leq 10r\cdot n_{k+1}\leq 10 c_2 \lambda r^{\gamma-1}$. Hence $g_{A_X}(r)\leq 10 c_2 \lambda \cdot r^\gamma$. It follows that ${\operatorname{GKdim}}(A_X)\leq \alpha$.
Moreover, for $r=5^{2^{e_i}}$ using $(*)$: $$ p_X(5^{2^{e_i}}) \geq \frac{2}{25}5^{2^{e_i}}\cdot n_{2^{e_i}} \geq \frac{2 \cdot (5^{2^{e_i}})^{\alpha-1}}{25\cdot 2^{3(\alpha-2)e_i}} $$ so: $$ g_{A_X}(2\cdot 5^{2^{e_i}}) \geq \frac{2 \cdot (5^{2^{e_i}})^{\alpha}}{25\cdot 2^{3(\alpha-2)e_i}} $$ which can be written as $g_{A_X}(r)\geq c_2' \frac{r^\alpha}{(\log_5 r)^{3(\alpha-2)}}$ for some $c_2'>0$ (and infinitely many $r$'s) which proves that ${\operatorname{GKdim}}(A_X)=\alpha$.
We need one additional calculation related to $A_X$, to be used in the sequel. For $r=5^{2^{d_i}}$ using $(*)$: $$ p_X(5^{2^{d_i}}) \geq \frac{2}{25}5^{2^{d_i}}\cdot n_{2^{d_i}} \geq \frac{2 \cdot (5^{2^{d_i}})^{\gamma-1}}{25\cdot 2^{3(\gamma-2)d_i}} $$ so: $$ g_{A_X}(2\cdot 5^{2^{d_i}}) \geq \frac{2 \cdot (5^{2^{d_i}})^{\gamma}}{25\cdot 2^{3(\gamma-2)d_i}} $$
We are now ready to calculated the growth of the tensor product $A_X\otimes_F A_Y$.
\textbf{Calculating ${\operatorname{GKdim}}(A_X\otimes_F A_Y)$.} Let $V_X,V_Y$ be the standard generating subspaces of $A_X,A_Y$, respectively. Then $V_X\otimes_F V_Y$ is a generating subspace of $A_X\otimes_F A_Y$. It holds that: $$\left(V_X\otimes_F V_Y\right)^n\subseteq V_X^n\otimes_F V_Y^n\subseteq \left(V_X\otimes_F V_Y\right)^{2n}$$ By the above calculations, for $r\in S$ we have: $$ g_{A_Y}(r) = O(r^\beta),\ \ \ g_{A_X}(r) = O(r^\gamma) $$ so: $$ \dim_F \left(V_X\otimes_F V_Y\right)^n\leq \dim_F \left(V_X^n\otimes_F V_Y^n\right) = \left( \dim_F V_X^n \right) \cdot \left( \dim_F V_Y^n \right) = O(r^{\beta+\gamma}) $$ And for $r\in S'$: $$ g_{A_Y}(r) = O(r^2),\ \ \ g_{A_X}(r) = O(r^\alpha) $$ so: $$ \dim_F \left(V_X\otimes_F V_Y\right)^n \leq \dim_F \left(V_X^n\otimes_F V_Y^n\right) = \left( \dim_F V_X^n \right) \cdot \left( \dim_F V_Y^n \right) = O(r^{\alpha+2}) $$ Since $\beta+\gamma \geq \alpha+2$, it follows that ${\operatorname{GKdim}}(A_X\otimes_F A_Y)\leq \beta+\gamma$.
For each $i$ let $r_i=5^{2^{d_i}}$ and notice that the above calculations also show that: $$ \dim_F V_Y^{2r_i} \geq \frac{2r_i^\beta}{25(\log_5 r_i)^{3(\beta-2)}},\ \ \ \dim_F V_X^{2r_i} \geq \frac{2r_i^\gamma}{25(\log_5 r_i)^{3(\gamma-2)}} $$ so: \begin{eqnarray*} g_{A_X\otimes_F A_Y}(4r_i) & \geq & \dim_F \left(V_X^{2r_i}\otimes_F V_Y^{2r_i} \right) \\ & = & \left( \dim_F V_X^{2r_i} \right) \cdot \left( \dim_F V_Y^{2r_i} \right) \geq \frac{4r_i^{\beta+\gamma}}{625(\log_5 r_i)^{3(\beta+\gamma-4)}} \end{eqnarray*} hence ${\operatorname{GKdim}}(A_X\otimes_F A_Y)=\beta+\gamma$.
\begin{proof}[{Proof of Theorem D}] Given $2\leq \gamma\leq \alpha\leq \beta$ construct $A_X,A_Y$ as above. Since $X,Y$ are Toeplitz sequences of and $A_X,A_Y$ are of {GK-dimension}\ greater than $1$, then by Lemma \ref{words} they are both primitive. By the above calculations: $$ {\operatorname{GKdim}}(A_X)=\alpha,\ \ {\operatorname{GKdim}}(A_Y)=\beta,\ \ {\operatorname{GKdim}}(A_X\otimes_F A_Y)=\beta+\gamma $$ as required. \end{proof}
\begin{rem} Using the growth analysis of convolution algebras associated with minimal subshifts presented in \cite{Nekrashevych}, one can use the above construction to construct examples of \textit{simple} algebras $A,B$ with ${\operatorname{GKdim}}(A)=\alpha,{\operatorname{GKdim}}(B)=\beta,{\operatorname{GKdim}}(A\otimes_FB)=\beta+\gamma$. \end{rem}
\end{document} | arXiv |
Let $x,$ $y,$ and $z$ be positive real numbers. Then the minimum value of
\[\frac{(x^4 + 1)(y^4 + 1)(z^4 + 1)}{xy^2 z}\]is of the form $\frac{a \sqrt{b}}{c},$ for some positive integers $a,$ $b,$ and $c,$ where $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of a prime. Enter $a + b + c.$
By AM-GM,
\begin{align*}
\frac{x^4 + 1}{x} &= x^3 + \frac{1}{x} \\
&= x^3 + \frac{1}{3x} + \frac{1}{3x} + \frac{1}{3x} \\
&\ge 4 \sqrt[4]{x^3 \cdot \frac{1}{3x} \cdot \frac{1}{3x} \cdot \frac{1}{3x}} \\
&= \frac{4}{\sqrt[4]{27}}.
\end{align*}Similarly,
\[\frac{z^4 + 1}{z} \ge \frac{4}{\sqrt[4]{27}}.\]Again by AM-GM,
\[\frac{y^4 + 1}{y^2} = y^2 + \frac{1}{y^2} \ge 2 \sqrt{y^2 \cdot \frac{1}{y^2}} = 2.\]Therefore,
\[\frac{(x^4 + 1)(y^4 + 1)(z^4 + 1)}{xy^2 z} \ge \frac{4}{\sqrt[4]{27}} \cdot 2 \cdot \frac{4}{\sqrt[4]{27}} = \frac{32 \sqrt{3}}{9}.\]Equality occurs when $x^3 = \frac{1}{3x},$ $y^2 = \frac{1}{y^2},$ and $z^3 = \frac{1}{3z}.$ We can solve, to get $x = \frac{1}{\sqrt[4]{3}},$ $y = 1,$ and $z = \frac{1}{\sqrt[4]{3}},$ so the minimum value is $\frac{32 \sqrt{3}}{9}.$ The final answer is $32 + 3 + 9 = \boxed{44}.$ | Math Dataset |
Unduloid
In geometry, an unduloid, or onduloid, is a surface with constant nonzero mean curvature obtained as a surface of revolution of an elliptic catenary: that is, by rolling an ellipse along a fixed line, tracing the focus, and revolving the resulting curve around the line. In 1841 Delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid.[1]
Formula
Let $\operatorname {sn} (u,k)$ represent the normal Jacobi sine function and $\operatorname {dn} (u,k)$ be the normal Jacobi elliptic function and let $\operatorname {F} (z,k)$ represent the normal elliptic integral of the first kind and $\operatorname {E} (z,k)$ represent the normal elliptic integral of the second kind. Let a be the length of the ellipse's major axis, and e be the eccentricity of the ellipse. Let k be a fixed value between 0 and 1 called the modulus.
Given these variables,
$\operatorname {x} (u)=-a(1-e)(\operatorname {F} (\operatorname {sn} (u,k),k)+\operatorname {F} (1,k))-a(1+e)(\operatorname {E} (\operatorname {sn} (u,k),k)+\operatorname {E} (1,k))\,$
$\operatorname {y} (u)=a(1+e)\operatorname {dn} (u,k)\,$
The formula for the surface of revolution that is the unduloid is then
$\operatorname {X} (u,v)=\langle \operatorname {x} (u),\operatorname {y} (u)\cos(v),\operatorname {y} (u)\sin(v)\rangle \,$
Properties
One interesting property of the unduloid is that the mean curvature is constant. In fact, the mean curvature across the entire surface is always the reciprocal of twice the major axis length: 1/(2a).
Also, geodesics on an unduloid obey the Clairaut relation, and their behavior is therefore predictable.
Occurrence in material science
Unduloids are not a common pattern in nature. However, there are a few circumstances in which they form. First documented in 1970, passing a strong electric current through a thin (0.16—1.0mm), horizontally mounted, hard-drawn (non-tempered) silver wire will result in unduloids forming along its length.[2] This phenomenon was later discovered to also occur in molybdenum wire.[3] Unduloids have also been formed with ferrofluids.[4] By passing a current axially through a cylinder coated with a viscous magnetic fluid film, the magnetic dipoles of the fluid interact with the magnetic field of the current, creating a droplet pattern along the cylinder’s length.
References
1. Delaunay, Ch. (1841). "Sur la surface de révolution dont la courbure moyenne est constante". Journal de Mathématiques Pures et Appliquées. 6: 309–314.
2. Lipski, T.; Furdal, A. (1970), "New observations on the formation of unduloids on wires", Proceedings of the Institution of Electrical Engineers, 117 (12): 2311-2314, doi:10.1049/piee.1970.0425
3. “Periodic Videos, Exploding wires” on YouTube
4. Weidner, D.E. (2017), "Drop formation in a magnetic fluid coating a horizontal cylinder carrying an axial electric current", Physics of Fluids, 29 (5): 052103, doi:10.1063/1.4982618
| Wikipedia |
A repeated measures study of phenol, paraben and Triclocarban urinary biomarkers and circulating maternal hormones during gestation in the Puerto Rico PROTECT cohort
Amira M. Aker1,
Kelly K. Ferguson1,2,
Zaira Y. Rosario3,
Bhramar Mukherjee4,
Akram N. Alshawabkeh5,
Antonia M. Calafat6,
José F. Cordero7 &
John D. Meeker1
Prenatal exposure to some phenols and parabens has been associated with adverse birth outcomes. Hormones may play an intermediate role between phenols and adverse outcomes. We examined the associations of phenol and paraben exposures with maternal reproductive and thyroid hormones in 602 pregnant women in Puerto Rico. Urinary triclocarban, phenol and paraben biomarkers, and serum hormones (estriol, progesterone, testosterone, sex-hormone-binding globulin (SHBG), corticotropin-releasing hormone (CRH), total triiodothyronine (T3), total thyroxine (T4), free thyroxine (FT4) and thyroid-stimulating hormone (TSH)) were measured at two visits during pregnancy.
Linear mixed models with a random intercept were constructed to examine the associations between hormones and urinary biomarkers. Results were additionally stratified by study visit. Results were transformed to hormone percent changes for an inter-quartile-range difference in exposure biomarker concentrations (%Δ).
Bisphenol-S was associated with a decrease in CRH [(%Δ -11.35; 95% CI: -18.71, − 3.33), and bisphenol-F was associated with an increase in FT4 (%Δ: 2.76; 95% CI: 0.29, 5.22). Butyl-, methyl- and propylparaben were associated with decreases in SHBG [(%Δ: -5.27; 95% CI: -9.4, − 1.14); (%Δ: -3.53; 95% CI: -7.37, 0.31); (%Δ: -3.74; 95% CI: -7.76, 0.27)]. Triclocarban was positively associated with T3 (%Δ: 4.08; 95% CI: 1.18, 6.98) and T3/T4 ratio (%Δ: 4.67; 95% CI: -1.37, 6.65), and suggestively negatively associated with TSH (%Δ: -10.12; 95% CI: -19.47, 0.32). There was evidence of susceptible windows of vulnerability for some associations. At 24–28 weeks gestation, there was a positive association between 2,4-dichlorophenol and CRH (%Δ: 9.66; 95% CI: 0.67, 19.45) and between triclosan and estriol (%Δ: 13.17; 95% CI: 2.34, 25.2); and a negative association between triclocarban and SHBG (%Δ: -9.71; 95% CI:-19.1, − 0.27) and between bisphenol A and testosterone (%Δ: -17.37; 95% CI: -26.7, − 6.87).
Phenols and parabens are associated with hormone levels during pregnancy. Further studies are required to substantiate these findings.
Exposure to phenols and parabens has been linked to various adverse health effects, including ovarian toxicity, cancer, and adverse neurodevelopmental outcomes [1,2,3,4]. Prenatal exposure to these chemicals, in particular, may have a long lasting effect on fetal health into adulthood. For example, prenatal exposure to phenols and parabens has been linked to adverse birth outcomes [5, 6], respiratory health effects in children [7], and cardiometabolic risk [8]. The exact mechanisms at play are still not fully understood; however, endocrine disruption is hypothesized to be one of the main toxicity pathways [3, 9,10,11].
Reproductive and thyroid hormones play an essential role in the maintenance of pregnancy and the development of the fetus [12,13,14,15,16], therefore pregnancy is a vulnerable window for endocrine disruption due to the varying levels of hormones involved in the growing organism [17]. Endocrine disrupting chemicals could act through several pathways, including hormone synthesis, regulation, transport and metabolism, and/or interference with receptors. Phenols and parabens have estrogenic and androgenic properties [1, 18,19,20], but few human studies have looked into the effect of these chemicals on maternal hormones during pregnancy. Most existing studies in this area use smaller study populations or only examined a single time point in pregnancy, which do not capture the changing hormone levels and high variability of phenols and paraben exposure during pregnancy. Furthermore, no or few studies explored the associations between these chemicals and maternal testosterone, corticotropin-releasing hormone (CRH), sex hormone-binding globulin (SHBG) and estriol, all of which play essential roles in maintaining healthy pregnancies.
Given the growing evidence of the endocrine disrupting effects of phenols and parabens [18, 21,22,23,24,25], our aim was to study the relationships between phenols and parabens on reproductive and thyroid hormones in our ongoing cohort of pregnant women in Puerto Rico. The study follows the women over multiple time points during pregnancy, providing more power than previous studies, and allows for the identification of potential windows of susceptibility. We previously reported early preliminary results on associations between select phenols and parabens with hormones in this Puerto Rican cohort [26]. This manuscript is an update of our previous results that utilizes a much larger sample size, includes additional hormones (estriol, testosterone, total triiodothyronine, and total thyroxine), as well as additional exposure biomarkers yet to be studied in detail (ethylparaben, BPS, BPF and triclocarban). Due to the lack of human health data, this study was exploratory in nature, with the exception of BPA, triclosan, methylparaben and propylparaben. We hypothesized a decrease in serum thyroid hormone levels in association with triclosan, methyl- and propyl-paraben, and an increase in serum thyroid hormones with BPA concentrations.
Participants for the present study were from an ongoing prospective cohort of pregnant women in Puerto Rico, named the Puerto Rico Testsite for Exploring Contamination Threats (PROTECT) cohort. Details on the recruitment and inclusion criteria have been described previously [27, 28]. The study participants included in the present analysis were recruited from 2012 to 2017 at 14 ± 2 weeks gestation from two hospitals and five affiliated prenatal clinics in Northern Puerto Rico. They were aged between 18 and 40 years. The exclusion criteria included women who lived outside the region, had multiple gestations, used oral contraceptives within 3 months prior to getting pregnant, got pregnant using in vitro fertilization, or had known medical health conditions (diabetes, hypertension, etc.). Three visits were conducted with the study participants to coincide with periods of rapid fetal growth and routine clinical visits (Visit 1: 16–20; Visit 2: 20–24; Visit 3: 24–28 gestation weeks). Demographic information was collected via questionnaires at the initial study visit. Spot urine samples were collected at the three study visits, whereas blood samples were collected during the first and third visits.
The present analysis includes 602 women recruited into the study (of the total 1311 women enrolled in the cohort to date) for whom both total phenol and paraben concentrations and hormone measurements from at least one study visit were available. This study was approved by the research and ethics committees of the University Of Michigan School Of Public Health, University of Puerto Rico, Northeastern University, and the University of Georgia. All study participants provided full informed consent prior to participation. The involvement of the Centers for Disease Control and Prevention (CDC) laboratory did not constitute engagement in human subjects research.
Quantification of urinary biomarkers
After collection, spot urine samples were divided into aliquots and frozen at -80 °C until they were shipped overnight with dry ice to the CDC for analysis. Urine samples were analyzed for seven phenols (2,4-dichlorophenol, 2,5-dichlorophenol, BPA, BPS, BPF, benzophenone-3, triclosan), triclocarban, and four parabens (ethylparaben, methylparaben, butylparaben, propylparaben) using online solid phase extraction-high-performance liquid chromatography-isotope dilution tandem mass spectrometry [29,30,31]. Biomarker concentrations below the limit of detection (LOD) were assigned a value of the LOD divided by √2 [32]. The LODs were as follows: 0.1 μg/L (2,4-dichlorphenol, 2,5-dichlorophenol, BPS, triclocarban, butylparaben, propylparaben); 0.2 μg/L (BPA, BPF); 0.4 μg/L (benzophenone-3); 1 μg/L (methylparaben, ethylparaben); and 1.7 μg/L (triclosan). Urinary dilution was accounted for by using urinary specific gravity (SG), and was measured using a digital handheld refractometer (AtagoCo., Ltd., Tokyo, Japan). For preliminary data analysis, urinary biomarker concentrations were corrected for SG using the following formula:
$$ {\mathrm{P}}_{\mathrm{C}}=\mathrm{M}\left[\left({\mathrm{SG}}_{\mathrm{m}}-1\right)/\left({\mathrm{SG}}_{\mathrm{i}}-1\right)\right] $$
where Pc is the SG-corrected concentration (μg/L), M is the measured concentration, SGm is the study population median urinary specific gravity (1.0196), and SGi is the individual's urinary specific gravity. The sample size for BPF, BPS, triclocarban and ethylparaben was smaller than the rest of the biomarkers because they were only quantified in a later sub-sample of the cohort.
Hormone measurement
Serum samples were collected during visits 1 and 3. Volume limitations resulted in differences in the number of samples analyzed by hormone. All hormone analyses were conducted at the Central Ligand Assay Satellite Services (CLASS) laboratory, Department of Epidemiology, School of Public Health, University of Michigan. Progesterone, SHBG, testosterone, total triiodothyronine (T3), total thyroxine (T4), free thyroxine (FT4), and thyroid-stimulating hormone (TSH) were measured in serum using a chemiluminescence immunoassay (ADVIA Centaur® CP Immunoassay System, Seimens Healthineers). Estriol and CRH were measured in serum using an enzyme immunoassay (Estriol ELISA Kit, ALPCO; CRH ELISA Kit, LifeSpan BioSciences). In addition to measured hormones, the ratio of progesterone to estriol (Prog/Estriol Ratio), and the ratio of T3 and T4 (T3/T4 ratio) were calculated for the purposes of this analysis. Hormone ratios may be a better indicator of adverse pregnancy outcomes (such as preterm birth) than the individual hormones alone [33,34,35]. Two samples had a TSH level below the LOD. Because this result was not biologically plausible, these two values were dropped from the analyses.
Distributions of key demographic characteristics were calculated. All urinary exposure biomarkers, and the serum hormones progesterone, estriol, CRH, TSH and progesterone/estriol ratio were positively-skewed, and were natural log-transformed. The distributions of SHBG, FT4, T3, T4 and T3/T4 ratio approximated normality and remained untransformed in all analyses. Geometric means and standard deviations were calculated for all SG-corrected exposure biomarkers, hormones, and the ratios of progesterone/estriol and T3/T4. We examined urinary exposure biomarkers concentrations and serum hormone levels by study visit, and calculated Spearman correlations between unlogged average SG-corrected exposure biomarkers. To assess differences in exposure biomarkers and hormones across study visits, we ran Linear Mixed Models (LMM) with a subject-specific random intercept regressing the biomarker or hormone against the study visit. Specific gravity was used as a covariate in the model instead of using the SG-corrected biomarker concentrations. The selection of a random intercept and slope was determined using BIC values. BPF and ethylparaben were detected in less than 50% of the samples. Therefore, we transformed BPF and ethylparaben into dichotomous variables, where 0 represented concentrations below the LOD, and 1 represented detectable concentrations. These categorical BPF and ethylparaben variables were used in all of the following regression analyses.
In our repeated measures analysis, we regressed one hormone or hormone ratio on one urinary biomarker using LMM, with a subject-specific random intercept for each model to account for intra-individual correlation of serial hormone measurements collected over the two study visits. The urinary biomarker concentrations at the two visits were treated as time-varying variables in the LMM models. Crude models included specific gravity and study visit as covariates. Potential confounders were selected a priori from the existing literature, and included as covariates if they were found to change the main effect estimate by > 10%. Final models were adjusted for specific gravity, study visit, body mass index (BMI) at the first study visit, maternal age, the number of hours of second-hand smoking exposure per day, and a socio-economic variable. All covariates, except for maternal age and specific gravity, were categorical. The socio-economic variable used in the model differed by the hormone regressed. Maternal education was a strong confounder for models regressing progesterone, estriol, and progesterone/estriol ratio against urinary biomarkers concentrations, and was used as the socio-economic index for those models. All other models used insurance type as the socio-economic status index. The selection of the socio-economic variable was based on the percent change in the main effect estimate, and the p value of the socio-economic variable in final models.
To assess windows of vulnerability, we ran two more analyses. First, we ran the same LMMs regressing hormones and urinary biomarkers concentrations with an interaction term between the urinary biomarker and the study visit. Second, we ran multiple linear regressions (MLR) stratified by study visit of sample collection. The MLR models were adjusted for the same covariates as those in the LMMs.
To increase interpretability of our results, we transformed regression coefficients to percent changes (and associated 95% confidence intervals, CIs) in hormone concentration in relation to the interquartile range (IQR) increase in urinary biomarker concentrations. Beta coefficients from models with categorical biomarkers (BPF and ethylparaben) were transformed to percent changes (and associated 95% confidence intervals) in hormone concentration at detectable vs non-detectable biomarker concentrations. The alpha level was set at 0.05. All statistical analyses were conducted in R Version 3.4.2.
As a sensitivity analysis, all models were re-run using specific gravity as a covariate in combination with exposure biomarkers corrected for specific gravity as was described by O'Brien et al. [36]. We observed no differences in our results, and therefore, retained our original models using un-corrected exposure biomarkers with specific gravity included as a covariate.
The 602 study participants had a mean age of 26.4 and approximately 60% had BMI levels below 30 kg/m2 (Table 1). Although the majority of women reported never smoking (75%), 4% reported currently smoking, and 7% reported exposure to second-hand smoking for more than an hour per day. Six percent reported consuming alcohol in the last few months. A quarter of the study participants reported a household income of less than $10,000, and only 11% reported a household income >$50,000. A quarter of the participants did not report their incomes. As compared to the overall PROTECT cohort, the study participants included in the present analysis had higher rates of smoking, and had overall lower household income and education levels.
Table 1 Summary demographics and differences between the PROTECT study participants included in present analysis versus participants not included because of missing urine and/or serum samples
The exposure biomarkers included in this analysis were highly detected in the study population, with the exception of ethylparaben and BPF (Table 2). BPF was detected in between 50 and 60% of the study sample; ethylparaben was detected in between 42 and 54% of the sample, depending on study visit. Concentrations of urinary biomarkers remained relatively consistent across the two study visits, with the exception of a decrease in BPA (p < 0.001) and butylparaben (p = 0.04). There was an increase in most hormones across the two study visits, particularly progesterone, estriol, SHBG and CRH. T4 levels remained consistent from 16 to 20 and 24–28 weeks gestation.
Table 2 Distribution of SG-corrected urinary biomarker concentrations and hormones and differences by study visit of sample collection in pregnancy
Methylparaben and propylparaben were strongly correlated [Spearman correlation of 0.8 (p < 0.001)] (Fig. 1). Ethylparaben and butylparaben showed moderate correlation with methylparaben and propylparaben with Spearman correlations between 0.33–0.47 (p values < 0.001). 2,4-Dichlorophenol and 2,5-dichlorophenol showed a fairly strong correlation (Spearman r = 0.6, p < 0.001). Triclosan was moderately correlated with 2,4-dichlorophenol (Spearman r = 0.5, p < 0.001), but not with 2,5-dichlorophenol (Spearman r = − 0.03). BPA, BPS and BPF showed low correlation (Spearman r = 0.11–0.21, p < 0.001).
Heat map of Spearman correlations between unlogged urinary triclocarban, phenols and parabens. Biomarkers concentrations were adjusted for urinary dilution. 2,4-DCP: 2,4-dichlorophenol; 2,5-DCP: 2,5-dichlorophenol; BP-3: Benzophenone; TCS: Triclosan; TCC: Triclocarban; EPB: ethylparaben; MPB: Methylparaben; BPB: Butylparaben; PPB: Propylparaben
Results from LMMs and MLRs are described in detail below by biomarker (Tables 3, Additional file 1: Table S1 and S2, and Additional file 2). There were few differences between most adjusted and unadjusted models, with the exception of associations with CRH. MPB and PPB were associated with CRH in our unadjusted models, but in the adjusted models, these associations disappeared, and CRH was associated with BPS and TCS. A further analysis of CRH concentrations across the covariate levels did not reveal any large differences to report.
Table 3 Results of the adjusted LMMs regressing hormones versus exposure biomarkers
There were no associations between 2,4-dichlorophenol and 2,5-dichlorophenol with hormones in LMMs. An IQR increase in 2,4-dichlorophenol was associated with a 10% increase in CRH at 24–28 weeks [9.66% change in hormone per IQR change in the biomarker/ percent change in hormone at detectable biomarker concentrations (%Δ); 95% CI: 0.67, 19.45], and a suggestive 2% decrease in T3 at 16–20 weeks (%Δ -2.22 95% CI -4.55, 0.10).
Associations across the bisphenols differed, and BPS had the strongest associations in LMM models. BPS was associated with an 11% decrease in CRH (%Δ -11.35; 95% CI: -18.71, − 3.33), and this association was stronger at 16–20 weeks gestation. At this time point, BPS was additionally associated with a 12% decrease in TSH (%Δ -11.93; 95% CI: -22.49, 0.07). BPF was associated with a 3% increase in FT4 (%Δ 2.76; 95% CI: 0.29, 5.22), and this association was also stronger at 16–20 weeks. BPA, on the other hand, had stronger associations at 24–28 weeks gestation. BPA was associated with a 17% decrease in testosterone, and 2–4% increases in FT4 and T3 at 24–28 weeks [(%Δ -17.37; 95% CI: -26.7, − 6.87); (%Δ 2.38; 95% CI: 0.04, 4.72); (%Δ4.33, 95% CI: 0.11, 8.55), respectively]. The increase in FT4 and T3 in relation to BPA was in line with our a priori hypothesis Benzophenone-3 was not significantly associated with any hormones.
Triclocarban was associated with a number of thyroid hormones and SHBG. An IQR increase in triclocarban is associated with a 4% increase in T3 (%Δ 4.08; 95% CI: 1.18, 6.98), a 5% increase in the T3/T4 ratio (%Δ 4.67; 95% CI: 1.24, 10.10), a suggestive 10% decrease in TSH (%Δ -10.12; 95% CI: -19.47, 0.32), and a 10% decrease in SHBG at 24–28 weeks (%Δ -9.71; 95% CI: -19.1, − 0.27).
Triclosan was associated with an increase in a number of reproductive hormones, however most were only suggestive with p values between 0.05 and 0.10. This includes a 9% increase in CRH (%Δ 9.20; 95% CI: -0.97, 20.42), a 7% increase in testosterone (%Δ 7.13; 95% CI: -0.60, 15.5), and 10–13% increases in progesterone and estriol at 24–28 weeks [(%Δ 9.72, 95% CI: -1.27, 21.9); (%Δ 13.2; 95% CI: 2.34, 25.2), respectively]. In addition, triclosan was associated with a 5.8% decrease in T3 at 24–28 weeks; this finding was in line with our a priori hypothesis.
IQR increases in butylparaben, methylparaben and propylparaben were associated with a decrease in SHBG [(%Δ -5.27; 95% CI:-9.40, − 1.14); (%Δ -3.53; 95% CI: -7.37, 0.31); (%Δ -3.74; 95% CI: -7.76, 0.27), respectively]. Methylparaben was also associated with decreases in reproductive hormones, including an 8% decrease in estriol, a suggestive 3% increase in the progesterone/estriol ratio, and a suggestive decrease in testosterone at 16–20 weeks [(%Δ -7.76; 95% CI: -15.4, 0.61); (%Δ 3.14; 95% CI: -2.95, 9.61); (%Δ -6.77; 95% CI: -13.13, 0.29), respectively]. Conversely, an IQR increase in propylparaben was associated with a 9–10% increase in progesterone and estriol at 24–28 weeks [(%Δ 9.67; 95% CI: -1.30, 21.85); (%Δ 8.92; 95% CI: -1.56, 20.52)]. Interaction terms between study visit*methylparaben and propylparaben had p values < 0.05 in models regressed against estriol. We expected to see a decrease in thyroid hormones in relation to methyl- and propyl- paraben, but only observed a decrease in TSH in association with methylparaben, particularly at 16–20 weeks (%Δ -11.69; 95% CI: -21.97, − 0.06). The decrease in TSH could indicate an increase in circulating thyroid hormones, in contrast to our hypothesis.
Associations differed by exposure biomarker and hormone, and there was little consistency within chemical classes with the exception of some parabens. There was evidence of a decrease up to 6% in T3 in association with 2,4-dichlorophenol, BPA and triclosan, whereas triclocarban was associated with a 4% increase in T3. In the case of bisphenols, BPS was more strongly related to decreases in hormones at 16–20 weeks, and BPA had stronger negative relationships at 24–28 weeks. Triclosan was associated with general increases in reproductive hormones of approximately 10%, and triclocarban was associated with 5–10% changes in thyroid hormones. Parabens were associated with a decreased level of SHBG.
While there may be structural similarities between BPA, BPS and BPF, the structural variations may be sufficient to alter receptor-binding affinities across the bisphenols [37]; therefore, the biological effects may vary among the bisphenols. To this, we found that the earlier time point (16–20 weeks gestation) may be a more vulnerable time of exposure to BPS and BPF, in contrast to the stronger relationships observed at the 24–28 weeks with respect to BPA. Our results were somewhat consistent with results from previous studies. BPA has been suspected to interfere with thyroid hormones, as evidenced by several epidemiological studies. We observed an increase in FT4 and T3, which was consistent with two previous studies our group conducted in a preliminary analysis in the PROTECT cohort, and a cohort of pregnant women in Boston, MA with four repeated measures during pregnancy [38, 39]. Two cross-sectional studies in the United States (N = 249 and 476 women) also looked at the association between maternal BPA and thyroid hormones during gestation [40, 41]. The only significant association reported was between maternal urinary BPA and a decrease in T4 [40], which we did not observe in the present study. A decrease in T4 could be indicative of an increase in FT4, in the case of thyroxine becoming less bound to thyroxine-binding globulin, however, the associations between BPA and T4 in the current study had p values ranging from 0.51–0.93. Furthermore, we did not observe a relationship between BPA and TSH that was reported in the Boston cohort study [42], and among adults from the Korean National Environmental Health Survey [43].
One of the strongest associations we observed was the 17% decrease in testosterone in relation to BPA. This is the first study that explores this association in pregnant women, and there is little correlation between maternal and fetal testosterone levels [44]. However, a decrease in testosterone was identified in an in vitro study on TM3 murine Leydig with BPA exposure [45], in the F2 generation after in-utero BPA exposure in mice [46], and in-utero BPA concentrations in young boys aged 8–14 [47]. These associations provide further evidence in support of our finding. Although the role of maternal testosterone in gestation is still unclear, evidence points to androgens playing an essential role in myometrial relaxation, cervical ripening and initiating parturition [48]. Therefore, BPA, via reduced testosterone, could increase gestational age, which we previously observed in this cohort [49]. Additionally, maternal testosterone has a role in gender role behaviors [50], indicating that maternal testosterone may impact fetal development.
No human studies have previously investigated the associations between triclocarban, phenols and parabens on CRH during pregnancy; however, CRH plays an important role in gestation. Maternal CRH levels during pregnancy largely originate from gestational tissues [51]. Evidence suggests CRH inhibits immune rejection processes by killing activated T cells [52], plays an important role in determining time of parturition, and an increase in CRH has been associated with the onset of miscarriage and preeclampsia [53,54,55,56,57]. CRH receptor expression is regulated by estrogen, and CRH gene expression in the placenta is mediated by ER-α [58, 59]. Given the endocrine disrupting potential of bisphenols via estrogen receptors [60], associations between CRH and bisphenols (and potentially other phenols and parabens) could be important to consider in pregnancy studies. Animal and in vitro studies showed an increase in CRH with exposure to BPA and BPS, contrary to our results of an inverse relationship between CRH and BPS. BPA increased plasma concentrations of CRH in pregnant mice [61] and CRH levels in human placenta primary trophoblast cells [62]. The differences in our results could be in part due to the unique role CRH plays in human pregnancies, as compared to animals [63].
Triclosan was suggestively associated with select hormones, but none reached statistical significance, including an increase in testosterone, an increase in CRH at 16–20 weeks gestation, and a decrease in T3 at 24–28 weeks gestation. There was a similar decrease in T3 with increased urinary triclosan concentrations in the Boston cohort, albeit the associations were stronger earlier in pregnancy, in contrast to our stronger associations at the later visit in the current study [39]. While larger human studies with more statistical power may be needed, the decrease in T3 in association with triclosan is consistent with animal studies [64], including in pregnant rats [65] and pregnant mice [66, 67], perhaps due to triclosan's structural similarities to thyroid hormones [64]. Animal studies also report a decrease in T4 with triclosan exposure, including rat and mice dams [65,66,67,68,69,70,71,72,73,74,75], but we did not find evidence of this in humans. Other population studies found no associations between triclosan and thyroid hormones [76,77,78], although there was evidence of vulnerable time points during gestation [76, 77]. Interestingly, a study in pregnant rats showed that the greatest accumulation of triclosan was in the placenta, indicating that pregnancy may be a sensitive time period for triclosan exposure [79]. Alternatively, maternal serum TSH and FT4 levels at > 28 weeks gestation (obtained from medical records) were negatively associated with urinary triclosan at 38 weeks gestation [80]. The differences in our results could be explained by the differences in the study population, exposure biomarker concentrations, and differences in the pregnancy time points examined.
No studies have looked at the effect of triclosan on maternal testosterone and CRH during pregnancy in humans. However, in contrast to our results, triclosan was found to reduce testosterone levels in male rats [81], and in pregnant rats [79]. An excess of maternal testosterone has been associated with restricted fetal growth [82], as well as an increased chance of developing Alzheimer disease [83] and anxiety like symptoms in the offspring.
Triclocarban was associated with thyroid hormone changes. We observed an increase in T3 and a decrease in TSH in association with triclocarban, which is in line with the negative feedback loop in maintaining thyroid hormone homeostasis. We also observed a decrease in SHBG. SHBG levels tend to rise with thyroid hormones, so this observed pattern was unexpected. This could be due to factors influencing the relationship between thyroid hormone and SHBG levels that have not been accounted for in the present study. Our previous Boston study also reported a negative association between triclocarban and TSH, but a negative association with T3. Triclocarban concentrations in this cohort were much higher than the exposure levels found in the Boston cohort. In fact, the triclocarban concentrations observed in PROTECT are 37 times larger than the concentration observed in NHANES women of reproductive age [84]. This difference in exposure levels may explain the differences in the associations observed.
All parabens were generally negatively associated with SHBG. In contrast to our current findings, our previous preliminary analysis in the PROTECT cohort showed that methylparaben was positively associated with SHBG [26]. However, the current study has a much larger sample size. Associations between parabens and some hormones appeared to be dependent on the timing of exposure. Associations between methylparaben and propylparaben and estriol changed direction from a negative association at 16–20 weeks to a positive association at 24–28 weeks gestation. We observed a similar change in direction in our preliminary analyses between methylparaben and propylparaben with estradiol [26]. Although not statistically significant, associations between methylparaben and propylparaben with progesterone followed a similar pattern to that of estriol. Given that the population urinary levels of methylparaben and propylparaben remained consistent between the two time points, the similar change of direction observed in associations with methylparaben and propylparaben in both of our previous analyses, and the significant interaction term between these parabens and visit in association with estriol, this lends confidence that these observations may not be occurring by chance and may be detected in future larger studies. The strong correlation between propyl- and methylparaben could indicate that their associations with estriol are being driven by only one of the parabens. However, given the differences in the associations between these two parabens and all hormones, there do seem to be unique relationships between the exposure and hormone levels. No previous studies have looked at the effect of parabens on estriol, SHBG or CRH; however, evidence suggests parabens have ER-β agonistic activity [85], and stimulate progesterone mRNA expression via ER-α signaling [86, 87]. This could suggest a potential mechanism by which reproductive hormone levels could be directly or indirectly altered in response to paraben exposure.
The present study also showed a general decrease in TSH in association with parabens, but only methylparaben reached a significant association with TSH. Additionally, methylparaben and propylparaben were associated with a decrease in the T3/T4 ratio, particularly at 24–28 weeks gestation. Results from our Boston cohort also showed a decrease in T3/T4 ratio, as well as T3, at median 26 weeks gestation [88]. In other research, human and animal studies reported a decrease in T4 and FT4 with paraben exposure in females [78, 89], and two small studies in men found no associations between parabens and thyroid hormones [90, 91]. The difference in the results is likely due to the different study populations; none of those studies looked specifically at prenatal exposure.
Our study had several limitations. We did not have data on the iodine status of the women; deficiency in this element could affect thyroid hormone function. However, iodine may act as mechanistic intermediate exposure between the exposure and thyroid hormone, and controlling for iodine status could lead to bias [92]. Furthermore, iodine had no effect on the associations between phenols and thyroid hormones in our previous study of NHANES data [78]. We also did not have data on thyroperoxidase antibodies nor human chorionic gonadotropin (hCG), which could potentially affect thyroid function as well [93, 94]. While data at two time points is a great improvement from the more common cross-sectional study design, the two time points may not be sufficient to understand the potential influence of these biomarkers on maternal hormones. The relatively high variation in urinary concentrations of the target biomarkers (particularly BPA) over time may also introduce potential bias stemming from random measurement error. Given the multiple comparisons conducted, there is a chance of Type I error, and caution must be used when interpreting our findings. Finally, although one of the strengths of the present study is our ability to investigate the relationships between these chemicals and hormone levels in a vulnerable population, our study population was based in a population in Puerto Rico of lower income who also had higher urinary concentrations of some of the exposure biomarkers; therefore, the results may not be fully generalizable to other populations.
Our study also had many strengths. Our robust sample size, and the collection of exposure biomarkers and hormone data at two time points during pregnancy helps account for the biomarkers' short lifespan in the body, and the varying levels of hormones throughout pregnancy. The repeated measures allow for the control of intra-individual variability, and increases statistical power. We were also able to explore potential windows of susceptibility for these associations.
Additionally, we were able to compare our results from this analysis to our own analyses that employed similar statistical methods in two other data sets, namely LMMs to capture biomarkers at various time points and allow subject-specific intercepts. While there were many similarities in the results across the three analyses, the differences in results may point to the importance of outside factors that may not be captured in our models that alter the associations between these chemicals and endocrine disruption through interaction with the chemicals. These outside factors could include other endocrine-altering variables, such as exposure to other unaccounted for chemicals, maternal stress, genetic, epigenetic, or other differences. It is imperative that future studies look beyond the association between a single chemical and singe hormone, and explore potential interactions with chemical exposure.
Our results provide suggestive human evidence for associations between select biomarkers with maternal thyroid and reproductive hormones during gestation. Of note, we report negative associations between parabens and SHBG, a negative association between BPS and CRH, and associations between triclocarban and triclosan with reproductive and thyroid hormones. Our stratified analyses show that some associations may be stronger at certain time points during pregnancy. Further studies in larger populations and with more repeated measures across pregnancy to will be useful to confirm our findings, and better understand if and how these hormone changes may affect downstream maternal and infant health outcomes.
BPA:
BPF:
Bisphenol-F
BPS:
Bishphenol-S
CRH:
Corticotropin-releasing hormone
FT4:
Free thyroxine
hCG:
LMM:
Linear Mixed Models
MLR:
NHANES:
National Health and Nutrition Examination Survey
PROTECT:
Puerto Rico Testsite for Exploring Contamination Threats
SHBG:
Sex-hormone-binding globulin
T3:
Total triiodothyronine
Total thyroxine
TSH:
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We gratefully acknowledge Antonia Calafat and Xiaoyun Ye at the Centers for Disease Control and Prevention for analysis of urinary phenol, paraben and triclocarban concentrations. We would like to thank and D. McConnell of the CLASS Lab at University of Michigan for assistance in hormone analysis.
This work was supported by the National Institute of Environmental Health Sciences, National Institutes of Health (Grants P42ES017198, P50ES026049, and UG3OD023251). Funding for Ferguson KK was provided by the Intramural Research Program of the National Institute of Environmental Health Sciences, NIH. The funding sources had no involvement in the study design, collection, analysis & interpretation of data or writing of the report. The findings and conclusions in this report are those of the authors and do not necessarily represent the official position of the Centers for Disease Control and Prevention. Use of trade names is for identification only and does not imply endorsement by the CDC, the Public Health Service, or the US Department of Health and Human Services.
Datasets analyzed for the current study are not publicly available because they contain sensitive and protected health information on our participants.
Department of Environmental Health Sciences, University of Michigan School of Public Health, Room 1835 SPH I, 1415 Washington Heights, Ann Arbor, MI, 48109-2029, USA
Amira M. Aker, Kelly K. Ferguson & John D. Meeker
Epidemiology Branch, Intramural Research Program, National Institute of Environmental Health Sciences, Durham, USA
Graduate School of Public Health, Medical Sciences Campus, University of Puerto Rico, San Juan, PR, USA
Zaira Y. Rosario
Department of Biostatistics, University of Michigan School of Public Health, Ann Arbor, MI, USA
Bhramar Mukherjee
College of Engineering, Northeastern University, Boston, MA, USA
Akram N. Alshawabkeh
Centers for Disease Control and Prevention, Atlanta, GA, USA
Antonia M. Calafat
College of Public Health, University of Georgia, Athens, GA, USA
José F. Cordero
Amira M. Aker
AMA: data analysis, AMA & JDM: data interpretation and drafting of document, KKF: data cleaning, JDM, ANA & JFC: conception and design, ZYR: acquisition of data, BM: methodology advisor; AMC: data measurement. All authors read and approved the final manuscript.
Correspondence to John D. Meeker.
This study was approved by the research and ethics committees of the University Of Michigan School Of Public Health, University of Puerto Rico, Northeastern University, and the University of Georgia. All study participants provided full informed consent prior to participation.
The authors declare that they have no competing interests. The involvement of the Centers for Disease Control and Prevention (CDC) laboratory did not constitute engagement in human subjects research.
Table S1. Results of the adjusted MLRs regressing reproductive hormones versus exposure biomarkers by visit. Table S2. Results of the adjusted MLRs regressing thyroid hormones versus exposure biomarkers by visit. Table S3. Result comparison between the common exposure biomarkers and hormones. (DOCX 32 kb)
Adjusted multiple linear regressions of hormones versus urinary concentrations of biomarkers stratified by study visit. Visit 1: 16-20 weeks; Visit 3: 24-28 weeks. EPB and BPF are categorical variables. * represents at least one marginal association between the urinary concentration and the hormone across the four time points. ** represents at least one significant association between the urinary biomarker concentration and the hormone across the four time points. BPF and EPB were dichotomous variables. 2,4-DCP: 2,4-dichlorophenol; 2,5-DCP: 2,5-dichlorophenol; BP-3: Benzophenone; TCS: Triclosan; TCC: Triclocarban; EPB: ethylparaben; MPB: Methylparaben; BPB: Butylparaben; PPB: Propylparaben (DOCX 664 kb)
Aker, A.M., Ferguson, K.K., Rosario, Z.Y. et al. A repeated measures study of phenol, paraben and Triclocarban urinary biomarkers and circulating maternal hormones during gestation in the Puerto Rico PROTECT cohort. Environ Health 18, 28 (2019). https://doi.org/10.1186/s12940-019-0459-5
Reproductive hormones
In-utero
Endocrine disruption | CommonCrawl |
\begin{document}
\title{Role of qubit-cavity entanglement for switching dynamics of quantum interfaces in superconductor metamaterials} \author{Sergey V. Remizov$^{1,2}$} \author{Dmitriy S. Shapiro$^{1,2,3}$} \email{[email protected]} \author{Alexey N. Rubtsov$^{1,3,4}$} \affiliation{$^1$Dukhov Research Institute of Automatics (VNIIA), 127055 Moscow, Russia} \affiliation{$^2$V. A. Kotel'nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, 125009 Moscow, Russia} \affiliation{$^3$National University of Science and Technology MISIS, Leninsky prosp. 4, Moscow, 119049, Russia} \affiliation{$^4$Russian Quantum Center, Skolkovo, 143025 Moscow Region, Russia}
\begin{abstract}
We study quantum effects of strong driving field applied to dissipative hybrid qubit-cavity system which are relevant for a realization of quantum gates in superconducting quantum metamaterials. We demonstrate that effects of strong and non-stationary drivings have significantly quantum nature and can not be treated by means of mean-field approximation. This is shown from a comparison of steady state solution of the standard Maxwell-Bloch equations and numerical solution of Lindblad equation on a density matrix. We show that mean-field approach provides very good agreement with the density matrix solution at not very strong drivings $f<f^*$ but at $f>f^*$ a growing value of quantum correlations between fluctuations in qubit and photon sectors changes a behavior of the system. We show that in regime of non-adiabatic switching on of the driving such a quantum correlations influence a dynamics of qubit and photons even at weak $f$.
\end{abstract}
\maketitle
\subsection{Introduction} Quantum metamaterials are hybrid systems consisting of arrays of qubits coupled to the photon modes of a cavity \cite{Macha,Astafiev,Rakhmanov, Fistul,ZKF,SMRU,Brandes,Zou}. In solid state structures the qubits are realized using nitrogen-vacancy (NV) centers in diamonds \cite{nv-centers-0,nv-centers, nv-centers-1}, and spins of $^{31}$P dopants in $^{28}$Si crystals \cite{Morton} or Cr$^{3+}$ in Al$_2$O$_3$ samples \cite{Schuster}, and superconducting Josephson qubits \cite{MSS, Orlando,mooij}. Among of others, the Josephson qubits are particularly perspective for an implementation of quantum gates \cite{DiCarlo, MSS, Nation, Clarke} due to their high degree of tunability. Frequency of excitation, given by an energy difference between ground and excited states, can be controllably tuned in a wide range using the external magnetic flux threading a loop of the qubit. Modern technology allows for a production of metematerial structures obeying sophisticated geometry and low decoherence effects.
High nonlinearity of the qubit excitation spectrum, combined with low decoherence, gives rise to unusual properties of quantum metamaterials, distinguishing them from the linear-optical metastructures. These unusual features are associated with intrinsic quantum dynamics of qubits and photon degrees of freedom. They are revealed in the optical response of a metamaterial to the external strong pump field, driving the system away from its ground state. A textbook example is the rotation on a Bloch sphere of the state of a single qubit subjected to an external field pulse. The well-understood solution for dynamics of a single qubit is commonly used as a key building block in the mean-field description of complex metamaterials containing a number of qubits and cavity modes. Assuming no correlations between the qubits and photons, one comes to the set of Maxwell-Bloch equations virtually describing qubits coupled to a classical field of the cavity and (or) external pump.
This article is devoted to the role of quantum entanglement between qubit and cavity modes of the superconducting metamaterial. Whereas it is generally clear that these correlation effects beyond Maxwell-Bloch scheme are revealed in strong-driving regimes, their quantitative role in an experimentally/technologically relevant situation is not yet studied. At the same time, such study is highly motivated by the quantum technology development, because a realization of qubit gates and operation of quantum simulators assume applying of driving fields of strengths comparable with qubit-cavity coupling energy. We argue that a quantitative description of an operation of realistic quantum metamaterials, which involve non-adiabatic and strong perturbations, assumes taking into account quantum correlations.
We present a study of the a simple yet realistic model of the quantum interface, defined as a dissipative hybrid system containing a resonant qubit being connected to the cavity mode and simultaneously subjected to the strong external field. We assume that the two-level system is highly anharmonic flux qubit, being a loop with several Josephson junctions, where highest levels are not excited by the external drivings.
Hybridization between the qubit and the cavity mode provides a transfer of the pump photons to the cavity mode via qubit excitations. Therefore the internal qubit dynamics is fingerprinted in the cavity field, and can be later read out or transferred to another qubit. We describe the evolution of the many-body density matrix of the system using the Lindblad equation, and compare the results with whose obtained using Maxwell-Bloch approximation. We observe, that for a constant driving field the two approaches give almost same results (that is, qubit-cavity correlations are negligible) up to certain threshold value of the pump $f^*$ depending mainly on relaxation rates in a cavity and in a qubit. For higher pump field, the effect of correlations rapidly grow, making Maxwell-Bloch approximation quite inaccurate. There is a remarkable artifact which follows from the Maxwell-Bloch approach, but is not present in the many-body description: a hysteresis in photon number as a function of $f$. This behavior shows up in a certain range around of the threshold $f^*$ if a coupling energy between photons and the qubit is large enough. Furthermore, we find that non-adiabatic switching on of the driving, from zero to a certain value, reveals discrepancy between mean-field and exact solutions even for drivings below the steady-state threshold $f^*$.
\section{Theoretical approach}
\label{theory}
Description of circuit quantum electrodynamics of superconducting metamaterials of qubits and transmission line are reduced to a Hamiltonian of Tavis-Cummings model \cite{Carmichael}. In our analysis we start from more simple situation of a single qubit which is coupled to photon mode. Quantum mechanical description is reduced to well-known Janes-Cummings model which is exactly integrable. Namely, Hamiltonian of an isolated qubit-cavity system is $$H_{JC}=\omega_R a^+a + \epsilon \sigma^+ \sigma^- +g (a\sigma^+ + a^+\sigma^-).$$ First term describes excitations in photon mode of the resonator, where bosonic operators $a,a^+$ obey commutation rules $[a,a^+]=1$. Second term is related to excitations in qubit where $\sigma^+ , \sigma^- $ are Pauli operators. External transversal driving applied to a qubit is accounted for by $$ H_{ext}=\frac{f(t)}{2} \left(e^{-i\omega t}\sigma^+ + e^{i\omega t}\sigma^-\right) $$ where $f(t)$ is slow envelope function and $\omega$ is fast reference frequency. In our studies we are limited by single non-adiabatic switching of the form \begin{equation} f(t)=f\theta(t). \label{f-t} \end{equation} The system under consideration is coupled to a dissipative environment, hence, we employ Lindblad equation on the density matrix $\rho(t)$ dynamics written in many-body basis of qubit and photon states. The Lindblad equation reads as \begin{equation} i(\partial_t\rho(t) - \Gamma[\rho(t)])=[H(t),\rho(t)]. \label{lindblad} \end{equation} where full Hamiltonian $$
H(t)=H_{JC}+H_{ext}(t) $$ and relaxations in qubit and cavity are taken into account by means of \begin{multline} \Gamma[\rho]=\frac{\kappa}{2}(2a\rho a^+ - a^+a \rho- \rho a^+a) + \\ +\frac{\Gamma_1}{2} (2\sigma^-\rho\sigma^+ - \sigma^+\sigma^-\rho- \rho\sigma^+\sigma^-). \end{multline}
In our numerical solution we calculate $\rho(t)$ by means of direct integration of Lindblad equation in a truncated Hilbert space. In approximate mean-field techniques we derive equations on averages from this equation (\ref{lindblad}) as well.
Note that everywhere below we perform transition into rotating frame basis related to the main frequency $\omega$ of driving signal which is tuned in resonance with cavity mode frequency $\omega=\omega_R$. The full Hamiltonian $H(t)$ in (\ref{lindblad}), given by $H_{JC}+H_{ext}(t)$, in this $\omega$-rotating frame basis reads \begin{equation} H(t)= \Delta \sigma^+\sigma^- +g (a\sigma^+ + a^+\sigma^-)+\frac{f(t)}{2} \sigma^x. \label{h} \end{equation} where qubit detuning is $\Delta=\epsilon-\omega_R$.
From the Lindblad equation $ i(\partial_t\rho(t) -\Gamma[\rho(t)])=[H(t),\rho(t)]$ we derive equations for averages $\langle a \rangle, \langle a^+ \rangle, \langle \sigma^{\pm} \rangle,\langle \sigma^z \rangle$. This is done with use of definitions, e.g. applied to $a$, by the following scheme \begin{equation} \partial_t \langle a (t) \rangle ={\rm Tr}( \partial_t\rho(t) a)={\rm Tr}( -i[H(t),\rho(t)]a-\Gamma[\rho(t)])a ). \end{equation} We apply the following standard mean-field approximation: we factorize the averages \begin{equation} \langle a\sigma^+ \rangle\rightarrow \langle a\rangle \langle \sigma^+ \rangle, \langle a\sigma^z \rangle\rightarrow \langle a\rangle \langle \sigma^z \rangle \label{factorization-m-b} \end{equation} which appears in r.h.s. parts of equations on $\langle a \rangle, \langle a^+ \rangle, \langle \sigma^{\pm} \rangle,\langle \sigma^z \rangle$. On the level of density matrix this corresponds to the introduction of the reduced density matrices $\rho_q$ and $\rho_{ph}$ and the full one $\rho_{mf}=\rho_q\otimes \rho_{ph}$. This factorization (\ref{factorization-m-b}) is an approximation where we neglect correlation between fluctuations in qubit and photon mode ($\delta\sigma^{\pm}, \delta\sigma^{z}$ and $\delta a, \delta a^+$) \begin{equation} \langle a\sigma^{z,\pm} \rangle=\langle a\rangle \langle \sigma^{z,\pm} \rangle+\langle \delta a\delta\sigma^{z,\pm}\rangle. \label{fluctuations-0} \end{equation}
After such the factorization (\ref{factorization-m-b}) we end up with Maxwell-Bloch non-linear equations (we do not write $\langle \rangle$ for brevity)
\begin{equation} \partial_t a (t) = -\frac{\kappa}{2}a(t)-i g \sigma^-(t), \quad c.c., \label{a} \end{equation} \begin{equation} \partial_t \sigma^+ (t) = (i\Delta-\Gamma_1/2) \sigma^+ (t) -i\left(\frac{f(t)}{2}+g a^+(t)\right) \sigma^z(t), c.c., \label{sigma_minus} \end{equation} \begin{multline} \partial_t \sigma^z (t) =- \Gamma_1 \left(\sigma^z(t)+1\right) + \\ +2 i g \left( a^+(t)\sigma^-(t)-a(t)\sigma^+(t)\right)+\\ +i f(t) \left(\sigma^-(t)-\sigma^+(t)\right). \label{sigma-z} \end{multline} Note, that photon number dynamics is found from solution for $a(t)$ in such a mean-field technique as \begin{equation}
n_{ph} (t) = |a(t)|^2. \label{n-mb-0} \end{equation}
\section{Results} \subsection{ Steady state regime} In this part of the paper we demonstrate a comparison between results obtained from solutions of Lindblad (full many-body density matrix) and Maxwell-Bloch (mean-field) equations in a wide range of drivings $f$. Here and below we are limited by fully resonant regime where $\epsilon=\omega_R$, i.e. the detuning is zero $\Delta=0$. We evaluate numerically the $f$-dependences for qubit excited state occupation number $n_q$, generated photon number $n$ in the cavity and correlators $\langle \delta a\delta\sigma^{z,\pm}\rangle$. All the data presented in the paper are obtained for the system with the parameters $\Gamma_1=0.5$ MHz, $\kappa=0.4$ MHz. This section is devoted to the steady-state regime emerging after a long evolution of the system subjected to the driving field having a constant amplitude and phase.
We observe from Figures \ref{result:stationary:nph} and \ref{result:stationary:nq} a very good agreement between the mean-field and the full density matrix solutions for $n_{ph}$ and $n_q$ at $f<f^*$ where the value of $f^*$ divides a ranges of weak and strong field steady state regimes. At $f> f^*$ we observe an agreement for qubit occupation number which is $n_q=1/2$ in both of solutions. Indeed there are significant distinctions in behavior of photon degree of freedom: in strong field limit of $f> f^*$ the photon number $n$ decays to zero in Maxwell-Bloch solution but saturates to a finite value in the Lindblad numerical calculation.
The steady state solution of Maxwell-Bloch equations can be analyzed to explain the observed differences.
Taken the l.h.s. parts of the equations (\ref{a},\ref{sigma_minus}) and their conjugates equal to zero, the following
relations between $\langle a \rangle, \langle a^+ \rangle, \langle \sigma^{\pm} \rangle$ and $n_q=(\langle \sigma^z \rangle +1)/2$ are derived \begin{equation} \left( \begin{array}{c} \langle \sigma^{-} \rangle \\ \\ \langle \sigma^{+} \rangle \\ \\ \langle a \rangle \\ \\ \langle a^+ \rangle \\ \end{array} \right)= \left( \begin{array}{c}
-\frac{i f (2n_q-1) \kappa }{4 g^2 (2n_q-1) -\Gamma_1 \kappa } \\ \\
\frac{i f (2n_q-1) \kappa }{4 g^2 (2n_q-1) -\Gamma_1 \kappa } \\ \\
-\frac{2 f g (2n_q-1) }{4 g^2 (2n_q-1) -\Gamma \kappa } \\ \\
-\frac{2 f g (2n_q-1) }{4 g^2 (2n_q-1) -\Gamma_1 \kappa } \\ \end{array} \right). \label{a-sigma-sol} \end{equation} Combining these results with (\ref{sigma-z}) with zero l.h.s. part we obtain the relation between $n_{ph}$ and $n_q$ \begin{equation} n_{ph}=-4n_q(2n_q-1)\frac{ g^2}{\kappa^2}. \label{nph} \end{equation}
The relation between qubit occupation number itself and driving amplitude $f$ is given by the implicit expression which can be found from (\ref{a-sigma-sol}) as well
\begin{equation}
f=\frac{|\Gamma_1 \kappa - 4 g^2 (2n_q-1)|}{\kappa}\sqrt{\frac{ n_q}{1-2n_q}}. \label{f}
\end{equation} \begin{figure}
\caption{Photon number vs driving amplitude $f$ in the steady state regime.}
\label{result:stationary:nph}
\end{figure} \begin{figure}
\caption{Qubit occupation number vs driving amplitude $f$ in the steady state regime}
\label{result:stationary:nq}
\end{figure}
Definitely, the zero value of $n_{ph}$ resulting from Eq. (\ref{nph}) at large $f$, when qubit occupation number is saturated to $n_q=1/2$ (see Fig. \ref{result:stationary:nq}), is wrong. A correct value for $n_{ph}$ can be easily found from the Hamiltonian (\ref{h}) in the limit of $f\gg g$. Namely, qubit ground state in such a limit is odd superposition $|\psi_{gs}\rangle=(|g\rangle-|e\rangle)/\sqrt{2}$, and, hence, $\sigma^\pm=1/2$. After that, we find perturbatively steady state $a=-2i(g/\kappa) \sigma^-$ from (\ref{a}) for the dissipative system, yielding $n_{ph}=|a|^2=(g/\kappa)^2$ from the mean-field definition of $n_{ph}$ (\ref{n-mb-0}). This result is in agreement with the tendency to saturation of photon number $n_{ph}$ at large $f$ observed in the numerical solution.
Other comment is about the bistability region in the Maxwell-Bloch result seen in Figure \ref{result:stationary:nph} and \ref{result:stationary:nq}. Mathematically it is due to the fact that (\ref{f}) is a 3-rd order equation with respect to $f$. There exists a range for $f$, where three solutions for $n_q$, and, consequently, three values of $n_{ph}$ at a given $f$ are possible. The condition for an existence of the three solutions in Maxwell-Bloch equations in this stationary regime is
\begin{equation}
g>\sqrt{2 \Gamma_1 \kappa}. \label{gc}
\end{equation} This condition follows from the expression for two extrema of the inverse relation between $n_q$ and $f$ (shown as dased curve in Fig. \ref{result:stationary:nq}): $$n_{q}^{(1,2)}= \frac{1}{8}\left( 3\pm \sqrt{1-\frac{2 \Gamma \kappa}{ g^2}} \right).$$ One of the three solutions appears unstable and does not show up in the curves obtained numerically. The two others are stable and give rise to a bistability regime similar to the one in \cite{SavageCarmichael} where a driving was applied to photon mode. We insist, however, that the solution of the Lindblad equation for the many-body density matrix does not contain such a bistable regime and we therefore interpret it as an artifact of the mean-field approximation.
It is important that non-zero correlators $\langle \delta a\delta\sigma^{z,\pm}\rangle$ demonstrate the increase of the effect of quantum fluctuations in the regime of strong driving $f>f^*$, see Fig. \ref{result:stationary:spl-a}. In the regime of strong coupling (\ref{gc}) the typical $f^*$ can be estimated from the mean-field relation (\ref{f}) as follows $$ f^*\sim {\rm max}[\Gamma_1,\frac{g^2}{\kappa}]. $$
As it is seen from the curve for $n_{ph}$ these fluctuations make a significant contribution in the photon sector of the system. Value of the fluctuations can be perturbatively estimated from the Maxwell-Bloch equations :
$$
\langle \delta a\delta\sigma^+\rangle=\frac{2i\kappa g (2n_q-1)(2g n_{ph}+f a)}{(2\kappa+\Gamma_1)\Gamma_1} - \frac{2 i g n_q}{2\kappa+\Gamma_1}
$$ This correlator saturates to a non-zero value of $$\langle \delta a\delta\sigma^+\rangle_{f\gg f^*} = \frac{-i g}{2\kappa+\Gamma_1}$$ in the limit of strong driving where the qubit occupation number is $n_q=1/2$. In the Figure \ref{result:stationary:spl-a} we present the results for the correlators obtained from the Lindblad solution for the full density matrix. The saturation of $\langle \delta a\delta\sigma^+\rangle$ at high $f$, appeared in the mean-field approach, is observed in these data as well. \begin{figure}
\caption{Correlator of fluctuations $ \langle \delta a\delta\sigma^{+}\rangle$ and $ \langle \delta a\delta\sigma^{z}\rangle$ extracted from solution of the Lindblad equation for the full density matrix.}
\label{result:stationary:spl-a}
\end{figure}
In Figure \ref{result:stationary:s} we show the numerical results for the von Neumann entropy $S=-{\rm Tr}\rho \ln \rho$. The solid curve demonstrates $S(f)$ calculated from the Lindblad approach while the dashed one is related to the mean-field approximation where the effective Hamiltonian include the values of $\langle a \rangle, \langle a^+ \rangle, \langle \sigma^{\pm} \rangle$ found from the solution of Maxwell-Bloch equations. The difference between them at $f>f^*$ shows again that there is a significant entanglement between the qubit and photon degrees of freedom in the strong driving domain. The mean-field solution assumes that the many-body density matrix is a direct product of the qubit and photon ones $\rho_{mf}=\rho_{ph} \otimes \rho_q$, where the elements responsible for the entanglement are zero. These non-diagonal elements of the density matrix, taken into account in the solution of the Lindblad equation, increase the entropy. \begin{figure}
\caption{Entropy vs driving amplitude $f$ in the stationary regime. The solid curve is related to the density matrix found from the solution of Lindblad equation. The dashed curve describes entropy calculated within the mean-field approximation. }
\label{result:stationary:s}
\end{figure}
\subsection{Non-stationary regime}
The second result of our paper is that quantum corrections $\langle \delta a\delta\sigma^{z,+}\rangle$ play a significant role in the non-stationary dynamics of the quantum interface even at drivings less than the steady state threshold $f^*$. This is demonstrated via time evolution of $n_{ph}(t)$ and $n_q(t)$ after the moment $t=0$ when the external driving is suddenly switched on. The threshold value, observed for the steady state regime, is estimated as $f\approx 1.5 g$ for our parameters of the system. We set the after-quench value of the driving at the smaller value $f=g$. Figures \ref{result:nonstationary:ns}, \ref{result:nonstationary:nph} and \ref{result:nonstationary:spl-a} demonstrate the distinctions between the qubit and photon occupation number dynamics obtained from non-stationary solutions of the Maxwell-Bloch (\ref{a},\ref{sigma_minus},\ref{sigma-z}) and Lindblad equation (\ref{lindblad}). \begin{figure}
\caption{Time evolution of the qubit occupation number $n_q(t)$ found from the solution on the full density matrix and the mean-field approach at $f=g$.}
\label{result:nonstationary:ns}
\end{figure} \begin{figure}
\caption{Time evolution of photon occupation number $n_{ph}(t)$ found from solution of the Lindblad equation on the full density matrix and the mean-field approach at $f=g$. }
\label{result:nonstationary:nph}
\end{figure} \begin{figure}
\caption{Time evolution of correlations $ \langle \delta a\delta\sigma^{+}\rangle$ and $ \langle \delta a\delta\sigma^{z}\rangle$ extracted from the solution of the Lindblad equation at $f=g$.}
\label{result:nonstationary:spl-a}
\end{figure} \begin{figure}
\caption{Time evolution of entropy $S$ at different amplitudes of the driving $f$.}
\label{result:nonstationary:s}
\end{figure}
In Figure \ref{result:nonstationary:s} we present the results for von Neumann entropy as function of time at different values of the driving $f$.
We observe a strong difference in values entropy found from solving of Lindblad (solid curves) and mean-field (dashed curves) equations. For $f>f^*$, the entropy grows almost monotonically, until the saturation at the steady-state value. Contrary, for $f<f^*$ there is a pronounced maximum at $t\approx 4 \mu s$. The peak is present and the full-$\rho$ result is different from the mean-field one even for a weak driving $f=0.1 g$, although the steady-state entropy is almost vanished for much larger $f=0.75 g$. This indicates an emergent entanglement between qubit and photon mode of the quantum interface being switched.
\section{Conclusions} We have studied the response of a dissipative hybrid qubit-cavity system to the applied strong driving field, having in mind the future possible realization of quantum operations in superconducting quantum metamaterials. We demonstrated that for the case studied the many-body effects (or, equally, the entanglement between the qubit and photon excitations) are important and that the system cannot be treated by means of a mean-field approximation. This is shown from a comparison of analytical steady state solution of the standard Maxwell-Bloch equations and numerical simulations based on Lindblad equation on the many-body density matrix. Speaking more concretely, we have shown that mean-field approach, where the density matrix of the system can be represented via direct product of isolated qubit and photon ones $\rho_{mf}=\rho_{ph} \otimes \rho_q$, provides a good steady state solution up to certain threshold $f^*$ but at $f>f^*$ the strong discrepancy from the many-body result is observed. It is related with a growing value of quantum correlations between fluctuations of qubit and photons fields which start to play a significant role in behavior of the system. We show in our analysis that at large enough coupling energy between cavity and qubit modes the solution of Maxwell-Bloch equations reveals an artifact being a hysteresis in number of photons as function of the driving amplitude in vicinity of the threshold $f^*$. Such a hysteresis has not been observed in the full density matrix solution. Also we have studied an effect of the non-adiabatic switching of the driving and show that there is a difference between mean-field and the density matrix solutions even for the drivings weaker than the steady state threshold $f^*$.
Our findings demonstrate quantitative limitations of standard mean-field description and show the crossover between the classical and many-body quantum regimes. In the classical regime, the qubit virtually acts as a linear (Gaussian) degree of freedom; this regime cannot reveal a difference between the quantum and linear-optical metamaterials. When the two-level nature of the qubit plays an essential role, its entanglement with the cavity mode is also large and should be accounted. We also point out that the effect of correlations is revealed while the number of photons in the cavity mode is not small and one could naively expect that the cavity operates in a classical regime.
In our solutions we have used parameters relevant for contemporary metamaterials involving highly anharmonic flux qubits, and we expect that the obtained results will find an application in realization of quantum gates in superconducting quantum circuits and metamaterials.
\section{Acknowledgments}
Authors thank Yuriy E. Lozovik, Andrey A. Elistratov, Evgeny S. Andrianov and Kirill V. Shulga for fruitful discussions. The study was funded by the Russian Science Foundation (grant No. 16-12-00095).
\end{document} | arXiv |
Rapid and quantitative uncertainty estimation of coseismic slip distribution for large interplate earthquakes using real-time GNSS data and its application to tsunami inundation prediction
Keitaro Ohno ORCID: orcid.org/0000-0002-2001-16261,7,
Yusaku Ohta ORCID: orcid.org/0000-0003-4818-477X1,2,3,
Ryota Hino1,3,
Shunichi Koshimura3,
Akihiro Musa4,5,
Takashi Abe3 &
Hiroaki Kobayashi6
This study proposes a new method for the uncertainty estimation of coseismic slip distribution on the plate interface deduced from real-time global navigation satellite system (GNSS) data and explores its application for tsunami inundation prediction. Jointly developed by the Geospatial Information Authority of Japan and Tohoku University, REGARD (REal-time GEONET Analysis system for Rapid Deformation monitoring) estimates coseismic fault models (a single rectangular fault model and slip distribution model) in real time to support tsunami prediction. The estimated results are adopted as part of the Disaster Information System, which is used by the Cabinet Office of the Government of Japan to assess tsunami inundation and damage. However, the REGARD system currently struggles to estimate the quantitative uncertainty of the estimated result, although the obtained result should contain both observation and modeling errors caused by the model settings. Understanding such quantitative uncertainties based on the input data is essential for utilizing this resource for disaster response. We developed an algorithm that estimates the coseismic slip distribution and its uncertainties using Markov chain Monte Carlo methods. We focused on the Nankai Trough of southwest Japan, where megathrust earthquakes have repeatedly occurred, and used simulation data to assume a Hoei-type earthquake. We divided the 2951 rectangular subfaults on the plate interface and designed a multistage sampling flow with stepwise perturbation groups. As a result, we successfully estimated the slip distribution and its uncertainty at the 95% confidence interval of the posterior probability density function. Furthermore, we developed a new visualization procedure that shows the risk of tsunami inundation and the probability on a map. Under the algorithm, we regarded the Markov chain Monte Carlo samples as individual fault models and clustered them using the k-means approach to obtain different tsunami source scenarios. We then calculated the parallel tsunami inundations and integrated the results on the map. This map, which expresses the uncertainties of tsunami inundation caused by uncertainties in the coseismic fault estimation, offers quantitative and real time insights into possible worst-case scenarios.
A rapid understanding of the magnitude and fault areas of large earthquakes is crucial for disaster response. Numerous studies have shown the advantages of the high-rate (typically 1 Hz) global navigation satellite system (GNSS) as a broadband sensor that can directly measure displacement without saturation (e.g., Larson et al. 2003; Ohta et al. 2006; Larson 2009). The use of onshore high-rate GNSS data enable the rapid estimation of offshore finite fault models that are expected to be used for tsunami prediction (e.g., Blewitt et al. 20062009; Ohta et al. 2012; Melgar et al. 2012; Tsushima et al., 2014; Tsushima and Ohta 2014; Ohta 2016). The Geospatial Information Authority of Japan (GSI) operates a nationwide continuous GNSS network called GEONET. After the massive tsunami caused by the 2011 Tohoku-Oki earthquake, the GSI and Tohoku University jointly developed the REGARD system (REal-time GEONET Analysis system for Rapid Deformation monitoring, Kawamoto et al. 2016, 2017), which consists of real-time GNSS positioning, automatic detection of coseismic displacement, and quasi-real-time finite fault model inversion routines.
The finite fault models estimated by REGARD are helpful for obtaining the initial sea surface distribution for tsunami forecasting. The consortium led by Tohoku University has developed a real-time damage estimation system for tsunami inundation using the REGARD fault model as an initial tsunami source model (Musa et al. 2018; Ohta et al. 2018). This system is expected to be used for the initial response of the government when a disaster occurs.
For the 2016 Kumamoto earthquake (Mw 7.0), REGARD successfully estimated a single rectangular fault model automatically in real time (Kawamoto et al. 2016). However, estimation using only onshore GNSS data does not necessarily provide accurate estimates of offshore fault areas. From this point of view, in the single rectangular fault model, Ohno et al. (2021) developed a new method to quantitatively evaluate the uncertainties of the estimated results using the Bayesian inversion approach. This study showed a tradeoff between the fault area and slip amount, especially for offshore earthquakes, which affects the initial wave field of the tsunami. In the slip distribution model of REGARD, quantitative evaluation of the uncertainty is also essential because of the effects of station arrangement, modeling errors, and constraints on slip smoothing. For example, in the case of the 2011 Tohoku-Oki earthquake, it was suggested that there was a large slip around the trench axis based on seafloor geodetic data (e.g., Iinuma et al. 2012), although such slip was not significantly suggested using only onshore GNSS data (e.g., Melgar and Bock 2015).
The quantification of the uncertainty of fault models enables us to consider the accuracy of tsunami predictions calculated based on them. With the underestimation of tsunami wave height in the 2011 Tohoku-Oki earthquake as a lesson, the danger of announcing tsunami predictions as a single value has been proposed. The Japan Meteorological Agency (JMA) introduced the announcement of tsunami warnings with more qualitative expressions (JMA 2012). In addition, several studies have shown a method for adding uncertainty information to tsunami prediction (e.g., Sraj et al. 2014; Fukutani et al. 2015; Takagawa and Tomita 2015; Dettmer et al. 2016; Goda and Song 2016; Gibbons et al. 2020; Goda et al. 2020; Mulia et al. 2020; Giles et al. 2021). Dettmer et al. (2016) evaluated the uncertainty of the initial sea surface displacement in the 2011 Tohoku-Oki earthquake deduced from tsunami waveforms using a trans-dimensional algorithm based on wavelet decomposition of the displacement field. Goda et al. (2020) presented an extensive tsunami hazard assessment for Nankai–Tonankai trough events using 1000 kinematic earthquake rupture models for Monte Carlo tsunami simulations. Giles et al. (2021) proposed a workflow that integrates the entire chain of components from the tsunami source to quantify hazard uncertainties by approximating the functionally complex and computationally expensive high-resolution tsunami simulations with a simple and cheap statistical emulator. On the other hand, in the damage estimation system due to tsunami inundation, it is currently difficult to evaluate the impact of the uncertainty of fault models on tsunami prediction.
With this background, the purpose of this study was to add uncertainty information to tsunami inundation estimation by quantifying the uncertainty of the coseismic slip distribution model estimated from onshore real-time GNSS data. We propose a "stepwise partitioning algorithm," for estimating coseismic slip distribution using Markov chain Monte Carlo methods (MCMC). Furthermore, we propose a "real-time tsunami inundation risk map," which visualizes the probability of risk of tsunami inundation based on the uncertainty of the estimated fault model. We verified the performance of this approach by applying it to simulated data in the Nankai Trough area.
Stepwise partitioning algorithm
Conventionally, the slip distribution model is estimated with a hyper-parameter governing the intensity of slip smoothing constraints determined with an index, such as ABIC (Akaike's Bayesian Information Criterion, Akaike 1980; e.g., Fukahata 2009). The error matrix obtained in this case is essential for interpreting the uncertainty of slip amount at each subfault. These methods are highly relevant in terms of obtaining a single most likely solution and its error. In other words, with these approaches, it is difficult to obtain multiple models that can explain the observed data, and the discussion of the errors remains a challenge. It is important to present several possible models with errors, while satisfying the observed data, to treat the obtained results as disaster information. Recently, many studies have adopted the Bayesian approach to geodetic studies because of its advantages in quantifying uncertainty (Ito et al. 2012; Dettmer et al. 2014; Minson et al. 2014a2014b; Jiang and Simons 2016; Ohno and Ohta 2018; Ohno et al. 2021). Owing to its sampling process, MCMC can generate a large number of model samples and quantify the uncertainty as probability density function (PDF), which is the ensemble of those samples. In this study, we regarded the diversity of samples as individual slip distribution models that can explain the observation data to evaluate the uncertainties of tsunami inundation caused by uncertainties in the estimation of coseismic slip distribution. Furthermore, to get the diversity of samples efficiently in real-time, essentially similar to smoothing constraint, but we tried a different approach to regularization.
We adopted simulated data based on the assumed Mw 8.75 1707 Hoei earthquake (Furumura et al. 2011; Todoriki et al. 2013; Inoue et al. 2016; Kawamoto et al. 2017). The assumed slip reached from Suruga Bay to the westernmost part of Shikoku (Fig. 1). Let \({\varvec{\theta}}\) be a model parameter vector that contains slip amounts on subfaults along the plate boundary in the Nankai Trough, and \({\varvec{d}}\) be a permanent displacement data vector with three components (horizontal and vertical components) based on real-time GNSS observations. We used the same method as Ohno et al. (2021) for MCMC sampling. Bayesian estimation of the unknown model parameters conditional on the observations is based on Bayes' theorem:
$$\begin{array}{*{20}c} {p\left( {{\varvec{\theta}}{|}{\varvec{d}}} \right) = \frac{{{\varvec{p}}\left( {{\varvec{d}}{|}{\varvec{\theta}}} \right) {\varvec{p}}\left( {\varvec{\theta}} \right)}}{{{\varvec{p}}\left( {\varvec{d}} \right)}}} \\ \end{array}$$
Inputted slip distribution for the 1707 Hoei earthquake in southwestern Japan (subdivision model of Inoue et al. 2016). The Mw is 8.75 and the color scale indicates the slip amount along each subfault. White dots indicate the 642 continuous Global Navigation Satellite System (GNSS) network (GEONET) stations, with displacements calculated using Okada (1992). The width of rectangular subfaults is approximately 8 km
where \({\varvec{p}}\left({\varvec{d}}\right)\) is a PDF of the observations, \({\varvec{p}}\left({\varvec{\theta}}\right)\) is a prior PDF of the model parameters, \({\varvec{p}}\left({\varvec{d}}|{\varvec{\theta}}\right)\) is a likelihood function, and \({\varvec{p}}\left({\varvec{\theta}}|{\varvec{d}}\right)\) is a posterior PDF of the model parameters. Note that \({\varvec{p}}\left({\varvec{d}}\right)\) is constant, because observations are fixed values and, hence, the posterior PDF is proportional to the product of the prior PDF and likelihood function. The likelihood function measures the degree of fit between the observed data \({\varvec{d}},\) and calculated data \(\widehat{{\varvec{d}}}\left({\varvec{\theta}}\right)\). The residuals are given by \({\varvec{r}}\left({\varvec{\theta}}\right)=\widehat{{\varvec{d}}}\left({\varvec{\theta}}\right)-{\varvec{d}}\). When the number of observation stations is \(N\), the dimension of \({\varvec{d}} \mathrm{is} 3N\). Assuming that the estimation error follows a normal distribution function, the likelihood function is defined as follows:
$$\begin{array}{*{20}c} {p\left( {{\varvec{d}}{|}{\varvec{\theta}}} \right) = \mathop \prod \limits_{i = E, N, U} \mathop \prod \limits_{j = 1}^{N} \frac{1}{{\sqrt {2\pi \sigma_{ij}^{2} } }}{\text{exp}}\left[ { - \frac{1}{{2\sigma_{ij}^{2} }}r_{ij}^{2} } \right]} \\ \end{array}$$
where \({\sigma }_{ij} (i=E (\mathrm{East}), N (\mathrm{North}), U (\mathrm{UpDown}))\) (which is identical at every GNSS station) is an event-dependent hyper parameter that includes modeling and observation errors. This study assumed the hyper parameters for each horizontal and vertical components as follows:
$$\begin{array}{*{20}c} \begin{aligned} \sigma_{Ej} & = \sigma_{Nj} = \max \left( {0.1\sqrt {d_{Ej}^{2} + d_{Nj}^{2} } , \sqrt {e_{Ej}^{2} + e_{Nj}^{2} } } \right), \\ \sigma_{Uj} & = \max \left( {0.1d_{Uj} , e_{Uj} } \right) \\ \end{aligned} \\ \end{array}$$
where \({e}_{ij}\) is the steady noise of real-time GNSS observations. We used \({0.1d}_{ij}\) as the proxy for the modeling and observation errors and assumed \({\sigma }_{ij}\) on each component at each station is whichever of those two error sources is larger. By assuming that \({\sigma }_{ij}\) depends on the displacement while keeping the real-time GNSS steady noise as a lower bound, we can prevent the model from overfitting the data. We used the Metropolis–Hasting method as the MCMC sampler (hereafter, M–H method; Metropolis et al. 1953; Hastings 1970). We adopted parallel tempering with eight parallel chains to improve the search efficiency (Geyer 1991; Jasra et al. 2007). For simplicity of the real-time analysis, we fixed rake angles to 90°; as such, \({\varvec{\theta}}\) contains only slip amounts, the element number of which is the same as that of the subfaults. In addition, we assumed the slip amounts to be non-negative, which is equivalent to assuming that the prior PDF formed a uniform distribution \(\mathrm{U}\left(0, \infty \right)\) for \({\varvec{\theta}}\). Incorporating non-negative slip constraint into the inversion as a search range setting is one of the advantages of the MCMC sampling process over conventional methods. There were 2951 subfaults on the plate boundary. We adopted the analytical solution of Okada (1992) as the Green's function for rectangular subfaults, where the fault width are approximately 8 km.
In the M–H method, we propose new transition candidates by applying random perturbations \(\Delta{\varvec{\theta}}\) with uniform distribution between \({-\Delta{\varvec{\theta}}}_{\mathrm{max}}\) and + \({\Delta{\varvec{\theta}}}_{\mathrm{max}}\). However, applying MCMC to such a problem with many unknown parameters takes a long time to converge. Therefore, in addition to parallel tempering, we improved the search efficiency using a multistage approach with stepwise partitioning of perturbation groups. Figure 2 shows an overview of the "stepwise partitioning algorithm". The Markov chain was divided into four stages (number of steps in each chain): stage 1 (\(3\times {10}^{6}\) steps); stage 2 (\(3\times {10}^{6}\) steps); stage 3 (\(3\times {10}^{6}\) steps); and stage 4 (\(3\times {10}^{6}\) steps). In each stage, we set spatial perturbation groups in advance, because grouping \(\Delta{\varvec{\theta}}\) around subfaults promoted convergence. Thus, we applied the same perturbation \(\Delta{\varvec{\theta}}\) to the same-colored subfaults, as shown in Fig. 2; we refer to these pre-determined areas with the same \(\Delta{\varvec{\theta}}\) as "perturbation groups". The number of perturbation groups are 80, 185, 388, and 1451 for stage 1, 2, 3, and 4, respectively. Increasing the number of perturbation groups as the stage progresses allows us to estimate the rough-to-detailed features. The utilization of such a grouping is equivalent to reducing the number of unknown parameters. It should be noted that we did not change the shapes of the 2951 rectangular subfaults; that is, only the perturbation groups were changed. Thus, the analytical displacements in each step were calculated for 2951 background subfaults using a constant each subfault's Green's function.
"Stepwise partitioning algorithm" overview, depicting transition from stage \(n\) to stage \(n+1\) (\(n=1, 2, 3\)). Colors indicate the perturbation groups of \(\Delta{\varvec{\theta}}\). Perturbation groups that differ from stage to stage enable search from the outline to the details without smoothing constraints. The number of perturbations \({\Delta{\varvec{\theta}}}_{\mathrm{max}}\) that differ for each perturbation group enables a flexible parameter search with large estimation uncertainty
As shown in Fig. 2, in the transition between stages, we used the median of the posterior PDFs of the previous stage as the initial value for the next stage and used the 95% confidence interval (CI) as the amount of perturbation \({\Delta{\varvec{\theta}}}_{\mathrm{max}}\) for the next stage. By inheriting the uncertainty as the transition amount, we searched for a broader slip amount with more uncertainty in a limited number of samples. Therefore, in the initial 10% of samples for each stage, we adjusted \({\Delta{\varvec{\theta}}}_{\mathrm{max}}\) by equal multiplication so that the acceptance rate would be 20–40% (e.g., Roberts and Rosenthal 2001), and the samples under adjustment were discarded during PDF generation as burn-in. In addition to the model parameters, we also calculated Variance Reduction (VR) for evaluation of the estimated model:
$$\begin{array}{*{20}c} {VR = 100\left( {1 - {\varvec{r}}^{{\text{T}}} {\varvec{r}} / {\varvec{d}}^{{\text{T}}} {\varvec{d}}} \right).} \\ \end{array}$$
In the later stage, the partitioning may be excessive for the sensitivity of the data and earthquake magnitude. We calculated the AIC (Akaike's Information Criterion, Akaike 1973) using the following equation to determine the optimum stage:
$$\begin{array}{*{20}c} {AIC = - 2\log \left( L \right) + 2M,} \\ \end{array}$$
where \(L\) indicates the maximum likelihood, and \(M\) indicates the number of unknown parameters, which is the number of perturbation groups in each stage.
The approach presented above is similar to the trans-dimensional approach (e.g., Dettmer et al. 2014), but our algorithm was designed to force change in one direction to simplify the problem for real-time purposes. Moreover, our procedure is predefined by a set of perturbation groups with different spatial resolutions. Then, an optimum stage is selected based on AIC corresponding to a regularization process similar to smoothing constraints with discrete search (i.e., 4 stages) of regularization parameters. We designed this approach to automatically generate various models with the appropriate spatial resolution for the magnitude of the target earthquake without smoothing constraints between adjacent subfaults. The subsequent tsunami calculation process ("Real-time tsunami inundation risk map" section) utilizes samples from one selected stage rather than multiple stages. In each stage with the constant spatial resolution, we can benefit from sampling without smoothing constraints as a diversity of multiple samples.
The computation time of this approach is within 30 min for each stage (\(3\times {10}^{6}\) steps in each chain) using an SX-Aurora TSUBASA Type20B processor with eight parallel chains in the case of 642 GNSS stations. In this study, we decided the number of samples with the highest priority to obtain a sufficient number to evaluate the differences between the stages. For real-time utilization, the convergence judge (e.g., Gelman 1996) should be introduced and must move to the next stage automatically. According to the estimated results of this paper, the number of chains in each stage seems to converge at about 10% of the number of advance settings. Based on this, if we could design convergence judges and AIC staging decisions properly, it is possible to complete the calculation up to stage 2 in approximately 6 min, which is a computation time that may be used in real time in the future.
Real-time tsunami inundation risk map
In general, because the tsunami inundation calculation is a nonlinear problem, it is difficult to evaluate the uncertainty of the tsunami inundation based on a single coseismic fault slip model and variation index, such as standard deviation of slip amounts. Therefore, to evaluate this uncertainty, we need to prepare multiple coseismic fault slip models that explain the data well. However, it is not realistic to calculate tsunami inundation for all MCMC samples, even if calculation speed were to improve in the future. Therefore, our algorithm aims to efficiently classify MCMC samples and integrate multiple tsunami inundation scenarios to probabilistically present tsunami inundation risk. Here, assuming that multiple calculations of \({10}^{2}\) orders will be possible within a time of several minutes in the future, we added "real-time" to the flow name in this study.
Figure 3 shows a flow chart of the real-time tsunami inundation risk map process.
Obtain sufficient MCMC samples to evaluate the uncertainty of the slip distribution model ("Stepwise partitioning algorithm" section). Samples from which VR was extracted are equivalent to a representative value based on the posterior PDF.
Classify these samples into small clusters (K) using the k-means approach using \({\varvec{\theta}}\) as the feature value.
Generate K representative slip distributions (using the median value of each subfault), and use them as inputs to calculate individual K tsunami inundations.
Count the number of inundations on the map.
Overview of the "real-time tsunami inundation risk map" process in the case of using MCMC samples in stage 2. (i) Obtain enough Markov chain Monte Carlo (MCMC) samples to evaluate the uncertainty of the slip distribution model ("Stepwise partitioning algorithm" section). Then, extract the samples in which variance reduction (VR) is equivalent to the representative value based on the posterior probability density function (PDF). (ii) Classify them into small clusters (K) by the k-means approach using \({\varvec{\theta}}\) as the feature value. (iii) Generate K representative slip distributions (using the median value in each subfault), and using them as inputs, calculate K individual tsunami inundations. (iv) Count the number of inundations on the map
The resulting map shows the possible tsunami arrival rate in each computation grid based on the uncertainty of fault slip estimated from real-time GNSS data.
The main feature of our algorithm is the clustering of MCMC samples, whereas generally utilize posterior PDFs. As discussed, the nonlinearity of tsunami inundation calculation is a problem, so we need to obtain sufficient fault models to evaluate the tsunami risk. In contrast, samples using the M–H method are not necessarily independent, because the M–H method proposes samples by perturbing the previous one. We adopted the k-means method (Steinhaus, 1956) to classify the samples into a predetermined number, with the number of clusters K determined by considering the computational cost of tsunami inundation, the availability of denominators for probability display, and the maintenance of the diversity of MCMC samples. The feature value used for clustering is the slip vector \({\varvec{\theta}}\), because the clustering phase does not depend on the specific target region of the tsunami calculation. Samples with the same VR were used to ensure fairness in terms of reproducibility of the observed data.
Here, the target of the tsunami inundation calculation was a 1563 km2 area between Tosa City and Aki City in Kochi Prefecture (red box in Additional file 1: Figure S1). Additional file 1: Table S1 lists the conditions of the tsunami inundation calculation. The grid size in the target area was 30 × 30 m, and the number of grids was 1,736,388 (2,082 grids along the east–west direction and 834 grids along the north–south direction). Tsunami propagation and inundation were calculated using the TUNAMI code (Tohoku University's numerical analysis model for investigating tsunami), which numerically solves the nonlinear shallow water equation using the staggered leap-flog finite difference method (Imamura 1995). The tsunami simulation was carried out from the time of the earthquake to 6 h later. The fracture propagation of the earthquake was not considered, and the slip was assumed to be instantaneous. We calculated seafloor displacement using the analytical result of Okada (1992), and the uplift of the seawater due to the horizontal movement of the seafloor was considered (Tanioka and Satake 1996). Coastal structures, such as breakwaters, were input as line data every 30 m and were not breached by earthquakes or tsunamis; buildings were not considered. We used the maximum inundation depth up to 6 h after each calculation grid as the inundated depth.
Uncertainty of slip distribution
We applied the stepwise partitioning algorithm ("Stepwise partitioning algorithm" section) to the simulated data—that is, displacement data at 642 GEONET stations calculated using the analytical result of Okada (1992) with Gaussian noise (standard deviation of 2 cm horizontally and 5 cm vertically).
Figure 4 shows the estimation results for four stages: median of the posterior PDFs and the 95% CI which was calculated from the difference between samples located at 2.5% and those at 97.5%. For all stages, the estimated fault model based on the median value well reproduced the observed data. However, the 95% CI was large, especially near the trough axis, and increased from near the land toward the offshore. This suggests that slip was well estimated in regions with large slip, while that offshore had low estimated resolution.
Estimation results for four stages. Left column: median model of posterior probability density functions (PDFs) of each subfault. Inserted values indicate the Mw and variance reduction (VR) of the median model. Black and white vectors indicate inputted and calculated horizontal displacements. Right column: 95% confidence intervals (CI) of posterior PDFs of each subfault, as calculated from the difference between the samples located at 2.5% and 97.5%
As the number of perturbation groups increased with the progress of the stage, the obtained slip distribution consisted of slips with shorter wavelengths (Fig. 4). In particular, in stage 4, a large amount of local slip occurred on certain small perturbation groups. Figure 5 shows the Markov chains and PDFs of Mw and VR for all samples except the burn-in. The VR increased with an increase in the model's freedom, which shows that the four-stage sampling successfully estimated models with high VR values without smoothing constraints. However, the amount of increase was small from stage 3 to stage 4. These results suggest that the spatial scale of the slip depended on that of the perturbation groups, because they were estimated without the smoothing constraint. The bottom chains in Fig. 5a show the calculated AIC values for each sample using Eq. (5). The mean values (± standard deviation) of AIC in each stage were 5502 ± 35, 5384 ± 47, 5455 ± 66, and 7572 ± 78, respectively. The AIC values of stage 2 and stage 3 were lower than those of stage 4, which objectively suggests the possibility of over-division. Based on this, we mainly use the samples of stage 2 for further analysis in "Uncertainty of tsunami inundation" section. Comparing the input (Fig. 1), the median slip is located more landward at all stages (Fig. 4). The Mw and VR values calculated based on the median models (inset in Fig. 4; vertical red lines in Fig. 5b) were differed slightly from those calculated based on each sample (PDFs in Fig. 5b). These indicate that the model obtained for each sample does not completely agree with the slip distribution by representative values because of the diversity of the slip in the near-trench area.
Markov chains and posterior probability density functions (PDFs). We divided the Markov chain into four stages (number of steps in each chain): stage 1 (\(3\times {10}^{6}\) steps); stage 2 (\(3\times {10}^{6}\) steps); stage 3 (\(3\times {10}^{6}\) steps); and stage 4 (\(3\times {10}^{6}\) steps). a Markov chains of Mw, variance reduction (VR), and Akaike's Information Criterion (AIC). In dotted lines, \(\Delta {{\varvec{\theta}}}_{\mathrm{max}}\) is adjusted for \(5\times {10}^{4}\) steps as burn-in, which samples are not shown in this figure. b Posterior PDFs of Mw and VR. The range on the horizontal axis is same with the range on the vertical axis of (a). Inserted values indicate mean, median, and mode (from top to bottom). Inserted vertical red lines indicate the Mw and VR values from the median models in Fig. 4
Figure 6 shows eight examples of posterior PDFs from stage 2. The point A, B and F are the area off Cape Ashizuri, where the estimated slip was large and arranged in the fault dip direction. The 95% CI at point A (close to land) was small, while the value at point B (farther offshore) was large. At point F, located further offshore, the search extends to approximately 10 m, while the maximum frequency value remains at zero. These tendencies were also observed for points C and G. In the case of point E, which corresponds to a region, where the assumed slip gradually decays, the PDF includes the assumed slip (red line in Fig. 6). This diversity of offshore slip suggests a tradeoff of slip in the fault dip direction. The diversity is more pronounced owing to the lack of real-time seafloor observation sites. Ideally, modeling should be done according to spatial resolution (we discuss in "Optimization of perturbation groups for each stage and background subfaults" section). As shown above, because the tradeoff between model parameters and the posterior PDF was not necessarily normally distributed, this study used the median as the representative value (cf. mean and mode). These results, focused on one stage, show that the MCMC can estimate significant slip distributions without smoothing constraints, and that the variability of the uncertainty based on data can be quantified as posterior PDFs.
Examples of posterior probability density functions (PDFs) in stage 2. A–G: slip amounts of eight perturbation groups. The limits of the horizontal axis are constant from 0 to 20 m. Solid and dotted black lines show the median and the 95% confidence intervals (CI). The solid red lines show the assumed slip amount in the background subfault of the perturbation group. Inserted values indicate mean, median, and mode (from the top to bottom). A, B, and F shows the area off Cape Ashizuri, where the estimated slip was large, arranged in the fault dip direction. These tendencies can also be confirmed for C and G. E and H shows the area, where the assumed slip amounts are diverse
Uncertainty of tsunami inundation
We applied the "real-time tsunami inundation risk map ("Real-time tsunami inundation risk map" section)" to the MCMC samples of stage 2, where had the lowest AIC (see "Uncertainty of slip distribution" section). First, among the \(3\times {10}^{6}\) MCMC samples of stage 2, we extracted 319,184 samples with VR of 99.52%, which is the representative value of the PDF (Fig. 3i). We then classified the extracted samples into 100 clusters using the k-means method. In each cluster, we generated one scenario using the median values for each subfault belonging to the cluster (Fig. 3ii). Using these scenarios as inputs, we calculated 100 tsunami inundation calculations for the area of Kochi Prefecture (Fig. 3iii). Finally, we counted the number of inundated points on each grid on the map (Fig. 3iv). The threshold of the inundated point was 1 cm or more during the 6 h since the earthquake. We did not apply any weights to obtain the frequencies.
Figure 7a shows the generated real-time tsunami inundation risk map. The envelope of the maximum inundation area and the arrival probability (risk level) are shown. The area where the risk of tsunami inundation is extremely high in all 100 cases (pink coloring) is similar to the inundation area due to the assumed slip (Fig. 8c). The areas with the highest probability of tsunami inundation are the Uranouchi Bay, Monobe River, and Yasu River. In addition, the southern half of Kochi Airport also has a high risk of inundation. These results show that tsunami inundation is strongly related to land topography and rivers. Tsunamis would likely run northward along large rivers (e.g., Niyodo River, Monobe River), and extend their inundation area around rivers and waterways (e.g., Shimoda River). The limit of tsunami run-up on land was defined in some cases by the fact that the kinetic energy was zero, and in other cases, because it could not overcome the difference in elevation of the terrain (rivers, channels, and embankments). For comparison, Fig. 7b shows an elevation map of the same area. High inundation risk is generally distributed in areas with elevations of 10 m or less. On the other hand, tsunami inundation does not spread to Urado Bay, despite the low elevation, owing to the structural conditions in the tsunami calculation. The elevation of area A in Fig. 7b varies gently (from 0 to 8 m), and Fig. 7a shows a clear gradation of 0–100%. Area B has an embankment of more than 1 m and this becomes the inundation boundary. Area C has an elevation of more than 10 m parallel to the coast; in about 50% of the scenarios inundation proceeded from the east side, and in a few scenarios, inundation overcame from the south to the north. Therefore, the results of inundation calculations vary depending on small differences in topographical data and arrival tsunami scenarios.
Real-time tsunami inundation risk maps for Kochi city. a Real-time tsunami inundation risk map for Kochi city (K = 100) overlaid on a shadow topography map (Geospatial Information Authority of Japan). Colors indicate normalized frequency of inundation from the 100 tsunami inundation calculations. b Elevation map (Geospatial Information Authority of Japan). Dotted rectangles indicate areas mentioned in the text
Slip distribution, seafloor vertical deformation (initial wave field), and tsunami inundation scenarios (K = 100). a Case of flooding across the largest area. b Case of flooding across the second largest area. c The case of the input slip model that was used to generate the synthetic observations (Fig. 1). Left: Slip distribution (tsunami scenario), for which the slip amount of each subfault is the median of the cluster. Inserted values indicate the Mw and frequency of the cluster and variance reduction (VR) with horizontal and vertical components. Black rectangles indicate the area of seafloor vertical deformation shown in the middle panels. Middle: seafloor vertical deformation calculated from the left slip distribution using Okada (1992). Inserted values indicate the maximum and minimum vertical deformation. Black rectangles indicate the area shown in the tsunami inundation map to the right. Right: Tsunami inundation, where colors indicate tsunami height. Inserted values indicate the inundation area
Figure 7a shows that there were few areas, where the probability of inundation was 50–90%. This indicates that the tendency of inundation is different between always inundated and rarely inundated areas. Figure 8 shows the cases with large inundation areas. The inundation area of the largest two cases was more than 40 \({\mathrm{km}}^{2}\), which is clearly larger than the assumed slip (16.784 \({\mathrm{km}}^{2}\)). In both cases, the initial wave fields had large uplift and subsidence in the offshore area of Kochi Prefecture. If these two cases were removed from the count, the appearance of the map changed (Additional file 1: Figure S2). In addition, samples with large vertical seafloor deformation for the entire slip area did not necessarily have large tsunami inundation areas (Additional file 1: Figure S3), because tsunami inundation was evaluated on a local scale.
It is also important to consider the method of editing the real-time tsunami inundation risk maps. Additional file 1: Figure S4 shows editing examples, where (a) the map is weighted by the label frequency distribution (Fig. 9b), and where (b) the threshold value of the inundation depth considered to be inundation was changed from 1 cm to 1 m. In Additional file 1: Figure S4(a), there was almost no change in the frequency tendency of the inundation area, because there was no strong correlation between the frequency and the inundation area. However, as shown in Additional file 1: Figure S4(b), there was a decrease in the inundation probabilities, because inundation of less than 1 m was not counted.
Clustering using k-means with the slip vector as the feature value. A Elbow method when the number of clusters is changed. b Frequency distribution of each cluster when K = 100 and K = 300. Red and light blue bins indicate clusters with inundation areas of 60 \({\mathrm{km}}^{2}\) or more, and 40 to 60 \({\mathrm{km}}^{2}\), respectively. c Correlation between the number of models included in each cluster and the variation of the slip distribution, which is the scalar value obtained by adding the slip ranges of all perturbation faults for each cluster. d Correlation between Mw and variance reduction (VR) of the median model (tsunami scenario) of each cluster. The dotted black line indicates the original VR of all clustered samples
These real-time tsunami inundation risk maps provide quantitative probabilistic information. As the concept of the target area is not included in the scenario classification method (Fig. 3i, ii), the same map can be generated for any region using the same MCMC samples.
Setting optimization in stepwise partition algorithm
Optimization of \({{\varvec{\sigma}}}_{{\varvec{i}}{\varvec{j}}}\) in the likelihood function
In this study, we assumed \({\sigma }_{ij}\) (see Eq. (3)). In contrast, Ohno et al. (2021) automatically adjusted \({\sigma }_{ij}\) according to the event using the maximum likelihood estimation scheme. Here, this adjustment was not applied to the slip distribution model, because the large number of parameters easily explained the observed data, resulting in a very small estimate of \({\sigma }_{ij}\). This dynamic adjustment is more effective for a single rectangular fault model; however, modeling errors exist in the slip distribution model owing to the uncertainty of Green's function. Therefore, in this study, we assumed \({\sigma }_{ij}\), which depends on the displacements and observation errors.
Optimization of a priori information
Ohno et al. (2021) used earthquake early warning values (EEW: latitude, longitude, depth, M) as a priori information. While our algorithm does not use such a priori information, it is important to appropriately utilize such information to achieve accurate and rapid estimation for real-time estimation. For example, the algorithm can use the EEW and independent estimated models, such as the single rectangular fault model and the slip distribution by REGARD, as a priori information.
Optimization of perturbation groups for each stage and background subfaults
As discussed in "Uncertainty of slip distribution" section, there was a tradeoff of slip in the dip direction. We applied a stepwise partitioning algorithm to various assumed slip distributions and found a similar tendency (Additional file 1: Figures S5–S10). The position and wavelength of the slip in the dip direction affected the tsunami calculation. Optimization of the perturbation group settings is especially important, because the empirical geophysical law (slip continuity) is not considered when a smoothing constraint is not used. For example, it is desirable to evaluate the spatial resolution depending on the location of the observation stations and to determine the division of the perturbation group accordingly. Kimura et al. (2019) developed an algorithm for automatically setting up the division of subfaults according to the arrangement of observation stations. It may be effective to use this algorithm for setting subfaults (smallest division) and arranging the perturbation group in each stage. The required size varies depending on the magnitude of the earthquake, and so we should determine the optimum stage objectively (i.e., using the AIC) and judge the convergence (e.g., using Gelman 1996) automatically for real-time utilization.
Optimum number of clusters
The optimum number of clusters K for k-means ("Uncertainty of tsunami inundation" section) should be determined considering the cost of real-time computation, the availability of denominators for probability display, and the availability of a sufficient number of clusters that do not distort the diversity of all samples. Here, we consider the validity of K = 100 in comparison with K = 300.
Figure 9 shows the results of clustering using k-means. According to the elbow method graph (Fig. 9a), a cluster number of K = 50–100 seems to be appropriate. On the other hand, according to the frequency distribution of the clusters (Fig. 9b), even if we increased the number of clusters from K = 100 to K = 300, it looks like the classification just was subdivided overall. Figure 9c shows the correlation between the number of models included in each cluster and the variability of the slip distribution, which is a scalar sum of the ranges of slip amounts on each subfault. According to Fig. 9c, both the number of models included (horizontal axis) and variability (vertical axis) are smaller for K = 300 than for K = 100. This may indicate that by increasing the number of clusters, clusters were formed among the slip distribution models that have the closest features to each other. Figure 9d shows the variability of the VRs for the K scenarios (the median model in each cluster). Although the used samples have the same VR (99.52%), those of the classified scenarios vary in the order of the second decimal place. Note that there are some scenarios, where VR is greater than the original VR (99.52%). This is probably because the outlier-like short-wavelength slip of each sample was smoothed by taking the median at each subfault, resulting in a model that better explains the data.
Figure 10 shows the correlation between the number of models included in each cluster and the inundation areas for K = 100 and K = 300. The frequency distribution of the tsunami inundation area (vertical axis) is asymmetric, with a peak near the tsunami inundation area due to the assumed slip (16.784 \({\mathrm{km}}^{2}\)) and long tail on the side of the large tsunami inundation. The maximum and minimum inundation areas for K = 100 and K = 300 are similar. On the other hand, 300 cases of K = 300 are distributed so that they filled in the gaps between the 100 cases of K = 100. The maximum inundation area for K = 300 is slightly larger than that for K = 100 (and the maps look slightly different; Additional file 1: Figure S11). These characteristics suggest that the larger the number of K, the more the tsunami scenario is subdivided, and the degree of smoothing of extreme slip decreases. However, for the present trial, K = 100 was sufficient to obtain the range of the inundation area.
Tsunami inundation calculation results for multiple tsunami scenarios. Correlation between the number of models included in each cluster and the tsunami inundation area. The dotted black line indicates the inundation area of assumed model (Fig. 1). The vertical and horizontal axes on the inserted diagrams indicate the frequency distributions. Inserted vectors indicate the scenarios of Fig. 9 (a, b)
In the case of K = 100, there was a negative correlation between the inundation area and the number of models included in each cluster (Fig. 10). However, as shown in Fig. 9b where colored the clusters with an inundation area of 40 \({\mathrm{km}}^{2}\) or more, there was not necessarily a relationship between the frequency of each cluster and the flooded area in Kochi City. This may be because the number of models included in each cluster shows a tendency for the "entire" slip distribution, whereas inundation was evaluated by focusing on a target area, which seems to be a natural result. On the other hand, samples with extreme slip patterns tend to be classified into clusters with small frequencies, and the possibility that these samples cause maximum inundation in some other areas cannot be excluded.
Utilization of real-time tsunami inundation risk maps
The risk of tsunami inundation is expressed by multiplying hazard (e.g., tsunami inundation) by exposure (e.g., population and buildings) and fragility (e.g., buildings), then dividing it by resiliency (e.g., Wood 2011). The real-time tsunami inundation risk maps evaluate the probability of tsunami inundation for each area, which corresponds to the probabilistic representation of hazards in the above equation.
For example, if we reflect the difference in day/night population (i.e., exposure), we can evaluate the difference in disaster risk depending on the time of day, and if we reflect the difference in resiliency, we can quantify the disaster risk according to the time scale we want to evaluate. In addition, to maximize the benefit of evaluating disaster triggers as probabilities in this study, it is easier to evaluate the risk level if other factors can be treated as probabilities in the same dimension. For easy-to-understand expression, qualitative evaluation may also be useful (e.g., three risk levels of large, medium, and small). Furthermore, if we take the envelope of the maximum inundation area, we can extract the worst-case scenario for the event (Fig. 7a).
As mentioned in "Introduction" section, the real-time tsunami inundation damage estimation system is expected to be utilized as a function of the disaster information system (DIS) of the Cabinet Office for the initial response of the government when a disaster occurs. Our developed framework for the quantitative evaluation of uncertainty in tsunami inundation prediction has the potential to improve this system. The risk level can be expressed as a probability instead of a single inundation prediction. This system has the potential to improve understanding of high disaster-risk regions and to support the decision-making process for the initial response in such areas.
In this study, we developed a new coseismic fault model estimation algorithm using Bayesian statistics to quantitatively evaluate the estimation uncertainty in the real-time estimation of slip distribution using real-time GNSS data. In addition, as an application of the quantified uncertainty, we investigated how the uncertainty of the fault model estimation may affect tsunami inundation prediction and proposed a new method to extract the uncertainty of tsunami inundation prediction.
One of the major problems in the estimation of slip distribution models with MCMC is that they take a long time to converge. To overcome this problem for real-time purposes, we developed a stepwise partitioning algorithm. We divided the entire Markov chain into four stages with different perturbing groups, and used the 95% CI in the previous stage as the amount of perturbation in the next stage. This enabled us to extract multiple models with different spatial patterns that explained the observed data well. We applied the algorithm to the numerical simulation data of the Nankai Trough region (assumed to be the Mw 8.75 1707 Hoei earthquake). The 95% CI of the obtained posterior PDFs increased with distance from the coast to the offshore, indicating the uncertainty in estimating plate boundary slip from onshore GNSS.
We developed a real-time tsunami inundation risk map process to fully utilize the advantages of the stepwise partitioning algorithm, wherein the uncertainty of slip distribution is quantified as a large number of samples. Specifically, we classified the slip distribution models into 100 clusters determined by the k-means method after extracting samples that have the same VR. We calculated the individual tsunami inundation using 100 scenarios. Furthermore, we counted the number of inundations in each grid on the map and displayed them probabilistically. We showed the envelope of the maximum inundation area and the probability of reaching each computational grid in Kochi City, Kochi Prefecture. However, to truly utilize the obtained tsunami inundation risk, it may be necessary to examine the kind of information needed not only from the viewpoint of disaster triggers but also from the viewpoint of society as a recipient.
The analyzed data for the current study are available from the corresponding author upon reasonable request.
ABIC:
Akaike's Bayesian Information Criterion
AIC:
Akaike's Information Criterion
Confidence Intervals
DIS:
Disaster Information System
EEW:
Earthquake Early Warning
MCMC:
Markov chain Monte Carlo methods
REGARD:
REal-time GEONET Analysis system for Rapid Deformation monitoring
TUNAMI:
Tohoku University's Numerical Analysis Model for Investigating tsunami
VR:
Variance Reduction
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The paper benefited from careful reviews by Dr. Sarah Minson and an anonymous reviewer. This study was supported by the Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (KAKENHI; grant no. 17H06108) and by the Toray Science Foundation (Toray Science and Technology Grant). This work was also supported by the JST FOREST Program (grant number: JPMJFR202P, Japan). This study was also supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, under its The Second Earthquake and Volcano Hazards Observation and Research Program (Earthquake and Volcano Hazard Reduction Research). This work was also supported by the Next Generation High-Performance Computing Infrastructures and Applications R&D Program by MEXT. This work was also supported by the Research Project for Disaster Prevention on the great Earthquakes along the Nankai trough by MEXT.
This study was supported by the Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research (KAKENHI; grant no. 17H06108) and by the Toray Science Foundation (Toray Science and Technology Grant). This work was also supported by the JST FOREST Program (grant number: JPMJFR202P, Japan). This study was also supported by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, under its The Second Earthquake and Volcano Hazards Observation and Research Program (Earthquake and Volcano Hazard Reduction Research). This work was also supported by the Next Generation High-Performance Computing Infrastructures and Applications R&D Program by MEXT. This work was also supported by the Research Project for Disaster Prevention on the great Earthquakes along the Nankai trough by MEXT.
Research Center for Prediction of Earthquakes and Volcanic Eruptions, Graduate School of Science, Tohoku University, Sendai, Japan
Keitaro Ohno, Yusaku Ohta & Ryota Hino
Division for the Establishment of Frontier Sciences of Organization for Advanced Studies, Tohoku University, Sendai, Japan
Yusaku Ohta
International Research Institute of Disaster Science, Tohoku University, Sendai, Japan
Yusaku Ohta, Ryota Hino, Shunichi Koshimura & Takashi Abe
Cyberscience Center, Tohoku University, Sendai, Japan
Akihiro Musa
NEC Corporation, Tokyo, Japan
Computer Architecture Laboratory, Graduate School of Information Sciences, Tohoku University, Sendai, Japan
Hiroaki Kobayashi
Geospatial Information Authority of Japan, 1 Kitasato, Tsukuba, 305-0811, Japan
Keitaro Ohno
Ryota Hino
Shunichi Koshimura
Takashi Abe
KO and YO designed the study, developed the programs, and analyzed the data. SK and TA performed the tsunami inundation calculations. AM and HK contributed to speeding up the programs developed. YO, SK, RH, and HK assisted in the interpretation of the results. All authors read and approved the final manuscript.
Authors' information
K.O. was a graduate student at the Graduate School of Science, Tohoku University (now at the Geospatial Information Authority of Japan). Y.O. and R.H. are associate professor and professor, respectively, at the Graduate School of Science, Tohoku University. S.K. and T.A. are professor and research fellow, respectively, at the International Research Institute for Disaster Science, Tohoku University. A.M. and H.K. are visiting professor and professor, respectively, at the Graduate School of Information Sciences, Tohoku University.
Correspondence to Keitaro Ohno.
Table S1 and Figures S1–S11.
Ohno, K., Ohta, Y., Hino, R. et al. Rapid and quantitative uncertainty estimation of coseismic slip distribution for large interplate earthquakes using real-time GNSS data and its application to tsunami inundation prediction. Earth Planets Space 74, 24 (2022). https://doi.org/10.1186/s40623-022-01586-6
Tsunami inundation
Tsunami prediction
Real-time GNSS
Bayesian inversion
Uncertainty estimation
6. Geodesy
Recent Advances in Scientific Application of GNSS Array Data | CommonCrawl |
For each of 8 colors, I have one shirt and one tie of that color. How many shirt-and-tie outfits can I make if I refuse to wear a shirt and a tie of the same color?
There are 8 options for the shirt, and only 7 options for the tie because one of the ties has the same color as the shirt, so the number of outfits is $8 \times 7 = \boxed{56}$. | Math Dataset |
Co-immobilization of laccase and TEMPO onto amino-functionalized magnetic Fe3O4 nanoparticles and its application in acid fuchsin decolorization
Zhen Gao1 na1,
Yunfei Yi2 na1,
Jia Zhao1,
Yongyang Xia1,
Min Jiang1,
Fei Cao1,
Hua Zhou1,
Ping Wei1,
Honghua Jia1 &
Xiaoyu Yong1
Laccase, a multicopper oxidase that catalyzes the oxidation of phenols, aromatic amines, and benzenethiols, has attracted much attention in applications of organic synthesis, bioremediation, and pulp/textile bleaching. However, free laccases cannot be recycled and are easily inactivated in diverse environmental conditions. Enzyme immobilization is a promising strategy to improve stability, resistance to extreme conditions, and reusability of laccase.
In this study, amino-functionalized magnetic Fe3O4 nanoparticles were synthesized for co-immobilization of 2,2,6,6-tetramethylpiperidine-1-oxyl (TEMPO) and laccase by glutaraldehyde cross-linking method. The magnetic nanoparticles were characterized with FTIR, XRD and VSM. Cyclic voltammetry was carried out to verify electrochemical behaviors of the co-immobilized laccase and TEMPO nanoparticles. When the co-immobilized laccase and TEMPO nanoparticles were used to decolorize acid fuchsin, the maximum decolorization rate of 77.41% was obtained with the ratio of TEMPO to laccase being 0.3 mM/g:120 U/g.
The co-immobilized nanoparticles retained above 50% residual activity after eight cycles of operation, which presented an approach to develop a co-immobilized laccase and mediator system for potential industrial application.
Laccase (benzenediol:oxygen oxidoreductase, EC 1.10.3.2) is a multicopper oxidase and widely discovers from many plants, insects, and fungi (Mate and Alcalde 2017). It contains four copper atoms in its catalytic center and catalyzes the oxidation of phenols, polyphenols, polyamines, and benzenethiols by reducing molecular oxygen to water (Chao et al. 2017; Su et al. 2018; Mate and Alcalde 2015). In the past years, laccases have attracted much attention and been used in various applications, including organic synthesis, bioremediation, pulp bleaching, and biofuel production (Jeon and Chang 2013; Kudanga and Le Roes-Hill 2014). However, free laccases cannot be recycled and are easily inactivated in diverse environmental conditions, which limit their further use in industry.
Immobilization is a promising strategy to improve stability, resistance to extreme conditions, and reusability of laccase (Ba and Kumar 2017). So far, laccase has been successfully immobilized with several methods, including entrapment, encapsulation, adsorption, covalent binding, and self-immobilization (Ba et al. 2013; Fernández-Fernández et al. 2013). Although the stability of the immobilized laccase against temperature, organic solvents, pH, storage, and operation has been greatly improved, the activity recovery is not always satisfactory. In addition, the ability of laccase to catalyze recalcitrant compounds is limited for its low redox potential. This can be overcome by establishing laccase mediator system (LMS) (Jeon and Chang 2013; Mogharabi and Faramarzi 2014). 2,2′-Azino-bis-(3-ethylbenzothiazoline)-6-sulfonic acid (ABTS), 1-hydroxybenzo-triazole (HBT), and 2,2,6,6-tetramethylpiperidine-1-oxyl (TEMPO) are usually used as mediators to transfer electrons from substrate to laccase. However, these mediators are usually expensive and difficult to recover from reaction mixture, which hampers the applications of LMS. Attempts have been made to immobilize mediators to recycle them. Machado et al. (2015) reported the immobilization of 4-hydroxy-TEMPO on mesoporous silica using 1,4-diisocyanatobutane as the linking agent. Tucker-Schwartz et al. (2010) employed iron oxide (Fe3O4) superparamagnetic nanoparticles to immobilize TEMPO with strong metal-oxide-chelating phosphonates and azide/alkyne "click" chemistry. The resultant TEMPO-coated nanoparticles, with good TEMPO loading, can be used to efficiently catalyze the oxidation of a wide range of alcohols.
In recent years, magnetic nanoparticles have attracted widespread attention because of their large surface areas, nontoxicity, magnetic properties, and biocompatibility. As a support material, magnetic nanoparticles can be rapidly separated from solution by the application of an external magnetic field rather than by centrifugation, and they have been deemed to be efficient carriers for enzyme immobilization (Ansari and Husain 2012; Can et al. 2009; Torres-Salas et al. 2011; Xin et al. 2010). Ren et al. have successfully immobilized lipase onto magnetic iron oxide nanoparticles via a biomimetic coating, which significantly improved its thermal and pH stability (Ren et al. 2011). Aminated magnetic mesoporous silica, (Fe3O4@MSS)-NH2, was synthesized to immobilize laccase using covalent cross-linking methods (Huang et al. 2014). 2,4-Dichlorophenol was degraded by the immobilized laccase with a removal efficiency of 88%, and 61.5% activity was remained after five cycles of operation. Zheng et al. (2012) prepared magnetic silica composite particles functionalized with 3-aminopropyltriethoxysilane (APTES) for laccase immobilization with the activity recovery of 83.9%.
In the present study, we reported a system that co-immobilizes laccase and mediator, allowing the simultaneous reuse of both laccase and the mediator. Magnetic nanoparticles were successfully prepared and modified with 3-amino-propyltriethoxysilane (APTES) to introduce abundant amine groups onto their surfaces. Then, laccase and 4-amino-TEMPO were efficiently co-immobilized onto the amino-functionalized Fe3O4 nanoparticles, using glutaraldehyde as the cross-linking agent. Finally, the decolorization of acid fuchsin, a triphenylmethane dye, by the co-immobilized laccase mediator system was conducted to evaluate the stability and efficiency for its potential future applications.
Laccase (EC 1.10.3.2 from Trametes versicolor, 13.6U/mg) and ABTS were purchased from Sigma-Aldrich Co. Ltd. (St. Louis, MO, USA). APTES, 4-amino-TEMPO, glutaraldehyde (25%, v/v, aqueous solution), Coomassie Brilliant Blue, bovine serum albumin, acid fuchsin and polyvinyl alcohol (PVA) with 1750 ± 50 of polymerization and 98% of degree of hydrolysis were from Aladdin Co. Ltd (Shanghai, China). All other chemicals were of analytical grade unless otherwise mentioned.
Preparation and characterization of magnetic Fe3O4 nanoparticles
Magnetic Fe3O4 nanoparticles were synthesized with the co-precipitation method and further modified with amino groups according to the method reported previously (Yamaura et al. 2004). First, 1.04 g of FeCl3·6H2O and 0.4 g of FeCl2·4H2O were dissolved in 10 mL deionized water, and 25 mL NaOH solution (3.0 M) was added to the mixture with vigorously stirring at 70 °C. After 30 min reaction, by applying an external magnetic field, the resultant Fe3O4 nanoparticles were collected and washed several times with deionized water until pH 7.0. The FTIR spectra of the nanoparticles were recorded on a Fourier transformed infrared spectroscopy (Thermo, USA). The XRD spectra of the nanoparticles were analyzed with D8 Advance XRD (Bruker, Switzerland). VSM (EV7, ADE Technologies) was used to record the magnetic susceptibility of the nanoparticles.
Co-immobilization of laccase and TEMPO
The procedure of co-immobilization of laccase and TEMPO onto amino-functionalized Fe3O4 nanoparticles was briefly described as follows (Scheme 1). The nanoparticles were modified with APTES via a silanization reaction to form the amino-functionalized Fe3O4 nanoparticles (Reza et al. 2010). The procedure involved the addition of 40 mL of 10% APTES in water to the magnetite suspension with glycerol (pH 4.0, adjusted with glacial acetic acid), and heating at 90 °C for 3.0 h with continuous stirring. The resultant amino-functionalized Fe3O4 nanoparticles were washed three times with deionized water, dried under vacuum, and stored before use. Then, 100 mg of nanoparticles was suspended in 10 mL of 15% glutaraldehyde solution in pH 6.0, 0.1 M acetate buffer, and the suspension was kept under mild stirring at 25 °C for 12 h. The suspension was collected and washed three times with deionized water and stored under 4 °C. Thereafter, 100 mg of nanoparticles activated with glutaraldehyde, as described above, was added to 10 mL of acetate buffer (0.1 M, pH 4.5) containing a certain amount of laccase and 4-amino-TEMPO. The suspension was shaken at 120 rpm for several hours to produce nanoparticles carrying co-immobilized laccase and TEMPO. The resultant nanoparticles were washed several times with acetate buffer and collected to store under 4 °C.
Scheme 1
The process of co-immobilization of laccase and TEMPO onto amino-functionalized Fe3O4 nanoparticles
The amount of TEMPO attached onto the nanoparticles was measured spectrophotometrically, monitoring the changes in the 4-amino-TEMPO concentration before and after the reaction as the changes in the absorbance at a wavelength of 230 nm (A230) with a UV–Vis spectrophotometer (Thermo/GENESYS 10S, USA).
Electrochemical analysis
The modified glassy carbon (GC) electrodes were used as a working electrode with an Ag/AgCl as reference electrode and a Pt wire as a counter electrode, respectively. The glassy carbon electrodes were modified as follows: 5 mg of immobilized laccase and co-immobilized laccase and TEMPO was mixed with 100 μL of 0.3% PVA solution to produce PVA/Fe3O4-Laccase and PVA/Fe3O4-Laccase–TEMPO colloids, respectively. The surface of GC electrodes (5 mm in diameter) was polished thoroughly with Al2O3 (0.05 μm) by rinsing thoroughly with distilled water. Then, 10 μL PVA/Fe3O4-Laccase and PVA/Fe3O4-Laccase–TEMPO colloids solutions were dropped on the surface of pretreated electrodes and allowed to dry under ambient condition at 4 °C, respectively. After the modified electrodes were rinsed with distilled water twice or thrice, PVA/Fe3O4-Laccase and PVA/Fe3O4-Laccase–TEMPO modified GC electrodes were obtained. Experiments of cyclic voltammetry were run in a three-electrode system (PGSTAT302N, Metrohm, Switzerland) at a scan rate of 50 mV/s in pH 4.5 acetate butter solution containing 0.1 mM catechol.
Decolorization of acid fuchsin
The reaction cycle catalyzed by the co-immobilized laccase and TEMPO is shown in Scheme 2. Acid fuchsin was selected as the model dye to test the activity of the co-immobilized laccase and TEMPO. Nanoparticles (100 mg) carrying co-immobilized laccase and TEMPO were added to an acid fuchsin solution (50 mg/L, pH 4.5, 100 mL), which was then stirred at 40 °C. During the process of decolorization, the concentrations of the sample solutions were determined every 2 h by detecting A540 with a UV–Vis spectrophotometer. The decolorization rate D was calculated with the equation:
$$D = (A_{ 0} - A_{\text{t}} ) /A_{ 0} \times 1 0 0$$
where A0 is the initial A540 and At is A540 after various time intervals. The nanoparticles used for dye decolorization were run for 10 cycles to test their reusability. At the end of each cycle, the nanoparticles were separated magnetically and washed several times with acetate buffer (0.1 M), then the next cycle was commenced.
The reaction cycle catalyzed by the co-immobilized laccase and TEMPO
Assay of enzyme activity
The activity of the immobilized laccase was determined with ABTS as the substrate (Bourbonnais et al. 1998). The change in the amount of ABTS was measured with a UV spectrophotometer at 420 nm. One unit of laccase activity was defined as the amount of enzyme required to oxidize 1 μM of substrate per minute. The activity recovery R (percent) of the immobilized laccase was calculated with the equation:
$$R = A_{\text{i}} /A_{\text{f}} \times 1 0 0$$
where Ai is the activity (units) of the immobilized laccase and Af is the activity (units) of the same amount of free laccase immobilized on the support.
Characterization of amino-functionalized Fe3O4 nanoparticles
The FTIR spectra of Fe3O4 nanoparticles and amino-functionalized Fe3O4 nanoparticles were determined to confirm the presence of terminal amino group on the functionalized nanoparticles. The peaks at 580, 3417, and 1626 cm−1 in Fig. 1a were assigned to Fe–O stretching vibration, O–H stretching, and O–H deformation, respectively (Alba et al. 1996; Lü et al. 2003; Peng et al. 2004). The significant peak appeared at 999 cm−1 in Fig. 1b, which was corresponded with Si–O stretching, was related to the silanization reaction (Hu et al. 2009). The strong absorption band at 3394 cm−1 in Fig. 1b was probably attributable to the free amino groups and overlapped by the O–H stretching vibration (Hu et al. 2015). Therefore, it can be speculated that amine groups have been successfully introduced onto the surfaces of the Fe3O4 nanoparticles by silanization reaction.
FTIR spectra of Fe3O4 nanoparticles (a) and amino-functionalized Fe3O4 nanoparticles (b). The strong absorption at 3394 cm−1 in b was probably attributable to the free amino groups and overlapped by the O–H stretching vibration
The XRD spectra of magnetic Fe3O4 nanoparticles and amino-functionalized Fe3O4 nanoparticles are shown in Fig. 2. The characteristic peaks of the Fe3O4 nanoparticles and amino-functionalized Fe3O4 nanoparticles were distinguishable and consistent with the standard pattern for Fe3O4 (Rocktotpal et al. 2009), which indicated that the crystalline structure was not altered by particle modification.
X-ray diffraction patterns of magnetic Fe3O4 nanoparticles (a) and amino-functionalized magnetic Fe3O4 nanoparticles (b)
As shown in Fig. 3, the hysteresis loop of the magnetic Fe3O4 nanoparticles and amino-functionalized Fe3O4 nanoparticles revealed a ferromagnetic behavior and the remanence and coercive force was very small, almost near to zero, closing to superparamagnetism with the saturated magnetic intensity value of 41.81 and 41.77 emu/g, respectively.
VSM images of magnetic Fe3O4 nanoparticles (a) and amino-functionalized Fe3O4 nanoparticles (b). The hysteresis loop of the both nanoparticles revealed a ferromagnetic behavior and the remanence and coercive force was near to zero
Cyclic voltammetry of PVA/Fe3O4-Laccase GC and PVA/Fe3O4-Laccase–TEMPO modified GC electrodes
Cyclic voltammetry was carried out to study the electrochemical behaviors of PVA/Fe3O4-Laccase and PVA/Fe3O4-Laccase–TEMPO modified GC electrodes (Fig. 4). They both had redox behaviors in pH 4.5 acetate buffer in the presence of 0.1 mM catechol. However, the anodic peak potential of PVA/Fe3O4-Laccase–TEMPO modified GC electrode shifted to a more positive value and the cathodic peak potential shifted to a lower negative direction. Moreover, the catalytic current of PVA/Fe3O4-Laccase–TEMPO modified GC electrode was larger than PVA/Fe3O4-Laccase modified GC electrode. These demonstrated that TEMPO immobilized on Fe3O4 played an important role during the electron transfer between electrode and immobilized enzyme, which indicated that TEMPO was successfully immobilized on carriers. It might lead electrons of the substrate to the redox centers of laccase and improve the catalytic activity of laccase (Wei et al. 2008; Zhao et al. 2002).
Cyclic voltammetry of PVA/Fe3O4-Laccase–TEMPO GC (a) and PVA/Fe3O4-Laccase GC (b). The anodic peak potential of PVA/Fe3O4-Laccase–TEMPO modified GC electrode shifted to a more positive value and the cathodic peak potential shifted to a lower negative direction
Effect of TEMPO on the decolorization of acid fuchsin by the immobilized laccase
As can be observed in Fig. 5, acid fuchsin cannot be decolorized well by the immobilized laccase or immobilized TEMPO separately. However, we found that the co-immobilized laccase and TEMPO showed a great capacity for decolorization of acid fuchsin, which showed a 68.8% decolorization rate after 16 h. Results further confirmed that the laccase and TEMPO had been successfully co-immobilized onto the magnetic nanoparticles. The possible reason why the immobilized laccase or immobilized TEMPO cannot decolorize acid fuchsin well is due to the substrate bearing a large structure or high redox potential. While in the case of the co-immobilized laccase and TEMPO, this limitation had been overcome with the addition of mediator TEMPO. TEMPO can act as an intermediate substrate for the laccase to react with acid fuchsin at a high redox potential oxidized radical form (Riva 2006).
Decolorization of acid fuchsin by different Fe3O4 nanoparticles. The co-immobilized laccase and TEMPO showed a 68.8% decolorization rate for acid fuchsin after 16 h of reaction
Effect of the ratio of TEMPO to laccase on the co-immobilized laccase and TEMPO
The investigations were conducted with various ratios of TEMPO to laccase to achieve the co-immobilization and the resulting co-immobilized beads were characterized by decolorizing acid fuchsin. The amount of laccase and TEMPO immobilized onto the support played a decisive role in the decolorization of acid fuchsin. TEMPO can be oxidized to stable radicals by laccase, oxidizing the dye molecules, which are not a natural substrate of laccase. The N–OH functionality of the mediator is related to the increase in dye decolorization by LMS system (Claus et al. 2002). As shown in Fig. 6, the decolorization rate of acid fuchsin increased with the rise of the amount of laccase and TEMPO. The maximum decolorization rate of 77.41% was obtained when the ratio of TEMPO to laccase was 0.3 mM/g:120 U/g. However, excess TEMPO did not improve the decolorization of acid fuchsin by co-immobilized laccase and TEMPO. This is probably because excess mediator would produce large amounts of free radicals, which might harm the enzyme and, thus, hinder decolorization (Mirzadeh et al. 2014).
Effect of the ratio of TEMPO to laccase on the co-immobilized laccase and TEMPO. The maximum decolorization rate of 77.41% was obtained when the ratio of TEMPO to laccase was 0.3 mM/g:120 U/g
Reusability of the co-immobilized laccase and TEMPO onto Fe3O4 nanoparticles
The reusability of the co-immobilized laccase and TEMPO onto Fe3O4 nanoparticles was also investigated (Fig. 7). As the number of repeated cycles increased, the decolorization rate of acid fuchsin by the co-immobilized laccase and TEMPO decreased gradually. It is probable that the immobilized laccase and TEMPO fell off the nanoparticles or that a proportion of the laccase molecules was inactivated during the cycling process. After the particles were used eight times repeatedly, the co-immobilized nanoparticles retained above 50% residual activity.
Reusability of the co-immobilized laccase and TEMPO. The co-immobilized nanoparticles retained above 50% residual activity after 8 cycles of operation
In summary, amino-functionalized magnetic Fe3O4 nanoparticles were prepared successfully by modifying magnetic nanoparticles to carry abundant amine groups on their surfaces. Laccase and 4-amino-TEMPO were co-immobilized onto the amino-functionalized magnetic Fe3O4 nanoparticles and used to decolorize acid fuchsin. The catalytic current of Fe3O4/PVA/Lac–TEMPO GC electrode is larger than Fe3O4/PVA/Lac GC electrode, which indicated that TEMPO was successfully immobilized onto carriers. When the ratio of TEMPO to laccase was 0.3 mM/g:120 U/g on the co-immobilized laccase and TEMPO nanoparticles, the maximum decolorization rate of acid fuchsin was 77.41%. Additionally, the co-immobilized nanoparticles retained above 50% residual activity after eight cycles of operation. This study offers a feasible approach to establish a co-immobilized laccase and mediator system, which could be useful in industrial applications.
2,2,6,6-tetramethylpiperidine-1-oxyl
FTIR:
Fourier transfer infrared spectroscopy
XRD:
VSM:
vibrating sample magnetometer
LMS:
laccase mediator system
ABTS:
2,2′-azino-bis-(3-ethylbenzothiazoline)-6-sulfonic acid
HBT:
1-hydroxybenzo-triazole
PVA:
APTES:
3-amino-propyltriethoxysilane
GC:
glassy carbon
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HJ and PW designed the experiments, ZG and YY conducted most of the experiments, JZ analyzed the data, YX and XY wrote the manuscript, MJ, FC and HZ provided advices on the experimental design and language. All authors read and approved the final manuscript.
The datasets supporting the conclusions of this article are included in the main manuscript.
The authors approved the consent for publishing the manuscript.
Ethics approval and consent participate
The authors have read and agreed the ethics for publishing the manuscript.
This research was financially supported by NSFC (21406114 and 20906048), the Natural Science Foundation of Jiangsu Higher Education Institutions of China (14KJB530002), the Science Foundation for Postdoctoral Research from Jiangsu Province of China (1401009A), the Jiangsu Synergetic Innovation Center for Advanced Bio-Manufacture, PCSIRT (IRT_14R28), and PAPD.
Zhen Gao and Yunfei Yi contributed equally to this work
College of Biotechnology and Pharmaceutical Engineering, Nanjing Tech University, No. 30 South Puzhu Road, Pukou District, Nanjing, 211800, China
Zhen Gao, Jia Zhao, Yongyang Xia, Min Jiang, Fei Cao, Hua Zhou, Ping Wei, Honghua Jia & Xiaoyu Yong
College of Pharmacy, Nanjing Tech University, Nanjing, 211800, China
Yunfei Yi
Zhen Gao
Jia Zhao
Yongyang Xia
Fei Cao
Hua Zhou
Ping Wei
Honghua Jia
Xiaoyu Yong
Correspondence to Honghua Jia.
Gao, Z., Yi, Y., Zhao, J. et al. Co-immobilization of laccase and TEMPO onto amino-functionalized magnetic Fe3O4 nanoparticles and its application in acid fuchsin decolorization. Bioresour. Bioprocess. 5, 27 (2018). https://doi.org/10.1186/s40643-018-0215-7
Amino-functionalized magnetic Fe3O4 nanoparticles
Co-immobilization
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List of things named after Eduard Heine
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\begin{document}
\baselineskip=17pt
\titlerunning{On Yang-Mills connections on compact K\"{a}hler surfaces} \title{On Yang-Mills connections on compact K\"{a}hler surfaces} \author{Teng Huang} \date{} \maketitle \address{T. Huang: School of Mathematical Sciences, University of Science and Technology of China; Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, Hefei, Anhui 230026, PR China; \email{[email protected]; [email protected]}}
\begin{abstract} We extend an $L^{2}$-energy gap of Yang-Mills connections on principal $G$-bundles $P$ over a compact Riemannian manfold with a $good$ Riemannian metric \cite{Feehan} to the case of a compact K\"{a}hler surface with a $generic$ K\"{a}hler metric $g$, which guarantees that all ASD connections on the principal bundle $P$ over $X$ are irreducible. \end{abstract} \section{Introduction}
Let $G$ be a compact, semisimple Lie group and $P$ be a principal $G$-bundle over a closed, smooth, Riemannian manifold with Riemannian metric $g$. Suppose that $A$ is a connection on $P$ and its curvature denote by $F_{A}\in\Omega^{2}(X)\otimes\mathfrak{g}_{P}$. Here $\Omega^{k}:=\Omega^{k}(T^{\ast}X)$ and $\mathfrak{g}_{P}$ is the real vector bundle associated to $P$ by the adjoint representation of $G$ on its Lie algebra $\mathfrak{g}$. We define the Yang-Mills energy function by $$YM(A):=\int_{X}|F_{A}|^{2}dvol_{g},$$ the fiber metric defined through the Killing form on $\mathfrak{g}$, see \cite[Section 2]{Feehan}. Then energy functional $YM(A)$ is gauge-invariant and thus descends to a function on the quotient space $\mathcal{B}(P,g):=\mathcal{A}_{P}/\mathcal{G}_{P}$, of the affine space $\mathcal{A}_{P}$ of connections on $P$ moduli the gauge transformation. A connection $A$ is called Yang-Mills connection when it gives a critical point of the Yang-Mills functional, that is, it satisfies the Yang-Mills equation $$d^{\ast}_{A}F_{A}=0.$$ From the Bianchi identity $d_{A}F_{A}=0$, a Yang-Mills connection is nothing but a connection whose curvature is harmonic with respect to the covariant exterior derivative $d_{A}$.
Over a $4$-dimensional Riemannnian manifold, $F_{A}$ is decomposed into its self-dual and anti-self-dual components, $$ F_{A}=F^{+}_{A}+F^{-}_{A}$$ where $F^{\pm}_{A}$ denotes the projection onto the $\pm1$ eigenspace of the Hodge star operator. A connection is called self-dual (respectively anti-self-dual) if $F_{A}=F^{+}_{A}$ (respectively $F_{A}=F^{-}_{A}$). A connection is called an instanton if it is either self-dual or anti-self-dual. On compact oriented $4$-manifolds, an instanton is always an absolute minimizer of the Yang-Mills energy \cite{Taubes1982,Taubes1984}. Not all Yang-Mills connections are instantons, in \cite{SS,SSU}, the authors given some examples for the $SU(2)$ Yang-Mills connections on $S^{4}$ which are neither self-dual nor anti-self-dual.
It is a natural question whether or not there is a positive uniform gap between the energy $YM(A)$ of points $[A]$ in the stratum $$M(P,g):=\{[A]\in\mathcal{B}(P,g): F_{A}^{+}=0\},$$ of absolute minimal of $YM(A)$ on $\mathcal{B}(P,g)$ and energies of points in the strata in $\mathcal{B}(P,g)$ of non-minimal critical points.
In \cite{BL,BLS}, Bourguignon-Lawson proved that if $A$ is a Yang-Mills on a principal $G$-bundle over $S^{4}$ with its standard round metric of radius one such that $\|F_{A}^{+}\|_{L^{\infty}(X)}<\sqrt{3}$, then $A$ is anti-self-dual. The result was significantly improved by Min-Oo \cite{Mi} and Parker \cite{Parker}, by replacing the preceding $L^{\infty}$ condition with an $L^{2}$-energy condition, $\|F_{A}^{+}\|_{L^{2}(X)}\leq\varepsilon$, where $\varepsilon=\varepsilon(g)$ is a small enough constant and by assume $X$ to be a closed, smooth, four-dimensional manifold endowed with a $positive$ Riemannian metric $g$. In \cite{Feehan}, Feehan extend the $L^{2}$-energy gap result from the case of positive Riemannian metrics \cite{Mi,Parker} to the more general case of $good$ Riemannian metrics. The key step in the proof of Feehan's \cite{Feehan} Theorem 1 is to derive an uniform positive lower bound for the lower eigenvalue of the operator $d_{A}^{+}d_{A}^{+,\ast}$ with respect to the connection $A$, the curvature $F_{A}$ obeying $\|F_{A}^{+}\|_{L^{2}(X)}\leq\varepsilon$ for a suitable small constant $\varepsilon=\varepsilon(g)$. In \cite{GKS}, the authors showed a sharp, conformally invariant improvement of these gap theorems which is nontrivial when the Yamabe invariant $Y([g])$ of $(X,g)$ is positive.
Denote by $\mathcal{A}_{YM}$ the space of Yang-Mills connections and $\mathcal{A}_{HYM}$ the space of connections whose curvature satisfies $$\sqrt{-1}\La_{\w}F_{A}=\la Id,$$
where $\la=\frac{2\pi deg P}{rank(P)vol(X)}$. There spaces are gauge invariant with respect to the group $\mathcal{G}_{P}$ of gauge transformations. In \cite{Huang}, the author proved that an open subset $$W=\{[A]:\|\sqrt{-1}\La_{\w}F_{A}-\la Id\|_{L^{2}(X)}<\de\}$$ in the orbit space $\mathcal{A}_{P}/\mathcal{G}_{P}$ of connections with property $\mathcal{A}_{HYM}/\mathcal{G}_{P}=W\cap\mathcal{A}_{YM}/\mathcal{G}_{P}$ under the scalar curvature $S$ of the metric is positive.
Now if we suppose the base manifold $X$ is a K\"{a}hler surface, $P$ is a principal $SU(N)$-bundle over $X$. An ASD connection $A$ on $P$ naturally induces the Yang-Mills complex $$\Omega^{0}(X,\mathfrak{g}_{P})\xrightarrow{d^{0}=d_{A}}\Omega^{1}(X,\mathfrak{g}_{P})\xrightarrow{d^{1}=d^{+}_{A}}{\Omega^{2,+}(X,\mathfrak{g}_{P})}.$$ The $i$-th cohomology group $H^{i}_{A}:=Ker d^{i}/ Im d^{i-1}$ of this complex if finite dimensional and the index $d=h^{0}-h^{1}+h^{2}$ ($h^{i}=dim H^{i}_{A}$ )is given by
$c(G)\kappa(P)-dim G(1-b_{1}+b^{+})$. Here $c(G)$ is a normalising constan t, $\kappa(P)$ is a characteristic number of $P$ obtained by evaluating a $4$-dimensional characteristic class on the fundamental cycle $[X]$, $b_{1}$ is the first Betti number of $X$ and $b^{+}$ is the rank of a maximal positive subspace for the intersection form on $H^{2}(X)$. $H^{0}_{A}$ is the Lie algebra of the stabilizer $\Gamma_{A}$, the group of gauge transformation of $P$ fixing by $A$. From \cite[Proposition 2.3]{Itoh} or \cite[Chapter IV]{Friedman-Morgan}, the second cohomology $H^{2}_{A}$ is $\mathbf{R}$-isomorphic to $H^{0}_{A}\oplus\mathbf{H}$, where $\mathbf{H}:=\ker(\bar{\partial}_{A}^{\ast})|_{\Omega^{0,2}(X,\mathfrak{g}_{P}^{\C})}$ It's difficult to addition certain mild conditions to ensure $\mathbf{H}$ and $H^{0}_{A}$ vanish at some time. The K\"{a}hler metric $g$ often could not be $good$. But one can see that $H^{0}_{A}=0$ is equivalent to the connection $A$ is irreducible.
In \cite{Feehan}, the author shown that if the close $4$-manifold $X$ admits a $good$ metric $g$, then the connection $A\in\mathcal{B}_{\varepsilon}(P,g):=\{[A]\in\mathcal{B}(P,g):\|F_{A}^{+}\|_{L^{2}(X)}\leq\varepsilon \}$ such that the last eigenvalue of $d_{A}^{+}d_{A}^{+,\ast}$ on $L^{2}(\Omega^{2,+}(X,\mathfrak{g}_{P}))$ has a lower positive bound, where $\varepsilon=\varepsilon(g)$ is a suitable small positive constant. The purpose of this article is to introduce the definition of $strong$ irreducible connection $A$ which only guarantees that $\la(A)$ has a low positive bound, See Definition \ref{D3}.
If the Riemannian metric is $good$, there is a well known gluing theorem for anti-self-dual connection which due to Taubes \cite{Taubes1982}. Following the idea of Taubes', if we suppose the connection $A\in\mathcal{A}_{P}$ which obeys $\|F_{A}^{+}\|_{L^{2}(X)}\leq \varepsilon$ for a suitable small positive constant and $\la(A)\geq\la_{0}>0$, then we could deform the connection $A$ to an other connection $A_{\infty}$ which satisfies $\La_{\w}F_{A_{\infty}}=0$, see Corollary \ref{C4}. The connection $A_{\infty}$ may be not an ASD connection, but the $(0,2)$-part $F_{A_{\infty}}^{0,2}$ of the curvature $F_{A_{\infty}}$ could estimated by $\La_{\w}F_{A}$. Following the priori estimate in Theorem \ref{T6} and the vanishing Theorem \ref{T2}, we have \begin{theorem}\label{T1} Let $X$ be a compact, simply-connected, K\"{a}hler surface with a K\"{a}hler metric $g$, $P$ be a principal $G$-bundle with $G$ being $SU(2)$ or $SO(3)$. Suppose that the connections $A\in M(P,g)$ are $strong$ irreducible ASD connections in the sense of Definition \ref{D3}, then there is a positive constant $\varepsilon=\varepsilon(g,P)$ with following significance. If the Yang-Mills connection $A$ on $P$ such that \begin{equation*}
\|F_{A}^{+}\|_{L^{2}(X)}\leq\varepsilon, \end{equation*} then $A$ is anti-self-dual with respect to $g$, i.e., $F_{A}^{+}=0$. \end{theorem} We may assume that any connection $A\in M(P,g)$ is $strong$ irreducible ASD connection in the sense of Definition \ref{D3} if the conditions in Theorem \ref{T4} are obeyed. \begin{corollary} Let $X$ be a compact, simply-connected, K\"{a}hler surface with a K\"{a}hler metric $g$, that is $generic$ in the sense of Definition \ref{D1}, $P$ be a $SO(3)$-bundle over $X$. Suppose that the second Stiefel-Whitney class, $\w_{2}(P)\in H^{2}(X;\mathbb{Z}/2\mathbb{Z})$, is non-trivial, then there is a positive constant $\varepsilon=\varepsilon(g,P)$ with following significance. If the curvature $F_{A}$ of a Yang-Mills connection $A$ on $P$ obeying
$$\|F_{A}^{+}\|_{L^{2}(X)}\leq\varepsilon,$$ then $A$ is anti-self-dual with respect to $g$, i.e., $F_{A}^{+}=0$. \end{corollary}
This paper is organised as follows. In Section 2, we first establish our notation and recall basic definitions in gauge theory over K\"{a}hler manifolds required for the remainder of this article. Following the idea on \cite{Feehan}, we prove that the least eigenvalue, $\la(A)$, of $d^{\ast}_{A}d_{A}$ has a positive lower bound $\la_{0}=\la_{0}(g,P)$ that is uniform with respect to $[A]\in\mathcal{B}(g,P)$ obeying $\|F^{+}_{A}\|_{L^{2}(X)}\leq\varepsilon$, for a small enough $\varepsilon=\varepsilon(g,P)\in (0, 1]$ and under the given sets of conditions on $g, G$. In Section 3, using the similar way of construct of ASD connection by Taubes \cite{Taubes1982}, we obtain that an approximate ASD connection $A\in\mathcal{A}_{P}$ could deform into an other approximate ASD connection $A_{\infty}$ which satisfies $\La_{\w}F_{A_{\infty}}=0$. Thus we can prove that if a Yang-Mills connection obeying $\|F^{+}_{A}\|_{L^{2}(X)}\leq\varepsilon$, then the curvature is harmonic and $\La_{\w}F_{A}=0$. In Section 4, we establish a vanishing theorem (Theorem \ref{T2}) on the space of $\tilde{\Omega}^{0,2}(X,\mathfrak{g}_{P}^{\C})$ Equ. (\ref{E3.10}). Thus we also can prove that the least eigenvalue, $\mu(A)$, of $\bar{\partial}_{A}\bar{\partial}^{\ast}_{A}$ on space $\tilde{\Omega}^{0,2}(X,\mathfrak{g}_{P}^{\C})$ has a positive lower bound $\mu_{0}=\mu_{0}(g,P)$ that is uniform with respect to $[A]\in\mathcal{B}(g,P)$ obeying $\|F^{+}_{A}\|_{L^{2}(X)}\leq\varepsilon$. Combining the curvature is harmonic, then we complete the proof of Theorem \ref{T1}. \section{Preliminaries} \subsection{ Weitzenb\"{o}ck formula} Let $X$ be a K\"{a}hler surface with K\"{a}hler form $\w$ and $P$ be a principal $G$-bundle over $X$. For any connection $A$ on $P$ we have the covariant exterior derivatives $d_{A}:\Omega^{k}(X, \mathfrak{g}_{P})\rightarrow\Omega^{k+1}(X,\mathfrak{g}_{P})$. Like the canonical splitting the exterior derivatives $d=\partial+\bar{\partial}$, decomposes over $X$ into $d_{A}=\partial_{A}+\bar{\partial}_{A}$. We denote also by $\Omega^{p,q}(X,\mathfrak{g}_{P}^{\C})$ the space of $C^{\infty}$-$(p,q)$ forms on $\mathfrak{g}_{P}^{\C}:=\mathfrak{g}_{P}\otimes\C$. Denote by $L_{\w}$ the operator of exterior multiplication by the K\"{a}hler form $\w$: $$L_{\w}\a=\w\wedge\a, \a\in\Omega^{p,q}(X,\mathfrak{g}_{P}^{\C}),$$ and, as usual, let $\La_{\w}$ denote its pointwise adjoint, i.e., $$\langle\La_{\w}\a,\b\rangle=\langle\a,L_{\w}\b\rangle.$$ Then it is well known that $\La_{\w}=\ast^{-1}\circ L_{\w}\circ \ast$. We could decompose the curvature, $F_{A}$, as $$F_{A}=F^{2,0}_{A}+F^{1,1}_{A0}+\frac{1}{2}\La_{\w}{F}_{A}\otimes\w+F^{0,2}_{A},$$ where $F^{1,1}_{A0}=F^{1,1}_{A}-\frac{1}{2}\La_{\w}F_{A}\otimes\w$.\ We can write Yang-Mills functional as \begin{equation}\nonumber \begin{split}
YM(A)&=4\|F^{0,2}_{A}\|^{2}+\|\La_{\w} F_{A}\|^{2}+\int_{X}tr(F_{A}\wedge F_{A}).\\ \end{split} \end{equation}
The energy functional $\|\La_{\w}F_{A}\|^{2}$ plays an important role in the study of Hermitian-Einstein connections, see \cite{DK,UY}. If the connection $A$ is an ASD connection, the Yang-Mills functional is minimum. We recall some identities on Yang-Mills connection over K\"{a}hler surface, see \cite[Proposition 3.1]{Itoh} or \cite[Proposition 2.1]{Huang} . \begin{proposition}\label{P1} Let $A$ be a Yang-Mills connection on a principal $G$-bundle $P$ over a K\"{a}hler surface $X$, then we have following identities: \begin{equation*} \begin{split} &(1)\ 2\bar{\partial}^{\ast}_{A}F^{0,2}_{A}=\sqrt{-1}\bar{\partial}_{A}\La_{\w}F_{A},\\ &(2)\ 2\partial_{A}^{\ast}F^{2,0}_{A}=-\sqrt{-1}\partial_{A}\La_{\w}F_{A}.\\ \end{split} \end{equation*} \end{proposition} We define a Hermitian inner product $\langle\cdot,\cdot\rangle$ on $\Omega^{p,q}(X,\mathfrak{g}_{P}^{\C})$ by $$\langle\a,\b\rangle_{L^{2}(X)}=\int_{X}\langle\a,\b\rangle(x)dvol_{g},$$ $$\langle\a,\b\rangle(x)dvol_{g}=\langle\a\wedge\ast\bar{\b} \rangle,$$ where $\ast$ is the $\C$-linearly extend Hodge operator over complex forms and $\bar{}$ is the conjugation on the bundle $\mathfrak{g}_{P}^{\C}$-forms which is defined naturally. One also can see \cite[Page 99]{Itoh2} or \cite{Huy}. We recall a Weitzenb\"{o}ck formula for Lie algebra-valued $(0,2)$-forms, see \cite[Proposition 2.3]{Itoh}, a self-dual operator denote by $\De_{\bar{\partial}_{A}}=\bar{\partial}_{A}\bar{\partial}^{\ast}_{A}+\bar{\partial}_{A}^{\ast}\bar{\partial}_{A}$. \begin{proposition}\label{P3} Let $X$ be a smooth K\"{a}hler surface with a K\"{a}hler metric $g$, $A$ be a connection on a principal $G$-bundle $P$ over $X$. For any $\phi\in\Omega^{0,2}(X,\mathfrak{g}_{P}^{\C})$, \begin{equation}\label{12} \De_{\bar{\partial}_{A}}\phi=\na^{\ast}_{A}\na_{A}\phi+[\sqrt{-1}\La_{\w}F_{A},\phi]+2S\phi \end{equation} where $S$ is the scalar curvature of the metric $g$. \end{proposition} Combining Weitzenb\"{o}ck formula on Proposition \ref{P3} with the identities on Proposition \ref{P1}, we have an identity for Yang-Mills connection. \begin{proposition}\label{P5} Let $X$ be a smooth K\"{a}hler surface with a K\"{a}hler metric $g$, $A$ be a Yang-Mills connection on a principal $G$-bundle $P$ over $X$. Then we have \begin{equation}\label{E14} \na_{A}^{\ast}\na_{A}F^{0,2}_{A}+\frac{3}{2}[\sqrt{-1}\La_{\w}F_{A},F^{0,2}_{A}]+2SF^{0,2}_{A}=0 \end{equation} Furthermore, if $X$ is compact, \begin{equation}\label{E12}
\frac{3}{4}\|\bar{\partial}_{A}\La_{\w}F_{A}\|^{2}_{L^{2}(X)}=\|\na_{A}F^{0,2}_{A}\|^{2}_{L^{2}(X)}+\int_{X}2S|F_{A}^{0,2}|^{2}dvol_{g}. \end{equation} \end{proposition} \begin{proof} Following Proposition \ref{P1}, we obtain that \begin{equation*} \begin{split}
\|\bar{\partial}_{A}\La_{\w}F_{A}\|_{L^{2}(X)}^{2}&=\langle\bar{\partial}^{\ast}_{A}\bar{\partial}_{A}\La_{\w}F_{A},\La_{\w}F_{A}\rangle_{L^{2}(X)}\\&=-\langle2\sqrt{-1}\ast [F^{0,2}_{A},\ast F_{A}^{0,2}],\La_{\w}F_{A}\rangle_{L^{2}(X)}\\ &=-\langle2\sqrt{-1} [\La_{\w}F_{A}, F^{0,2}_{A}], F_{A}^{0,2}\rangle_{L^{2}(X)}\\ \end{split} \end{equation*} The Weizenb\"{o}ck formula for $F^{0,2}_{A}$ yields $$\na_{A}^{\ast}\na_{A}F_{A}^{0,2}+2SF_{A}^{0,2}+\frac{3}{2}[\sqrt{-1}\La_{\w}F_{A},F_{A}^{0,2}]=0.$$ If $X$ is closed, taking the $L^{2}$-inner product of above identity with $F_{A}^{0,2}$ and integrating by parts, we then obtain (\ref{E12}). \end{proof}
\subsection{Irreducible connections} In this section, we first recall a definition of irreducible connection on a principal $G$-bundle $P$, where $G$ being a compact, semisimple Lie group. Given a connection $A$ on a principal $G$-bundle $P$ over $X$. We can define the stabilizer $\Gamma_{A}$ of $A$ in the gauge group $\mathcal{G}_{P}$ by
$$\Gamma_{A}:=\{g\in\mathcal{G}_{P}|g(A)=A\},$$
one also can see \cite{DK}. A connection $A$ called reducible if the connection $A$ whose stabilizer $\Gamma_{A}$ is larger than the centre $C(G)$ of $G$. Otherwise, the connections are irreducible, they satisfy $\Gamma_{A}\cong C(G)$. It's easy to see that a connection $A$ is irreducible when it admits no nontrivial covariantly constant Lie algebra-value $0$-form, i.e., $\ker d_{A}|_{\Omega^{0}(X,\mathfrak{g}_{P})}=0$. Taubes introduced the number in \cite[Equation 6.3]{Taubes1988} to measure the irreducibility of $A$. We can defined the least eigenvalue $\la(A)$ of $d^{\ast}_{A}d_{A}$ as follow. \begin{definition}\label{D2} The least eigenvalue of $d_{A}^{\ast}d_{A}$ on $L^{2}$ section of $\Gamma(\mathfrak{g}_{P})$ is \begin{equation}\label{E3}
\la(A):=\inf_{v\in\Gamma(\mathfrak{g}_{P})\backslash\{0\}}\frac{\|d_{A}v\|^{2}}{\|v\|^{2}}.
\end{equation} \end{definition} It is easy to see that the function $\la(A)$ depends only on the connection $A$. We introduce the definition of $strong$ irreducible connection on a principal $G$-bundle $P$. \begin{definition}\label{D3} We call $A$ a smooth $strong$ irreducible connection on $G$-bundle $P$ over a smooth $n$-dimensional, Riemannian manifold $X$, ($n\geq2$), if the least eigenvalue of $d_{A}^{\ast}d_{A}$ on $L^{2}$ section of $\Gamma(\mathfrak{g}_{P})$ has a positive lower bound, i.e, there is a constant $\la_{0}=\la_{0}(P,g)\in(0,\infty)$ such that $\la(A)\geq\la_{0}$. \end{definition}
The Sobolev norms $L^{p}_{k,A}$,\ where $1\leq p<\infty$ and $k$ is an integer, with respect to the connections defined as: \begin{equation}\nonumber
\|u\|_{L^{p}_{k,A}(X)}:=\big{(}\sum_{j=0}^{k}\int_{X}|\na^{j}_{A}u|^{p}dvol_{g}\big{)}^{1/p}, \end{equation} where $\na_{A}$ is the covariant derivative induced by the connection $A$ on $P$ and the Levi-Civita connection defined by the Riemannian metric $g$ on $T^{\ast}X$ and $\na^{j}_{A}:=\na_{A}\circ\ldots\circ\na_{A}$ (repeated $j$ times for $j\geq0$). \begin{remark}\label{R2.5}
Let $A$ be a irreducible connection on the principal $G$-bundle over a compact manifold $X$, i.e., $\ker d_{A}|_{\Omega^{0}(X,\mathfrak{g}_{P})}=0$. Then we can assume that $\ker d_{A}|_{\Omega^{0}(X,\mathfrak{g}_{P}^{\C})}=0$, where $\mathfrak{g}_{P}^{\C}:=\mathfrak{g}_{P}\otimes\C$. We denote $s$ by a section of $\Gamma(\mathfrak{g}_{P}^{\C})$, i.e., $s$ can be seen as a function over $X$ which takes value in the Lie algebra $\mathfrak{g}\otimes\C$. Here $\mathfrak{g}\otimes\C$ is the complexification of Lie algebra $\mathfrak{g}$, See \cite[Pages 11--12]{Samelson} . Thus in a local coordinate, there exists two $s_{1},s_{2}\in\mathfrak{g}$ such that $s=s_{1}+\rm{i}s_{2}$. Then we have two identities,
$|\na_{A}s|^{2}=|\na_{A}s_{1}|^{2}+|\na_{A}s_{2}|^{2}$ and $|s|^{2}=|s_{1}|^{2}+|s_{2}|^{2}$. Since $A$ is irreducible, $\|\na_{A}s\|^{2}_{L^{2}(X)}\geq\la\|s\|^{2}_{L^{2}(X)}$. \end{remark} By the similar method of the proof of \cite[Proposition A.3]{Feehan} or \cite[Lemma 7.1.24]{DK}, the least eigenvalue $\la(A)$ of $d^{\ast}_{A}d_{A}$ with respect to connection $A$ is continuous in $L^{p}_{loc}$-topology for $2\leq p<4$. Due to a result of Sedlacek \cite[Theorem 4.3]{Sedlacek}, we then have \begin{proposition}\label{P2.6}
Let $G$ be a compact,semisimple Lie group and P be a principal $G$-bundle over a closed, smooth, oriented, four-dimensional Riemannian manifold $X$ with a Riemannian metric $g$. If $\{A_{i}\}_{i\in\mathbb{N}}$ is a sequence $C^{\infty}$ connection on $P$ and the curvatures $\|F^{+}_{A_{i}}\|_{L^{2}(X)}$ converge to zero as $i\rightarrow\infty$, then there are a finite set of points, $\Sigma=\{x_{1},\ldots,x_{L}\}\subset X$ and a subsequence $\Xi\subset\N$, we also denote by $\{A_{i}\}$, a sequence gauge transformatin $\{g_{i}\}_{i\in\Xi}$ such that, $g_{i}(A_{i})\rightarrow A_{\infty}$, a anti-self-dual connection on $P\upharpoonright_{X\backslash\Sigma}$ in the $L^{2}_{1}$-topology over $X\backslash\Sigma$. Furthermore, we have $$\lim_{i\rightarrow\infty}\la(A_{i})=\la(A_{\infty}),$$ where $\la(A)$ is as in Definition \ref{D2}. \end{proposition} \begin{theorem} Let $X$ be a closed, Riemannian 4-manifold, $P$ be a principal $G$-bundle with $G$ being compact, semisimple Lie group. Suppose that the ASD connections $A\in M(P,g)$ are $strong$ irreducible connections in the sense of Definition \ref{D3}, then there are positive constants $\la_{0}$ and $\varepsilon$ such that \begin{equation*} \begin{split} &\la(A)\geq\la_{0},\ \forall [A]\in M(P,g),\\ &\la(A)\geq\frac{\la_{0}}{2},\ \forall [A]\in\mathcal{B}_{\varepsilon}(P,g).\\ \end{split} \end{equation*} \end{theorem} \begin{proof} By the definition of $strong$ irreducible connection, there is a positive constant $\la_{0}=\la_{0}(P,g)$ such that $\la(A)\geq\la_{0}$, $\forall [A]\in M(P,g)$.
Suppose that the constant $\varepsilon\in(0,1]$ does not exist. We may then choose a sequence $\{A_{i}\}_{i\in\N}$ of connection on $P$ such that $\|F^{+}_{A_{i}}\|_{L^{2}(X)}\rightarrow0$ and $\la(A_{i})\rightarrow0$ as $i\rightarrow\infty$. Since $\lim_{i\rightarrow\infty}\la(A_{i})=\la(A_{\infty})$ and $\la(A_{\infty})\geq\la_{0}$, then it is contradict to our initial assumption regarding the sequence $\{A_{i}\}_{i\in\N}$. In particular, the preceding argument shows that the desired constant $\varepsilon$ exists. \end{proof} Friedman and Morgan introduced the $generic$ K\"{a}hler metric which ensures the connections on the compactification of moduli space of ASD connections on $E$, $\bar{M}(P,g)$, are irreducible, See \cite[Chapter IV]{Friedman-Morgan}. The authors also given an example on a K\"{a}hler surface which admits a $generic$ K\"{a}hler metric. Fix an algebraic surface $S$ and an ample line bundle $L$ on $S$. For every integer $c$ we have defined the moduli space $\mathcal{M}_{c}(S,L)$ of $L$-stable rank two holomorphic vector bundles $V$ on $S$ and such that $c_{1}(V)=0$, $c_{2}(V)=c$. Next let us determine when a Hodge metric with K\"{a}hler form $\w$ admits reducible ASD connections. Corresponding to such a connection is an associated ASD harmonic $1$-form $\a$, well-defined up to $\pm1$, representing an integral cohomology class, which by the description of $\Omega^{2,-}(X)$ is of type $(1,1)$ and orthogonal to $\w$. Thus Friedman-Morgan proved for an integer $c>0$, there is an open dense subset $\mathcal{D}$ of the cone of ample divisors on $S$ such that if $g$ is a Hodge metric whose K\"{a}hler form lies in $\mathcal{D}$, $g$ is a $generic$ metric in the sense of Definition \ref{D1}, See \cite[Chapter IV, Proposition 4.8]{Friedman-Morgan}. \begin{definition}\label{D1} Let $X$ be a compact K\"{a}hler surface, $P$ be a principal $G$-bundle over $X$ with $c_{2}(P)=c$. We say that a K\"{a}hler metric $g$ on $X$ is $generic$ if for every $G$-bundle $\tilde{P}$ over $X$ with $0<c_{2}(\tilde{P})\leq c$, there are no reducible ASD connections on $\tilde{P}$. \end{definition}
For a compact K\"{a}hler surface $X$ we have a moduli space of ASD connections $M(P,g)$. The Donaldson-Uhlenbeck compactification $\bar{M}(P,g)$ of $M(P,g)$ contained in the disjoint union \begin{equation}\nonumber \bar{M}(P,g)\subset\cup(M(P_{l},g)\times Sym^{l}(X)), \end{equation} Following \cite[Theorem 4.4.3]{DK}, the space $\bar{M}(P,g)$ is compact. \begin{lemma} Let $X$ be a compact, simply-connected, K\"{a}hler surface with a $generic$ K\"{a}hler metric, $P$ be a $SO(3)$-bundle over $X$. If the second Stiefel-Whitney classes $w_{2}\in H^{2}(X;\mathbb{Z}/2\mathbb{Z})$ is non-trivial, then $\la(A)>0$ for any $[A]\in\bar{M}(P,g)$. \end{lemma} \begin{proof} The only reducible anti-self-dual connection on a principal $SO(3)$-bundle over $X$ is the product connection on the product bundle $P=X\times SO(3)$ by \cite[Corollary 4.3.15]{DK} and the latter possibility is excluded by our hypothesis in this case that $w_{2}(P)\neq 0$. Then the conclusion is a consequence of the definition of $generic$ K\"{a}hler metric. \end{proof} Following Feehan's idea, combining $\bar{M}(P,g)$ is compact and $\la(A)$ is continuous under the Uhlenbeck topology for the connection $[A]\in\bar{M}(P,g)$, then we have \begin{theorem}\label{T4} Let $X$ be a compact, simply-connected, K\"{a}hler surface with a $generic$ K\"{a}hler metric, $P$ be a $SO(3)$-bundle over $X$. If the second Stiefel-Whitney classes $w_{2}(P)\in H^{2}(X;\mathbb{Z}/2\mathbb{Z})$ is non-trivial, then there are positive constants $\mu_{0}=\mu_{0}(P,g)$ and $\varepsilon=\varepsilon(P,g)$ such that \begin{equation*} \begin{split} &\la(A)\geq\la_{0},\ \forall [A]\in M(P,g),\\ &\la(A)\geq\frac{\la_{0}}{2},\ \forall [A]\in\mathcal{B}_{\varepsilon}(P,g).\\ \end{split} \end{equation*} \end{theorem} \subsection{An estimate on Yang-Mills connections}\label{S1} \begin{lemma}\label{L1} Let $X$ be a closed, smooth, four-manifold with a Riemannian metric $g$, $A$ be a connection on a principal $G$-bundle $P$ over $X$ with $G$ being a compact, semisimple Lie group. There are positive constants $\la=\la(g,P)$ and $C=C(\la,g,P)$ with following significance. If $\la(A)\geq\la$, where $\la(A)$ is as in (\ref{E3}), then for any section $v$ on $\Gamma(\mathfrak{g}_{P})$, \begin{equation}\label{E6}
\|v\|_{L^{2}_{2}(X)}\leq C\|\na_{A}^{\ast}\na_{A}v\|_{L^{2}(X)}. \end{equation} \end{lemma} \begin{proof} Since $\na_{A}^{\ast}\na_{A}$ is a elliptic operator of degree $2$, from a priori estimate of elliptic operator, for $p\geq1$ and $k\geq 0$, we have \begin{equation}\label{E7}
\|v\|_{L^{p}_{k+2}(X)}\leq c\|\na_{A}^{\ast}\na_{A}v\|_{L^{p}_{k}(X)}+\|v\|_{L^{p}(X)} \end{equation} We take $k=0$ and $p=2$,
$$\|v\|_{L^{2}_{2}(X)}\leq c\|\na_{A}^{\ast}\na_{A}v\|_{L^{2}(X)}+\|v\|_{L^{2}(X)}.$$ By the definition of $\la(A)$, we also have
$$\|v\|_{L^{2}(X)}\leq \la^{-1}\|\na_{A}^{\ast}\na_{A}v\|_{L^{2}(X)}.$$ Combining the preceding inequalities yields (\ref{E6}). \end{proof} We construct a priori on the Yang-Mills connection under the condition of $\La_{\w}F_{A}$ is sufficiently small in $L^{2}$-norm. \begin{proposition}\label{P4} Let $X$ be a compact K\"{a}hler surface, $A$ be a Yang-Mills connection on a principal $G$-bundle $P$ over $X$ with $G$ being a compact, semisimple Lie group, $p\in(4,\infty)$. Let $q\in(4/3,2)$ be defined by $1/q=1/2+1/p$. There are positive constants $\la=\la(g,P)$ and $\varepsilon=\varepsilon(\la,g,P)$ with following significance. If the curvature $F_{A}$ of the connection $A$ on $P$ obeying
$$\|\La_{\w}F_{A}\|_{L^{2}(X)}\leq\varepsilon,$$ and $\la(A)\geq\la$, where $\la(A)$ is as in (\ref{E3}), then \begin{equation}\label{E4}
\|F_{A}^{0,2}\|_{L^{p}(X)}\leq C\|F_{A}^{0,2}\|_{L^{q}(X)}, \end{equation} where $C=C(\la,g,P,p)$ is a positive constant. \end{proposition} \begin{proof} We apply the estimate (\ref{E7}) and Sobolev embedding $L^{q}_{2}\hookrightarrow L^{p}$, then
$$\|F_{A}^{0,2}\|_{L^{p}(X)}\leq c\|F_{A}^{0,2}\|_{L^{q}_{2}(X)}\leq \|\na_{A}^{\ast}\na_{A}F_{A}^{0,2}\|_{L^{q}(X)}+\|F^{0,2}_{A}\|_{L^{q}(X)}.$$ By using the Weizenb\"{o}ck formula of $F^{0,2}_{A}$, see equation (\ref{E14}), we then observe that \begin{equation*} \begin{split}
\|\na_{A}^{\ast}\na_{A}F_{A}^{0,2}\|_{L^{q}(X)}&\leq c\|F^{0,2}_{A}\|_{L^{q}(X)}+\|\{F_{A}^{0,2},\La_{\w}F_{A}\}\|_{L^{q}(X)}\\
&\leq c\|F^{0,2}_{A}\|_{L^{q}(X)}+c\|\La_{\w}F_{A}\|_{L^{2}(X)}\|F_{A}^{0,2}\|_{L^{p}(X)}.\\ \end{split} \end{equation*} where $c=c(g,G,p)$ is a positive constant.
Combining the preceding inequalities gives
$$\|F_{A}^{0,2}\|_{L^{p}(X)}\leq c\|F^{0,2}_{A}\|_{L^{q}(X)}+c\|\La_{\w}F_{A}\|_{L^{2}(X)}\|F_{A}^{0,2}\|_{L^{p}(X)}+\|F^{0,2}_{A}\|_{L^{q}(X)}.$$
where $c=c(g,G,p)$ is a positive constant. Provide $c\|\La_{\w}F_{A}\|_{L^{2}(X)}\leq1/2$, rearrangement gives (\ref{E4}). \end{proof} \section{Yang-Mills connection on K\"{a}hler surface} \subsection{Approximate Hermitian-Yang-Mills connections} In this section we will give a general criteria under which an approximate ASD connection $A\in\mathcal{A}_{P}$ could deform into an other approximate ASD connection $A_{\infty}$ which satisfies \begin{equation} \label{E3.1} \La_{\w}F_{A_{\infty}}=0. \end{equation} One also can see \cite[Section 4.2]{Huang2019}. Let $A$ be a connection on a principal $G$-bundle over $X$. The equation (\ref{E3.1}) for a second connection $A+a$, where $a\in\Omega^{1}(X,\mathfrak{g}_{P})$ is a bundle valued $1$-form, could be rewritten to \begin{equation}\label{E1} \La_{\w}(d_{A}a+a\wedge a)=-\La_{\w}F_{A}. \end{equation} We seek a solution of the equation (\ref{E1}) in the form $$a=d_{A}^{\ast}(s\otimes\w)=\sqrt{-1}(\partial_{A}s-\bar{\partial}_{A}s)$$ where $s\in\Omega^{0}(X,\mathfrak{g}_{P})$ is a bundle value $0$-form. Then Equation (\ref{E1}) becomes the second order equation: \begin{equation}\label{E2} -d_{A}^{\ast}d_{A}s+\La_{\w}(d_{A}s\wedge d_{A}s)=-\La_{\w}F_{A}. \end{equation} For convenience, we define a map $$B(u,v):=\frac{1}{2}\La_{\w}[d_{A}u\wedge d_{A}v].$$ It's easy to check, we have the pointwise bound:
$$|B(u,v)|\leq C|\na_{A}u||\na_{A}v|,$$ where $C$ is a uniform positive constant.
We would like to prove that if $\La_{\w}F_{A}$ is small in an appropriate sense, there is a small solution $s$ to equation (\ref{E2}). \begin{theorem}\label{T6}(\cite[Theorem 4.8]{Huang2019}) Let $X$ be a compact, K\"{a}hler surface with a K\"{a}hler metric $g$, $P$ be a principal $G$-bundle over $X$ with $G$ being a compact, semisimple Lie group. There are positive constant $\la=\la(g,P)$ and $\varepsilon=\varepsilon(\la,g,P)$ with following significance. If the curvature $F_{A}$ of a connection $A$ on $P$ obeying \begin{equation*} \begin{split}
&\|\La_{\w}F_{A}\|_{L^{2}(X)}\leq\varepsilon,\\ &\la(A)\geq\la,\\ \end{split} \end{equation*} where $\la(A)$ is as in (\ref{E3}), then there is a section $s\in\Gamma(\mathfrak{g}_{P})$ such that the connection $$A_{\infty}:=A+\sqrt{-1}(\partial_{A}s-\bar{\partial}_{A}s)$$ satisfies\\ (1) $\La_{\w}F_{A_{\infty}}=0$\\
(2) $\|s\|_{L^{2}_{2}(X)}\leq C\|\La_{\w}F_{A}\|_{L^{2}(X)}$,\\ where $C=C(\la,g)\in[1,\infty)$ is a positive constant. Furthermore, let $p\in(2,\infty)$,
$$\|F^{0,2}_{A_{\infty}}-F_{A}^{0,2}\|_{L^{2}(X)}\leq C(\|\La_{\w}F_{A}\|_{L^{2}(X)}+\|F_{A}^{0,2}\|_{L^{p}(X)})\|\La_{\w}F_{A}\|_{L^{2}(X)},$$ for a positive constant $C=C(\la,g,p)$. \end{theorem} Now, we begin to prove Theorem \ref{T6}, the proof of above theorem is base on Taubes' ideas \cite{Taubes1982} and \cite{Donaldson1993}. At first, suppose $s$ and $f$ are sections of $\Gamma(\mathfrak{g}_{P)}$ with \begin{equation}\label{E8} d^{\ast}_{A}d_{A}s=f,\ i.e.,\ \na_{A}^{\ast}\na_{A}s=f, \end{equation} the first observation is \begin{lemma}\label{L3}(\cite[Lemma 4.9]{Huang2019}) If $\la(A)\geq\la>0$, then there exists a unique $C^{\infty}$ solution to equation (\ref{E8}). Furthermore, we have \begin{equation*} \begin{split}
&\|s\|_{L^{2}_{2}(X)}\leq c\|f\|_{L^{2}(X)},\\
&\|B(s,s)\|_{L^{2}(X)}\leq c\|f\|^{2}_{L^{2}(X)},\\ \end{split} \end{equation*} where $c=c(\la,g)$ is a positive constant. \end{lemma} \begin{proof} Following the estimate on Lemma \ref{L1}, we have
$$\|s\|_{L^{2}_{2}(X)}\leq \|\na_{A}^{\ast}\na_{A}s\|_{L^{2}(X)}\leq c\|f\|_{L^{2}(X)},$$ for a positive constant $c=c(\la,g,P)$. By the Sobolev inequality in four dimension, \begin{equation*}
\|B(s,s)\|_{L^{2}(X)}\leq C\|\na_{A}s\|^{2}_{L^{4}(X)}\leq C\|\na_{A}s\|^{2}_{L^{2}_{1}(X)}\leq C\|s\|^{2}_{L^{2}_{2}(X)}, \end{equation*} for a positive constant $C=C(\la,g,P)$. \end{proof} \begin{lemma}(\cite[Lemma 4.10]{Huang2019}) If $d_{A}^{\ast}d_{A}s_{1}=f_{1}$, $d_{A}^{\ast}d_{A}s_{2}=f_{2}$, then \begin{equation*}
\|B(s_{1},s_{2})\|_{L^{2}(X)}\leq c\|f_{1}\|_{L^{2}(X)}\|f_{2}\|_{L^{2}(X)}. \end{equation*} \end{lemma} We will prove the existence of a solution of (\ref{E2}) by the contraction mapping principle. We write $s=(d_{A}^{\ast}d_{A})^{-1}f$ and (\ref{E2}) becomes an equation for $f$ of the from \begin{equation}\label{E10} f-S(f,f)=\La_{\w}F_{A}, \end{equation} where $S(f,g):=B((d_{A}^{\ast}d_{A})^{-1}f,(d_{A}^{\ast}d_{A})^{-1}g)$. Following Lemma \ref{L3}, \begin{equation*} \begin{split}
\|S(f_{1},f_{1})-S(f_{2},f_{2})\|_{L^{2}(X)}&=\|S(f_{1}+f_{2},f_{1}-f_{2})\|_{L^{2}(X)}\\
&\leq c\|f_{1}+f_{2}\|_{L^{2}(X)}\|f_{1}-f_{2}\|_{L^{2}(X)}.\\ \end{split} \end{equation*} We denote $g_{k}=f_{k}-f_{k-1}$ and $g_{1}=f_{1}$, then $$g_{1}=\La_{\w}F_{A},\ g_{2}=S(g_{1},g_{1})$$ and $$g_{k}=S(\sum_{i=1}^{k-1}g_{i},\sum_{i=1}^{k-1}g_{i})-S(\sum_{i=1}^{k-2}g_{i},\sum_{i=1}^{k-2}g_{i}),\ \forall\ k\geq 3.$$ It is easy to show that, under the assumption of $\La_{\w}F_{A}$, the sequence $f_{k}$ defined by $$f_{k}=S(f_{k-1},f_{k-1})+\La_{\w}F_{A},$$ starting with $f_{1}=\La_{\w}F_{A}$, is Cauchy with respect to $L^{2}$, and so converges to a limit $f$ in the completion of $\Gamma(\mathfrak{g}_{P})$ under $L^{2}$. \begin{proposition}(\cite[Proposition 4.11]{Huang2019})
There are positive constants $\varepsilon\in(0,1)$ and $C\in(1,\infty)$ with following significance. If $$\|\La_{\w}F_{A}\|_{L^{2}(X)}\leq\varepsilon,$$ then each $g_{k}$ exists and is $C^{\infty}$. Further for each $k\geq1$, we have \begin{equation}\label{E9}
\|g_{k}\|_{L^{2}(X)}\leq C^{k-1}\|\La_{\w}F_{A}\|^{k}_{L^{2}(X)}. \end{equation} \end{proposition} \begin{proof} The proof is by induction on the integer $k$. The induction begins with $k=1$, one can see $g_{1}=\La_{\w}F_{A}$. The induction proof if completed by demonstrating that if (\ref{E9}) is satisfied for $j<k$, then it also satisfied for $j=k$. Indeed, since \begin{equation}\nonumber \begin{split}
\|S(\sum_{i=1}^{k-1}g_{i},\sum_{i=1}^{k-1}g_{i})-S(\sum_{i=1}^{k-2}g_{i},\sum_{i=1}^{k-2}g_{i})\|_{L^{2}(X)}&\leq c\|\sum_{i=1}^{k-1}g_{i}+\sum_{i=1}^{k-2}g_{i}\|_{L^{2}(X)}\|g_{k-1}\|_{L^{2}(X)},\\
&\leq 2c\sum\|g_{i}\|_{L^{2}(X)}\|g_{k-1}\|_{L^{2}(X)},\\
&\leq \frac{2cC^{k-2}\|\La_{w}F_{A}\|^{k}_{L^{2}(X)}}{1-C\|\La_{\w}F_{A}\|_{L^{2}(X)}}.\\ \end{split} \end{equation}
Now, we provide $\varepsilon$ sufficiently small and $C$ sufficiently large to ensure $\|\La_{\w}F_{A}\|_{L^{2}(X)}\leq C^{-2}(C-2c)$, i.e., $\frac{2c}{1-C\|\La_{w}F_{A}\|_{L^{2}(X)}}\leq C$, hence we complete the proof of this Proposition. \end{proof} \begin{proof}[\textbf{Proof of Theorem \ref{T6}}] The sequence $g_{k}$ is Cauchy in $L^{2}$, the limit $f:=\lim_{i\rightarrow}f_{k}$ is a solution to (\ref{E10}). Following Lemma \ref{L3}, we have \begin{equation*}
\|s\|_{L^{2}_{2}(X)}\leq c\|f\|_{L^{2}(X)}\leq c\sum_{k=1}^{\infty}\|g_{k}\|_{L^{2}(X)}\leq \frac{c\|\La_{\w}F_{A}\|_{L^{2}(X)}}{1-C\|\La_{\w}F_{A}\|_{L^{2}(X)}}, \end{equation*} for a positive constant $c$. We provide $\varepsilon$ and $C$ to ensure $C\varepsilon\leq\frac{1}{2}$, hence
$$\|s\|_{L^{2}_{2}(X)}\leq 2c\|\La_{\w}F_{A}\|_{L^{2}(X)}.$$ We denote $A_{\infty}:=A+\sqrt{-1}(\partial_{A}s-\bar{\partial}_{A}s)$, and $r\in(2,\infty)$ defined by $1/r=1/2-1/p$, then \begin{equation*} \begin{split}
\|F^{0,2}_{A_{\infty}}-F_{A}^{0,2}\|_{L^{2}(X)}&=\|-\sqrt{-1}\bar{\partial}_{A}\bar{\partial}_{A}s-\bar{\partial}_{A}s\wedge\bar{\partial}_{A}s\|_{L^{2}(X)}\\
&=\|-\sqrt{-1}[F_{A}^{0,2},s]-\bar{\partial}_{A}s\wedge\bar{\partial}_{A}s\|_{L^{2}(X)}\\
&\leq 2\|\bar{\partial}_{A}s\|^{2}_{L^{4}(X)}+2\|F_{A}^{0,2}\|_{L^{p}(X)}\|s\|_{L^{r}(X)}\\
&\leq c\|\na_{A}s\|^{2}_{L^{2}_{1}(X)}+c\|F_{A}^{0,2}\|_{L^{p}(X)}\|s\|_{L^{2}_{2}(X)}\\
&\leq c(\|\La_{\w}F_{A}\|_{L^{2}(X)}+\|F_{A}^{0,2}\|_{L^{p}(X)})\|\La_{\w}F_{A}\|_{L^{2}(X)}.\\ \end{split} \end{equation*} where $c$ is a positive constant. We complete the proof of Theorem \ref{T6}. \end{proof} \begin{corollary}\label{C4} Let $X$ be a compact, simply-connected, K\"{a}hler surface with a K\"{a}hler metric $g$, that is $generic$ in the sense of Definition \ref{D1}, $P$ be a $SO(3)$-bundle over $X$. Suppose that the second Stiefel-Whitney class, $\w_{2}(P)\in H^{2}(X;\mathbb{Z}/2\mathbb{Z})$, is non-trivial, then there is a positive constant $\varepsilon=\varepsilon(g,P)$ with following significance. If the curvature $F_{A}$ of a connection $A$ on $P$ obeying
$$\|F_{A}^{+}\|_{L^{2}(X)}\leq\varepsilon,$$ then there is a section $s\in\Gamma(\mathfrak{g}_{P})$ such that the connection $A_{\infty}:=A+\sqrt{-1}(\partial_{A}s-\bar{\partial}_{A}s)$ satisfies\\ (1) $\La_{\w}F_{A_{\infty}}=0$\\
(2) $\|s\|_{L^{2}_{2}(X)}\leq C\|\La_{\w}F_{A}\|_{L^{2}(X)}$,\\ where $C=C(g,P)$ is a positive constant. \end{corollary} \subsection{Yang-Mills connection with harmonic curvature} Suppose that an integrable connecion $A\mathcal{A}_{P}^{1,1}$ on a holomorphic bundle over a K\"{a}hler surface is Yang-Mills, then $\La_{\w}F_{A}$ is parallel, that is equivalent to $F_{A}^{0,2}$ being harmonic with respect to Laplacian operator $\De_{\bar{\partial}_{A}}$. For a general case, we introduce the definition of a connection with harmonic curvature, See \cite[p. 96]{Itoh2}. \begin{definition} A connection $A$ on a compact K\"{a}hler surface is said to be with a harmonic curvature if $(0,2)$-part of curvature is harmonic, i.e., $\bar{\partial}_{A}^{\ast}F_{A}^{0,2}=0$. \end{definition} \begin{lemma}\label{L5} Let $X$ be a compact K\"{a}hler surface, $P$ be a $G$-bundle over $X$ with $G$ being a compact, semisimple Lie group, $A$ be a Yang-Mills connection on $P$. There are positive constant $\la=\la(g,P)$ and $\varepsilon=\varepsilon(\la,g,P)$ with following significance. If the curvature $F_{A}$ of the connection $A$ obeying \begin{equation*} \begin{split}
&\|F_{A}^{+}\|_{L^{2}(X)}\leq \varepsilon,\\ &\la(A)\geq\la,\\ \end{split} \end{equation*} where $\la(A)$ is as in (\ref{E3}), then the curvature is harmonic and $\La_{\w}F_{A}=0$. \end{lemma} \begin{proof} For a suitable constant $\varepsilon$, from Theorem \ref{T6}, there exist a connection $A_{\infty}$ such that
$$\|A-A_{\infty}\|_{L^{2}_{1}(X)}\leq c\|\La_{\w}F_{A}\|_{L^{2}(X)}$$ and $\La_{\w}F_{A_{\infty}}=0$. We apply the Weizenb\"{o}ck formula (\ref{E14}) to $F_{A}^{0,2}$,
$$\|\bar{\partial}_{A_{\infty}}F_{A}^{0,2}\|^{2}_{L^{2}(X)}+\|\bar{\partial}_{A_{\infty}}^{\ast}F_{A}^{0,2}\|^{2}_{L^{2}(X)}=\|\na_{A_{\infty}}F_{A}^{0,2}\|^{2}_{L^{2}(X)}+\int_{X}2S|F_{A}^{0,2}|^{2}dvol_{g}.$$ We observe that $\bar{\partial}_{A_{\infty}}F_{A}^{0,2}=0$ and \begin{equation*} \begin{split}
\|\bar{\partial}^{\ast}_{A_{\infty}}F_{A}^{0,2}\|^{2}_{L^{2}(X)}
&\leq c\|\{A-A_{\infty},F_{A}^{0,2}\}\|^{2}_{L^{2}(X)}+\|\bar{\partial}_{A}^{\ast}F_{A}^{0,2}\|^{2}_{L^{2}(X)}\\
&\leq c\|\{A-A_{\infty},F_{A}^{0,2}\}\|^{2}_{L^{2}(X)}+\frac{1}{4}\|\bar{\partial}_{A}\La_{\w}F_{A}\|^{2}_{L^{2}(X)},\\
&\leq c\|A-A_{\infty}\|^{2}_{L^{4}(X)}\|F_{A}^{0,2}\|^{2}_{L^{4}(X)}+\frac{1}{4}\|\bar{\partial}_{A}\La_{\w}F_{A}\|^{2}_{L^{2}(X)},\\
&\leq c\|A-A_{\infty}\|^{2}_{L^{2}_{1}(X)}\|F_{A}^{0,2}\|^{2}_{L^{2}(X)}+\frac{1}{4}\|\bar{\partial}_{A}\La_{\w}F_{A}\|^{2}_{L^{2}(X)},\\
&\leq c\|\La_{\w}F_{A}\|^{2}_{L^{2}(X)}\|F_{A}^{0,2}\|^{2}_{L^{2}(X)}+\frac{1}{4}\|\bar{\partial}_{A}\La_{\w}F_{A}\|^{2}_{L^{2}(X)}.\\ \end{split} \end{equation*} Here we use the estimates on Proposition \ref{P4} and Theorem \ref{T6} and Sobolev embedding $L^{2}_{1}\hookrightarrow L^{4}$. Combining the preceding inequalities with integrable identity (\ref{E12}) on Proposition \ref{P5}, gives \begin{equation*} \begin{split}
\frac{3}{4}\|\bar{\partial}_{A}\La_{\w}F_{A}\|^{2}_{L^{2}(X)}&= \|\na_{A}F_{A}^{0,2}\|^{2}_{L^{2}(X)}+\int_{X}2S|F_{A}^{0,2}|^{2}dvol_{g}\\
&\leq \|\na_{A_{\infty}}F^{0,2}_{A}\|^{2}_{L^{2}(X)}+\int_{X}2S|F_{A}^{0,2}|^{2}dvol_{g}+\|\{A-A_{\infty},F_{A}^{0,2}\}\|^{2}_{L^{2}(X)}\\
&\leq c\|\La_{\w}F_{A}\|^{2}_{L^{2}(X)}\|F_{A}^{0,2}\|^{2}_{L^{2}(X)}+\frac{1}{4}\|\bar{\partial}_{A}\La_{\w}F_{A}\|^{2}_{L^{2}(X)}.\\ \end{split} \end{equation*} for a positive constant $c=c(\la,g)$. Thus, we have \begin{equation}\label{E15}
\|\bar{\partial}_{A}\La_{\w}F_{A}\|^{2}_{L^{2}(X)}\leq c\|\La_{\w}F_{A}\|^{2}_{L^{2}(X)}\|F_{A}^{0,2}\|^{2}_{L^{2}(X)}. \end{equation} We apply Weitzenb\"{o}ck formula to $\La_{\w}F_{A}$, See \cite[Lemma 6.1]{DK} . $$\bar{\partial}^{\ast}_{A}\bar{\partial}_{A}\La_{\w}F_{A}=\frac{1}{2}\na_{A}^{\ast}\na_{A}\La_{\w}F_{A}+[\sqrt{-1}\La_{\w}F_{A},\La_{\w}F_{A}],$$ thus
$$\|\na_{A}\La_{\w}F_{A}\|^{2}_{L^{2}(X)}=2\|\bar{\partial}_{A}\La_{\w}F_{A}\|_{L^{2}(X)}^{2}.$$ Combining above identity with estimate (\ref{E15}) yields,
$$\|\La_{\w}F_{A}\|^{2}_{L^{2}(X)}\leq c\|\na_{A}\La_{\w}F_{A}\|^{2}_{L^{2}(X)}\leq c\|\La_{\w}F_{A}\|^{2}_{L^{2}(X)}\|F_{A}^{0,2}\|^{2}_{L^{2}(X)},$$
where $c=c(\la,g)$ is a positive constant. Provide $c\|F_{A}^{0,2}\|^{2}_{L^{2}(X)}\leq\frac{1}{2}$, thus $\La_{\w}F_{A}\equiv0$. \end{proof}
Following the Lemma \ref{L5} and the eigenvalue $\la(A)$ has a uniform positive lower bounded under the hypothesis of K\"{a}hler metric $g$ is generic and the curvature $F_{A}$ of the connection $[A]$ obeys $\|F_{A}^{+}\|_{L^{2}(X)}\leq\varepsilon$ for a small enough constant $\varepsilon$, then we have \begin{corollary}\label{C2} Let $X$ be a compact, simply-connected, K\"{a}hler surface with a K\"{a}hler metric $g$, that is $generic$ in the sense of Definition \ref{D1}, $P$ be a $SO(3)$-bundle over $X$. Suppose that the second Stiefel-Whitney class, $\w_{2}(P)\in H^{2}(X;\mathbb{Z}/2\mathbb{Z})$, is non-trivial, then there is a positive constant $\varepsilon=\varepsilon(g,P)$ with following significance. If the curvature $F_{A}$ of a Yang-Mills connection $A$ on $P$ obeying
$$\|F_{A}^{+}\|_{L^{2}(X)}\leq\varepsilon,$$ then the curvature is harmonic and $\La_{\w}F_{A}=0$. \end{corollary} \subsection{A vanishing theorem} Let $(X,\w)$ be a compact K\"{a}hler surface. Given an orthonormal coframe $\{e_{0},e_{1},e_{2},e_{3}\}$ on $X$ for which $\w=e^{01}+e^{23}$, where $e^{ij}=e^{i}\wedge e^{j}$. We define $dz^{1}=e^{0}+\rm{i}e^{1}$, $dz^{2}=e^{2}+\rm{i}e^{3}$ and $d\bar{z}^{1}=e^{0}-\rm{i}e^{1}$, $d\bar{z}^{2}=e^{2}-\rm{i}e^{3}$, so that $\w=\frac{\rm{i}}{2}(dz^{1}\wedge d\bar{z}^{1}+dz^{2}\wedge d\bar{z}^{2})$. \begin{proposition} Let $A$ be a connection on a principal $SU(2)$ or $SO(3)$ bundle over a compact K\"{a}hler surface. If the curvature $F_{A}$ of connection $A$ is harmonic, then $F_{A}^{0,2}$ has at most rank one. \end{proposition} \begin{proof} Since $\bar{\partial}_{A}^{\ast}F_{A}^{0,2}=0$, we have \begin{equation}\label{E3.7} 0=\bar{\partial}^{\ast}_{A}\bar{\partial}^{\ast}_{A}F_{A}^{0,2}=-\ast[F_{A}^{0,2}\wedge\ast F_{A}^{0,2}]. \end{equation} In an orthonormal coframe, we can written $F_{A}^{0,2}$ as $$F_{A}^{0,2}=(B_{1}+{\rm{i}}B_{2})d\bar{z}^{1}\wedge d\bar{z}^{2},$$ where $B_{1},B_{2}$ take value in Lie algebra $\mathfrak{su}(2)$ or $\mathfrak{so}(3)$. Thus $$\ast F_{A}^{0,2}=(-B_{1}+{\rm{i}}B_{2})dz^{1}\wedge dz^{2}.$$ Following Equation (\ref{E3.7}), we obtain that \begin{equation}\label{E3.8} 0=[B_{1},B_{2}]. \end{equation} Thus $F_{A}^{0,2}$ has most rank one. For the details of the calculation, one also can see \cite{Mares} Chapter 4. \end{proof} We define $\b$ as follows, if $$B=B_{1}(e^{01}+e^{23})+B_{2}(e^{02}+e^{31})+B_{3}(e^{03}+e^{12}),$$ then $$\b:=\frac{1}{2}(B_{2}-{\rm{i}}B_{3})dz^{1}\wedge dz^{2},\ \b^{\ast}:=-\frac{1}{2}(B_{2}+{\rm{i}}B_{3})d\bar{z}^{1}\wedge d\bar{z}^{2}.$$ It follows that $B:=B_{1}\w+\b-\b^{\ast}$. We define a bilinear map $$[\bullet.\bullet]:\Omega^{2,+}(X,\mathfrak{g}_{P})\otimes\Omega^{2,+}(X,\mathfrak{g}_{P})\rightarrow\Omega^{2,+}(X,\mathfrak{g}_{P})$$ by $\frac{1}{2}[\cdot,\cdot]_{\Omega^{2,+}}\otimes[\cdot,\cdot]_{\mathfrak{g}_{P}}$, see \cite{Mares} Section B.4. In a direct calculate, see \cite{Mares} Section 7.1, $$-\frac{1}{4}[B.B]=[B_{2},B_{3}](e^{01}+e^{23})+[B_{3},B_{1}](e^{02}+e^{31})+[B_{1},B_{2}](e^{03}+e^{12}).$$ \begin{proposition} Let $G$ be a compact, semisimple Lie group, $A$ be a connection on a principal $G$-bundle over a compact K\"{a}hler surface. If the curvature of the connection $A$ satisfies $\La_{\w}F_{A}=0$ and $\bar{\partial}_{A}^{\ast}F_{A}^{0,2}=0$, then $[F_{A}^{+}. F_{A}^{+}]=0$. \end{proposition} \begin{proof} We can written $F_{A}^{+}$ as $F_{A}^{+}:=F_{A}^{0,2}+F_{A}^{2,0}+\frac{1}{2}\La_{\w}F_{A}\otimes\w$. By the hypothesis of curvature and Equation (\ref{E3.8}), we then have $[F_{A}^{+}. F_{A}^{+}]=0$. \end{proof} Before the prove of vanishing theorem \ref{T2}, we should recall a useful lemma proved by Donaldson \cite{DK} Lemma 4.3.21. \begin{lemma}\label{L4} If $A$ is an irreducible $SU(2)$ or $SO(3)$ ASD connection on a bundle $P$ over a simply connected four-manifold $X$, then the restriction of $A$ to any non-empty open set in $X$ is also irreducible. \end{lemma} We recall the following simple but powerful corollary of unique continuation for ASD connections which proved in \cite{Mares} Theorem 4.2.1. For the convenience of the readers, we give a proof of this theorem. \begin{theorem}\label{T3.12} Let $X$ be a simply-connected, oriented, smooth Riemannian four-manifold, $P$ be a principal $SU(2)$ or $SO(3)$ bundle over $X$ and $A$ be an irreducible ASD connection on $P$. If $B\in\Omega^{2,+}(X,\mathfrak{g}_{P})$ satisfies $$d_{A}^{\ast}B=0\ and\ [B. B]=0,$$ then $B=0$. \end{theorem} \begin{proof}
Let $Z^{c}$ denote the complement of the zero set of $B$. By unique continuation of the elliptic equation $d_{A}^{\ast}B=0$, $Z^{c}$ is eithor empty or dense. On $Z^{c}$ write $B=f\otimes\sigma$ for $\sigma\in\Omega^{0}(Z^{c},\mathfrak{g}_{P})$ with $|\sigma|^{2}=1$ and $f\in\Omega^{2,+}(Z^{c})$. We compute $$0=d_{A}^{\ast}B=-\ast d_{A}(f\otimes\sigma)=-\ast(df\otimes\sigma+f\otimes d_{A}\sigma).$$
Taking the inner product with $\sigma$ and using the consequence of $|\sigma|^{2}=1$ that $\langle\sigma,d_{A}\sigma\rangle=0$, we get $df=0$. It follows that $f\otimes d_{A}\sigma=0$. Since $f$ is nowhere zero along $Z^{c}$, we have $d_{A}\sigma=0$ along $Z^{c}$. Therefore, $A$ is reducible along $Z^{c}$. However according to Lemma \ref{L4}, $A$ is irreducible along $Z^{c}$. This is a contradiction unless $Z^{c}$ is empty. Therefore $Z=X$, so $B=0$.
\end{proof} \begin{corollary}\label{T2} Let $X$ be a compact, simply-connected, K\"{a}hler surface, $P$ be a principal $G=SU(2)$ or $SO(3)$ bundle over $X$ and $A$ be an irreducible ASD connection on $P$. If $\phi\in\Omega^{0,2}(X,\mathfrak{g}_{P}^{\C})$ satisfies $$[\phi,\ast\phi]=0\ and\ \bar{\partial}_{A}^{\ast}\phi=0,$$ then $\phi$ vanish. \end{corollary} \begin{proof} By the hypothesis of $\phi$, it follows that $B:=\phi-\phi^{\ast}$ satisfies $[B.B]=0$ and $d_{A}^{\ast}B=0$. Following vanishing theorem \ref{T3.12}, we obtain that $B=0$, i.e., $\phi=0$. \end{proof} At first, we define a subset of $\Omega^{0,2}(X,\mathfrak{g}_{P}^{\C})$ as follow: \begin{equation}\label{E3.10} \tilde{\Omega}^{0,2}(X,\mathfrak{g}_{P}^{\C})=\{\phi\in\Omega^{0,2}(X,\mathfrak{g}_{P}^{\C}): [\phi,\ast\phi]=0\}. \end{equation} \begin{definition}\label{D3.13} The least eigenvalue of $\bar{\partial}_{A}\bar{\partial}_{A}^{\ast}$ on $L^{2}(\tilde{\Omega}^{0,2}(X,\mathfrak{g}_{P}^{\C}))$ is \begin{equation}
\mu(A):=\inf_{v\in\tilde{\Omega}^{0,2}(X,\mathfrak{g}_{P}^{\C})\backslash\{0\}}\frac{\|\bar{\partial}^{\ast}_{A}v\|^{2}}{\|v\|^{2}}. \end{equation} \end{definition} \begin{proposition}\label{P3.12} Let $X$ be a simply-connected, compact K\"{a}hler surface with a K\"{a}hler metric $g$, $P$ be a principal $SU(2)$ or $SO(3)$ bundle over $X$. If $A$ is an irreducible anti-self-dual connection on $P$, then $\mu(A)>0$. \end{proposition} \begin{proof}
If not, the eigenvalue $\mu(A)=0$. We may then choose a sequence $\{v_{i}\}_{i\in\N}\subset\tilde{\Omega}^{0,2}\backslash\{0\}$ such that $$\|\bar{\partial}^{\ast}_{A}v_{i}\|^{2}_{L^{2}(X)}\leq\mu_{i}\|v_{i}\|^{2}_{L^{2}(X)}$$ and $$\mu_{i}\rightarrow0^{+}\ as\ i\rightarrow\infty.$$
Since $[\frac{v}{\|v\|_{L^{2}}}\wedge\ast\frac{v}{\|v\|_{L^{2}}}]=0$ for $v\in\tilde{\Omega}^{0,2}\backslash\{0\}$, we then noting $\|v_{i}\|_{L^{2}(X)}=1$, $\forall i\in\mathbb{N}$. Following the Weizenb\"{o}ck formula, we have
$$\|\na_{A}v_{i}\|^{2}_{L^{2}(X)}=-\langle Sv_{i},v_{i}\rangle_{L^{2}(X)}+\|\bar{\partial}_{A}^{\ast}v_{i}\|^{2}_{L^{2}(X)}.$$ Thus
$$\|v_{i}\|^{2}_{L^{2}_{1}}\leq(C+\la_{i})<\infty,$$ where $C$ is a positive constant only dependence on the metric. Therefore, there exist a subsequence $\Xi\subset N$ such that $\{v_{i}\}_{i\in\Xi}$ weakly convergence to $v_{\infty}$ in $L^{2}_{1}$, we also have $\bar{\partial}_{A}^{\ast}v_{i}$ converge weakly in $L^{2}$ to a limit $\bar{\partial}_{A}^{\ast}v_{\infty}=0$. On the other hand, $L^{2}_{1}\hookrightarrow L^{p}$, for $2\leq p<4$, we may choose $p=2$, then \begin{equation}\nonumber \begin{split}
\|[v_{\infty}\wedge\ast v_{\infty}]\|_{L^{1}(X)}&=\|[(v_{\infty}-v_{i})\wedge \ast v_{\infty}+v_{i}\wedge\ast(v_{\infty}-v_{i}) ]\|_{L^{1}(X)}\\
&\leq\|v_{i}-v_{\infty}\|_{L^{2}(X)}(\|v\|_{L^{2}(X)}+\|v_{i}\|_{L^{2}(X)})\rightarrow0\ as\ i\rightarrow\infty,\\ \end{split} \end{equation} Hence $$[v_{\infty}\wedge\ast v_{\infty}=0],\ i.e.,\ v_{\infty}\in\tilde{\Omega}^{0,2}.$$
Therefore the corollary \ref{T2} implies that $\ker{\bar{\partial}_{A}^{\ast}}|_{\tilde{\Omega}^{0,2}(X,\mathfrak{g}_{P}^{\C})}=0$. Thus $v_{\infty}$ vanish. It's contradicting to $\|v_{\infty}\|_{L^{2}(X)}=1$. In particular, the preceding arguments shows that the $\mu(A)>0$. \end{proof} \begin{lemma}(\cite{DK} Lemma 7.2.10)\label{L3.15} There is a universal constant $C$ and for any $N\geq2$, $R>0$, a smooth radial function $\b=\b_{N,R}$ on $\mathbb{R}^{4}$, with $$0\leq\b(x)\leq1$$ $$\b(x)=\left\{ \begin{aligned}
1& &|x|\leq R/N \\
0& &|x|\geq R \end{aligned} \right.$$ and
$$\|\na\b\|_{L^{4}}+\|\na^{2}\b\|_{L^{2}}<\frac{C}{\sqrt{\log N}}.$$ Assuming $R<R_{0}$, the same holds for $\b(x-x_{0})$ on any geodesic ball $B_{R}(x_{0})\subset X$. \end{lemma} Following the idea in \cite{Feehan}, we can prove that the least eigenvalue of $\bar{\partial}_{A}\bar{\partial}_{A}^{\ast}$ on the space $\tilde{\Omega}^{0,2}(X,\mathfrak{g}_{P}^{\C})$ with respect to the connection $A$ is continuity in the sense of $L^{4}_{loc}$. \begin{proposition}\label{P3.16} Let $X$ be a compact K\"{a}hler surface, $\Sigma=\{x_{1},x_{2},\ldots,x_{L}\}\subset X$ ($L\in\N^{+}$) and $0<\rho\leq\min_{i\neq j}dist_{g}(x_{i},x_{j})$ and $U\subset X$ be the open subset give by $$U:=X\backslash\bigcup_{l=1}^{L}\bar{B}_{\rho/2}(x_{l}).$$ Let $G$ be a compact, semisimple Lie group, $A_{0}$ be a $C^{\infty}$ connection on a principal $G$-bundle $P_{0}$ over $X$ obeying the curvature bounded \begin{equation}\label{E3.9}
\|F^{+}_{A_{0}}\|_{L^{2}(X)}\leq\varepsilon \end{equation} where $\varepsilon\in(0,1)$ is a sufficiently small positive constant. Let $P$ be a principal $G$-bundle over $X$ such that there is an isomorphism of principal $G$-bundles, $u:P\upharpoonright X\backslash\Sigma\cong P_{0}\upharpoonright X\backslash\Sigma$, and identify $P\upharpoonright X\backslash\Sigma$ with $P_{0}\upharpoonright X\backslash\Sigma$ using this isomorphism. Then there are positive constants $c=c(\rho,g)\in(0,1]$, $c\in(1,\infty)$ and $\de\in(0,1]$ with the following significance. Let $A$ be a $C^{\infty}$ connection on $P$ obeying the curvature bounded (\ref{E3.9}) with constant $\varepsilon$ such that
$$\|A-A_{0}\|_{L^{4}(U)}\leq\de.$$ Then $\mu(A)$ and $\mu(A_{0})$ satisfy \begin{equation}\label{E3.2} \mu(A)\leq(1+\eta)\mu(A_{0})+c\big{(}(1+\eta)(C+\de^{2})+(1+\eta^{-1})L\rho^{2}\mu(A)\big{)}(1+\mu(A_{0})) \end{equation} and \begin{equation}\label{E3.3} \mu(A_{0})\leq(1+\eta)\mu(A)+c\big{(}(1+\eta)(C+\de^{2})+(1+\eta^{-1})L\rho^{2}\mu(A_{0})\big{)}(1+\mu(A)) \end{equation} where $\eta\in(0,\infty)$ is a positive constant. \end{proposition} \begin{proof} Assume first that $supp(v)\subset U$, write $a:=A-A_{0}$. We then have
$$\big{|}\|\bar{\partial}^{\ast}_{A}v\|^{2}-\|\bar{\partial}^{\ast}_{A_{0}}v\|^{2}\big{|}\leq 2\|a\|^{2}_{L^{4}}\|v\|^{2}_{L^{4}}.$$ On the other hand, if $supp(v)\subset\bigcup_{l=1}^{L}\bar{B}_{\rho/2}(x_{l})$, then
$$\|v\|^{2}_{L^{2}}\leq cL\rho^{2}\|v\|^{2}_{L^{4}}.$$
Let $\psi=\sum\b_{N,\rho}(x-x_{i})$ be a sum of the logarithmic cut-off functions of Lemma \ref{L3.15}, and $\tilde{\psi}=1-\psi$. We now choose $v\in\tilde{\Omega}^{0,2}$ with $\|v\|_{L^{2}(X)}=1$. At last, we observe that $$[\tilde{\psi}v\wedge\ast\tilde{\psi}v]=0,\ i.e.,\ \tilde{\psi}v\in\tilde{\Omega}^{0,2}.$$ By the definition of $\mu(A)$, we have
$$\mu(A)\|\tilde{\psi}v\|^{2}\leq\|\bar{\partial}^{\ast}_{A}(\tilde{\psi}v)\|^{2}.$$ Following the Weitzenb\"{o}ck formula for $v\in\Omega^{0,2}(X,\mathfrak{g}_{P})$, we have \begin{equation*} \begin{split}
\|\na_{A}v\|^{2}_{L^{2}(X)}&\leq C\|v\|^{2}_{L^{2}(X)}+\|\bar{\partial}_{A}^{\ast}v\|^{2}_{L^{2}(X)}+\|F^{+}_{A}\|_{L^{2}(X)}\|v\|^{2}_{L^{4}(X)}\\
&\leq C\|v\|^{2}_{L^{2}(X)}+\|\bar{\partial}_{A}^{\ast}v\|^{2}_{L^{2}(X)}+C\|F^{+}_{A}\|_{L^{2}(X)}(\|v\|^{2}_{L^{2}(X)}+\|\na_{A}v\|^{2}_{L^{2}(X)}),\\ \end{split} \end{equation*}
where $C$ is a positive constant dependence on $X,g$. Provided $C\|F^{+}_{A}\|_{L^{2}(X)}\leq\frac{1}{2}$, we then have a priori estimate for $v\in\Omega^{0,2}(X,\mathfrak{g}_{P}^{\C})$,
$$\|\na_{A}v\|^{2}_{L^{2}(X)}\leq C(\|v\|^{2}_{L^{2}(X)}+\|\bar{\partial}_{A}^{\ast}v\|^{2}_{L^{2}(X)}).$$ Combining the above observations, we have \begin{equation}\label{E3.11} \begin{split}
\mu(A)\|v\|^{2}_{L^{2}(X)}&\leq\mu(A)(\|\psi v\|^{2}_{L^{2}(X)}+\|\tilde{\psi}v\|^{2}_{L^{2}(X)}+2\langle\psi v,\tilde{\psi}v\rangle_{L^{2}(X)})\\
&\leq\mu(A)(1+\eta^{-1})\|\psi v\|^{2}_{L^{2}(X)}+\mu(A)(1+\eta)\|\tilde{\psi}v\|^{2}_{L^{2}(X)},\\ \end{split} \end{equation} where $\eta\in(0,\infty)$ is a positive constant.\\ For the first term on right-hand of (\ref{E3.11}), \begin{equation*}
\mu(A)(1+\eta^{-1})\|\psi v\|^{2}_{L^{2}(X)}\leq c(1+\eta^{-1})L\rho^{2}\mu(A)\|v\|^{2}_{L^{4}(X)}, \end{equation*} for some positive constant $c=c(g)$.\\ For the second term on right-hand of (\ref{E3.7}), \begin{equation*} \begin{split}
\mu(A)(1+\eta)\|\tilde{\psi}v\|^{2}_{L^{2}(X)})&\leq (1+\eta)\|\bar{\partial}^{\ast}_{A}(\tilde{\psi}v)\|^{2}_{L^{2}(X)}\\
&\leq(1+\eta)(\|\bar{\partial}^{\ast}_{A_{0}}(\tilde{\psi}v)\|^{2}_{L^{2}(X)}+2\|a\|^{2}_{L^{4}(U)}\|v\|^{2}_{L^{4}(X)})\\
&\leq(1+\eta)(\|\tilde{\psi}\bar{\partial}^{\ast}_{A_{0}}v\|^{2}_{L^{2}(X)}+\|(\na\tilde{\psi})v\|^{2}_{L^{2}(X)}+2\|a\|^{2}_{L^{4}(U)}\|v\|^{2}_{L^{4}(X)})\\
&\leq(1+\eta)\big{(}\|\bar{\partial}^{\ast}_{A_{0}}v\|^{2}_{L^{2}(X)}
+(\|\na\tilde{\psi}\|^{2}_{L^{4}(X)}+2\|a\|^{2}_{L^{4}(U)})\big{)}\|v\|^{2}_{L^{4}(X)}\\ \end{split} \end{equation*} Combining the preceding inequalities, \begin{equation}\nonumber \begin{split}
\mu(A)&\leq(1+\eta)\|\bar{\partial}^{\ast}_{A_{0}}v\|^{2}_{L^{2}(X)}\\
&+\big{(}(1+\eta)(\|\na\tilde{\psi}\|^{2}_{L^{4}(X)}+\|a\|^{2}_{L^{4}(U)})+c(1+\eta^{-1})L\rho^{2}\mu(A)\big{)}\|v\|^{2}_{L^{4}(X)}\\
&\leq(1+\eta)\|\bar{\partial}^{\ast}_{A_{0}}v\|^{2}_{L^{2}(X)}\\
&+c\big{(}(1+\eta)(\|\na\tilde{\psi}\|^{2}_{L^{4}(X)}+\|a\|^{2}_{L^{4}(U)})+c(1+\eta^{-1})L\rho^{2}\mu(A)\big{)}(1+\|\bar{\partial}^{\ast}_{A_{0}}v\|^{2}_{L^{2}(X)})\\ \end{split} \end{equation}
In the space $\tilde{\Omega}^{0,2}(X,\mathfrak{g}_{P}^{\C})$, we can choose a sequence $v_{\tilde{\varepsilon}}\in\tilde{\Omega}^{0,2}$, $\tilde{\varepsilon}\rightarrow0$, such that $$\|\bar{\partial}^{\ast}_{A_{0}}v_{\tilde{\varepsilon}}\|^{2}_{L^{2}(X)}\leq(\mu(A_{0})+\tilde{\varepsilon})\|v_{\tilde{\varepsilon}}\|^{2}\ and \ \|v_{\tilde{\varepsilon}}\|^{2}=1.$$ Therefore, \begin{equation}\nonumber \begin{split}
\mu(A)&\leq c(1+\eta)(\|\na\tilde{\psi}\|^{2}_{L^{4}(X)}+\|a\|^{2}_{L^{4}(U)})+c(1+\eta^{-1})L\rho^{2}\mu(A)\big{)}(1+\mu(A_{0})+\tilde{\varepsilon})\\ &+(1+\eta)(\mu(A_{0})+\tilde{\varepsilon}),\\ \end{split} \end{equation} Let $\tilde{\varepsilon}\rightarrow0^{+}$, we then have
$$\mu(A)\leq (1+\eta)\mu(A_{0})+c(1+\eta)\big{(}(\|\na\tilde{\psi}\|^{2}_{L^{4}(X)}+\|a\|^{2}_{L^{4}(U)})+(1+\eta^{-1})L\rho^{2}\mu(A)\big{)}(1+\mu(A_{0})).$$
Since $\|\na\tilde{\psi}\|^{2}_{L^{4}(X)}\leq\frac{C'}{\log N}$ for a uniform constant $C'$, we denote $C=\frac{C'}{\log N}$, (see Lemma \ref{L3.15}), we then have $$\mu(A)\leq(1+\eta)\mu(A_{0})+c\big{(}(1+\eta)(C+\de^{2})+(1+\eta^{-1})L\rho^{2}\mu(A)\big{)}(1+\mu(A_{0}))$$ Therefore, exchange the roles of $A$ and $A_{0}$ in the preceding derivation yields the inequality (\ref{E3.3}) for $\mu(A)$ and $\mu(A_{0})$. \end{proof}
We then have the convergence of the least eigenvalue of $\bar{\partial}_{A_{i}}\bar{\partial}^{\ast}_{A_{i}}|_{\tilde{\Omega}^{0,2}}$ for a sequence of connections $\{A_{i}\}_{i\in\mathbb{N}}$ converging strongly in $L^{2}_{1,loc}(X\backslash\Sigma)$. \begin{corollary}\label{C3.9} Let $G$ be a compact, semisimple Lie group and $P$ be a principal $G$-bundle over a compact K\"{a}hler surface $X$ and $\{A_{i}\}_{i\in\mathbb{N}}$ a sequence of smooth connections on $P$ that converges strongly in $L^{2}_{1,loc}(X\backslash\Sigma)$, moduli a sequence $\{u_{i}\}_{i\in\mathbb{N}}: P_{\infty}\upharpoonright X\backslash\Sigma\cong P\upharpoonright X\backslash\Sigma$ of class $L^{3}_{1,loc}(X\backslash\Sigma)$ to a connection $A_{\infty}$ on a principal $G$-bundle $P_{\infty}$ over $X$. Then $$\lim_{i\rightarrow\infty}\mu(A_{i})=\mu(A_{\infty}),$$ where $\mu(A)$ is as in Definition \ref{D3.13}. \end{corollary} \begin{proof} By the Sobolev embedding $L^{2}_{1}\hookrightarrow L^{4}$ and Kato inequality, we have
$$\|u^{\ast}_{i}(A_{i})-A_{\infty}\|_{L^{4}(U)}\rightarrow0\ strongly\ in\ L^{2}_{1,A_{\infty}}(U,\Omega^{1}\otimes \mathfrak{g}_{P_{\infty}})\ as\ i\rightarrow\infty.$$ Hence from the inequalities on Proposition \ref{P3.16}, we have \begin{equation}\label{E3.5} \mu(A_{\infty})\leq(1+\eta)\liminf_{i\rightarrow\infty}\mu(A_{i})+c\big{(}(1+\eta)C+(1+\eta^{-1})L\rho^{2}\mu(A_{\infty})\big{)} (1+\liminf_{i\rightarrow\infty}\mu(A_{i})) \end{equation} and \begin{equation}\label{E3.6} \limsup_{i\rightarrow\infty}\mu(A_{i})\leq(1+\eta)\mu(A_{\infty})+c\big{(}(1+\eta)C+(1+\eta^{-1})L\rho^{2}\limsup_{i\rightarrow\infty}\mu(A_{i})\big{)}(1+\mu(A_{\infty})) \end{equation} The inequalities (\ref{E3.5}) and (\ref{E3.6}) about $\liminf_{i\rightarrow\infty}\mu(A_{i})$ and $\limsup_{i\rightarrow\infty}\mu(A_{i})$ hold for every $\rho\in(0,\rho_{0}]$ and $\eta\in(0,\infty)$. It's easy to see that $C\rightarrow0^{+}$ while $\rho\rightarrow0^{+}$. At first, let $\rho\rightarrow0^{+}$, we then have $$\mu(A_{\infty})\leq(1+\eta)\liminf_{i\rightarrow\infty}\mu(A_{i}) \leq(1+\eta)\limsup_{i\rightarrow\infty}\mu(A_{i})\leq(1+\eta)^{2}\mu(A_{\infty}).$$ Next, let $\eta\rightarrow0^{+}$, thus the conclusion follows. \end{proof} We then have \begin{proposition}
Let $G$ be a compact, semisimple Lie group and $P$ a principal $G$-bundle over a compact K\"{a}hler surface $X$. If $\{A_{i}\}_{i\in\mathbb{N}}$ is a sequence $C^{\infty}$ connection on $P$ and the curvatures $\|F^{+}_{A_{i}}\|_{L^{2}(X)}$ converge to zero as $i\rightarrow\infty$, then there are a finite set of points, $\Sigma=\{x_{1},\ldots,x_{L}\}\subset X$ and a subsequence $\Xi\subset\N$, we also denote by $\{A_{i}\}$, a sequence gauge transformation $\{g_{i}\}_{i\in\Xi}$ such that, $g_{i}(A_{i})\rightarrow A_{\infty}$, a anti-self-dual connection on $P\upharpoonright_{X\setminus\Sigma}$ in the $L^{2}_{1}$-topology over $X\setminus\Sigma$ . Furthermore, we have $$\lim_{i\rightarrow\infty}\mu(A_{i})=\mu(A_{\infty}),$$ where $\mu(A)$ is as in Definition \ref{D3.13}. \end{proposition} Following Feehan's idea, we then have \begin{theorem}\label{T3.14} Let $X$ be a compact, simply-connected, K\"{a}hler surface with a K\"{a}hler metric $g$, $P$ be a principal $G$-bundle with $G$ being $SU(2)$ or $SO(3)$. Suppose that the ASD connections $A\in M(P,g)$ are $strong$ irreducible connections in the sense of Definition \ref{D3}. Then there are positive constants $\mu_{0}=\mu_{0}(P,g)$ and $\varepsilon=\varepsilon(P,g)$ such that \begin{equation*} \begin{split} &\mu(A)\geq\mu_{0},\ \forall [A]\in M(P,g),\\ &\mu(A)\geq\frac{\mu_{0}}{2},\ \forall [A]\in\mathcal{B}_{\varepsilon}(P,g).\\ \end{split} \end{equation*} \end{theorem} \begin{proof} Combining $\bar{M}(P,g)$ is compact, $\la(A)$ is continuous under the Uhlenbeck topology, $\forall [A]\in\bar{M}(P,g)$ and $A$ is a $strong$ irreducible connection, then any connection $A\in\bar{M}(P,g)$ is $strong$ in the sense of Definition \ref{D3}. Following the Proposition \ref{P3.12} , we then have $\mu(A)\geq\mu_{0}$.
Suppose that the constant $\varepsilon\in(0,1]$ does not exist. We may then choose a sequence $\{A_{i}\}_{i\in\N}$ of connection on $P$ such that $\|F^{+}_{A_{i}}\|_{L^{2}(X)}\rightarrow0$ and $\mu(A_{i})\rightarrow0$ as $i\rightarrow\infty$. Since $\lim_{i\rightarrow\infty}\mu(A_{i})=\mu(A_{\infty})$ and $\mu(A_{\infty})\geq\mu_{0}$, then it is contradict to our initial assumption regarding the sequence $\{A_{i}\}_{i\in\N}$. In particular, the preceding argument shows that the desired constant $\varepsilon$ exists. \end{proof} \begin{corollary} Let $X$ be a compact, simply-connected, K\"{a}hler surface with a K\"{a}hler metric $g$, that is $generic$ in the sense of Definition \ref{D1}, $P$ be a $SO(3)$-bundle over $X$. If the second Stiefel-Whitney class, $\w_{2}(P)\in H^{2}(X;\mathbb{Z}/2\mathbb{Z})$, is non-trivial, then there are positive constants $\mu_{0}=\mu_{0}(P,g)$ and $\varepsilon=\varepsilon(P,g)$ such that \begin{equation*} \begin{split} &\mu(A)\geq\mu_{0},\ \forall [A]\in M(P,g),\\ &\mu(A)\geq\frac{\mu_{0}}{2},\ \forall [A]\in\mathcal{B}_{\varepsilon}(P,g).\\ \end{split} \end{equation*} \end{corollary} \begin{proof}[\textbf{Proof of Theorem \ref{T1}}]
For a Yang-Mills connection $A$ on $P$ with $\|F_{A}^{+}\|_{L^{2}(X)}\leq\varepsilon$, where $\varepsilon\in(0,1)$ is as in the hypothesis of Corollary \ref{C2}, then the curvature $F_{A}$ satisfies $\bar{\partial}_{A}^{\ast}F_{A}^{0,2}=0$, $\La_{\w}F_{A}=0$. Following the Definition of $\mu(A)$ and Theorem \ref{T3.14}, we have
$$\frac{\mu_{0}}{2}\|v\|^{2}_{L^{2}(X)}\leq\|\bar{\partial}_{A}^{\ast}v\|^{2}_{L^{2}(X)},\ \forall v\in\tilde{\Omega}^{0,2}(X,\mathfrak{g}_{P}^{\C}).$$ where $\mu_{0}$ is the uniform positive lower bound in Theorem \ref{T3.14}. Since $F_{A}^{0,2}$ is harmonic, $F_{A}^{0,2}=0$ on $X$. Thus we complete this proof. \end{proof} Suppose $\phi\in\Omega^{0,2}(X,\mathfrak{g}^{\C}_{P})$ takes values in a 1-dimensional subbundle of $\mathfrak{g}_{P}^{\C}$, i.e., suppose that $$\phi=f\otimes\sigma,$$
where $f$ is a $(0,2)$-form and where $\sigma$ is a section of $\mathfrak{g}_{P}^{\C}$ with $|\sigma|^{2}=1$, following the idea of Bourguignon-Lawson \cite[Proposition 3.15]{BL}, we then have a useful \begin{lemma}\label{L2} Suppose that the curvature $F_{A}$ of the connection $A$ obeying $\La_{\w}F_{A}=0$. The $\phi$ is harmonic with respect to Laplacian operator $\De_{\bar{\partial}_{A}}$ if only if $f$ is harmonic form and $\sigma$ is parallel aways from the zeros of $f$. \end{lemma} \begin{proof}
The Weizenb\"{o}ck formula for any $\phi\in\Omega^{0,2}(X,\mathfrak{g}^{\C}_{P})$ yields, see (\ref{E14})
$$\na_{A}^{\ast}\na_{A}\phi+ 2S\phi=0.$$
A direct computation shows that
$$\na_{A}^{\ast}\na_{A}\phi=(\na^{\ast}\na f)\otimes\sigma-\sum_{j}(\na_{e_{j}}f)\otimes(\na_{A_{e_{j}}}\sigma)+f\otimes(\na_{A}^{\ast}\na_{A}\sigma),$$
and
$$\langle S\phi,\phi\rangle=\langle Sf,f\rangle.$$
Taking the derivative of the condition $|\sigma|^{2}=1$, we find that $\langle\na_{A}\sigma,\sigma\rangle=0$. Consequently,
$$\langle\na_{A}^{\ast}\na_{A}\sigma,\sigma\rangle\equiv-\langle\na_{A_{e_{i},e_{i}}}^{2}\sigma,\sigma\rangle\equiv\sum_{i}\langle\na_{A_{e_{i}}}\sigma,\na_{A_{e_{i}}}\sigma\rangle\equiv|\na_{A}\sigma|^{2}.$$
We then have
\begin{equation}\label{E11}
\langle\na_{A}^{\ast}\na_{A}\phi,\phi\rangle=\langle\na^{\ast}\na f,f\rangle+|f|^{2}|\na_{A}\sigma|^{2}.
\end{equation}
The Weitzenb\"{o}ck formula applied to $(0,2)$-forms on $\Omega^{0,2}(X)$ , states that
$$(d^{\ast}d+dd^{\ast})f=\na^{\ast}\na f+2Sf.$$
Therefore (\ref{E11}) can be rewritten as
$$\langle\De f,f \rangle+|f|^{2}|\na_{A}\sigma|^{2}=0.$$
Since $\De\geq 0$ on $X$ we conclude that $\De f=0$ and that $\na_{A}\sigma=0$ away from the zero of $f$. We complete the proof this lemma. \end{proof} We apply the proof of Lemma \ref{L4} to ASD connections for group $S^{1}$. If $A$ is an ASD $S^{1}$-connection which is flat in the some ball, then in a radial gauge the connection matrix vanishes over the ball and we deduce that $A$ must be flat everywhere. This is a local argument, so applies to any closed ASD $2$-form. Of course, we have just the same results for self-dual forms. We obtain then: \begin{corollary}\label{C3}(\cite{DK} Corollary 4.3.23) Suppose $\w$ is a closed two-form on $X$ which satisfies $\ast\w=\pm\w$. Then if $\w$ vanishes on a non-empty open set in $X$ it is identically zero. \end{corollary} Furthermore, if $\w$ is a harmonic two from on $X$, then $\w+\ast\w$ or $\w-\ast\w$ is self-dual or ASD closed two-from. Then if $\w\pm\ast\w$ all vanish on a non-empty open set in $X$ it is identically zero, i.e., $\w$ is identically zero. \begin{proposition}\label{P3.24} Suppose that $\w$ is a smooth harmonic $2$-form on a closed, simply-connected, four-manifold $X$. Then if $\w$ vanishes on a non-empty open set in $X$ it is identically zero. \end{proposition} We then have \begin{theorem} Let $X$ be a compact, simply-connected, K\"{a}hler surface, $P$ be a principal $G$-bundle over $X$ with $G$ being a compact, semisimple Lie group, $A$ be an irreducible connection on $P$. If the curvature $F_{A}$ of the connection $A$ obeying $\La_{\w}F_{A}=0$. Then the harmonic forms take values in a 1-dimensional subbundle of $\mathfrak{g}_{P}^{\C}$ on $\Omega^{0,2}(X,\mathfrak{g}_{P}^{\C})$ with respect to Laplacian operator $\De_{\bar{\partial}_{A}}$ are zero. \end{theorem} \begin{proof}
We set $\phi:=f\otimes\sigma$ for any section $\phi$ on $\Omega^{0,2}(X,\mathfrak{g}_{P}^{\C})$. We suppose $\phi$ is harmonic with respect to $\De_{\bar{\partial}_{A}}$, then $|f||\na_{A}\sigma|=0$ and $df=d^{\ast}f=0$. We denote a closed set $$Z:=\{x\in X:f(x)=0\}\subset X$$ by the the zero of harmonic form $f$, i.e. the zero of $\phi$. We then have $\na_{A}\sigma=0$ along $X\backslash Z$, thus the set $$\tilde{Z}:=\{x\in X:\na_{A}\sigma\neq 0\}\subset Z.$$
Since the connection $A$ is irreducible, $\|\na_{A}\sigma\|_{L^{2}(X)}>0$, (See Remark \ref{R2.5}), thus the set $\tilde{Z}$ is non-empty. We could choose a point $p\in\tilde{Z}$ such that $\na_{A}\sigma\neq0$. Then there is a constant $\rho$ such that the geodesic ball $B_{\rho}(p)\subset\tilde{Z}\subset Z$. Since $f$ is harmonic $(0,2)$-from and $f(x)=0$ for any $x\in B_{\rho}(p)$, $\ast f$ is a harmonic $(2,0)$-from and $\ast f$ also vanishes on $B_{\rho}(p)$. Following Proposition \ref{P3.24}, $f$ is identically zero. Therefore $Z=X$, so $\phi$ is identically zero. \end{proof} \section*{Acknowledgment} We would like to thank Feehan for kind comments regarding his article \cite{Feehan}. This work was partially supported by Nature Science Foundation of China No. 11801539.
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\end{document} | arXiv |
Infrastructure (number theory)
In mathematics, an infrastructure is a group-like structure appearing in global fields.
Historic development
In 1972, D. Shanks first discovered the infrastructure of a real quadratic number field and applied his baby-step giant-step algorithm to compute the regulator of such a field in ${\mathcal {O}}(D^{1/4+\varepsilon })$ binary operations (for every $\varepsilon >0$), where $D$ is the discriminant of the quadratic field; previous methods required ${\mathcal {O}}(D^{1/2+\varepsilon })$ binary operations.[1] Ten years later, H. W. Lenstra published[2] a mathematical framework describing the infrastructure of a real quadratic number field in terms of "circular groups". It was also described by R. Schoof[3] and H. C. Williams,[4] and later extended by H. C. Williams, G. W. Dueck and B. K. Schmid to certain cubic number fields of unit rank one[5][6] and by J. Buchmann and H. C. Williams to all number fields of unit rank one.[7] In his habilitation thesis, J. Buchmann presented a baby-step giant-step algorithm to compute the regulator of a number field of arbitrary unit rank.[8] The first description of infrastructures in number fields of arbitrary unit rank was given by R. Schoof using Arakelov divisors in 2008.[9]
The infrastructure was also described for other global fields, namely for algebraic function fields over finite fields. This was done first by A. Stein and H. G. Zimmer in the case of real hyperelliptic function fields.[10] It was extended to certain cubic function fields of unit rank one by R. Scheidler and A. Stein.[11][12] In 1999, S. Paulus and H.-G. Rück related the infrastructure of a real quadratic function field to the divisor class group.[13] This connection can be generalized to arbitrary function fields and, combining with R. Schoof's results, to all global fields.[14]
One-dimensional case
Abstract definition
A one-dimensional (abstract) infrastructure $(X,d)$ consists of a real number $R>0$, a finite set $X\neq \emptyset $ together with an injective map $d:X\to \mathbb {R} /R\mathbb {Z} $.[15] The map $d$ is often called the distance map.
By interpreting $\mathbb {R} /R\mathbb {Z} $ as a circle of circumference $R$ and by identifying $X$ with $d(X)$, one can see a one-dimensional infrastructure as a circle with a finite set of points on it.
Baby steps
A baby step is a unary operation $bs:X\to X$ on a one-dimensional infrastructure $(X,d)$. Visualizing the infrastructure as a circle, a baby step assigns each point of $d(X)$ the next one. Formally, one can define this by assigning to $x\in X$ the real number $f_{x}:=\inf\{f'>0\mid d(x)+f'\in d(X)\}$; then, one can define $bs(x):=d^{-1}(d(x)+f_{x})$.
Giant steps and reduction maps
Observing that $\mathbb {R} /R\mathbb {Z} $ is naturally an abelian group, one can consider the sum $d(x)+d(y)\in \mathbb {R} /R\mathbb {Z} $ for $x,y\in X$. In general, this is not an element of $d(X)$. But instead, one can take an element of $d(X)$ which lies nearby. To formalize this concept, assume that there is a map $red:\mathbb {R} /R\mathbb {Z} \to X$; then, one can define $gs(x,y):=red(d(x)+d(y))$ to obtain a binary operation $gs:X\times X\to X$, called the giant step operation. Note that this operation is in general not associative.
The main difficulty is how to choose the map $red$. Assuming that one wants to have the condition $red\circ d=\mathrm {id} _{X}$, a range of possibilities remain. One possible choice[15] is given as follows: for $v\in \mathbb {R} /R\mathbb {Z} $, define $f_{v}:=\inf\{f\geq 0\mid v-f\in d(X)\}$; then one can define $red(v):=d^{-1}(v-f_{v})$. This choice, seeming somewhat arbitrary, appears in a natural way when one tries to obtain infrastructures from global fields.[14] Other choices are possible as well, for example choosing an element $x\in d(X)$ such that $|d(x)-v|$ is minimal (here, $|d(x)-v|$ is stands for $\inf\{|f-v|\mid f\in d(x)\}$, as $d(x)$ is of the form $v+R\mathbb {Z} $); one possible construction in the case of real quadratic hyperelliptic function fields is given by S. D. Galbraith, M. Harrison and D. J. Mireles Morales.[16]
Relation to real quadratic fields
D. Shanks observed the infrastructure in real quadratic number fields when he was looking at cycles of reduced binary quadratic forms. Note that there is a close relation between reducing binary quadratic forms and continued fraction expansion; one step in the continued fraction expansion of a certain quadratic irrationality gives a unary operation on the set of reduced forms, which cycles through all reduced forms in one equivalence class. Arranging all these reduced forms in a cycle, Shanks noticed that one can quickly jump to reduced forms further away from the beginning of the circle by composing two such forms and reducing the result. He called this binary operation on the set of reduced forms a giant step, and the operation to go to the next reduced form in the cycle a baby step.
Relation to $\mathbb {R} /R\mathbb {Z} $
The set $\mathbb {R} /R\mathbb {Z} $ has a natural group operation and the giant step operation is defined in terms of it. Hence, it makes sense to compare the arithmetic in the infrastructure to the arithmetic in $\mathbb {R} /R\mathbb {Z} $. It turns out that the group operation of $\mathbb {R} /R\mathbb {Z} $ can be described using giant steps and baby steps, by representing elements of $\mathbb {R} /R\mathbb {Z} $ by elements of $X$ together with a relatively small real number; this has been first described by D. Hühnlein and S. Paulus[17] and by M. J. Jacobson, Jr., R. Scheidler and H. C. Williams[18] in the case of infrastructures obtained from real quadratic number fields. They used floating point numbers to represent the real numbers, and called these representations CRIAD-representations resp. $(f,p)$-representations. More generally, one can define a similar concept for all one-dimensional infrastructures; these are sometimes called $f$-representations.[15]
A set of $f$-representations is a subset $fRep$ of $X\times \mathbb {R} /R\mathbb {Z} $ such that the map $\Psi _{fRep}:fRep\to \mathbb {R} /R\mathbb {Z} ,\;(x,f)\mapsto d(x)+f$ is a bijection and that $(x,0)\in fRep$ for every $x\in X$. If $red:\mathbb {R} /R\mathbb {Z} \to X$ is a reduction map, $fRep_{red}:=\{(x,f)\in X\times \mathbb {R} /R\mathbb {Z} \mid red(d(x)+f)=x\}$ is a set of $f$-representations; conversely, if $fRep$ is a set of $f$-representations, one can obtain a reduction map by setting $red(f)=\pi _{1}(\Psi _{fRep}^{-1}(f))$, where $\pi _{1}:X\times \mathbb {R} /R\mathbb {Z} \to X,\;(x,f)\mapsto x$ is the projection on $X$. Hence, sets of $f$-representations and reduction maps are in a one-to-one correspondence.
Using the bijection $\Psi _{fRep}:fRep\to \mathbb {R} /R\mathbb {Z} $, one can pull over the group operation on $\mathbb {R} /R\mathbb {Z} $ to $fRep$, hence turning $fRep$ into an abelian group $(fRep,+)$ by $x+y:=\Psi _{fRep}^{-1}(\Psi _{fRep}(x)+\Psi _{fRep}(y))$, $x,y\in fRep$. In certain cases, this group operation can be explicitly described without using $\Psi _{fRep}$ and $d$.
In case one uses the reduction map $red:\mathbb {R} /R\mathbb {Z} \to X,\;v\mapsto d^{-1}(v-\inf\{f\geq 0\mid v-f\in d(X)\})$, one obtains $fRep_{red}=\{(x,f)\mid f\geq 0,\;\forall f'\in [0,f):d(x)+f'\not \in d(X)\}$. Given $(x,f),(x',f')\in fRep_{red}$, one can consider $(x'',f'')$ with $x''=gs(x,x')$ and $f''=f+f'+(d(x)+d(x')-d(gs(x,x')))\geq 0$; this is in general no element of $fRep_{red}$, but one can reduce it as follows: one computes $bs^{-1}(x'')$ and $f''-(d(x'')-d(bs^{-1}(x'')))$; in case the latter is not negative, one replaces $(x'',f'')$ with $(bs^{-1}(x''),f''-(d(x'')-d(bs^{-1}(x''))))$ and continues. If the value was negative, one has that $(x'',f'')\in fRep_{red}$ and that $\Psi _{fRep_{red}}(x,f)+\Psi _{fRep_{red}}(x',f')=\Psi _{fRep_{red}}(x'',f'')$, i.e. $(x,f)+(x',f')=(x'',f'')$.
References
1. D. Shanks: The infrastructure of a real quadratic field and its applications. Proceedings of the Number Theory Conference (Univ. Colorado, Boulder, Colo., 1972), pp. 217-224. University of Colorado, Boulder, 1972. MR389842
2. H. W. Lenstra Jr.: On the calculation of regulators and class numbers of quadratic fields. Number theory days, 1980 (Exeter, 1980), 123–150, London Math. Soc. Lecture Note Ser., 56, Cambridge University Press, Cambridge, 1982. MR697260
3. R. J. Schoof: Quadratic fields and factorization. Computational methods in number theory, Part II, 235–286, Math. Centre Tracts, 155, Math. Centrum, Amsterdam, 1982. MR702519
4. H. C. Williams: Continued fractions and number-theoretic computations. Number theory (Winnipeg, Man., 1983). Rocky Mountain J. Math. 15 (1985), no. 2, 621–655. MR823273
5. H. C. Williams, G. W. Dueck, B. K. Schmid: A rapid method of evaluating the regulator and class number of a pure cubic field. Math. Comp. 41 (1983), no. 163, 235–286. MR701638
6. G. W. Dueck, H. C. Williams: Computation of the class number and class group of a complex cubic field. Math. Comp. 45 (1985), no. 171, 223–231. MR790655
7. J. Buchmann, H. C. Williams: On the infrastructure of the principal ideal class of an algebraic number field of unit rank one. Math. Comp. 50 (1988), no. 182, 569–579. MR929554
8. J. Buchmann: Zur Komplexität der Berechnung von Einheiten und Klassenzahlen algebraischer Zahlkörper. Habilitationsschrift, Düsseldorf, 1987. PDF
9. R. Schoof: Computing Arakelov class groups. (English summary) Algorithmic number theory: lattices, number fields, curves and cryptography, 447–495, Math. Sci. Res. Inst. Publ., 44, Cambridge University Press, 2008. MR2467554 PDF
10. A. Stein, H. G. Zimmer: An algorithm for determining the regulator and the fundamental unit of hyperelliptic congruence function field. In "Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, ISSAC '91," Association for Computing Machinery, (1991), 183–184.
11. R. Scheidler, A. Stein: Unit computation in purely cubic function fields of unit rank 1. (English summary) Algorithmic number theory (Portland, OR, 1998), 592–606, Lecture Notes in Comput. Sci., 1423, Springer, Berlin, 1998. MR1726104
12. R. Scheidler: Ideal arithmetic and infrastructure in purely cubic function fields. (English, French summary) J. Théor. Nombres Bordeaux 13 (2001), no. 2, 609–631. MR1879675
13. S. Paulus, H.-G. Rück: Real and imaginary quadratic representations of hyperelliptic function fields. (English summary) Math. Comp. 68 (1999), no. 227, 1233–1241. MR1627817
14. Fontein, F. (2011). "The Infrastructure of a Global Field of Arbitrary Unit Rank". Math. Comp. 80 (276): 2325–2357. arXiv:0809.1685. doi:10.1090/S0025-5718-2011-02490-7. S2CID 14352393.
15. F. Fontein: Groups from cyclic infrastructures and Pohlig-Hellman in certain infrastructures. (English summary) Adv. Math. Commun. 2 (2008), no. 3, 293–307. MR2429459
16. S. D. Galbraith, M. Harrison, D. J. Mireles Morales: Efficient hyperelliptic arithmetic using balanced representation for divisors. (English summary) Algorithmic number theory, 342–356, Lecture Notes in Comput. Sci., 5011, Springer, Berlin, 2008. MR2467851
17. D. Hühnlein, S. Paulus: On the implementation of cryptosystems based on real quadratic number fields (extended abstract). Selected areas in cryptography (Waterloo, ON, 2000), 288–302, Lecture Notes in Comput. Sci., 2012, Springer, 2001. MR1895598
18. M. J. Jacobson Jr., R. Scheidler, H. C. Williams: The efficiency and security of a real quadratic field based key exchange protocol. Public-key cryptography and computational number theory (Warsaw, 2000), 89–112, de Gruyter, Berlin, 2001 MR1881630
| Wikipedia |
Einstein field equations to the Alcubierre metric
I was wondering how Alcubierre derived the metric for the warp drive? Sources have said it's based on Einstein's field equations, but how did he go from this to the metric?
general-relativity metric-tensor tensor-calculus warp-drives
Qmechanic♦
111k1212 gold badges214214 silver badges13091309 bronze badges
Lord Farquaad - ELord Farquaad - E
$\begingroup$ You can read the gory details on Arxiv where the paper is archived. $\endgroup$ – StephenG May 15 '18 at 10:33
Alcubierre started with the metric and used the Einstein equation to calculate what stress energy tensor was required.
The Einstein equation tells us:
$$ R_{\mu\nu} - \tfrac{1}{2}R g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu} $$
Normally we start with a known stress-energy tensor $T_{\mu\nu}$ and we're trying to solve the equation to find the metric. This is in general exceedingly hard. However if you start with a metric it's easy to calculate the Ricci tensor and scalar so the left hand side of the equation is easy to calculate, and therefore the matching stress-energy tensor is easy to calculate.
The only trouble is that doing things this way round will usually produce an unphysicial stress-energy tensor e.g. one that involves exotic matter. And indeed this is exactly what happens for the Alcubierre metric - it requires a ring of exotic matter.
John RennieJohn Rennie
The metric for the Alcubierre warp drive was constructed by considering the properties that it should obey, and not the matter source (which is why it's fairly unphysical).
The two ingredients used in it are :
A bump function, so that the warp drive is localized in a specific region (and that bump function moves, so that the inside may move along with it)
A widening of the lightcone in that bump function, so that, compared to the outside, the speed of light is "larger".
Given these two characteristics, we get the properties we want for a warp bubble. It is possible to also get variants by changing them, for instance the Krasnikov tunnel does not have a travelling bump function, but still has a widening of the light cone. This is why it is "static", compared to the warp drive.
SlereahSlereah
Not the answer you're looking for? Browse other questions tagged general-relativity metric-tensor tensor-calculus warp-drives or ask your own question.
Harold White's work on the Alcubierre warp drive
Alcubierre warp bubble effect on gravity and space
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Space Warps & Frame Invariance
Can the Alcubierre Drive be explained by Gravitoelectromagnetism?
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How does a world line of an Alcubierre drive look like?
Einstein field equations and SEM tensor + Alcubierre
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Derivation for Alcubierre Drive | CommonCrawl |
\begin{document}
\title[Left-invariant metrics on six-dimensional nilpotent Lie groups]{The moduli space of left-invariant metrics of a class of
six-dimensional nilpotent Lie groups}
\dedicatory{Dedicated to Carlos Olmos on the occasion of his 60th
birthday}
\author{Silvio Reggiani} \address{CONICET and Universidad Nacional de Rosario, ECEN-FCEIA,
Departamento de Ma\-te\-má\-ti\-ca. Av. Pellegrini 250, 2000
Rosario, Argentina.} \email{\href{mailto:[email protected]}{[email protected]}} \urladdr{\url{http://www.fceia.unr.edu.ar/~reggiani}}
\author{Francisco Vittone} \address{CONICET and Universidad Nacional de Rosario, ECEN-FCEIA,
Departamento de Ma\-te\-má\-ti\-ca. Av. Pellegrini 250, 2000
Rosario, Argentina.} \email{\href{mailto:[email protected]}{[email protected]}} \urladdr{\url{http://www.fceia.unr.edu.ar/~vittone}}
\date{\today}
\thanks{Supported by CONICET. Partially supported by SeCyT-UNR}
\keywords{Left-invariant metrics, Nilpotent Lie groups, Complex
structures, Abelian structures, Hermitian structures}
\subjclass[2010]{53C30, 53B35}
\maketitle
\begin{abstract}
In this paper we determine the moduli space, up to isometric
automorphism, of left-invariant metrics on a $6$-dimensional Lie
group $H$, such that its Lie algebra $\mathfrak{h}$ admits a complex
structure and has first Betti number equal to four. We also
investigate which of these metrics are Hermitian and classify the
corresponding complex structures. \end{abstract}
\section{Introduction}
The present work concerns the study of the moduli space of left-invariant metrics on nilpotent Lie groups up to diffeomorphism. As it was proved by Wolf in \cite{wolf-1963}, and later generalized in \cite{alekseevskii-1971,gordon-wilson-1988} for Riemannian solvmanifolds, this is equivalent to the study of the moduli space of left-invariant metrics up to isometric automorphism. Notice that by Mal'cev criterion, every compact nilmanifold is the quotient of a simply connected nilpotent Lie group by a discrete subgroup. So, the problem we approach is closely related with the problem of determining the moduli space of invariant metrics on compact nilmanifolds, up to diffeomorphism. There is some previous work addressing the problem of determining the moduli space of left-invariant metrics on Lie groups. One can mention the work of Ha-Lee \cite{ha-lee-2009} which solves the problem in dimension 3. In \cite{lauret-2003}, Lauret classifies all Lie groups with only one left-invariant metric up to isometry and scaling. Later, in \cite{kodama-takahara-tamaru-2011}, Kodama, Takahara and Tamaru give another proof of Lauret's theorem and also study the case when the moduli space up to isometry and scaling has dimension $1$. In \cite{Di_Scala_2012}, Di Scala classifies the moduli space of left-invariant metrics up to isometric isomorphism on the Iwasawa manifold. For semi-definite metrics, some progress has also been made, for instance Vukmirović classifies in \cite{vukmirovic-2015} the pseudo-Riemannian left-metrics metrics on Heisenberg groups. In a recent paper \cite{kondo2020classification}, Kondo and Tamaru determine the moduli space up to isometry and scaling of Lorentzian left-invariant metrics on certain nilpotent Lie groups. Some of the above results use the so-called Milnor frames, a concept that dates back to the famous paper of Milnor \cite{milnor-1976}, for describing the Lie bracket on $3$-dimensional Lie groups in terms of an orthonomal basis. In \cite{hashinaga2016}, it is showed a closed retationship between the moduli space of left-invariant metrics and Milnor frames by giving several examples of Milnor-type theorems.
Geometric structures associated to low-dimensional Lie groups with left-invariant metrics have been widely studied. For the case of $6$-dimensional nilpotent Lie groups, particular attention has been paid to the Iwasawa manifold $\mathcal{I}=\Gamma\backslash H$, which is a compact quotient of the $3$-dimensional complex Heisenberg group $H$. The Hermitian geometry of $\mathcal{I}$, with a standard metric, was studied in \cite{abbena-1997,abbena-2001} and \cite{Ketsetzis_2004}. The classification of Di Scala \cite{Di_Scala_2012}, of the moduli space of metrics up to isometric automorphism, relies on fixing a distinguished complex structure on the Lie algebra $\mathfrak{h}$ of $H$, which allows to determine the automorphism group $\Aut(\mathfrak{h})$ in an elegant way on the canonical basis.
In \cite{salamon-2001}, Salamon classified all $6$-dimensional Lie algebras $\mathfrak{g}$ which admit a complex structure. Such Lie algebras are grouped according to the first Betti number of $\mathfrak{g}$. In particular, $3$-dimensional complex Heisenberg Lie algebra belongs to the class whose first Betti number is equal to $4$. This class contains five Lie algebras that are characterized by the property that $\textrm{dim}[\mathfrak{g},\mathfrak{g}]=2$. In the notation of \cite{salamon-2001}, they are $\mathfrak{h}_2, \mathfrak{h}_4, \mathfrak{h}_5, \mathfrak{h}_6$ and $\mathfrak{h}_9$ (see Section \ref{nildim6}). The complex Heisenberg Lie algebra is $\mathfrak{h}_5$. All these Lie algebras are $2$-step nilpotent, with the exception of $\mathfrak{h}_9$ which is $3$-step nilpotent.
As a natural continuation of the work in \cite{Di_Scala_2012}, in this paper we deal with $6$-dimensional, simply connected, nilpotent Lie groups $H$ which admit a left-invariant complex structure and their Lie algebras $\mathfrak{h}$ have first Betti number equal to $4$. Our main goal is to classify the moduli space of left-invariant metrics, up to isometric automorphism, for this particular family. Left-invariant metrics on $H$ are in a one-to-one correspondence with the inner products on $\mathfrak{h}$. Moreover, $\Aut(H)$ is isomorphic to $\Aut(\mathfrak{h})$ and the classification of left-invariant metrics on $H$ up to automorphism reduces to the classification of inner products of $\mathfrak{h}$, up to an automorphism of $\mathfrak{h}$.
It is important to observe that the methods developed in \cite{Di_Scala_2012} cannot be directly adapted to any of the other Lie algebras studied here. In fact, while $\Aut(\mathfrak{h}_5)$ is a complex Lie group, this is not true for any of the other Lie algebras we are dealing with. For each case, we explicitly find the automorphism group $\Aut(\mathfrak{h})$ (for the case of $\mathfrak{h}_4$ and $\mathfrak{h}_5$ this was also done in \cite{magnin-2007} by means of computational methods, and in \cite{saal-1996} for $H$-type Lie algebras). In this way, we are able to describe the moduli space $\mathcal M(H)/{\sim}$ of left-invariant metrics on $H$ up to isometric automorphisms. It is interesting to notice that the only case in which $\mathcal M(H)/{\sim}$ is a differentiable manifold is when the Lie algebra of $H$ is $\mathfrak{h}_9$. Our classification also shows that the moduli space up to isometry and scaling of the Lie algebra $\mathfrak h_6$ has dimension $1$. This extends the result in \cite{kodama-takahara-tamaru-2011}, as the Lie algebra $\mathfrak h_6$ is not present among the examples there of Lie groups with $1$-dimensional moduli space up to isometry and scaling. We also obtain the full isometry group $\I(H, g)$ associated to each left-invariant metric $g$ on $H$.
Another interesting problem is to determine which of the metrics $g$ are Hermitian, that is, when there exists an invariant complex structure $J$ on $H$ such that $(g, J)$ is an Hermitian structure. This turns out to be a very difficult computational problem. Even though the complex structures on $6$-dimensional nilpotent Lie algebras $\mathfrak{h}$ were classified in \cite{ceballos-2014}, it is very difficult to explicitly obtain the form of a particular complex structure $J$ on a given basis of $\mathfrak{h}$. For the case of the Iwasawa manifold, the set $\mathcal{C}$ of complex structures compatible with a standard metric and orientation of $\mathcal{I}$ was described in \cite{abbena-2001} by means of topological methods. The authors show there that $\mathcal{C}$ is the disjoint union of the standard complex structure $J_0$ and a $2$-sphere.
In this paper we give a complete classification of the Hermitian structures for $\mathfrak h_4, \mathfrak h_5$ and $\mathfrak h_6$. The problem becomes wild for $\mathfrak h_2$ and $\mathfrak h_9$, however we obtain some interesting partial results. For $\mathfrak h_2$ we prove that every left-invariant metric admits a finite number of compatible complex structures, and for $\mathfrak h_9$ we include a qualitative analysis and prove that there are left-invariant metrics which are not Hermitian.
We hope that the methods developed here will be useful to study the remaining cases in the classification of \cite{salamon-2001}.
The authors would like to thank Antonio Di Scala for suggesting the problem and for very useful discussions about it. They also thank Isolda Cardoso for her suggestions which helped to simplify some computations.
\section{Preliminaries}
\subsection{The moduli space of left-invariant metrics}
Let $H$ be a simply connected Lie group with Lie algebra $\mathfrak h$. Every left-invariant metric on $H$ is uniquely determined by a (positive definite) inner product on $\mathfrak h$, so, the set $\mathcal M(H)$ of left-invariant metrics on $H$ is identified, after the choice of a basis of $\mathfrak h$, with the symmetric space $\Sym_n^+ = \GL_n(\mathbb R)/{\OO(n)}$, where $n = \dim H$. Recall that the group $\Aut(H)$ of automorphisms of $H$ acts on the right on $\mathcal M(H)$ by \begin{equation}
\label{eq:24}
g \cdot \varphi = \varphi_*(g), \end{equation} for $g \in \mathcal M(H)$ and $\varphi \in \Aut(H)$, where $\varphi_*(g)(u,v)=g(d\varphi(u),d\varphi(v))$, i.e, $\varphi_*(g)$ is the pullback of $g$ by $\varphi$. The moduli space of left-invariant metrics of $H$ up to isometric automorphisms is $\mathcal M(H)/{\sim}$, where $\sim$ is the equivalence relation induced by the action given in (\ref{eq:24}). Since $H$ is simply connected, $\Aut(H)$ is isomorphic to the group $\Aut(\mathfrak h)$ of automorphisms of its Lie algebra $\mathfrak h$, which we can identify with a subgroup of $\GL_n(\mathbb R)$. If we think of $\Sym_n^+$ as the set of symmetric positive definite matrices of size $n \times n$, then the action of $\Aut(H)$ on $\mathcal M(H)$ is equivalent to the action of $\Aut(\mathfrak h)$ on $\Sym_n^+$ given by \begin{equation*}
X \cdot A = A^TXA, \end{equation*} for $X \in \Sym_n^+$, $A \in \Aut(\mathfrak h)$.
\subsection{Complex structures}
In the same spirit as in the previous paragraphs, one can identify the set $\mathcal C(H)$ of left-invariant complex structures on $H$ with \begin{equation*} \mathcal C(\mathfrak h) = \{J \in \End_{\mathbb R}(\mathfrak h): J^2 = -\id_{\mathfrak h} \text{ and } N_J = 0\} \end{equation*} where $N_J$ is the so-called Nijenhuis tensor of $J$, which is given for $X, Y \in \mathfrak h$ by \begin{equation*}
N_J(X, Y) = [JX, JY] - J[JX, Y] - J[X, JY] - [X, Y]. \end{equation*}
In the same manner, left-invariant abelian structures on $H$ are identify with the subset \begin{equation*} \mathcal A(\mathfrak h) = \{J \in \mathcal C(\mathfrak h): [JX, JY] = [X, Y] \text{ for all } X, Y \in \mathfrak h\} \end{equation*} of $\mathcal{C}(\mathfrak h)$. We say that two complex (resp.\ abelian) structures are equivalent if they are conjugated by an element of $\Aut(\mathfrak h)$. It is customary to consider the left-action of $\Aut(\mathfrak h)$ on $\mathcal C(\mathfrak h)$, which is given by \begin{equation*}
\varphi \cdot J = \varphi J \varphi^{-1} \end{equation*} for $J \in \mathcal C(\mathfrak h)$ and $\varphi \in \Aut(\mathfrak h)$. Recall however that the pullback action of $\Aut(H)$ on $\mathcal C(H)$ induces the right-action of $\Aut(\mathfrak h)$ on $\mathcal C(\mathfrak h)$ given by $J \cdot \varphi = \varphi^{-1} J \varphi$. These two actions have the same orbits and leave $\mathcal A(\mathfrak h)$ invariant. This is not true, in general, for the left- and right-actions of $\Aut(\mathfrak h)$ on $\Sym_n^+$.
\subsection{Nilpotent Lie algebras of dimension $6$} \label{nildim6}
In this section we shall recall some relevant notation and useful properties of $6$-dimensional nilpotent Lie algebras which will be used in the whole paper. For further details we refer the reader to \cite{salamon-2001}. Let $\mathfrak{h}$ be a $6$-dimensional Lie algebra, $\mathcal{B} = \{e_1, \ldots, e_6\}$ a basis of $\mathfrak{h}$ and $\mathcal{B}^* = \{e^1, \ldots, e^6\}$ the dual basis of $\mathfrak{h}^*$. For each $i = 1, \ldots, 6$, we write \begin{equation*}
de^k = \sum_{i<j} c^k_{ij} \, e^{ij}, \end{equation*} where $e^{ij}$ denotes the exterior product $e^i \wedge e^j$. Since $\mathfrak{h}$ is nilpotent and $6$-dimensional), one can choose $\mathcal{B}$ in such a way that $c^{k}_{ij} \in \{0,1\}$ for $i, j, k \in \{1, \ldots, 6\}$ and such that $c^k_{ij}=0$ for $i, j < k$. In this way, one can completely determine $\mathfrak{h}$ by knowing the differentials \begin{equation*} de^1, de^2, \ldots, de^6 \end{equation*} since this information together with the formula $d\theta(X, Y) = - \theta([X, Y])$, for $\theta \in \Lambda^1(\mathfrak h)$, allow us to reconstruct all the Lie brackets of $\mathfrak{h}$. Following Salamon's notation, if $de^k = e^{i_1 j_1}+ \cdots + e^{i_lj_l}$ we shall simply denote it by $i_1j_1+\cdots+ i_lj_l$. In this way, for example, we will write $$\mathfrak{h} = (0, 0, 0, 0, 0, 12 + 34)$$ for the Lie algebra that admits a basis $\mathcal{B}$ such that $de^6 = e^{12} + e^{34}$, and hence on which the only non trivial brackets are $[e_1,e_2]=[e_3,e_4]=-e_6$.
As we indicated in the Introduction, we are interested on those $6$-dimensional Lie algebras which admit a complex structure and has their first Betti number equal to $4$. These are the Lie algebras which, in the classification of Salamon, belong to the same class of the Lie algebra of the Iwasawa manifold. With the notation presented above, there are exactly five $6$-dimensional nilpotent Lie algebras with these properties: \begin{equation}\label{liealgs}
\begin{array}{l}
\mathfrak{h}_2 = (0, 0, 0, 0, 12, 34)\\
\mathfrak{h}_4 = (0, 0, 0, 0, 12, 14 + 23)\\
\mathfrak{h}_5 = (0, 0, 0, 0, 13 + 42, 14 + 23)\\
\mathfrak{h}_6 = (0, 0, 0, 0, 12, 13)\\
\mathfrak{h}_9 = (0, 0, 0, 0,
12, 14 +
25).
\end{array} \end{equation} Observe that in all cases $[\mathfrak{h},\mathfrak{h}]=\textrm{span}\{e_5,e_6\}$. The Lie algebra $\mathfrak h_5$ corresponds to the Iwasawa manifold, which was studied in \cite{Di_Scala_2012}.
In order to find the moduli spaces $\mathcal M(H)/{\sim}$, for a nilpotent simply connected $6$-dimensional Lie group $H$ whose Lie algebra $\mathfrak{h}$ is one of the Lie algebras listed above, we will determine in the following sections the corresponding full automorphism groups. The following lemma picks some common behaviour present in most of these groups.
\begin{lemma}
\label{lemaaut}
Let $\mathfrak h$ be a $2$-step nilpotent Lie algebra of dimension
$6$ with first Betti number equal to $4$. Let $e_1, \ldots, e_6$ be
a basis of $\mathfrak h$ such that $[\mathfrak h, \mathfrak h]$ is
spanned by $e_5, e_6$. Then there exist an algebraic subgroup
$G \subset \GL_4(\mathbb R)$ and a representation
$\Delta: G \to \GL_2(\mathbb R)$ such that
$\Aut(\mathfrak h) \simeq \mathbb R^8 \rtimes G$. More precisely, in
the above basis, every automorphism of $\mathfrak h$ has the form
\begin{equation}
\label{eq:64}
\begin{pmatrix}
A & 0 \\
M & \Delta(A)
\end{pmatrix}
\end{equation}
for some $A \in G$ and $M \in \mathbb R^{2 \times 4} \simeq \mathbb
R^8$. \end{lemma}
\begin{proof}
Clearly every automorphism of $\mathfrak h$ leaves $[\mathfrak h,
\mathfrak h]$ invariant. The group $G$ is induced by $\Aut(\mathfrak
h)$ via the projection $\mathfrak h \to \mathfrak h / [\mathfrak h,
\mathfrak h]$. So we can write any automorphism as \(
\begin{pmatrix}
A & 0 \\
M & B
\end{pmatrix}
\). Since
$[\mathfrak h, \mathfrak h] = \operatorname{span}_{\mathbb R}\{e_5,
e_6\}$, $B$ depends only on $A$, say $B = \Delta(A)$, and the group
structure of $\Aut(\mathfrak h)$ forces $\Delta$ to be a
representation of $G$ in $\mathbb R^2$. Finally, it is easy to see
that every linear map of the form (\ref{eq:64}) preserves the Lie
bracket of $\mathfrak h$. Recall that with these identifications,
$\mathbb R^{2 \times 4} \simeq \mathbb R^8$ is an abelian normal
subgroup of $\Aut(\mathfrak h)$. \end{proof}
\section{The case of $\mathfrak h_5 = (0, 0, 0, 0, 13 + 42, 14 + 23)$}
Let $e_1, \ldots, e_6$ be the basis of the Lie algebra $\mathfrak h_5$ such that the only non trivial Lie brackets are \begin{align*}
[e_1, e_3] = [e_4, e_2] = -e_5, && [e_1, e_4] = [e_2,e_3] = -e_6. \end{align*} It was shown in \cite{Di_Scala_2012} that $\Aut_0(\mathfrak{h}_5)$, the connected component of the identity of $\Aut(\mathfrak{h}_5)$, is isomorphic to a twisted (in the sense of Lemma \ref{lemaaut}) semi-direct product \begin{equation*}
\mathbb C^{2 \times 2} \rtimes \GL_2(\mathbb{C}) \end{equation*} and that the the moduli space $\mathcal{M}(\mathfrak{h}_5)$ is homeomorphic to the product $T \times \Sym_2^+/\sigma$, where $T$ is the triangle $\{(r, s): 0 < s \le r \le 1\}$ and \( \sigma \begin{pmatrix}
E & F \\
F & G \end{pmatrix} = \begin{pmatrix}
E & -F\\
-F & G \end{pmatrix} \). More precisely, in the standard basis $e_1, \ldots, e_6$, every left-invariant metric is represented by a unique matrix of the form \begin{equation}
\label{eq:69}
g =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & r & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & s & 0 & 0 \\
0 & 0 & 0 & 0 & E & F \\
0 & 0 & 0 & 0 & F & G
\end{pmatrix}, \end{equation} where $0 < s \le r \le 1$, $EG - F^2 > 0$ and $F \ge 0$.
In this section we will find the whole isometry group of each of the metrics (\ref{eq:69}). We start by recalling the following well known fact that will be used in the sequel.
\begin{remark}
\label{sec:nilp-lie-algebr}
Let $H$ be a connected nilpotent Lie group endowed with a
left-invariant metric $g$. Let us denote by $\mathfrak h$ the Lie
algebra de $H$. Then by \cite{wolf-1963} (see also
\cite{Wilson_1982}), the full isometry group of $H$ is given by
$\I(H, g) = H \rtimes K$, where
$K = \Aut(\mathfrak h) \cap \OO(\mathfrak h, g)$ under the usual
identifications. \end{remark}
\begin{theorem}
\label{sec:nilp-lie-algebr-1}
Let $g$ be the left-invariant metric on $H_5$ given in
(\ref{eq:69}). Then the full isometry group group of $g$ is given by
\begin{equation*}
\I(H_5, g) = H_5 \rtimes K
\end{equation*}
where $K \simeq \Aut(\mathfrak h) \cap \OO(\mathfrak h, g)$. The
different subgroups $K$, according to $r, s, E, F, G$ are listed in
Table \ref{tab:isometric-automorphisms}. \end{theorem}
\begin{table}[ht]
\caption{Isotropy subgroups of $\I(H_5, g)$.}
\centering
\begin{tabular}[ht]{|l|l|l|}
\hline
$K$ & $(r, s)$ & $E, F, G$ \\
\hline \hline
$\mathbb Z_2 \times \mathbb Z_2$ & $0 < s < r < 1$ & $F \neq 0$
\\
\hline
$\mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_2$
& $0 < s < r < 1$ & $F = 0$ \\
\hline
$\mathbb Z_2 \times \mathbb Z_2$ & $0 < s < r = 1$ & $F \neq 0$
\\
\hline
$\mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_2$
& $0 < s < r = 1$ & $F = 0, \, G \neq E$ \\
\hline
$\OO(2)$ & $0 < s < r = 1$ & $F = 0, \, G = E$ \\
\hline
$\OO(2)$ & $0 < s = r < 1$ & $F \neq 0$ \\
\hline
$\OO(2) \times \mathbb Z_2$ & $0 < s = r < 1$ & $F = 0$ \\
\hline
$\SU(2) \rtimes \mathbb Z_2$ & $s = r = 1$ & $F \neq 0$ \\
\hline
$(\SU(2) \rtimes \mathbb Z_2) \rtimes \mathbb Z_2$
& $s = r = 1$ & $F = 0, \, G \neq E$ \\
\hline
$\U(2) \rtimes \mathbb Z_2$ & $s = r = 1$ & $F = 0,\, G = E$ \\
\hline
\end{tabular}
\label{tab:isometric-automorphisms} \end{table}
It is important to note that the two cases when $K = \OO(2)$ in Table \ref{tab:isometric-automorphisms} correspond to different subgroups of $\Aut(\mathfrak h_5)$. These inclusions will become clear in the proof of Theorem \ref{sec:nilp-lie-algebr-1}.
\begin{proof}
Let $g$ be given as in (\ref{eq:69}). By using Remark
\ref{sec:nilp-lie-algebr}, in order to determine the full isometry
group, we only need to compute the automorphisms of $\mathfrak h_5$
which are isometric with respect to $g$. Recall that from
\cite{Di_Scala_2012}, every $\varphi \in \Aut_0(\mathfrak h_5)$ has,
in the standard basis, the form
\begin{equation}
\label{eq:65}
\varphi =
\begin{pmatrix}
A & 0 \\
M & \Delta(A)
\end{pmatrix},
\end{equation}
where, under de usual identifications,
$A \in \GL_2(\mathbb C) \subset \GL_4(\mathbb R)$,
$M \in \mathbb R^{2 \times 4}$ and
$\Delta(A) = \det A \in \GL_1(\mathbb C) \subset \GL_2(\mathbb
R)$. Moreover, the full automorphism group of $\mathfrak h_5$ has
two connected components:
\begin{equation*}
\Aut(\mathfrak h_5) = \Aut_0(\mathfrak h_5) \cup \psi
\Aut_0(\mathfrak h_5),
\end{equation*}
where $\psi = \diag(1, -1, 1, -1, 1, -1)$.
Notice that if $\varphi$ as in (\ref{eq:65}) preserves $g$, then we
have that $M = 0$, and since $\GL_2(\mathbb C)$ is connected,
$A \in \SO_{r, s}(4)$ and $\Delta(A) \in \SO_{E, F, G}(2)$, where
these are the orthogonal groups determined by the $4 \times 4$ and
$2 \times 2$ nontrivial blocks in $g$. Moreover, we must have
$\det_{\mathbb R} \Delta(A) = 1$ and so
$\Delta(A) \in \SO(2) \cap \SO_{E,F,G}(2)$, which implies that
either $\Delta(A) = \pm I_2$ or $F = 0$ and $G = E$. So, the
difficult part of the proof is describing
$\GL_2(\mathbb C) \cap \SO_{r, s}(4)$. Let us write
\begin{equation}
\label{eq:68}
A =
\begin{pmatrix}
a_1 & -a_2 & b_1 & -b_2 \\
a_2 & a_1 & b_2 & b_1 \\
c_1 & -c_2 & d_1 & -d_2 \\
c_2 & c_1 & d_2 & d_1
\end{pmatrix}
=
\begin{pmatrix}
z_1 & z_2 \\
z_3 & z_4
\end{pmatrix}
\in \GL_2(\mathbb C)
\end{equation}
and $g' = \diag(1, r, 1, s) = \diag(R, S)$, where $R = \diag(1, r)$
and $S = \diag(1, s)$. With these identifications, we can write
the orthogonality condition $A^T g' A = g'$ as
\begin{equation}
\label{eq:67}
\begin{pmatrix}
\bar z_1 R z_1 + \bar z_3 S z_3 & \bar z_1 R z_2 + \bar z_3 S
z_4 \\
\bar z_2 R z_1 + \bar z_4 S z_3 & \bar z_2 R z_2 + \bar z_4 S z_4
\end{pmatrix}
=
\begin{pmatrix}
R & 0 \\
0 & S
\end{pmatrix}
\end{equation}
After a close inspection, we notice that the last two entries on the
diagonal of the left side are $b_1^2 + r b_2^2 + d_1^2 + s d_2^2$
and $r b_1^2 + b_2^2 + s d_1^2 + d_2^2$. We then equal these values
to the corresponding entries on the diagonal of $S$ in order to get
that
\begin{equation}
\label{eq:66}
(r - s) b_1^2 + (1 - rs)b_2^2 + (1 - s^2)d_2^2 = 0.
\end{equation}
So we have to study several cases according to the values of $r, s$.
\textbf{Case $0 < s < r < 1$.} This is the generic case and
according to (\ref{eq:66}) we have $b_1 = b_2 = d_2 = 0$, which
forces $d_1 = \pm 1$ and $z_3 = 0$. Therefore, $\bar z_1 R z_1 = R$,
and as we noticed before, since $r \neq 1$, this implies
$z_1 = \pm 1$ (i.e.\ $a_1 = \pm 1$ and $a_2 = 0$). Now we check for
isometric automorphisms in the other connected component. Recall
that these are all of the form $\psi \varphi$ with
$\varphi \in \Aut_0(\mathfrak h_5)$. If we keep the notation
(\ref{eq:65}) and call $\psi' = \diag(1,-1,1,-1)$, then we find that
$\psi' A$ preserves $g'$. Since $\psi'$ preserves $g'$, we conclude
that $A$ also preserves $g'$. Thus, if $\psi \varphi$ is an
isometric automorphism then $F = 0$.
\textbf{Case $0 < s < r = 1$.} We use (\ref{eq:66}) again in order
to conclude that $z_2 = z_3 = 0$ and $z_4 = \pm 1$. Since $r = 1$,
we must have $\bar z_1 z_1 = 1$, which with our identifications
means that $z_1 \in \SO(2)$. Also, since
$\Delta(A) = \pm z_1 \in \SO_{E, F, G}(2)$ we see that $F \neq 0$ or
$G \neq E$ imply $z_1 = \pm 1$. When $F = 0$ and $G = E$, we
trivially have $\SO_{E,0,E}(2) = \SO(2)$. Finally, with the same
argument as in the previous case, we can find isometric
automorphisms outside the connected component of the identity of
$\Aut(\mathfrak h_5)$ if and only if $F = 0$. Moreover, if in
addition $G = E$, then $\psi$ is an isometric automorphism which
lies outside the connected component of the identity of
$\OO(2) = \OO_{E,0,E}(2)$.
\textbf{Case $0 < s = r < 1$.} In this case equation \eqref{eq:66}
becomes $(1 - r^2)(b_2^2 + d_2^2) = 0$, which means
$z_2, z_4 \in \mathbb R$. With the same idea we used to derive
\eqref{eq:66}, we can also show that $z_1, z_3 \in \mathbb R$. Now
looking back to \eqref{eq:67}, with $R = S$ we get that
\begin{equation*}
a_1^2 + c_1^2 = b_1^2 + d_1^2 = 1.
\end{equation*}
From this, it is not hard to see that the subgroup of
$\Aut_0(\mathfrak h_4)$ preserving the metric is the intersection of
$\GL_2(\mathbb R) \subset \GL_2(\mathbb C) \subset \GL_4(\mathbb R)$
with $\OO(4)$, which is isomorphic to $\OO(2)$. Finally, we will
have isometric automorphisms other that the ones in
$\Aut_0(\mathfrak h_5)$ if and only if $\psi$ is isometric, which
only happens when $F = 0$. Notice that in this case $\psi$ commutes
with $\OO(2)$, which gives us that the isotropy group of the full
isometry group is isomorphic to $\OO(2) \times \mathbb Z_2$.
\textbf{Case $0 < s = r = 1$.} This is the case with most
symmetries. It is immediate that $A$ as in \eqref{eq:68} belongs to
$\U(2) = \GL_2(\mathbb C) \cap \OO(4)$. Since $\Delta(A) \in \U(1)$,
if $F \neq 0$, then $\Delta(A) = \pm 1$ and so
$A \in \SU(2) \rtimes \mathbb Z_2$. If $F = 0$ and $G \neq E$, we
also have that $A \in \SU(2) \rtimes \mathbb Z_2$. But $\psi$ is an
isometric automorphism, then we have two connected component for the
isometric automorphisms. Finally, if $F = 0$ and $G = E$, then every
automorphism in $\U(2)$ preserves the metric, and hence the
isometric automorphisms are isomorphic to
$\U(2) \rtimes \mathbb Z_2$. \end{proof}
\section{The case of $\mathfrak h_6 = (0, 0, 0, 0, 12, 13)$} \label{sec:case-mathfrak-h_6}
\subsection{Automorphism group} \label{sec:automorphism-group}
Let $\mathfrak h_6$ be the $6$-dimensional $2$-step nilpotent real Lie algebra corresponding to $(0, 0, 0, 0, 12, 13)$ in Salamon notation \cite{salamon-2001}. That is, we have a canonical basis $e_1, \ldots, e_6$ such that the only non-trivial brackets are $[e_1, e_2] = -e_5$ and $[e_1, e_3] = -e_6$. Equivalently, if $d: \mathfrak h_6^* \to \Lambda^2(\mathfrak h_6^*)$ is the exterior derivative on left-invariant forms, then $\ker d$ is spanned by $e^1, \ldots, e^4$ and $de^5 = e^{12}$, $de^6 = e^{13}$.
It is known from \cite{salamon-2001} that $\mathfrak h_6$ admits an invariant complex structure. Moreover, according to \cite{ceballos-2014} there is a unique invariant complex structure up to equivalence on $\mathfrak h_6$. This means that $\Aut(\mathfrak h_6)$ acts transitively by conjugation on the set $\mathcal C(\mathfrak h_6)$ of invariant complex structures. Recall that the standard almost complex structure associated to the multiplication by $\sqrt{-1}$ via the identification $\mathfrak h_6 \simeq \mathbb R^6 \simeq \mathbb C^3$ is not integrable.
\begin{lemma}
\label{sec:autom-group-mathfr}
The invariant almost complex structure $J: \mathfrak h_6 \to
\mathfrak h_6$ determined by the equations $Je_1 = e_4$, $Je_2 =
e_3$ and $Je_5 = e_6$ is integrable. \end{lemma}
\begin{proof}
Let us denote $\Lambda^{1,0} = \Lambda^{1,0}(\mathfrak
h_6^*)_{\mathbb C}$ the $i$-eigenspace of $J^*$ on the
complexification of $\mathfrak h_6$. Notice that $J ^*$ is the
transpose of $J$, so the equations determining $J^*$ are $J^*e^1 =
-e^4$, $J^*e^2 = -e^3$ and $J^*e^5 = -e^6$. According to
\cite{ceballos-2014}, $J$ is integrable if and only if there exists
a basis $\omega^1, \omega^2, \omega^3$ of $\Lambda^{1, 0}$ such that
$d\omega^1 = d\omega^2 = 0$ and
\begin{equation}
\label{eq:1}
d\omega^3 = \omega^1 \wedge \omega^2 + \omega^1 \wedge
\bar\omega^1 + \omega^1 \wedge \bar\omega^2=\omega^1\wedge(\bar\omega^1+2\operatorname{Re}(\omega^2)).
\end{equation}
The standard basis of $\Lambda^{1,0}$ associated with the canonical
basis of $\mathfrak h_6$ is given by
\begin{align*}
\eta^1 & = e^1 - iJ^*e^1 = e^1 + ie^4, \\
\eta^2 & = e^2 - iJ^*e^2 = e^2 + ie^3, \\
\eta^3 & = e^5 - iJ^*e^5 = e^5 + ie^6.
\end{align*}
Suppose that there exist $\omega^1, \omega^2, \omega^3$ as in
\eqref{eq:1}. We can assume that $\omega^3 = \eta^3$, and so
$d\omega^3 = e^{12} + ie^{13}$. We can further assume that
$\omega^1, \omega^2$ belong to the subspace spanned by
$\eta^1, \eta^2$. If we write $\omega^1 = A\eta^1 + B\eta^2$ then,
taking the imaginary part of both sides of equation \eqref{eq:1},
we get that that $A = 0$ and $B \neq 0$. One can also assume that
$B = 1$ and so, $\omega^1 = \eta^2$. Let us write
$\omega^2 = C\eta^1 + D\eta^2$. Then replacing it in \eqref{eq:1}
one gets
\begin{align*}
e^{12} + ie^{13} & = (e^2 + ie^3) \wedge (2Ce^1 + (2D+ 1)e^2 -
ie^3) \\
& = -2C(e^{12} + ie^{13}) - 2i(D + 1)e^{23}
\end{align*}
So $C = -\frac12$, $D = -1$ and $\omega^1 = \eta^2$,
$\omega^2 = -\frac12\eta^1 - \eta^2$, $\omega^3 = \eta^3$ is the
basis of $\Lambda^{1,0}$ we were looking for. \end{proof}
\begin{lemma}
\label{sec:autom-group-mathfr-1}
If $f \in \Aut(\mathfrak h_6)$ then:
\begin{enumerate}
\item \label{item:1} $e^1(f(e_j)) = 0$ for $j = 2, \ldots, 6$;
\item \label{item:2} $e^2(f(e_j)) = 0$ for $j = 4, 5, 6$;
\item \label{item:3} $e^3(f(e_j)) = 0$ for $j = 4, 5, 6$;
\item \label{item:4} $e^4(f(e_j)) = 0$ for $j = 5, 6$;
\item \label{item:5} $e^5(f(e_j)) = e^1(f(e_1))e^2(f(e_{j -3}))$,
for $j = 5, 6$;
\item \label{item:6} $e^6(f(e_j)) = e^1(f(e_1))e^3(f(e_{j - 3})),$
for $j = 5, 6$.
\end{enumerate} \end{lemma}
\begin{proof}
Since the center of $\mathfrak h_6$ is spanned by $e_4, e_5, e_6$
and $f$ leaves the center invariant, we get that $e^k(f(e_j)) = 0$
for all $k = 1, 2, 3$ and $j = 4, 5, 6$. Also, since
$\dim(\ker \ad_{e_j})$ is preserved under automorphisms,
$e^1(f(e_j)) = 0$ if $j \ge 2$. These two observations together
prove (\ref{item:1}), (\ref{item:2}) and (\ref{item:3}). Part
(\ref{item:4}) follows from Lemma \ref{lemaaut}. For parts
(\ref{item:5}) and (\ref{item:6}) recall that $e_5 = -[e_1,
e_2]$. Then
\begin{align*}
f(e_5) & = -[f(e_1), f(e_2)] \\
& = -[e^1(f(e_1))e_1, e^2(f(e_2))e_2 + e^3(f(e_2))e_3] \\
& = e^1(f(e_1))e^2(f(e_2))e_5 + e^1(f(e_1))e^3(f(e_2))e_6.
\end{align*}
With the same argument we can see that
\begin{equation*}
f(e_6) = e^1(f(e_1))e^2(f(e_3))e_5 + e^1(f(e_1))e^3(f(e_3))e_6. \qedhere
\end{equation*}
\end{proof}
\begin{lemma}
\label{sec:autom-group-mathfr-2}
Let $J$ be the complex structure of Lemma
\ref{sec:autom-group-mathfr}. Then the isotropy subgroup at $J$ of
$\Aut(\mathfrak h_6)$ is isomorphic to
\begin{equation*}
\mathbb R^4 \rtimes_\varphi (\GL_1(\mathbb R) \times
\GL_1(\mathbb C)),
\end{equation*}
where $\varphi: \GL_1(\mathbb R) \times \GL_1(\mathbb C) \to
\GL_4(\mathbb R)$ is the representation given by
\begin{equation*}
\varphi(r, a + ib) =
\begin{pmatrix}
a & 0 & b & 0 \\
0 & r & 0 & 0 \\
-b & 0 & a & 0 \\
0 & 0 & 0 & r
\end{pmatrix},
\end{equation*}
for $r \neq 0$ and $a^2 + b^2 \neq 0$. \end{lemma}
\begin{proof}
Let $f \in \Aut(\mathfrak h_6)$ and identify it with its matrix
$(a_{ij})$ in the basis $e_1, \ldots, e_6$. From Lemma
\ref{sec:autom-group-mathfr-1} we must have
\begin{equation}
\label{eq:2}
\begin{aligned}
& a_{1j} =0 \text{ for } j \geq 2,
&& a_{2j} = a_{3j} =0 \text{ for } j\geq 4,
&& a_{45} = a_{46} = 0,
&& \\
& a_{55} = a_{11} a_{22},
&& a_{56} = a_{11} a_{23},
&& a_{65} = a_{11} a_{32},
&& a_{66}= a_{11}a_{33}.
\end{aligned} \end{equation}
If, in addition, we ask $f$ to be in the isotropy of $J$, then $f$
must commute with the matrix associated to $J$, and thus it has the
form
\begin{equation}
\label{eq:3}
f =
\begin{pmatrix}
a_{11} & 0 & 0 & 0 & 0 & 0 \\
0 & a_{22} & a_{23} & 0 & 0 & 0 \\
0 & -a_{23} & a_{22} & 0 & 0 & 0 \\
0 & 0 & 0 & a_{11} & 0 & 0 \\
a_{51} & a_{52} & -a_{62} & -a_{61} & a_{11}a_{22} & a_{11}a_{23} \\
a_{61} & a_{62} & a_{52} & a_{51} & -a_{11}a_{23} & a_{11}a_{22}
\end{pmatrix}.
\end{equation}
with $a_{11} \neq 0$ and $a_{22}^2 + a_{23}^2 \neq 0$. Moreover,
every linear map of the form \eqref{eq:3} is an automorphism of
$\mathfrak h_6$. In order to see this, we can show that
$f^* \circ d = d \circ f^*$. Since $\ker d$ is spanned by
$e^1, e^2, e^3, e^4$, clearly $f^*(de^j) = d(f^*(e^j))$ for
$1 \le j \le 4$. Also,
\begin{align*}
d(f^*(e^5)) & = a_{11}a_{22}e^{12} + a_{11}a_{23}e^{13} \\
& = a_{11}e^1 \wedge (a_{22}e^2 + a_{23}e^3) \\
& = f^*(e^1) \wedge f^*(e^2) \\
& = f^*(e^{12}) = f^*(de^5)
\end{align*}
and similarly $d(f^*(e^6)) = f^*(de^6)$. Hence $f$ commutes with $J$
if and only it has the form~\eqref{eq:3}.
Finally, notice that $\Aut(\mathfrak h_6)_J = K \rtimes H$ is the
inner semi-direct product of the normal subgroup
\begin{equation*}
K = \left\{
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
a_{51} & a_{52} & -a_{62} & -a_{61} & 1 & 0 \\
a_{61} & a_{62} & a_{52} & a_{51} & 0 & 1
\end{pmatrix}: a_{51}, a_{52}, a_{61}, a_{62} \in \mathbb R
\right\} \simeq \mathbb R^4
\end{equation*}
and the subgroup
\begin{align*}
H & = \left\{
\begin{pmatrix}
a_{11} & 0 & 0 & 0 & 0 & 0 \\
0 & a_{22} & a_{23} & 0 & 0 & 0 \\
0 & -a_{23} & a_{22} & 0 & 0 & 0 \\
0 & 0 & 0 & a_{11} & 0 & 0 \\
0 & 0 & 0 & 0 & a_{11}a_{22} & a_{11}a_{23} \\
0 & 0 & 0 & 0 & -a_{11}a_{23} & a_{11}a_{22}
\end{pmatrix}: a_{11} \neq 0, a_{22}^2 + a_{23}^2 \neq 0
\right\} \\
& \simeq \GL_1(\mathbb R) \times \GL_1(\mathbb C).
\end{align*}
Now one can easily check that
$\Aut(\mathfrak h_6)_{J} \simeq \mathbb R^4
\rtimes_{\varphi}(\GL_1(\mathbb R) \times \GL_1(\mathbb C))$ as
stated. \end{proof}
It is convenient to introduce some notation before stating the main result of this section. Let us consider the presentation of the $5$-dimensional Heisenberg Lie group $\Heis_2 = \{(x, y, z): x, y \in \mathbb R^2,\, z \in \mathbb R\}$ with the multiplication given by \begin{equation*}
(x, y, z)(x', y', z') = (x + x', y + y', z + z' + y \cdot x') \end{equation*} and let $G$ be the subgroup of $\GL_4(\mathbb R)$ consisting of all the matrices in block form \begin{equation}
\label{eq:5}
A =
\begin{pmatrix}
r & & \\
x & \tilde A & \\
z & y^T & s
\end{pmatrix} \end{equation} where $r, s \in \mathbb R - \{0\}$, $\tilde A \in \GL_2(\mathbb R)$, $x, y \in \mathbb R^2 \simeq \mathbb R^{2 \times 1}$ and $z \in \mathbb R$. It is not hard to see that \begin{equation}
\label{eq:4}
G \simeq (\Heis_2 \rtimes_{\varphi_1} \GL_2(\mathbb R))
\rtimes_{\varphi_2} (\GL_1(\mathbb R) \times \GL_1(\mathbb R)) \end{equation} where $\varphi_1: \GL_2(\mathbb R) \to \Aut(\Heis_2)$ and $\varphi_2: \GL_1(\mathbb R) \times \GL_1(\mathbb R) \to \Aut(\Heis_2 \rtimes_{\varphi_1} \GL_2(\mathbb R))$ are the Lie groups morphisms given by \begin{align*}
\varphi_1(\tilde A)(x, y, z) & = (\tilde Ax, (\tilde A^{-1})^Ty, z)
\\
\varphi_2(r, s)(x, y, z, \tilde A) & = \left(\frac x r, s(\tilde
A^{-1})^Ty, \frac {s z} r, \tilde
A\right) \end{align*}
Let us also consider the Lie groups epimorphism $\Delta: G \to \GL_2(\mathbb R)$ defined by \begin{equation}
\label{eq:6}
\Delta(A) = r\tilde A. \end{equation} Recall that after the identification of $G$ given by (\ref{eq:4}), the kernel of $\Delta$ is a normal subgroup isomorphic to $\Heis_2 \rtimes \GL_1(\mathbb R)$.
\begin{theorem}
\label{sec:autom-group-mathfr-3}
Let $G$ be the Lie subgroup of $\GL_4(\mathbb R)$ defined in
(\ref{eq:5}). There exists an isomorphism of Lie groups
\begin{equation*}
\Aut(\mathfrak h_6) \simeq \mathbb R^{2 \times 4}
\rtimes_{\varphi} G,
\end{equation*}
where $\mathbb R^{2 \times 4}$ is the abelian Lie group of $2 \times
4$ matrices and $\varphi: G \to \GL(\mathbb R^{2 \times 4})$ is given
by $\varphi(A)M = \Delta(A)MA^{-1}$, being $\Delta$ defined as in
(\ref{eq:6}). Moreover, every automorphism of $\mathfrak h_6$ is
represented in the canonical basis by a matrix of the form
\begin{equation}
\label{eq:7}
\begin{pmatrix}
A & 0 \\
M & \Delta(A)
\end{pmatrix}
\end{equation}
where $A \in G$ and $M \in \mathbb R^{2 \times 4}$. \end{theorem}
\begin{proof}
Let $\tilde G$ the subgroup consisting of all the matrices of the
form (\ref{eq:7}). Recall that this subgroup agrees with the one
defined by the equations (\ref{eq:2}). So, from Lemma
\ref{sec:autom-group-mathfr-1} and the above paragraphs,
\begin{equation*}
\Aut(\mathfrak h_6) \subset \tilde G \simeq \mathbb R^{2 \times 4}
\rtimes_{\varphi} G.
\end{equation*}
Now, it follows from \cite{salamon-2001} that
$\mathcal C(\mathfrak h_6)$ has real dimension $12$. Since
$\Aut(\mathfrak h_6)$ is transitive on $\mathcal C(\mathfrak h_6)$,
it follows from Lemma \ref{sec:autom-group-mathfr-2} that
$\dim \Aut(\mathfrak h_6) = \dim \tilde G = 19$. So, the identity
components of $\Aut(\mathfrak h_6)$ and $\tilde G$ coincide. In
order to see $\Aut(\mathfrak h_6) = \tilde G$, it is enough to see
that there is an automorphism of $\mathfrak h_6$ in each of the
other seven connected components of $\tilde G$. Let us choose the
following representatives for the connected components of
$\tilde G$:
\begin{align*}
f_1 & = I_6 & f_5 & = \diag(-1,1,1,1,-1,-1), \\
f_2 & = \diag(1,1,1,-1,1,1), & f_6 & = \diag(-1,1,1,-1,-1,-1), \\
f_3 & = \diag(1,1,-1,1,1,-1), & f_7 & = \diag(-1,1,-1,1,-1,1), \\
f_4 & = \diag(1,1,-1,-1,1,-1), & f_8 & = \diag(-1,1,-1,-1,-1,1). \\
\end{align*}
Since the $f_j$'s form a subgroup of $\tilde G$ and every $f_j$ but
$f_1$ has order $2$, it is enough to show that three out of $f_2,
\ldots, f_8$ are in $\Aut(\mathfrak h_6)$. Moreover, from Lemma
\ref{sec:autom-group-mathfr-2}, $f_6 \in \Aut(\mathfrak h_6)$. Let
us verify that also $f_2, f_3 \in \Aut(\mathfrak h_6)$. Reasoning as
in the proof of Lemma \ref{sec:autom-group-mathfr-2},
\begin{align*}
d(f_2^*(e^5)) & = de^5 = e^{12} = f_2^*(e^{12}) = f_2^*(de^5) \\
d(f_2^*(e^6)) & = de^6 = e^{13} = f_2^*(e^{13}) = f_2^*(de^6) \\
d(f_3^*(e^5)) & = de^5 = e^{12} = f_3^*(e^{12}) = f_3^*(de^5) \\
d(f_3^*(e^6)) & = -de^6 = -e^{13} = f_3^*(e^{13}) = f_3^*(de^6)
\end{align*}
This concludes the proof of the theorem. \end{proof}
The following result is an immediate consequence of the proof of Theorem \ref{sec:autom-group-mathfr-3}.
\begin{corollary}
$\Aut(\mathfrak h_6)$ has $8$ connected components. Moreover,
\begin{equation*}
\Aut(\mathfrak h_6)/\Aut_0(\mathfrak h_6) \simeq \mathbb Z_2
\oplus \mathbb Z_2 \oplus \mathbb Z_2.
\end{equation*} \end{corollary}
\subsection{Left-invariant metrics}
Consider an inner product $g$ on $\mathfrak h_6$. Then in the canonical basis, $g$ can be represented by a symmetric positive definite matrix of the form \begin{equation}
\label{metrica}
g = \left(
\begin{array}{cc}
B & C^T\\
C & D
\end{array}
\right) \end{equation} with $B \in \Sym_4^+$, $D \in \Sym_2^+$ and $C \in \mathbb R^{2\times 4}$.
\begin{lemma}
\label{sec:left-invar-metr-6}
Let $G$ be the subgroup of $\GL_4(\mathbb R)$ defined in
(\ref{eq:5}). Then:
\begin{enumerate}
\item \label{item:7} $G$ acts transitively on $\Sym_4^+$.
\item \label{item:8} Any metric $g$ on $\mathfrak h_6$ is
equivalent, by an automorphism in
$G \subset \Aut(\mathfrak{h_6})$, to a metric of the form
\(
\begin{pmatrix}
I_4 & \tilde{C}^T \\
\tilde{C} & \tilde{D}
\end{pmatrix}
\).
\item \label{item:9} Any metric $g$ on $\mathfrak h_6$ is
equivalent, by an automorphism in
$\mathbb{R}^{8}\subset \Aut(\mathfrak h_6)$ to a metric of the
form
\(
\begin{pmatrix}
\tilde{A} & 0 \\
0 & \tilde{D}
\end{pmatrix}
\).
\end{enumerate} \end{lemma}
Recall that the inclusions $G \subset \Aut(\mathfrak h_6)$ and $\mathbb R^8 \simeq \mathbb R^{2 \times 4} \subset \Aut(\mathfrak h_6)$ are the ones provided by Theorem \ref{sec:autom-group-mathfr-3}.
\begin{proof}
Observe that any element of $\Sym_4^+$ can be written as $X^{T}X$,
where $X$ is a lower-triangular matrix. Since the set of
lower-triangular matrices is contained in $G$, we conclude that any
element of $\Sym_4^{+}$ is in the orbit of the identity. This proves
part (\ref{item:7}). Item~(\ref{item:8}) follows from the first
property. In fact, if $g$ has the form given in
equation~(\ref{metrica}), one only needs to choose an element
$A \in G \subset \Aut(\mathfrak h_6)$ such that $A^TBA =
I_4$. Finally, for (\ref{item:9}), let
$M \in \mathbb R^{2\times 4} \simeq \mathbb R^8 \subset
\Aut(\mathfrak h_6)$. Then if $g$ is as in equation (\ref{metrica}),
\begin{equation*}
M^T g M =
\begin{pmatrix}
\tilde B & \tilde C^T \\
\tilde C & D
\end{pmatrix}
\end{equation*}
where $\tilde B = B + C^TM + M^TC + M^TDM$ and $\tilde C = C +
DM$. Choosing $M = -D^{-1}C$ we get the desired result. \end{proof}
\begin{corollary}\label{coro}
Any inner product $g$ on $\mathfrak h_6$ is equivalent via an
element of $\Aut(\mathfrak h_6)$ to one of the form
\(
\begin{pmatrix}
I_4 & 0 \\
0 & \tilde{D}
\end{pmatrix},
\)
with $\tilde{D}\in \Sym_2^+$. \end{corollary}
\begin{remark}\label{rem}
Observe that any two inner products $g$, $g'$ given by matrices
$\tilde{D}$, $\tilde{D}'$ as in Corollary~\ref{coro} are equivalent
by an automorphism of $\mathfrak h_6$ if and only if the matrices
$\tilde{D}$ and $\tilde{D'}$ are conjugated by an element of
$\OO(2) \subset G \subset \Aut(\mathfrak h_6)$. Since
$\Sym_2^+ = \GL_2(\mathbb R)/\OO(2)$, each family of equivalent
metrics can be identified with an orbit of the isotropy action in
this symmetric space. \end{remark}
\begin{theorem}
\label{sec:left-invar-metr-2}
Let $H_6$ be the simply connected Lie group with Lie algebra
$\mathfrak h_6$. Each left-invariant metric on $H_6$ is equivalent
by an automorphism to a metric of the form
\begin{equation}
\label{eq:17}
g = \sum_{i = 1}^4 e^i \otimes e^i + a e^5 \otimes e^5 + b e^6 \otimes e^6,
\end{equation}
with $a,b>0$. Moreover, the moduli space $\mathcal M(H_6)/{\sim}$ is
homeomorphic to
$$
\{(a,b) \in \mathbb R^2: \,0<a\leq b\}.
$$ \end{theorem}
\begin{proof}
By Remark \ref{rem}, we only need to find a section to the orbits of
the $\OO(2)$-action on the symmetric space
$\Sym_2^+ = \GL_2(\mathbb R)/\OO(2)$. Observe that
$\mathfrak{gl}_2(\mathbb R) = \mathfrak{so}(2) + \Sym_2$ is a Cartan
decomposition of $\mathfrak{gl}_2(\mathbb R)$, where $\Sym_2$ denote
the subspace of symmetric matrices. So, a section of the
$\OO(2)$-action on $\Sym_2^+$ is the exponential of a maximal
abelian subalgebra of $\Sym_2$, which is given by the $2 \times 2$
diagonal matrices. This proves the first assertion. The second one
is a consequence of the fact that conjugation by $J_0 \in \OO(2)$ of
a diagonal matrix interchanges the diagonal entries, where $J_0$
denotes the multiplication by $\sqrt{-1}$ in
$\mathbb C \simeq \mathbb R^2$. \end{proof}
\begin{corollary}
\label{sec:left-invar-metr-1}
Let $g_{a, b}$ be the left-invariant metric on $H_6$ given by
\eqref{eq:17}. Then the full isometry group of $g_{a, b}$ is given by
\begin{equation*}
\I(H_6, g_{a, b}) =
\begin{cases}
H_6 \rtimes (\OO(2) \times \mathbb Z_2 \times \mathbb Z_2), & a
= b \\
H_6 \rtimes (\mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_2)
& a \neq b
\end{cases}
\end{equation*} \end{corollary}
\begin{proof}
According to Remark \ref{sec:nilp-lie-algebr}, we only need to
compute the isometric automorphisms of $\mathfrak h_6$. From
Theorem \ref{sec:autom-group-mathfr-6}, an automorphism $f$ of
$\mathfrak h_6$ has the form
\begin{equation*}
\begin{pmatrix}
r & & & \\
x & A & & \\
z & y^T & s \\
M_1 & M_2 & M_3 & rA
\end{pmatrix}
\end{equation*}
in the canonical basis, where $r, s \in \mathbb R - \{0\}$,
$z \in \mathbb R$, $x, y, M_1, M_3 \in \mathbb R^{2 \times 1}$,
$A \in \GL_2(\mathbb R)$ and $M_2 \in \mathbb R^{2 \times 2}$. If
$f$ leaves $g_{a, b}$ invariant, then $x, y, z, M_1, M_2, M_2$ all
vanish, $r, s \in \{\pm 1\}$ and $A \in \OO(2)$. Moreover, if
$a \neq b$ then $A = \pm I_2$. This implies the result. \end{proof}
\section{The case of $\mathfrak h_4 = (0, 0, 0, 0, 12, 14 + 23)$} \label{sec:case-mathfrak-h_4}
\subsection{Automorphism group}
Let us consider the basis of $\mathfrak h_4$ whose only non vanishing are differentials are $de^5 = e^{12}$ and $de^6 = e^{14} + e^{23}$. In terms of the Lie bracket, we can assume that the only non trivial brackets in the above basis are \begin{align*}
[e_1, e_2] = -e_5, && [e_1, e_4] = [e_2, e_3] = -e_6. \end{align*}
\begin{lemma}
\label{sec:autom-group-mathfr-4}
Let $f \in \Aut(\mathfrak h_4)$, then:
\begin{enumerate}
\item $e^i(f(e_j)) = 0$ for $i = 1,2$ and $j = 3, 4$;
\item $e^i(f(e_5)) = 0$ for $i = 1, \ldots, 4$;
\item $e^i(f(e_6)) = 0$ for $i = 1, \ldots, 5$.
\end{enumerate} \end{lemma}
\begin{proof}
Since $f$ is an automorphism, it leaves invariant $\dim(\ker \ad_x)$
for all $x \in \mathfrak h_4$. In particular, if $j = 3, 4$, then
$\dim(\ker \ad_{f(e_j)}) = 1$ and hence
$e^1(f(e_j)) = e^2(f(e_j)) = 0$. Now if $j = 5, 6$ then
$e^i(f(e_j)) = 0$ for $i = 1, \ldots 4$, since $f$ leaves the center
of $\mathfrak h_4$ invariant. Moreover, since
$f(e_6) = -[f(e_1), f(e_4)]$ and the $e_1$- and $e_2$-components of
$f(e_4)$ are zero, it follows that $e^{5}(f(e_6)) = 0$. \end{proof}
In order to compute the full automorphism group of $\mathfrak h_4$, it is easier to determine first the connected component of the identity. Recall that the Lie algebra of $\Aut(\mathfrak h_4)$ is given by the derivations of $\mathfrak h_4$, \begin{equation*}
\Der(\mathfrak h_4) = \{D \in \mathfrak{gl}(\mathfrak h_4):
D[X,Y] = [DX,Y] + [X, DY] \text{ for all } X, Y \in \mathfrak
h_4\}. \end{equation*} Identifying, as usual, $D$ with its matrix in the basis $e_1, \ldots, e_6$, the conditions $D[e_i, e_j] = [De_i, e_j] + [e_i, De_j]$, for $i < j$, define a linear system in the entries of $D$. A straight-forward computation, together with Lemma \ref{sec:autom-group-mathfr-4}, allows us to prove the following fact.
\begin{lemma}
\label{sec:autom-group-mathfr-5}
The Lie algebra $\Der(\mathfrak h_4)$, after the usual
identification, is given by the Lie subalgebra of
$\mathfrak{gl}_6(\mathbb R)$ which consists of the matrices of the
following form
\begin{equation}\label{eq:8}
D =
\begin{pmatrix}
d_{11} & d_{12} & 0 & 0 & 0 & 0 \\
d_{21} & d_{22} & 0 & 0 & 0 & 0 \\
d_{31} & d_{32} & d_{11} + x & -d_{12} & 0 & 0 \\
d_{41} & d_{42} & -d_{21} & d_{22} + x & 0 & 0 \\
d_{51} & d_{52} & d_{53} & d_{54} & d_{11} + d_{22} & 0 \\
d_{61} & d_{62} & d_{63} & d_{64} & -d_{31} + d_{42} & d_{11} +
d_{22} + x \\
\end{pmatrix},
\end{equation}
where $x, d_{ij} \in \mathbb R$. In particular,
$\dim \Aut(\mathfrak h_4) = 17$. \end{lemma}
In order to describe the full automorphism group, we introduce the following notation. Let $\sigma: \GL_2(\mathbb R) \to \GL_2(\mathbb R)$ be the Lie involution given by \begin{equation*}
\sigma
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
=
\begin{pmatrix}
a & -b \\
-c & d
\end{pmatrix} \end{equation*} and let $(\cdot,\cdot)$ the semi-definite inner product on $\mathfrak{gl}_2(\mathbb R)$ defined as \begin{equation}
\label{eq:13}
(A, B) = a_{11}b_{22} - a_{12}b_{21} + a_{22}b_{12} - a_{22}b_{11}. \end{equation}
Consider the closed Lie subgroup $G \subset \GL_4(\mathbb R)$ of matrices of the form \begin{equation}\label{eq:9}
\begin{pmatrix}
A & 0 \\
B & x\sigma(A)
\end{pmatrix} \end{equation} where $A \in \GL_2(\mathbb R)$, $B \in \mathbb R^{2 \times 2}$ and $x \in \mathbb R - \{0\}$. It follows that $G$ is isomorphic to the semi-direct product \begin{equation*}
G \simeq \mathbb R^{2 \times 2} \rtimes_{\varphi_1} (\GL_2(\mathbb R)
\times \GL_1(\mathbb R)) \end{equation*} where \begin{equation*}
\varphi_1(A, x) B = x\sigma(A)BA^{-1}. \end{equation*}
Finally, consider the representation $\Delta: G \to \GL_2(\mathbb R)$ given by \begin{equation}
\label{eq:12}
\Delta
\begin{pmatrix}
A & 0 \\
B & x\sigma(A)
\end{pmatrix}
=
\begin{pmatrix}
\det A & 0 \\
(A, B) & x \det A
\end{pmatrix}. \end{equation}
\begin{theorem}
\label{sec:autom-group-mathfr-6}
Let $G$ be the Lie subgroup of $\GL_4(\mathbb R)$ defined in
(\ref{eq:9}). There exists an isomorphism of Lie groups
\begin{equation*}
\Aut(\mathfrak h_4) \simeq \mathbb R^{2 \times 4} \rtimes_\varphi G,
\end{equation*}
where $\mathbb R^{2 \times 4}$ is the abelian Lie group of
$2 \times 4$ real matrices and
$\varphi: G \to \GL(\mathbb R^{2 \times 4})$ is the representation
given by $\varphi(A)M = \Delta(A)MA^{-1}$, with $\Delta$ defined as
in (\ref{eq:12}). Moreover, any automorphism of $\mathfrak h_4$ is
represented in the canonical basis by a matrix of the form
\begin{equation}
\label{eq:10}
\begin{pmatrix}
A & 0 \\
M & \Delta(A)
\end{pmatrix},
\end{equation}
where $A \in G$ and $M \in \mathbb R^{2 \times 4}$. \end{theorem}
\begin{proof}
Let $\tilde G$ the Lie subgroup of $\GL_6(\mathbb R)$ which consists
of all the matrices of the form~\eqref{eq:10}. It follows from Lemma
\ref{sec:autom-group-mathfr-5} that
$\tilde G \subset \Aut(\mathfrak h_4)$. Moreover, since these two
groups have dimension $17$, their connected components coincide. It
only remains to show that $\Aut(\mathfrak h_4)$ has no other
connected components apart from the ones given by $\tilde G$. From
Lemma \ref{sec:autom-group-mathfr-4}, we know that any
$f \in \Aut(\mathfrak h_4)$, with matrix $(a_{ij})$ in the basis
$e_1, \ldots, e_6$, is such that
\begin{align}
\label{eq:11}
a_{1j} = a_{2j} = 0 \text{ for } j\geq 3,
&&
a_{3j} = a_{4j} = 0 \text{ for } j \geq 5, && a_{56}=0.
\end{align}
Of course, some of these parameters are dependent on the
others. Lets call
\begin{align*}
A =
\begin{pmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{pmatrix},
&&
B =
\begin{pmatrix}
a_{31} & a_{32} \\
a_{41} & a_{42}
\end{pmatrix},
&&
C =
\begin{pmatrix}
a_{33} & a_{34} \\
a_{43} & a_{44}
\end{pmatrix}.
\end{align*}
Since $[f(e_1), f(e_2)] = -f(e_5)$, we easily check that
$a_{55} = \det A$ and $a_{65} = (A, B)$ where $(\cdot, \cdot)$ is
the bilinear form defined in (\ref{eq:13}). Using that
$[f(e_1), f(e_4)] = [f(e_2), f(e_3)] = -f(e_6)$ and
$[f(e_1), f(e_3)] = [f(e_2), f(e_4)] = 0$, we obtain the following
equations:
\begin{align*}
a_{11}a_{43} + a_{21}a_{33} = a_{22}a_{34} + a_{12}a_{44} & = 0 \\
a_{11}a_{44} + a_{21}a_{34} = a_{22}a_{33} + a_{12}a_{43} & = -a_{66}
\end{align*}
We can rewrite the above system as
\begin{equation*}
\operatorname{adj}(\sigma(C)) A = -a_{66}I_2
\end{equation*}
and hence the only possible solution is
\begin{align*}
C = x \sigma(A), && a_{66} = x \det(A)
\end{align*}
for some $x \neq 0$, as we wanted to show. \end{proof}
\begin{corollary}
$\Aut(\mathfrak h_4)$ has $4$ connected components. Moreover,
\begin{equation*}
\Aut(\mathfrak h_4)/\Aut_0(\mathfrak h_4) \simeq \mathbb Z_2
\oplus \mathbb Z_2.
\end{equation*}
\end{corollary}
Recall that $\mathcal C(\mathfrak h_4)$ is an algebraic variety and according to \cite{salamon-2001}, $\dim \mathcal C( \mathfrak h_4) = 12$.
\begin{corollary}
\label{sec:automorphism-group-3}
$\mathcal A(\mathfrak h_4)$ is a $9$-dimensional smooth manifold. \end{corollary}
\begin{proof}
According to \cite{Andrada_2011}, $\Aut(\mathfrak h_4)$ is
transitive on $\mathcal C(\mathfrak h_4)$. Moreover, every abelian
structure on $\mathfrak h_4$ is conjugated by an automorphism to the
one given, in the canonical basis, by the matrix
\begin{equation*}
J =
\begin{pmatrix}
0 & -1 & 0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 & 1 & 0
\end{pmatrix}.
\end{equation*}
Let $f$ be an automorphism of $\mathfrak h_4$. Assume that $f$ is
represented in the canonical basis by the matrix
\begin{equation*}
\begin{pmatrix}
A & 0 & 0 \\
B & x \sigma(A) & 0 \\
M_1 & M_2 & \Delta(A)
\end{pmatrix}
\end{equation*}
with $A \in \GL_2(\mathbb R)$,
$B, M_1, M_2 \in \mathbb R^{2 \times 2}$ and $x \neq 0$. Notice that
we are keeping the notation of Theorem
\ref{sec:autom-group-mathfr-6} but replacing the matrix $M$ by the
square matrices $M_1$ and $M_2$. It follows that $f$ commutes with
$J$ if and only if, after the usual identifications, $x = 1$,
$A \in \GL_1(\mathbb C) \subset \GL_2(\mathbb R)$, and
$B, M_1, M_2 \in \mathbb C \subset \mathbb R^{2 \times 2}$. In
particular,
\begin{equation*}
\mathcal A(\mathfrak h_4) \simeq \frac{\mathbb R^8 \rtimes (\mathbb
R^4 \rtimes (\GL_2(\mathbb R) \times \GL_1(\mathbb R)))}{\mathbb
C^2 \rtimes (\mathbb C \rtimes \GL_1(\mathbb C))}
\end{equation*}
and the corollary follows. \end{proof}
\subsection{Left-invariant metrics}
We have seen that $\Aut(\mathfrak h_4) = \mathbb R^{2 \times 4} \rtimes G$, where $G$ is the subgroup of $\GL_4(\mathbb R)$ defined in (\ref{eq:9}). We shall study first the action of $G$ on the symmetric space $\Sym_4^+ = \GL_4(\mathbb R)/{\OO(4)}$. Recall, as usual, that a generic element $g \in \Sym_4^+$ has the form \begin{equation*}
\begin{pmatrix}
P & Q \\
Q^T & R
\end{pmatrix} \end{equation*} where $P, R \in \Sym_2^+$. On the other hand, any element $\varphi \in G$ has the form \begin{equation}
\label{eq:14}
\begin{pmatrix}
A & 0 \\
B & x\sigma(A)
\end{pmatrix} \end{equation} where $A \in \GL_2(\mathbb R)$, $B \in \mathbb R^{2 \times 2}$ and $x \in \mathbb R - \{0\}$. The action of $\varphi$ on $g$ is the restriction of the right-action of $\GL_4(\mathbb R)$, which is given by $g \cdot \varphi = \varphi^T g \varphi$. In particular, if we take $A = I_2$ and $x = 1$ in \eqref{eq:14}, there exists a unique $B \in \mathbb R^{2 \times 2}$ such that $g \cdot \varphi$ is block diagonal. More precisely, $B = -R^{-1}Q^T$.
We can now make an element of $G$ with $B=0$ act on a block diagonal representative of $g$ and, since $\GL_2(\mathbb R)$ is transitive on $\Sym_2^+$, we conclude that every orbit of $G$ meets an element of the form \begin{equation*}
\begin{pmatrix}
I_2 & 0 \\
0 & R
\end{pmatrix} \end{equation*} with $R \in \Sym_2^+$. Now, to fully determine the action of $G$ on $\Sym_4^+$ we only need to look at the action of $\OO(2) \times \GL_1(\mathbb R)$ on $\Sym_2^+$, since any element of $G$ which leaves a representative of $g$ of the previous form must have $A\in \OO(2)$. Note that this action is the one given as follows \begin{equation*}
R \cdot (A, x) = x^2A^TRA = x^2A^{-1}RA. \end{equation*} On the other hand, $R$ is conjugated by an orthogonal matrix to a diagonal matrix. So one can choose $x$ in such a way that the first diagonal element of the conjugated matrix is $1$. Hence we obtain the following result.
\begin{lemma}
\label{sec:left-invar-metr}
Every orbit of $G$ on $\Sym_4^+$ intersects exactly once the
subset
\begin{equation*}
\{\diag(1,1,1,r): 0 < r \le 1\}.
\end{equation*} \end{lemma}
\begin{proof}
Let $r > 0$ and denote $g_r = \diag(1, 1,1, r)$. We have seen that
each orbit has an element of the form $g_r$. Suppose that there
exists $\varphi \in G$ and $r' > 0$ such that
$g_r \cdot \varphi = g_{r'}$. Assuming that $\varphi$ has the form
\eqref{eq:14}, a simple calculation shows that
\begin{equation*}
g_r \cdot \varphi =
\begin{pmatrix}
A^TA + B^T \diag(1, r) B & x B^T \diag(1, r) \sigma(A) \\
x \sigma(A)^T \diag(1, r) B & x^2 \sigma(A)^T \diag(1, r)
\sigma(A)
\end{pmatrix}
\end{equation*}
It follows that $B = 0$ and $A \in \OO(2)$. Without loss of
generality, we can assume that $A \in \SO(2)$ rotates an angle
$\theta$ around the origin. Now
\begin{equation*}
x^2 \sigma(A)^T
\begin{pmatrix}
1 & 0 \\
0 & r
\end{pmatrix}
\sigma(A) = x^2
\begin{pmatrix}
\cos^2\theta + r \sin^2\theta & (1 - r) \sin \theta
\cos \theta \\
(1 - r) \sin \theta \cos\theta & \sin^2\theta + r
\cos^2\theta
\end{pmatrix}.
\end{equation*}
So, the only possibilities for $g_r \cdot \varphi = g_{r'}$ are
$r = 1$, which implies $r' = 1$; $\sin \theta = 0$, which implies
$r' = r$; and $\cos \theta = 0$ which implies $r' = 1/r$. From this
the lemma follows. \end{proof}
\begin{remark}
Recall that in the symmetric space
$\Sym^{+}_4= \GL_4(\mathbb R)/{\OO(4)}$ the symmetry, at an element
$p \in \Sym^+_4$ is given by $s_p(q) = p q^{-1} p$. Set
$S' = \{\diag(1, 1, 1, r): r \in \mathbb{R}^+\}$. Then it is
straightforward to see that $s_p(S')=S'$ for each $p\in S'$ and
hence, $S'$ is a totally geodesic submanifold. In fact, if $\alpha$
denotes the second fundamental form of $S'$, for each $p\in S'$ and
$v,w\in S'$ we have that
$-\alpha(v, w) = (ds_p)_p(\alpha(v,w)) = \alpha((ds_p)_p v,(ds_p)_p
w) = \alpha(v, w)$. So $\alpha \equiv 0$. \end{remark}
\begin{theorem}
The moduli space $\mathcal M(H_4)/{\sim}$ of left-invariant metrics
on $H_4$ up to isometric automorphism is homeomorphic to the space
\begin{equation*}
(0, 1] \times \Sym_2^+/\mathbb Z_2,
\end{equation*}
where $\mathbb Z_2$ is the subgroup of $\I(\Sym_2^+)$ generated by
$\sigma
\begin{pmatrix}
a & b \\
b & c
\end{pmatrix}
=
\begin{pmatrix}
a & -b \\
-b & c
\end{pmatrix}
$. Moreover, every left-invariant metric is conjugated by an
automorphism to a unique metric of the form
\begin{equation}
\label{eq:15}
g = \sum_{i = 1}^3 e^i \otimes e^i + r e^4
\otimes e^4 + a e^5 \otimes e^5 + 2b e^5 \otimes e^6 + c e^6 \otimes e^6
\end{equation}
where $0 < r \le 1$, $a, b, c \ge 0$ and $ac - b^2 > 0$. \end{theorem}
\begin{proof}
Let $g$ be a left-invariant metric on $H_4$. Identify $g$ with the
inner product on $\mathfrak h_4$ which in the canonical basis is
represented by the matrix $
\begin{pmatrix}
P & Q \\
Q^T & R
\end{pmatrix}
$ where $P \in \Sym_4^+$, $R \in \Sym_2^+$ and
$Q \in \mathbb R^{4 \times 2}$. With a similar argument as the one
given for Lemma \ref{sec:left-invar-metr} one can assume that
$Q = 0$ and $P = \diag(1, 1, 1, r)$ with $0 < r \le 1$. Denote
$g = g_{r, R}$ to indicate that, up to automorphism, $g$ only
depends on $0 < r \le 1$ and $R \in \Sym_2^+$. Let $\varphi$ be an
automorphism of $\mathfrak h_4$ and let us write $\varphi$ in the
canonical basis as
\begin{equation}
\label{eq:16}
\begin{pmatrix}
A & 0 & 0 \\
B & x\sigma(A) & 0 \\
M_1 & M_2 & \Delta(A, B, x)
\end{pmatrix}
\end{equation}
(see Theorem \ref{sec:autom-group-mathfr-6}). As it follows from
the proof of Lemma \ref{sec:left-invar-metr},
$g_{r, R} \cdot \varphi= g_{r', R'}$ if and only if $A \in \OO(2)$,
$x = \pm 1$ and $B = M_1 = M_2 = 0$. So,
$\Delta(A, B, x) \in \{I_2, \diag(1, -1)\} \cup \{-I_2, \diag(-1,
1)\}$. Since $-I_2$ acts trivially on $\Sym_2^+$, we can assume
that $\Delta(A, B, x) \in \{I_2, \diag(1, -1)\} \simeq \mathbb Z_
2$. Since conjugation by $\diag(1, -1)$ acts as the involution
$\sigma$, we conclude that any left-invariant metric is equivalent
to one of the form $g_{r, R}$, and such a metric is unique if we
require $0 < r \le 1$ and that all the entries on $R$ are non
negative. \end{proof}
\begin{corollary}
Let $g_{r, a, b, c}$ be the left-invariant metric on $H_4$ given in
\eqref{eq:15}. Then the full isometry group of $g_{r, a, b, c}$ is
given by
\begin{equation*}
\I(H_4, g_{r, a, b, c}) =
\begin{cases}
H_4 \rtimes (\OO(2) \rtimes \mathbb Z_2) & r = 1 \text{ and } b = 0 \\
H_4 \rtimes \OO(2) & r = 1 \text{ and } b \neq 0 \\
H_4 \rtimes (\mathbb Z_2 \times \mathbb Z_2) & r \neq 1 \text{
and } b = 0 \\
H_4 \rtimes \mathbb Z_2 & r \neq 1 \text{ and } b \neq 0
\end{cases}
\end{equation*} \end{corollary}
\begin{proof}
We use the same argument as in the proof of Corollary
\ref{sec:left-invar-metr-1}. A generic automorphism $\varphi$ of
$\Aut(\mathfrak h_4)$ can be written as \eqref{eq:16}. Recall that
if $\varphi$ preserves the metric then $B$, $M_1$ and $M_2$ must
vanish and $A \in \OO(2)$. This implies that
\begin{equation*}
\Delta(A, B, x) = \Delta(A, x) =
\begin{pmatrix}
\varepsilon & 0 \\
0 & x \varepsilon
\end{pmatrix}
\end{equation*}
with $\varepsilon \in \{\pm 1\}$. Hence $|x| = 1$ and $x = -1$ is
only possible if $b = 0$. Since
$\sigma: \GL_2(\mathbb R) \to \GL_2(\mathbb R)$ leaves $\OO(2)$
invariant, if $r \neq 1$, then $A \in \{\pm I_2\}$. From the
previous comments the corollary follows. \end{proof}
\section{The case of $\mathfrak h_2 = (0, 0, 0, 0, 12, 34)$}
\subsection{Automorphism group}
Let $e_1, \ldots, e_6$ be the basis of $\mathfrak h_2$ such that the only non trivial brackets are \begin{align*}
[e_1, e_2] = -e_5, && [e_3, e_4] = -e_6. \end{align*} Clearly, $\mathfrak h_2$ is isomorphic to the direct product of two copies of the $3$-dimensional Heisenberg Lie algebra $\mathfrak{heis}_1$. Recall that the only ideals of $\mathfrak h_2$ isomorphic to $\mathfrak{heis}_1$ are the ones corresponding to factors in the decomposition \begin{equation}
\label{eq:18}
\mathfrak h_2 \simeq \mathfrak{heis}_1 \oplus \mathfrak{heis}_1 \end{equation} modulo $[\mathfrak h_2, \mathfrak h_2]$. More precisely, $\mathfrak k$ is such an ideal if and only if \begin{equation*}
\mathfrak k = \spann_{\mathbb R}\{e_1 + Z_1, e_2 + Z_2, e_5\} \qquad
\text{or} \qquad \mathfrak k = \spann_{\mathbb R}\{e_3 + Z_1, e_4 + Z_2, e_6\} \end{equation*} where $Z_1, Z_2$ are two fixed elements in $[\mathfrak h_2, \mathfrak h_2]$. In fact, let $\mathfrak k$ be an ideal of $\mathfrak h_2$ isomorphic to $\mathfrak{heis}_1$. There must exist at least one element $x\in \mathfrak k$ such that $e^i(x)\neq 0$ for some $i=1,\ldots,4$. This implies that either $e_5$ or $e_6$ belongs to $\mathfrak{k}$. Assume first that $e_5\in \mathfrak{k}$. Since the center of $\mathfrak k$ is one dimensional, $e_6\notin \mathfrak{k}$ and so $e^3(\mathfrak k) = e^4(\mathfrak k) = 0$. The other case is analogous. Now if $\varphi: \mathfrak h_2 \to \mathfrak h_2$ is an automorphism, then the induced linear map $\tilde \varphi: \mathfrak h_2/[\mathfrak h_2, \mathfrak h_2] \to \mathfrak h_2/[\mathfrak h_2, \mathfrak h_2]$ either preserves or swaps the factors of the decomposition $\mathfrak h_2/[\mathfrak h_2, \mathfrak h_2] \simeq \spann_{\mathbb
R}\{e_1, e_2\} \oplus \spann_{\mathbb R}\{e_3, e_4\}$. Notice that the involution $\varphi_0: \mathfrak h_2 \to \mathfrak h_2$, which is given by \begin{equation}
\label{eq:20}
\begin{pmatrix}
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 0
\end{pmatrix}, \end{equation} in the canonical basis, is an automorphism of $\mathfrak h_2$ that reverses the decomposition \eqref{eq:18}.
\begin{theorem}
\label{sec:automorphism-group-1}
There exists an isomorphism of Lie groups
\begin{equation}
\label{eq:19}
\Aut(\mathfrak h_2) \simeq \mathbb R^8 \rtimes ((\GL_2(\mathbb R)
\times \GL_2(\mathbb R)) \rtimes \mathbb Z_2).
\end{equation}
More precisely, every automorphism of $\mathfrak h_2$ can be
represented in the canonical basis by a matrix of the form
\begin{equation}
\label{eq:21}
\begin{pmatrix}
A & 0 & 0 \\
0 & B & 0 \\
M_1 & M_2 & \Delta(A, B)
\end{pmatrix}
\qquad \text{or} \qquad
\begin{pmatrix}
0 & A & 0 \\
B & 0 & 0 \\
M_1 & M_2 & \Delta'(A, B)
\end{pmatrix}
\end{equation}
where $A, B \in \GL_2(\mathbb R)$, $M_1, M_2 \in \mathbb R^{2 \times
2}$, and
\begin{equation*}
\Delta(A, B) =
\begin{pmatrix}
\det A & 0 \\
0 & \det B
\end{pmatrix},
\qquad \qquad
\Delta'(A, B) =
\begin{pmatrix}
0 & \det A \\
\det B & 0
\end{pmatrix}.
\end{equation*}
In particular, $\Aut(\mathfrak h_2)$ has $8$ connected components
and
\begin{equation*}
\Aut(\mathfrak h_2) / {\Aut_0(\mathfrak h_2)} \simeq (\mathbb Z_2
\times \mathbb Z_2) \rtimes \mathbb Z_2 \simeq \mathbb Z_4 \rtimes
\mathbb Z_2
\end{equation*}
is isomorphic to the dihedral group $D_4$. \end{theorem}
Recall the following identifications. The subgroup $\mathbb Z_2$ is identified with the subgroup of $\Aut(\mathfrak h_2)$ generated by $\varphi_0$ in \eqref{eq:20}. The subgroups isomorphic to $\GL_2(\mathbb R)$ correspond to the automorphisms with $M_1 = M_2 = 0$ and $B = I_2$ or $A = I_2$. Finally the normal subgroup isomorphic to $\mathbb R^8$ is obtained in the connected component of the identity with $A = B = I_2$.
\begin{proof}
[{\proofname} of Theorem \Ref{sec:automorphism-group-1}] It follows
from the discussion at the beginning of this subsection. In fact,
any Lie algebra automorphism $\mathfrak h_2 \to \mathfrak h_2$,
which preserves or swaps the factors of the decomposition
\eqref{eq:18} modulo $[\mathfrak h_2, \mathfrak h_2]$, has one of
the forms described in \eqref{eq:21} and it is easy to verify that
all of these maps are automorphisms. \end{proof}
\subsection{Left-invariant metrics}
We follow the same approach as in the previous cases, so let us first study the action of $(\GL_2(\mathbb R) \times \GL_2(\mathbb R)) \rtimes \mathbb Z_2$ on $\Sym_4^+ = \GL_4(\mathbb R)/{\OO(4)}$. We do not lose generality by considering the action of the diagonal subgroup $\GL_2(\mathbb R) \times \GL_2(\mathbb R) \subset \GL_4(\mathbb R)$. Recall that if we write a generic element $g \in \Sym_4^+$ as \begin{equation*}
\begin{pmatrix}
P & Q \\
Q^T & R
\end{pmatrix}, \end{equation*} with $P, R \in \Sym_2^+$ and $Q \in \mathbb R^{2 \times 2}$, then the above action is given by \begin{equation*}
g \cdot (A, B) =
\begin{pmatrix}
A^T P A & A^T Q B \\
B^T Q^T A & B^T R B
\end{pmatrix} \end{equation*} and so the orbit of every $g$ meets an element of the form $g = \begin{pmatrix}
I_2 & Q \\
Q^T & I_2 \end{pmatrix} $. Therefore we can restrict our attention to the action of $\OO(2) \times \OO(2)$ on $\mathbb R^{2 \times 2}$ given by \begin{equation}
\label{eq:23}
Q \cdot (A, B) = A^T Q B. \end{equation}
Recall that the positive definite inner product \begin{equation}
\label{eq:22}
\langle Q_1, Q_2\rangle = \frac12\tr(Q_1Q_2^T) \end{equation} makes $\mathbb R^{2 \times 2}$ an Euclidean space and moreover, the action \eqref{eq:23} is isometric.
\begin{lemma}
\label{sec:left-invar-metr-3}
Let $Q \in \mathbb R^{2 \times 2}$ and let $\mathcal O_Q$ be the
orbit of $Q$ under the action of $\OO(2) \times \OO(2)$ given by
$Q \cdot (A, B) = A^T Q B$. Then:
\begin{enumerate}
\item \label{item:10} $\mathcal O_Q$ intersect the subspace of
diagonal matrices.
\item \label{item:11} Moreover, $\mathcal O_Q$ contains exactly one
element of the form $\diag(a, b)$ with $0 \le a \le b$.
\end{enumerate} \end{lemma}
\begin{proof}
The decomposition $\mathbb R^{2 \times 2} = \mathfrak{so}(2) \oplus \Sym_2$ is
orthogonal with respect to the metric given in \eqref{eq:22}. Let us
consider first the isometric action of $\SO(2)$ on
$\mathbb R^{2 \times 2}$ given by the restriction to the connected
component of the first factor: $Q \cdot A = A^T Q$ and let
$\mathcal O'_Q$ be the orbit of $Q$ under this action. It follows
that $\mathcal O'_Q \cap \Sym_2 \neq \varnothing$. In fact, we can
assume, by multiplying by a multiple of $I_2$ that $\|Q\| =
1$. Hence $\mathcal O'_Q$ is a great circle in the unit sphere
$S^3 \subset \mathbb R^{2 \times 2}$ and it must intersect the
$3$-dimensional subspace $\Sym_2$. Item \eqref{item:10} follows
by noticing that if $Q \in \Sym_2$, then there is $A \in \OO(2)$
such that $A^T Q A$ is diagonal.
Now suppose that $\diag(a, b) \in \mathcal O_Q$. This implies that
$\diag(\varepsilon_1 a, \varepsilon_2 b)$ and
$\diag(\varepsilon_1 b, \varepsilon_2 a)$ also belong to
$\mathcal O_Q$, for any
$\varepsilon_1, \varepsilon_2 \in \{\pm 1\}$. So, $\mathcal O_Q$ has
an element of the form $\diag(a, b)$ with $0 \le a \le b$. Suppose
that there exists $A, B \in \OO(2)$ such that
$A^T \diag(a, b) B = \diag(a', b')$, for some $0 \le a' \le
b'$. Since the action is isometric we can assume that
$a^2 + b^2 = (a')^2 + (b')^2 = 1$. Moreover, we do not lose
generality by assuming that $A, B \in \SO(2)$, say
\begin{align}
\label{eq:25}
A =
\begin{pmatrix}
\cos t & -\sin t \\
\sin t & \cos t
\end{pmatrix}, && B =
\begin{pmatrix}
\cos s & -\sin s \\
\sin s & \cos s
\end{pmatrix},
\end{align}
which yields to the equations
\begin{align*}
a \cos t & = a' \cos s, &
b \sin t & = a' \sin s, \\
a \sin t & = b' \sin s, &
b \cos t & = b' \cos s.
\end{align*}
We can further assume that
$a, b, a', b', \cos t, \sin t, \cos s, \sin s$ are all non zero,
otherwise the result holds trivially. This implies $ab' = ba'$ and
$aa' = bb'$. From this it is easy to see that $a = b$, and hence
$a' = b'$, which proves \eqref{item:11}. \end{proof}
\begin{remark}
Observe that the right $(\OO(2) \times \OO(2))$-action on
$\mathfrak{gl}_2(\mathbb R)$ of Lemma~\ref{sec:left-invar-metr-3}
coincides with the isotropy representation of the symmetric space
$\OO(2,2) / (\OO(2) \times \OO(2))$, i.e., the Grassmannian of
positive definite $2$-planes in $\mathbb{R}^4$ with the metric of
signature $2$. This readily implies that there must always exist a
section given by the diagonal matrices. \end{remark} \begin{remark}
We can also give a geometric argument for the proof of part
\eqref{item:11} of Lemma~\ref{sec:left-invar-metr-3}. Consider the
geodesics $\gamma(t) = A^T \diag(a, b)$ and
$\beta(s) = \diag(a', b') B^T$ of
$S^3 \subset \mathbb R^{2 \times 2}$, where $A$ and $B$ are as in
(\ref{eq:25}). The image of $\gamma(t)$ is the intersection of $S^3$
with the plane $\pi_1 \subset \mathbb R^{2 \times 2}$ generated by
$\diag(a, b)$ and $
\begin{pmatrix}
0 & b \\
-a & 0
\end{pmatrix}
$. Similarly, the image of $\beta(s)$ is the intersection of $S^3$
with the plane $\pi_2$ generated by $ \diag(a', b')$ and $
\begin{pmatrix}
0 & a' \\
-b' & 0
\end{pmatrix}
$. So, assuming $0 \le a \le b$, $0 \le a' \le b'$ and
$a^2 + b^2 = (a')^2 + (b')^2 = 1$, the condition
$(a, b) \neq (a', b')$ implies $\pi_1 \cap \pi_2 = \{0\}$ and
therefore $A^T\diag(a, b) \neq \diag(a', b') B^T$ for all
$A, B \in \OO(2)$. \end{remark}
\begin{theorem}
\label{sec:left-invar-metr-5}
Let $H_2$ be the simply connected Lie group with Lie algebra
$\mathfrak h_2$. The moduli space $\mathcal M(H_2)/{\sim}$ of
left-invariant metrics on $H_2$ up to isometric automorphism is
homeomorphic to the space
\begin{equation*}
\{(a, b) \in \mathbb R^2: 0 \le a \le b < 1\} \times \Sym_2^+/\mathbb Z_2,
\end{equation*}
where $\mathbb Z_2$ is the subgroup of $\I(\Sym_2^+)$ generated by
the involution
\begin{equation*}
\sigma
\begin{pmatrix}
E & F \\
F & G
\end{pmatrix}
=
\begin{pmatrix}
E & -F \\
-F & G
\end{pmatrix}.
\end{equation*}
Moreover, every left-invariant metric on $H_2$ is conjugated by an
automorphism to a unique metric of the form
\begin{equation}
\label{eq:26}
g = \sum_{i = 1}^4 e^i \otimes e^i + 2a e^1 \otimes e^3 + 2b
e^2 \otimes e^4 + E e^5 \otimes e^5 + 2F e^5 \otimes e^6 + G e^6
\otimes e^6,
\end{equation}
where $0 \le a \le b$, $E, F, G \ge 0$ and $EG - F^2 > 0$. \end{theorem}
\begin{proof}
From Theorem \ref{sec:automorphism-group-1},
$\Aut_0(\mathfrak h_2) \simeq \mathbb R^8 \rtimes (\GL_2(\mathbb R)
\times \GL_2(\mathbb R))_0$, and hence we can use a similar argument
as in Lemma \ref{sec:left-invar-metr-6} to conclude that any
left-invariant metric on $H_2$ is equivalent to a block diagonal
metric $g = \diag(g_1, g_2)$ with $g_1 \in \Sym_4^+$ and
$g_2 \in \Sym_2^+$. From the discussion at the beginning of this
subsection and Lemma \ref{sec:left-invar-metr-3} we know that there
exist unique $0 \le a \le b < 1$ such that $g_1$ is equivalent under
the action of $\GL_2(\mathbb R) \times \GL_2(\mathbb R)$ to
\begin{equation}
\label{eq:27}
\begin{pmatrix}
1 & 0 & a & 0 \\
0 & 1 & 0 & b \\
a & 0 & 1 & 0 \\
0 & b & 0 & 1
\end{pmatrix}.
\end{equation}
Recall that we need to impose the additional condition $b < 1$ in
order to get that the matrix \eqref{eq:27} is positive definite. To
complete the proof of the theorem, one only needs to note that the
isotropy of $(\GL_2(\mathbb R) \times \GL_2(\mathbb R))_0$ is
isomorphic to $\mathbb Z_ 2 \times \mathbb Z_2$. This group induces,
after making the action effective, the action of $\mathbb Z_2$ on
$\Sym_2^+$ given by the involution $\sigma$. \end{proof}
\begin{corollary}
Let $g = g_{a, b, E, F, G}$ be the left-invariant metric on $H_2$
defined in (\ref{eq:26}). Then the full isometry group of $g$ is
given by
\begin{equation*}
\I(H_2,g) =
\begin{cases}
H_2 \rtimes ((\OO(2) \times \OO(2)) \rtimes \mathbb Z_2), & a =
b = 0, \, F = 0, \, E = G \\
H_2 \rtimes (\OO(2) \times \OO(2)), & a = b = 0, \, F = 0, \, E
\neq G \\
H_2 \rtimes (\mathrm{S}(\OO(2) \times \OO(2)) \rtimes \mathbb
Z_2), & a =
b = 0, \, F \neq 0, \, E = G \\
H_2 \rtimes \mathrm{S}(\OO(2) \times \OO(2)), & a = b = 0, \, F
\neq 0, \,
E \neq G \\
H_2 \rtimes (\diag(\OO(2) \times \OO(2)) \rtimes \mathbb Z_2), &
a = b \neq 0, \, E = G \\
H_2 \rtimes \diag(\OO(2) \times \OO(2)), & a = b \neq 0, \, E
\neq G \\
H_2 \rtimes D_4, & 0 \le a < b, \, E = G \\
H_2 \rtimes (\mathbb Z_2 \times \mathbb Z_2), & 0 \le a < b, E
\neq G. \\
\end{cases}
\end{equation*} \end{corollary}
\begin{proof}
From Theorem \ref{sec:left-invar-metr-5}, we can identify the
left-invariant metric $g = g_{a, b, E, F, G}$ with the symmetric
positive definite matrix
\begin{equation*}
\label{eq:28}
g =
\begin{pmatrix}
1 & 0 & a & 0 & 0 & 0 \\
0 & 1 & 0 & b & 0 & 0 \\
a & 0 & 1 & 0 & 0 & 0 \\
0 & b & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & E & F \\
0 & 0 & 0 & 0 & F & G
\end{pmatrix}.
\end{equation*}
On the other hand, from Theorem \ref{sec:automorphism-group-1}, the
discrete group $\Aut(\mathfrak h_2) / {\Aut_0(\mathfrak h_2)}$ is
isomorphic to the dihedral group
$D_4 \simeq (\mathbb Z_2 \times \mathbb Z_2) \rtimes \mathbb Z_2$,
where each $\mathbb Z_2$ it is generated by the projection of the
involutive automorphisms given by
\begin{align*}
\varphi_1 & = \diag(-1,1,1,1,-1,1) \\
\varphi_2 & = \diag(1,1,-1,1,1,-1) \\
\varphi_3 & =
\begin{pmatrix}
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 0
\end{pmatrix}
\end{align*}
Moreover, $\varphi_1$ and $\varphi_2$ generate the two connected
components on each factor of $\OO(2) \times \OO(2)$, after their
natural inclusion into $\Aut(\mathfrak h_2)$, and $\varphi_3$ gives
the bijection between the block diagonal and anti-diagonal
automorphisms described in Theorem \ref{sec:automorphism-group-1}.
The following facts are easy to verify:
\begin{enumerate}
\item $g \cdot (\varphi_1 \varphi_2) = g \cdot (\varphi_2 \varphi_1)
= g$;
\item $g \cdot \varphi_1 = g$ if and only if $a = 0$ and $F = 0$;
\item $g \cdot \varphi_2 = g$ if and only if $a = 0$ and $F = 0$;
\item $g \cdot \varphi_3 = g$ if and only if $E = G$.
\end{enumerate}
So, in order to compute all the isometric automorphisms we can
restrict our attention to the action of the connected component
$\SO(2) \times \SO(2)$. This raises three possibilities. Firstly, if
$a = b = 0$, then it is clear that $g \cdot \varphi = g$ for all
$\varphi \in \SO(2) \times \SO(2)$. Secondly, if $a = b \neq 0$,
and $\varphi = (A, B)$ is such that $g \cdot \varphi = A^{-1} g B$,
then $A = B$. Finally, if $a < b$, the only $\varphi \in \SO(2)
\times \SO(2)$ such that $g \cdot \varphi$ are $\varphi = \pm
I_6$. This completes the proof of the corollary. \end{proof}
\section{The case of $\mathfrak h_9 = (0, 0, 0, 0, 12, 14 + 25)$}
This is the most difficult case to describe, since as we will see, $\Aut(\mathfrak h_9)$ does not admit a normal abelian subgroup such that $\Aut(\mathfrak h_9)$ is the semi-direct product of this subgroup and an algebraic subgroup which descends down to the quotient $\mathfrak h_9/[\mathfrak h_9, \mathfrak h_9]$. According to our notation $\mathfrak h_9$, has a basis $e_1, \ldots, e_6$ such that $de^1 = \cdots = de^4 = 0$, $de^5 = e^{12}$ and $de^6 = e^{14} + e^{25}$, or equivalently, the non trivial brackets are \begin{align*}
[e_1, e_2] = -e_5 && [e_1, e_4] = [e_2, e_5] = -e_6. \end{align*} In particular, $\mathfrak h_9$ is $3$-step nilpotent. We find it convenient change to the basis $\hat e_1 = e_2$, $\hat e_2 = e_1$, $\hat e_3 = e_4$, $\hat e_4 = e_3$, $\hat e_5 = e_5$ and $\hat e_6 = e_6$ where the non trivial brackets are \begin{align}
\label{eq:29}
[\hat e_1, \hat e_2] = \hat e_5 && [\hat e_1, \hat e_5] = [\hat e_2,
\hat e_3] = -\hat e_6. \end{align}
Notice that with respect to this basis we have $\mathfrak z(\mathfrak h_9) = \spann_{\mathbb R}\{\hat e_4, \hat e_6\}$ and $[\mathfrak h_9, \mathfrak h_9] = \spann_{\mathbb R}\{\hat e_5, \hat e_6\}$.
\subsection{Automorphism group}
\begin{theorem}
\label{sec:automorphism-group-2}
With respect to the basis $\hat e_1, \ldots, \hat e_6$ every
automorphism $\varphi \in \Aut(\mathfrak h_9)$ has the form
\begin{equation}
\label{eq:30}
\begin{pmatrix}
a_{11} & 0 & 0 & 0 & 0 & 0 \\
a_{21} & a_{22} & 0 & 0 & 0 & 0 \\
a_{31} & a_{32} & a_{11}^{2} & 0 & 0 & 0 \\
a_{41} & a_{42} & a_{43} & a_{44} & 0 & 0 \\
a_{51} & a_{52} & -a_{11} a_{21} & 0 & a_{11} a_{22} & 0 \\
a_{61} & a_{62} & a_{63} & a_{64} & a_{22} a_{31} - a_{21}
a_{32} - a_{11} a_{52} & a_{11}^{2} a_{22}
\end{pmatrix}. \end{equation}
In particular, $\Aut(\mathfrak h_9)$ is a $15$-dimensional solvable Lie group, which has $8$ connected components and $\Aut(\mathfrak h_9) / {\Aut_0(\mathfrak h_9)} \simeq \mathbb Z_2 \times \mathbb Z_2 \times \mathbb Z_2$. \end{theorem}
\begin{proof}
We show first that the matrix $(a_{ij})$ of $\varphi$ in the given
basis is lower triangular. It follows from \eqref{eq:29} that
$\hat e^1(\varphi(\hat e_2)) = 0$. In fact,
$(\ad_{\varphi(\hat e_2)})^2 = (\ad_{\hat e_2})^2 = 0$ and so the
$\hat e_1$-component of $\varphi(\hat e_2)$ must vanish. Similarly,
since
$\dim(\ker \ad_{\varphi(\hat e_3)}) = \dim(\ker \ad_{\hat e_i}) =
5$, then
$\hat e^1(\varphi(\hat e_3)) = \hat e^2(\varphi(\hat e_3)) =
0$. Recall that $\varphi$ preserves the subalgebras
$\mathfrak z(\mathfrak h_9)$, $[\mathfrak h_9, \mathfrak h_9]$,
$\mathfrak z(\mathfrak h_9) + [\mathfrak h_9, \mathfrak h_9]$ and
$\mathfrak z(\mathfrak h_9) \cap [\mathfrak h_9, \mathfrak
h_9]$. This implies that $\hat e^i(\varphi(\hat e_j)) = 0$ for all
$i = 1, 2, 3$, $j = 4, 5, 6$; $\hat e^4(\varphi(\hat e_j)) = 0$ for
$j = 5, 6$, and $\hat e^5(\varphi(\hat e_6)) = 0$. Moreover, since
$\hat e^4 \in \mathfrak z(\mathfrak h_9)$, one has
$\hat e^5(\varphi(\hat e_4)) = 0$.
On the other side,
\begin{align*}
\hat e^5(\varphi(\hat e_5)) & = \hat e^5([\varphi(\hat e_1),
\varphi(\hat e_2)]) = a_{11} a_{22} \\
\hat e^6(\varphi(\hat e_5)) & = \hat e^6([\varphi(\hat e_1),
\varphi(\hat e_2)]) = a_{22} a_{31}
- a_{21} a_{32} - a_{11} a_{52},
\end{align*}
and since
$a_{11}^2a_{22} = [\varphi(\hat e_1), \varphi(\hat e_5)] =
[\varphi(\hat e_2), \varphi(\hat e_3)] = -\varphi(\hat e_6)$ we
conclude that
\begin{align*}
\hat e^3(\varphi(\hat e_3)) & = a_{11}^2, &
\hat e^5(\varphi(\hat e_3)) & = -a_{11} a_{21}, &
\hat e^6(\varphi(\hat e_6)) & = a_{11}^2 a_{22}.
\end{align*}
This proves that $\varphi$ has the form \eqref{eq:30}. It is easy to
see that any linear map of this form is an automorphism of
$\mathfrak h_9$. (For instance, one can compute the dimension of
$\Der(\mathfrak h_9)$ and check that it equals $15$. The
automorphisms with $a_{11} = a_{22} = a_{44} = 1$ are the ones in the
exponential of the nilradical of $\Der(\mathfrak h_9)$.) \end{proof}
\begin{remark}
Observe that from the previous theorem, $\Aut(\mathfrak h_9)$ is not
the semi-direct product of an abelian normal subgroup. \end{remark}
\subsection{Left-invariant metrics}
Let $T_6$ be the subgroup of $\GL_6(\mathbb R)$ of lower triangular matrices, and denote by $T_6^+$ the normal subgroup of $T_6$ of matrices whose diagonal entries are all positive. It is known that $T_6^+$ acts simply transitively on $\Sym_6^+$ with the restriction of the action of $\GL_6(\mathbb R)$. Moreover, this action is proper since it is equivalent to the action by right multiplication of $T_6^+$ on itself.
\begin{theorem}
Let $H_9$ be the simply connected Lie group with Lie algebra
$\mathfrak h_9$. The moduli space $\mathcal M(H_9)/{\sim}$ is a
$6$-dimensional smooth manifold. Moreover, $\mathcal M(H_9)/{\sim}$
is diffeomorphic to the homogeneous manifold
$T_6^+/\Aut_0(\mathfrak h_9)$ and every left-invariant metric on
$H_9$ is equivalent to a unique metric of the form
\begin{align}\label{eq:32}
g & = \hat e^1 \otimes \hat e^1 + \hat e^2 \otimes \hat e^2 + (A^2 + D^2) \hat e^3 \otimes
\hat e^3 + DE \hat e^3 \otimes \hat e^4 \notag \\
& \qquad + BD \hat e^3 \otimes \hat e^5 + (E^2 + 1)\hat e^4 \otimes\hat e^4 + BE
\hat e^4 \otimes \hat e^5 + (B^2 + F^2)\hat e^5 \otimes\hat e^5 \\
& \qquad\qquad + CF\hat e^5 \otimes\hat e^6 + C^2\hat e^6 \otimes\hat e^6, \notag
\end{align}
where $A, B, C > 0$, $D, E, F \in \mathbb R$, and
$\hat e^1,\ldots,\hat e^6$ is the dual basis of
$\hat e_1,\ldots,\hat e_6$. \end{theorem}
\begin{proof}
Since $\mathcal M(H_9) \simeq \Sym_6^+$ is connected, it is enough
to consider the orbits of $\Aut_0(\mathfrak h_9)$ in $\Sym_6^+$. Let
$\Phi: T_6^+ \to \Sym_6^+$ given by $\Phi(X) = X^tX$. As we mention
above, the action of $\Aut_0(\mathfrak h_9)$ on $\Sym_6^+$ is
equivalent via $\Phi$ to the action of $\Aut_0(\mathfrak h_9)$ by
right multiplication. Note that the submanifold $\Sigma$ of
$T_6^+$ given by
\begin{equation}
\label{eq:31}
\Sigma = \left\{
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & A & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & D & E & B & 0 \\
0 & 0 & 0 & 0 & F & C
\end{pmatrix}: A, B, C > 0, \, D, E, F \in \mathbb R
\right\}
\end{equation}
is a slice for the action of $\Aut_0(\mathfrak h_6)$. Moreover, the
map $\Sigma \times \Aut_0(\mathfrak h_9) \to T_6^+$ given by
$(S, \varphi) \mapsto S \varphi$ is a diffeomorphism. In fact, lets
see that for any $X \in T_6^+$, there exist unique $S \in \Sigma$
and $\varphi \in \Aut_0(\mathfrak h_9)$ such that $S \varphi =
X$. Denote $X = (x_{ij})$ and assume that $\varphi$ and $S$ are as
in \eqref{eq:30} and \eqref{eq:31} respectively. It is clear that
$A, B, C$ and the elements on the diagonal of $\varphi$ are uniquely
determined by $X$ and so we can assume that $A, B, C$ and all the
elements on the diagonals of $X$ and $\varphi$ are equal to
$1$. Moreover, since the principal $4 \times 4$ block of $S \varphi$
coincides with the one of $\varphi$, we can assume further that
$a_{ij} = x_{ij} = 0$ for all $2 \le i \le 4$ and $1 \le j \le
3$. The equation $S \varphi = X$ has now the form
\begin{equation*}
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
a_{51} & a_{52} & D & E & 1 & 0 \\
F a_{51} + a_{61} & F a_{52} + a_{62} & a_{63} & a_{64} & F - a_{52} & 1
\end{pmatrix} =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
x_{51} & x_{52} & x_{53} & x_{54} & 1 & 0 \\
x_{61} & x_{62} & x_{63} & x_{64} & x_{65} & 1
\end{pmatrix}
\end{equation*}
which clearly has a unique solution in $S, \varphi$.
So, $\Phi(\Sigma)$ is a full slice in $\mathcal M(H_9)$ for the
action of the action of the automorphism group of $\mathfrak h_9$
and every left-invariant metric on $H_9$ is equivalent to a unique
metric which in the basis given by \eqref{eq:29} is represented by
the matrix
\begin{equation}
\label{eq:70}
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & A^{2} + D^{2} & D E & B D & 0 \\
0 & 0 & D E & E^{2} + 1 & B E & 0 \\
0 & 0 & B D & B E & B^{2} + F^{2} & C F \\
0 & 0 & 0 & 0 & C F & C^{2}
\end{pmatrix},
\end{equation}
with $A, B, C > 0$, as we wanted to show. \end{proof}
\begin{corollary}
Let $g$ be the left-invariant metric on $H_9$ given by
\eqref{eq:32}. Then the full isometry group of $g$ is given by
\begin{equation*}
\I(H_9, g) \simeq H_9\rtimes \mathbb{Z}_2^k,
\end{equation*}
where $k$ is the number of null parameters among $D, E$ and $F$. \end{corollary}
\begin{proof}
Let $\varphi$ be an automorphism of $\mathfrak{h}_9$ and let
$\varphi = N + S$ be Jordan-Chevalley decomposition of $\varphi$,
that is, $N$ is nilpotent and $S$ is semisimple such that $NS =
SN$. Since $N$ and $S$ can be obtained as polynomials on $\varphi$,
it follows from Theorem \ref{sec:automorphism-group-2} that, in the
basis $\hat e_1, \ldots, \hat e_6$, $N$ is a strictly lower
triangular matrix and $S$ is a lower triangular matrix, such that
its diagonal elements coincide with the diagonal elements of
$\varphi$. Since the isometric automorphisms of $H_9$ (which are
induced by isometric automorphisms of $\mathfrak{h}_9$) constitute
a compact subgroup of the isometry group of $H_9$, the matrix $S$
must have all its diagonal entrances equal to $\pm 1$.
Assume first that $S = I_6$. Since the coefficients
$(N^k)_{i + 1, i} = 0$ for all $k \geq 2$, $i = 1, \ldots, 5$ and
\begin{equation*}
\varphi^n = \sum_{k = 0}^n \binom{n}{k} N^k,
\end{equation*}
we get that $(\varphi^n)_{i+1, i} = n N_{i+1, i}$. Since $\varphi^n$
is an isometric automorphism (which must be bounded), we conclude
that $N_{i+1,i}=0$ for all $i = 1, \ldots, 5$. Applying the same
argument one can prove that $N=0$.
If $S\neq I_6$, we consider the isometric automorphism
$\varphi^2=S^2+2SN+N^2$. Since all the diagonal entrances of
$\varphi^2$ are equal to $1$, its Jordan-Chevalley decomposition is
given by $\varphi^2=N'+I_6$. We can apply the previous reasoning to
$\varphi^2$ and conclude that $N'=0$. On the other hand, $S^2$ is
semisimple and $2SN+N^2$ is nilpotent. Hence $S^2=I_6$ and
\begin{equation}
\label{eqaux1}
2SN+N^2=N'=0.
\end{equation}
But $SN$ and $N^2$ have minimal polynomials of different degrees,
unless $N=0$. So, equation \eqref{eqaux1} holds only if $N=0$.
We conclude that an isometric automorphism of $\mathfrak{h_9}$ must
be a subgroup of
\begin{equation*}
\mathbb{Z}_2^3 = \{\diag(\varepsilon_1, \varepsilon_2,1,
\varepsilon_3, \varepsilon_1 \varepsilon_2, \varepsilon_2):
\varepsilon_1,\varepsilon_2,\varepsilon_3=\pm1\}.
\end{equation*}
Now a straightforward computation concludes the proof. \end{proof}
\section{Hermitian metrics} \label{sec:hermitian-metrics}
\subsection{Hermitian structures on $\mathfrak h_5$} \label{sec:hermitian-iwasawa}
Recall that, from \cite{Di_Scala_2012}, every left-invariant metric on $H_5$ is equivalent by an automorphism to one and only one metric which is represented in the standard basis $e_1, \ldots, e_6$ by the symmetric positive definite matrix \begin{equation}
\label{eq:33}
g =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & r & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & s & 0 & 0 \\
0 & 0 & 0 & 0 & E & F \\
0 & 0 & 0 & 0 & F & G
\end{pmatrix}, \end{equation} where $0 < s \le r \le 1$, $0 < E$, $0 \le F$, $0 < G$ and $0 < E G - F^2$. It is not difficult to see that the left-invariant almost Hermitian structures, with respect to $g$, which preserve the orientation induced by the standard basis, are parameterized by two $2$-spheres. More precisely, any orientation preserving, left-invariant almost Hermitian structure (in the standard basis) has either the form \begin{equation}
\label{eq:34}
J_1 =
\begin{pmatrix}
0 & -a \sqrt{r} & -b & -c \sqrt{s} & 0 & 0 \\[.4pc]
\frac{a}{\sqrt{r}} & 0 & -\frac{c}{\sqrt{r}} & \frac{b
\sqrt{s}}{\sqrt{r}} & 0 & 0 \\[.4pc]
b & c \sqrt{r} & 0 & -a \sqrt{s} & 0 & 0 \\[.4pc]
\frac{c}{\sqrt{s}} & -\frac{b \sqrt{r}}{\sqrt{s}} &
\frac{a}{\sqrt{s}} & 0 & 0 & 0 \\[.4pc]
0 & 0 & 0 & 0 & -\frac{F}{\sqrt{-F^{2} + E G}} &
-\frac{G}{\sqrt{-F^{2} + E G}} \\[.4pc]
0 & 0 & 0 & 0 & \frac{E}{\sqrt{-F^{2} + E G}} &
\frac{F}{\sqrt{-F^{2} + E G}}
\end{pmatrix} \end{equation} or the form \begin{equation}
\label{eq:35}
J_2 =
\begin{pmatrix}
0 & -a \sqrt{r} & -b & -c \sqrt{s} & 0 & 0 \\[.4pc]
\frac{a}{\sqrt{r}} & 0 & \frac{c}{\sqrt{r}} & -\frac{b
\sqrt{s}}{\sqrt{r}} & 0 & 0 \\[.2pc]
b & -c \sqrt{r} & 0 & a \sqrt{s} & 0 & 0 \\[.4pc]
\frac{c}{\sqrt{s}} & \frac{b \sqrt{r}}{\sqrt{s}} &
-\frac{a}{\sqrt{s}} & 0 & 0 & 0 \\[.4pc]
0 & 0 & 0 & 0 & \frac{F}{\sqrt{-F^{2} + E G}} &
\frac{G}{\sqrt{-F^{2} + E G}} \\[.4pc]
0 & 0 & 0 & 0 & -\frac{E}{\sqrt{-F^{2} + E G}} &
-\frac{F}{\sqrt{-F^{2} + E G}}
\end{pmatrix}, \end{equation} where $(a, b, c) \in S^2$. So, we distinguish two cases in order to determine when $J_1$ and $J_2$ are integrable.
\subsubsection{The case of $J_1$}
In order to simplify some calculations we denote \begin{align*}
\Delta = E G - F^2, && \alpha = \frac{\sqrt r + \sqrt s}{1 + \sqrt{r
s}}. \end{align*} With much patience and after long computations one can see that the non-trivial equations on the integrability of $J_1$ are given by \begin{align}
a & = \frac{(1 - b^2) \sqrt \Delta}{G \alpha} \label{eq:37}, \\
a & = \frac{(1 - c^2) \alpha \sqrt \Delta}{E} \label{eq:38}, \\
Fa & = b c \sqrt \Delta \label{eq:39}, \\
Fb & = \frac{Ec}{\alpha} - a c \sqrt \Delta \label{eq:40}, \\
Fc & = G \alpha b - a b \sqrt \Delta \label{eq:41}. \end{align}
It follows from (\ref{eq:37}) and (\ref{eq:38}) that $a > 0$ and, in the generic case $F \neq 0$, (\ref{eq:39}) says that $b$ and $c$ must have the same sign. Moreover, it is not hard to see that (\ref{eq:39}) follows from (\ref{eq:37}) and (\ref{eq:38}) by multiplying these two equations. With a similar argument, one can see that (\ref{eq:40}) and (\ref{eq:41}) also follow from (\ref{eq:37}) and (\ref{eq:38}). Now we can combine (\ref{eq:37}), (\ref{eq:38}) and the condition $a^2 + b^2 + c^2 = 1$ to obtain the following quadratic equation in $a$: \begin{equation}
\label{eq:42}
a^2 - a \frac1{\sqrt \Delta} \left(\frac{E}{\alpha} + G
\alpha\right) + 1 = 0. \end{equation} With a little algebra we see that the condition on (\ref{eq:42}) for having real roots is equivalent to the tautology $\left(\frac E \alpha - G \alpha\right)^2 + 4 F^2 \geq 0$. So the solution between $0$ and $1$ is \begin{equation}
\label{eq:44}
a = \frac12 \left( \frac1{\sqrt \Delta} \left(\frac{E}{\alpha} + G
\alpha \right) - \sqrt{\frac 1 \Delta \left(\frac E \alpha + G
\alpha\right)^2 - 4}\right), \end{equation} which gives the following two values for $b$ and $c$: \begin{align}
b = \pm \sqrt{1 - \frac{G a \alpha}{\sqrt \Delta}},
&& c = \pm \sqrt{1 - \frac{E a}{\alpha \sqrt \Delta}}. \end{align}
Finally, notice that when $F = 0$ equations (\ref{eq:39}), (\ref{eq:40}) and (\ref{eq:41}) reduces to \begin{equation}
\label{eq:45}
b c = \left(\frac{\sqrt E}{\sqrt G \alpha} - a\right) c =
\left(\frac{\sqrt G \alpha}{\sqrt E} - a\right) b = 0. \end{equation} Hence, if $F = 0$ we get \begin{equation}
\label{eq:46}(a, b, c) =
\begin{cases}
\displaystyle \left(\frac{\sqrt E}{\sqrt G \alpha}, 0, \pm \sqrt{1
- \frac{E}{G \alpha^2}}\right), & \displaystyle \text{if }
\frac E G \le \alpha^2, \\[1.2pc]
\displaystyle \left(\frac{\sqrt G \alpha}{\sqrt E}, \pm \sqrt{1 -
\frac{G \alpha^2}{E}}, 0\right), & \displaystyle \text{if }
\alpha^2 \le \frac E G.
\end{cases} \end{equation}
\subsubsection{The case of $J_2$}
The equations for the integrability of $J_2$ are somewhat more delicate as they behave differently depending on the values of $r, s$. The general form for such equations is \begin{align}
a {\left(\sqrt{r} - \sqrt{s}\right)}
& = -\frac{{\left(1 - b^{2}\right)} {\left(1 - \sqrt{r s}\right)}
\sqrt{\Delta}}{G} \label{eq:36}, \\
a {\left(1 - \sqrt{r s}\right)}
& = -\frac{{\left(1 - c^{2}\right)} {\left(\sqrt{r} -
\sqrt{s}\right)} \sqrt{\Delta}}{E} \label{eq:43}, \\
F a {\left(\sqrt{r} - \sqrt{s}\right)}
& = b c {\left(\sqrt{r} - \sqrt{s}\right)}
\sqrt{\Delta} \label{eq:48}, \\
F a {\left(1 - \sqrt{r s}\right)}
& = b c {\left(1 - \sqrt{r s}\right)} \sqrt{\Delta} \label{eq:49}, \\
F c {\left(1 - \sqrt{r s}\right)}
& = - G b {\left(\sqrt{r} - \sqrt{s}\right)} - a b
{\left(1 - \sqrt{r s}\right)} \sqrt{\Delta} \label{eq:47}, \\
F b {\left(\sqrt{r} - \sqrt{s}\right)}
& = -E c {\left(1 - \sqrt{r s}\right)} -a c {\left(\sqrt{r} -
\sqrt{s}\right)} \sqrt{\Delta} \label{eq:51}, \end{align} where $\Delta = E G - F^2$ as in the previous case.
Recall that when $s = r = 1$ all the equations hold trivially, which means that $J_2$ is a complex structure for all $(a, b, c) \in S^2$. This is a result already known (see \cite{abbena-1997,abbena-2001}). If $r = s < 1$, equations (\ref{eq:36}) to (\ref{eq:51}) reduce to \begin{align}
\label{eq:52}
a = 0, && b = \pm 1, && c = 0. \end{align} So, we can assume that $0 < s < r < 1$. Let us denote \begin{equation}
\label{eq:53}
\beta = \frac{\sqrt r - \sqrt s}{1 - \sqrt{r s}} \end{equation} and notice that $\beta$ is always positive. Now we can rewrite equations (\ref{eq:36}) to (\ref{eq:51}) as \begin{align}
a & = - \frac{(1 - b^2) \sqrt \Delta}{G \beta} \label{eq:54}, \\
a & = - \frac{(1 - c^2) \beta \sqrt \Delta}{E} \label{eq:55}, \\
Fa & = b c \sqrt \Delta \label{eq:56}, \\
Fb & = - \frac{Ec}{\beta} - a c \sqrt \Delta \label{eq:57}, \\
Fc & = - G \beta b - a b \sqrt \Delta \label{eq:58}. \end{align}
Notice that equations (\ref{eq:54}) to (\ref{eq:58}) are formally equal to equations (\ref{eq:37}) to (\ref{eq:41}) if we replace $\alpha$ by $-\beta$. The only difference is that in this case $a < 0$ and $b, c$ have opposite signs. Since we never actually use the value of $\alpha$ when solving (\ref{eq:37}) to (\ref{eq:41}) we can conclude that when $F \neq 0$ \begin{align}
a & = \frac12 \left( \frac{-1}{\sqrt \Delta} \left(\frac{E}{\beta} + G
\beta \right) + \sqrt{\frac 1 \Delta \left(\frac E \beta + G
\beta\right)^2 - 4}\right), \\
b & = \pm \sqrt{1 + \frac{G a \beta}{\sqrt \Delta}}, \\
c & = \mp \sqrt{1 + \frac{E a}{\beta \sqrt \Delta}} \end{align} and if $F = 0$, \begin{equation}
(a, b, c) =
\begin{cases}
\displaystyle \left(- \frac{\sqrt E}{\sqrt G \beta}, 0, \pm \sqrt{1
- \frac{E}{G \beta^2}}\right), & \displaystyle \text{if } \frac E G \le
\beta^2 \\[1.2pc]
\displaystyle \left(-\frac{\sqrt G \beta}{\sqrt E}, \pm \sqrt{1 -
\frac{G \beta^2}{E}}, 0\right), & \displaystyle \text{if } \beta^2 \le
\frac E G
\end{cases} \end{equation}
Summarizing, we obtained the following result.
\begin{theorem}
\label{sec:case-j_2}
Consider in $H_5$ the left-invariant metric $g$ given in
(\ref{eq:33}). Then $(g, J)$ is an Hermitian structure on $H_5$ if
and only if
\begin{enumerate}
\item $J = \pm J_1$, as in (\ref{eq:34}) with $a, b, c$ given as in
Table \ref{tab:J1} or
\item $J = \pm J_2$, as in (\ref{eq:35}) with $a, b, c$ given as in
Table \ref{tab:J2}.
\end{enumerate}
In particular, every left-invariant metric on $H_5$ is Hermitian
with respect to some left-invariant complex structure. \end{theorem}
\begin{table}[ht]
\caption{Case $\mathfrak h_5$: $J_1$ where $\Delta = E G - F^2$, $\alpha = \frac{\sqrt r +
\sqrt s}{1 + \sqrt{r s}}$, $\gamma = \frac E \alpha + G \alpha$}
\centering {\tabulinesep=1.2mm
\begin{tabu}{|c|c|c|c|c|}
\hline
$a$ & $b$ & $c$ & $F$ & $(r, s)$ \\
\hline\hline
$\frac{\gamma - \sqrt{ {\gamma^2} - 4 \Delta}}{2 \sqrt \Delta}$
& $ \pm\sqrt{1 - \frac{G a \alpha}{\sqrt \Delta}}$
& $\pm\sqrt{1 - \frac{E a}{\alpha \sqrt \Delta}}$
& $> 0$ & any \\
\hline
$\frac{\sqrt E}{\sqrt G \alpha}$ & $0$
& $\pm \sqrt{1 - \frac{E}{G \alpha^2}}$
& $= 0$ & $\frac E G \le \alpha^2$ \\
\hline
$\frac{\sqrt G \alpha}{\sqrt E}$ & $\pm \sqrt{1 - \frac{G
\alpha^2}{E}}$
& $0$ & $= 0$ & $\alpha^2 \le \frac E G$ \\
\hline
\end{tabu}}
\label{tab:J1} \end{table}
\begin{table}[ht]
\caption{Case $\mathfrak h_5$: $J_2$ where $\Delta = E G - F^2$, $\beta = \frac{\sqrt r -
\sqrt s}{1 - \sqrt{r s}}$, $\delta = \frac E \beta + G \beta$}
\centering
{\tabulinesep=1.2mm
\begin{tabu}{|c|c|c|c|c|}
\hline
$a$ & $b$ & $c$ & $F$ & $(r, s)$ \\
\hline\hline
\multicolumn{3}{|c|}{ $a^2 + b^2 + c^2 = 1$ } & any & $s = r = 1$\\
\hline
$0$ & $\pm 1$ & 0 & any & $s = r < 1$ \\
\hline
$-\frac{\delta - \sqrt{ {\delta^2} - 4 \Delta}}{2 \sqrt \Delta}$
& $ \pm\sqrt{1 + \frac{G a \beta}{\sqrt \Delta}}$
& $\mp\sqrt{1 + \frac{E a}{\beta \sqrt \Delta}}$
& $> 0$ & $s < r < 1$ \\
\hline
$-\frac{\sqrt E}{\sqrt G \beta}$ & $0$
& $\pm \sqrt{1 - \frac{E}{G \beta^2}}$
& $= 0$ & $s < r < 1$ and $\frac E G \le \beta^2$ \\
\hline
$-\frac{\sqrt G \beta}{\sqrt E}$
& $\pm \sqrt{1 - \frac{G \beta^2}{E}}$ & $0$ & $= 0$
& $s < r < 1$ and $\beta^2 \le \frac E G$ \\
\hline
\end{tabu}
}
\label{tab:J2} \end{table}
\begin{proof}
It follows from the above discussion. Notice that we introduce the
$\pm$ sign in the statement of the theorem so that our
classification also includes the Hermitian structures which reverse
the orientation. \end{proof}
\subsection{Hermitian structures on $\mathfrak h_4$}
We follow the same approach as in the previous case. Remember that for $H_4$, the left-invariant metrics, up to isometric automorphism, are given, in the standard basis $e_1, \ldots, e_6$ defined in Section \ref{sec:case-mathfrak-h_4}, by \begin{equation}
\label{eq:61}
g =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & r & 0 & 0 \\
0 & 0 & 0 & 0 & E & F \\
0 & 0 & 0 & 0 & F & G
\end{pmatrix} \end{equation} where $0 < r \le 1$, $E, G > 0$, $F \ge 0$ and $\Delta = E G - F^2 > 0$. In this case the orientation preserving almost Hermitian structures, with respect to this metric, are given by \begin{equation}
\label{eq:62}
J_1 = \begin{pmatrix}
0 & -a & -b & -c \sqrt r & 0 & 0 \\[.2pc]
a & 0 & -c & b \sqrt r & 0 & 0 \\[.2pc]
b & c & 0 & -a \sqrt r & 0 & 0\\[.2pc]
\frac c {\sqrt r} & - \frac b {\sqrt r} & \frac a {\sqrt r} & 0 &
0 & 0 \\[.2pc]
0 & 0 & 0 & 0 & - \frac F {\sqrt \Delta} & - \frac G {\sqrt
\Delta} \\[.4pc]
0 & 0 & 0 & 0 & \frac E {\sqrt \Delta} & \frac F {\sqrt \Delta}
\end{pmatrix} \end{equation} and \begin{equation}
\label{eq:63}
J_2 = \begin{pmatrix}
0 & -a & -b & -c \sqrt{r} & 0 & 0 \\[.2pc]
a & 0 & c & -b \sqrt{r} & 0 & 0 \\[.2pc]
b & -c & 0 & a \sqrt{r} & 0 & 0 \\[.2pc]
\frac{c}{\sqrt{r}} & \frac{b}{\sqrt{r}} & -\frac{a}{\sqrt{r}} & 0
& 0 & 0 \\[.2pc]
0 & 0 & 0 & 0 & \frac F {\sqrt \Delta} & \frac G {\sqrt
\Delta} \\[.4pc]
0 & 0 & 0 & 0 & - \frac E {\sqrt \Delta} & - \frac F {\sqrt
\Delta}
\end{pmatrix} \end{equation} where \begin{equation}
\label{eq:59}
a^2 + b^2 + c^2 = 1. \end{equation} With the same ideas as in Subsection \ref{sec:hermitian-iwasawa}, we can find conditions on $a, b, c$ for the integrability of $J_1$, $J_2$. For the sake of completeness we present the non-trivial equations but we omit the calculations that are too similar to the ones in the case of $\mathfrak h_5$.
\subsubsection{The case of $J_1$}
Let us denote \begin{equation}
\label{eq:60}
\alpha = \frac{1 + \sqrt r}{\sqrt r}. \end{equation} Then, the non-trivial equation for the vanishing of Nijenhuis tensor of $J_1$ are \begin{align*}
b & = - \frac{(1 - a^2) \sqrt \Delta}{G \alpha}, \\
b & = -\frac{(1 - c^2) \alpha \sqrt \Delta}{E}, \\
F b & = - a c \sqrt \Delta, \\
F a & = \frac{E c}{\alpha} + b c \sqrt \Delta, \\
F c & = G \alpha a + a b \sqrt \Delta, \end{align*} which are the same equations as (\ref{eq:37}) to (\ref{eq:41}) under the symmetry $(a, b, c) \mapsto (-b, a, c)$.
\subsubsection{The case of $J_2$}
The general form for the meaningful equations for the integrability of $J_2$ is \begin{align*}
b (1 - \sqrt r) & = - \frac{(1 - a^2) \sqrt r \sqrt \Delta}{G}, \\
b & = - \frac{(1 - c^2) (1 - \sqrt r) \sqrt \Delta}{E \sqrt r}, \\
F b & = - a c \sqrt \Delta, \\
F a (1 - \sqrt r) & = E c \sqrt r + b c (1 - \sqrt r) \sqrt \Delta, \\
F c & = \frac{G a (1 - \sqrt r)}{\sqrt r} + a b \sqrt \Delta, \end{align*} which resembles equations (\ref{eq:36}) to (\ref{eq:51}). Moreover, if $r \neq 1$ and we denote \begin{equation*}
\beta = \frac{1 - \sqrt r}{\sqrt r}, \end{equation*} then these equations simplify to \begin{align*}
b & = - \frac{(1 - a^2) \sqrt \Delta}{G \beta}, \\
b & = - \frac{(1 - c^2) \beta \sqrt \Delta}{E}, \\
F b & = - a c \sqrt \Delta, \\
F a & = \frac{E c}{\beta} + b c \sqrt \Delta, \\
F c & = G \beta a + a b \sqrt \Delta. \end{align*}
Again, these equations behave exactly as in the previous case, after replacing $\alpha$ with $\beta$. So, with no extra effort we achieve the following classification of the left-invariant Hermitian structures on $H_4$.
\begin{theorem}
\label{sec:case-j_2-1}
Consider in $H_4$ the left-invariant metric induced by $g$, given as
in (\ref{eq:61}). Then $(g, J)$ is an Hermitian structure on $H_4$
if and only if
\begin{enumerate}
\item $J = \pm J_1$, as in (\ref{eq:62}) with $a, b, c$ given as in
Table \ref{tab:J1-h4} or
\item $J = \pm J_2$, as in (\ref{eq:63}) with $a, b, c$ given as in
Table \ref{tab:J2-h4}.
\end{enumerate}
In particular, every left-invariant metric on $H_4$ is Hermitian
with respect to some left-invariant complex structure. \end{theorem}
\begin{table}[ht]
\caption{Case $\mathfrak h_4$: $J_1$ where $\Delta = E G - F^2$, $\alpha = \frac{1 + \sqrt
r}{\sqrt r}$, $\gamma = \frac E \alpha + G \alpha$}
\centering
{\tabulinesep=1.2mm
\begin{tabu}{|c|c|c|c|c|}
\hline
$a$ & $b$ & $c$ & $F$ & $r$ \\
\hline\hline
$ \pm\sqrt{1 + \frac{G b \alpha}{\sqrt \Delta}}$
& $-\frac{\gamma - \sqrt{ {\gamma^2} - 4 \Delta}}{2 \sqrt \Delta}$
& $\pm\sqrt{1 - \frac{E b}{\alpha \sqrt \Delta}}$
& $> 0$ & any \\
\hline
$0$ & $-\frac{\sqrt E}{\sqrt G \alpha}$
& $\pm \sqrt{1 - \frac{E}{G \alpha^2}}$
& $= 0$ & $\frac E G \le \alpha^2$ \\
\hline
$\pm \sqrt{1 - \frac{G \alpha^2}{E}}$
& $-\frac{\sqrt G \alpha}{\sqrt E}$
& $0$ & $= 0$ & $\alpha^2 \le \frac E G$ \\
\hline
\end{tabu}}
\label{tab:J1-h4} \end{table}
\begin{table}[ht]
\caption{Case $\mathfrak h_4$: $J_2$ where $\Delta = E G - F^2$,
$\beta = \frac{1 - \sqrt r}{\sqrt{r}}$,
$\delta = \frac E \beta + G \beta$}
\centering
{\tabulinesep=1.2mm
\begin{tabu}{|c|c|c|c|c|}
\hline
$a$ & $b$ & $c$ & $F$ & $r$ \\
\hline\hline
$\pm 1$ & $0$ & $0$ & any & $r = 1$ \\
\hline $\pm\sqrt{1 + \frac{G b \beta}{\sqrt \Delta}}$ &
$-\frac{\delta - \sqrt{ {\delta^2} - 4 \Delta}}{2 \sqrt \Delta}$
& $\pm\sqrt{1 + \frac{E b}{\beta \sqrt \Delta}}$
& $> 0$ & $0 < r < 1$ \\
\hline $0$ & $-\frac{\sqrt E}{\sqrt G \beta}$ &
$\pm \sqrt{1 - \frac{E}{G \beta^2}}$
& $= 0$ & $0 < r < 1$ and $\frac E G \le \beta^2$ \\
\hline $\pm \sqrt{1 - \frac{G \beta^2}{E}}$ &
$-\frac{\sqrt G \beta}{\sqrt E}$ & $0$ & $= 0$
& $0 < r < 1$ and $\beta^2 \le \frac E G$ \\
\hline
\end{tabu}
}
\label{tab:J2-h4} \end{table}
\begin{remark}
Recall that in $H_4$ there is, up to automorphism, a unique
left-invariant abelian structure, the one given in the proof of
Corollary \ref{sec:automorphism-group-3}, and it can be obtained as
$J_2$ from (\ref{eq:63}) with $a = 1$, $b = c = 0$, $F = 0$ and
$G = E$. \end{remark}
\subsection{Hermitian structures on $\mathfrak h_6$}
The case of $\mathfrak h_6$ can be treated in the same way as $\mathfrak h_5$ and $\mathfrak h_4$. And as a matter of fact, calculations are much simpler for $\mathfrak h_6$. We do no repeat such calculations but only state the theorem of classification of Hermitian structures on $\mathfrak h_6$. Recall from Theorem \ref{sec:left-invar-metr-2} that any left-invariant metric on $H_6$ is equivalent to one, and only one, of the form \begin{equation}
\label{eq:50}
g = \diag(1, 1, 1, 1, E, G), \end{equation} where $0 < E \le G$ and with respect to the standard basis $e_1, \ldots, e_6$ given at the beginning of Section \ref{sec:case-mathfrak-h_6}.
\begin{theorem}
Consider in $H_6$ the left-invariant metric induced by $g$, as in
(\ref{eq:50}) and let us denote $\alpha = \sqrt{E / G}$. Then
$(g, J)$ is a Hermitian structure on $H_6$ if and only if, in the
standard basis, $J = \pm J_1^\pm$ or $J = \pm J_2^\pm$ where
\begin{equation*}
J_1^\pm =
\begin{pmatrix}
0 & 0 & \pm \sqrt{1 - \alpha^2} & - \alpha & 0 & 0 \\
0 & 0 & - \alpha & \mp \sqrt{1 - \alpha^2} & 0 & 0 \\
\mp \sqrt{1 - \alpha^2} & \alpha & 0 & 0 & 0 & 0 \\
\alpha & \pm \sqrt{1 - \alpha^2} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & - \frac 1 \alpha \\
0 & 0 & 0 & 0 & \alpha & 0
\end{pmatrix}
\end{equation*}
and
\begin{equation*}
J_2^\pm =
\begin{pmatrix}
0 & 0 & \pm \sqrt{1 - \alpha^2} & - \alpha & 0 & 0 \\
0 & 0 & \alpha & \pm \sqrt{1 - \alpha^2} & 0 & 0 \\
\mp \sqrt{1 - \alpha^2} & - \alpha & 0 & 0 & 0 & 0 \\
\alpha & \mp \sqrt{1 - \alpha^2} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \frac 1 \alpha \\
0 & 0 & 0 & 0 & - \alpha & 0
\end{pmatrix}.
\end{equation*}
In particular, $J_1^+ = J_1^-$ and $J_2^+ = J_2^-$ if and only if
$E = G$. \end{theorem}
\begin{corollary}
Every left-invariant metric on $H_6$ is Hermitian. \end{corollary}
\subsection{Hermitian structures on $\mathfrak h_2$}
The cases of $\mathfrak h_2$ and $\mathfrak h_9$ are significantly harder to treat by the above methods. In fact, the generic metrics in our classification for both Lie algebras are not diagonal, with respect to the standard basis, when restricted to the orthogonal complement of the commutator. So the polynomial equations describing the integrability of an arbitrary almost Hermitian structure, by means of the Cholesky decomposition, become wild. However, one can still recover some information from these polynomials in order to estimate the amount of Hermitian structures.
In order to simplify the exposition, let us change slightly the notation for the metrics computed in Theorem \ref{sec:left-invar-metr-5} for the Lie group $H_2$. We denote such metrics by \begin{equation}
\label{eq:71}
g = \sum_{i = 1}^4 e^i \otimes e^i + 2 A e^1 \otimes e^3 + 2 B
e^2 \otimes e^4 + E e^5 \otimes e^5 + 2F e^5 \otimes e^6 + G e^6
\otimes e^6, \end{equation} where $0 \le A \le B < 1$, $E, F, G \ge 0$ and $EG - F^2 > 0$. As in the previous cases, we denote $\Delta = EG - F^2$ and in addition we introduce the notations \begin{align*}
\alpha = \sqrt{1 - A^2}, && \beta = \sqrt{1 - B^2}. \end{align*} Observe that these parameters satisfy \begin{equation*}
A^2 + \alpha^2 = B^2 + \beta^2 = 1. \end{equation*} In order to simplify some calculations, we further define \begin{align*}
\phi = B \alpha - A \beta, && \psi = A B + \alpha \beta. \end{align*} It is easy to see that $\psi$ is always positive and $\phi$ is non negative, and positive if $A \neq B$.
The orientation-preserving almost Hermitian structures with respect to (\ref{eq:71}) are homeomorphic to the disjoint union of two $2$-spheres. Since the computation in the general case become very complicated, we illustrate the procedure only for the connected component given by \begin{equation}
\label{eq:72}
J =
\begin{pmatrix}
-\frac{A b}{\alpha} & -\frac{a \alpha + A c}{\alpha} &
-\frac{b}{\alpha} & -\frac{a \phi + c \psi}{\alpha} & 0 & 0
\\[0.4pc]
\frac{a \beta - B c}{\beta} & \frac{B b}{\beta} & -\frac{a \phi
+ c \psi}{\beta} & \frac{b}{\beta} & 0 & 0 \\[0.4pc]
\frac{b}{\alpha} & \frac{c}{\alpha} & \frac{A b}{\alpha} &
-\frac{a \beta - B c}{\alpha} & 0 & 0 \\[0.4pc]
\frac{c}{\beta} & -\frac{b}{\beta} & \frac{a \alpha + A
c}{\beta} & -\frac{B b}{\beta} & 0 & 0 \\[0.4pc]
0 & 0 & 0 & 0 & -\frac{F}{\sqrt{\Delta}} &
-\frac{G}{\sqrt{\Delta}} \\[0.4pc]
0 & 0 & 0 & 0 & \frac{E}{\sqrt{\Delta}} &
\frac{F}{\sqrt{\Delta}}
\end{pmatrix} \end{equation} where \begin{equation}
\label{eq:75}
a^2 + b^2 + c^2 = 1. \end{equation}
After some long computations, which were verify using the computer software SageMath, we obtain that $J$ as in (\ref{eq:72}) is integrable if the following equations hold:
\begin{align}
0 & = -a^{2} \beta \phi + b^{2} A + c^{2} B \psi + a c {\left(B \phi - \beta
\psi\right)} - \frac{b {\left(F \alpha + G \beta\right)}}{
\sqrt{\Delta}}, \\
0 & = a b \sqrt{\Delta} + a E \phi+ c {\left(E \psi + F\right)}, \\
0 & = (1 - a^{2}) \sqrt{\Delta} + b E \phi \label{eq:73}, \\
0 & = (a \phi + c \psi)^2 + b^{2} + \frac{G b \phi}{\sqrt{\Delta}}, \\
0 & = a c \alpha + (1 - a^{2}) A - \frac{b {\left(F \alpha + E
\beta\right)}}{\sqrt{\Delta}}, \\
0 & = a c \phi + (1 - a^{2}) \psi - \frac{b F
\phi}{\sqrt{\Delta}} \label{eq:74}, \\
0 & = (c \phi - a \psi) b \sqrt{\Delta} + a F \phi + c {\left(F \psi
+ G\right)}, \\
0 & = a^{2} \alpha \phi + b^{2} B + c^{2} A \psi + a c {\left(A \phi + \alpha
\psi\right)} + \frac{b {\left(G \alpha + F
\beta\right)}}{\sqrt{\Delta}}, \\
0 & = a c \beta - {\left(1 - a^{2}\right)} B - \frac{b {\left(E \alpha + F
\beta\right)}}{\sqrt{\Delta}}. \end{align}
We can fully solve the following important particular case. \begin{proposition}
If $A = B$, or equivalently $\phi = 0$, then $J$ is integrable if and
only if $a = \pm 1$ and $b = c = 0$. Moreover, in this case $J$ is abelian. \end{proposition}
\begin{remark}
We can approach the abelian case from the classification given in
\cite{Andrada_2011} using a similar argument as the one we will use
in the next subsection. In fact, one can prove that if an abelian
structure is hermitian with respect to a metric of the form
(\ref{eq:71}), then $A = B$. \end{remark}
One can argue that the same approach mentioned in the above remark could be use in the non-abelian case by using the classification of \cite{ceballos-2014}, but due to the complexity of the problem, the calculations in this case turn extremely difficult to be solve explicitly. Instead, to treat the case $\phi \neq 0$ we note that from (\ref{eq:73}), \begin{equation*}
b = -\frac{(1 - a^2) \sqrt \Delta}{E \phi}. \end{equation*} Observe that, since $a \neq \pm 1$, we get that $b$ is negative and from (\ref{eq:74}) we get \begin{align*}
c & = \frac{1}{a \alpha}\left( \frac{F b \phi}{\sqrt \Delta} - (1 -
a^2)\psi \right) \\
& = - \frac{1 - a^2}{a \alpha} \left( \frac{F}{E \phi} + \psi \right). \end{align*}
By replacing these values of $b$ and $c$ in (\ref{eq:75}) we obtain a quartic equation for $a$, which is in fact quadratic in $a^2$. Disregarding the complex solutions we get the following result.
\begin{proposition}
There exists at most two different values of $(a, b, c)$ such that
$J$ is a complex structure. \end{proposition}
Note that one still needs to check that the almost complex structure given by this construction satisfies the integrability equations other that (\ref{eq:73}) and (\ref{eq:74}). However, it is our belief that these equations always admit a solution, as we could check by solving them numerically for random parameters.
\subsection{Hermitian structures on $\mathfrak h_9$}
As in the above case, the problem of determining the Hermitian metrics on $H_9$ is very hard, since the moduli space of left-invariant metrics is described by six real parameters. However, we can perform a qualitative analysis by using an appropriate decomposition of a subgroup of automorphisms which preserve the metrics given in (\ref{eq:32}). At some point, calculations become extremely tedious and we use SageMath to check some computations.
Recall that in the basis $\hat e_1, \ldots, \hat e_6$, every left-invariant metric on $H_9$ is equivalent, via an automorphism, to one in the slice $\Phi(\Sigma)$, which consist of all the metrics induced by the matrices given in (\ref{eq:70}), where $A, B, C >0$ and $D, E, F \in \mathbb R$. Notice also that form \cite{Andrada_2011} there exists a unique complex structure (which is also abelian) on $\mathfrak h_9$ up to automorphism. After rearranging the basis, we get that such complex structure, say $J_0$, is given by the relations \begin{align*}
J_0 \hat e_1 = - \hat e_2,
&& J_0 \hat e_3 = \hat e_5,
&& J_0 \hat e_4 = - \hat e_6. \end{align*}
Given a complex structure $J$ on $H_9$ and $\varphi \in \Aut(\mathfrak h_9)$ we denote by $\varphi \cdot J = \varphi J \varphi^{-1}$ the standard action of the automorphism group on the space on complex structures. Let \begin{equation*}
\Aut_0(\mathfrak h_9)^\Sigma = \{\varphi \in \Aut_0(\mathfrak h_9):
(g, \varphi \cdot J_0) \text{ is an Hermitian structure for some } g
\in \Phi(\Sigma)\}. \end{equation*} As we will see later, $\Aut_0(\mathfrak h_9)^\Sigma$ is not a subgroup of $\Aut_0(\mathfrak h_9)$, but this set is relevant to understand the problem of the existence of Hermitian metrics. In fact, since every complex structure has the form $\varphi \cdot J_0$, if $g$ is a Hermitian metric then there must exist $g' \in \Phi(\sigma)$ and $\varphi' \in \Aut_0(\mathfrak h_9)^\Sigma$ such that $g'$ is isometric to $g$ and $(g', \varphi' \cdot J_0)$ is a Hermitian structure.
Lets keep the notation of (\ref{eq:30}) for a generic $\varphi \in \Aut_0(\mathfrak h_9)$. Even though $\Aut_0(\mathfrak h_9)^\Sigma$ is not a group, it is not hard to see that it is contained in \begin{equation*}
G = \{\varphi \in \Aut_0(\mathfrak h_9): a_{21} = 0, a_{22} =
a_{11}\}, \end{equation*} which is a closed subgroup of the automorphism group. The group structure is easier to understand than the one of the full automorphism group. In fact, consider the following subgroups of $G$: \begin{align*}
G_1 & = \left\{
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
a_{51} & a_{52} & 0 & 0 & 1 & 0 \\
a_{61} & a_{62} & a_{63} & a_{64} & -a_{52} & 1
\end{pmatrix}
\right\},
\\
G_2 & = \left\{
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
a_{31} & a_{32} & 1 & 0 & 0 & 0 \\
a_{41} & a_{42} & a_{43} & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & a_{31} & 1
\end{pmatrix}
\right\}, \\
G_3 & = \{\diag(a_{11}, a_{11}, a_{11}^2, a_{44}, a_{11}^2,
a_{11}^3)\}. \end{align*}
With a routine calculation, one can see that $G_1$ is a normal subgroup of $G$. The group $G_2$ is not a normal subgroup of $G$, however, $G_1 \rtimes G_2$ is a normal subgroup of $G$ and therefore $G$ is the triple semi-direct product \begin{equation*}
G = (G_1 \rtimes G_2) \rtimes G_3. \end{equation*}
Let us denote \begin{equation*}
G_i^{\Sigma} = G_i \cap \Aut_0(\mathfrak h_9)^\Sigma \end{equation*} for $i = 1, 2, 3$ and \begin{equation*}
\Phi(\Sigma_i) = \{g \in \Phi(\Sigma): (g, \varphi \cdot J_0) \text{
is a Hermitian structure for some }\varphi \in G_i\}. \end{equation*}
The following results are straightforward.
\begin{lemma}
\label{sec:herm-struct-mathfr}
\begin{enumerate}
\item
\begin{equation*}
\Phi(\Sigma_1) = \left\{
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & A^{2} & 0 & 0 & 0 \\
0 & 0 & 0 & E^{2} + 1 & \sqrt{E^{2} + 1} A E & 0 \\
0 & 0 & 0 & \sqrt{E^{2} + 1} A E & {\left(E^{2} + 1\right)} A^{2} & 0 \\
0 & 0 & 0 & 0 & 0 & E^{2} + 1
\end{pmatrix}: A > 0, E \in \mathbb R\right\}.
\end{equation*}
\item
\begin{equation*}
G_1^\Sigma = \left\{\varphi =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & a_{63} & 0 & 0 & 1
\end{pmatrix}: a_{63} \in \mathbb R
\right\}
\end{equation*}
is a subgroup of $G$. Moreover, if $\varphi \in G_1^\Sigma$, then
$\varphi \cdot J_0$ is Hermitian with respect to any metric in
$\Phi(\Sigma_1)$ such that
$a_{63} = \frac{A E}{\sqrt{E^{2} + 1}}$.
\end{enumerate} \end{lemma}
\begin{lemma} \label{sec:herm-struct-mathfr-1}
\begin{enumerate}
\item
\begin{equation*}
\Phi(\Sigma_2) = \left
\{
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & A^{2} & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & A^{2} + F^{2} & F \\
0 & 0 & 0 & 0 & F & 1
\end{pmatrix}: A > 0, F \in \mathbb R\right\}.
\end{equation*}
\item
\begin{equation*}
G_2^\Sigma = \left\{\varphi =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & a_{43} & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{pmatrix}: a_{43} \in \mathbb R
\right\}
\end{equation*}
is a subgroup of $G$. Moreover, if $\varphi \in G_2^\Sigma$, then
$\varphi \cdot J_0$ is Hermitian with respect to any metric in
$\Phi(\Sigma_2)$ such that $a_{43} = -F$.
\end{enumerate} \end{lemma}
\begin{lemma}
\label{sec:herm-struct-mathfr-2}
\begin{enumerate}
\item
$\Phi(\Sigma_3) = \left\{\diag(1,1,A^2,1,A^2,C^2): A, C > 0
\right\}$.
\item $G_3^\Sigma = G_3$. Moreover, if $\varphi \in G_3$ then
$\varphi \cdot J_0$ is Hermitian with respect to any metric in
$\Phi(\Sigma_3)$ such that $C = \frac{a_{44}}{a_{11}^3}$.
\end{enumerate} \end{lemma}
\begin{remark}
\label{sec:herm-struct-mathfr-3}
It follows from Lemmas \ref{sec:herm-struct-mathfr} and
\ref{sec:herm-struct-mathfr-1} that $G_1^\Sigma$ commutes with
$G_2^\Sigma$ and their product is normalized by $G_3$. Moreover, the
four dimensional Lie group
\begin{equation*}
G' = (G_1^\Sigma \times G_2^\Sigma) \rtimes G_3
\end{equation*}
is contained in $\Aut_0(\mathfrak h_9)^\Sigma$. In fact, an arbitrary
element in $G'$ has the form \begin{equation*}
\varphi' =
\begin{pmatrix}
a_{11} & 0 & 0 & 0 & 0 & 0 \\
0 & a_{11} & 0 & 0 & 0 & 0 \\
0 & 0 & a_{11}^{2} & 0 & 0 & 0 \\
0 & 0 & a_{43} & a_{44} & 0 & 0 \\
0 & 0 & 0 & 0 & a_{11}^{2} & 0 \\
0 & 0 & a_{63} & 0 & 0 & a_{11}^{3}
\end{pmatrix}
\end{equation*}
where $a_{11}, a_{44} > 0$ and $a_{43}, a_{63} \in \mathbb R$, and
after long computations one can check that $(g', \varphi' \cdot
J_0)$ is a Hermitian structure for the metric $g' \in \Phi(\Sigma)$
given by
\begin{align*}
B & = \frac{A^2 a_{11}^5}{\sqrt{A^2 a_{11}^{10} - a_{44}^2
a_{63}^2}},
&& C = \frac{A a_{11}^{2} a_{44}}{\sqrt{A^2 a_{11}^{10} - a_{44}^2 a_{63}^2}},
&& D = 0, \\
E &= \frac{a_{44} a_{63}}{\sqrt{A^{2} a_{11}^{10} - a_{44}^{2}
a_{63}^{2}}},
&& F = -\frac{A a_{11}^{3} a_{43}}{\sqrt{A^2 a_{11}^{10} -
a_{44}^2 a_{63}^2}}
&&
\end{align*}
for sufficiently large values of $A$. However, notice that $G' \neq
\Aut_0(\mathfrak h_9)^\Sigma$. In order to prove this we can note
that the automorphisms in
\begin{equation*}
G'' = \left\{\varphi_{s, t} =
\begin{pmatrix}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & s & 1 & 0 & 0 & 0 \\
t & -t & 0 & 1 & 0 & 0 \\
s & 0 & 0 & 0 & 1 & 0 \\
t & t & 0 & 0 & 0 & 1
\end{pmatrix}: s, t \in \mathbb R
\right\}
\end{equation*}
form an abelian group which is contained in
$\Aut_0(\mathfrak h_9)^\Sigma$ such that $G'' \cap G' = \{I_6\}$. In
fact, it is easy to see that $\varphi_{s, t} \cdot J_0$ is hermitian
with respect to any metric of the form $g = \diag(1, 1, A, 1, A, 1)$
(recall that all of these metrics belong to $\Phi(\Sigma_3)$, in the
notation of Lemma \ref{sec:herm-struct-mathfr-2}). \end{remark}
\begin{remark}
$\Aut_0(\mathfrak h_9)^\Sigma$ is not a subgroup of
$\Aut_0(\mathfrak h_9)$. In fact, one can check by a direct
calculation that if $\varphi_2 \in G_2^\Sigma$ is not the identity,
then $\varphi_{s, t} \varphi_2 \in \Aut_0(\mathfrak h_9)^\Sigma$ if
and only if $s = 0$ (we are keeping the notation of Remark
\ref{sec:herm-struct-mathfr-3}). \end{remark}
Finally, using the previous results, we were able to detect a family of left-invariant metrics on $H_9$ which are not Hermitian.
\begin{proposition}
Let us consider the left-invariant metrics on $H_9$ given by \begin{equation*}
g_{A, B} = \hat e^1 \otimes \hat e^1 + \hat e^2 \otimes \hat e^2
+ A^2 \hat e^3 \otimes \hat e^3 + \hat e^4 \otimes \hat e^4 + B^2
\hat e^5 \otimes \hat e^5 + \hat e^6 \otimes \hat e^6
\end{equation*}
with $A, B > 0$. Then $g_{A, B}$ is Hermitian if and only if
$B = A$. \end{proposition}
\begin{proof} It follows form a direct computation. Notice that one only needs to deal with complex structures obtained by conjugating $J_0$ by an element of the group $G$. \end{proof}
\end{document} | arXiv |
Abstract: We consider the map of three-dimensional $\mathcal N=4$ superfields to the $\mathcal N=3$ harmonic superspace. The left and right representations of the $\mathcal N=4$ superconformal group are constructed on $\mathcal N=3$ analytic superfields. These representations are convenient for describing $\mathcal N=4$ superconformal couplings of Abelian gauge superfields to hypermultiplets. We investigate the $\mathcal N=4$ invariance in the non-Abelian $\mathcal N=3$ Yang–Mills theory. | CommonCrawl |
Find $AX$ in the diagram if $CX$ bisects $\angle ACB$. [asy]
import markers;
real t=1-.32;
pair A=(-4.07,0);
pair B=(40.008,0);
pair C=(0,20.616);
pair X=t*A+(1-t)*B;
draw(C--A--B--C--X);
label("$A$",A,SW);
label("$B$",B,E);
label("$C$",C,N);
label("$X$",X,S);
//markangle(n=1,radius=15,A,C,X,marker(markinterval(stickframe(n=1),true)));
//markangle(n=1,radius=15,X,C,B,marker(markinterval(stickframe(n=1),true)));
label("$30$",.5*(B+X),S);
label("$45$",.5*(B+C),NE);
label("$21$",.5*(A+C),NW);
[/asy]
The Angle Bisector Theorem tells us that \[\frac{AC}{AX}=\frac{BC}{BX}\]so \[AX=\frac{AC\cdot BX}{BC}=\frac{21\cdot30}{45}=\boxed{14}.\] | Math Dataset |
Research and application of personalized human body simplification and fusion method
Lulu Ma1,
Ke Zhang2 &
Dianjie Lu3
In crowd simulation, 3D (three-dimensional) character modeling is an important topic since the appropriate character models are helpful to improve the efficiency and realism of crowd simulation. The reconstructed 3D character model based on Kinect has a strong sense of reality and low cost. However, these models are all complex and cannot be applied to large-scale crowd simulation directly. In this paper, we propose a novel personalized human body modeling method for mass crowd simulation based on Kinect. The human modeling process is divided into head modeling and torso modeling, and then they are fused into each other to build a personalized human body model. This method can be divided into two parts: In the first part, a simplified method is presented based on edge curvature and area of error. In addition, to preserve the detail characteristics of model, the way of interactive operation is introduced. In the second part, the automatic fusion for the simplified head model and body model is made by using the improved FCF (fusion control function) fusion method. Finally, a hierarchical database for the personalized human body models is built. The experiment results show that the method proposed in this paper has high efficiency and good robustness in practical applications.
Traditional 3D (three-dimensional) human body models usually are captured by the structured light or laser scanner. Although it can obtain high-precision personalized 3D human body model, the cost is high and the operation is complicated [1]. The Kinect device launched by Microsoft can make use of infrared technology to achieve the fast acquisition of three-dimensional information at low cost. This breakthrough has greatly promoted some applications using 3D technology, such as human action recognition based on Kinect, skeleton modeling, face recognition, and 3D reconstruction of the scene, which have become a research hotspot in the related fields [2,3,4]. Because of its low cost and simple operation, Kinect depth camera is also used as a scanner to rapidly build personalized 3D human body models in real time [5, 6]. However, the point cloud density of the model of Kinect scanning is too large, so the 3D human model constructed by it is difficult to be widely applied. For example, in the crowd simulation scene, the real-time rendering speed of human body is required to be fast due to the large scale of the crowd. If the 3D human body model scanned by Kinect is applied directly to the group simulation, it will undoubtedly increase the system cost and reduce the efficiency of crowd simulation. Therefore, how to preserve the necessary details of the model while simplifying the model with high similarity to original model is a very meaningful problem.
At present, the simplified method commonly used for three-dimensional models is the simplified method in CGAL (computational geometry algorithms library). CGAL is a computational geometry algorithms library written in C++. In the simplification of the 3D mesh, the simplified method of edge collapse is used [7, 8], and the Lindstrom-Turk method is used to calculate the collapse cost of each side. Although the efficiency is high, the accuracy is not high, and the effective detail features of the model cannot be retained. The simplified method based on the fillet surface reconstruction proposed by Peng et al. [9] realizes the simplification of the fillet surface, which is applicable to the model with fillet characteristics but not universal. Other simplified methods for 3D models, such as the 3D model simplification algorithm with texture proposed by Feng and Zhou [10], consider the geometric information of the model and the geometric error of the texture information. Zhou and Chen [11] put forward a mesh model simplification algorithm based on polygon vertex normal vector, which is a simplified method of visual feature optimization. Quan et al. [12] propose the geometric model simplification method based on the region segmentation. This method needs to keep the detail features and introduce the region segmentation principle of the image. A mesh model region segmentation method based on curvature for region growing is proposed, and then it is simplified according to the ratio of the number of triangles in the region. Zhang et al. [13] put forward a new simplification method of terrain model based on divergence function. This method combines the discrete particle swarm theory to simplify the terrain model based on the hierarchical structure of the implicit quadtree. An improved mesh simplification algorithm based on edge collapse was introduced in [14]. In this algorithm, the quadric error metric was utilized to compute the vertex significance and control the sequence of edge collapse. Sanchez et al. [15] use an estimated local density of the point cloud to simplify the point cloud. In this method, the point clouds are clustered by using the expectation maximization algorithm according to the local distribution of the points. Then, a linear programming model is applied to reduce the cloud. Han et al. [16] propose a point cloud simplification method with preserved edge points. In this work, the authors try their best to retain the edge points since these points have more significant properties than non-edge points. First, a least square plane is constructed by using the topology relationship of the points and normal vectors and then each point in the vicinity is projected to the fitted plane. Next, the edge points can be extracted according to the homogeneity of projection. These detected edge points are retained in the decimating process. As For the non-edge points, the authors delete the least important points until the predesigned simplification rate is satisfied. In [17], the incremental segmentation and triangulation of planar segments from dense point clouds are studied to enhance the quality and efficiency. In this paper, the authors proposed a point-based triangulation algorithm to improve the planar segment decimation and triangulation in a gradually expanding point cloud map as well as a polygon-based triangulation algorithm. Both of the two algorithms can produce more accurate and simpler planar triangulations. Although the above method can simplify the model, it cannot keep the effective details of the model, and it is inefficient when dealing with the models with large amount of points.
When the whole 3D body model is collected by Kinect, the amount of data is huge and the later processing cost is big instead of its convenience. The huge amount of 3D information and data is mainly manifested in three aspects: dense point, dense edge, and dense surface. Thus, the simplification of 3D model is mainly from the three aspects of point, edge, and surface. In this paper, we first use Kinect to get the head with the most personalized features, and then use the edge collapse simplification method based on edge curvature and area error. In addition, the interactive approach is used to preserve the detail characteristics and simplify the model. Furthermore, we also use the improved FCF (fusion control function) [18] model fusion method to realize the seamless integration between models automatically. With this model, the complete character models can be reconstructed to build a database of LOD (levels of detail) character model. The simplified 3D body models can reduce system overhead and improve system efficiency when they are applied to crowd simulation.
The simplification and fusion method
The related work of 3D human reconstruction algorithm proposed in this paper was three-dimensional scanning technology, model denoising, model simplification, model fusion, and simulation experiment. Firstly, the head model was obtained by using a cheap Kinect depth camera, and then a smooth model was obtained by removing the fragments in the model, the model denoising, and other preprocessing operations. Secondly, we simplified the complex head model by using the simplified method of preserving model features based on edge curvature and area error proposed in this paper. Finally, the improved FCF model fusion method was used to fuse the head model and the body model, and the personalized hierarchical model library was built, which was finally applied to group simulation.
The denoising of the initial model can be divided into two steps: One is to delete the fragments generated during scanning. In this paper, we used DFS (depth first search) algorithm to delete fragments. The other is to use the weighted Laplace smoothing algorithm [19] to remove the tiny noise on the model and make the model as smooth as possible. The Laplace smoothing algorithm has a low computational complexity and can control the details of the model very well in operation. Therefore, this paper used the Laplace smoothing algorithm to remove the small noise in the model.
The topological structure of the model was mapped to a graph, and the connectivity of the whole graph can be obtained by using the method of DFS. The largest connected graph was the structure that needed to be retained, and the rest of the disconnected graph was deleted. Figure 1a is the graph before deleting the fragment, and Fig. 1b is the graph after the fragment is deleted.
Remove fragment
The specific operation of the Laplace algorithm is as follows: The 3D position of the vertex is moved to the center of gravity position of the surrounding vertexes to minimize the difference between the vertex and the surrounding vertexes. For every point in the model, the space position of the point is recalculated according to the position information of the surrounding vertexes. The Laplace smoothing formula is as follows:
$$ \overline{x_i}=\frac{1}{N}\sum \limits_{j=1}^N{x}_j $$
N is the number of vertexes around the current point and \( \overline{x_i} \)is the new coordinate of the i-th vertex.
In practice, N cannot be too large. If it is too large, the details may be lost. If N is too small, the smooth effect will not be achieved. Therefore, an improved Laplace smoothing algorithm was used in this paper, which was weighted by Gauss weighted method. For the point far away from the center point, its influence on the center point was reduced, and for the point near the center point, its influence on the center point was increased. Figure 2a is a model before denoising, and Fig. 2b is a model after denoising. It is obvious that the preprocessing method proposed in this paper can effectively remove the small noise on the model.
Detail features preserving
According to the different geometric elements, mesh model simplification method is divided into vertex deletion, edge collapse, triangle deletion, and patch deletion. Edge collapse operation is based on half edge structure and an edge with the smallest triangular patch cost is collapsed in order to achieve the purpose of deleting the simplified model of triangular patch.
When the mesh model is simplified by the simplification method based on geometric elements, the higher the simplification rate, the more serious loss of the details of the model, and the detail features of the model cannot be preserved. The purpose of model simplification is to maintain the detail features of the model to the maximum extent on the basis of reducing the scale of the model. Based on the edge collapse method, an interactive method was proposed in this paper, which can effectively retain the detail features of the model and make the model have higher identifiability.
The detail features of the model can be reflected by the edge curvature and the area change of the triangular patch on the model. If the curvature of the edge is large in a region, the features of the model are obvious here. The change of the area of the related triangle patch caused by the collapse edge also reflects the features of the model here. The smaller the change, the smoother. Therefore, the edge collapse cost can be defined as two errors: one is the curvature of the edge, and the other is the change of the area brought by the deletion point.
As shown in Fig. 3, the way to calculate the edge curvature is:
$$ {E}_c=\beta \left(1-\mathit{\cos}\alpha \right)=\left(\frac{l}{h_1}+\frac{l}{h_2}\right)\left(1-{n}_1{n}_2\right) $$
Edge curvature
Among them, l = ‖q − p‖, hi = d{vi, e} 。.
As shown in Fig. 4, t′ is the reproduced triangular patch while deleting the triangular patch t. This process can be achieved by rotating the triangular patch T by an angle. When the angle of triangle rotation is set as θ, the error Qt resulting from the deletion of one surface t is as follows:
$$ {Q}_t={l}_t\times \theta $$
Area error
Among them, lt = (A1 + A2)/2,A1 is the area of the triangle t = ( v0, v1, v2), A2 is the area of the triangle t′ = (v, v1, v2) . The computation angle θ involves trigonometric operation, so the computation speed is slow. In order to improve the computation speed, let \( \theta =1-{n}_t{n}_{t^{\prime }} \), in which ni is a normal vector of the surface i. As shown in Fig. 4, the area error Ea caused by the deletion edgy is:
$$ {E}_s=\sum \limits_{t\in P}{Q}_t $$
P is the set of all triangle patches connected to v0. Therefore, the error of the edge collapse is as follows:
$$ E={E}_c+{E}_s $$
Although the detail features of the model can be retained by using this method, with the improvement of the rate of simplification, the detail features of the model will also be lost gradually. In order to keep the detail features of the model, an interactive method was used in this paper to retain the detail features of the model. The simplified operation to preserve the detail features is shown in Algorithm 1:
The complexity of the algorithm is the calculation of the folding cost of each edge. The folding cost of each edge includes two parts: the edge curvature and the area difference. For each edge, the value of both of the two parts should be calculated. Generally, the simplification of the whole model can be realized if the order of all edges is sorted by the size of all edges according to the folding cost of all edges and the edge folding operation. However, the specific details on the model cannot be preserved and it cannot guarantee the real sense of the model through the simple model can be obtained according the simplification method mentioned above. Therefore, we introduce an interactive method to mark the area that needs to be preserved by a manual way when choosing the folding edge, so as to retain the local features with special significance. In this paper, a random weight is set up for the folding cost of the edge of the reserved region, that is (rand() + a). The average folding cost of all edges is added to obtain the edges of the reserved area with a larger folding cost, so that the edges of the feature area can be retained effectively. For the head model collected by Kinect, we can adjust the value of a to control the simplification rate of the reserved area. We introduce a function rand() to reduce the effect of noise on the model. When a takes a large value, it can completely preserve the details of the feature area; when a is taken for smaller values, the simplification ratio of the reserved area becomes larger, which cannot achieve the purpose of preserving the details of the model. When a is taken between the two values, the reserved area can be simplified to a suitable level.
The proposed algorithm can effectively retain the detail features of the model. Experiments show that this way of processing can effectively retain the important features of the model, and it can also retain the features of the model well when the simplification ratio is high.
Fusion of models
The simplified head model also needs to be fused with the body model to get a personalized human model. Kanai et al. propose a FCF-based mesh model fusion method. This method needs to map the model to two-dimensional space and then map it to 3D space according to FCF function, so as to achieve the integration between models. However, the high computational complexity does not apply to the fusion of large models, and it needs to choose the fusion area manually, which cannot realize the automatic integration between models. Therefore, in this paper, an improved FCF fusion method based on boundary edge was applied to achieve the automatic fusion between two large-scale models. The first step was to determine the fusion area according to the boundary edge of the two models. As shown in Fig. 5, the fusion areas F1, F2 were determined and then mapped to the two-dimensional space H1, H2to achieve the mapping from three dimensions to two dimensions. By using the method in literature [20], we can approximately calculate the locations of all points on H1 and H2. The second step was to merge H1 and H2 and get Hc. Hc included two kinds of vertexes: one was the original vertex that is directly inherited from F1, F2, and the other was the new point generated by cross computing. The third step was to reconstruct the blending surface Fc according to every point in Hc. Each point in Hccontained two parts of three-dimensional information, one was the three-dimensional information on F1, and the other was the three-dimensional information on F2. The transition curved surface Fc of F1 and F2can be obtained by FCF method. As shown in Fig. 6, we define a non-uniform cubic B-spline curve interpolation algorithm, called f(s) (0 ≤ f(s) ≤ 1), to control the fusion models.
An overview of mesh fusion
Fusion control function
The coordinate of the vertex vc in Fcis as follows:
$$ {v}^c=f(s){v}^1+\left(1-f(s)\right){v}^2 $$
Among them, v1 represents the coordinate of vc onF1, v2 represents the coordinate of vc on F2, s = 1 − l/L,l. The definition of L is shown in Fig. 6. l is the distance from point to bottom boundary of Hc, and L is the distance between upper and lower boundaries. It can be concluded that the value of s is closer to 1 when the distance between a point and the corresponding lower boundary is smaller. On the other hand, when the distance between a point and the lower boundary is larger, s is closer to 0.
Discussions and result analysis
The algorithm has been implemented on Microsoft Visual Studio platform by using C++ programming. The test computer's CPU (central processing unit) is Intel Core i5-2520M with the basic frequency 2.5 GHz. In addition, the memory is 8 GB and the GPU (graphics processing unit) is Intel HD Graphics 3000.
Figure 7 shows the original model. Figure 7b, c shows the points and triangles of model in Fig. 7a which has 123,801 points and 41,267 triangles. Figure 8 shows the comparison results between the original model and the simplified models. Figure 8a is the original model, with 20,921 points and 41,261 triangle patches. Figure 8b–d use the simplified method preserving the detail features of the model proposed in this paper, retaining the detail features around the eyes and nose. Figure 8b has 3640 points and 7986 triangle patches, and the rate of simplification is 80%. Figure 8c has 1916 points and 3799 triangle patches, and the rate of simplification is 90%. Figure 8d has 1014 points and 1995 triangle patches, and the rate of simplification is 95%. Figure 8e is the figure that does not adopt the simplified method of preserving detail features proposed in this paper, and it has 1243 points and 1905 triangles, and the rate of simplification is 95%. Through Fig. 8b–d, we can see that when the simplification ratio is very low, the detail features of the eyes and nose of the model can be also effectively retained. As can be seen from Fig. 8d, e, although the simplification ratio of the two models is the same, the detail features of the eyes and nose of Fig. 8d are more obvious than Fig. 8e and have higher identifiability.
The original model
The simplified model
As shown in Fig. 9, we use our proposed algorithm and the cost-driven algorithm to simplify the original model shown in Fig. 7. Figure 9a is a model after simplification of our algorithm, with 818 points and 1248 triangular patches. Figure 9b is a model after the use of cost-driven simplification, with 1036 points and 1491 triangular patches. Obviously, the boundary of the model obtained by using our algorithm is smoother than that obtained by cost-driven algorithm. Moreover, we can see from Fig. 9 that the simplified algorithm in this paper can retain the detail features of the model more effectively.
Model fusion
Figure 10 is the efficiency comparison when different simplification algorithms are used to simplify the head models of different sizes. Cost-driven method algorithm refers to the literature [7, 8], and JADE (just another decimator) algorithm refers to the literature [21]. From Fig. 10, we can see that the algorithm efficiency in this paper is better than that of JADE algorithm, and slightly better than that of cost-driven method algorithm. For the head models of different sizes, a simplified algorithm for constructing hierarchical models proposed in this paper is very efficient.
Comparison of different simplification algorithms
As shown in Fig. 11, this paper uses a semicircle simulated human body with a hole at the top to fuse with a simplified head model, which is shown in Fig. 11c. After fusion, the detail features of the model will not be lost, and the original boundary edge area will be very smooth after fusion. For the fusion method based on the boundary edge, the size of the fusion region of the model is controllable, because the new vertex produced by the fusion has little effect on the complexity of the model and can be ignored. As shown in Fig 11c, there are 3740 points before the fusion of the model, and after fusion, there are 3934 points, which only less than 200 points are generated. The fusion method in this paper will not change the complexity of the model too much and still enables the model to maintain its scale before fusion.
As shown in Table 1, the mesh model fusion method based on FCF is compared with the improved FCF method. When there are 420 and 398 points in F1, F2, respectively, 2058 points can be generated through fusion by using the method before the improvement, and 1023 points can be generated through fusion by using the improved method, which is far less than the points produced by using the method before improvement. According to the comparison of data in Table 1, we can see that compared with the original fusion algorithm, the improved fusion algorithm can get a simpler transition model. The effect diagram of fusion experiment in Fig. 12 shows that, although the improved fusion algorithm making the transition model simpler, there is little difference in the effect of fusion.
Table 1 Comparison of different fusion algorithms
Comparison of different fusion results
The hierarchical human body model library can be constructed by this method and applied to crowd simulation evacuation [22,23,24,25,26]. As shown in Fig. 13, the scene of an emergency evacuation of the crowd in a teaching building is simulated. In Fig. 13a), the LOD method is used. When the viewpoint distance is less than 40, a fine human body model (with 43,261 triangular patches) is used, and a medium precision model (with 3216 triangular patches) is used when the viewpoint distance is greater than 40 less than 100, and the rough model (with 1046 triangular patches) is used when the viewpoint distance is greater than 100. The most complex model is used in the Fig. 13b scene. The average frame rate using the LOD method is 22 FPS (frames per second), while the frame rate using the most complex model is only 7.5 FPS. It can be seen that building hierarchical model library in crowd simulation can effectively improve the performance of the system.
The application of simplified model in school scene
The 3D human body model is single, and the production process is complex, which is time-consuming and laborious. The Kinect depth camera launched by Microsoft can quickly scan the human body model with human body construction features, so as to reconstruct the 3D model of human body. However, the amount of data of 3D human body model reconstructed with Kinect is large, and the computational complexity is high, which cannot be directly applied to the group simulation and needs to be simplified. At the same time, we need to keep the effective detail features of the model while simplifying it. Therefore, the main contributions of this paper are as follows: An interactive method for preserving detail features of models based on the simplified methods of edge curvature and area error is proposed. The fusion method based on FCF function is improved, which realizes the automatic fusion between models. The simplified method and improved fusion method proposed in this paper have good robustness for different models. Through the above methods, the personalized character model suitable for group simulation can be reconstructed, and a hierarchical character model database can be constructed, which improves the sense of reality and efficiency of the group simulation.
3D:
CGAL:
Computational geometry algorithms library
Depth first search
FCF:
FPS:
Graphics processing unit
JADE:
Just another decimator
LOD:
Levels of detail
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We gratefully thank the anonymous reviewers and Associate Editor for the constructive and detailed comments that helped improve the paper.
This work is supported by Shandong Provincial Natural Science Foundation of China under Grant No. ZR2014FQ009.
Please request authors.
Department of Finance, Shandong Normal University, Jinan, 250014, China
Lulu Ma
Shandong Pingyin Limited by Share Ltd of the rural commercial bank, Jinan, 250400, China
Ke Zhang
School of Information Science and Engineering, Shandong Normal University, Shandong Provincial Key Laboratory for Distributed Computer Software Novel Technology, Jinan, 250014, China
Dianjie Lu
The original idea of the research was proposed by LM but was largely inspired by the discussions with DL. KZ contributed to the experiment analysis. All three authors worked closely during the preparation and writing of the manuscript. All authors read and approved the final manuscript.
Correspondence to Dianjie Lu.
Author 1: Lulu Ma received her Master's degree in accounting from the Shandong Economic University, Jinan, China, in 2006.After that, she joined in the Shandong Normal University, Jinan, China. Currently, she is a lecture of the department of finance at Shandong Normal University. Her research interests include CSCW and CAD.
Author 2: Ke Zhang received his B. S. and M. S. degrees in Shandong Normal University, Jinan, China, in 2013 and 2016, respectively. He is now working at Shandong Pingyin Limited by Share Ltd. of the rural commercial bank. His research interests include CAD and swarm intelligence.
Author 3: Dianjie Lu received his Ph.D. degree in computer science from the Institute of Computing Technology, Chinese Academy of Science, in 2012. Currently, he is an associate professor at Shandong Normal University. His research interests include CSCW, swarm intelligence, and cloud computing. His contact information is [email protected].
Ma, L., Zhang, K. & Lu, D. Research and application of personalized human body simplification and fusion method. J Image Video Proc. 2018, 55 (2018). https://doi.org/10.1186/s13640-018-0293-7
Model simplification
Detail features
Visual Information Learning and Analytics on Cross-Media Big Data | CommonCrawl |
Article (Russian)
Concerning the solution of certain boundary problems with discontinuous coefficients of the boundary condition
Kozlov O. M.
↓ Abstract | Full text (.pdf)
Ukr. Mat. Zh. - 1964. - 16, № 2. - pp. 143-156
For a certain class of coefficients a, including $\sigma \in L_p$ ($p$ depending on the number of variables) a generalized solution of (I), (II) is constructed satisfying the boundary condition (II) in a certain «strong» sense.
Brief Communications (Russian)
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Fodchuk V. I. | CommonCrawl |
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A study on lump solutions to a (2+1)-dimensional completely generalized Hirota-Satsuma-Ito equation
Yufeng Zhang 1, , Wen-Xiu Ma 2,3,4,5,6,7,, and Jin-Yun Yang 8,
School of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, China
Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia
Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, USA
School of Mathematics, South China University of Technology, Guangzhou 510640, China
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China
International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa
School of Mathematics and Physical Science, Xuzhou Institute of Technology, Xuzhou 221008, Jiangsu, China
* Corresponding author: Wen-Xiu Ma
We aim to generalize the (2+1)-dimensional Hirota-Satsuma-Ito (HSI) equation, passing the three-soliton test, to a new one which still has diverse solution structures. We add all second-order derivative terms to the HSI equation but demand the existence of lump solutions. Such lump solutions are formulated in terms of the coefficients, except two, in the resulting generalized HSI equation. As an illustrative example, a special completely generalized HSI equation is given, together with a lump solution, and three 3d-plots and contour plots of the lump solution are made to elucidate the characteristics of the presented lump solutions.
Keywords: Symbolic computation, lump solution, interaction solution.
Mathematics Subject Classification: Primary: 35Q51, 35Q53; Secondary: 37K40.
Citation: Yufeng Zhang, Wen-Xiu Ma, Jin-Yun Yang. A study on lump solutions to a (2+1)-dimensional completely generalized Hirota-Satsuma-Ito equation. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020167
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Figure 1. Profiles of $ u $ when $ x = 0, 25, 50 $: 3d plots (top) and contour plots (bottom)
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Yufeng Zhang Wen-Xiu Ma Jin-Yun Yang | CommonCrawl |
Ethnobotanical study and phytochemical profiling of Heptapleurum hypoleucum leaf extract and evaluation of its antimicrobial activities against diarrhea-causing bacteria
S. M. Rakib-Uz-Zaman1 na1,
Asif Iqbal1 na1,
Sadrina Afrin Mowna1,
Mst Gitika Khanom2,
Mohammad Mastak Al Amin1 &
Kashmery Khan1
Due to the development of superbugs as a result of unprescribed and frequent use of antibiotics in recent years, an alternate form of medicine had to be introduced. In light of this global threat, researchers all over the world have been gravitating towards herbal medicines. In order to find out new ways of saving the planet using medicinal plants, ethnobotanical studies must be carried out. Concerning this, an ethnobotanical study has been done in this paper to identify potential medicinal plants in Rangamati, Bangladesh.
For the ethnobotanical survey, randomized 104 people were interviewed and 62 different plant species were found to treat 19 different kinds of diseases and 84% of people reported to be completely recovered. Furthermore, among the 19 diseases found, the majority of them were common cold, abdominal pain or gastric, diarrhea, and dysentery. From the 62 different plant species, Heptapleurum hypoleucum, used for the treatment of diarrhea, was selected for conducting further studies due to its heavy use as reported by the tribal people. In this study, the aqueous, ethanol, and methanol extracts of Heptapleurum hypoleucum were subjected to microbial susceptibility assays using the agar well diffusion method. The test microorganisms were Salmonella typhi, Streptococcus pneumoniae, Staphylococcus aureus, Shigella flexneri, and Escherichia coli. Among these, the most susceptible organisms were Staphylococcus aureus (21 mm) and Salmonella typhi (19 mm) in the ethanolic extract. Also, the methanolic extract showed an inhibition zone of 13 mm against E. coli, which was more than that of the antibiotic's (11 mm). Phytochemical screening of the plant revealed that it contains alkaloids, phenols, steroids, and flavonoids, but lacks saponins and tannins.
To combat the rising threat of antibiotic resistance, ethnoscience needs to be consolidated with modern biotechnological techniques to make the most use of the vast amount of natural resources. The findings of this study indicate that Heptapleurum hypoleucum, an ethnobotanical medicinal plant, has shown comparable antimicrobial activity with commercial antibiotics against several diarrhea-causing pathogens and also contains several medically important phytochemicals.
The discovery of antibiotics has been a blessing to the modern world. Sadly, the uncontrolled and unprescribed use of antibiotics has turned them into the enemy of human existence as microorganisms are becoming multi-drug resistant. Given this, researchers and scientists agree that medicinal plants are a substantial alternative to this global crisis [1]. However, to discover new species of medicinally important plants, ethnobotanical surveys are needed to be conducted. In respect to that, an ethnobotanical study has been done as a part of this research paper to identify some of the plants used by tribal people in Rangamati, Bangladesh.
Medicinal plants are a common phenomenon in various continents and are also an essential source of different medicines [2]. Being a developing country, the use of medicinal plants in Bangladesh is notable in many regions. Therefore, medicinal plants have become one of the leading choices for treating different injuries and diseases. A significant portion of the plant species has been recognized as valuable resources of natural antimicrobial compounds which can be used as an alternative for the treatment of antibiotic-resistant bacterial infections [3]. They are the most suitable solution to the problems regarding antibiotic resistance, as they are cost-effective and have the least amount of side effects [4]. Medicinal plants exhibit antibacterial activities because of the phytochemicals produced during the secondary metabolism of the plants [5,6,7]. Plants are rich in a wide variety of secondary metabolites, including tannins, alkaloids, phenolic compounds, and flavonoids [8, 9]. These phytochemicals are known to exhibit anti-inflammation, antimicrobial, and antifungal activities [5]. Reports suggest that phenolic and flavonoid compounds prevent diseases such as cancer, heart problems, cataracts, eye disorders, and Alzheimers [6]. The most vital properties of flavonoids include their ability to protect against oxidative diseases, reduce the oxidation of low-density lipoproteins to provide protection against cardiovascular diseases, and activate or inhibit various enzymes bind specific receptors [10].
Moreover, it is suggested that herbal extracts may be an effective alternative to antibiotics in order to cure common recurrent diseases like diarrhea, skin disease, throat, or ear infection [11]. Surprisingly, medicinal plants have also shown their efficiency in treating diabetes due to the presence of carotenoids, flavonoids, terpenoids, alkaloids, and glycosides [12]. The anti-hyperglycemic effects that result from treatment with plants are often due to their ability to improve the performance of pancreatic tissue, which is done by increasing insulin secretions or reducing the intestinal absorption of glucose [13].
A survey conducted among the Santal tribe of Joypurhat District, Bangladesh, found 33 plant species that were used by the tribal people for treating diabetes (A.H.M [14]).
Ethnobotanical surveys in our country [15, 16] have brought plants having effective medicinal characteristics into attention. Some studies have been done specifically to search for different medicinal plants, which are being used for the treatment of diarrhea and dysentery [17, 18]. After conducting the survey, a medicinal plant—Heptapleurum hypoleucum, locally known as Jharobbo hogoeya (Bangladesh Ethnobotany Online Database), was collected to evaluate its antibacterial activity against some pathogenic microorganisms responsible for causing diarrhea. This plant grows in the forest regions of the Rangamati, and people from the survey have claimed to use this plant for the treatment of diarrhea and dysentery. Tribal people are extremely experienced as they have been treating diseases using medicinal plants since prehistoric times and thus are knowledgeable about herbal medicine including those used for the treatment of diarrhea (M.S. [16]).
In this study, the test microorganisms were Salmonella typhi, Streptococcus pneumoniae, Staphylococcus aureus, Shigella flexneri, and Escherichia coli. The aqueous, ethanol, and methanol extracts of Heptapleurum hypoleucum were subjected to microbial susceptibility assays using the agar well diffusion method. After the phytochemical screening of plant extract, we have successfully detected alkaloids, phenolics, steroids, and flavonoids in different solvents. However, no saponins and tannins were identified in our extract. This study reveals the antibacterial effects of ethnobotanical medicinal plants as well as the presence of phytochemicals that may have antimicrobial activity against diarrhea-causing pathogenic microorganisms.
Survey site selection
The region selected for this research purpose has an area of 6116.13 km2 and is located in between 22°27′ and 23°44′ north latitudes and in between 91°56′ and 92°33′ east longitudes. Approximately 5,08,182 people live here and the main indigenous communities that live here are Chakma, Monipuri, Tripura, Khumi, Marma, Tanchangya, Santal, Mro, and many more.
Ethics and consent to participate
The purpose of our study was elaborately explained to each informant and verbal consent was taken to avoid any misunderstanding. We have also taken permission from the local authority to conduct the survey. As our study only dealt with the medicinal plants with no intention to conduct any trials on human, formal institutional consent for this study is not required. In addition, Bangladesh Medical Research Council Ethical Guidelines for Conducting Research Studies Involving Human Subjects deemed ethics approval unnecessary for this kind of study on page 39, section 10.5.2.
All the data were listed alphabetically and ordered by the plant's scientific name, local name, plant part used, name of the disease, and mode of preparation of the plant. Also, the data were analyzed in the "IBM SPSS Statistics 25" software and graphical presentations were made.
Antibacterial activity test
The pathogenic test organisms used in this assay are as follows: Salmonella typhi, Streptococcus pneumoniae, Staphylococcus aureus, Shigella flexneri, and Escherichia coli, all are known diarrhea-causing bacteria. The bacterial cultures were obtained from the Biotechnology and Molecular Biology Laboratory of the Department of Mathematics and Natural Sciences at BRAC University and the International Centre for Diarrheal Disease Research, Bangladesh (ICDDR, B).
Sample collection and processing
With the help of some local practitioners, parts of the Heptapleurum hypoleucum plant were taken and identified. The stem of the plant was used as the sample for this research work, obtained from the rural areas of Rangamati district. The stem was washed, cleaned, and air dried under open light for several days. As soon as it dried, it was mashed into powder.
Preparation of extracts using different solvents
Ethanolic extraction
Ten grams of powdered sample was dissolved in 100 ml of absolute ethanol in a conical flask, covered with aluminum foil and then kept at 37 °C in a shaker incubator at 120 rpm for 24 h. Using an autoclaved filter paper, the filtrate was collected slowly in a conical flask and then evaporated using a rotary evaporator till the final volume was reduced to one-fourth of the original volume of the solvent used. This concentrated extract solution was poured on a sterile petri dish lid and kept in the incubator at 55 °C for 20 min. Finally, a sticky semi-solid extract appeared on the surface of the plate when all the solvent was evaporated. The extract was collected and stored in a McCartney bottle that was autoclaved and weighed. The extract inside the bottle was weighed, recorded, and the exact amount of the extract was calculated by subtracting the mass of the empty bottle. Then the bottle was labeled and stored at 4 °C in the refrigerator.
Methanolic extraction
Ten grams of powdered sample was dissolved in 100 ml of absolute methanol in a conical flask, covered with aluminum foil and then kept at 37 °C in a shaker incubator at 120 rpm for 24 h. The filtrate was collected slowly in a conical flask using an autoclaved filter paper. It was then evaporated using a rotary evaporator till the final volume was reduced to one-fourth of the original volume of the solvent used. Then the concentrated extract solution was poured on a sterile petri dish lid and kept in the incubator at 55 °C for 20 min. Finally, a sticky semi-solid extract appeared on the surface of the plate when all the solvent was evaporated. The extract was collected and stored in a McCartney bottle that was autoclaved and weighed. The extract inside the container was weighed, recorded, and the exact amount of the extract was calculated by subtracting the mass of the empty bottle. Then the bottle was labeled and stored at 4 °C in the refrigerator.
Aqueous extraction
Ten grams of powdered sample was weighed and mixed with 100 ml of distilled water in a conical flask, covered with aluminum foil. Then kept at 37 °C in a shaker incubator at 120 rpm for 24 h. Using an autoclaved Whatman No.1 filter paper, the filtrate was collected slowly in a conical flask and then stored inside autoclaved falcon tubes at 4 °C in the refrigerator.
Preparation of extract solution for antibacterial activity test
The following formula was used to determine the amount of solvent to be added for making the extract solution for antibacterial activity test.
$$ \mathrm{Amount}\ \mathrm{of}\ \mathrm{solvent}\ \mathrm{to}\ \mathrm{be}\ \mathrm{added}=\frac{100\times \mathrm{amount}\ \mathrm{of}\ \mathrm{extract}\ \mathrm{obtained}}{\mathrm{amount}\ \mathrm{of}\ \mathrm{plant}\ \mathrm{powder}\ \mathrm{used}} $$
Agar well diffusion
Antibacterial activity of aqueous and solvent extracts was determined based on agar well diffusion method developed by Clinical and Laboratory Standards Institute, USA with some modifications depending on our laboratory conditions [19].
Plates of bacterial strains were taken in the laminar hood. A loop was sterilized in the Bunsen flame and was used to scrape off the bacteria and dipped into the test tubes containing saline solution (each containing 9 ml of NaCl) to make a suspension. The test tubes were vortexed and the turbidity of the suspension was visually compared with the 0.5% MacFarland standard solution in order to keep the number of bacteria in the saline suspension within a given range for standardizing the lawn culture of antimicrobial tests. Then, an autoclaved cotton swab was dipped into the suspension and pressed against the inner walls of test tubes to remove excess liquid before taking them out. The cotton swab was rubbed horizontally across the surface of the labeled Mueller Hinton Agar (MHA) plates to conduct the lawn culture of the bacterial strains. A cork borer was heated to sterilize, cooled, and then pressed onto the MHA plates to create the required number of wells on the quadrants of the agar. After that, each well was labeled and filled with 60 μL of diluted methanolic, ethanolic, and aqueous extracts, respectively. Different antibiotic disks were used as a positive control and placed onto one quadrant. Then the MHA plates were kept in the incubator for 24 h at 37 °C and the results were observed and recorded the next day.
All the tests were conducted 3 times to obtain the average value of zones of inhibition. The zones were measured using a millimeter scale and then the activity index for each extract was calculated.
The following formula was used for calculating the activity index:
$$ \mathrm{Activity}\ \mathrm{index}=\frac{\mathrm{zone}\ \mathrm{of}\ \mathrm{inhibition}\ \mathrm{of}\ \mathrm{plant}\ \mathrm{extract}}{\mathrm{zone}\ \mathrm{of}\ \mathrm{inhibition}\ \mathrm{of}\ \mathrm{antibiotic}\ \mathrm{disc}} $$
Biochemical assays and phytochemical analysis
The plant extracts in methanolic, ethanolic, and aqueous solutions were assessed for the existence of the phytochemical compounds based on the following standard methods [20,21,22,23].
Tests for alkaloids
The methanolic extract was diluted in an acidic solution of HCl. This test solution was used for the detection of alkaloids using various reagents.
Hager's test: 1 ml of extract was carefully mixed with 3 drops of freshly prepared Hager's reagent in a test tube. The formation of yellow precipitates represents a positive result and the presence of alkaloids in the extract.
Wagner's test: 1 ml of extract was mixed in a test tube with 3 drops of Wagner's reagent prepared beforehand. The formation of brown precipitate will indicate the presence of alkaloids.
Dragendraff's test: 2 ml of extract was taken in a test tube with 0.2 ml diluted HCl and 1 ml of Dragendraff's reagent and left for a few minutes. A positive result is indicated by the presence of an orange-brown precipitate.
Test for steroidal compounds
Salkowaski's test: 0.5 g of the extract was dissolved in 2 ml chloroform in a test tube. Concentrated sulfuric acid was carefully added on the wall of the test tube to form a lower layer. A reddish-brown color at the interface will indicate the presence of a steroid ring.
Test for phenolic compounds
Equal amounts of 1% ferric chloride solution and 1% potassium ferrocyanide were mixed. Then, 3 drops of this freshly prepared mixture were added to 2 ml extract. The formation of a bluish-green color will represent a positive result.
Test for flavonoids
Reaction with sodium hydroxide: 2 ml diluted NaOH solution was added to 3 ml of extract. The mixture was inspected for the production of yellow color, which is considered positive.
Tests for saponins
Froth test: 0.5 g of the extract was dissolved in 10 ml distilled water. The test tube was stoppered and then shaken vigorously for 30 s. It was then allowed to stand for 30 min. Formation and retention of honey-comb froth on the surface for 30 min is considered positive for saponins.
Tests for tannins
Lead acetate test: 5 ml of extract and a few drops of freshly prepared 1% lead acetate were dissolved together. The formation of a yellow precipitate is considered a positive result.
Ethnobotanical survey
In this survey, it was found that 101 people out of 104 have used or been using medicinal plants. Eighty-five people reported to be fully recovered using the respective medicinal plants, 15 people recovered partially, and 7 people faced some side effects during their treatment. From Table 1, we can see that the percentage of people using medicinal plants was the highest (30.77%) for the age group 30–40 and the percentage of people aged above 60 was the least (6.73%). The second-highest percentage (26.92%) of people using plants as medicine was for the age group 51–60, while the second least percentage (11.54%) of people using herbs was aged less than 30. Medicinal plants were used more by men (74.04%) than women (25.96%). Most people using herbal medicines had a job or a business (20.19% both) and the second-highest percentage of people using medicinal plants were housewives (17.31%). Farmers (14.42%) and workers (11.54%) seemed to use herbal medicines more than unemployed people (8.65%), although, still less than housewives. Lastly, the lowest number of people (1.92%) utilizing herbs were guide by profession (Table 1).
Table 1 Demographic data of the informants. The percentage of people using medicinal plants was highest (30.77%) for the age group 30–40 and the percentage of people aged above 60 was the least (6.73%). In addition, considering the profession, students (5.77%) and guides (1.92%) used the least amount of herbal medicine, while local businessmen and job holders used the highest amount (both 20.19%) of medicinal plants to treat different diseases and illness
In this survey, 42.86% of the plant part used were the leaves, confirming the fact that leaves are the most used part of a plant (Fig. 1). The second most used parts were fruits and whole plants, each having 15.65% of the total count. The least used parts were the seed and bulb, having just 1.36% each. The second least used parts were the rhizome and stem, being only 2.04% each. The percentage of bark used (8.84%) can be termed as the third least part used as it has a higher percentage of use than the seed, rhizome, bulb, and stem. The root (10.20%) is utilized more than the bark but still less than the fruit, whole plant, and leaves (Fig. 1).
Part of the plant used as medicines. The most used part of the plant was leaves (42.86%) and the least used parts were the seed and bulb (1.36%)
In this survey, 33 people preferred to make "paste" from the plant parts (Fig. 2), which is also the highest count and shows the highest percentage (23.91%). From Fig. 2, we can see that plant parts were also taken in the form of "juice" by 31 people, which is the second-highest form (22.46%). In addition, 23 and 20 people used the "decoction" (16.67%) and "extraction" (14.49%) method respectively. Twenty-one people took the whole plant as medicine or processed in some way like cooking (15.22%), which is lower than for the decoction method and higher than for the extraction method. The lowest number of people, 3, took the plant in a powdered form (2.17%) and the second-lowest number of people, 7, used the plant as an infusion (5.07%). In summary, the highest percentage of people used medicinal plants in the form of "paste," while the least number of people used the plant in the "powdered" form.
Mode of preparation of plant parts. The most used form of preparation was "paste" (23.91%) and the least used was "powder" (2.17%)
A wide range of diseases was cured using different kinds of plants. The diseases include minor injuries, common viral infections, and also bacterial diseases, all common throughout our country, Bangladesh. In this survey, common cold, abdominal pain, diarrhea, dysentery, and allergy or rashes are the most common ones, including 19 types of different diseases in total. From Fig. 3, we can conclude that anemia, piles, and measles were the least common. The highest count of occurrence was for common cold, while the second-highest count was for diarrhea among the participants of this study. The least count of occurrence was for piles and the second least count was for both anemia and measles. Although the count of occurrence was quite high for dysentery and abdominal pain and gastric, it was still lower than for common cold, throat pain, and diarrhea. The cases of arthritis, diabetes, worms, hypertension, fever/dengue, constipation, snake-bite, and asthma were very low, but they were still higher than the occurrence of piles, allergy, jaundice, eczema/skin disease, and bleeding from wounds (Fig. 3).
Diseases for which different plants are used. The highest count of occurrence was for common cold, while the least was for piles
Ethnomedicinal plants used by the tribal people of Rangamati
A total of 62 medicinal plants were found to treat several diseases (Table 2). Out of the 26 plants treating gastrointestinal ailments (GIA), some plants are Azadirachta indica, Canna indica, and Heptapleurum hypoleucum. Similarly, 12 plants including Allium sativum were found to treat respiratory system disorders (RSD). Poisonous bites (PB) and skeleton muscular system disorders (SMSD) were treated by 2 plants, including Alocasia cucullate and Brassica nigra respectively. Endocrinal disorders (ED), throat pain, and hemorrhoids (HEM) were healed by Andrographis paniculate, Leea macrophylla, and Nelumbo nucifera respectively. Four plants, together with Stephania japonica, were used as a remedy for different types of fever (Fvr). To treat liver problems (LP), five plants were used, including Carica papaya and Hibiscus sabdariffa. Dermatological infections/diseases (DID) were treated using nine different plants, like, Bridelia retusa, Cassia alata, and Curcuma longa. Three plants were used to treat hypertension (HTN) and two were used to treat Anemia, along with Stephania japonica and Basella alba respectively (Table 2). Heptapleurum hypoleucum was the most frequently used plant for the treatment of diarrhea as reported by the participant of this study which motivated us to choose this plant to further investigate its antimicrobial potentials against diarrhea-causing pathogens.
Table 2 Ethnomedicinal plants used by the tribal people of Rangamati for treating different diseases (source: Medicinal Plants Database of Bangladesh [24,25,26]). A total of 62 medicinal plant species were found out of which 26 plants treat gastrointestinal ailments (GIA), 12 plants treat respiratory system disorders (RSD). Poisonous bites (PB) and skeleton muscular system disorders (SMSD) were treated by 2 plants. Endocrinal disorders (ED), throat pain, and hemorrhoids (HEM) were healed by 3 plants. Four plants were used to cure different types of fever (Fvr). For liver problems (LP), 5 plants were used. Dermatological infections/diseases (DID) were treated using 9 different plants. Three plants were used to treat hypertension (HTN) and 2 were used to treat anemia
Antibacterial assay
The antibiotic disks used were ampicillin and gentamicin (for S. typhii and S. aureus), gentamycin (for E. coli), chloramphenicol (for S. pneumonae), and cefoxitin (for S. flexneri) (Fig. 4). The tests were repeated thrice to ensure the accuracy of the results. Among the three different types of extracts, the distilled water extract showed hardly any positive result. However, ethanolic and methanolic extracts were able to show some positive results, but not against all the organisms (Fig. 5). Ethanolic extract showed the highest zone of inhibition against Staphylococcus aureus, which was 21 mm (Table 3). The average result of the three trials are given below (Fig. 5).
Antibacterial effects of ethanolic and methanolic extracts of Heptapleurum hypoleucum against. aS. pneumonae.bS. aureus. cS. typhii. dE. coli. eShigella flexneri. Ethanolic extract showed the highest zone of inhibition against Staphylococcus aureus (21 mm). Each of the plates is also showing the zone (white) produced by the commercial antibiotics against respective bacteria (as mentioned in Table 3)
Graphical representation zone of inhibition (mm) of different extracts of Heptapleurum hypoleucum in comparison with antibiotic disks against some disease-causing microorganisms. Ethanolic extract showed the highest zone of inhibition against Staphylococcus aureus (21 mm)
Table 3 Antibacterial test results of the ethanolic, methanolic, and aqueous extracts of Heptapleurum hypoleucum. Ethanolic extract showed the highest zone of inhibition against Staphylococcus aureus (21 mm)
The activity index of E. coli was the highest in the ethanolic extract (value 0.81) as well as in the methanolic extract (value 1.18), while the lowest was seen for Shigella flexneri (value 0 for all solvents) (Fig. 6). The activity index for aqueous extract was 0 for E. coli, Salmonella typhi, and Shigella flexneri but had a value of 0.28 for Staphylococcus aureus and a value of 0.25 for Streptococcus pneumoniae. Here, the activity index of ethanolic extract for Streptococcus pneumoniae (0.73) is seen to be lower than for Staphylococcus aureus (0.78) and higher than Salmonella typhi (0.63) (Fig. 6).
Activity index of ethanolic, methanolic, and aqueous extracts against different microorganism. E. coli had the highest activity index in the ethanolic extract (value 0.81) as well as in the methanolic extract (value 1.18), while the lowest was seen for Shigella flexneri (value 0 in all solvents)
Summary of the phytochemical screening
We have performed 6 tests for screening the phytochemical compounds in our extract. The tests are for alkaloids, phenolic compounds, flavonoids, saponins, steroids, and tannins. We got positive results for alkaloids, phenolics, flavonoids, and steroids. However, negative results were observed for saponins and tannins (Fig. 7). The result of the phytochemical screening has been summarized in Table 4.
Figures are showing phytochemical test results for different plant compounds. a Alkaloids (Hager's test): from left to right—Hager's reagent, test result, and ethanolic extract. There was a visible color change of the sample which indicates a positive result for alkaloids. b Alkaloids (Wagner's test): from the left side is the test sample, Wagner's reagent mixture, Wagner's reagent, and ethanol. The visible color change of the sample indicates a positive result for alkaloids. c Test for the identification of phenolic compounds. From the left side, the first two are sample extracts showing the desired color change which indicate positive result and the 3rd one is the control, FeCl3. A dark green color indicates the presence of phenolic compounds in the sample. d Tests for tannins (lead acetate test): no yellow precipitation in the test sample indicates a negative result. e Test for flavonoids: from the left side—1st one is test sample showing yellow color, 2nd one ethanol, 3rd one NaOH, and 4th one is the mixture of ethanol and NaOH. There was a visible color change of the sample which indicates a positive result for flavonoids. f Tests for saponins (froth test): the left one is the test sample and the other one is distilled water. No froth formed in the test sample indicates a negative result for saponins
Table 4 Phytochemical analysis showing positive results for alkaloids, flavonoids, and steroids. In contrast, our study also indicates the absence of saponins and tannins in the sample
Tribal people have been using plants to treat different kinds of diseases for a long time. Years of experience has made them a "knowledge house" on herbal remedies. An ethnobotanical survey of medicinal plants in the Chittagong Hill Tracts has revealed that 56 plant species were used by the tribal people only to treat dermatological disorders [27]. In another ethnobotanical study, conducted among Pangkhua community in Bilaichari, Bangladesh, revealed the traditional use of 117 plant species to treat 11 categories of ailments, recorded from 218 traditional healers and elderly men and women [28]. Our study also found that 62 plant species were used by the tribal people of Rangamati to treat almost 20 different diseases, namely abdominal pain, common cold and sneezing, injury, diarrhea, and dysentery. In this study, about 77% of the participants reported that they had recovered from the above-mentioned diseases within 1 to 10 days using only the medicinal plants as a remedy. In contrast, only 2–3% of the participants took additional medications (besides the medicinal plants) which might contribute to their quick recovery. Our study also found that age has significant influence when choosing medicinal plants over traditional medications. Older people naturally ought to have a better understanding and greater knowledge about the traditional use of medicinal plants as pharmaceuticals or modern medicines were unavailable in their era. The fact of having superior knowledge about the use of medicinal plants is reflected to some extent in this survey as about 34% of the total people questioned aged above 50 years old. While using plants as remedies against diseases, leaves (42.68%) were found to be the most used part as medicine as they are most easily accessible can also be used in different ways very easily. Different plants and their parts may have a distinct mode of preparations and are preferred over one another according to the user's flexibility and satisfaction. The simplest and most common modes of preparation are extraction, juice, paste, and decoction. In this study, most people used paste (33%).
Knowledge regarding medicinal plants is obtained from different sources such as family, the local Kaviraaj, and the local people. Consequently, the information gained is discrete, given that, it was found that several plants or plant parts were used for treating the same disease. For instance, for treating the common cold, "Tulsi" (Ocimum tenuiflorum), "Bashok" (Justicia adhatoda), "Dhan-shabarang" (Cymbopogan citratus) etc. were used. Supporting this fact, a study conducted in the Lushai community of Bandarban district of Bangladesh, pointed out that 37 different plant species were used to treat diarrhea and 40 plant species were used to treat dysentery [16]. On the other hand, one particular plant was used to treat many different diseases. The plant "Thankuni" (Centella asiatica) was used to treat both diarrhea and gastric. Similarly, "Ghritkumari" (aloe vera) is used as a remedy for both constipation, due to its laxative property which relieves constipation by promoting intestinal motility, as well as diabetes, as aloe vera is believed to improve insulin secretion and enhance pancreatic β cell function [29, 30].
The plant Heptapleurum hypoleucum was selected for further study as this has been used most commonly to treat diarrhea in the tribal areas. The stem part of the plant was collected and processed into powder and examined to assay its antibacterial activity against several diarrhea-causing bacteria. From the trials, the highest inhibition zone was seen by ethanolic extract against Staphylococcus aureus, which was 21 mm. In contrast, another research work with medicinal plants showed that the chloroform extract of P. sagitatta plant showed an inhibition zone of 16 mm against Staphylococcus aureus [31]. Comparing the two, the 21 mm inhibition zone from the ethanolic extract seems very significant. However, no zone was observed against Shigella flexneri both in methanolic and ethanolic extract. Additionally, the aqueous extract showed results only against S. aureus and S. pneumoniae. Although the activity for commercial antibiotics is much higher than herbal plants, the use of medicinal plants should be encouraged as the overuse of antibiotics will eventually lead to the development of superbugs which will hinder proper treatment. As the extract of our medicinal plant was not purified and yet we found a comparable value (21 mm in ethanolic extract and 27 mm in commercial antibiotic against Staphylococcus aureus), we believe purification of the plant extract will increase its activity significantly.
Medicinal plants certainly contain various types of minerals and both primary and secondary metabolites and for these, they confer antimicrobial effects [32,33,34]. In this study, phytochemical analysis of the plant extract showed both positive and negative results for several compounds. The plant extract has shown to contain alkaloids, phenolics, steroids, and flavonoids, whereas a negative result was observed for saponins and tannins, which indicates that Heptapleurum hypoleucum lack these two phytochemicals.
Ethnoscience and indigenous knowledge need to be consolidated with modern biotechnological techniques and approaches to achieve the desired end products with scientific validation and to make the most use of the vast amount of natural resources around us. This study suggests that various medicinal plants can serve as an alternative to the mainstream drugs to which bacteria are gradually becoming resistant. From the results, Heptapleurum hypoleucum has not only shown significant antibacterial effect against several diarrhea-causing pathogens, but we also found some medically important constituents after conducting the phytochemical tests; thus, further research is recommended to purify these phytochemicals to understand the full potentials of this medicinal plant.
All data generated or analyzed during this study are included in this published article.
MHA:
Mueller Hinton Agar
Activity index
GIA:
Gastrointestinal ailments
RSD:
Respiratory system disorders
PB:
Poisonous bites
SMSD:
Skeleton muscular system disorders
ED:
Endocrinal disorders
HEM:
LP:
DID:
Dermatological infections and diseases
HTN:
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The authors are thankful to the BRAC University, Bangladesh, for the financial and infrastructural support provided for this study. The authors are cordially grateful to the tribal communities inhabiting in different localities of Rangamati district because of their kind support and cooperation during the field surveys. A lot of thanks to all the traditional health practitioners and people involved in the interviews for providing information about the medicinal applications of the plants.
This research received no external funding.
S. M. Rakib-Uz-Zaman and Asif Iqbal contributed equally to this work.
Biotechnology Program, Department of Mathematics and Natural Sciences, School of Sciences, BRAC University, Dhaka, Bangladesh
S. M. Rakib-Uz-Zaman, Asif Iqbal, Sadrina Afrin Mowna, Mohammad Mastak Al Amin & Kashmery Khan
Department of Biotechnology and Genetic Engineering, Islamic University of Kushtia, Kushtia, Bangladesh
Mst Gitika Khanom
S. M. Rakib-Uz-Zaman
Asif Iqbal
Sadrina Afrin Mowna
Mohammad Mastak Al Amin
Kashmery Khan
AI and RUZ were involved in conception and design of the experiments. AI conducted the survey. AI, RUZ, and KK contributed to perform the experiments. AI, MMAA, and GK analyzed the data and contributed to drafting the article. SAM helped preparing and revising the manuscript. RUZ made the final approval of the version to be published. All authors contributed to revising it critically for important intellectual content. All authors read and approved the final manuscript.
Correspondence to S. M. Rakib-Uz-Zaman.
Prior informed verbal consent was always obtained before interview and comply with local guidelines.
Rakib-Uz-Zaman, S.M., Iqbal, A., Mowna, S.A. et al. Ethnobotanical study and phytochemical profiling of Heptapleurum hypoleucum leaf extract and evaluation of its antimicrobial activities against diarrhea-causing bacteria. J Genet Eng Biotechnol 18, 18 (2020). https://doi.org/10.1186/s43141-020-00030-0
Ethnobotanical study
Phytochemical screening
Heptapleurum hypoleucum
Antimicrobial activity | CommonCrawl |
The sensitivity of global temperature to CO2
September 13, 2011 / geoenergymath
Doing exploratory analysis with the CO2 perturbation data at the Wood For Trees data repository and the results are very interesting. The following graph summarizes the results:
Rapid changes in CO2 levels track with global temperatures, with a variance reduction of 30% if d[CO2] derivatives are included in the model. This increase in temperature is not caused by the temperature of the introduced CO2, but is likely due to a reduced capacity for the biosphere to take-up the excess CO2.
This kind of modeling is very easy if you have any experience with engineering controls development. The model is of the type called Proportional-Derivative, and it essentially models a first-order equation
$$\Delta T = k[CO_2] + B \frac{d[CO_2]}{dt}$$
The key initial filter you have to apply is to average the Mauna Loa CO2 data over an entire year. This gets rid of the seasonal changes and the results just pop out.
The numbers I used in the fit are B=1.1 and k=0.0062. The units are months. The B coefficient is large because the CO2 impulse response has got that steep downslope which then tails off:
$$ d[CO_2] \sim \frac{1}{\sqrt{t}}$$
Is this a demonstration of causality, +CO2 => +Temperature?
If another causal chain supports the change in CO2, then likely. We have fairly good records of fossil fuel (FF) emissions over the years. The cross-correlation of the yearly changes, d[CO2] and d[FF], show a zero-lag peak with a significant correlation (below right). The odds of this happening if the two time-series were randomized is about 1 out of 50.
Left chart from Detection of global economic fluctuations in the atmospheric co2 record. This is not as good a cross-correlation as the d[CO2] and dTemperature data — look at year 1998 in particular, but the zero-lag correlation is clearly visible in the chart..
This is the likely causality chain:
$$d[FF] \longrightarrow d[CO_2] \longrightarrow dTemperature$$
If it was the other way, an increase in temperature would have to lead to both CO2 and carbon emission increases independently. CO2 could happen because of outgassing feedbacks (CO2 in oceans is actually increasing despite the outgassing), but I find it hard to believe that the world economy would increase FF emissions as a result of a warmer climate.
What happens if there is a temperature forcing CO2 ?
With the PD model in place the of d[CO2] against Temperature cross-correlation looks like the following:
The Fourier Transform set looks like the following:
This shows the two curves have the same slope in spectrum and just a scale shift. The upper curve is the ratio between the two curves and is essentially level.
The cross-correlation has zero lag and a strong correlation of 0.9. The model again is
$$ \Delta T = k[CO_2] + B \frac{d[CO_2]}{dt}$$
The first term is a Proportional term and the second is the Derivative term. I chose the coefficients to minimize the variance between the measured Temperature data and the model for [CO2]. In engineering this is a common formulation for a family of feedback control algorithms called PID control (the I stands for integral). The question is what is controlling what.
When I was working with vacuum deposition systems we used PID controllers to control the heat of our furnaces. The difference is that in that situation, the roles are reversed, with the process variable being a temperature reading off a thermocouple and the forcing function is power supplied to a heating coil as a PID combination of T. So it is intuitive for me to immediately think that the [CO2] is the error signal, yet that gives a very strong derivative factor which essentially amplifies the effect. The only way to get a damping factor is by assuming that Temperature is the error signal and then we use a Proportional and an Integral term to model the [CO2] response. Which would then give a similar form and likely an equally good fit.
It is really a question of causality, and the controls community have a couple of terms for this. There is the aspect of Controllability and that of Observability (due to Kalman).
Controllability: In order to be able to do whatever we (Nature) want
with the given dynamic system under control input, the system must be controllable.
Observability: In order to see what is going on inside the system under observation, the system must be observable.
So it gets to the issue of two points of view:
1. The people that think that CO2 is driving the temperature changes have to assume that nature is executing a Proportional/Derivative Controller on observing the [CO2] concentration over time.
2. The people that think that temperature is driving the CO2 changes have to assume that nature is executing a Proportional/Integral Controller on observing the temperature change over time, and the CO2 is simply a side effect.
What people miss is that it can be potentially a combination of the two effects.
Nothing says that we can't model something more sophisticated like this:
$$ c \Delta T + M \int {\Delta T}dt = k[CO_2] + B \frac{d[CO_2]}{dt}$$
The Laplace transfer function Temperature/CO2 for this is:
$$ \frac {s(k + B s)}{c s + M} $$
Because of the s in the numerator, the derivative is still dominating but the other terms can modulate the effect.
This blog post did the analysis a while ago, the image below is fascinating because the overlay between the dCO2 and Temperature anomaly matches to the point that the noise even looks similar . This doesn't go back to 1960.
If CO2 does follow Temperature, due to the Causius-Clapeyron and the Arrhenius rate law, a positive feedback will occur — as the released CO2 will provide more GHG which will then potentially increase temperature further. It is a matter of quantifying the effect. It may be subtle or it may be strong.
From the best cross-correlation fit, the perturbation is either around (1) 3.5 ppm change per degree change in a year or (2) 0.3 degree change per ppm change in a year.
(1) makes sense as a Temperature forcing effect as the magnitude doesn't seem too outrageous and would work as a perturbation playing a minor effect on the 100 ppm change in CO2 that we have observed in the last 100 years.
(2) seems very strong in the other direction as a CO2 forcing effect. You can understand this if we simply made a 100 ppm change in CO2, then we would see a 30 degree change in temperature, which is pretty ridiculous, unless this is a real quick transient effect as the CO2 quickly disperses to generate less of a GHG effect.
Perhaps this explains why the dCO2 versus Temperature data has been largely ignored. Even though the evidence is pretty compelling, it really doesn't further the argument on either side. On the one side interpretation #1 is pretty small and on the other side interpretation #2 is too large, so #1 may be operational.
One thing I do think this helps with is providing a good proxy for differential temperature measurements. There is a baseline increase of Temperature (and of CO2), and accurate dCO2 measurements can predict at least some of the changes we will see beyond this baseline.
Also, and this is far out, but if #2 is indeed operational, it may give credence to the theory that that we may be seeing the modulation of global temperatures the last 10 years because of a plateauing in oil production. We will no longer see huge excursions in fossil fuel use as it gets too valuable to squander, and so the big transient temperature changes from the baseline no longer occur. That is just a working hypothesis.
I still think that understanding the dCO2 against Temperature will aid in making sense of what is going on. As a piece in the jigsaw puzzle it seems very important although it manifests itself only as a second order effect on the overall trend in temperature. In summary, as a feedback term for Temperature driving CO2 this is pretty small but if we flip it and say it is 3.3 degrees change for every ppm change of CO2 in a month, it looks very significant. I think that order of magnitude effect more than anything else is what is troubling.
One more plot of the alignment. For this one, the periodic portion of the d[CO2] was removed by incorporating a sine wave with an extra harmonic and averaging that with a kernel function for the period. This is the Fourier analysis with t=time starting from the beginning of the year.
$$ 2.78 \cos(2 \pi t – \theta_1) + 0.8 \cos(4 \pi t – \theta_2) $$
$$ \theta_1 = 2 $$
$$ \theta_2 = -0.56 $$
phase shift in radians. The yearly kernel function is calculated from this awk function:
BEGIN {
n[I++] = $1
END {
Groups = int(I / 12)
## Kernel function
for(i=0; i<=12; i++) {
for(j=0; j<Groups; j++) {
x += n[j*12+i]
G[i] = x/Groups
Scale = (G[12]-G[0])
Y[i] = (G[i]-G[0]) -i*Scale/12
for(i=0; i<12; i++) {
Diff = n[j*12+i] – Y[i]
print Diff
This was then filtered with a 12 month moving average. It looks about the same as the original one from Wood For Trees, with the naive filter applied at the source, and it has the same shape for the cross-correlation. Here it is in any case; I think the fine structure is a bit more apparent(the data near the end points is noisy because I applied the moving average correctly).
How can the derivative of CO2 track the temperature so closely? My working theory assumes that new CO2 is the forcing function. An impulse of CO2 enters the atmosphere and it creates an impulse response function over time. Let's say the impulse response is a damped exponential and the atmospheric temperature responds quickly to this profile.
The CO2 measured at the Mauna Loa station takes some time to disperse over from the original source points. This implies that a smearing function would describe that dispersion, and we can model that as a convolution. The simplest convolution is an exponential with an exponential, as we just need to get the shape right. But what the convolution does is eliminate the strong early impulse response, and thus create a lagged response. As you can see from the Alpha plot below, the way we get the strong impulse back is to take the derivative. What this does is bring the CO2 signal above the sample-and-hold characteristic caused by the fat-tail. The lag disappears and the temperature anomaly now tracks the d[CO2] impulses.
If we believe that CO2 is a forcing function for Temperature, then this behavior must happen as well; the only question is whether the effect is strong enough to be observable.
If you realize that the noisy data below is what we started with, and we had to extract a non-seasonal signal from the green curve, one realizes that detecting that subtle a shift in magnitude is certainly possible.
← Explaining the "Missing Carbon"
Fat-Tail Impulse Response of CO2 → | CommonCrawl |
\begin{definition}[Definition:Injection/Definition 3]
Let $f$ be a mapping.
Then $f$ is an injection {{iff}}:
:$f^{-1} {\restriction_{\Img f} }: \Img f \to \Dom f$ is a mapping
where $f^{-1} {\restriction_{\Img f} }$ is the restriction of the inverse of $f$ to the image set of $f$.
\end{definition} | ProofWiki |
\begin{document}
\begin{frontmatter}
\title{{\bfseries Some estimates for commutators of the fractional maximal function on stratified Lie groups }}
\author[mymainaddress]{Jianglong Wu\corref{mycorrespondingauthor}} \cortext[mycorrespondingauthor]{Corresponding author} \ead{[email protected]}
\author[mysecondaryaddress]{Wenjiao Zhao}
\address[mymainaddress]{Department of Mathematics, Mudanjiang Normal University, Mudanjiang 157011, China} \address[mysecondaryaddress]{School of Mathematics, Physics and Finance, Anhui Polytechnic University, Wuhu 241000, China}
\begin{abstract} In this paper, the main aim is to consider the boundedness of the fractional maximal commutator $M_{\alpha,b}$ and the nonlinear commutator $[b, M_{\alpha}]$ on the Lebesgue spaces over some stratified Lie group $\mathbb{G}$ when $b$ belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces on Lie group are given. \end{abstract}
\begin{keyword} stratified Lie group, fractional maximal function, Lipschitz function, commutator
\MSC[2020] Primary 42B35, 43A80, 26A16, 26A33 \end{keyword}
\end{frontmatter}
\section{Introduction and main results} \label{sec:introduction}
During the last several decades, stratified groups appear in quantum physics and many parts of mathematics, including several complex variables, Fourier analysis, geometry, and topology \cite{folland1982hardy,varopoulos2008analysis}. The geometry structure of stratified groups is so good that it inherits a lot of analysis properties from the Euclidean spaces \cite{stein1993harmonic,grafakos2009modern}. Apart from this, the difference between the geometry structures of Euclidean spaces and stratified groups makes the study of function spaces on them more complicated. However, many harmonic analysis problems on stratified Lie groups deserve a further investigation since most results of the theory of Fourier transforms and distributions in Euclidean spaces cannot yet be duplicated.
It is worthwhile to note that the fractional maximal operator plays an important role in real and harmonic analysis and applications, such as potential theory and partial differential equations (PDEs), since it is intimately related to the Riesz potential operator, which is a powerful tool in the study of the smooth function spaces (see \cite{folland1982hardy,bonfiglioli2007stratified,carneiro2017derivative}). On the other hand, there are two major reasons why the study of the commutators has got widespread attention. The first one is that the boundedness of commutators can produce some characterizations of function spaces \cite{janson1978mean,paluszynski1995characterization}. The other one is that the theory of commutators is intimately related to the regularity properties of the solutions of certain PDEs \cite{chiarenza1993w2,difazio1993interior,ragusa2004cauchy,bramanti1995commutators}.
Let $T$ be the classical singular integral operator. The Coifman-Rochberg-Weiss type commutator $[b, T]$ generated by $T$ and a suitable function $b$ is defined by \begin{align} \label{equ:commutator-1}
[b,T]f & = bT(f)-T(bf). \end{align} A well-known result shows that $[b,T]$ is bounded on $L^{p}(\mathbb{R}^{n})$ for $1<p<\infty$ if and only if $b\in \bmo(\mathbb{R}^{n})$ (the space of bounded mean oscillation functions). The sufficiency was provided by Coifman et al.\cite{coifman1976factorization} and the necessity was obtained by Janson \cite{janson1978mean}. Furthermore, Janson \cite{janson1978mean} also established some characterizations of the Lipschitz space $\Lambda_{\beta}(\mathbb{R}^{n})$ via commutator \labelcref{equ:commutator-1} and proved that $[b,T]$ is bounded from $L^{p}(\mathbb{R}^{n})$ to $L^{q}(\mathbb{R}^{n})$ for $1<p<n/\beta$ and $1/p-1/q=\beta/n$ with $0<\beta<1$ if and only if $b\in \Lambda_{\beta}(\mathbb{R}^{n})$ (see also Paluszy\'{n}ski \cite{paluszynski1995characterization}).
Denote by $\mathbb{G}$ and $\mathbb{R}$ the sets of groups and real numbers separately. Let $0\le \alpha <Q$ and $f: \mathbb{G} \to \mathbb{R}$ be a locally integrable function, the fractional maximal function is given by
\begin{align*}
M_{\alpha}(f)(x) &= \sup_{B\ni x \atop B\subset \mathbb{G}} \dfrac{1}{|B|^{1-\alpha/Q}} \displaystyle\int_{B} |f(y)| \mathrm{d} y,
\end{align*}
where the supremum is taken over all $ \mathbb{G}$-balls $B\subset \mathbb{G}$ containing $x$ with radius $r>0$ , and $|B|$ represents the Haar measure of the $ \mathbb{G}$-ball $B$ (for the notations and notions, see \cref{sec:preliminary} below). When $\alpha=0$, we simply write $M $ instead of $M_{0}$, which is the Hardy-Littlewood maximal function defined as
\begin{align*}
M(f)(x) &= \sup_{B\ni x\atop B\subset \mathbb{G}} \dfrac{1}{|B|} \displaystyle\int_{B} |f(y)| \mathrm{d} y.
\end{align*}
Similar to \labelcref{equ:commutator-1}, we can define two different kinds of commutator of the fractional maximal function as follows.
\begin{definition} \label{def.commutator-frac-max}
Let $0 \le \alpha<n$ and $b$ be a locally integrable function on $\mathbb{G}$.
\begin{enumerate}[label=(\roman*)]
\item The maximal commutator of $M_{\alpha}$ with $b$ is given by
\begin{align*}
M_{\alpha,b} (f)(x) &= \sup_{B\ni x \atop B\subset \mathbb{G}} \dfrac{1}{|B|^{1-\alpha/Q}} \displaystyle\int_{B} |b(x)-b(y)| |f(y)| \mathrm{d} y,
\end{align*}
where the supremum is taken over all $ \mathbb{G}$-balls $B\subset\mathbb{G}$ containing $x$.
\item The nonlinear commutators generated by $M_{\alpha}$ and $b$ is defined by
\begin{align*}
[b,M_{\alpha}] (f)(x) &= b(x) M_{\alpha} (f)(x) -M_{\alpha}(bf)(x).
\end{align*}
\end{enumerate}
\end{definition}
When $\alpha=0$, we simply denote by $[b,M]=[b,M_{0}]$ and $M_{b} =M_{0,b} $.
We call $[b,M_{\alpha}] $ the nonlinear commutator because it is not even a sublinear operator, although the commutator $[b,T]$ is a linear one. It is worth noting that the nonlinear commutator $[b,M_{\alpha}] $ and the maximal commutator $M_{\alpha,b}$ essentially differ from each other. For example, $M_{\alpha,b}$ is positive and sublinear, but $[b,M_{\alpha}] $ is neither positive nor sublinear.
In 1990, by using real interpolation techniques, Milman and Schonbek\cite{milman1990second} established a commutator result that applies to the Hardy-Littlewood maximal function as well as to a large class of nonlinear operators.
In 2000, Bastero et al.\cite{bastero2000commutators} proved the necessary and sufficient condition for the boundedness of the nonlinear commutators $[b,M]$ and $[b,M^{\sharp}]$ on $L^{p}$ spaces. In 2009, Zhang and Wu\cite{zhang2009commutators} studied the same problem for $[b,M_{\alpha}]$. In 2017, Zhang\cite{zhang2017characterization} considered some new characterizations of the Lipschitz spaces via the boundedness of maximal commutator $M_{b}$ and the (nonlinear) commutator $[b, M]$ in Lebesgue spaces and Morrey spaces on Euclidean spaces. In 2018, Zhang et al.\cite{zhang2018commutators} gave necessary and sufficient conditions for the boundedness of the nonlinear commutators $[b,M_{\alpha}]$ and $[b,M^{\sharp}]$ on Orlicz spaces when the symbol $b$ belongs to Lipschitz spaces, and obtained some new characterizations of non-negative Lipschitz functions. Recently, Guliyev\cite{guliyev2022some} extended the mentioned results to Orlicz spaces $L^{\Phi} (\mathbb{G})$ over some stratified Lie group when the symbols belong to $\bmo(\mathbb{G})$. And Liu et al.\cite{liu2022characterisation} established the characterization of BMO spaces by the boundedness of some commutator in variable Lebesgue spaces. Meanwhile, Wu and Zhao \cite{wu2022characterization} extended some results of \cite{zhang2017characterization} to stratified Lie group when the symbols belong to the Lipschitz spaces.
Motivated by the papers mentioned above, the purpose of this paper is to study the boundedness of the fractional maximal commutator $M_{\alpha,b}$ and the nonlinear commutator $[b, M_{\alpha}]$ on the Lebesgue spaces in the context of some stratified Lie group $\mathbb{G}$ when $b\in \Lambda_{\beta}(\mathbb{G})$, by which some new characterizations of the Lipschitz spaces are given.
To state the results, we also give the following notations.
Let $\alpha\ge 0$, for a fixed $\mathbb{G}$-ball $B^{*}$, the fractional maximal function with respect to $B^{*}$ of a locally integrable function $f$ is given by
\begin{align*}
M_{\alpha,B^{*}} (f)(x) &= \sup_{ B\ni x \atop B\subset B^{*}} \dfrac{1}{|B|^{1-\alpha/Q}} \displaystyle\int_{B} |f(y)| \mathrm{d} y,
\end{align*}
where the supremum is taken over all $\mathbb{G}$-balls $B$ such that $x\in B\subset B^{*}$. When $\alpha= 0$, we simply write $M_{B^{*}}$ instead of $M_{0,B^{*}}$.
Our main results can be stated as follows.
\begin{theorem} \label{thm:nonlinear-frac-max-lip}
Let $0 <\beta <1$, $0 <\alpha <Q$, $0 <\alpha+\beta <Q$ and let $b$ be a locally integrable function on $\mathbb{G}$. Then the following assertions are equivalent:
\begin{enumerate} [label=(A.\arabic*) ]
\item $b\in \Lambda_{\beta}(\mathbb{G})$ and $b\ge 0$.
\label{enumerate:nonlinear-frac-max-lip-1}
\item $[b,M_{\alpha} ]$ is bounded from $L^{p}(\mathbb{G})$ to $L^{q}(\mathbb{G})$ for all $p$ and $q$ satisfy $1 <p < \frac{Q}{\alpha+\beta}$ and $\frac{1}{q} =\frac{1}{p} - \frac{\alpha+\beta}{Q}$.
\label{enumerate:nonlinear-frac-max-lip-2}
\item $[b,M_{\alpha} ]$ is bounded from $L^{p}(\mathbb{G})$ to $L^{q}(\mathbb{G})$ for some $p$ and $q$ such that $1 <p < \frac{Q}{\alpha+\beta}$ and $\frac{1}{q} =\frac{1}{p} - \frac{\alpha+\beta}{Q}$.
\label{enumerate:nonlinear-frac-max-lip-3}
\item There exists $s\in [1,\infty)$ such that
\begin{align} \label{inequ:nonlinear-frac-max-lip-4}
\sup_{B} \dfrac{1}{|B|^{\beta/Q}} \left( \dfrac{1}{|B|} \displaystyle\int_{B} |b(x) -|B|^{-\alpha/Q}M_{\alpha,B}(b)(x) |^{s} \mathrm{d} x \right)^{1/s} < \infty.
\end{align}
\label{enumerate:nonlinear-frac-max-lip-4}
\item For all $s \in [1,\infty)$ such that \labelcref{inequ:nonlinear-frac-max-lip-4} holds.
\label{enumerate:nonlinear-frac-max-lip-5}
\end{enumerate}
\end{theorem}
\begin{remark} \label{rem.nonlinear-frac-max-lip}
\begin{enumerate}[ label=(\roman*)]
\item For the case $\alpha=0$, the equivalence of \labelcref{enumerate:nonlinear-frac-max-lip-1}, \labelcref{enumerate:nonlinear-frac-max-lip-2} and \labelcref{enumerate:nonlinear-frac-max-lip-4} was proved in \cite{wu2022characterization} (see Theorem 1.3).
\item Moreover, it was proved in Theorem 1.3 of \cite{wu2022characterization}, see also \cref{lem:non-negative-max-lip} below, that $b\in \Lambda_{\beta}(\mathbb{G})$ and $b\ge 0$ if and only if
\begin{align} \label{inequ:1.3-wu2022characterization}
\sup_{B} |B|^{-\beta/Q} \left( |B|^{-1} \displaystyle\int_{B} |b(x) -M_{B}(b)(x) |^{q} \mathrm{d} x \right)^{1/q} <\infty.
\end{align}
Compared with \labelcref{inequ:1.3-wu2022characterization}, \labelcref{inequ:nonlinear-frac-max-lip-4} gives a new characterization for nonnegative Lipschitz functions.
\end{enumerate}
\end{remark}
Next, we consider some necessary and sufficient conditions for the boundedness of $M_{\alpha,b}$ when $b$ belongs to a Lipschitz space.
\begin{theorem} \label{thm:frac-max-lip} Let $0 <\beta <1$, $0 <\alpha <\alpha+\beta <Q$ and $b$ be a locally integrable function on $\mathbb{G}$. Then the following statements are equivalent:
\begin{enumerate}[label=(B.\arabic*)]
\item $b\in \Lambda_{\beta}(\mathbb{G})$.
\label{enumerate:thm-frac-max-lip-1}
\item $ M_{\alpha,b} $ is bounded from $L^{p}(\mathbb{G})$ to $L^{q}(\mathbb{G})$ for all $p, q$ with $1<p<\frac{Q}{\alpha+\beta} $ and $\frac{1}{q} = \frac{1}{p} -\frac{\alpha+\beta}{Q}$.
\label{enumerate:thm-frac-max-lip-2}
\item $ M_{\alpha,b} $ is bounded from $L^{p}(\mathbb{G})$ to $L^{q}(\mathbb{G})$ for some $p, q$ with $1<p<\frac{Q}{\alpha+\beta} $ and $\frac{1}{q} = \frac{1}{p} -\frac{\alpha+\beta}{Q}$.
\label{enumerate:thm-frac-max-lip-3}
\item There exists $s\in [1,\infty)$ such that
\begin{align} \label{inequ:frac-max-lip-4}
\sup_{B} \dfrac{1}{|B|^{\beta/Q}} \Big( \dfrac{1}{|B|} \displaystyle\int_{B} |b(x)-b_{B}|^{s} \mathrm{d} x \Big)^{1/s} &< \infty.
\end{align}
\label{enumerate:thm-frac-max-lip-4}
\item \labelcref{inequ:frac-max-lip-4} holds for all $s\in [1,\infty)$.
\label{enumerate:thm-frac-max-lip-5}
\end{enumerate}
\end{theorem}
\begin{remark}
\begin{enumerate}[label=(\roman*)]
\item The equivalence of \labelcref{enumerate:thm-frac-max-lip-1}, \labelcref{enumerate:thm-frac-max-lip-2} and \labelcref{enumerate:thm-frac-max-lip-3} was proved in \cite{wu2022characterization} (see Theorem 1.1 for $\alpha=0$). The equivalence of \labelcref{enumerate:thm-frac-max-lip-1}, \labelcref{enumerate:thm-frac-max-lip-4} and \labelcref{enumerate:thm-frac-max-lip-5} is contained in \cref{lem:2.2-li2003lipschitz} below.
\item When $\mathbb{G} =\mathbb{R}^{n}$, the above equivalence was proved in \cite{zhang2019some} (see Corollary 1.3).
\item For the case $\alpha= 0$ and $\mathbb{G} =\mathbb{R}^{n}$, similar results were given in \cite{zhang2017characterization} for Lebesgue spaces with constant exponents, and in \cite{zhang2019some,zhang2019characterization} for the variable case.
\end{enumerate}
\end{remark}
This paper is organized as follows. In \cref{sec:preliminary}, we will recall some basic definitions and known results. In \cref{sec:proof-mab}, we will prove main results.
Throughout this paper, the letter $C$ always stands for a constant independent of the main parameters involved and whose value may differ from line to line. In addition, we give some notations. Here and hereafter $L^{p} ~(1\le p\le \infty)$ will always denote the standard $L^{p} $-space with respect to the Haar measure $\mathrm{d} x$, with the $L^{p} $-norm $\|\cdot\|_{p}$. And let $WL^{p}$ be weak-type $L^{p} $-space. Denote by $\chi_{E}$ the characteristic function of a measurable set $E$ of $\mathbb{G}$.
\section{Preliminaries and lemmas} \label{sec:preliminary}
To prove the main results of this paper, we first recall some necessary notions and remarks. Firstly, we recall some preliminaries concerning stratified Lie groups (or so-called Carnot groups). We refer the reader to \cite{folland1982hardy,bonfiglioli2007stratified,stein1993harmonic}.
\subsection{Lie group $\mathbb{G}$}
\begin{definition}
\label{def:stratified-Lie-algebra-krantz1982lipschitz}
Let $m\in \mathbb{Z}^{+}$, $\mathcal{G}$ be a finite-dimensional Lie algebra, $[X, Y] = XY - YX \in \mathcal{G}$ be Lie bracket with $X,Y \in \mathcal{G}$.
\begin{enumerate}[label=(\roman*)]
\item If $Z \in \mathcal{G}$ is an $m^{\text{th}}$ order Lie bracket and $W \in \mathcal{G}$, then $[Z,W]$ is an $(m + 1)^{\text{st}}$ order Lie bracket.
\item We say $\mathcal{G}$ is a nilpotent Lie algebra of step $m$ if $m$ is the smallest integer for which all Lie brackets of order $m+1$ are zero.
\item We say that a Lie algebra $\mathcal{G}$ is stratified if there is a direct sum vector space decomposition
\begin{align}\label{equ:lie-algebra-decomposition}
\mathcal{G} =\oplus_{j=1}^{m} V_{j} = V_{1} \oplus \cdots \oplus V_{m}
\end{align}
such that $\mathcal{G}$ is nilpotent of step $m$, that is,
\begin{align*}
[V_{1},V_{j}] =
\begin{cases}
V_{j+1} & 1\le j \le m-1 \\
0 & j\ge m
\end{cases}
\end{align*}
holds.
\end{enumerate}
\end{definition}
It is not difficult to find that the above $V_{1}$ generates the whole of the Lie algebra $\mathcal{G}$ by taking Lie brackets since each element of $ V_{j}~(2\le j \le m)$ is a linear combination of $(j-1)^{\text{th}}$ order Lie bracket of elements of $ V_{1}$.
With the help of the related notions of Lie algebra (see \cref{def:stratified-Lie-algebra-krantz1982lipschitz}), the following definition can be obtained.
\begin{definition}\label{def:stratified-Lie-group}
Let $\mathbb{G}$ be a finite-dimensional, connected and simply-connected Lie group associated with Lie algebra $\mathcal{G}$. Then
\begin{enumerate}[label=(\roman*)]
\item $\mathbb{G}$ is called nilpotent if its Lie algebra $\mathcal{G}$ is nilpotent.
\item $\mathbb{G}$ is said to be stratified if its Lie algebra $\mathcal{G}$ is stratified.
\item $\mathbb{G}$ is called homogeneous if it is a nilpotent Lie group whose Lie algebra $\mathcal{G}$ admits a family of dilations $\{\delta_{r}\}$, namely, for $r>0$, $X_{k}\in V_{k}~(k=1,\ldots,m)$,
\begin{align*}
\delta_{r} \Big( \sum_{k=1}^{m} X_{k} \Big) = \sum_{k=1}^{m} r^{k} X_{k},
\end{align*}
which are Lie algebra automorphisms.
\end{enumerate}
\end{definition}
\begin{remark} \label{rem:lie-algebra-decom-zhu2003herz}
Let $\mathcal{G} = \mathcal{G}_{1}\supset \mathcal{G}_{2} \supset \cdots \supset \mathcal{G}_{m+1} =\{0\}$ denote the lower central series of $\mathcal{G}$, and $X=\{X_{1},\dots,X_{n}\}$ be a basis for $V_{1}$ of $\mathcal{G}$.
\begin{enumerate}[label=(\roman*) ]
\item (see \cite{zhu2003herz}) The direct sum decomposition \labelcref{equ:lie-algebra-decomposition} can be constructed by identifying each $\mathcal{G}_{j}$ as a vector subspace of $\mathcal{G}$ and setting $ V_{m}=\mathcal{G}_{m}$ and $ V_{j}=\mathcal{G}_{j}\setminus \mathcal{G}_{j+1}$ for $j=1,\ldots,m-1$.
\item (see \cite{folland1979lipschitz}) The number $Q={\rm{trace}}~ A =\sum\limits_{j=1}^{m} j\dim(V_{j})$ is called the homogeneous dimension of $\mathcal{G}$, where $A$ is a diagonalizable linear transformation of $\mathcal{G}$ with positive eigenvalues.
\item (see \cite{zhu2003herz} or \cite{folland1979lipschitz}) The number $Q$ is also called the homogeneous dimension of $\mathbb{G}$ since $\mathrm{d}(\delta_{r}x)=r^{Q}\mathrm{d} x$ for all $r>0$, and
\begin{align*}
Q = \sum_{j=1}^{m} j \dim(V_{j}) = \sum_{j=1}^{m} \dim(\mathcal{G}_{j}).
\end{align*}
\end{enumerate}
\end{remark}
By the Baker-Campbell-Hausdorff formula for sufficiently small elements $X$ and $Y$ of $\mathcal{G}$ one has
\begin{align*}
\exp X \exp Y= \exp H(X,Y)= X+Y +\frac{1}{2}[X,Y]+\cdots
\end{align*}
where $\exp : \mathcal{G} \to \mathbb{G}$ is the exponential map, $H(X, Y )$ is an infinite linear combination of $X$ and $Y$ and their Lie brackets, and the dots denote terms of order higher than two. And the above equation is finite in the case of $\mathcal{G}$ is a nilpotent Lie algebra.
The following properties can be found in \cite{ruzhansky2019hardy}(see Proposition 1.1.1, or Proposition 2.1 in \cite{yessirkegenov2019function} or Proposition 1.2 in \cite{folland1982hardy}).
\begin{proposition}\label{pro:2.1-yessirkegenov2019}
Let $\mathcal{G}$ be a nilpotent Lie algebra, and let $\mathbb{G}$ be the corresponding connected and simply-connected nilpotent Lie group. Then we have
\begin{enumerate}[label=(\roman*) ]
\item The exponential map $\exp: \mathcal{G} \to \mathbb{G}$ is a diffeomorphism. Furthermore, the group law $(x,y) \mapsto xy$ is a polynomial map if $\mathbb{G}$ is identified with $\mathcal{G}$ via $\exp$.
\item If $\lambda$ is a Lebesgue measure on $\mathcal{G}$, then $\exp\lambda$ is a bi-invariant Haar measure on $\mathbb{G}$ (or a bi-invariant Haar measure $\mathrm{d} x$ on $\mathbb{G}$ is just the lift of Lebesgue measure on $\mathcal{G}$ via $\exp$).
\end{enumerate}
\end{proposition}
Thereafter, we use $Q$ to denote the homogeneous dimension of $\mathbb{G}$, $y^{-1}$ represents the inverse of $y\in \mathbb{G}$, $y^{-1}x$ stands for the group multiplication of $y^{-1}$ by $x$ and the group identity element of $\mathbb{G}$ will be referred to as the origin denotes by $e$.
A homogenous norm on $\mathbb{G}$ is a continuous function $x\to \rho(x)$ from $\mathbb{G}$ to $[0,\infty)$, which is $C^{\infty}$ on $\mathbb{G}\setminus\{0\}$ and satisfies
\begin{align*}
\begin{cases}
\rho(x^{-1}) = \rho(x), \\
\rho(\delta_{t}x) = t\rho(x) \ \ \text{for all}~ x \in \mathbb{G} ~\text{and}~ t > 0, \\
\rho(e) = 0.
\end{cases}
\end{align*}
Moreover, there exists a constant $c_{0} \ge 1$ such that $\rho(xy) \le c_{0}(\rho(x) + \rho(y))$ for all $x,y \in \mathbb{G}$.
With the norm above, we define the $\mathbb{G}$ ball centered at $x$ with radius $r$ by $B(x, r) = \{y \in \mathbb{G} : \rho(y^{-1}x) < r\}$, and by $\lambda B$ denote the ball $B(x,\lambda r)$ with $\lambda>0$, let $B_{r} = B(e, r) = \{y \in \mathbb{G} : \rho(y) < r\}$ be the open ball centered at $e$ with radius $r$, which is the image under $\delta_{r}$ of $B(e, 1)$.
And by $\sideset{^{\complement}}{} {\mathop {B(x,r)}} = \mathbb{G}\setminus B(x,r)= \{y \in \mathbb{G} : \rho(y^{-1}x) \ge r\}$ denote the complement of $B(x, r)$. Let $|B(x,r)|$ be the Haar measure of the ball $B(x,r)\subset \mathbb{G}$, and
there exists $c_{1} =c_{1} (\mathbb{G})$ such that
\begin{align*}
|B(x,r)| = c_{1} r^{Q}, \ \ \ \ x\in \mathbb{G}, r>0.
\end{align*}
In addition, the Haar measure of a homogeneous Lie group $\mathbb{G}$ satisfies the doubling condition (see pages 140 and 501,\cite{fischer2016quantization}), i.e. $\forall ~ x\in \mathbb{G}$, $r>0$, $\exists~ C$, such that
\begin{align*}
|B(x,2r)| \le C |B(x,r)|.
\end{align*}
The most basic partial differential operator in a stratified Lie group is the sub-Laplacian associated with $X=\{X_{1},\dots,X_{n}\}$, i.e., the second-order partial differential operator on $\mathbb{G}$ given by
\begin{align*}
\mathfrak{L} = \sum_{i=1}^{n} X_{i}^{2}
\end{align*}
The following lemma is known as the H\"{o}lder's inequality on Lebesgue spaces over Lie groups $\mathbb{G}$, it can also be found in \cite{rao1991theory} or \cite{guliyev2022some}, when Young function $\Phi(t)=t^{p}$ and its complementary function $\Psi(t)=t^{q}$ with $\frac{1}{p}+\frac{1}{q}=1$.
\begin{lemma}[H\"{o}lder's inequality on $\mathbb{G}$]\label{lem:holder-inequality-Lie-group}
Let $1\le p,q \le\infty$ with $\frac{1}{p}+\frac{1}{q}=1$, $\Omega\subset \mathbb{G}$ be a measurable set and measurable functions $f\in L^{p}(\Omega)$ and $g\in L^{q}(\Omega)$. Then there exists a positive constant $C$ such that
\begin{align*}
\displaystyle\int_{\Omega} |f(x)g(x)| \mathrm{d}x \le C \|f\|_{L^{p}(\Omega)} \|g\|_{L^{q}(\Omega)}.
\end{align*}
\end{lemma}
By elementary calculations we have the following property. It can also be found in \cite{guliyev2022some}, when Young function $\Phi(t)=t^{p}$.
\begin{lemma}[Norms of characteristic functions]\label{lem:norm-characteristic-functions-Lie-group}
Let $0<p<\infty$ and $\Omega\subset \mathbb{G}$ be a measurable set with finite Haar measure. Then
\begin{align*}
\|\scalebox{1.2}{$\chi$}_{\Omega}\|_{L^{p}(\mathbb{G})} = \|\scalebox{1.2}{$\chi$}_{\Omega}\|_{WL^{p}(\mathbb{G})} = |\Omega|^{1/p}.
\end{align*}
\end{lemma}
\subsection{Lipschitz spaces on $\mathbb{G}$}
Next we give the definition of the Lipschitz spaces on $\mathbb{G}$, and state some basic properties and useful lemmas.
\begin{definition}[Lipschitz-type spaces on $\mathbb{G}$] \label{def.lip-space} \
\begin{enumerate}[ label=(\roman*)]
\item Let $0<\beta <1$, we say a function $b$ belongs to the Lipschitz space $\Lambda_{\beta}(\mathbb{G}) $ if there exists a constant $C>0$ such that for all $x,y\in \mathbb{G}$,
\begin{align*}
|b(x)-b(y)| &\le C(\rho(y^{-1}x))^{\beta},
\end{align*}
where $\rho$ is the homogenous norm. The smallest such constant $C$ is called the $\Lambda_{\beta}$ norm of $b$ and is denoted by $\|b\|_{\Lambda_{\beta}(\mathbb{G})}$.
\label{enumerate:def-lip-1}
\item (see \cite{macias1979lipschitz} ) Let $0<\beta <1$ and $1\le p<\infty$. The space $\lip_{\beta,p}(\mathbb{G}) $ is defined to be the set of all locally integrable functions $b$, i.e., there exists a positive constant $C $, such that
\begin{align*}
\sup_{B\ni x} \dfrac{1}{ |B|^{\beta/Q}}\Big( \dfrac{1}{|B|} \displaystyle\int_{B} |b(x)- b_{B}|^{p}\mathrm{d} x \Big)^{1/p} \le C
\end{align*}
where the supremum is taken over every ball $B\subset \mathbb{G}$ containing $x$ and $b_{B}=\frac{1}{|B|} \int_{B} b(x) \mathrm{d} x$. The least constant $C$ satisfying the conditions above shall be denoted by $\|b\|_{\lip_{\beta,p}(\mathbb{G})}$.
\label{enumerate:def-lip-2}
\end{enumerate}
\end{definition}
\begin{remark} \label{rem.Lipschitz-def}
\begin{enumerate}[label=(\roman*)]
\item Similar to the definition of Lipschitz space $\Lambda_{\beta}(\mathbb{G}) $ in \labelcref{enumerate:def-lip-1}, we also have the definition form as following (see \cite{krantz1982lipschitz,chen2010lipschitz,fan1995characterization} et al.)
\begin{align*}
\|b\|_{\Lambda_{\beta}(\mathbb{G})}&= \sup_{x,y\in \mathbb{G}\atop y\neq e} \dfrac{|b(xy)- b(x)|}{(\rho(y))^{\beta}} = \sup_{x,y\in \mathbb{G} \atop x\neq y} \dfrac{|b(x)-b(y)|}{(\rho(y^{-1}x))^{\beta}}.
\end{align*}
And $\|b\|_{\Lambda_{\beta}(\mathbb{G})} =0$ if and only if $b$ is constant.
\item In \labelcref{enumerate:def-lip-2}, when $p=1$, we have
\begin{align*}
\|b\|_{\lip_{\beta,1}(\mathbb{G})} =\sup_{B\ni x} \dfrac{1}{ |B|^{\beta/Q}}\Big( \dfrac{1}{|B|} \displaystyle\int_{B} |b(x)- b_{B}| \mathrm{d} x \Big) :=\|b\|_{\lip_{\beta}(\mathbb{G})}
\end{align*}
\item There are two basically different approaches to Lipschitz classes on the $n$-dimensional Euclidean space. Lipschitz classes can be defined via Poisson (or Weierstrass) integrals of $L^{p}$-functions, or, equivalently, by means of higher order difference operators (see \cite{meda1988lipschitz}).
\end{enumerate}
\end{remark}
\begin{lemma} (see \cite{macias1979lipschitz,chen2010lipschitz,li2003lipschitz} ) \label{lem:2.2-li2003lipschitz} Let $0<\beta<1$ and the function $b(x)$ integrable on bounded subsets of $\mathbb{G}$.
\begin{enumerate}[label=(\roman*)]
\item When $1\le p<\infty$, then
\begin{align*}
\|b\|_{\Lambda_{\beta}(\mathbb{G})} &= \|b\|_{\lip_{\beta}(\mathbb{G})} \approx \|b\|_{\lip_{\beta,p}(\mathbb{G})}.
\end{align*}
\item Let balls $B_{1}\subset B_{2}\subset \mathbb{G}$ and $b\in \lip_{\beta,p}(\mathbb{G})$ with $p\in [1,\infty]$. Then there exists a constant $C$ depends on $B_{1}$ and $B_{2}$ only, such that
\begin{align*}
|b_{B_{1}}- b_{B_{2}} | &\le C \|b\|_{\lip_{\beta,p}(\mathbb{G})} |B_{2}|^{\beta/Q}
\end{align*}
\item When $1\le p<\infty$, then there exists a constant $C$ depends on $\beta$ and $p$ only, such that
\begin{align*}
| b(x)- b(y) | &\le C \|b\|_{\lip_{\beta,p}(\mathbb{G})} |B|^{\beta/Q}
\end{align*}
holds for any ball $B$ containing $x$ and $y$.
\end{enumerate}
\end{lemma}
\subsection{Some pointwise estimates and auxiliary lemmas}
Hereafter, for a function $b$ defined on $\mathbb{G}$, we denote
\begin{align*}
b^{-}(x) :=- \min\{b, 0\} =
\begin{cases}
0, & \text{if}\ b(x) \ge 0 \\
|b(x)|, & \text{if}\ b(x) < 0
\end{cases}
\end{align*}
and $b^{+}(x) =|b(x)|-b^{-}(x)$. Obviously, $b(x)=b^{+}(x)-b^{-}(x)$.
From the proof of Theorem 1.3 in \cite{wu2022characterization}, we can obtain the following characterization of nonnegative Lipschitz functions.
\begin{lemma} \label{lem:non-negative-max-lip}
Let $0 <\beta <1$ and $b$ be a locally integrable function on $\mathbb{G}$. Then the following assertions are equivalent:
\begin{enumerate}[label=(\roman*)]
\item $b\in \Lambda_{\beta}(\mathbb{G})$ and $b\ge 0$.
\label{enumerate:Lem-non-negative-max-lip-1}
\item For all $1\le s<\infty$, there exists a positive constant $C$ such that
\begin{align} \label{inequ:non-negative-max-lip}
\sup_{B} |B|^{-\beta/Q} \left( |B|^{-1} \displaystyle\int_{B} |b(x) -M_{B}(b)(x) |^{s} \mathrm{d} x \right)^{1/s} \le C.
\end{align}
\label{enumerate:Lem-non-negative-max-lip-2}
\item \labelcref{inequ:non-negative-max-lip} holds for some $1\le s<\infty$.
\label{enumerate:Lem-non-negative-max-lip-3}
\end{enumerate}
\end{lemma}
\begin{proof}
Since the implication \labelcref{enumerate:Lem-non-negative-max-lip-2} $\xLongrightarrow{\ \ }$ \labelcref{enumerate:Lem-non-negative-max-lip-3} follows readily, and the implication \labelcref{enumerate:Lem-non-negative-max-lip-3} $\xLongrightarrow{\ \ }$ \labelcref{enumerate:Lem-non-negative-max-lip-1} was proved in \cite[Theorem 1.3]{wu2022characterization}, we only need to prove \labelcref{enumerate:Lem-non-negative-max-lip-1} $\xLongrightarrow{\ \ }$ \labelcref{enumerate:Lem-non-negative-max-lip-2}.
If $b\in \Lambda_{\beta}(\mathbb{G})$ and $b\ge 0$, then it follows from \cite[Theorem 1.3]{wu2022characterization} that \labelcref{inequ:non-negative-max-lip} holds for all $s$ with $Q/(Q-\beta)<s<\infty$. Applying H\"{o}lder's inequality, we see that \labelcref{inequ:non-negative-max-lip} holds for $1\le s\le Q/(Q-\beta)$ as well.
So, the implication \labelcref{enumerate:Lem-non-negative-max-lip-1} $\xLongrightarrow{\ \ }$ \labelcref{enumerate:Lem-non-negative-max-lip-2} is proven.
\end{proof}
The following strong-type estimate for the fractional maximal function $M_{\alpha}$ is well known, which can be obtained from \cite[Proposition A]{kokilashvili1989fractional} or \cite[Theorem 1.6]{bernardis1994two} when the weights are constant 1, see \cite{macias1981well}, \cite{kokilashvili1989fractional} or \cite{bernardis1994two} for more details.
\begin{lemma}\label{lem:frac-maximal-kokilashvili1989fractional} Let $0<\alpha<Q$, $1< p< Q/\alpha$ and $1/q=1/p-\alpha/Q$. If $f\in L^{p}(\mathbb{G})$.
then there exists a positive constant $C$ such that
\begin{align*}
\|M_{\alpha}(f)\|_{L^{q}(\mathbb{G})} &\le C \|f\|_{L^{p}(\mathbb{G})}.
\end{align*}
\end{lemma}
\begin{remark} \label{rem.a.e.-frac-maximal}
\begin{enumerate}[label=(\roman*)]
\item By \cref{lem:frac-maximal-kokilashvili1989fractional}, if $0 <\alpha<Q$, $1< p< Q/\alpha$ and $f\in L^{p}(\mathbb{G})$, then $M_{\alpha}(f)(x)<\infty $ for almost everywhere $x\in \mathbb{G}$.
\item The above lemma can also refer to Theorem 3.3 in \cite{guliyev2022some} when Young function $\Phi(t) = t^{p}$ and its complementary function $\Psi(t)=t^{q}$ with $1/q=1/p-\alpha/Q$.
\end{enumerate}
\end{remark}
Now, we give the following pointwise estimate for $[b,M_{\alpha}] $ on $\mathbb{G}$ when $b\in \Lambda_{\beta}(\mathbb{G})$.
\begin{lemma} \label{lem:frac-maximal-pointwise} Let $0\le\alpha<Q$, $0<\beta <1$, $0<\alpha+\beta<Q$ and $f: \mathbb{G} \to \mathbb{R}$ be a locally integrable function. If $b\in \Lambda_{\beta}(\mathbb{G})$ and $b\ge 0$, then, for arbitrary $x\in \mathbb{G} $ such that $M_{\alpha} (f)(x) <\infty$, we have
\begin{align*}
\big|[b,M_{\alpha}] (f)(x)\big| &\le \|b\|_{\Lambda_{\beta}(\mathbb{G})} M_{\alpha+\beta} (f)(x).
\end{align*}
\end{lemma}
\begin{proof}
Similar to the discussion of lemma 2.11 in \cite{zhang2019some}. For any fixed $x \in \mathbb{G}$ such that $M_{\alpha}(f)(x) <\infty$, if $b\in \Lambda_{\beta}(\mathbb{G})$ and $b\ge 0$, then we have
\begin{align*}
\big|[b,M_{\alpha}] (f)(x) \big| &= \big|b(x)M_{\alpha}(f)(x)-M_{\alpha}(bf)(x) \big| \\
&= \bigg| \sup_{B\ni x \atop B\subset \mathbb{G}} \dfrac{1}{|B|^{1-\alpha/Q}} \displaystyle\int_{B} b(x)|f(y)| \mathrm{d} y \\
&\;\qquad -\sup_{B\ni x \atop B\subset \mathbb{G}} \dfrac{1}{|B|^{1-\alpha/Q}} \displaystyle\int_{B} b(y)|f(y)| \mathrm{d} y \bigg| \\
&\le \sup_{B\ni x \atop B\subset \mathbb{G}} \dfrac{1}{|B|^{1-\alpha/Q}} \displaystyle\int_{B} |b(x)-b(y)| |f(y)| \mathrm{d} y \\
&\le \|b\|_{\Lambda_{\beta}(\mathbb{G})} \sup_{B\ni x \atop B\subset \mathbb{G}} \dfrac{1}{|B|^{1-(\alpha+\beta)/Q}} \displaystyle\int_{B} |f(y)| \mathrm{d} y \\
&\le \|b\|_{\Lambda_{\beta}(\mathbb{G})} M_{\alpha+\beta} (f)(x).
\end{align*}
\end{proof}
Similar to Lemma 2.3 in \cite{zhang2009commutators}, we get the following result.
\begin{lemma} \label{lem:frac-maximal-pointwise-ralation}
Let $0 \le \alpha<Q$, $B\subset \mathbb{G}$ be a ball, and $f$ be a locally integrable function. Then, for all $x\in B$, we have
\begin{align} \label{equ:frac-maximal-pointwise-ralation}
M_{\alpha} (f\scalebox{1.2}{$\chi$}_{B})(x) &= M_{\alpha,B}(f)(x).
\end{align}
\end{lemma}
\begin{proof}
Some ideas are taken from \cite{bastero2000commutators} and \cite{zhang2009commutators}. Reasoning as the discussion of lemma 2.3 in \cite{zhang2009commutators}. For any $x \in B$, it is easy to verify that
\begin{align} \label{inequ:frac-maximal-pointwise-ralation-r}
M_{\alpha} (f\scalebox{1.2}{$\chi$}_{B})(x) \ge M_{\alpha,B}(f)(x)
\end{align}
from the definitions of $M_{\alpha} (f\scalebox{1.2}{$\chi$}_{B})(x)$ and $M_{\alpha,B}(f)(x)$.
So, in order to prove the equality \labelcref{equ:frac-maximal-pointwise-ralation} is true, we only need to prove the following realtion, namely, for any $\mathbb{G}$-ball $B^{*}\ni x$ with radius $r^{*}$, there exist $\mathbb{G}$-ball $B'\ni x$ with radius $r'$ and $B'\subset B$, such that
\begin{align} \label{inequ:frac-maximal-pointwise-ralation-l}
\dfrac{1}{|B^{*}|^{1-\alpha/Q}} \displaystyle\int_{B^{*}} |f(y)\scalebox{1.2}{$\chi$}_{B}(y)| \mathrm{d} y \le \dfrac{1}{|B'|^{1-\alpha/Q}} \displaystyle\int_{B'} |f(y)| \mathrm{d} y.
\end{align}
Indeed, for the case $B^{*} \cap B =\emptyset$, it is clear that \labelcref{inequ:frac-maximal-pointwise-ralation-l} is true since $f(y)\scalebox{1.2}{$\chi$}_{B}(y)=0$ for any $y\in B^{*}$.
For the case $B^{*} \cap B \neq\emptyset$. If $B^{*} \cap B $ is a $\mathbb{G}$-ball, then we take $B'=B^{*} \cap B $. If $B^{*} \cap B $ is not $\mathbb{G}$-ball, then, there exists a $\mathbb{G}$-ball $B'$ such that $B\supset B' \supset B^{*} \cap B \ni x$ and $r' \le r^{*}$, thereby, we obtain that $|B'|\le |B^{*}|$. Consequently, we infer that
\begin{align*}
\dfrac{1}{|B^{*}|^{1-\alpha/Q}} \displaystyle\int_{B^{*}} |f(y)\scalebox{1.2}{$\chi$}_{B}(y)| \mathrm{d} y
&= \Big( \dfrac{|B'|}{|B^{*}|}\Big)^{1-\alpha/Q} \dfrac{1}{|B'|^{1-\alpha/Q}} \displaystyle\int_{B^{*}} |f(y)\scalebox{1.2}{$\chi$}_{B}(y)| \mathrm{d} y \\
&\le \dfrac{1}{|B'|^{1-\alpha/Q}} \displaystyle\int_{B'} |f(y)\scalebox{1.2}{$\chi$}_{B}(y)| \mathrm{d} y .
\end{align*}
Summarizing the discussion above we find that \labelcref{inequ:frac-maximal-pointwise-ralation-l} is valid.
\labelcref{inequ:frac-maximal-pointwise-ralation-r} and \labelcref{inequ:frac-maximal-pointwise-ralation-l} together give \labelcref{equ:frac-maximal-pointwise-ralation}, this completes the proof.
\end{proof}
\begin{remark} \label{rem.frac-maximal-pointwise-ralation}
\begin{enumerate}[label=(\roman*)]
\item Further, by applying \cref{lem:frac-maximal-pointwise-ralation} and the definition of $M_{\alpha,B}(\scalebox{1.2}{$\chi$}_{B})(x)$, we have that
\begin{align*}
M_{\alpha} (\scalebox{1.2}{$\chi$}_{B})(x) &= M_{\alpha,B}(\scalebox{1.2}{$\chi$}_{B})(x)=|B|^{\alpha/Q}.
\end{align*}
\item For the case $\alpha=0$, the following results are also valid, namely
\begin{align*}
M (\scalebox{1.2}{$\chi$}_{B})(x) &= M_{B}(\scalebox{1.2}{$\chi$}_{B})(x)=\scalebox{1.2}{$\chi$}_{B}(x), \ \ M (f\scalebox{1.2}{$\chi$}_{B})(x) = M_{B}(f)(x).
\end{align*}
\end{enumerate}
\end{remark}
Referring to \cite[page 3331]{bastero2000commutators} or \cite{zhang2009commutators}, through elementary calculations and derivations, it is easy to check that the following assertions are true.
\begin{lemma} \label{lem:frac-max-pointwise-assert}
Let $b$ be a locally integrable function on $\mathbb{G}$ and $B \subset \mathbb{G}$ be an arbitrary given ball.
\begin{enumerate}[label=(\roman*)]
\item If $E=\{x\in B: b(x)\le b_{B}\}$ and $F= B\setminus E =\{x\in B: b(x)> b_{B}\}$. Then the following equality
\begin{align*}
\displaystyle\int_{E} |b(x)-b_{B}| \mathrm{d} x &= \displaystyle\int_{F} |b(x)-b_{B}| \mathrm{d} x
\end{align*}
is trivially true.
\item Then for any $x\in B$, we have
\begin{align*}
|b_{B}| &\le |B|^{-\alpha/Q} M_{\alpha,B}(b)(x).
\end{align*}
\end{enumerate}
\end{lemma}
\section{Proof of the principal results } \label{sec:proof-mab}
We now give the proof of the principal results.
\subsection{Proof of \cref{thm:nonlinear-frac-max-lip}}
To prove \cref{thm:nonlinear-frac-max-lip}, we first prove the following lemma.
\begin{lemma} \label{lem:frac-Lie-lip-norm}
Let $0 <\beta <1$ and $0 <\alpha <Q$. If $b$ is a locally integrable function on $\mathbb{G}$ and satisfies
\begin{align} \label{inequ:lem-frac-Lie-lip-norm}
\sup_{B} \dfrac{1}{|B|^{\beta/Q}} \left( \dfrac{1}{|B|} \displaystyle\int_{B} |b(x) -|B|^{-\alpha/Q}M_{\alpha,B}(b)(x) |^{s} \mathrm{d} x \right)^{1/s} < \infty
\end{align}
for some $s\in [1,\infty)$, then $b\in \Lambda_{\beta}(\mathbb{G})$.
\end{lemma}
\begin{proof}
Some ideas are taken from \cite{bastero2000commutators,zhang2009commutators,zhang2014commutators} and \cite{zhang2019some}.
For any $\mathbb{G}$-ball $B\subset \mathbb{G}$, let $E=\{x\in B: b(x)\le b_{B}\}$ and $F= B\setminus E =\{x\in B: b(x)> b_{B}\}$. Noticing from \cref{lem:frac-max-pointwise-assert}(ii) that
\begin{align*}
|b_{B}| &\le |B|^{-\alpha/Q} M_{\alpha,B}(b)(x) \qquad \forall ~ x\in B.
\end{align*}
Then, for any $x\in E\subset B$, we have $b(x)\le b_{B}\le |b_{B}| \le |B|^{-\alpha/Q} M_{\alpha,B}(b)(x)$. It is clear that
\begin{align*}
|b(x)- b_{B}| &\le \Big| b(x) - |B|^{-\alpha/Q} M_{\alpha,B}(b)(x) \Big|, \qquad \forall~ x\in E.
\end{align*}
Therefore, by using \cref{lem:frac-max-pointwise-assert}(i), we get
\begin{align*}
\dfrac{1}{|B|^{1+\beta/Q}} \displaystyle\int_{B} \big| b(x)-b_{B}) \big| \mathrm{d} x &= \dfrac{1}{|B|^{1+\beta/Q}} \displaystyle\int_{E\cup F} \big| b(x)-b_{B}) \big| \mathrm{d} x \\
&= \dfrac{2}{|B|^{1+\beta/Q}} \displaystyle\int_{E} \big| b(x)-b_{B}) \big| \mathrm{d} x \\
&\le \dfrac{2}{|B|^{1+\beta/Q}} \displaystyle\int_{E} \Big| b(x) - |B|^{-\alpha/Q} M_{\alpha,B}(b)(x) \Big| \mathrm{d} x \\
&\le \dfrac{2}{|B|^{1+\beta/Q}} \displaystyle\int_{B} \Big| b(x) - |B|^{-\alpha/Q} M_{\alpha,B}(b)(x) \Big| \mathrm{d} x .
\end{align*}
By using \cref{lem:holder-inequality-Lie-group}, \labelcref{inequ:lem-frac-Lie-lip-norm} and \cref{lem:norm-characteristic-functions-Lie-group}, we have
\begin{align*}
\dfrac{1}{|B|^{1+\beta/Q}} & \displaystyle\int_{B} \big| b(x)-b_{B}) \big| \mathrm{d} x \\
&\le \dfrac{2}{|B|^{1+\beta/Q}} \displaystyle\int_{B} \Big| b(x) - |B|^{-\alpha/Q} M_{\alpha,B}(b)(x) \Big| \mathrm{d} x \\
&\le \dfrac{C}{|B|^{1+\beta/Q}} \left( \displaystyle\int_{B} |b(x) -|B|^{-\alpha/Q}M_{\alpha,B}(b)(x) |^{s} \mathrm{d} x \right)^{1/s} \|\scalebox{1.2}{$\chi$}_{B}\|_{L^{s'}(\mathbb{G}) } \\
&\le \dfrac{C}{|B|^{\beta/Q}} \left( \dfrac{1}{|B|} \displaystyle\int_{B} |b(x) -|B|^{-\alpha/Q}M_{\alpha,B}(b)(x) |^{s} \mathrm{d} x \right)^{1/s} \\
&\le C.
\end{align*}
So, the proof is completed by applying \cref{lem:2.2-li2003lipschitz} and \cref{def.lip-space}.
\end{proof}
\begin{figure}
\caption{Proof structure \\ where $w_{ij}$ denotes $i\Longrightarrow j$}
\label{fig:ps-equivalent-non}
\end{figure}
\begin{proof}[Proof of \cref{thm:nonlinear-frac-max-lip}]
Since the implications \labelcref{enumerate:nonlinear-frac-max-lip-2} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:nonlinear-frac-max-lip-3} and \labelcref{enumerate:nonlinear-frac-max-lip-5} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:nonlinear-frac-max-lip-4} follows readily,
we only need to prove \labelcref{enumerate:nonlinear-frac-max-lip-1} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:nonlinear-frac-max-lip-2},
\labelcref{enumerate:nonlinear-frac-max-lip-3} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:nonlinear-frac-max-lip-4},
\labelcref{enumerate:nonlinear-frac-max-lip-4} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:nonlinear-frac-max-lip-1}, and \labelcref{enumerate:nonlinear-frac-max-lip-2} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:nonlinear-frac-max-lip-5}
(see \Cref{fig:ps-equivalent-non} for the proof structure).
\labelcref{enumerate:nonlinear-frac-max-lip-1} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:nonlinear-frac-max-lip-2}:\ Let $b\in \Lambda_{\beta}(\mathbb{G})$ and $b\ge 0$. We need to prove that $[b,M_{\alpha} ]$ is bounded from $L^{p}(\mathbb{G})$ to $L^{q}(\mathbb{G})$ for all $p$ and $q$ satisfy $1 <p < \frac{Q}{\alpha+\beta}$ and $\frac{1}{q} =\frac{1}{p} - \frac{\alpha+\beta}{Q}$. For such $p$ and any $f\in L^{p}(\mathbb{G})$, it follows from \cref{rem.a.e.-frac-maximal}(i) that $M_{\alpha}(f)(x)<\infty $ for almost everywhere $x\in \mathbb{G}$. By \cref{lem:frac-maximal-pointwise}, we have
\begin{align*}
\big|[b,M_{\alpha}] (f)(x)\big| &\le \|b\|_{\Lambda_{\beta}(\mathbb{G})} M_{\alpha+\beta} (f)(x).
\end{align*}
Then, assertion \labelcref{enumerate:nonlinear-frac-max-lip-2} follows from \labelcref{lem:frac-maximal-kokilashvili1989fractional}.
\labelcref{enumerate:nonlinear-frac-max-lip-3} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:nonlinear-frac-max-lip-4}:\
Let $(p,q)$ be such that $[b,M_{\alpha} ]$ is bounded from $L^{p}(\mathbb{G})$ to $L^{q}(\mathbb{G})$. We will verify \labelcref{inequ:nonlinear-frac-max-lip-4} for $s=q$.
For any fixed $\mathbb{G}$-ball $B\subset \mathbb{G}$ and any $x \in B$, it follows from \cref{lem:frac-maximal-pointwise-ralation} and \cref{rem.frac-maximal-pointwise-ralation} that the pointwise estimates
\begin{align*}
M_{\alpha} (b\scalebox{1.2}{$\chi$}_{B})(x) = M_{\alpha,B} (b)(x)
\ \text{and} \
M_{\alpha} (\scalebox{1.2}{$\chi$}_{B})(x) = M_{\alpha,B} (\scalebox{1.2}{$\chi$}_{B})(x)= |B|^{\alpha/Q}.
\end{align*}
Then, for any $x \in B$, we have
\begin{align*}
b(x) -|B|^{-\alpha/Q}M_{\alpha,B}(b)(x) &= |B|^{-\alpha/Q} \Big( b(x)|B|^{\alpha/Q} - M_{\alpha,B}(b)(x) \Big) \\
&= |B|^{-\alpha/Q} \Big( b(x) M_{\alpha} (\scalebox{1.2}{$\chi$}_{B})(x) - M_{\alpha} (b\scalebox{1.2}{$\chi$}_{B})(x) \Big) \\
&= |B|^{-\alpha/Q} [b,M_{\alpha}] (\scalebox{1.2}{$\chi$}_{B})(x).
\end{align*}
Noting that $ [b,M_{\alpha}] $ is bounded from $L^{p}(\mathbb{G})$ to $L^{q}(\mathbb{G})$ with $\frac{1}{q} = \frac{1}{p} -\frac{\alpha+\beta}{Q}$. For any ball $B\subset \mathbb{G}$, applying \cref{lem:norm-characteristic-functions-Lie-group}, we obtain
\begin{align*}
\begin{aligned}
\dfrac{1}{|B|^{\beta/Q}} &\left( \dfrac{1}{|B|} \displaystyle\int_{B} |b(x) -|B|^{-\alpha/Q}M_{\alpha,B}(b)(x) |^{q} \mathrm{d} x \right)^{1/q} \\
&\le |B|^{-(\alpha+\beta)/Q-1/q} \big\|[b,M_{\alpha}] (\scalebox{1.2}{$\chi$}_{B}) \big\|_{L^{q}(\mathbb{G})} \\
&\le C |B|^{-(\alpha+\beta)/Q-1/q} \| \scalebox{1.2}{$\chi$}_{B} \|_{L^{p}(\mathbb{G})} \\
& \le C,
\end{aligned}
\end{align*}
which gives \labelcref{inequ:nonlinear-frac-max-lip-4} for $s=q$ since the ball $B\subset \mathbb{G}$ is arbitrary and $C$ is independent of $B$.
\labelcref{enumerate:nonlinear-frac-max-lip-4} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:nonlinear-frac-max-lip-1}:\ By \cref{lem:non-negative-max-lip}, it suffices to prove
\begin{align} \label{inequ:proof-lem-non-negative-max-lip-41}
\sup_{B} \dfrac{1}{|B|^{1+\beta/Q} } \displaystyle\int_{B} \big|b(x) -M_{B}(b)(x) \big| \mathrm{d} x <\infty.
\end{align}
For any fixed ball $B\subset \mathbb{G}$, we have
\begin{align} \label{inequ:proof-lem-non-negative-max-lip-41-2} \begin{aligned}
\dfrac{1}{|B|^{1+\beta/Q} } & \displaystyle\int_{B} \big|b(x) -M_{B}(b)(x) \big| \mathrm{d} x \\
&\le \dfrac{1}{|B|^{1+\beta/Q} } \displaystyle\int_{B} \Big| b(x)-|B|^{-\alpha/Q} M_{\alpha,B}(b)(x) \Big| \mathrm{d} x \\
&\qquad +\dfrac{1}{|B|^{1+\beta/Q} } \displaystyle\int_{B} \Big| |B|^{-\alpha/Q} M_{\alpha,B}(b)(x) -M_{B}(b)(x) \Big| \mathrm{d} x \\
&:=I_{1}+I_{2}.
\end{aligned} \end{align}
For $ I_{1}$, by applying statement \labelcref{enumerate:nonlinear-frac-max-lip-4}, \cref{lem:holder-inequality-Lie-group} (H\"{o}lder's inequality) and \cref{lem:norm-characteristic-functions-Lie-group}, we get
\begin{align*}
I_{1} &= \dfrac{1}{|B|^{1+\beta/Q} } \displaystyle\int_{B} \Big| b(x)-|B|^{-\alpha/Q} M_{\alpha,B}(b)(x) \Big| \mathrm{d} x \\
&\le \dfrac{1}{|B|^{1+\beta/Q} } \bigg( \displaystyle\int_{B} \Big| b(x)-|B|^{-\alpha/Q} M_{\alpha,B}(b)(x) \Big|^{s} \mathrm{d} x \bigg)^{1/s} \|\scalebox{1.2}{$\chi$}_{B}\|_{L^{s'}(\mathbb{G}) } \\
&\le \dfrac{C}{|B|^{\beta/Q} } \bigg( \dfrac{1}{|B| }\displaystyle\int_{B} \Big| b(x)-|B|^{-\alpha/Q} M_{\alpha,B}(b)(x) \Big|^{s} \mathrm{d} x \bigg)^{1/s} \\ &\le C,
\end{align*}
where the constant $C$ is independent of ball $B$.
Now we consider $ I_{2}$. For all $x\in B$, it follows from \cref{lem:frac-maximal-pointwise-ralation} and \cref{rem.frac-maximal-pointwise-ralation} that the pointwise estimates
\begin{align*}
M_{\alpha} (\scalebox{1.2}{$\chi$}_{B})(x) &= |B|^{\alpha/Q} \ \text{and}\ M_{\alpha} (b\scalebox{1.2}{$\chi$}_{B})(x) = M_{\alpha,B} (b)(x),
\\ \intertext{and}
M(\scalebox{1.2}{$\chi$}_{B})(x) &= \scalebox{1.2}{$\chi$}_{B}(x) = 1 \ \text{and}\ M (b\scalebox{1.2}{$\chi$}_{B})(x) = M_{B} (b)(x).
\end{align*}
Then, for any $x\in B$, we get
\begin{align} \label{inequ:proof-lem-non-negative-max-lip-41-3}
\begin{split}
&\Big| |B|^{-\alpha/Q} M_{\alpha,B}(b)(x) -M_{B}(b)(x) \Big| \\
&= \Big| |B|^{-\alpha/Q} M_{\alpha,B}(b)(x) -|b(x)|+ |b(x)| -M_{B}(b)(x) \Big| \\
&\le |B|^{-\alpha/Q} \Big| M_{\alpha,B}(b)(x) - |B|^{\alpha/Q}|b(x)| \Big| + \Big| |b(x)|-M_{B}(b)(x) \Big| \\
&\le |B|^{-\alpha/Q} \big| M_{\alpha} (b\scalebox{1.2}{$\chi$}_{B})(x) - |b(x)| M_{\alpha} (\scalebox{1.2}{$\chi$}_{B})(x) \big| \\
&\qquad + \big| |b(x)| M(\scalebox{1.2}{$\chi$}_{B})(x) -M (b\scalebox{1.2}{$\chi$}_{B})(x) \big| \\
&\le |B|^{-\alpha/Q} \big|[|b|,M_{\alpha}](\scalebox{1.2}{$\chi$}_{B})(x) \big| + \big| [ |b|, M] (\scalebox{1.2}{$\chi$}_{B})(x) \big|.
\end{split}
\end{align}
Since statement \labelcref{enumerate:nonlinear-frac-max-lip-4} along with \cref{lem:frac-Lie-lip-norm} gives $b\in \Lambda_{\beta}(\mathbb{G})$, which implies $|b|\in \Lambda_{\beta}(\mathbb{G})$. Hence, we can apply \cref{lem:frac-maximal-pointwise} to $[|b|,M_{\alpha}]$ and $[ |b|, M]$ since $|b|\in \Lambda_{\beta}(\mathbb{G})$ and $|b|\ge 0$.
By using \cref{lem:frac-maximal-pointwise}, \cref{lem:frac-maximal-pointwise-ralation} and \cref{rem.frac-maximal-pointwise-ralation}, for any $x\in B$, we have
\begin{align*}
\big|[|b|,M_{\alpha}](\scalebox{1.2}{$\chi$}_{B})(x) \big| &\le \|b\|_{\Lambda_{\beta}(\mathbb{G})} M_{\alpha+\beta} (\scalebox{1.2}{$\chi$}_{B})(x) \le C \|b\|_{\Lambda_{\beta}(\mathbb{G})} |B|^{(\alpha+\beta)/Q}
\\ \intertext{and}
\big| [ |b|, M] (\scalebox{1.2}{$\chi$}_{B})(x) \big| &\le \|b\|_{\Lambda_{\beta}(\mathbb{G})} M_{\beta} (\scalebox{1.2}{$\chi$}_{B})(x) \le C \|b\|_{\Lambda_{\beta}(\mathbb{G})} |B|^{\beta/Q}.
\end{align*}
Thus, it follows from \labelcref{inequ:proof-lem-non-negative-max-lip-41-3} that
\begin{align*}
I_{2}&= \dfrac{1}{|B|^{1+\beta/Q} } \displaystyle\int_{B} \Big| |B|^{-\alpha/Q} M_{\alpha,B}(b)(x) -M_{B}(b)(x) \Big| \mathrm{d} x \\
&\le \dfrac{C}{|B|^{1+(\alpha+\beta)/Q} } \displaystyle\int_{B} \big|[|b|,M_{\alpha}](\scalebox{1.2}{$\chi$}_{B})(x) \big| \mathrm{d} x \\
&\qquad +\dfrac{C}{|B|^{1+\beta/Q} } \displaystyle\int_{B} \big| [ |b|, M] (\scalebox{1.2}{$\chi$}_{B})(x) \big| \mathrm{d} x \\
&\le C\|b\|_{\Lambda_{\beta}(\mathbb{G})}.
\end{align*}
Putting the above estimates for $I_{1}$ and $I_{2}$ into \labelcref{inequ:proof-lem-non-negative-max-lip-41-2}, we obtain \labelcref{inequ:proof-lem-non-negative-max-lip-41}.
\labelcref{enumerate:nonlinear-frac-max-lip-2} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:nonlinear-frac-max-lip-5}:\ Assume statement \labelcref{enumerate:nonlinear-frac-max-lip-2} is true. Reasoning as in the proof of \labelcref{enumerate:nonlinear-frac-max-lip-3} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:nonlinear-frac-max-lip-4}, we have
\begin{align} \label{inequ:proof-lem-non-negative-max-lip-25}
\begin{aligned}
\sup_{B} \dfrac{1}{|B|^{\beta/Q}} &\left( \dfrac{1}{|B|} \displaystyle\int_{B} |b(x) -|B|^{-\alpha/Q}M_{\alpha,B}(b)(x) |^{q} \mathrm{d} x \right)^{1/q} <\infty
\end{aligned}
\end{align}
for any $q$ for which there exists a $p$ such that $\frac{1}{q} =\frac{1}{p} - \frac{\alpha+\beta}{Q}$.
For any $s \in [1,\infty)$, choosing an $r>Q/(Q-\beta)>1$, we have $1<rs(Q-\beta)/Q< rs$. Set $q=rs$ and define $p$ by $\frac{1}{p} =\frac{1}{q} + \frac{\alpha+\beta}{Q}$. Noting that
\begin{align*}
\dfrac{1}{s} &= \dfrac{1}{rs} + \dfrac{1}{r's} = \dfrac{1}{q} + \dfrac{1}{r's},
\end{align*}
it follows from \cref{lem:holder-inequality-Lie-group}, \labelcref{inequ:proof-lem-non-negative-max-lip-25} and \cref{lem:norm-characteristic-functions-Lie-group} that
\begin{align*}
\dfrac{1}{|B|^{\beta/Q}} & \left( \dfrac{1}{|B|} \displaystyle\int_{B} \big |b(x) -|B|^{-\alpha/Q}M_{\alpha,B}(b)(x) \big |^{s} \mathrm{d} x \right)^{1/s} \\
&\le \dfrac{C}{|B|^{1/s+\beta/Q}} \left( \displaystyle\int_{B} \big |b(x) -|B|^{-\alpha/Q}M_{\alpha,B}(b)(x) \big |^{q} \mathrm{d} x \right)^{\frac{1}{q}} \|\scalebox{1.2}{$\chi$}_{B}\|_{L^{r's}(\mathbb{G}) } \\
&\le \dfrac{C}{|B|^{1/s-1/q-\frac{1}{r's}}} \dfrac{1}{|B|^{\beta/Q}} \left( \dfrac{1}{|B|} \displaystyle\int_{B} \big |b(x) -|B|^{-\alpha/Q}M_{\alpha,B}(b)(x) \big |^{q} \mathrm{d} x \right)^{\frac{1}{q}} \\
&\le C,
\end{align*}
which is what we want.
The proof is completed.
\end{proof}
\subsection{Proof of \cref{thm:frac-max-lip}}
\begin{proof}[Proof of \cref{thm:frac-max-lip}]
Since the implications \labelcref{enumerate:thm-frac-max-lip-2} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:thm-frac-max-lip-3} and \labelcref{enumerate:thm-frac-max-lip-5} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:thm-frac-max-lip-4} follows readily,
we only need to prove \labelcref{enumerate:thm-frac-max-lip-1} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:thm-frac-max-lip-2}, \labelcref{enumerate:thm-frac-max-lip-3} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:thm-frac-max-lip-4}, \labelcref{enumerate:thm-frac-max-lip-4} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:thm-frac-max-lip-1} and \labelcref{enumerate:thm-frac-max-lip-2} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:thm-frac-max-lip-5} (the proof structure is also shown in \Cref{fig:ps-equivalent-non}).
\labelcref{enumerate:thm-frac-max-lip-1} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:thm-frac-max-lip-2}:\ Let $b\in \Lambda_{\beta}(\mathbb{G})$, then, using \cref{def.lip-space} \labelcref{enumerate:def-lip-1}, we have
\begin{align} \label{inequ:proof-frac-main-1-1}
\begin{aligned}
M_{\alpha,b} (f)(x) &= \sup_{B\ni x} \dfrac{1}{|B|^{1-\alpha/Q}} \displaystyle\int_{B} |b(x)-b(y)| |f(y)| \mathrm{d} y \\
&\le C\|b\|_{\Lambda_{\beta}(\mathbb{G})} \sup_{B\ni x} \dfrac{1}{|B|^{1-\alpha/Q}} \displaystyle\int_{B}|\rho(y^{-1}x)|^{\beta} |f(y)| \mathrm{d} y \\
&\le C\|b\|_{\Lambda_{\beta}(\mathbb{G})} \sup_{B\ni x} \dfrac{1}{|B|^{1-(\alpha+\beta)/Q}} \displaystyle\int_{B} |f(y)| \mathrm{d} y \\
&\le C\|b\|_{\Lambda_{\beta}(\mathbb{G})} M_{\alpha+\beta} (f)(x).
\end{aligned}
\end{align}
Therefore, assertion \labelcref{enumerate:thm-frac-max-lip-2} follows from \cref{lem:frac-maximal-kokilashvili1989fractional} and \labelcref{inequ:proof-frac-main-1-1}.
\labelcref{enumerate:thm-frac-max-lip-3} $\xLongrightarrow[]{\ \ \ \ }$ \labelcref{enumerate:thm-frac-max-lip-4}:\
For any fixed ball $B\subset \mathbb{G}$, we have
\begin{align*}
|b(x)-b_{B}| &\le \dfrac{1}{ |B| } \displaystyle\int_{B} |b(x)-b(y)| \mathrm{d} y \\
&= \dfrac{1}{ |B| } \displaystyle\int_{B} |b(x)-b(y)| \scalebox{1.2}{$\chi$}_{B} (y) \mathrm{d} y \\
&\le \dfrac{1}{ |B|^{\alpha/Q} } M_{\alpha,b}(\scalebox{1.2}{$\chi$}_{B})(x)
\end{align*}
for all $x\in B$. Then, for all $x\in \mathbb{G}$, we get
\begin{align*}
|(b(x)-b_{B})\scalebox{1.2}{$\chi$}_{B} (x)| \le \dfrac{1}{ |B|^{\alpha/Q} } M_{\alpha,b}(\scalebox{1.2}{$\chi$}_{B})(x).
\end{align*}
Since $ M_{\alpha,b} $ is bounded from $L^{p}(\mathbb{G})$ to $L^{q}(\mathbb{G})$. Then, by using assertion \labelcref{enumerate:thm-frac-max-lip-3} and \cref{lem:norm-characteristic-functions-Lie-group}, for any ball $B\subset \mathbb{G}$, one obtains
\begin{align*}
\dfrac{1}{|B|^{\beta/Q}} \Big( \dfrac{1}{|B|} \displaystyle\int_{B} |b(x)-b_{B}|^{q} \mathrm{d} x \Big)^{1/q}
&\le \dfrac{1}{ |B|^{(\alpha+\beta)/Q}} \Big( \dfrac{1}{|B|} \displaystyle\int_{B} \big( M_{\alpha,b}(\scalebox{1.2}{$\chi$}_{B})(x) \big)^{q} \mathrm{d} x \Big)^{1/q} \\
&\le \dfrac{C}{ |B|^{1/q+(\alpha+\beta)/Q}} \|\scalebox{1.2}{$\chi$}_{B}\|_{L^{p}(\mathbb{G})} \\ &\le C.
\end{align*}
which gives \labelcref{inequ:frac-max-lip-4} for $s = q $ since $B$ is arbitrary and $C$ is independent of $B$.
\labelcref{enumerate:thm-frac-max-lip-4} $\xLongrightarrow[]{\ \ \ \ }$ \labelcref{enumerate:thm-frac-max-lip-1}:\ For any ball $B \subset \mathbb{G}$, it follows from H\"{o}lder's inequality (see \cref{lem:holder-inequality-Lie-group}), \cref{lem:norm-characteristic-functions-Lie-group} and assertion \labelcref{enumerate:thm-frac-max-lip-4} that
\begin{align*}
\dfrac{1}{ |B|^{1+\beta/Q}} \displaystyle\int_{B} |b(x)- b_{B}| \mathrm{d} x
&\le \dfrac{C}{|B|^{1+\beta/Q}} \Big( \displaystyle\int_{B} |b(x)-b_{B}|^{q} \mathrm{d} x \Big)^{1/q} \Big(\displaystyle\int_{B} \scalebox{1.2}{$\chi$}_{B}(x) \mathrm{d} x \Big)^{1/q'} \\
&\le \dfrac{C}{|B|^{\beta/Q}} \Big( \dfrac{1}{|B|} \displaystyle\int_{B} |b(x)-b_{B}|^{q} \mathrm{d} x \Big)^{1/q} \\ &\le C.
\end{align*}
It follows from \cref{lem:2.2-li2003lipschitz} and \cref{def.lip-space} that $b\in \Lambda_{\beta}(\mathbb{G})$ since $B$ is an arbitrary ball in $\mathbb{G}$.
\labelcref{enumerate:thm-frac-max-lip-2} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:thm-frac-max-lip-5}:\ Similar to the course of the proof of \labelcref{enumerate:nonlinear-frac-max-lip-2} $\xLongrightarrow[]{\ \ }$ \labelcref{enumerate:nonlinear-frac-max-lip-5}, thus, we omit it.
The proof of \cref{thm:frac-max-lip} is completed.
\end{proof}
\subsubsection*{Funding information:}
This work was financially supported by the Scientific Research Fund of APU (No.S022022177, for Zhao), Project of Heilongjiang Province Science and Technology Program (No.2019-KYYWF-0909, for Wu), the National Natural Science Foundation of China (No.11571160, for Wu), the Reform and Development Foundation for Local Colleges and Universities of the Central Government(No.2020YQ07, for Wu) and the Scientific Research Fund of Mudanjiang Normal University (No.D211220637, for Wu).
\subsubsection*{Data availability statement:}
This manuscript has no associate data.
\phantomsection \addcontentsline{toc}{section}{References}
\end{document} | arXiv |
The block-constrained configuration model
Giona Casiraghi ORCID: orcid.org/0000-0003-0233-57471
Applied Network Science volume 4, Article number: 123 (2019) Cite this article
We provide a novel family of generative block-models for random graphs that naturally incorporates degree distributions: the block-constrained configuration model. Block-constrained configuration models build on the generalized hypergeometric ensemble of random graphs and extend the well-known configuration model by enforcing block-constraints on the edge-generating process. The resulting models are practical to fit even to large networks. These models provide a new, flexible tool for the study of community structure and for network science in general, where modeling networks with heterogeneous degree distributions is of central importance.
Stochastic block-models (SBMs) are random models for graphs characterized by group, communities, or block structures. They are a generalization of the classical G(n,p) Erdős-Rènyi model (1959), where vertices are separated into B different blocks, and different probabilities to create edges are then assigned to each block. This way, higher probabilities correspond to more densely connected groups of vertices, capturing the structure of clustered graphs (Fienberg et al. 1985; Holland et al. 1983; Peixoto 2012).
SBMs are specified by defining a B×B block-matrix of probabilities B such that each of its elements \(\omega _{b_{i}b_{j}}\) is the probability of observing an edge between vertices i and j, where bi denotes the block to which vertex i belongs. Most commonly, block-matrices are used to encode community structures. This is achieved by defining a diagonal block-matrix, with the inclusion of small off-diagonal elements.
Thanks to its simple formulation, the edge generating process of the standard SBM can retain the block structure of the graph that needs to be modeled (Karrer and Newman 2011). However, it fails to reproduce empirical degree sequences. The reason for this is that in the G(n,p) model and its extensions, edges are sampled independently from each other with fixed probabilities, generating homogeneous degree-sequences across blocks. This issue impairs the applicability of the standard SBM to most real-world graphs. Because of the lack of control on the degree distributions generated by the model, SBMs are not able to reproduce the complex structures of empirical graphs, resulting in poorly fitted models (Karrer and Newman 2011).
Different strategies have been formulated to overcome this issue. Among others, one approach is that of using exponential random graph models (Krivitsky 2012). These models are very flexible in terms of the kind of patterns they can incorporate. However, as soon as their complexity increases, they lose the analytical tractability that characterizes the standard SBM. This is due to the need for computing the partition function that defines the underlying probability distribution (Park and Newman 2004). Another, more prominent, approach taken to address the issue of uniform degree-sequences in SBMs are degree-corrected block models (DC-SBM) (e.g. Peixoto (2014); Newman and Peixoto (2015); Karrer and Newman (2011); Peixoto (2015)). Degree-corrected block-models address this problem by extending standard SBMs with degree corrections, which serve the purpose of enforcing a given expected degree-sequence within the block structures. Moreover, this is achieved without hampering the simplicity of the standard SBM. For this reason, DC-SBMs are widely used for community detection tasks (Newman and Reinert 2016; Peixoto 2015). Recently, they have further been extended to a Bayesian framework, allowing non-parametric model estimation (Peixoto 2017; Peixoto 2018).
One of the main assumptions of G(n,p) models, SBMs, and DC-SBMs as well, is that the probability of creating edges for each pair of vertices are independent of each others (Karrer and Newman 2011). While such a modeling assumption allows defining distributions whose parameters are in general easy to estimate, for many real-world graphs, this is a strong assumption that should be verified, and which is possibly unrealistic (Squartini et al. 2015). Many social phenomena studied through empirical graphs, such as triadic closure (Granovetter 1973), or balance theory (Newcomb and Heider 1958), are based on the assumption that edges between vertices are not independent. Similarly, for graphs arising from the observation of constrained systems, like financial and economic networks, it is unreasonable to assume that edge probabilities are independent of each other. This is because the observed edges in the graph, which are the representation of interactions between actors in a system, are driven by optimization processes characterized by limited resources and budget constraints, which introduce correlations among different edge probabilities (Caldarelli et al. 2013; Nanumyan et al. 2015).
Moreover, one of the consequences of the assumption of independence of edge probabilities is the fact that the total number of edges of the modelled graph is preserved only in expectation. In the case of SBMs and DC-SBMs, the total number of edges is assumed to follow a Poisson distribution. For a Poisson process to be the appropriate model for an empirical graph, the underlying edge generating process needs to meet the following conditions (Consul and Jain 1973): (i) the observation of one edge should not affect the probability that a second edge will be observed, i.e., edges occur independently; (ii) the rate at which edges are observed has to be constant; (iii) two edges cannot be observed at precisely the same instant. However, it is often hard to evaluate whether these conditions are verified because the edge generating process may not be known, or these conditions are not met altogether.
Melnik et al. (2014) have proposed an alternative approach to the problem of preserving degree distributions and the independence of edges. Such an approach is a generalisation of the configuration model that allows constructing modular random graphs, characterised by heterogeneous degree-degree correlations between each block. The model, in particular, relies on specifying different values \(P_{k,k\prime }^{b_{i},b_{i\prime }}\) for the probability that a randomly chosen edge connects a degree-k node from block bi to a degree- k′ node from block bi′. The so-called \(P_{k,k\prime }^{i,i\prime }\)-model, though, only considers unweighted and undirected graphs (Melnik et al. 2014).
Similarly to the approach discussed in Melnik et al. (2014), we address the problem of incorporating degree distributions generalising the configuration model. Doing so, we propose a family of block-models that preserves the number of edges exactly, instead of in expectation. This circumvents the issue of assuming a given model for the number of edges in the graph, treating it merely as an observed datum. The configuration model of random graphs (Chung and Lu 2002a; 2002b; Bender and Canfield 1978; Molloy and Reed 1995) is, in fact, the simplest random model that can reproduce heterogeneous degree distributions given a fixed number of edges. It achieves this by randomly rewiring edges between vertices and thus preserving the degree-sequence of the original graph. Doing so, it keeps the number of edges in the graph fixed.
Differently from what proposed by Melnik et al. (2014), though, we extend the standard configuration model to reproduce arbitrary block structures by introducing block constraints on its rewiring process by means of the formalism provided by the generalised hypergeometric ensemble of random graphs. While this approach is not as general as the one proposed by Melnik et al. (2014) in terms of how degree-degree correlations can be incorporated, it allows us to deal with multi-edge, directed graphs. We refer to the resulting model as block-constrained configuration model (BCCM). Significant advantages of our approach are (i) the natural degree-correction provided by BCCM, and (ii) the preservation of the exact number of edges.
Generalised hypergeometric ensembles of random graphs (gHypEG)
Our approach builds on the generalized hypergeometric ensemble of random graphs (gHypEG) (Casiraghi et al. 2016; Casiraghi and Nanumyan 2018). This class of models extends the configuration model (CM) (Molloy and Reed 1995; 1998) by encoding complex topological patterns, while at the same time preserving degree distributions. Block constraints fall into the larger class of patterns that can be encoded by means of gHypEG. For this reason, before introducing the formulation of the block-constrained configuration model, we provide a brief overview of gHypEG. More details, together with a more formal presentation, are given in Casiraghi et al. (2016); Casiraghi and Nanumyan (2018).
In the configuration model of random graphs, the probability of connecting two vertices depends only on their (out- and in-) degrees. In its most common formulation, the configuration model assigns to each vertex as many out-stubs (or half-edges) as its out-degree, and as many in-stubs as its in-degree. It then proceeds connecting random pairs of vertices joining out- and in-stubs. This is done by sampling uniformly at random one out- and one in-stub from the pool of all out- and in-stubs respectively and then connecting them, until all stubs are connected. The left side of Fig. 1 illustrates the case from the perspective of a vertex A. The probability of connecting vertex A with one of the vertices B, C or D depends only on the abundance of stubs, and hence on the in-degree of the vertices themselves. The higher the in-degree, the higher the number of in-stubs of the vertex. Hence, the higher the probability to randomly sample a stub belonging to the vertex.
Probabilities of connecting different stubs. Graphical illustration of the probability of connecting two vertices as a function of degrees (left figure), and degree and propensities (right figure)
Generalized hypergeometric ensembles of random graphs provide an expression for the probability distribution underlying this process, where the degrees of the vertices are preserved in expectations. This result is achieved by mapping the process described above to an urn problem. Edges are represented by balls in an urn, and sampling from the configuration model is described by sampling balls (i.e., edges) from an urn appropriately constructed. For each pair of vertices (i,j), we can denote with \(k^{\text {out}}_{i}\) and \(k^{\text {in}}_{j}\) their respective out- and in-degrees. The number of combinations of out-stubs of i with in-stubs of j which could be connected to create an edge is then given by kioutkjin. To map this process to an urn, for each dyad (i,j) we should place exactly kioutkjin balls of a given colour in the urn (Casiraghi and Nanumyan 2018). The process of sampling m edges from the configuration model is hence described by sampling m balls from this urn, and the probability distribution of observing a graph \(\mathcal {G}\) under the model is given by the multivariate hypergeometric distribution with parameters Ξ={kioutkjin}i,j:
$$ \Pr(\mathcal{G}|\mathbf\Xi)=\dbinom{\sum_{ij}\Xi_{ij}}{m}^{-1}\prod_{i,j\in V}{\dbinom{\Xi_{ij}}{A_{ij}}}, $$
where Aij denotes the element ij of the adjacency matrix of \(\mathcal {G}\), and the probability of observing \(\mathcal {G}\) is non-zero only if \(\sum _{ij}A_{ij}=m\).
Generalized hypergeometric ensembles of random graphs further extend this formulation. In gHypEG, the probability of connecting two vertices depends not only on the degree (i.e., number of stubs) of the two vertices but also on an independent propensity of the two vertices to be connected, which captures non-degree related effects. Doing so allows constraining the configuration model such that given edges are more likely than others, independently of the degrees of the respective vertices. The right side of Fig. 1 illustrates this case, where A is most likely to connect with vertex D, belonging to the same group, even though D has only one available stub.
In generalized hypergeometric ensembles the distribution over multi-graphs (denoted \(\mathcal {G}\)) is formulated such that it depends on two sets of parameters: the combinatorial matrix Ξ, and a propensity matrix Ω that captures the propensity each pair of vertices to be connected. Each of these two matrices has dimensions n×n where n is the number of vertices in \(\mathcal {G}\). The contributions of the two matrices to the model are as follows. The combinatorial matrix Ξ encodes the configuration model as described above. The propensity matrix Ω encodes dyadic propensities of vertices that go beyond what prescribed by the combinatorial matrix Ξ. The ratio between any two elements Ωij and Ωkl of the propensity matrix is the odds-ratio of observing an edge between vertices i and j instead k and l, independently of the degrees of the vertices.
As for the case of the configuration model, this process can be seen as sampling edges from an urn. Moreover, specifying a propensity matrix Ω allows to bias the sampling in specified ways, so that some edges are more likely to be sampled than others. The probability distribution over a graph \(\mathcal {G}\) given Ξ and Ω is then described by the multivariate Wallenius' noncentral hypergeometric distribution (Wallenius 1963; Chesson 1978).
We further denote with A the adjacency matrix of the multi-graph \(\mathcal {G}\) and with V its set of vertices, the probability distribution underlying a gHypEG \(\mathbb X(\mathbf \Xi,\mathbf \Omega,m)\) with parameters Ξ,Ω, and with m edges is defined as follows:
$$ \Pr(\mathcal{G}|\mathbf\Xi,\mathbf\Omega)=\left[\prod_{i,j\in V}{\dbinom{\Xi_{ij}}{A_{ij}}}\right] \int_{0}^{1}{\prod_{i,j\in V}{\left(1-z^{\frac{\Omega_{ij}}{S_{\mathbf{\Omega}} }}\right)^{A_{ij}}}dz} $$
$$ S_{\mathbf{\Omega}}= \sum_{i,j\in V} \Omega_{ij}(\Xi_{ij}-A_{ij}). $$
In Eq. 2, the first term on the right-hand side represents combinatorial effects encoding degrees, inherited from the configuration model. The second term, constituted by the integral, encodes the biases that need to be enforced on top of the process defined by the configuration model. Note that, if Ωij=c for all i,j and for any constant c, i.e., if no biases are enforced on the configuration model, Eq. 2 corresponds to Eq. 1 (Casiraghi and Nanumyan 2018). The probability distribution for undirected graphs and graphs without self-loops are defined similarly: by excluding the lower triangular entries of the adjacency matrix or by excluding its diagonal entries respectively (we refer to Casiraghi and Nanumyan (2018) for more details).
In the case of large graphs, sampling from an urn without replacement can be approximated by sampling with replacement from the same urn. Under this assumption, the approximation allows to estimate the probability given in Eq. 2 by means of a multinomial distribution with parameters \(p_{ij}=\Xi _{ij}\Omega _{ij}/\sum _{kl}\Xi _{kl}\Omega _{kl}\).
Block-constrained configuration model
The main modelling assumption that differentiate gHypEGs from SBMs is in the dependence/independence of edge probabilities. In particular, while SBMs assume independent edge probabilities, and specifies a Poisson process for the edge generating process, gHypEG fixes the total number of edges m in the model and removes the assumption of independence between edge probabilities. This assumption has the conceptual advantage of not assuming an arbitrary edge generating process, such as the Poisson process considered by DC-SBMs.
We hence define the block-constrained configuration model (BCCM) building on the framework provided by generalized hypergeometric ensembles of random graphs. We achieve so by utilizing a particular form of the propensity matrix Ω. Specifically, we need to encode the block structure that we observe in the propensity matrix Ω. We do so by specifying a block propensity matrix Ω(B) where each of its elements \({\Omega ^{(B)}}_{ij}=\omega _{b_{i}}\) if the vertices i and j are in the same block bi, and \({\Omega ^{(B)}}_{ij}=\omega _{b_{i}b_{j}}\) if the vertices i and j are in different blocks bi and bj respectively. Figure 2 shows a block-propensity matrix characterised by three blocks. Similarly to SBMs, in the presence of B blocks, we can specify a B×B block-matrix B that captures the block structure through its parameters \(\omega _{b_{i}b_{j}}\). However, in the case of a BCCM, the entries \(\omega _{b_{i}b_{j}}\) capture the deviations in terms of edge propensities from the configuration model defined by the matrix Ξ, constraining edges into blocks.
Block-matrix. Structure of a block propensity matrix with 3 different blocks (blue, green, yellow). The entries along the diagonal capture the within-block propensities, those away from the diagonal capture the between-block propensities
The block-matrix B can be specified to generate various structures, extending those naturally produced by degrees only, such as a diagonal block-matrix can model graphs with disconnected components. The inclusion of small off-diagonal elements gives rise to standard community structures, with densely connected clusters of vertices. By specifying different types of block-matrices, it is also possible to model core-periphery, hierarchical, or multipartite structures.
The block-constrained configuration model \(\mathbb X(\mathbf \Xi,\mathbf B,m)\) with m edges is thus completely defined by the combinatorial matrix Ξ, and by the block-matrix B generating the propensity matrix Ω(B). We can then rewrite the general probability for a gHypEG given in Eq. 2 for BCCM:
$$ \Pr(\mathcal{G}|\mathbf\Xi,\mathbf B)=\left[\prod_{i,j\in V}{\dbinom{\Xi_{ij}}{A_{ij}}}\right] \int_{0}^{1}{\prod_{i,j\in V}{\left(1-z^{\frac{\omega_{b_{i}b_{j}}}{S_{\mathbf{B}} }}\right)^{A_{ij}}}dz} $$
$$ S_{\mathbf{B}}= \sum_{i,j\in V} \omega_{b_{i}b_{j}}(\Xi_{ij}-A_{ij}). $$
Table 1 summarises the differences between the distributions underlying the DC-SBM and the BCCM.
Table 1 Comparison of the properties of DC-SBMs and BCCMs
Despite its complicated expression, the probability distribution in Eq. 4 allows computing probabilities for large graphs, without the need to resort to Monte-Carlo simulations (Fog 2008a). This permits the study of large graphs and provides simple model selection methods based on the comparison of likelihoods, such as likelihood-ratio tests, or those based on information criteria. In this article, we will consider model selection based on the comparison of information criteria.
We will adopt the two most commonly used ones: Akaike information criterion (AIC) (Akaike 1974), and Schwarz or Bayesian information criterion (BIC) (Schwarz and et al 1978). Both criteria depend on the likelihood function of the models to be compared and penalize for the number of parameters estimated by the model. The model with the lowest score is the preferred one, as it best fits the data without overfitting it. In particular, it is not the absolute size of the score, but it is the difference between values that matters for model selection.
Information-theoretic methods considered here provide a simple way to select the best-approximating model from a candidate set of models. The concept of information criterion has allowed major practical and theoretical advances in model selection and the analysis of complex data sets (Stone 1982;Bozdogan 1987;DeLeeuw 1992). In particular, AIC and BIC allow performing model selection without the need of simulations, nor the assumption of specific asymptotic behaviors of the probability distribution of the model (although BIC assumes that the priors for the parameters estimated are asymptotically normal). Moreover, the aim of model selection by means AIC and BIC is not to identify exactly the 'true model,' i.e., the actual process generating the data, but to propose simpler models that are good approximations of it (Kuha 2004). They only allow the selection of the best model among those within a specified set. This means that, if all models in the set are very poor, information criteria will select the best model, but even that relatively best model might be poor in the absolute sense (Burnham and Anderson 2004).
The Akaike information criterion for a model \(\mathbb X\) given a graph \(\mathcal {G}\) is formulated as follows:
$$ \text{AIC}(\mathbb X|\mathcal{G})=2k-2\log\left[\hat L(\mathbb X|\mathcal{G})\right], $$
where k is the number of parameters estimated by X and \(\hat L(\mathbb X|\mathcal {G})\) is the likelihood of model \(\mathbb X\) given the graph \(\mathcal {G}\). AIC gives an estimate of the expected, relative Kullback-Leibler distance between the fitted model and the unknown true mechanism generating the observed data. Hence, the best model among a set of models is the one that has the minimal distance from the true process, and thus the one that minimizes AIC.
The Bayesian information criterion for a model \(\mathbb X\) given a graph \(\mathcal {G}\) is given by:
$$ \text{BIC}(\mathbb X|\mathcal{G})=\log(m)k-2\log\left[\hat L(\mathbb X|\mathcal{G})\right], $$
where k is the number of parameters estimated by \(\mathbb X\), m is the number of observations, i.e., edges, and \(\hat L(\mathbb X|\mathcal {G})\) is the likelihood of model \(\mathbb X\) given the graph \(\mathcal {G}\). Similarly to AIC, the best model in a set according to BIC is the one which minimizes the criterion. Because of the presence of a higher penalty for model size, BIC tends to select models with lower parameters compared to AIC.
As mentioned above, what matters for model selection is the difference between the value of AICs or BICs and not their absolute values. For this reason, it is helpful to rank models in terms of their differences from the model which minimizes a given criterion. Suppose that there are R models, and we want to find the best one according to either AIC or BIC. Let AICmin be the model which minimizes AIC for a given dataset. Then we can define the AIC differences \(\Delta _{i}^{\text {AIC}{}}\) as the difference AICi−AICmin of the AIC score for model i∈{1,…,R}, and the model which minimizes AIC. BIC differences are defined in a similar manner. While AIC and BIC differences are useful in ranking models, it is possible to quantify the plausibility of each model by defining relative likelihoods for the models. Specifically, the quantity \(\phantom {\dot {i}\!}e^{-1/2\Delta _{i}}\) defines the relative likelihood of model i given the data (Burnham and Anderson 2004). To better interpret relative likelihoods, statisticians usually normalize relative likelihoods to be a set of positive weights wi defined as
$$ w_{i}:=\frac{e^{-1/2\Delta_{i}}}{\sum_{r=1}^{R} e^{-1/2\Delta_{r}}}. $$
In the case of AIC, such model weights are usually referred to as Akaike weights and are considered to be the weight of evidence in favor of model i being the best model. In the case of BIC, instead, the weights define the posterior model probabilities. The bigger Δi is, the smaller wi and the less plausible is model i as being the actual best model based on the design and sample size used. These weights provide an effective way to scale and interpret the Δi values and hence select the best model (Burnham and Anderson 2004).
In the next sections, we describe how BCCM can be used to generate graphs and how to fit its parameters to an observed graph. Because the absolute values of AIC and BIC are not important, and only relative Δis matter, in the following we will usually report only the value of the relative differences.
Generating realizations from the BCCM.BCCM is a practical generative model that allows the creation of synthetic graphs with complex structures by drawing realizations from the multivariate Wallenius non-central hypergeometric distribution. The process of generating synthetic graphs can be divided into two tasks. First, it is needed to specify the degree sequences for the vertices. It can be accomplished by, e.g., sampling the degree sequences from a power-law or exponential distributions. From the degree sequences we can generate the combinatorial matrix Ξ, specifying its elements \(\Xi _{ij}=k_{i}^{\text {out}}k_{j}^{\text {in}}\), where \(k_{i}^{\text {out}}\) is the out-degree of vertex i. Second, we need to define a block-matrix B, whose elements specify the propensities of observing edges between vertices, between and within the different blocks.
The block-matrix B takes the form given in Eq. 9:
$$ \mathbf B= \left[\begin{array}{ccc} \omega_{1} & \dots & \omega_{1B} \\ & \vdots & \\ \omega_{B1} & \dots & \omega_{B} \end{array}\right]. $$
Elements ωkl, with k,l∈{1,…,B}, should be specified such that the ratio between any two elements corresponds to the chosen odds-ratios of observing an edge in the block corresponding to the first element instead of the block corresponding to the second element, given the degrees of the corresponding vertices were the same. For example, ω1/ω32 corresponds to the odds-ratio of observing an edge between vertices in block 1 compared to an edge between block 2 and block 3. Note that in the case of an undirected graph, ωkl=ωlk∀k,l∈{1,…,B}. On the other hand, in the case of a directed graph, blocks may have a preferred directionality, i.e., edges between blocks may be more likely in one direction. In this case, we may choose ωkl≠ωlk.
Once the parameters of the model are defined, we sample graphs with m edges from the BCCM \(\mathbb X(\mathbf \Xi,\mathbf \Omega _{B},m)\) defined by the combinatorial matrix Ξ, and the block-propensity matrix ΩB defined by B. As described in the previous section, sampling a graph from \(\mathbb X(\mathbf \Xi,\mathbf \Omega _{B},m)\) corresponds to sample m edges according to the multivariate Wallenius non-central hypergeometric distribution.
ExamplesWe can specify different types of clustered graphs using this construction. As a demonstrative example, we define a block-matrix with five blocks connected in a ring. Each block is as dense as the others, and blocks are weakly connected with only their closest neighbors. The block-matrix quantifying these specification is given as
$$ \mathbf B= \left[\begin{array}{lllll} 1 & 0.1 & 0 & 0 & 0 \\ 0 & 1 & 0.1 & 0 & 0 \\ 0 & 0 & 1 & 0.1 & 0 \\ 0 & 0 & 0 & 1 & 0.1 \\ 0.1 & 0 & 0 & 0 & 1 \\ \end{array}\right]. $$
According to the choice made in Eq. 10, edges within diagonal blocks are 10 times more likely than edges within off-diagonal blocks.
After fixing this block-matrix, we can define different degree sequences for the vertices. We highlight here the results obtained when fixing three different options in a directed graph without self-loops, with n=50 vertices and m=500 edges. We generate realizations by specifying the combinatorial matrix Ξ and the block propensity matrix and exploiting the random number generator provided byFog (2008b) in the R library BiasedUrn.
The first degree-sequence we can set is the most straightforward option, corresponding to the standard non-degree-corrected stochastic block-model. This model corresponds to setting each entry in the combinatorial matrix Ξ equal to m2/(n(n−1)) (Casiraghi et al. 2016). If we assign the same number of vertices to each block, we expect the model to generate graphs with homogeneous blocks. Figure 3a shows a realisation from this model. The second degree-sequence we can set is defined such that the degrees of the vertices of each block are drawn from a power-law distribution. We expect that each block shows the same structure, with few vertices with high degrees, and many with low degrees. Because of this, we expect that most blocks are connected with directed edges starting from high-degree vertices. Figure 3b shows a realization from this model where this is visible. Finally, we set a degree sequence where the degrees of all vertices are drawn from a power-law distribution. Figure 3c shows a realization from this model. The combinatorial matrices corresponding to Fig. 3b and c are included with this article as additional files.
Realisations from a block-constrained configuration model obtained by fixing the block-matrix B and varying the out-degree distribution. Each realisation is obtained from a BCCM with N=50 vertices and m=500 directed edges. The vertices are separated into 5 equally sized blocks and the block-matrix B is given by Eq. 10. On left side, a is a realisation from a BCCM where the degree distributions are uniform. It corresponds to a realisation from a standard SBM. In the center, b is a realisation obtained by drawing the out-degree distribution of the vertices in each block from a power-law distribution with parameter α=1.8. On the right side, c is a realisation obtained by drawing the out-degree distribution of all vertices from the same power-law. All graphs are visualised using the force-atlas2 layout with weighted edges. Out-degrees determine vertex sizes, and edge widths the edge counts
Instead of varying the degree sequences of the underlying configuration model, we can as well alter the strength of the block structure, changing the block-matrix B. Similarly to what we did above, we show three different combinations of parameters. First, we set the within group parameters \(\omega _{b_{i}}\) equal to the between group parameters \(\omega _{b_{i}b_{j}} \forall i,j\). Second, we set the parameters \(\omega _{b_{1}}=10\) so that the more edges are concentrated in the first block. Third, we set the parameter to reconstruct a hierarchical structure. We modify the parameters \(\omega _{b_{1}b_{2}}=\omega _{b_{3}b_{4}}=\omega _{b_{4}b_{5}}=0.8\) to model graphs with two macro clusters weakly connected, where the one is split into two clusters strongly connected and the other into three clusters strongly connected. Realizations drawn from each of these three models are shown in Fig. 4.
Realisations from a block-constrained configuration model obtained by fixing the out-degree distribution and varying the parameters within the block-matrix B. Each realisation is obtained from a BCCM with N=50 vertices and m=500 directed edges. The out-degree distribution of the vertices in each block follows a power-law distribution with parameter α=1.8 The vertices are separated into 5 equally sized blocks and the structure of the block-matrix B is given by Eq. 10, but in each graph the values of some of the parameters \(\omega _{b_{i}b_{j}}\) are changed. On left side, a is a realisation from a BCCM where the between-block parameters are increased to 1. In the center, b is a realisation obtained by increasing the parameter \(\omega _{b_{1}}\) that controls for the internal cohesion of the first block. On the right side, c is a realisation obtained by increasing to 0.8 the between-block parameters \(\omega _{b_{1}b_{2}}, \omega _{b_{3}b_{4}}\), and \(\omega _{b_{4}b_{5}}\), to create a hierarchical block structure where the first two blocks are part of a macro cluster, and the last three blocks are part of another. All graphs are visualised using the force-atlas2 layout with weighted edges. Out-degrees determine vertex sizes, and edge widths the edge counts
Fitting the block-matrix.In DC-SBMs the number of edges between each pair (i,j) of vertices are assumed to be drawn from independent Poisson distributions, with parameters \(\phantom {\dot {i}\!}\theta _{i}\theta _{j}\omega _{b_{i}b_{j}}\). Let \(A_{b_{\alpha } b_{\beta }}=\sum _{i\in b_{\alpha },j\in b_{\beta }}A_{ij}\) denote the number of edges between all vertices i that are in the block bα and j in block bβ. We further denote bi the partition of vertex i. Exploiting the independence of probabilities, the maximum likelihood estimates \(\widehat \theta _{i}\) and \(\widehat \omega _{b_{i}b_{j}}\) of the parameters θi and \(\omega _{b_{i}b_{j}}\) are given by \(\widehat \theta _{i}:=k_{i}/\kappa _{b_{i}}\) and \(\widehat \omega _{b_{i}b_{j}}:=A_{b_{i} b_{j}}\) (Karrer and Newman 2011).
Because BCCM does not assume independence among edge probabilities, the parameter estimation is necessarily more complicated than that of DC-SBMs. However, the formulation of the block-constrained configuration model allows for the fast estimation of the parameters of the block-matrix. Similarly to what is done with SBMs, we fit the BCCM by preserving in expectation the observed number of edges between and within different blocks. To estimate the entries ωb of the block-matrix \(\hat {\mathbf {B}}\), we exploit the properties of the generalized hypergeometric ensemble of random graphs.
In gHypE, the entries of the expected adjacency matrix 〈Aij〉 are obtained by solving the following system of equations (Casiraghi and Nanumyan 2018):
$$\begin{array}{*{20}l} \left(1-\frac{\langle A_{11}\rangle}{\Xi_{11}}\right)^{\frac{1}{\Omega_{11}}} = \left(1-\frac{\langle A_{12}\rangle}{\Xi_{12}}\right)^{\frac{1}{\Omega_{12}}} = \ldots \end{array} $$
with the constraint \(\sum _{i,j \in V} \langle A_{ij}\rangle = m\).
Because to estimate BCCM we need to fix the expectation of the number of edges between blocks and not between dyads, we proceed as described below. We denote with \(\Xi _{b_{\alpha }}=\sum _{i,j\in b_{\alpha }}\Xi _{ij}\) the sum of all the elements of the matrix Ξ corresponding to those dyads. Then, we fix the expectations of the ensemble such that the number of edges between and within blocks is given by \(A_{b_{\alpha }}\)s. Hence, in the case of the block-constrained configuration model with B blocks we estimate the B·(B+1)/2 parameters \(\omega _{b_{\alpha } b_{\beta }}\)s constituting the block-matrix \(\hat {\mathbf {B}}\) solving the following set of independent equations, defined up to an arbitrary constant k:
$$\begin{array}{*{20}l} \left\{\begin{array}{ll} \left(1-\frac{A_{b_{1}}}{\Xi_{b_{1}}}\right)^{\frac{1}{\omega_{b_{1}}}} &= k\\ &\vdots\\ \left(1-\frac{A_{b_{B}}}{\Xi_{b_{B}}}\right)^{\frac{1}{\omega_{b_{B}}}} &= k\,.\\ \end{array}\right. \end{array} $$
Solving for \(\omega _{b_{\alpha } b_{\beta }}\), we find that the entries of the block-matrix \(\hat {\mathbf {B}}\) that preserve in expectation the observed number of edges between and within blocks are given by
$$ \omega_{b_{\alpha} b_{\beta}} := -\log\left(1-\frac{A_{b_{\alpha} b_{\beta}}}{\Xi_{b_{\alpha} b_{\beta}}}\right). $$
The estimation of the parameters scales quadratically only with the number of blocks.
When the parameters of the BCCM are estimated as described here, the block-constrained configuration model has the advantageous property of asymptotic consistency. It means that, if the method described here is applied to synthetic graphs generated from a BCCM, the technique introduced in this article can correctly recover the original model.
Estimating the Ξ matrix.In the case of the configuration model defined by Eq. 1, the elements Ξij of the combinatorial matrix are defined as \(k_{i}^{\text {in}}k_{j}^{\text {out}}\). This definition generates a model that preserves the degree sequences of the observed graph (Casiraghi and Nanumyan 2018). By generalizing the model according to Eq. 4, where the propensity matrix is estimated as in Eq. 13, we introduce constraints on the edge sampling process that allows preserving the observed number of edges in each block. The estimated parameters can hence be interpreted as the bias needed to modify the configuration model to reproduce block structures.
To preserve the degrees of the observed graph in the BCCM, we need to update the combinatorial matrix such that it defines the degree-sequences of the corresponding configuration model like there were no block constraints. We achieve this by redefining the combinatorial matrix elements as \(\Xi _{ij}=k_{i}^{\text {in}}k_{j}^{\text {out}}\theta _{i}^{\text {in}}\theta _{j}^{\text {out}}\). The estimation of Ξ and Ω is then performed by an expectation-maximization algorithm that iteratively estimates Ξ and Ω such that degrees and blocks are preserved in expectation. A pseudo-code for the algorithm estimating the parameters of a BCCM model for directed graphs is provided in Algorithm 1. In the case of undirected graphs, the algorithm is adapted according to the fact that Ξ and Ω are upper-triangular matrices.
We conclude the article with a case study analysis of synthetic and empirical graphs. We highlight the interpretability of the resulting block-constrained configuration models in terms of deviations from the classical configuration model. In particular, a weak community structure in a graph is reflected in a small contribution to the likelihood of the estimated block-matrix. On the other hand, a strong community structure is reflected in a substantial contribution to the likelihood of the estimated block-matrix. Here, we quantify this difference employing AIC or BIC. However, other information criteria may also be used. Moreover, studying the relative values of the estimated parameters in the block matrices quantifies how much the configuration model has to be biased towards a block structure to fit the observed graph optimally. The more different are the values of the parameters, the stronger is the block structure compared to what is expected from the configuration model.
We start by analyzing synthetic graphs generated according to different rules, and we show that fitting the block-constrained configuration model parameters allows selecting the correct, i.e., planted, partition of vertices, among a given set of different partitions. We perform three experiments with large directed graphs with clusters of different sizes. Finally, we conclude by employing the BCCM to compare how well different partitions obtained by different clustering algorithms fit popular real-world networks.
Analysis of synthetic graphs.We generate synthetic graphs incorporating 'activities' of vertices in a classical SBM, to be able to plant different out-degree sequences in the synthetic graphs. First, we need to assign the given activity to each vertex. Higher activity means that the vertex is more likely to have a higher degree. Second, we need to assign vertices to blocks and assign a probability of sampling edges to each block. Densely connected blocks have a higher probability than weakly connected blocks. The graph is then generated by a weighted sampling of edges with replacement from the list containing all dyads of the graph. The product between the activity corresponding to the from-vertex and the weight corresponding to the block to which the dyad belongs gives sampling-weights for each dyad. The probabilities of sampling edges correspond to the normalized weights so that their sum is 1.
For example, let us assume we want to generate a 3-vertices graph with two clusters. We can fix the block weights as follows: edges in block 1 or 2 have weight w1 and w2 respectively; edges between block 1 and block 2 have weight w12. Table 2 shows the list of dyads from which to sample together with their weights, where the activity of vertices is fixed to (a1,a2,a3), and the first two vertices belong to the first block. Note that if the activities of the vertices were all set to the same value, this process would correspond to the original SBM. In the following experiments, we generate different directed graphs with N=500 vertices, m=40000 edges, and different planted block structures and vertex activities.
Table 2 Edge list with weights for the generation of synthetic graphs with given vertex activities and block structure
In the first experiment, we show the difference between estimating the parameters for an SBM and the BCCM when the block structure is given. To do so, we first generate the activities of vertices from an exponential distribution with parameter λ=N/m (such that the expected sum of all activities is equal to the number of edges m we want to sample). After sorting the activity vector in decreasing order, we assign it to the vertices. In this way, the first vertex has the highest activity, and hence the highest out-degree, and so on. In the first experiment, we do not assign block weights so that the graphs obtained do not show any consistent cluster structure, and have a skewed out-degree distribution according to the fixed vertex activity (correlation ∼1).
First, we assign the vertices randomly to two blocks. We proceed by estimating the parameters for an SBM and a BCCM, according to the blocks to which the vertex has been assigned. Since no block structure has been enforced and the vertex has been assigned randomly to blocks, we expect that the estimated parameters for the block matrices \(\hat {\mathbf {B}}_{\text {SBM}}\) and \(\hat {\mathbf {B}}_{\text {BCCM}}\) will all be close to 1 (when normalized by the maximum value), reflecting the absence of a block structure. The resulting estimated parameters for an exemplary realisation are reported in Eq. 14.
$$ \hat{\mathbf{B}}_{\text{SBM}}= \left[\begin{array}{ll} 1.0000000 & 0.9992577\\ 0.9992577 & 0.9603127 \end{array}\right] \hfill\quad \hat{\mathbf{B}}_{\text{BCCM}}= \left[\begin{array}{ll} 0.9808935 & 1.0000000\\ 1.0000000 & 0.9805065 \end{array}\right] $$
As expected, the estimated values for both models are close to 1.
After changing the way vertices are assigned to blocks, we repeat the estimation of the two models. Now, we separate the vertices into two blocks such that the first 250 vertices ordered by activity are assigned to the first block and the last 250 to the second one. We expect that the SBM will assign different parameters to the different blocks because now the first block contains all vertices with high degree, and the second block all vertices with low degree. Hence, most of the edges are found between vertices in the first block or between the two blocks. Differently, from the SBM, the BCCM corrects for the observed degrees. Hence, we expect that the parameters found for the block-matrix will be all close to 1 again, as no structure beyond that one generated by degrees is present. Thus the block assignment does not matter for the estimated parameter. The block matrices for the two models, estimated for the same realisation used above, are provided in Eq. 15.
$$ \hat{\mathbf{B}}_{\text{SBM}}= \left[\begin{array}{ll} 1.000000 & 0.597866\\ 0.597866 & 0.194896 \end{array}\right] \hfill\quad \hat{\mathbf{B}}_{\text{BCCM}}= \left[\begin{array}{ll} 0.997024 & 0.995108\\ 0.995108 & 1.000000 \end{array}\right] $$
We observe that the SBM assigns different values to each block, impairing the interpretability of the result. In particular, the parameters of \(\hat {\mathbf {B}}_{\text {SBM}}\) show the presence of a core-periphery structure which cannot be distinguished from what obtained naturally from skewed degree distributions. The estimation of \(\hat {\mathbf {B}}_{\text {BCCM}}\), on the contrary, highlights the absence of any block structure beyond that one generated by the degree sequence, and we can correctly conclude that the degree distributions entirely generate the core-periphery structure of the observed graph.
In the second synthetic experiment, we highlight the model selection features of the BCCM. Thanks to the fact that we are able to compute the likelihood of the model directly, we can efficiently compute information criteria such as AIC or BIC to perform model selection. We generate directed graphs with self-loops with N=500 vertices, m=40000 edges, and two equally sized clusters. Again, we generate vertex activities from an exponential distribution with rate λ=N/m. We fix the block weights to be w1=1,w2=3, and w12=0.1. Using this setup, we can generate synthetic graphs with two clusters, one of which is denser than the other. If we fit a BCCM to the synthetic graph with the correct assignment of vertices to blocks, we obtain the following block-matrix \(\hat {\mathbf {B}}_{\text {BCCM}}\) for an exemplary realization:
$$ \hat{\mathbf{B}}_{\text{BCCM}}= \left[\begin{array}{ll} 1.1760878 & 0.1108463\\ 0.1108463 & 3.0000000 \end{array}\right] $$
We note that we approximately recover the original block weights used to generate the graph.
We can now compare the AIC obtained for the fitted BCCM model, AICBCCM=662060, to that obtained from a simple configuration model (CM) with no block assignment, AICCM=693540. The CM model is formulated in terms of a gHypEG where the propensity matrix Ω≡1. The AIC for the BCCM is considerably smaller, confirming that the model with block structure fits better the observed graph. In terms of AIC differences, ΔBCCMAIC=0 and ΔCMAIC=31480. This corresponds to model weights wBCCM∼1 and wCM∼0. That means that there is no evidence for model CM. As a benchmark, we compute the AIC for BCCM models where the vertices have been assigned randomly to the two blocks.
Table 3 reports the AIC differences obtained for 1000 random assignment of vertices to the blocks, computed on the same observed graph. We observe that this usually results in values close to that of the simple configuration model, as the block assignments do not reflect the structure of the graph. In a few cases, a small number of vertices are correctly assigned to blocks, showing a slight reduction in AIC, which is however far from that of the correct assignment.
Table 3 \(\Delta _{i}^{\text {AIC}{}}\) values from the model with the correct assignment vertices-blocks, obtained for 1000 random assignment of vertices to the blocks, computed on the same observed graph
BCCM also allows comparing models with a different number of blocks. To do so, we separate the vertices in one of the blocks of the model above into two new blocks. Because we add more degrees of freedom, we expect an increase in the likelihood of the new BCCM with three blocks, but this should not be enough to give a considerable decrease in AIC. Since the synthetic graph has been built planting two blocks, the AIC should allow us to select as an optimal model the BCCM with two blocks. The resulting block-matrix \(\hat {\mathbf {B}}_{\text {BCCM}}^{(3)}\) with three blocks is reported in Eq. 17.
$$ \hat{\mathbf{B}}_{\text{BCCM}}^{(3)}= \left[\begin{array}{lll} 1.1739475 & 1.1797875 & 0.1088987\\ 1.1797875 & 1.1706410 & 0.1129094\\ 0.1088987 & 0.1129094 & 3.0000000 \end{array}\right] $$
We see that the estimated model fits different parameter values for the two sub-blocks, since the added parameters can now accommodate for random variations generated by the edge sampling process. However, as expected, there is no (statistical) evidence to support the more complex model. In fact, comparing the AIC values we obtain \(\text {AIC}_{\text {BCCM}}^{(3)} = 662065 > 662060 = \text {AIC}_{\text {BCCM}}\). This corresponds to ΔBCCMAIC=0 and \(\Delta ^{\text {AIC}{}}_{\text {BCCM}^{(3)}}=5\). In terms of model weights, we get wBCCM∼0.92 and \(w_{\text {BCCM}}^{(3)}\sim 0.08\). That means that there is strong evidence against the more complex model, as the probability that the more complex model is closer to the real process is only 0.08, given the data used to estimate the model.
To provide more evidence in support of this selection procedure, we can repeat this experiment on 100 samples from the same model used before. The results provide median AIC differences of ΔBCCM=0 and \(\Delta _{\text {BCCM}}^{(3)}=4.32\). Moreover, out of the 100 samples only 7 have \(\text {AIC}_{\text {BCCM}}^{(3)} < \text {AIC}_{\text {BCCM}}\). This is aligned with the probability of 0.08 estimated employing model weights. We can thus successfully use BCCM to perform model selection, both when a different number of clusters or various vertex assignments are used.
In the third experiment, instead of two clusters, we plant three clusters of different sizes (|B1|=250, |B2|=125, |B3|=125). We choose the block parameters such that one of the smaller clusters is more densely connected with the bigger cluster, and the smaller cluster is relatively denser than the others. To do so we choose the block weights as follows: w1=w2=1,w3=3,w13=w23=0.1,w12=0.8. As before, we draw vertex activities from an exponential distribution with parameter λ=N/m. One exemplary realisation is plotted in Fig. 5. The plot clearly shows the separation into three clusters, with cluster 1 (purple) and 2 (orange) more densely connected to each other than to cluster 3 (green). Fitting the same BCCM as before allows comparing the AICs for the three-blocks BCCM to the 2-block BCCM. In this case, we expect that the model with three blocks will fit considerably better the graph. Results of the fitting for the realisation plotted in Fig. 5 give \(\text {AIC}_{\text {BCCM}}^{(3)} = 673585 < 699765 = \text {AIC}_{\text {BCCM}}^{(2)}\), correctly selecting the more complex model. This corresponds to \(\Delta ^{\text {AIC}{}}_{\text {BCCM}^{(2)}}=26180\) and \(\Delta ^{\text {AIC}{}}_{\text {BCCM}^{(3)}}=0\). In terms of model weights, we get \(w_{\text {BCCM}}^{(2)}\sim 0\) and \(w_{\text {BCCM}}^{(3)}\sim 1\). That means that there is strong evidence against the simpler model.
Visualisation of a synthetic graph with N=500 vertices and m=40000 directed edges, obtained with the force-atlas2 layout. Vertices are separated into three blocks of different sizes, such that the largest block (250 vertices, in purple) is strongly connected with one of the smaller blocks (125 vertices, in orange). Both blocks are weakly connected to the third block, that is clearly separated (125 vertices, in green). The out-degree sequence of the graph follows an exponential distribution with parameter λ=N/m. The joint effects of the non-uniform degree sequence together with the asymmetric block structure makes the task of community detection on this graph particularly hard for standard algorithms
It is known that AIC does not punish model complexity as much as BIC. For this reason, in this case, we also compare the values of BIC obtained for the two models. Also in this case, with \(\text {BIC}_{\text {BCCM}}^{(3)} = 2822787 < 2848941 = \text {BIC}_{\text {BCCM}}^{(2)}\), the information criterion allows to correctly select the model with 3 blocks. Comparing posterior probabilities for the two models, we get again \(w_{\text {BCCM}}^{(2)}\sim 0\) and \(w_{\text {BCCM}}^{(3)}\sim 1\).
Finally, we can use AIC and BIC to evaluate and rank the goodness-of-fit different block assignments that are obtained from various community detection algorithms. This allows choosing the best block assignment in terms of deviations from the configuration model, i.e., which of the detected block assignment better captures the block structure that goes beyond that generated by the degree sequence of the observed graph. We compare the result obtained from 5 different algorithms run using their igraph implementation for R. In the following we use: cluster_fast_greedy, a greedy optimisation of modularity (Clauset et al. 2004); cluster_infomap, the implementation of infomap available through igraph (Rosvall and Bergstrom 2008); cluster_label_prop, label propagation algorithm (Raghavan et al. 2007); cluster_spinglass, find communities in graphs via a spin-glass model and simulated annealing (Reichardt and Bornholdt 2006); cluster_louvain, the Louvain multi-level modularity optimisation algorithm (Blondel et al. 2008). As the modularity maximization algorithms are implemented only for undirected graphs, we apply them to the undirected version of the observed graph. The results of the application of the 5 different algorithms on the realisation shown in Fig. 5 are reported in the table in Table 4.
Table 4 Comparison of the goodness-of-fit of 5 different block structures detected by five different community detection algorithms
The five different community detection algorithms find three different block structures. Three of them are not able to detect the third block, while the other two algorithms split the vertices into too many blocks. AIC ranks best infomap even though it detects one block too many. BIC punishes for the number of parameters more, so ranks best the 2-blocks. These results are consistent when repeating the experiment with different synthetic graphs generated from the same model. It is worth noting that none of the community detection algorithms was able to detect the planted block structure correctly. However, both the AIC and BIC of the BCCM fitted with the correct block structure are lower than those found by the different algorithms. This shows that information criteria computed using BCCM have the potential to develop novel community detection algorithms that are particularly suited for applications where degree correction is crucial. However, the development of such algorithms is beyond the scope of this article and is left to future investigations.
Analysis of empirical graphsWe conclude this article by providing a comparison of the BCCM obtained by fitting the block structures detected by the five community detection algorithms described above on five different real-world networks. The results show that different algorithm performs better for different graphs, highlighting the non-trivial effect that degrees have on block structure and community detection in general.
We study five well-known graphs with heterogeneous characteristics and sizes. All graphs are multi-edge, and are freely available as dataset within the igraphdataR package. The first graph analyzed is rfid: hospital encounter network data. It consists of 32424 undirected edges between 75 individuals (Vanhems et al. 2013). The second graph analyzed is karate: Zachary's Karate Club. It consists of 231 undirected edges between 34 vertices (Zachary 1977). The third graph analyzed is UKfaculty: Friendship network of a UK university faculty. It consists of 3730 directed edges between 81 vertices (Nepusz et al. 2008). The fourth graph is USairports: US airport network of December 2010. It consists of 23473 directed edges between 755 airports (Von Mering et al. 2002). It has self-loops. The graph is plotted in Fig. 6, using the force-atlas2 layout (Jacomy et al. 2014). The four different plots are colored according to the block structures detected by four of the five algorithms (cluster_spinglass cannot be applied as the graph is disconnected). They are ordered by increasing AIC. From the visualization, we can see that the best block structure is the one who can separate three different blocks within the largest cluster of vertices (top of the visualizations). In particular, it is essential to note that the largest cluster consists of high- and low-degree vertices. If these vertices are belonging to the same block, the configuration model predicts then high-degree vertices should be connected by many edges (similarly to the first synthetic experiment described above). However, we observe then some of these high-degree vertices are separated and mainly connected to low-degree vertices. For this reason, block structures that can separate these high-degree vertices into different blocks rank higher than others. The fifth graph analyzed is enron: Enron Email Network. It consists of 125409 directed edges between 184 individuals (Priebe et al. 2005). It has self-loops.
USairports graph visualisation. USairports graph visualisation. The graph is plotted by means of the force-atlas2 layout with weighted edges, and the size of the vertices reflects their out-degrees. Only the largest connected component of the graph is shown. The visualisations clearly show the block structure that characterises this graph. The vertices in the four visualisations are coloured according to the labels detected applying four community detection algorithms, as described in Table 5. The visualisations are ordered from left to right according to the AIC of the BCCM fitted to observed graph according to the corresponding block structure. From left to right, we see the colours corresponding to the labels obtained from louvain, fast_greedy, infomap and label_propagation detection algorithms respectively. We highlight the fact that the ranking according to AIC corresponds approximately to the ability of the algorithms to detect the separation between high-degree (and low-degree) vertices within the largest cluster, at the top of the visualisations. The reason for this is that within the largest cluster there are clear deviations from what the configuration model predicts, i.e., high-degree vertices tend to connect to each other, and the best BCCMs captures more of these deviations
Table 5 Results of the fitting of BCCM to five real-world graphs, with vertex blocks given obtained from five different community detection algorithms
Each of these graphs has a clear block structure that could be detected. The different algorithms provide different results, both in the number of blocks detected and in the assignment of vertices. Ranking the different results employing the goodness-of-fit of BCCM fitted according to the different block partitions shows that the best results are not necessarily those with fewer or more blocks, nor those obtained from a specific algorithm, as the results change with the graph studied. The results of this analysis are provided in Table 5, where the smallest AICs and BICs for each graph are highlighted in bold, together with the algorithm that provides the smallest number of blocks. The algorithm that provides the largest number of blocks is highlighted in italic.
In this article we have presented a novel generative model for clustered graphs: the block-constrained configuration model. It generalizes the standard configuration model of random graphs by constraining edges within blocks, preserving degree distributions. The BCCM builds on the generalized hypergeometric ensemble of random graphs, by giving the propensity matrix Ω a block structure. The framework provided by gHypEG allows for a fast estimation of the parameters of the model. Moreover, thanks to the fact that the probability distribution underlying gHypEG is known, it allows for the generation of random realizations, as well as to the effortless computation of likelihoods, and hence various kinds of information criteria and goodness-of-fit measures, such as AIC and BIC.
There are many advantages of the formulation highlighted above. Firstly, the proposed model seamlessly applies to directed and undirected graphs with or without self-loops. Moreover, the BCCM preserves exactly the number of edges in the graph, avoiding the need for assuming an arbitrary edge generating process. This also allows dropping the assumption of independence between edge probabilities, which characterizes degree-corrected stochastic block models. Finally, model selection, facilitated by the gHypE framework, provides a natural method to quantify the optimal number of blocks needed to model given real-world graph. The statistical significance of a block structure can be studied performing likelihood-ratio tests (Casiraghi et al. 2016), or comparing information criteria such as AIC, BIC, or the description length of the estimated models. Furthermore, within the framework of generalized hypergeometric ensembles block-constrained configuration models can be extended, including heterogeneous properties of vertices or edges (see Casiraghi (2017)).
The more complicated expression and estimation of BCCM compared to DC-SBMs arises from dropping the assumption of independence between edge probabilities. However, thanks to the formulation provided in this article, BCCM is still practicable and can be applied to empirical graphs of various sizes. BCCM opens new routes to develop community detection algorithms suitable for applications where degree correction is particularly valuable, and where the assumption of an arbitrary edge generating process is not acceptable.
The datasets generated and/or analysed during the current study are available as a gitHub repository, https://github.com/gi0na/BCCM--Supporting-Material.git. The combinatorial matrices corresponding to Fig. 3b and c are included within the article (and its additional file(s)). A software implementation of the BCCM can be found as part of the R package ghypernet available at https://github.com/gi0na/ghypernet.git.
AIC:
Akaike information criterion
BCCM:
Bayesian information criterion
DC-SBM:
Degree-corrected stochastic block model
gHypEG:
Generalised hypergeometric ensemble of random graphs
SBM:
Stochastic block model
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The author thanks Frank Schweitzer for his support and valuable comments, and Laurence Brandenberger, Giacomo Vaccario and Vahan Nanumyan for useful discussions.
Chair of Systems Design, ETH Zürich, Weinbergstrasse 56/58, Zürich, 8092, Switzerland
Giona Casiraghi
The author read and approved the final manuscript.
Correspondence to Giona Casiraghi.
Casiraghi, G. The block-constrained configuration model. Appl Netw Sci 4, 123 (2019). https://doi.org/10.1007/s41109-019-0241-1
Block model
Random graphs
Configuration model
gHypEG | CommonCrawl |
communications biology
The reliability and heritability of cortical folds and their genetic correlations across hemispheres
Fabrizio Pizzagalli ORCID: orcid.org/0000-0003-4582-02241,
Guillaume Auzias ORCID: orcid.org/0000-0002-0414-56912,
Qifan Yang1,
Samuel R. Mathias3,4,
Joshua Faskowitz ORCID: orcid.org/0000-0003-1814-72061,
Joshua D. Boyd ORCID: orcid.org/0000-0002-6751-16281,
Armand Amini ORCID: orcid.org/0000-0002-5070-52661,
Denis Rivière ORCID: orcid.org/0000-0002-1902-22135,6,
Katie L. McMahon ORCID: orcid.org/0000-0002-6357-615X7,
Greig I. de Zubicaray ORCID: orcid.org/0000-0003-4506-05798,
Nicholas G. Martin ORCID: orcid.org/0000-0003-4069-80209,
Jean-François Mangin5,6,
David C. Glahn3,4,
John Blangero ORCID: orcid.org/0000-0001-6250-572310,
Margaret J. Wright ORCID: orcid.org/0000-0001-7133-497011,12,
Paul M. Thompson1,
Peter Kochunov13 &
Neda Jahanshad ORCID: orcid.org/0000-0003-4401-89501
Communications Biology volume 3, Article number: 510 (2020) Cite this article
Heritable quantitative trait
Cortical folds help drive the parcellation of the human cortex into functionally specific regions. Variations in the length, depth, width, and surface area of these sulcal landmarks have been associated with disease, and may be genetically mediated. Before estimating the heritability of sulcal variation, the extent to which these metrics can be reliably extracted from in-vivo MRI must be established. Using four independent test-retest datasets, we found high reliability across the brain (intraclass correlation interquartile range: 0.65–0.85). Heritability estimates were derived for three family-based cohorts using variance components analysis and pooled (total N > 3000); the overall sulcal heritability pattern was correlated to that derived for a large population cohort (N > 9000) calculated using genomic complex trait analysis. Overall, sulcal width was the most heritable metric, and earlier forming sulci showed higher heritability. The inter-hemispheric genetic correlations were high, yet select sulci showed incomplete pleiotropy, suggesting hemisphere-specific genetic influences.
Genetic drivers of brain structural and functional differences are important to identify as potential risk factors for heritable brain diseases, and targets for their treatment. Large-scale neuroimaging consortia, including the ENIGMA1 consortium, have identified common genetic variants that have small but significant associations with variations in brain morphology2. Studies have even identified genetic correlations between human brain structure and risk for disease3,4.
Enriched in neuronal cell bodies, the cortical gray matter plays an important role in human cognitive functions and behavior, including sensory perception and motor control5. Macroscale anatomical features of the human cortex can be reliably extracted from structural magnetic resonance imaging (MRI) scans, and among the most common are regional thickness and surface area measures. These MRI-based features show robust alterations in several neurological, neurodevelopmental, and psychiatric disorders6, and are influenced by both environmental and genetic variation7.
Gyrification of the cortical surface occurs in an orchestrated pattern8 during fetal life and into adolescence9, forming sulci (fissures) and gyri (ridges) in the cortical gray matter. The mechanisms of brain folding are not fully understood10,11, but the process is largely preserved among humans and nonhuman primates. The brain sulci delimit cortical areas with specific functionalities and are generally consistent across subjects12,13,14,15. The complexity and intersubject variability of brain gyrification are influenced by developmental, aging, and pathological processes, all of which are genetically and environmentally influenced16,17.
Large-scale neuroimaging studies have begun to discover common and rare genetic variants that contribute to brain variability as estimated using in vivo brain scans, such as MRI18; genome-wide association studies (GWAS) find that, as with other complex traits, individual common variants typically explain <1% of the population variance in MRI derived measures; still, common genetic factors account for a large fraction of the variance in aggregate2,19,20,21. Successful efforts to discover common variants that affect cortical structure require tens of thousands of scans, as well as independent samples for replication and generalization. Large-scale biobanks have amassed tens of thousands of MRI scans22. Even so, to replicate effects and ensure the generalizability of findings to other scanned populations, we must first assess that the brain measures are reliably extracted across a variety of possible MRI scanning paradigms. This reliability is the basis for pooling statistical effects across individual studies in multisite consortia such as ENIGMA1 and CHARGE23.
Sulcal-based morphometry provides in-depth analyses of the cortical fissures, or folds, as seen on MRI. Measures of sulcal morphometry—including length, depth, width, and surface area—among others—have been associated with brain maturation in adolescents24, neurodegenerative changes in the elderly24,25, and neuropsychiatric disorders such as schizophrenia26,27, bipolar disorder28, and autism spectrum disorder29; altered fissuration is also found in several genetic disorders, such as Williams syndrome30,31. Effects on sulcal patterns have been reported as being partially independent of those on cortical thickness or surface area24,32.
Effects on sulcal patterns have been reported being partially independent of those on cortical thickness or surface area24,32. Previous studies have investigated the genetics of cortical folds, but not across the full brain, and without ensuring the reliability of the measures themselves, or the heritability estimates. Kochunov et al.33 analyzed the effects of age on sulcal shape descriptors in a subset of 14 sulci finding wider sulci with older age in the adult human brain. The central sulcus has been the focus of many earlier publications34,35,36,37. Its depth has been reported to be highly heritable, with the degree of heritability varying along its profile37. In recent works, La Guen et al.38 studied the heritability of sulcal pits in the Human Connectome Project (HCP) and the genetic correlation of sulcal width across ten sulci in the UK Biobank39. The heritability of the depth, length, and surface area of primary sulci has been studied in baboons40. It has been suggested that deeper, earlier forming, sulci have higher heritability41, although this hypothesis has not been confirmed. The reliability of the findings across populations, and the extent to which heritability depends on the reliability of the measures, has not been investigated.
Here we: (a) estimate the reliability of four shape descriptors extracted from sulci across the whole brain; (b) evaluate heritability of these measures across four independent cohorts (three family-based cohorts and one cohort of unrelated participants); (c) determine the extent to which the heritability estimates depend on reliability; and (d) provide insights into the relationship between early forming sulci and higher heritability as well as cortical lateralization.
We performed an extensive reliability (N = 110) and heritability (N = 13,113) analysis. Reliability was estimated from four cohorts, totaling 110 participants (19–61 years of age, 47% females) who underwent two T1-weighted brain MRI scans across different brain imaging sessions. We included data sets for which we would expect minimal or no structural changes between scans, so we limited the analysis to healthy individuals aged 18–65, with an inter-scan interval < 90 days. See Table 1 for more details.
Table 1 Cohorts analyzed for the test–retest study.
We analyzed heritability in four independent cohorts, three with a family-based design and one using single-nucleotide polymorphism (SNP)-based heritability estimates. The cohorts included two twin-based samples (Queensland Twin Imaging study (QTIM) and HCP), one cohort of extended pedigrees (the Genetics of Brain Structure and Function; GOBS), and another of over 9000 largely unrelated individuals (the UK Biobank) (Table 2). Heritability estimates are population specific, but here our aim was to understand the heritability pattern across populations and estimate the degree to which genetic effects are consistently observed. We pooled information from all twim and family-based cohorts to estimate the generalized heritability values using meta- and mega-analytic methods42,43.
Table 2 Genetic analysis: demographics for the four cohorts analyzed in this study.
We estimated reliability and heritability for measures of each sulcus in the left and right hemispheres, separately. As there is limited evidence for genetic lateralization across most of the human brain44,45,46, we also evaluated the heritability estimates of the measures for each sulcus averaged across the two hemispheres. This may lead to more stable measurements and, if the bilateral measures are influenced by similar genetic factors, then more stable measures could lead to better powered genetic studies. We also assessed the genetic correlation between the measures across hemispheres. Sulci with limited genetic correlations between hemispheres may reveal novel insight into the brain's lateralization and identify key biomarkers for relating lateralized traits, such as language and handedness, to brain structure47.
Measurement reliability and its relationship to heritability
Supplementary Data 1 reports the sulcal nomenclature, including the abbreviation and full name for each sulcus. Reliability estimates may be found in Supplementary Data 2–4 for intraclass correlation (ICC) and Supplementary Data 5–8 for the bias evaluation; heritability estimates are reported in Supplementary Data 9–20 for the univariate analysis and Supplementary Data 21–24 for the bivariate analysis. We summarize the results below.
Intraclass correlation (ICC)
The ICC meta-analysis resulted in an ICC interquartile range of 0.59–0.82 Sulcal mean depth, width, and surface area showed similar reliability estimates, while the length showed lowest ICC (Table 3 and Fig. 1). For all descriptors other than mean depth, a higher fraction of sulci had "good" reliability, defined as ICC > 0.7548, after averaging metrics across corresponding left and right hemispheres, for all the descriptors (Fig. 1b) The fraction of sulci reaching ICC > 0.75 went from 24 (before averaging) to 39% (after averaging) for sulcal length, from 37 to 48% for the width, and from 48 to 59% for the surface area; mean depth remained consistant at 57%.
Table 3 Meta-analysis of ICC estimated from four independent cohorts for sulcal length, mean depth, width, and surface area.
Fig. 1: Intraclass correlation reliability estimates for sulcal length, depth, width and surface area.
a Sulcal-based meta-analysis of intraclass correlation (ICC) for bilaterally averaged sulcal measures (N = 110). Sulcal length showed generally "good" reproducibility, although no regions had ICC > 0.959. Mean depth showed "excellent" reproducibility (ICC > 0.9) for: the inferior frontal sulcus (S.F.inf.) and the superior frontal sulcus (S.F.sup.); sulcal width showed "excellent" reproducibility for: intraparietal sulcus (F.I.P.), superior postcentral intraparietal superior sulcus (F.I.P.Po.C.inf.), central sulcus (S.C.), superior postcentral sulcus (S.Po.C.sup.). Surface area showed "excellent" reproducibility for the central sulcus (S.C.), subcallosal sulcus (S.Call.), and the anterior occipito-temporal lateral sulcus (S.O.T.lat.ant.). b The intraclass correlation (ICC) for left, right, and bilaterally averaged sulcal length, mean depth, width, and surface area across the whole brain is plotted for the four test–retest cohorts. KKI showed the highest ICC across sulci.
The meta-analysis of ICC captures the consensus in the reliability across cohorts for each sulcus. Reliability measures depend to some extent on the cohort examined, or the scanning acquisition parameters. For example, for QTIM, which was collected at 4 T, the ICC is classified as "good" (ICC > 0.75) for the left sulcal surface area of the collateral sulcus (F.Coll.), but "poor or moderate" (ICC < 0.75) in OASIS for the same trait. Figure 1a shows the meta-analysis of ICC across the four cohorts, and highlights patterns for "good" and "excellent" (ICC > 0.9) reliability.
For a detailed breakdown of the ICC for measures of sulci morphometry per cohort, please see Supplementary Fig. 1 for the left hemisphere, Supplementary Fig. 2 for the right, and Supplementary Fig. 3 for bilaterally averaged measures.
For the complete meta-analyzed ICC results, please see Supplementary Figs. 4–7 for length, depth, width, and surface area respectively, all of which are tabulated in Supplementary Data 4.
For each sulcus, we averaged the reliability estimates across all four sulcal descriptors to find the most reliable sulci overall. The central sulcus (S.C.) gave the most reliable sulcal measures, followed by the median frontal sulcus (S.F.median), the intraparietal sulcus (F.I.P.), the occipito-temporal lateral sulcus (S.O.T.lat.ant.), the Sylvian sulcus (S.C.Sylvian), the sub-parietal sulcus (S.s.P.), the occipital lobe, and the superior temporal sulcus (S.T.s.) (Supplementary Fig. 8).
Bias (b)
We explored test–retest (TRT) consistency in terms of the "bias" (b, Eq. (4)), with Bland–Altman analyses. As in ref. 49, the generally low bias values showed high TRT consistency of sulcal shape measures (Supplementary Data 5). Bias values ≥ 0.1 are considered high, and were noted mainly for length estimates—e.g., for the length of the left and right anterior/posterior sub-central ramus of the lateral fissure (F.C.L.r.sc.ant./post.), and the length of the left and right insula (See Supplementary Data 6–8 for bias estimates across the left, right and bilaterally averaged sulcal metrics). Paralleling the higher ICC in bilaterally averaged measures, lower "bias" estimates were obtained with individual sulcal measures averaged across the left and right hemispheres (Supplementary Data 8).
ICC and bias (b) of bilaterally averaged sulcal metrics were significantly negatively correlated for all metrics except for length, in particular rlength = −0.11 [pval = 0.07], rmean-depth = −0.14 [pval = 0.02], rwidth = −0.25 [pval = 4.6 × 10−5], rsurface-area = −0.25 [pval = 1.2 × 10−5], suggesting, as expected, that a lower bias between test and retest measurements relates to higher reproducibility as estimated by ICC.
Heritability estimates for the cortical folding patterns
Across descriptors and sulci, heritability estimates (h2) showed a similar pattern across the three family-based cohorts, QTIM, HCP, GOBS (Supplementary Figs. 9–11); the GOBS cohort shows lower heritability, (h2 = 0.3 ± 0.1), compared to QTIM (h2 = 0.4 ± 0.1) and HCP (h2 = 0.4 ± 0.1); GOBS is a cohort with an extended pedigree design and a wide age range (18–85 years of age), while both HCP and QTIM are twin-based cohorts of young adults aged 25–35 years and 20–30 years, respectively.
The generalized heritability profile of cortical folding was obtained by meta-analyzing the estimates across these three independent family-design cohorts, and is highlighted in Fig. 2a. Aggregate heritability estimates were also calculated in a mega-analytic manner, where 3030 subjects from the family-based cohorts (QTIM, HCP, and GOBS) were pooled (after adjusting for covariates within cohort and normalizing across cohorts) before computing heritability estimates as in prior work42,43. As expected, we found similarities between meta- and mega-analysis derived heritability estimates as indicated by a significant Pearson's correlation between these two approaches (r ∼ 0.84, p = 10−3–10−7; see Supplementary Fig. 13 for more details). Individual heritability estimates, standard errors (SE), and p values for bilaterally averaged sulcal length, mean depth, width, and surface area are tabulated in Supplementary Data 9–18 for each cohort, and in Supplementary Data 19–20 for the meta- and mega-analyses.
Fig. 2: Heritability estimates.
Heritability estimates (h2) are mapped, for each bilaterally averaged sulcal descriptor. a The results of the inverse-variance weighted meta-analysis of the heritability estimates across three family-based cohorts QTIM, HCP, and GOBS highlight an overall heritability profile across 3030 individuals. b Heritability estimates (h2) calculated from sulcal features extracted from MRI scans of 10,083 unrelated individuals scanned as part of the UK Biobank were calculated using the genome-wide complex trait analysis (GCTA) package. The regional sulcal metrics that were found to be significantly heritable in the large population sample largely overlap with those found to be most highly heritable across the family-based studies. We highlight only regions that had significant heritability estimates in sulci that had an ICC > 0.75 (see Supplementary Data 2–4 for sulcal-based values of ICC). Significant regions survived Bonferroni correction for multiple comparisons across all bilateral traits and regions (p < 0.05/(61 × 4)); darker red colors indicate higher heritability estimates. The left hemisphere was used for visualization purposes.
For many sulcal features in the UK Biobank, the SNP-based heritability estimates were ~25% of the estimates derived from the family-based studies (h2 = 0.2 ± 0.1; Fig. 2b). The heritability estimates for the UK Biobank are reported in Supplementary Data 18.
Across the cortex, the global sulcal descriptors were significantly heritable for all cohorts. The patterns of heritability estimates were largely coherent between the family-based and large-scale population studies. The width was the most heritable measurement, while the length was the least, showing significant heritability estimates for only sparse regions of the cortex. The heritability of sulcal length was more frequently significant when not adjusting for ICV; we find minimal differences in the overall h2 estimates for sulcal depth and width before and after covarying for ICV (Supplementary Fig. 14).
The overall meta-analyzed reliability was significantly correlated with the heritability estimates meta-analyzed across the family-based cohorts: r = 0.36 (pval = 1 × 10−7) for sulcal length, r = 0.31 (pval = 4.1 × 10−6) for mean depth, r = 0.26 (pval = 7 × 10−5) for sulcal width, and r = 0.25 (pval = 1 × 10−4) for surface area (Supplementary Fig. 15); the reliability estimates were also correlated with heritability estimates in the UK Biobank for mean depth (r = 0.43, pval = 2 × 10−3) and sulcal width (r = 0.38, pval = 4 × 10−3) (Supplementary Fig. 16).
A few bilaterally averaged sulcal regions and metrics with "poor" reliability (ICC < 0.75) showed significant heritability estimates. These included the length of the parieto-occipital fissure (F.P.O.) [ICC = 0.66, h2 = 0.18 (pval = 1 × 10−5)], the mean depth of the ascending ramus of the lateral fissure (F.C.L.r.asc.) [ICC = 0.74, h2 = 0.2 (pval = 2.2 × 10−6)], the surface area of the anterior inferior frontal sulcus (S.F.inf.ant.) [ICC = 0.65, h2 = 0.17 (pval = 4.7 × 10−6)], and the width of the calloso-marginal ramus of the lateral fissure (F.C.M.ant.) [ICC = 0.63, h2 = 0.34 (pval = 1 × 10−16)] (Supplementary Data 4 and 19). For UK Biobank, the length of S.T.pol. [ICC = 0.70, h2 = 0.14 (pval = 6 × 10−5)], the width and the surface area for the insula [ICC = 0.65, h2 = 0.14 (pval = 2.6 × 10−5)] and [ICC = 0.65, h2 = 0.16 (pval = 3.8 × 10−6)], respectively (Supplementary Data 18).
The heritability estimates for the global measures (i.e., the sum across sulci) of sulcal length, mean depth, width, and surface area (covarying for ICV, age, and sex variables) are also reported in Supplementary Fig. 17. QTIM, HCP, and GOBS showed similar trends across descriptors and hemispheres; only QTIM had generally higher heritability estimates for sulci in the right hemisphere compared to those in the left (paired t-test: pval = 1.5 × 10−10).
Thirty-three percent (36% for mega-analysis) of the total number of bilaterally averaged sulci showed significant h2 for sulcal length, 57% (59% for mega-analysis) for mean depth, 67% (65% for mega-analysis) for width, and 62% (60% for mega-analysis) for the surface area. Six sulci were significantly heritable for only one of the four descriptors (one for mega-analysis). No sulcus show significant heritability for length only. Sulci that were significantly heritable across descriptors included the intraparietal sulcus, occipital lobe, subcallosal sulcus, internal frontal sulcus, orbital sulcus, anterior inferior temporal sulcus, and the polar temporal sulcus, among others; in total 15 sulci were significantly heritable across all four descriptors in the meta-analysis, and 19 for the mega-analysis (see Supplementary Data 19–20).
A significant Pearson's correlation was identified between heritability estimates averaged across sulcal descriptors and the approximate appearance of sulci (in weeks) during development50 (Supplementary Fig. 17, r = −0.62, p = 0.0025).
Genetic correlations between sulcal shape descriptors of the left and right cortical hemispheres
Genetic correlation across the hemispheres: Averaging brain-imaging derived traits across the left and right hemispheres, as above, has been shown to reduce noise due to measurement error in large scale, multi-cohort efforts2,20,51,52. Improvements in the signal-to-noise ratio may be essential for discovering single common genetic variants that explain < 1% of the overall population variability in a trait. However, by assessing left and right separately, we may be able to discover lateralized genetic effects, if they exist.
Bivariate variance components models confirmed that the genetic correlations between the same global sulcal descriptor on the left and right hemispheres of the brain were significant (ρG ∼ 0.92 ± 0.10) (Supplementary Data 21–24).
Fig. 3: Genetic correlations between sulcal shape descriptors of the left and right cortical hemispheres.
Left: the genetic correlations (ρG) between corresponding sulcal descriptors on the left and right hemispheres were assessed in three family based cohorts and meta-analyzed correlation values are mapped onto the brain. Right: the −log10 of the p value comparing the resulting genetic correlation to a perfect overlap (ρG = 1) are mapped. Significant values here suggest that the genetic components of variance may be partially unique across the left and right homologous sulcal metrics; i.e, despite a genetic correlation between hemispheres, lateralized genetic effects may be detectable. Sulci are mapped to the left hemisphere for visualization purposes.
The genetic correlation (ρG) between left and right homologous regions was computed for sulcal metrics that showed both "good" reliability estimates (ICC > 0.75) and significant univariate heritability estimates for both the left and right metrics; (Fig. 3, Supplementary Data 21–24). The genetic correlations between the left and right sulcal metrics was generally highest for the sulcal width metric. The width of the central sulcus, the inferior frontal sulcus, intermediate frontal sulcus, superior frontal sulcus, posterior lateral sulcus, the superior postcentral intraparietal superior sulcus, and the intraparietal sulcus, and the surface of the occipital lobe showed significant genetic correlations across all tested cohorts.
Two sets of p values are obtained when performing genetic correlations with bivariate variance components models: a more traditional p value comparing the correlation to the null hypothesis of no correlation (ρG = 0), and another p value comparing the genetic correlation to the hypothesis of a perfect overlap (ρG = 1), assessing the difference between the genetic correlation obtained and a correlation of 1. Bonferroni correction for the more traditional p values was conducted by correcting for the number of traits tested so p < 0.05/[N], where N = 11 heritable sulci with ICC > 0.75 for length, +31 for mean depth +36 for width +37 for surface area, for a total of 115 traits. The second set of p values comparing the genetic correlation (ρG) between hemisphere homologs to the indistinguishable value of 1 are listed in Supplementary Data 24, and the −log10(p values) of those significantly different than 1 are mapped in Fig. 3. For these regions, the 95% confidence interval surrounding the correlation estimate did not contain 1. These sulci represent regions and descriptors that may have diverging genetic influences across hemispheres. The meta-analysis revealed evidence for lateralized genetic effects in sulcal width or surface area of the occipital lobe, the intraparietal lobe, the median frontal sulcus, the intermediate left frontal sulcus, and the collateral sulcus; no evidence for lateralization of genetic influences was detected with either sulcal length or depth metrics.
Phenotypic correlations (ρP) between the left and right indices were on average less than 0.5 in each cohort (Supplementary Fig. 18). Sulcal width showed the highest (ρP = 0.38 ± 0.15) meta-analyzed correlation between left and right homologs compared to the other sulcal descriptors (0.29 ± 0.07 for sulcal length, 0.30 ± 0.11 for mean depth, and 0.33 ± 0.12 for surface area).
Our study has four main findings: (1) many of the sulci common across individuals were reliably extracted across a variety of MRI acquisition parameters, with some sulcal shape descriptors being more reliable than others; (2) cortical folding patterns were highly heritable and sulcal shape descriptors such as sulcal width may be promising phenotypes for genetic analysis of cortical gyrification; (3) the proportion of variance attributed to additive genetic factors varied regionally, with the earlier forming sulci having higher heritability estimates than later forming sulci; (4) incomplete pleotropy was identified between select left and right sulcal descriptors, suggesting sulcal analyses may provide insights into genetic factors underlying the lateralization of brain structure.
Cortical sulci may serve as prominent landmarks for identifying homologous functional regions across individuals35,36. BrainVISA offers the ability to automatically extract and characterize the sulci at high spatial resolutions, by segmenting and labeling 123 sulci across the cerebral cortex. Here, we analyzed four sulcal shape descriptors: length, mean depth, width, and surface area. Sulcal length has been associated with neurodevelopmental processes29,53,54, while sulcal depth and width have been correlated with aging and neurodegenerative processes33,55,56,57. Sulcal surface area represents a combination of depth, width and length features.
A primary goal of this work was to identify the sulci and the corresponding shape metrics that may be reliably extracted irrespective of the specific MRI scanner or scan acquisition protocol, to ensure a globally viable trait for disease related biomarker and genetic association analyses. Poor reliability may be attributable to measurement errors, which could lead to a ceiling effect on heritability estimates. This is because highly heritable traits can only be detected if the traits are robustly measured58 and low reliability could lead to an underestimation of the true heritability59. While heritability is a population-specific estimate, one main goal of imaging genetics consortia such as ENIGMA1 and CHARGE23 is to identify genetic variants that affect brain structure and function in populations around the world. Therefore, it is of utmost importance to ensure that measures are reliably extracted across different data sets, and furthermore, are heritable across different populations. Even beyond imaging genetics, the reliability of the measurements and the reproducibility of any set of results are essential for reproducible science at large.
Here we identified the most reliable sulcal regions using test-retest (TRT) data from four cohorts with independent samples and different scanning protocols to ensure the robustness of results. We assessed bias, a subject-based index of consistency49 as well as ICC, which compares the within-subject variance to the between-subjects variance. ICC may be affected by the homogeneity of the population under study; when variability in the population is low, for example, if age range is limited, then lower ICC values may be expected, while bias would be unaffected. Our results show high consistency between test and retest ("bias" < 0.149 on average). Furthermore, in considering the number of sulci that had ICC estimates greater than 0.9, sulcal width was the most reliable metric among the descriptors analyzed. Although some visual quality control was conducted on individual sulcal extractions, we did not ensure the anatomical validity of the entire set of sulcal labels for each of the individual MRI scans used in this study. Our reliability results are therefore more a reflection of methodological consistency, rather than anatomical accuracy.
A study examining the relationship between reproducibility and heritability of different brain structures in the QTIM cohort60 found a correlation between ICC and heritability, with a large percentage of traits showing low reliability (ICC < 0.75)60. Here we showed that most of the reliable sulcal shape descriptors were also highly heritable. This trend might be due to the lower variance across subjects for more robust anatomical regions, such as the central sulcus, which are easier to identify with automated image processing pipelines and less prone to segmentation errors. However, even in regions with "excellent" reliability (ICC > 0.9), we identified a range of heritability estimates, suggesting that not all reliable traits are necessarily highly heritable59.
Many earlier works have focused exclusively on the central sulcus37,40,61. We have replicated findings of significant heritability in the central sulcus and further, showed that it is indeed the sulcus with the highest heritability estimate across the entire cortex. However, out of 61 total bilateral cortical sulci, it is only one of 34 that showed significant heritability estimates across all four shape descriptors.
Our results indicated significant heritability estimates for sulcal surface area and width in several medial frontal regions, partially confirming findings in ref. 40. Our results also confirmed prior findings of sulcal heritability in the temporal lobe62 and the corpus callosum area63 and are also in line with studies showing high estimated heritability in prefrontal and temporal lobes for cortical thickness and surface area64,65,66,67,68,69,70, especially for sulcal mean depth and width.
The sulcal descriptors identified as being heritable in this work may serve as phenotypes for large-scale genome-wide association studies, or GWAS, enhancing our ability to identify specific genomic variants that influence brain structure and disease risk. These reliable and heritable sulcal measures may also serve as biomarkers for understanding genetically mediated brain disorders. The significant correlation identified between heritability estimates averaged across sulcal descriptors and the appearance of sulci (in weeks) during development50 implies that sulci appearing early in brain development71,72, including the central sulcus, Sylvian fissure, parieto-occipital lobes, and superior temporal sulcus50 may be under stronger genetic control. However, some regions including the frontal lobe and the temporal sulcus also had high heritability, even though these regions are reported to develop later72, suggesting more work is needed to identify the developmental role in the regional genetic architecture.
Across three independent family-based cohorts, QTIM—an Australian cohort of young adult twins and siblings—HCP, a North American cohort of twins and siblings, and GOBS—a Mexican-American cohort of extended pedigrees, we found similar patterns of heritability for four descriptors of sulcal morphometry. Globally, we found sulcal heritability estimates of ~0.3–0.4, similar to estimates in other species, including Papio baboons40. Heritability estimates from GOBS were lower than for QTIM or HCP, as may be expected for an extended pedigree design when compared to twin designs73. It has also been proposed that higher image quality, and therefore lower measurement error, could lead to higher heritability estimates74. GOBS and HCP MRI volumes were acquired with a 3T scanner and HCP has higher spatial resolution compared to GOBS. QTIM was acquired with slightly lower spatial resolution but at higher magnetic field strength (4T). Further analyses will be needed to investigate how the signal-to-noise estimates (SNR) vary across cohorts and how this affects heritability estimation74. SNP-based heritability estimated in the UK Biobank showed a similar h2 pattern (Supplementary Fig. 20) across the brain, but with lower h2 values compared to the family-based cohorts. This may be partially due to the "missing heritability" effect in the SNP-based heritability estimation75. We note that recent work has found less discrepancy between twin-based heritability estimates and those derived from large-scale population studies of approximately 20,000 individuals76, therefore, larger population samples may be required to better power our SNP-based heritability estimates and help determine the true extent of the missing heritability.
Apart from work by the ENIGMA Laterality group77, many published ENIGMA studies2,20,43,78 performed analyses on pooled bilateral measures of brain structure, averaging data from the left and right hemispheres. We indeed found that for most sulcal descriptors, averaging the sulcal measures across hemispheres provided more regions with reliable estimates, and more consistent heritability estimates across cohorts. The bivariate genetic analysis used to estimate the genetic correlation between left and right sulcal measures, further confirmed strong and significant genetic correlations between hemispheres.
A genetic correlation between measures across the right and left hemispheres indicates pleiotropy, suggesting that genetic influences underlying the structure and variability in the measures tends to overlap. In family-based studies, bivariate variance components analysis may be used to determine the genetic correlation between traits as in this work. When a significant genetic correlation is identified, the confidence interval around the genetic correlation often includes one, suggesting the underlying genetic influences of the measures were not statistically distinguished from each other. Incomplete pleiotropy is suggested when genetic correlations are significant, but the confidence intervals around the correlations do not include one. While in SNP-based genetic correlation models, incomplete pleiotropy may be suggested over complete pleiotropy in the presence of measurement error, in a bivariate polygenic model, measurement error falls into the environmental component of variance and the environmental correlation, and therefore, does not influence the maximum-likelihood estimate of the genetic correlation; i.e, measurement error makes it more difficult to reject the null hypothesis that the genetic correlation is one. Features that exhibit unique genetic influences in one hemisphere may reveal insights into the biological causes of brain lateralization that may play an important role in neurodevelopmental or psychiatric disorders. Evidence of less genetic control in the left hemisphere has been found in refs. 50,62, where the authors found higher cortical gyrification complexity in the right hemisphere at an early development stage.
Here we found incomplete pleiotropy, or suggested asymmetrical genetic influences, in the frontal lobe (width). This may relate to disorder-specific abnormalities seen in brain folding patterns, for example, as reported in a postmortem study on schizophrenia79. Incomplete pleiotropy was also detected in sulci of the occipital lobe, a highly polygenic region80; structural abnormalities in this region have been associated with Parkinson's disease81,82, posterior cortical atrophy, a disorder causing visual dysfunction, and logopenic aphasia83.
Some regions that showed this suggested lateralization of genetic effects for sulcal descriptors, showed the same effect for other measures extracted from the cortex; for example, the effect seen with the sulcal surface area of the collateral fissure was also detected with the corresponding gyral surface area (Supplementary Fig. 21). However, for the occipital lobe, we found evidence for lateralization of genetic effects with sulcal width, but not with either cortical thickness or surface area of corresponding gyri. This suggests that sulcal descriptors may offer additional insights into cortical development and lateralization, beyond more commonly analyzed metrics of gyral morphometry. As a larger than expected portion of our study population was right-handed, our findings may be biased towards right handed individuals and may not be fully representive; the degree of cerebral volume asymmetry has been shown to be lower for non-right-handed twins than right-handed pairs62 and future investigations focusing on the genetics of brain gyrification and lateralization across handedness are needed to confirm these findings.
The genetic influences on brain cortical structure are regionally dependent, and differ according to the metric, or descriptor, being evaluated. For example, the genetic correlation between average cortical thickness and total surface area has been shown to be weak and negative, with largely different genetic compositions32. Different metrics are often used to describe and quantify different biological processes such as those such as length and surface area with potentially more developmental orgins, and others including sulcal width that may capture more degenerative processes. In nonhuman primates, brain cortical folding was also found to be influenced by genetic factors largely independent of those underlying brain volume84,85. Measuring cortical folding through sulcal-based morphometry could therefore highlight brain metrics beyond thickness and surface area, and may complement these more traditional measures to reveal a deeper understanding of the processes underlying variation in human brain structure, its association with disease and the underlying genetic risk factors. Our findings suggest that conducting a GWAS of sulcal features may be particularly informative for the sulcal width—the most heritable of the four tested metrics. Although for most sulci, the genetic components of variances were largely indistinguishable (i.e., highly correlated) across the two hemispheres, our results suggest that conducting a separate GWAS of sulcal measures in select frontal, temporal, and occipital regions may provide added insight into the biological mechanisms that drive hemispheric specialization. The discovery and replication of specific genetic influences on brain structure require very highly powered analyses, achievable through large-scale studies and collaboration. Harmonized imaging and genetic analysis protocols, rigorous quality assurance, reproducibility assessments, along with statistical rigor are vital in the collaborative endeavors such as those proposed by the ENIGMA consortium. To allow for a variety of such international collaborations, the customized MRI image processing protocol using and extending the BrainVISA toolkit as in this work, has been made freely available at: http://enigma.ini.usc.edu/protocols/imaging-protocols/.
Participants and MRI imaging
Queensland Twin Imaging study (QTIM): Brain MRI from 1008 right-handed participants86, 370 females and 638 males, were used in this study. This included 376 dizygotic (DZ) and 528 monozygotic (MZ) twins (one set of DZ triplets) and 104 siblings, with an average age of 22.7 ± 2.7 years [range: 18–30]. T1-weighted images were acquired on a 4T Bruker Medspec scanner with an inversion recovery rapid gradient echo sequence. Acquisition parameters were inversion/repetition/echo time (TI/TR/TE) = 700/1500/3.35 ms; flip angle = 8°; with an acquisition matrix of 256 × 256; voxel size = 0.94 × 0.90 × 0.94 mm3.
Human Connectome Project (HCP): 816 participants87, 362 females and 454 males, average age 29.1 ± 3.5 years [range: 22–36]. These included 412 siblings, 205 DZ and 199 MZ twins, including triplets. T1-weighted images were acquired using a 3T Siemens scanner. MRI parameters: (TI/TR/TE) = 1000/2400/2.14 ms; flip angle = 8°; voxel size = 0.7 mm isotropic; acquisition matrix = 224 × 224. The subset of TRT scans includes all right-handed subjects.
Genetics of Brain Structure and Function (GOBS): A total of 1205 individuals of Mexican-American ancestry from extended pedigrees (71 families, average size 14.9 [1–87] people) were included in the analysis. Sixty-four percent of the participants were female and ranged in age from 18 to 97 (mean ± SD: 47.1 ± 14.2) years. Individuals in this cohort have actively participated in research for over 18 years and were randomly selected from the community with the constraints that they are of Mexican-American ancestry, part of a large family, and live within the San Antonio, Texas region. Imaging data were acquired at the UTHSCSA Research Imaging Center on a Siemens 3T Trio scanner (Siemens, Erlangen, Germany). Isotropic (800 µm) 3D Turbo-flash T1-weighted images were acquired with the following parameters: TE/TR/TI = 3.04/2100/785 ms, flip angle = 13°. Seven images were acquired consecutively using this protocol for each subject and the images were then co-registered and averaged to increase the signal-to-noise ratio and reduce motion artifacts88.
UK Biobank: Analyses were conducted on the 2017 imputed genotypes restricted to variants present in the Haplotype Reference Consortium89,90. UK Biobank bulk imaging data were made available under application #11559 in July 2017. We analyzed 10,083 participants (4807 females), mean age = 62.4 ± 7.3 years [range: 45–79]. Voxel matrix: 1.0 × 1.0 × 1.0 mm—acquisition matrix: 208 × 256 × 256. 3D MP-RAGE, TI/TR = 880/2000 ms, sagittal orientation, in-plane acceleration factor = 2. Raw MRI data were processed using the ENIGMA FreeSurfer and sulcal analysis protocols. Following processing, all images were visually inspected for quality control of FreeSurfer gray/white matter classifications. For all subjects, the central sulcus segmented and labeled by BrainVISA was visually assessed for labeling accuracy.
KKI (Kennedy Krieger Institute—Multi-Modal MRI Reproducibility Resource): 21 healthy volunteers with no history of neurological conditions (10 F, 22–61 years old) were recruited. All data were acquired using a 3T MRI scanner (Achieva, Philips Healthcare, Best, The Netherlands) with body coil excitation and an eight-channel phased array SENSitivity Encoding (SENSE) head-coil for reception. All scans were completed during a 2-week interval. The resulting data set consisted of 42 "1-h" sessions of 21 individuals. MP-RAGE T1-weighted scans were acquired with a 3D inversion recovery sequence: (TR/TE/TI = 6.7/3.1/842 ms) with a 1.0 × 1.0 × 1.2 mm3 resolution over a field of view of 240 × 204 × 256 mm acquired in the sagittal plane. The SENSE acceleration factor was 2 in the right–left direction. Multi-shot fast gradient echo (TFE factor = 240) was used with a 3-s shot interval and the turbo direction being in the slice direction (right–left). The flip angle was 8°. No fat saturation was employed91, https://www.nitrc.org/projects/multimodal/.
OASIS: This TRT reliability data set contains 20 right-handed subjects (19–34 years old) without dementia imaged on a subsequent visit within 90 days of their initial session. MP-RAGE T1-weighted scans were acquired on a 1.5-T Vision scanner (Siemens, Erlangen, Germany): (TR/TE/TI = 9.7/4.0/20 ms) with an in-plane resolution of 1.0 × 1.0 × mm2 resolution over a FOV of 256 × 256 mm acquired in the sagittal plane. Thickness/gap = 1.25/0 mm; flip angle = 10° (https://www.oasis-brains.org/)92.
MRI image processing and sulcal extraction. Anatomical images (T1-weighted) were corrected for intensity inhomogeneities and segmented into gray and white matter tissues using FreeSurfer (http://surfer.nmr.mgh.harvard.edu/); segmentations and regional labels were quality controlled using ENIGMA protocols for outlier detection and visual inspection (http://enigma.ini.usc.edu/protocols/imaging-protocols/). BrainVISA (http://brainvisa.info) was run for sulcal extraction, identification, and sulcal-based morphometry. Morphologist 2015, an image processing pipeline included in BrainVISA, was used to quantify sulcal parameters. Briefly, the Morphologist 2015 segmentation pipeline computes left and right hemisphere masks, performs gray and white matter classification, reconstructs a gray/white surface and a spherical triangulation of the external cortical surface, independently for both hemispheres. Sulcal labeling has been performed using BrainVISA suite which implements the algorithm fully described in the cited paper by Perrot et al.93. It relies on a probabilistic atlas of sulci. The sulcal recognition is made by combining localization and shape information. The atlas is described in detail and freely accessible here: http://brainvisa.info/web/morphologist.html and can be visualized online here: http://brainvisa.info/web/webgl_demo/webgl.html.
To improve sulcal extraction and build on current protocols used by hundreds of collaborators within ENIGMA, quality controlled FreeSurfer outputs (orig.mgz, ribbon.mgz, and talairach.auto) were directly imported into the pipeline to avoid recomputing several steps, including intensity inhomogeneity correction and gray/white matter classification. Sulci were then automatically labeled according to a predefined anatomical nomenclature of 62 sulcal labels for the left hemisphere and 61 sulcal labels for the right hemisphere94,95. The protocol developed for this work is available at http://enigma.ini.usc.edu/protocols/imaging-protocols/ (ENIGMA-Sulci).
Sulci descriptors and quality control. Analyzing the shape of the cortex through sulcal-based morphometry allows us to quantify the geometry of a sulcus in terms of several distinct and complementary descriptors, consisting of length, mean depth, surface area, and width (or fold opening) of all extracted and labeled sulci. Cortical thickness and surface area have both been found to be moderately to highly heritable, yet with largely independent and even negatively correlated genetic influences7,80,96. Cortical thickness, surface area, and folding tend to exhibit different age-related trajectories97,98. In particular, cortical thickness represents the laminar organization of the cerebral cortex, which contains about 14 billion neurons99. Each of the layers forming the cortex100 has a different cellular organization, mostly distinguished on the basis of pyramidal cells in the various laminae100. Surface area may reflect the number of radial columns perpendicular to the pial surface98 and sulcal morphometry may additionally relate to the microstructure of the neuronal sheets and to the local axonal connectivity within a cortical region, which may influence the degree of folding84.
The length of a sulcus is measured in millimeters as the geodesic length of the junction between a sulcus and the hull of the brain. The mean depth corresponds to the average of the depth across all the vertices along the bottom of a sulcus (the depth of a vertex located at the bottom of a sulcus is defined as the geodesic distance along the sulcus to the brain hull). The surface area is the total area of the sulcal surface. The enclosed cerebrospinal fluid (CSF) volume divided by the sulcal surface area gives the width, a gross approximation of the average width of the CSF in the fold61 (see Supplementary Fig. 32 for a representation of sulcal shape descriptors).
To further quality control the extracted sulcal measures and identify subjects whose sulci were not optimally identified, we consider as outliers those subjects showing abnormal values for at least one of the descriptors for each sulcus. That is, for a given sulcus, the z-score across subjects is computed for each descriptor. The set of subjects showing an absolute z-score > 2.5 for one or more descriptors was discarded from further analysis101. Therefore, if the length of the central sulcus for a given subject was an outlier but width, depth, and surface area were not, that subject's central sulcus was removed from further evaluation; this ensured that the same set of subjects were used for all analyses across descriptors. This led to discarding ∼3% of subjects for each sulcus.
Statistics and reproducibility
Univariate and bivariate quantitative genetic analyses: The relative influences of genetic and environmental factors on human traits can be estimated by modeling the known genetic relationship between individuals and relating it to observed covariance in measured traits; in twin studies, MZ twin pairs—who typically share all their common genetic variants—are compared to DZ twin pairs, who share, on average, 50%. The same principle can be used for extended pedigrees, in which many individuals have varying degrees of relatedness. Here, we used both twins and extended pedigrees to estimate the heritability of these in-depth cortical sulcal measures. For a given cohort of participants, the narrow-sense heritability (h2) is defined as the proportion of the observed variance in a trait (σ2p) that can be attributed to additive genetic factors (σ2g):
$$h^2 = {{\sigma _g^2} \over {\sigma _p^2}}.$$
Variance components methods, implemented in the Sequential Oligogenic Linkage Analysis Routines (SOLAR) software package102, were used for all genetic analyses. Heritability (h2) is the proportion of total phenotypic variance accounted for by additive genetic factors and is assessed by contrasting the observed phenotypic covariance matrix with the covariance matrix predicted by kinship. High heritability indicates that the covariance of a trait is greater among more closely related (genetically similar) individuals; here, for example, MZ twins as compared to DZ twins and siblings. Using SOLAR-ECLIPSE imaging genetics tools (http://www.nitrc.org/projects/se_linux)102, we investigated the heritability profile of four sulcal descriptors for sulci across the whole brain: 62 on the left and 61 on the right hemisphere.
Prior to testing for the significance of heritability, sulcal descriptor values for each individual are adjusted for a series of covariates. We estimated the influence of specific variables (additive genetic variation and covariates including intracranial volume, sex, age, age2, age × sex interaction, age2 × sex interaction) to calculate the sulcal trait heritability and its significance (p value) for accounting for a component of each trait's variance within this population.
The significance threshold for heritability analysis of individual sulci was set to be p ≤ (0.05/m*4), where m = 61 (number of bilateral sulci), and the times 4 corresponding to the number of shape descriptors assessed. We set m = 123 when left and right sulcal heritability were estimated separately. This reduced the probability of Type 1 errors associated with multiple measurements.
For bivariate genetic correlation estimates, classical quantitative genetic models were used to partition the phenotypic correlation (ρP) between the left and the corresponding right sulcal measures into the genetic (ρG), and a unique environmental (ρE) components, for each pair of traits. Just as with the univariate model, the bivariate phenotype of an individual is modeled as a linear function of kinship coefficients that express relatedness among all individuals within the cohort (MZ twins share all their additive genetic information and DZ twins and siblings share on average 50%). The significance of ρG and ρE was estimated from the likelihood ratio test when comparing the model to ones where the correlation components are constrained to be 0102,103,104. This estimates ρG and ρE and their standard error (SE). The significance of these coefficients is determined by a z-test of their difference from 0. If ρG differs significantly from 0, then a significant proportion of the traits' covariance is influenced by shared genetic factors.
In this case, we tested another model where the genetic correlation factor ρG is fixed to 1. Fixing ρG to 1 suggests that the additive genetic components comprising the two traits overlap completely, and there is no detectable unique genetic composition for the individual traits. Once again, the log-likelihood of this model is compared to one where the parameters are freely optimized. If ρG is not found to significantly differ from 1, then we cannot reject the hypothesis that both heritable traits are driven by the same set of genetic factors. If ρG is significantly different from 0 and significantly different from 1, then the traits share a significant portion of their variance, however, each is also likely to be partially driven by a unique set of genetic factors.
Some considerations should be made regarding the measurement error of the traits analyzed here: ρG is the correlation between the latent genetic effects on the two traits irrespective of the proportion of phenotypic variance these latent effects explain (i.e., heritability). Measurement error, which is uncorrelated between individuals regardless of their relatedness, falls into the environmental component and environmental correlations. Measurement error therefore influences h2, ρE, ρP, but not ρG.
In practice, measurement error does make ρG harder to estimate, because low heritability means that the underlying genetic effects cannot be estimated with precision. This causes the SE of the ρG estimate to increase, but critically, does not change its maximum-likelihood estimate systematically. So measurement error makes it harder to reject the null hypothesis that ρG = 1.
Moreover, the bivariate polygenic model used here to estimate the left–right genetic correlation is a linear function of laterality (L–R). Indeed, the genetic variance of L–R is:
$$\sigma _g^2\left( L \right) + \sigma _g^2\left( R \right) - 2 \times \rho _g \times \sqrt {\sigma _g^2\left( L \right) \times \sigma _g^2\left( R \right)},$$
where \(\sigma _g^2\left( L \right)\) and \(\sigma _g^2\left( R \right)\) are the genetic variance for the left and right traits. The phenotypic variance is similarly defined so that the heritability of L–R can be obtained. But if L–R shows significant heritability, it could be because: (1) genetic overlap is incomplete and/or (2) L and R have unequal genetic variances. So studying laterality is not recommended here because (1) and (2) are confounded.
Meta-analysis of additive genetic variance
Meta-analysis calculates weighted mean heritability (h2) and SE estimates based on measurements from individual cohorts42,43. We weighted the heritability estimate from each cohort by the heritability SE, as extracted from the variance component model of SOLAR. The heritability weighted by SE42,43 is:
$$h_{{\rm{MA - SE}}}^2\left( S \right) = \frac{{\mathop {\sum }\nolimits_j se_j^{ - 2} \times h_j^2(S)}}{{\mathop {\sum }\nolimits_j se_j^{ - 2}}},$$
where S = 1 to Ns indexes the sulci and j = 1,2,3 indexes the cohorts.
Mega-analysis of additive genetic variance
While meta-analyses compute first the heritability independently for each cohort and then combine the results, mega-analyses combine first different cohorts and then run a single computation for heritability evaluation. We use a program (polyclass), developed for SOLAR105 for mega-analysis of heritability on sulci descriptors43,106. This function fits the model after combining the pedigrees of QTIM, HCP, and GOBS into a single pedigree (for more details see refs. 42,43).
Meta-analysis of genetic correlation
A meta-analysis of genetic correlation is calculated weighting the genetic correlation computed for each cohort by its sample size:
$$\rho _{G - {\rm{MA}}}\left( S \right) = \frac{{\mathop {\sum }\nolimits_j \rho _{G_j}^2(S) \times N_{\rm{sub}}}}{{\mathop {\sum }\nolimits_j N_{{\rm{sub}}(j)}}},$$
where S = 1 to Ns indexes the sulci, j = 1, 2, 3 indexes the cohorts, and \(N_{{\rm{sub}}(j)}\) is the sample size of cohort j.
To combine p values in a meta-analysis, we used the Edgington's method that represents a compromise between methods more sensitive to largest p values (e.g., Pearson's method) and methods more sensitive to smallest p values (e.g., Fisher's method)107,108:
$${\rm{Meta}}\,p {\hbox{-}} {\rm{value}} = \frac{{S^k}}{{k!}} - \left( {k - 1} \right)C1\frac{{\left( {S - 1} \right)^k}}{{k!}} + \left( {k - 2} \right)C2\frac{{\left( {S - 2} \right)^k}}{{k!}},$$
where S is the sum of o values and k the number of tests (i.e., k = 3 cohorts in our study). The corrective additional terms are used if the number subtracted from S in the numerator is <S. All the p values in the meta-analyses estimated were computed using this method.
SNP-based heritability analysis
We used genome-wide complex trait analysis (GCTA)109 to estimate the heritability from the individual genotypes. Genotypes on the autosomal chromosomes were used to calculate the genetic relationship matrix with GCTA109. Heritability was calculated using a linear mixed model, with age, sex, ICV, and the first four genetic components from multidimensional scaling analysis as fixed covariates. We also covaried for the presence of any diagnosed neurological or psychiatric disorder. In our analysis, we excluded participants with non-European ancestry, missing genotypes, or phenotypes, and mismatched sex information.
Reliability analysis
Sulcal measurement reliability: To evaluate the reliability of the sulcal shape descriptors, we analyzed their variability, or reproducibility error, across the TRT sessions for each of the four TRT cohorts. For each MRI scan there are several sources of variability, including variability from hydration status, variability due to slightly different acquisitions in the two sessions (head position change in the scanner, motion artifacts, scanner instability, etc.), and finally variability due to the imaging processing methods themselves.
There could also be variability in the reliability estimates depending on the type of MRI system used (vendor, model, and acquisition parameters), so it is important to address the issue of reliability across a variety of platforms. We used two indices of reliability: (1) the dimensionless measure of absolute percent bias of descriptor, b (sulcal length, mean depth, width, and surface area) of a sulcus with respect to its average and (2) the ICC coefficient. b is computed as follows:
$$b = 100 \times \frac{{{\rm{test - retest}}}}{{\left( {{\rm{test + retest}}} \right)/2}}.$$
The estimation of the means is more robust than the estimation of the variance from the signed differences, in particular for smaller sets of subjects. The distributions of sulcal measurement differences plotted the mean across sessions were examined with a Bland–Altman analysis110. These plots show the spread of data, the bias (i.e., mean difference), and the limits of agreement (±1.96 SD), and were used to confirm that the distributions were approximately symmetric around 0 and to check for possible outliers. While the ICC estimates the relation between within-subject variance and between-subjects variance, b offers a subject-based index that might be used to find outliers. If scan and rescan are perfectly reliable, b should be equal to 0. The cases where b is >0.1, as in ref. 49, are considered unreliable.
The ICC coefficient was computed to quantify the reproducibility for sulcal-based measurements. ICC is defined as follows:
$${\rm{ICC}} = \frac{{\sigma _{\rm{BS}}^2}}{{\sigma _{\rm{BS}}^2 + \sigma _{\rm{WS}}^2}},$$
providing an adequate relation of within-subject (σ2WS) and between-subject (σ2BS) variability111,112,113.
The ICC estimates the proportion of total variance that is accounted for by the σ2BS. Values below 0.4 are typically classified as "poor" reproducibility, between 0.4 and 0.75 as "fair to good," and higher values as "excellent" reproducibility48.
Equation (5) was used to estimate the ICC for each sulcal descriptor, independently for each cohort. The four cohorts were then combined into a meta-analysis (ICCMA−SE), similar to Eq. (1), in order to account for intra-site variability end to better estimate the sulcal reliability:
$${{\rm{ICC}}_{{\rm{MA}} - {\rm{SE}}}}(S) = \frac{{\mathop {\sum }\nolimits_j se_j^{ - 2} \times {\rm{ICC}}(S)}}{{\mathop {\sum }\nolimits_j se_j^{ - 2}}},$$
where j = 1, 2, 3, 4 indexes the cohorts. The SE was computed like SE = ICC/Z, where Z is obtained from a normal distribution knowing the p value. \({{\rm{ICC}}_{{\rm{MA}} - {\rm{SE}}}}\) was computed only if the cohort-based ICC computed with Eq. (5) was estimated for at least 3/4 cohorts.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
OASIS: the OASIS data are distributed to the greater scientific community under the Creative Commons Attribution 4.0 license. All data are available via www.oasis-brains.org92. KKI (Kennedy Krieger Institute—Multimodal MRI Reproducibility Resource): open access: https://www.nitrc.org/projects/multimodal/91. QTIM: data from the QTIM cohort used in this paper can be applied for by contacting M.J.W. ([email protected]). Access to data by qualified investigators are subject to scientific and ethical review. Summary results from cohort QTIM are available as part of the supplementary data52. HCP: family status and other potentially sensitive information are part of the Restricted Data that is available only to qualified investigators after signing the Restricted Data Use Terms. Open access data (all imaging data and most of the behavioral data) are available to those who register and agree to the Open Access Data Use Terms. Restricted data elements that could be potentially used to identify subjects include family structure (twin or non-twin status and number of siblings); birth order; age by year; handedness; ethnicity and race; body height, weight, and BMI; and a number of other categories. Each qualified investigator wanting to use restricted data must apply for access and agree to the Restricted Data Use Terms (https://humanconnectome.org/study/hcp-young-adult/data-use-terms)87. GOBS: data from the GOBS cohort used in this paper can be applied for by contacting D.C.G. ([email protected]) or J. Blangero ([email protected]). Access to data by qualified investigators are subject to scientific and ethical review and must comply with the European Union General Data Protection Regulations (GDPR)/all relevant guidelines. The completion of a material transfer agreement (MTA) signed by an institutional official will be required. Summary results from cohort GOBS are available as part of the supplementary data. UK Biobank: access to data from the UK Biobank can be obtained by approved scientists through application with UK Biobank (www.ukbiobank.ac.uk/researchers)90.
Code availability
The image processing protocol developed for this work is available at http://enigma.ini.usc.edu/protocols/imaging-protocols/ (ENIGMA-Sulci).
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This research was funded in part by NIH ENIGMA Center grant U54 EB020403, supported by the Big Data to Knowledge (BD2K) Centers of Excellence program funded by a cross-NIH initiative. Additional grant support was provided by: R01 AG059874, R01 MH117601, R01 MH121246, and P41 EB015922. QTIM was supported by NIH R01 HD050735 and the NHMRC 486682, Australia; GOBS: financial support for this study was provided by the National Institute of Mental Health grants MH078143 (PI: D.C.G.), MH078111 (PI: J. Blangero), and MH083824 (PI: D.C.G. and J. Blangero); HCP data were provided [in part] by the Human Connectome Project, WU-Minn Consortium (Principal Investigators: David Van Essen and Kamil Ugurbil; 1U54MH091657) funded by the 16 NIH Institutes and Centers that support the NIH Blueprint for Neuroscience Research and by the McDonnell Center for Systems Neuroscience at Washington University; UK Biobank: this research was conducted using the UK Biobank Resource under Application Number "11559"; BrainVISA's Morphologist software development received funding from the European Union's Horizon 2020 Framework Programme for Research and Innovation under Grant Agreement Nos. 720270 and 785907 (Human Brain ProjectSGA1 & SGA2), and by the FRM DIC20161236445. OASIS: Cross-Sectional: Principal Investigators: D. Marcus, R. Buckner, J. Csernansky, J. Morris; P50 AG05681, P01 AG03991, P01 AG026276, R01 AG021910, P20 MH071616, U24 RR021382. KKI was supported by NIH grants NCRR P41 RR015241 (Peter C.M. van Zijl), 1R01NS056307 (Jerry Prince), 1R21NS064534-01A109 (Bennett A. Landman/Jerry L. Prince), 1R03EB012461-01 (Bennett A. Landman). N.J. and P.M.T. are joint principal investigators for a research grant from Biogen, Inc. (Boston, USA) for processing of imaging data, some of which was used in this paper. No funding sources were involved in the design, analysis or outcomes of the study. We thank Anderson M. Winkler for comments and corrections on our initial biorxiv preprint, and Alessandra Griffa for helping to map the FreeSurfer results.
Imaging Genetics Center, Mark and Mary Stevens Neuroimaging and Informatics Institute, Keck School of Medicine of USC, Marina del Rey, CA, USA
Fabrizio Pizzagalli, Qifan Yang, Joshua Faskowitz, Joshua D. Boyd, Armand Amini, Paul M. Thompson & Neda Jahanshad
Institut de Neurosciences de la Timone, UMR7289, Aix-Marseille Université & CNRS, Marseille, France
Guillaume Auzias
Department of Psychiatry, Boston Children's Hospital and Harvard Medical School, Boston, MA, USA
Samuel R. Mathias & David C. Glahn
Yale University School of Medicine, New Haven, CT, USA
Université Paris-Saclay, CEA, CNRS, Neurospin, Baobab, Gif-sur-Yvette, France
Denis Rivière & Jean-François Mangin
CATI, Multicenter Neuroimaging Platform, Paris, France
School of Clinical Sciences and Institute of Health and Biomedical Innovation, Queensland University of Technology, Brisbane, QLD 4000, Australia
Katie L. McMahon
Faculty of Health, Queensland University of Technology (QUT), Brisbane, QLD, 4000, Australia
Greig I. de Zubicaray
QIMR Berghofer Medical Research Institute, Brisbane, QLD, Australia
Nicholas G. Martin
South Texas Diabetes and Obesity Institute, University of Texas Rio Grande Valley School of Medicine, Brownsville, TX, USA
John Blangero
Queensland Brain Institute, University of Queensland, Brisbane, QLD, 4072, Australia
Margaret J. Wright
Centre for Advanced Imaging, University of Queensland, Brisbane, QLD, 4072, Australia
Maryland Psychiatric Research Center, Department of Psychiatry, University of Maryland School of Medicine, Baltimore, MD, USA
Peter Kochunov
Fabrizio Pizzagalli
Qifan Yang
Samuel R. Mathias
Joshua Faskowitz
Joshua D. Boyd
Armand Amini
Denis Rivière
Jean-François Mangin
David C. Glahn
Paul M. Thompson
Neda Jahanshad
F.P. and N.J. designed the research, analyzed and reviewed the data and code, and drafted the initial paper; G.A., P.K., S.R.M., D.R., and J.F.M. provided analytical support and expertise; Q.Y., J.D.B., J.F., and A.A. processed and quality controlled the data; D.C.G., J.B., N.G.M., K.L.M., G.I.Z., M.J.W., and P.M.T. collected data and assisted with research design. All authors reviewed, edited, and provided critical feedback on the paper.
Correspondence to Fabrizio Pizzagalli or Neda Jahanshad.
N.J. and P.M.T. are joint principal investigators for a research grant from Biogen, Inc. (Boston, USA) for processing of imaging data, some of which was used in this paper. Biogen, Inc had no role in the conceptualization, design, data collection, analysis, decision to publish, or preparation of the manuscript. The remaining authors declare no competing financial or non-financial interests.
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How long do Red Queen dynamics survive under genetic drift? A comparative analysis of evolutionary and eco-evolutionary models
Hanna Schenk ORCID: orcid.org/0000-0002-7427-22901,2,
Hinrich Schulenburg1,3 &
Arne Traulsen1
BMC Evolutionary Biology volume 20, Article number: 8 (2020) Cite this article
Red Queen dynamics are defined as long term co-evolutionary dynamics, often with oscillations of genotype abundances driven by fluctuating selection in host-parasite systems. Much of our current understanding of these dynamics is based on theoretical concepts explored in mathematical models that are mostly (i) deterministic, inferring an infinite population size and (ii) evolutionary, thus ecological interactions that change population sizes are excluded. Here, we recall the different mathematical approaches used in the current literature on Red Queen dynamics. We then compare models from game theory (evo) and classical theoretical ecology models (eco-evo), that are all derived from individual interactions and are thus intrinsically stochastic. We assess the influence of this stochasticity through the time to the first loss of a genotype within a host or parasite population.
The time until the first genotype is lost ("extinction time"), is shorter when ecological dynamics, in the form of a changing population size, is considered. Furthermore, when individuals compete only locally with other individuals extinction is even faster. On the other hand, evolutionary models with a fixed population size and competition on the scale of the whole population prolong extinction and therefore stabilise the oscillations. The stabilising properties of intra-specific competitions become stronger when population size is increased and the deterministic part of the dynamics gain influence. In general, the loss of genotype diversity can be counteracted with mutations (or recombination), which then allow the populations to recurrently undergo negative frequency-dependent selection dynamics and selective sweeps.
Although the models we investigated are equal in their biological motivation and interpretation, they have diverging mathematical properties both in the derived deterministic dynamics and the derived stochastic dynamics. We find that models that do not consider intraspecific competition and that include ecological dynamics by letting the population size vary, lose genotypes – and thus Red Queen oscillations – faster than models with competition and a fixed population size.
Diversity, induced by continuous co-evolution can theoretically be maintained by the intense antagonistic relationship of hosts and parasites. This is the central part of the Red Queen hypothesis, verbally first formulated by van Valen in 1973 [1]. The hypothesis has been mathematically formulated in many models. However, owing to the modern usage of the term 'Red Queen' for different but related phenomena [2–9], the models have diverging foci and many lack the implementation of stochastic forces and ecological dynamics. A common synonym for the term Red Queen dynamics is fluctuating selection dynamics (FSD). Such fluctuations can be induced by co-evolving hosts and parasites and, as one possibility, be driven by negative frequency-dependent selection (NFDS), where host and parasite genotype abundances oscillate in time and every genotype can temporally be best adapted. In detail, since parasites are selected to target the most common resource, being a rare host genotype is advantageous. This temporary high fitness makes the genotype grow in relative abundance, but before it can take over the whole population, it is severely diminished by the profiting parasites genotypes, which target this now common host type. By contrast, in arms race dynamics (ARD) novel favoured genotypes spread in the entire population by recurrent selective sweeps. The terms NFDS and ARD are both referred to as Red Queen dynamics [10–12] and describe an ongoing co-evolutionary change without approaching an equilibrium. In this paper, we use the term Red Queen dynamics for NFDS, as is commonly done in the literature, but return to other definitions of the Red Queen in the discussion.
Although Red Queen dynamics is a well-known and frequently cited concept, there is only little evidence for the ubiquitous prevalence of long term Red Queen dynamics in nature – empirical challenges preclude the observation of more than a few subsequent oscillations, as these require a major amounts of intensive and challenging lab work [13–17]. Thus, most work on the actual long term temporal dynamics is theoretical, often dealing with evolutionary dynamics or epidemiological dynamics in a deterministic fashion. We have summarised some of the literature in the context of these assumptions in Table 1 (methods in the Additional file 1). Similar literature summaries exist with a focus on sexual vs. asexual reproduction [8] or host-parasite coevolution models [18]. Many theoretical studies build on evolutionary game theory [19] and a zero-sum assumption, where the harm done to the host equals the benefit for the parasite, which was already envisioned by van Valen at the time. Some of the models are implemented with equations that describe both species' dynamics (explicit host-parasite HP dynamics), other studies, especially on the evolution and maintenance of sexual reproduction (Red Queen Hypothesis) revert to epidemiological models (susceptible-infected SI models), sometimes in the pursuit of including population dynamics. The present work focuses on evolutionary host-parasite models in comparison with eco-evolutionary models that include population dynamics without using SI models.
Table 1 Literature overview
While many studies assess the occurrence of oscillating selection dynamics and show under what assumptions oscillations dominate [18, 20–27], only few studies include both ecological population dynamics and stochastic noise, although the combination of the two has been shown to result in a fast loss of genotypes in either population [28]. It has been difficult to derive a stochastic model that easily switches between constant and changing population size using a single parameter. For example Gokhale et al. [28] artificially normalised population size every few generations. Here, we take a different approach and compare the modelling framework of evolutionary game theory, where population size is constant by design, to eco-evolutionary dynamics from the field of theoretical ecology, where population size is inherently free to change over time. Our goal is not to present the one model that is the best description of reality, but to illustrate how different modelling assumptions can drive the results from such models.
Specifically, we use individual-based models, since ecological and evolutionary dynamics of populations are driven by events on the individual level. The models are based on haploid and asexual populations that live in a well-mixed environment where encounters are density dependent. Individuals are born, interact with other individuals of their own or opposing species and die. Generally, we will consider at least two genotypes and track the associated abundances H1,H2,P1,P2 and the total population sizes NH,NP of hosts and parasites over time. We simulate the dynamics using a uniformly distributed initial standing genetic variation and the simple matching allele interaction profile, where parasites are highly specialised [29, 30] on a particular host genotype and identical in all other aspects. Yet, the way this interaction profile enters in the dynamical equations and thus defines fitness for the individual genotypes is very different between the models. In population dynamics models these events happen at constant rates and depending on the density of the interacting individuals. A similarly simple, yet completely different approach is the stochastic birth-death process which tracks only the evolutionary dynamics. In each time step one individual is born, proportional to its current 'fitness' and another individual dies proportional to its density.
These models all produce Red Queen dynamics (NFDS) and we assess the robustness of those by measuring the time to extinction, which we define as the earliest time that any genotype from the initial genetic variation is lost in either population. Further, we consider the impact of the derived deterministic dynamics and the influence of ecology in the form of a population-size-change on this extinction time. The time to extinction of a genotype represents the durability of the stochastic oscillations. Without the immigration or re-emergence of extinct genotypes, the diversity of both populations declines in the long run.
Evolutionary dynamics depict the change of relative genotype abundances over time and can be examined without keeping track of population size changes. However, it is well known that ecological dynamics can feed back on evolutionary dynamics. We want to understand this feedback in the context of Red Queen dynamics. To this end we compare models from evolutionary game theory, that do not include population size changes and theoretical ecology models that do. The models have been widely used in the literature and represent the simplest case of Red Queen dynamics with a matching allele interaction profile (for details see the "Methods" section below and Additional file 1).
The matching-allele host-parasite Red Queen dynamics in evolutionary and eco-evolutionary models
In an evolutionary birth-death process one individual is born and another dies in each population, here host or parasite, and in each time step. Thereby, population size remains constant and the focus lies on the genotypic composition of a population. The Evo + and Evo processes (see Table 2 and in the "Methods" section for a definition) are such birth-death processes [64, and references therein; 65]. Individuals are chosen to die randomly, but the individual that reproduces is chosen proportional to the fitness advantages of that genotype relative to other genotypes in the population. The fitness effects are imposed by the current state of the antagonist population and an interaction matrix. In the Evo + process, the fitness effect is normalised by the average fitness effect over the whole host population, which leads to a kind of intra-specific competition (+) while in the Evo process the difference in fitness effects is compared between a genotype-one individual and a genotype-two individual, thus competition is pairwise. Because of the population size constraint, both processes can be analytically treated (see Additional file 1) when implemented in discrete time (prefix dt).
Table 2 Model overview
In models adapted from theoretical ecology the events of birth, death and interaction happen independently with external rates and, importantly, between populations (EcoEvo, comparable with the Lotka-Volterra dynamics in [28]). Host and parasite individuals encounter one another based on their densities and if they match, an interaction is carried out with a constant rate upon which a host dies or a parasite reproduces. When competition between hosts (+) is included, the host population grows logistically with a carrying capacity K. The host population size NH reaches the carrying capacity in the absence of parasites. However, in the presence of parasites the population size is smaller than K because of the additional mortality from parasites (see Table 3 for details on the parameters in all models).
Table 3 Model parameters
Both Evo and EcoEvo modelling approaches are combined in an intermediate model with self-controlled, but not fixed, population size (Hybrid). The model is implemented as an individual interactions model, where reactions take place also between populations, but the rates of these events are taken from the game theory models: Host death and parasite birth happen according to the fitness effects, host death and parasite birth rates are then adapted dynamically to keep population size nearly constant.
From the derivations of the models (details in the "Methods" section and Additional file 1) some basic properties of the dynamics are obtained and summarised in Table 2. The evolutionary game theory models have a constant population size by design, whereas population size can change in all other models. The average behaviour of the individual-based stochastic processes is captured in the deterministic selection term and the noise term, which together determine the stochastic dynamics. The noise term is discussed in the "Results" section (Fig. 1). The role of intra-specific competition in the deterministic part is discussed in the "Discussion" section (Fig. 2).
Example run illustrating that extinction is faster with ecological dynamics. Oscillations of host and parasite genotype abundances in the Evo + process with constant population size and EcoEvo + process with changing population size. The simulations start with an equal abundance of both genotypes H1(0)=H2(0)=NH/2 and P1(0)=P2(0)=NP/2. Method: Simulation of the stochastic processes with the Gillespie algorithm. Parameters: Total population sizes NH=50,NP=150 (only initially for the EcoEvo + model), selection strengths wH=0.5,wP=1, matching allele parameters α=1,β=0, death rate of the parasite dP=1, birth rate of the host bH=6, carrying capacity K=100, interaction rate \(\lambda _{0}=4, \lambda = \frac {\lambda _{0}}{K}\), intra-specific competition rate \(\mu = \frac {b_{H}}{K}\). See the "Methods" section and Additional file 1 for method and parameter details
Large population size limit. Relative abundances of two genotypes of host h1 and h2 and parasite p1 and p2 over time (left) and 2D representation (right) in the deterministic equivalents of the Evo + and Evo process with constant population size. Top: Intraspecific competition within the whole population (+) results in an attracting fixed point which is reached eventually and does not changed once reached, leading to stasis (also EcoEvo +). Bottom: Pairwise competition between individuals allows for a neutrally stable fixed point which neither attracts nor repulses the dynamics resulting in continuous co-evolution in the form of negative frequency-dependent selection dynamics (NFDS) around the internal fixed point (also EcoEvo). Method: integration of ordinary differential equations, the adjusted replicator dynamics (Evo +) and the replicator dynamics (Evo), which are the deterministic limits of the respective stochastic processes. Parameters: selection strength wH=wP=1, matching allele parameters α=1 and β=0
In models with ecological dynamics genotypes die out faster
It is clear that populations with low total population sizes are more prone to genetic drift and the loss of genotypes than large populations. We now show that it is not only the population size but the possibility of population size change that speeds up this process. We exemplify our argument here, but provide a more detailed analysis below. As an example with two host genotypes and two parasite genotypes we select the Evo + process and the EcoEvo + process (Fig. 1). Starting with an equal abundance of genotypes we measure the time to the first loss of a genotype. When a genotype has died the population becomes monoclonal and oscillations are no longer possible. With the fixed population size in the Evo + process oscillations survive longer than in the EcoEvo + process with a changing population size. The evolutionary dynamics are similar and defined through the relative abundance of the types, but the population size change can speed up the frequency of event occurrences and increase the probability of extinction through the bottleneck effect when population sizes are low.
Intraspecific competition stabilises negative frequency-dependent selection
The equations that define the stochastic process consist of a deterministic selection term and a noise term and represent the mean and variance of many individual simulations. Therefore, it is impossible to understand the stochastic model without making the deterministic dynamics clear. Furthermore, when population size is large, the stochastic process approaches the more manageable deterministic dynamics (details in the Additional file 1). The deterministic equations for all models from Table 2 have an internal co-existence fixed point, where both genotypes exist in a fixed ratio, which does not change over time. This point is only attractive, if starting with suitable initial compositions of genotypes the dynamics approach the state, in this case in the form of damped oscillations. The intraspecific competition (+) in the Evo + process and the EcoEvo + process result in such an attractive pull (Fig. 2). A second possibility is neutral stability, where genotype abundances oscillate with a constant amplitude and period, which depend on the initial abundances. These neutral cycles are produced by models where individuals only compete with other genotypes locally like in the Evo process with pairwise competition or the EcoEvo model with no intraspecific competition and the Hybrid model (Fig. 2).
In our models, the noise in the stochastic dynamics always leads to extinction, while deterministic dynamics never do. When population size is large enough to be impacted by the deterministic behaviour but stochastic noise still plays a role, the global competition models (+) show persisting Red Queen oscillations. The deterministic 'pull' and the stochastic 'push' balance [66], prolonging extinction times. For models with neutral oscillations (NFDS) in the deterministic dynamics stochastic effects will on average increase the amplitudes and push the trajectories to the edges of the space towards a faster extinction of genotypes.
The single simulations (Fig. 1) are only a snapshot and one specific realisation of the stochastic processes. Ideally, we would analytically derive general extinction times depending on the parameters of the model. Yet, to derive an exact analytical solution for this problem is extremely challenging. In addition to simulations, we have calculated the numerical (but exact) extinction times for low population sizes and provide an approximative method based on the averaged noise (see Additional file 1 for further details). These methods are limited to a subset of the models and can thus not be used for a comparison of all models, but only to support the computationally costly simulations which provide the now following main result.
The strength of random effects depends on the model properties
We simulate 1000 replicates for a set of parameter combinations of the models with two genotypes in each population and record the time it takes until one genotype has died out. As a general trend, the more constrained a population size is, the longer oscillations survive (higher extinction times in Fig. 3). This holds true for small to intermediate population sizes – note a similar vertical order of extinction times to the ordering of models by population size constraint in Table 2. When population sizes become larger and the deterministic model properties gain influence, models with competition terms (+, stasis, compare with Fig. 2) have higher extinction times and therefore more stable Red Queen oscillations.
Extinction time of either genotype of either host or parasite population for different initial population sizes of the parasite NP for all models. We show the mean extinction time of any genotype over 1000 independent simulations (fat dots) and the distribution of those extinction times (shaded histogram area around the mean). The simulations start with equal abundance of both genotypes H1(0)=H2(0)=NH/2 and P1(0)=P2(0)=NP/2. Lines denote approximate results based on the average noise (see Additional file 1). The discrete time processes are simulated for values of NP for which analytical results are valid. The Evo + process is not simulated for high parasite population sizes since the computation becomes extremely time-consuming and the trend is already clear. Parameters as in Fig. 1 except NH=250,K=500, birthrate bh∈{0.24,0.32,...,1.6} in the EcoEvo + and for the EcoEvo model bh∈{0.12,0.16,...,0.8} and μ=0 to achieve the population sizes NP displayed
By design, the discrete time (dt) processes have much higher extinction times and are thus not directly comparable to the continuous time simulations. A scaling would be possible for equal population sizes, but with different extinction routes and NH≠NP no such factor can be derived. The dtEvo + and dtEvo extinction times in Fig. 3 can therefore only be compared between them. For growing NP, the dtEvo + process has an increased extinction time because of the stabilising attractive fixed point. This trend is even more pronounced in the approximate analytic solution (solid lines), inspired by Claussen [67, 68] (see Additional file 1). The error of the analytical approach cannot be neglected, but the qualitative trend is clearly visible and the result is fully analytical.
Due to the challenges of employing an exact analytical approach, we cannot perfectly tune the models for the same amplitudes, fluctuations and frequencies/periods of oscillations. The specific choice of the parameters is not necessarily directly comparable, but we have made an effort to choose them in a meaningful way, such that the fixed points are exactly the same and amplitudes comparable. We choose strong selection for the parasite wP=1 and weaker selection for the host wH=0.5 in the models derived from game theory, because the EcoEvo + model is built in a similar way: Parasite birth can only occur through the antagonistic interaction, but host mortality is also influenced by the competition term. While the parasite is obligate and thus completely dependent on the host, the host suffers, but does not always die from an infection.
The impact of selection intensities on the extinction times is further explored in the Additional file 1. We find that strongly diverging host and parasite selection intensities can counter-intuitively lead to more stable dynamics in the Evo process than in the Evo + process.
Diversity inflow results in sequential negative frequency-dependent selection dynamics and arms race dynamics
So far we have compared models with two genotypes in each species. We now provide an outlook of how diversity changes for many genotypes. We simulate one possible example with an initial uniform distribution of twenty genotypes in each species (see Additional file 1). Diversity, simply defined as the number of genotypes present in the population, declines exponentially with time at a constant rate. The manual re-introduction of an extinct, but temporarily best adapted parasite genotype can result in a selective sweep that leaves the parasite population monoclonal.
In reality, our genotypes are not as static in their traits as described here, but one of our 'genotypes' can actually be seen as an average of several individuals with slightly different traits. We now add a form of mutation or recombination to the model so that reproduction does not necessarily result in a clonal daughter, but a new individual with different traits. For example, parasites could evolve quickly by allowing beneficial mutations to produce other, even extinct, genotypes. Depending on the model system, a sexually reproducing host could also store genetic material to revive long extinct phenotypes by recombination. We abstract this by inserting a conversion rate μ from one genotype to the neighbouring genotype. For example with five pre-defined genotypes we have \(H_{1} \xrightarrow {\mu /2} H_{5}\) and \(H_{1} \xrightarrow {\mu /2} H_{2}\) and so on. The dynamics we observe now (Fig. 4) are not pure negative frequency-dependent selection dynamics, but a mixture of oscillations and arms race dynamics, where selective sweeps can make a population monoclonal in a very short time, but a re-introduction of extinct genotypes allows for oscillations to re-emerge.
Negative frequency-dependent selection and arms race dynamics. Revival of genotypes and evolution of host (top) and parasite (middle) populations with five possible genotypes each. With the rate μH=0.005 and μP=0.01 genotypes convert to neighbouring genotypes through mutation or recombination. Stacked plots – evolutionary dynamics: the area covered by one colour is proportional to the relative abundance of that genotype of host or parasite at that time. Lower panel – ecological dynamics: total abundance of hosts and parasites. Method: the example is a stochastic simulation (Gillespie algorithm) of an EcoEvo + process. The simulations start with equal abundance of all five genotypes Hi(0)=NH/5 and Pi(0)=NP/5 for i=1,2,3,4,5. Parameters: NH=300,NP=900 (both initially), bH=6,dP=1,K=600,λ0=10
In this paper we compare evolutionary models from evolutionary game theory to eco-evolutionary models from theoretical ecology to understand the impact of ecology and other model properties on the long term co-evolutionary Red Queen oscillations of host and parasite genotypes. The models are individual-based and intrinsically stochastic, thereby allowing genetic drift and the loss of genotypes from a population. Starting with an initially uniform distribution of genotypes, we define the extinction time as the first time that any genotype is lost from any of the two populations, and use this extinction time to measure the robustness of the Red Queen cycles and therefore, the maintenance of diversity. Our main result is that including ecology in models, in the form of a changing population size, leads to a faster loss of genotypes, when stochastic dynamics are considered. This result is similar to the simulation results by Gokhale et al. [28], where ecological dynamics were artificially removed from the simulations, in an attempt to make a straightforward comparison of eco-evo and evo dynamics. In contrast, we compare two modelling frameworks with historically developed differences between them. The models presented here are all based on the same widely used biological assumptions – haploid well-mixed host and parasite genotypes that interact through the matching-alleles infection matrix – but with differences in their mathematical properties: discrete and continuous time models with attractive or neutral deterministic dynamics. The models are further intrinsically stochastic, since they are derived from interactions between individuals. This inherent noise, genetic drift, also impacts the models within a given framework. The mean outward pull by noise that increases amplitudes and thus makes extinction more probable can be counteracted by intraspecific competition that pulls the dynamics back, decreases amplitudes and thus stabilises negative frequency-dependent selection dynamics, resulting in longer extinction times. Finally, we provide a snapshot of what happens when standing genetic variation is large initially. If no inflow of genotypes via mutation or migration is provided, the number of genotypes in a population will decline exponentially. However, when conversions between neighbouring types are allowed with a small mutation rate, negative frequency-dependent selection dynamics and selective sweeps can occur sequentially.
Previous theoretical studies have similarly examined the persistence of Red Queen oscillations under ecological feedbacks. For example Goméz et al. [24] found fluctuating selection and arms race dynamics in an epidemic model (host-focussed) with explicitly modelled parasite populations. MacPherson and Otto [27] also combined epidemiological and neutrally stable host-parasite dynamics and showed that this can dampen allele frequency oscillations, which leads to stasis in their deterministic model but would return to oscillations under stochasticity (see Table 2). Recently, the game theoretical replicator dynamics were mathematically tuned for population size influence using a single parameter [18] resulting in damped oscillations and thus stable polymorphism for both matching-alleles and gene-for-gene infection matrices. While we argue that eco-evolutionary feedbacks increase oscillation amplitudes, Ashby et al. argue that oscillation amplitudes are decreased over time. However, the population dynamics in their model were dampened by a maximal value which resembles our intra-specific competition leading to stasis (Fig. 2). Furthermore, their models are deterministic which closer resembles our models when population size is large, where stabilising effects have a larger influence. In the more theoretical literature, it is now well established that assumptions such as population size fluctuations and stochasticity can result in more rapid extinction ([69, 70], and many more).
The stabilising property of intra-specific competition is documented in the literature [71]. Intra-specific competition (+) enters in our evolutionary models as part of a genotype's fitness effect that is compared to the focal population's average fitness, whereas in the eco-evolutionary models it is implemented as an ecological intra-specific competition term. Both the evolutionary and the ecological implementation of this intra-specific competition stabilise the dynamics and lead to stasis following damped oscillations. The more commonly used host-parasite co-evolution models result in neutrally stable oscillations whereas damped oscillations are often seen as a termination of Red Queen dynamics. Yet, exactly this stasis shows similar oscillation patterns when stochasticity perturbs dynamics away from the stable fixed point (noise induced oscillations [66]). In a stochastic world, pure host-parasite dynamics therefore result in fast extinction, which would only be stabilised by intraspecific competition. For larger population sizes, when the stability of the fixed point gains in importance, the dynamics are pulled more towards the inner equilibrium state, making stochasticity less influential. Thus, only for organisms with large population sizes and good mixing, intraspecific competition would not be necessary for sustained Red Queen oscillations.
Although this study does not explicitly analyse modes of reproduction, our final result shows how reviving extinct genotypes can restore Red Queen dynamics. If parasites can evolve more quickly due to shorter generation times and larger numbers, then hosts are given an advantage by being able to "store" genotypes through recombination. Also, if clonal reproduction accumulates mutations (Muller's ratchet), this could impact population sizes and sexual reproduction would be even more important [72, 73]. Ashby and King [8] devised a stochastic individual based susceptible-infected model with diploid sexual hosts and showed that high diversity cannot maintain sexual reproduction when parasite transmission rates are low. Although our models are more abstract concerning reproduction, we do explicitly model parasites. If parasite populations are well mixed and diverse with high mutation rates, this can again select for higher diversity through sex, like in [24], where fluctuating selection dynamics, and thus high diversity, is more likely when hosts encounter a diverse parasite population and the disease load is high. Furthermore, our models can include global competition in both species or resource competition in the host, which stabilise the oscillating dynamics. More support for recombination during parasite infection was shown in [48], where hosts could optionally switch between two modes of reproduction. See also [4] for a comprehensive connection to the Red Queen Hypothesis for sexual reproduction.
We have shown that in the same setting and with the exact same parameters sequential occurrences of oscillating selection and arms race dynamics are possible. We show only a snapshot and we do not quantify dynamics as is done in [24], but we find it to be an interesting aspect that the dynamics can occur temporarily in the same simulation, with the same settings and assumptions. The more complete picture could include all possibilities discussed in the Red Queen literature: there can be constant extinction, as suggested by van Valen on a taxonomic level and there can be oscillations and arms race dynamics as suggested by host-parasite interactions and the resulting co-evolution. With our preliminary results we might be going too far if we also justify sexual reproduction, yet, without recombination or mutation, diversity decline is inevitable theoretically.
Our models explore stochasticity under different restrictions of population size, while other modelling aspects are kept relatively plain. In the present work, the infection pattern is restricted to the matching alleles model, and the zero-sum assumption, yet this is necessary for oscillations [21]. Other infectivity patterns that result in Red Queen dynamics have not been examined here. The gene-for-gene infection matrix could show similar results since the oscillations are also neutrally stable, yet including ecological dynamics changes the complexity of the cycles [25]. In general the robustness of those cycles under stochasticity would depend on the details of how different infection mechanisms are translated into mathematical equations. Further limitations are the haploidy of both hosts and parasites and thereby asexual reproduction, the lack of life history or infection history and there is no spatial structure and evolution in the values of resistance or infectiousness. We do, however, briefly explore the effects of including more genotypes and mutation as a means to revive genotypes. There is an increasing effort to openly discuss how verbal models and biological assumptions enter into models [27, 74]. Making the assumptions clear and readily available should be the standard for future publications. For stochastic processes the analogous deterministic dynamics should be stated to provide the reader with a more complete picture of stochastic dynamics.
The model predictions presented here although quite abstract may nevertheless apply to the real world. Bottlenecks are likely more common in natural host-parasite associations [75] than usually assumed and, therefore, the interaction dynamics are likely shaped by genetic drift and, thus, stochastic effects. Eco-evolutionary feedbacks have been confirmed to impact the form of co-evolution in bacteria-virus experiments [76]. Increasing diversity in the parasite or higher exposure lead to a shift from negative frequency-dependent selection to arms race dynamics in two bacteria-phage systems [24, 77]. Oscillations alongside incomplete selective sweeps were recently even documented in a nematode-bacteria interaction [17]. It would now be of particular interest to assess the occurrence of bottlenecks, drift and competition in natural host-parasite associations and relate them to the resulting allele frequency dynamics. Such empirical data would help us to obtain a more general understanding of host-parasite co-evolution and potentially question the importance of sustained Red Queen oscillations in this context.
We have shown that models equal in their verbal biological description can be quite different in their mathematical details, with great consequences for both deterministic and stochastic dynamics. The loss of genotypes is inevitable in stochastic models without mutation or immigration. This extinction is faster when ecological dynamics are considered in an evolutionary model. Competition between genotypes within a species stabilises the dynamics and slows down extinction thus sustaining Red Queen dynamics. The applicability of models to the real world thus depends on the system of interest and mathematical details should be carefully considered for each particular case study. When bottlenecks, drift and competition are observed, the model needs to be adapted accordingly.
The following method descriptions are short explanations of the stochastic processes used in this manuscript. The precise equations and methods of analysis can be found in the Additional file 1. The simulation code is provided at https://github.com/HannaSchenk/ShortLifeRQ.
The discrete time Evo+ process (dtEvo+), also discrete time Moran process, is a stochastic birth-death process, with a constant population size [78], often used in evolutionary game theory (see for example [79, 80] or [81]). Each birth-death reaction has a reaction probability (or transition probability), depending on the state of the system in each discrete time step Δt=1. The original definition ensures that the probabilities sum up to one so that one reaction (also reactions where no transition happens - when birth and death event happen within the same genotype) takes place in each time step. In the Moran process, a 'payoff' π is what a genotype gains from interactions with others. The interaction matrix is
$$ \left(\begin{array}{cc} \alpha & \beta \\ \beta & \alpha \end{array}\right), $$
where α is the fitness gain for the parasite and the fitness loss for a host if the genotypes match, and β is the fitness gain or loss for mismatching pairs (here α=1 and β=0). For example, the probability of a P1 birth and thus a subsequent P2 death is proportional to \(\pi _{P_{1}}=\alpha h_{1} +\beta h_{2}\), where h1 and h2,p1 and p2 are relative abundances. How much this effects the so-called 'fitness' f is controlled with the selection intensity wP (or wH for the host) so that \(f_{P_{1}}=1-w_{P}+w_{P} \, \pi _{P_{1}}\). This per capita 'fitness' is then normalised by a dynamically changing population average \(\overline {f_{P}}=p_{1} \, f_{P_{1}} + p_{2} \, f_{P_{2}}\) (this is the intraspecific competition step) and multiplied with the current abundance of the genotype. Thus \(p_{1} \, f_{P_{1}} / \overline {f_{P}}\) is then the birth probability of a genotype one parasite. The death probability is simply density dependent, thus the total probability of replacing a genotype-two (death) by a genotype-one (birth) parasite is \(p_{2} \, p_{1} \, f_{P_{1}} / \overline {f_{P}}\). Since we are modelling two populations (host and parasite), we choose to update both populations simultaneously instead of sequentially, such that frequency changes of host genotypes and parasite genotypes can happen at once. The deterministic limit (population sizes NH,NP→∞ and time steps Δt→0) of the Moran process in a single population is usually the differential equation of the replicator dynamics, however, in a two-population model the average fitness within each population is different and thus the adjusted replicator dynamics become the deterministic analogue [64, 82]. The adjusted replicator dynamics for host-parasite interactions have a globally attractive inner fixed point, in the symmetric matching alleles case this is the equal abundance of all genotypes.
The discrete time Evo process (dtEvo) [64, called pairwise comparison process or local update process in evolutionary game theory], is another birth death process, nearly equivalent to the Moran process but here competition is strictly local and pairwise, not normalised by a global average fitness. What is \(f_{P_{1}}/\overline {f_{P}}\) in the Moran process is here \(0.5+0.5 \, (f_{P_{1}}-f_{P_{2}})/ max(\Delta \pi _{P})\). The 'fitness' of parasite 1 only depends on the difference in 'fitness' to parasite 2 which depends on the abundances of host genotypes (see equation for \(f_{P_{1}}\)), but not, as when normalising with \(\overline {f_{P}}\), on the relative abundances of the parasite genotypes. Thus, in the pairwise comparison process, the antagonist influences globally (since there is no spatial structure), but within a species the competition is local. This results in the recovery of the replicator dynamics with neutral cycles in the deterministic limit.
A Gillespie algorithm [83] can be employed to simulate the above stochastic processes. In this case reaction rates (not probabilities) are calculated for each species and, using random numbers, the shortest waiting time for each reaction is determined under the assumption of exponential waiting times. The reaction with the shortest time takes place and time is updated accordingly. This makes time continuous and time steps unequal. In contrast to the discrete time models, the Gillespie algorithm only updates one species at a time. The now following processes are also implemented using a Gillespie algorithm.
The EcoEvo process uses independent reactions of host birth, parasite death and host-parasite interactions similar to the individual-based equivalent of the Lotka Volterra equations. This results in a microscopic process often believed to be a more natural approach because the reactions describe individual and independent events on the 'microscopic' level rather than population dynamics on the 'macroscopic' level. Host birth reactions are density dependent with constant rate bH, parasite death is density dependent with constant rate dP and a density-dependent interaction of matching host-parasite pairs can result in the death of the host or the birth of a parasite with constant rate λ. The population size has no restrictions in this case and freely follows the evolutionary dynamics. The deterministic analogue has a neutrally stable fixed point.
The EcoEvo+ process are like the EcoEvo independent reactions, but with additional competition in the host population. Density-dependent interactions of two host individuals, independent of the genotype, result in the death of one of the individuals with constant rate μ. This model, when reduced to the deterministic limit, is an antagonistic interaction model with logistic growth in the host (carrying capacity K=bH/μ) and an attractive inner fixed point.
The Hybrid model is a process with self-controlled population size. It is built on the EcoEvo model but with constrained birth and death rates adapted from the Evo model and dynamically varied to balance birth and death events on average. Thus, the population size is tightly controlled, yet it is not strictly constant. Building on the stochastic processes from evolutionary game theory above, one can set up a process that utilises the infection matrix for death events in the host and birth events in the parasite in explicit individual reactions. For example the dynamic reaction rate of a death event of a host genotype one is \(d_{H_{1}}=1-w_{H}+w_{H} \, \frac {\alpha \, P_{1}+ \beta \, P_{2}}{N_{P}}\). The birth rates for the host and the death rates for the parasite are then dynamically adjusted to equal the total rates of host death and parasite birth. The deterministic limit is the replicator dynamics with a neutrally stable fixed point, as in the pairwise comparison process.
The code that produces the simulated data is available at https://github.com/HannaSchenk/ShortLifeRQ
ARD:
Arms race dynamics
Coevolutionary stable strategy
FSD:
Fluctuating selection dynamics
GfG:
Gene-for-gene
Host-parasite
IBM:
Individual based model
Matching alleles
NFSD:
Negative frequency-dependent selection
ODE:
Ordinary differential equation
PDE:
Partial differential equation
Recursion equation
RK:
SDE:
Stochastic differential equation
SI:
Susceptible-infected
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We want to thank Peter Czuppon, Jens Christian Claussen and Andreas Rößler for detailed and enlightening discussions on methods, approximations and stochastic differential equations. We would like to thank Michael Raatz, Hye-Jin Park, Stefano Giaimo and Yuriy Pichugin for helpful comments on the manuscript.
We gratefully acknowledge generous funding by the Max Planck Society, which takes no part in the design, analysis or publication of research.
Max Planck Institute for Evolutionary Biology, Plön, Germany
Hanna Schenk
, Hinrich Schulenburg
& Arne Traulsen
German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Leipzig, Germany
Department of Evolutionary Ecology and Genetics, University of Kiel, Kiel, Germany
Hinrich Schulenburg
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H.Schulenburg and AT designed the research question. H.Schenk and AT developed and adapted the models. H.Schenk conducted the analysis. All authors discussed and interpreted the results. H.Schenk wrote the initial draft. All authors revisited the manuscript critically and approved the final version. All authors read and approved the final manuscript.
Correspondence to Hanna Schenk.
Arne Traulsen is an editorial board member of BMC Evolutionary Biology but he was not involved in handling this manuscript.
Additional file 1 Supplementary files. The additional pdf file supplementaryFile.pdf includes mathematical derivations and further descriptions of the models, explanation of methods used, additional results and extra figures.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Schenk, H., Schulenburg, H. & Traulsen, A. How long do Red Queen dynamics survive under genetic drift? A comparative analysis of evolutionary and eco-evolutionary models. BMC Evol Biol 20, 8 (2020) doi:10.1186/s12862-019-1562-5
Eco-evolutionary dynamics
Stochastic models
Extinction time | CommonCrawl |
Residual feedback suppression with extended model-based postfilters
Marco Gimm ORCID: orcid.org/0000-0003-1390-82801 na1,
Philipp Bulling2 &
Gerhard Schmidt1
When designing closed-loop electro-acoustic systems, which can commonly be found in hearing aids or public address systems, the most challenging task is canceling and/or suppressing the feedback caused by the acoustic coupling of the transducers of such systems. In many applications, feedback cancelation based on adaptive filters is used for this purpose. However, due to computational complexity such a feedback canceler is often limited in the length of the filter's impulse response. Consequently, a residual feedback, which is still audible and may lead to system instability, remains in most cases. In this work, we present enhancements for model-based postfilters based on a priori knowledge of the feedback path which can be used cooperatively with the adaptive filter-based feedback cancelation system to suppress residual feedback and improve the overall feedback reduction capability. For this, we adapted an existing reverberation model such that our model additionally considers the presence and the performance of the adaptive filter. We tested the effectiveness of our approach by means of both objective and subjective evaluations.
Signal processing in a closed electro-acoustic loop is a challenging task. It occurs in various applications such as hearing aids [1, 2], public address (PA) systems [3, 4] or so-called in-car communication (ICC) systems [5, 6]. In all these systems, feedback occurs because the signal that is played back using a loudspeaker is recorded by a microphone, processed, and then played back again using the same loudspeaker. This may lead to an instability of the system, namely when the loop gain for at least one frequency is larger than 0 dB and the phase is a multiple of 2π. Even if the system is in fact stable, the additional reverberation may make the signals sound unnatural or—more generally—degraded with respect to quality. To reduce the described effects, different methods already exist. The state of the art approach is to use an adaptive filter to estimate the acoustic path utilizing methods like the normalized least mean square (NLMS) algorithm or a Kalman filter [7]. However, besides the fact that operating in a closed acoustic loop requires a sophisticated control mechanism for a robust application of adaptive filters, there are some more limitations. One is that the filter converges towards a bias due to the high correlation between the local and the excitation signal. This makes an additional decorrelation stage essential for many approaches. Another limitation is that usually a filter with limited length will be used when implementing the adaptive filter. Consequently, the filter must be designed in such a way that its length covers at least the most important part of the room impulse response (RIR). Sometimes this is not possible, especially in cases of multichannel applications where a multitude of filters have to be implemented or in cases when the reverberation time is long, e.g., for large rooms like concert halls. The other limitation that leads to feedback never being completely removed is that there will always be a residual misalignment in the adaptive filter, which in turn leads to an error in the estimated feedback signal. Figure 1 depicts an example for a true impulse response hLM,i as well as the part \({\hat h}_{\text {LM},i}\) that an adaptive filter has estimated. The bottom plot shows the difference between the impulse response hLM,i and \({\hat h}_{\text {LM},i}\). The estimation error hΔ,i is the impulse response which causes the residual feedback in such a system.
Impulse response of a car cabin in the top graph, estimated impulse response of an adaptive filter with length 1024 taps in the middle graph, residual impulse response in the bottom graph
A different method to increase the stability gain in electro-acoustic loops is to estimate the short-term power spectral density (PSD) of the feedback by using the energy envelopes of the room's subband impulse responses. These envelopes can be obtained by a priori or online measurements as well as simulations. With this information, a model can be derived which is then used for a convolution with the loudspeaker subband power signal. This results in an estimate of the feedback's short-term PSD. This estimate is then used within a so-called Wiener filter (or a variant of it) to attenuate the feedback components within the microphone signal. Except from online measurements, the envelope is assumed to be constant. However, it can be shown that the model-based methods are robust against room changes and that the envelopes vary only slightly over time. The main advantage of this method is that the model can be implemented recursively and, thus, very efficiently in terms of computational complexity. There will not be any length limitations as described when using adaptive finite impulse response (FIR) filters. However, there are disadvantages, too. The main one lies in the derivation of the Wiener filter, which assumes that both the desired and the undesired signals are orthogonal. In the presented application this is not the case, since the feedback signal (undesired) is only a delayed and processed version of the local speech signal (desired). This means that not only feedback will be reduced, the model-based approach will also affect the desired signal. However, due to the fact that speech is assumed to be short-time stationary and there is a delay in the processing and also in the path between loudspeaker and microphone, it is usually observed that the attenuation of the desired signal is small compared to the attenuation of the feedback signal. Hence, this method is able to increase the stability in closed-loop systems.
In [8], we presented a method that makes use of the advantages of both described systems. Therefore, we introduced three ways to estimate the residual feedback PSD recursively, taking an adaptive filter into account. We also compared this with the model-based feedback suppression which was presented in [9]. In this work we made some improvements regarding the models. Furthermore, we show more implementation details. The objective evaluation was improved by adjusting the features. Additional simulations were also performed to investigate the performance during room changes and the convergence of the adaptive filter in the presence of a postfilter. In addition, further acoustic paths were simulated.
Organization of this paper
The paper is organized as follows: after this introduction, previous research work is summarized in Section 2. Afterwards the model-based feedback suppression approach is explained in Section 3. In Section 4, we show how we adapted the model-based approach to use it as a postfilter. After that, we present different methods to derive the required model parameters in Section 5. Finally, we show the evaluation procedure in Section 6 before a conclusion is provided in Section 7.
Throughout this contribution the notation will follow some basic rules:
Scalar quantities such as time-domain signals are written in lowercase, non-bold letters such as s(n) for a signal at time index n.
Short-term frequency-domain quantities are described by upper case letters such as X(μ,k), with k being the frame index and μ as frequency index.
Vectors are noted as bold letters, e.g., H(μ,k) represents a vector containing filter coefficients in subband μ at frame index k.
Smoothed signals are noted by over-lined letters such as \(\overline {x}(n)\) and estimated signals are written as letters with a hat such as \(\hat {x}(n)\).
All signals are represented in discrete time.
Previous and related work
Electro-acoustic feedback is a challenge in various technical systems. The most prominent ones are hearing aids, public address systems, and in-car communication systems. Therefore, lots of research has been done in those domains in recent years. A comprehensive overview of different approaches regarding feedback suppression can be found in [10]. In this work, we will focus on room modeling methods.
To fully erase the feedback and, therefore, to allow arbitrary gains, the impulse response of the feedback path must be estimated by means of an adaptive filter. Early approaches use a standard echo canceler to fulfill this task [11–13]. Here, the impulse response is estimated e.g. with a normalized least mean square (NLMS) algorithm in the time domain. If the local signal and the excitation signal are correlated, the problem is that adaptive filters converge to a biased solution. This is strongly the case in closed-loop electro-acoustic systems [14].
One solution to overcome this problem is to decorrelate the signals. This can, for example, be realized by frequency shifting. It is shown in [15–17] that a slight frequency shift within the frequency range of speech is sufficient to improve the convergence of the adaptive filter. The signals can also be decorrelated with linear prediction or pre-whitening [18]. In addition to the decorrelation of the signals, a special step-size control can further improve the convergence of the adaptive filter. In [1, 19], the decorrelation methods frequency shift and pre-whitening are compared and combined with a step-size control, based on a derivation of the so-called pseudo-optimal step size. Another step-size control that is able to improve the convergence of the adaptive filter without the need of any decorrelation method is described in [20, 21]. Here, the reverberation of the system is exploited to adapt the filter, since signals are not correlated during reverberation. With this step-size control, both stability and speed of convergence can be improved also for high system gains.
One drawback of the feedback cancelation approaches is that the adaptive filter must cover the relevant length of the room's impulse response. Otherwise, residual feedback is audible and may even cause the system to become instable. Since long filters increase the computational complexity as well as the convergence time, short filter lengths are often preferred.
In the field of acoustic echo cancelation (AEC), postfilters based on frequency-domain Wiener filters are commonly used [22–25]. The idea is that the residual echo is nothing but the undisturbed error signal which is the signal after the subtraction of the AEC took place assuming the absence of any local speech and noise signals. A very similar approach was already used for residual feedback suppression [13]. The downside of this technique is that the PSD estimation should only be done in remote single talk conditions [26]. Such a situation does not exist in case of closed-loop systems. There is however one exception which is the end of a speech segment when there is still some power in the loop due to the loop delay.
In [9], the authors present a feedback suppression method based on well known speech dereverberation techniques [27, 28]. Here, the feedback path is modeled with an statistical model. Based on this the feedback's PSD is estimated.
In [8], the model-based feedback suppression is tailored in a way that it can be used as a residual feedback suppression in combination with an adaptive feedback canceler. Therefore, three adapted statistical models have been proposed which can be used to model the feedback path taking an adaptive filter into account. Model-based approaches have already been used in adaptive echo cancelation systems [29, 30]. In [31], the authors also use a model-based approach as a postfilter for adaptive echo cancelation. The idea is to use adaptive approaches to model the residual echo power spectral density. However, in all of these approaches the adaptive filter is assumed to work perfectly and only the acoustic path, which is not covered by the filter is taken into account. In AEC applications, this might be sufficient as reasonable steady-state performance can be reached. However, this is not the case in adaptive feedback cancelation.
In the presented paper, the adapted models for residual feedback suppression are further investigated and a more detailed insight, as well as more simulations and results are given.
Model-based feedback suppression
In [9] it was shown that room dereverberation techniques as they were introduced in e.g. [27, 28] can be used to increase the stability of closed electro-acoustic loops as we face them in ICC systems. In this section, the model-based feedback suppression will be described before adapting it for a system with feedback cancelation. We will start with linear, time-invariant systems with coefficient index i. Of course, we can assume here only short-term stationarity. Therefore, we will introduce time-variance (by adding also a frame index k) after this generic view on the entire system.
A simple example of a time-domain system operating in a closed electro-acoustic loop can be seen in Fig. 2.
Structure of a single-channel, closed-loop system with feedback suppression, which is first of all assumed to be time invariant
The signal y(n) is the microphone signal at time index n and g is a Wiener filter with coefficients based on the estimated feedback which is used to suppress the recorded feedback that is present in y(n). hSE is the impulse response that belongs to the system of the individual application, where SE stands for signal enhancement. It differs with the individual application and may include noise suppression in case of an ICC system or an equalization filter in case of a public address (PA) system. After the signal enhancement stage, x(n) is played back using a loudspeaker resulting in a feedback r(n). The latter is obtained by a convolution of x(n) with the room impulse response hLM. As mentioned above, the room impulse response is assumed to be constant for now. Thus, the time index n can be dropped and we obtain the feedback signal as
$$\begin{array}{*{20}l} r(n) &= \sum\limits_{i=0}^{\infty} x(n-i)\, h_{\textrm{LM,{i}}}. \end{array} $$
r(n) is recorded again by the microphone together with additional background noise b(n) and local speech s(n), yielding the microphone signal
$$\begin{array}{*{20}l} y(n) &= s(n) + b(n) + r(n). \end{array} $$
The transfer function of the entire electro-acoustic loop can be described as follows:
$$\begin{array}{*{20}l} H_{\mathrm{L}}\left(e^{j\Omega}\right) = \frac{H_{\text{SE}}\left(e^{j\Omega}\right) \, G\left(e^{j\Omega}\right)}{1 - H_{\text{SE}}\left(e^{j\Omega}\right) \, G\left(e^{j\Omega}\right)\, H_{\text{LM}}\left(e^{j\Omega}\right)}. \end{array} $$
Here, Ω=2πf/fS∈[0,2π) is the frequency f normalized with respect to the sampling frequency fS. Furthermore,
$$\begin{array}{*{20}l} C\left(e^{j\Omega}\right) = H_{\text{SE}}\left(e^{j\Omega}\right)\,G\left(e^{j\Omega}\right)\, H_{\text{LM}}\left(e^{j\Omega}\right) \end{array} $$
is the so-called open-loop gain. For stability reasons
$$\begin{array}{*{20}l} \left|C\left(e^{j\Omega}\right)\right| < 1~~~ \forall\Omega \end{array} $$
$$\begin{array}{*{20}l} \angle {C\left(e^{j\Omega}\right)} \neq \eta 2\pi~~~ \forall\Omega, ~ \eta \in \mathbb{Z}_{0}^{+} \end{array} $$
should hold. G(ejΩ) can be used to fulfill this condition. One possibility would be to choose the transfer function based on the estimated feedback
$$\begin{array}{*{20}l} G\left(e^{j\Omega}\right) &= 1 - \frac{\hat{S}_{{rr}}\left(e^{j\Omega}\right)}{\hat{S}_{{yy}}\left(e^{j\Omega}\right)}. \end{array} $$
Therefore, an estimate of the feedback PSD \(\hat {S}_{{rr}}(e^{j\Omega })\) is required, which can be derived from Eq. (1).
To make this method more robust against small variation in hLM and to get the ability to save computational complexity we define a model of hLM based on its so-called reverberation time T60 and some other parameters, which will be explained next.
This finally results in an exponentially decaying model of the power envelope of the subband version of HLM(μ)
$$\begin{array}{*{20}l} \big| {H}&_{\text{LM,mod,A}}(\mu,k)\big|^{2} \end{array} $$
$$\begin{array}{*{20}l} &= \left\{\begin{array}{ll} 0, & \text{for }k<P(\mu),\\ A(\mu) \, e^{-\gamma(\mu)\left(k-P(\mu)\right)}, & \text{for } k \geq P(\mu),\end{array}\right. \end{array} $$
where μ is the discrete subband index, k is the frame index and A(μ) are coupling factors that describe the coupling properties of the acoustic path. P(μ) is the delay of the acoustic path in frames and
$$\begin{array}{*{20}l} \gamma(\mu) = \frac{2 \cdot 3 \ln(10)L}{T_{60}(\mu) \,f_{\text{S}}} \end{array} $$
describes the decay behavior, where L denotes the frameshift in samples. Using Eq. (8) the estimated short-time PSD of the current feedback \(\hat {S}_{rr,\mathrm {A}}(\mu,k)\) can be calculated as the convolution of the short-time PSD of the loudspeaker signal \(\hat {S}_{{xx}}(\mu,k)\) with the magnitude square of the modeled subband impulse response
$$\begin{array}{@{}rcl@{}} && \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\notag\hat{S}_{rr,\mathrm{A}}(\mu,k) \\ &=&\notag \!\!\!\sum_{i =P(\mu)}^{\infty}\hat{S}_{{xx}}(\mu,k-i)\,A(\mu)\, e^{-\gamma(\mu)\left(i-P(\mu)\right)}\\ &=&\notag A(\mu) \hat{S}_{{xx}}\left(\mu,k-P(\mu)\right) \\ && + \,e^{-\gamma(\mu)} \, \hat{S}_{rr,\mathrm{A}}(\mu,k-1). \end{array} $$
On the top left of Fig. 3, the energy envelope of the modeled subband impulse response for a single subband is depicted. This can now be used for a subband version
$$\begin{array}{*{20}l} G_{\mathrm{X}}(\mu,k) = \max\left\{ 0,\ 1 - \frac{\hat{S}_{\textrm{{rr}},X}(\mu,k)}{\hat{S}_{{yy}}(\mu,k)}\right\} \end{array} $$
Overview of the four different models
of Eq. (7), with X indicating the individual model type, e.g., X=A.
Model-based feedback suppression as postfilter
Due to stability reasons, FIR filters are commonly used in adaptive filter applications like echo- or feedback cancelation. If this kind of method is used in a closed electro-acoustic loop system, it is capable of subtracting parts of the feedback signal r(n) from the microphone signal y(n) depending on how good it is adapted to the true room impulse response. However, there are some limitations in the steady-state performance. One of them is that there will always be a residual system mismatch which is caused by non-optimal control or estimation errors, even if robust adaptive control schemes are used. If an efficient implementation in the subband domain is chosen, the performance is also limited due to aliasing effects caused by the filter banks. The other limitation is due to the part of the true room impulse response which cannot be covered by the adaptive filter. This happens because the FIR filter needs to be implemented with a fixed length, which is often restricted by computational complexity.
Here, a postfilter is usually used to suppress the parts of the feedback which remain after a feedback cancelation approach as it is depicted in Fig. 4.
Structure of a single-channel system in a closed loop with feedback canceler and residual feedback suppression
The idea is to use the method proposed in the previous section and adapt it, so it can be used as a postfilter. Because of its recursive nature, the model will also cover rooms with a long reverberation time without significant impact on the complexity. However, it has to be adapted with respect to the presence of the adaptive filter. Therefore, the effective impulse response, which is a combination of the true impulse response and the one estimated by the adaptive filter, needs to be computed to derive a new model. The effective impulse response can be derived using the signal e(n) from Fig. 4. For this signal holds:
$$ {}\begin{aligned} e(n) &= s(n) + b(n) + \sum\limits_{i=0}^{\infty} x(n-i)\, h_{\textrm{LM,{i}}}(n) - \hat{r}(n)\\ &= s(n) + b(n) + \sum\limits_{i=0}^{\infty} x(n-i)\, h_{\textrm{LM,{i}}}(n) - {\boldsymbol {x}}_{m}^{\textrm{\scriptsize{T}}}(n)\, \hat{{\boldsymbol{h}}}_{\text{LM}}(n)\\ &= s(n) + b(n) + \sum\limits_{i=0}^{\infty} x(n-i)\, h_{\textrm{LM,Res,{i}}}(n). \end{aligned} $$
$$\begin{array}{*{20}l} {}\hat{\boldsymbol{h}}_{\text{LM}}(n) = \left[ {\hat{h}}_{\textrm{LM,0}}(n),\,{\hat{h}}_{\textrm{LM,1}}(n),\,\cdots,\, {\hat{h}}_{\textrm{LM,m-1}}(n)\right]^{\textrm{\scriptsize{T}}} \end{array} $$
is a vector containing the adaptive filter coefficients and
$$\begin{array}{*{20}l} \!\!\!\!\!\!\!\!{\boldsymbol {x}}_{m}(n) = \left[x(n),\, x(n-1),\, \cdots,\, x(n-m+1)\right]^{\textrm{\scriptsize{T}}} \end{array} $$
contains the m latest samples of x(n), where m is the number of filter coefficients in the adaptive filter. With this the effective impulse response can be derived as:
$$\begin{array}{*{20}l} \notag {h}&_{\text{LM,Res},i}(n) \\ \notag &=\left\{\begin{array}{ll} {h}_{\text{LM},i}(n) - {\hat{h}}_{{\text{LM}},i}(n), & \text{for }0 \leq i < m,\\ {{h}}_{\text{LM},i}(n),& \text{else}, \end{array}\right.\\ &=\left\{\begin{array}{ll} {h}_{\text{LM},\Delta,i}(n), & \text{for } 0\leq i < m,\\ {{h}}_{\text{LM},i}(n),& \text{else.} \end{array}\right. \end{array} $$
As one may expect, knowledge about the actual system mismatch for each subband is necessary as it has an influence on the amount of residual feedback. Another difference compared to model A is that there is a direct connection from the loudspeaker signal to the error signal now which is caused by the misalignment in the adaptive filter.
However, this is unknown and has to be estimated. Furthermore, assumptions about the shape of the system misalignment over filter taps has to be made.
During the time when adaptive filters were being studied very extensively—decades ago—two different ideas of the progress of the filter coefficients during an adaptation period co-existed—and still do so today.
One idea is that adaptive algorithms spread the error more or less equally over all coefficients. As a consequence, the system mismatch vector can be modeled as a white process with zero mean and a time-variant, but lag-independent variance. Our investigations showed that this seems to be correct if the filter is well converged. So the first approach is to model a constant system mismatch for all filter taps, yielding a power envelope of the residual subband impulse response:
$$\begin{array}{*{20}l} \notag &\left| {H}_{\text{LM,mod,B}}(\mu,k)\right|^{2}\\ &=\left\{\begin{array}{ll}M \left\lVert {\boldsymbol {H}}_{\Delta}(\mu,k)\right\rVert^{2}, & \text{for }0\leq k \leq M-1,\\ A(\mu)\, e^{-\gamma(\mu)(k-P(\mu))}, & \text{for } M \leq k. \end{array}\right. \end{array} $$
In this work, this approach is named model B. M is the filter length in frames. In case of subband processing, HΔ(k) is a vector containing the system mismatch vector in every subband.
$$\begin{array}{*{20}l} \lVert&{\boldsymbol {H}}_{\Delta}(\mu,k)\rVert^{2} = \lVert{\boldsymbol {H}}(\mu,k)-\hat{{\boldsymbol{H}}}(\mu,k)\rVert^{2} \end{array} $$
$$\begin{array}{*{20}l} &= \sum_{j = 0}^{M-1}\lvert {H}(\mu,k-j)-\hat{{H}}(\mu,k-j)\rvert^{2} \end{array} $$
is a vector containing the norms of the system mismatch vectors. This is often estimated within adaptive control schemes. An overview about several estimation procedures can be found in [32]. The estimated residual feedback \(\hat {S}_{rr,\mathrm {B}}(\mu,k)\) can be obtained by convolving this modeled subband impulse response with the PSD of the loudspeaker signal yielding a solution consisting of two parts:
$$\begin{array}{@{}rcl@{}} & & \notag \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \hat{S}_{rr,\text{Res,B}}(\mu,k)\\ \notag &=& \!\!\sum_{i = 0}^{M-1}\hat{S}_{{xx}}(\mu,k-i) \, M\left\lVert {\boldsymbol {H}}_{\Delta}(\mu,k)\right\rVert^{2} \\ \notag && + \sum_{i =M}^{\infty}\hat{S}_{{xx}}(\mu,k-i) \, A(\mu) \, e^{-\gamma(\mu)\left(i-P(\mu)\right)}\notag\\ \notag &=& \!\!\sum_{i = 0}^{M-1}\hat{S}_{{xx}}(\mu,k-i) \, M\left\lVert {\boldsymbol{H}}_{\Delta}(\mu,k)\right\rVert^{2} \\ \notag & & + \sum_{i =0}^{\infty}\hat{S}_{{xx}}(\mu,k-i-M)\,A(\mu) \,e^{-\gamma(\mu)(i-P(\mu)+M)}\\ &=&\hat{S}_{\textrm{{xx}},rec}(\mu,k) \, M\left\lVert {\boldsymbol{H}}_{\Delta}(\mu,k)\right\rVert^{2}+\hat{S}_{{mm}}(\mu,k). \end{array} $$
Each of these two parts can be calculated recursively, leading to a very low computational complexity:
$$\begin{array}{*{20}l} \notag \hat{S}_{\textrm{{xx}},rec}(\mu,k) &=& \hat{S}_{\textrm{{xx}},rec}(\mu,k-1)\\ & + \hat{S}_{{xx}}(\mu,k)-\hat{S}_{{xx}}(\mu,k-M) \end{array} $$
$$\begin{array}{*{20}l} \notag\hat{S}_{{mm}}(\mu,k) &=& e^{-\gamma(\mu)}\,\hat{S}_{{mm}}(\mu,k-1)\\ \notag & + \hat{S}_{{xx}}(\mu,k\,-\,M)\,A(\mu)\,e^{-\gamma(\mu)(M-P(\mu))}.\\ \end{array} $$
In the top right of Fig. 3, the energy envelope of the subband-impulse response for a single subband of model B compared to model A (introduced in the previous section) can be seen.
The second idea is that the system mismatch of the individual coefficients is more or less proportional to the magnitude of the room impulse response of the system that should be identified. This behavior is also observable, but mainly at the beginning of adaptation processes or—in general—whenever the filter is not well adapted. Since feedback cancelation approaches for ICC systems face generally hard conditions e.g. permanent double-talk and high background noise levels, this model would be an option here.
To model this we assume the first interval for the direct part of the residual impulse response to be zero. This is usually the case only when the adaptive filter is initialized. After some iterations the coefficients will differ from zero. However, the system mismatch in this interval will always stay small compared to the interval between the largest coupling and the rest of the adaptive filter. Here, the system mismatch is modeled as exponentially decaying with the same T60 as it is used in all other approaches. Furthermore, we introduce Q(μ) which is the power of the maximum value in the system mismatch vector. This leads to the following power envelope of the model
$$\begin{array}{*{20}l} \notag &\left| H_{\text{LM,mod,C}}(\mu,k)\right|^{2}\\ &= \left\{\begin{array}{ll} 0, & \text{for }k < P(\mu),\\ Q(\mu,k) \, e^{-\gamma(\mu)(k-P(\mu))}, & \text{for }P(\mu)\leq k< M,\\ A(\mu)\, e^{-\gamma(\mu)(k-P(\mu))}, & \text{for }M\leq k. \end{array}\right. \end{array} $$
Using this model, the PSD of the residual feedback can be calculated as follows:
$$\begin{array}{*{20}l} & \!\!\!\!\!\!\!\!\! \hat{S}_{rr,\text{Res,C}}(\mu,k)\notag\\ &= \!\!\sum_{i =P(\mu)}^{M-1}\hat{S}_{{xx}}(\mu,k-i)\,Q(\mu)\, e^{-\gamma(\mu)\big(i-P(\mu)\big)}\notag\\ & \quad+ \sum_{i =M}^{\infty}\hat{S}_{{xx}}(\mu,k-i)\, A(\mu)\, e^{-\gamma(\mu)\big(i-P(\mu)\big)} \end{array} $$
$$\begin{array}{*{20}l} &= e^{-\gamma(\mu)}\, \hat{S}_{rr,\text{Res,C}}(\mu,k-1)\notag\\ & \quad+ \hat{S}_{{xx}}\big(\mu,k-P(\mu)\big)\,\notag Q(\mu,k)\\ & \quad+ \hat{S}_{{xx}}(\mu,k\,-\,M)\,\Big(A(\mu)\,-\,Q(\mu)\Big)e^{-\gamma(\mu)\big(M\,-\,P(\mu)\big)}.\notag \\ \end{array} $$
Model D
The drawback of the two proposed models is that the performance will depend on how accurate the estimation of the system distance is. An easier method compared to models B and C is to assume that the adaptive filter operates perfectly well and there is consequently only the length limitation which has to be covered by the postfilter. This would be advantageous, but it is not very realistic in practical approaches. Models B and C are better in this regard. In this case, ∥HΔ(μ,k)∥2 can be set to zero and the estimated PSD of the feedback signal simplifies to
$$\begin{array}{@{}rcl@{}} & & \notag \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \hat{S}_{rr,\text{Res,D}}(\mu,k) \\ \notag &=& e^{-\gamma(\mu)\left(M-P(\mu)\right)} \, A(\mu) \, \hat{S}_{{xx}}(\mu,k-M)\\ & & + e^{-\gamma(\mu)}\, \hat{S}_{rr,\text{Res,D}}(\mu,k-1), \end{array} $$
which corresponds to \(\hat {S}_{{mm}}(\mu,k)\) in Eq. (22).
Model parameters
To use the proposed model-based approach, a priori knowledge about the room is needed. In this work, the room is assumed to be power stationary, meaning that the power envelope of the room impulse response does not vary much over time. Depending on the particular control mechanism used for the adaptive filter, a correction of the model parameters based on the adaptive filter is also possible. The parameters can be extracted using a measured impulse response like it was proposed in [33].
To follow this approach, the time-domain impulse response has to be transformed into the subband domain
$$\begin{array}{*{20}l} H_{\text{LM}}(\mu,k) = \sum_{\kappa = 0}^{N-1}{h}_{\text{ana},\kappa}~{h}_{\text{LM},kL+\kappa}~ e^{-j2\pi \frac{\mu \kappa}{N}}, \end{array} $$
where N is the window length and, thus, the length of the DFT, and hana is the window function used in the filter bank. The absolute value of the subband impulse response is smoothed along the frequency axis in both positive (Eq. (28)) and negative (Eq. (29)) direction for every frame with the smoothing constant ζ:
$$\begin{array}{*{20}l} & {\tilde{H}}_{\text{LM}}(\mu,k) (1-\zeta) \, {\tilde{H}}{_{\text{LM}}(\mu-1,k)} + \zeta \, \lvert {H}_{\text{LM}}(\mu,k)\rvert \end{array} $$
$$\begin{array}{*{20}l} & {\bar{H}}_{\text{LM}}(\mu,k) (1-\zeta) \, {\bar{H}}_{\text{LM}}(\mu+1,k) + \zeta \, {\tilde{H}}_{\text{LM}}(\mu,k). \end{array} $$
This way, a zero-phase low-pass filter is realized to reduce the variance along the frequency axis. As a first step, the delay can be determined for each subband by finding the index of the first maximum value of the smoothed magnitude subband impulse response in each subband
$$\begin{array}{*{20}l} P(\mu) = {\underset{k\in[0,1,\cdots,L_{\text{LM}}-1]}{\arg\!\max}} \left\{ {\bar{H}}{_{\text{LM}}(\mu,k)} \, \lambda^{k} \right\}, \end{array} $$
with LLM representing the considered length of the impulse response in frames and λk representing an exponentially decaying series with λ∈(0,1), which can be used to avoid choosing late constructive interferences as maxima.
Next, a vector of length MLM is defined for every subband. It contains the logarithmic impulse responses, starting at the delay which was found before:
$$\begin{array}{*{20}l} \notag {\breve{{\boldsymbol {h}}}}&_{\text{LM}}(\mu) = \Big[\ln\left({\bar{H}}_{\text{LM}}(\mu,P(\mu))\right),\cdots,\\ &\ln\left({\bar{H}}_{\text{LM}}(\mu,M_{\text{LM}}+P(\mu))\right) \Big]^{T}. \end{array} $$
This can be modeled linearly for each subband
$$\begin{array}{*{20}l} \breve{h}_{\text{LM}}(\mu,k') &= \tilde{A}(\mu) +\tilde{\gamma}(\mu)k'\\ \notag k' &\in \left\{0,1,\cdots,M_{\text{LM}}-P(\mu)-1\right\}. \end{array} $$
Written in matrix notation, the logarithmic subband impulse responses can be combined to
$$\begin{array}{*{20}l} {\breve{{\boldsymbol{h}}}}_{\text{LM}} = {{\boldsymbol{\Gamma}}}\,{{\boldsymbol{\Theta}}}, \end{array} $$
where Γ is the observation matrix and Θ represents the parameter vector
$$\begin{array}{*{20}l} \mathbf{{\Gamma}} = \left[ {\begin{array}{ll} 0&1\\ 1&1\\ \vdots&\vdots\\ M_{\text{LM}}-P(\mu)-1&1 \end{array}}\right], {\boldsymbol {\Theta}} =\left[ {\begin{array}{l} \tilde{\gamma}(\mu)\\ \tilde{A}(\mu)\\ \end{array}}\right]. \end{array} $$
To find the unknown parameters in Θ, Eq. (33) can be modified according to
$$\begin{array}{*{20}l} {\boldsymbol {\Theta}} = \left[{\boldsymbol {\Gamma}}^{\textrm{\scriptsize{T}}}\,{\boldsymbol {\Gamma}}\right]^{-1} \, {\boldsymbol {\Gamma}}^{\textrm{\scriptsize{T}}} \, {\breve{{\boldsymbol{h}}}}_{\text{LM}}, \end{array} $$
with [ΓTΓ]−1ΓT=Γ† being the pseudo inverse of Γ. The last step is to convert the logarithmic values to linear values
$$\begin{array}{*{20}l} \gamma(\mu) &= e^{\tilde{\gamma}(\mu)}, \end{array} $$
$$\begin{array}{*{20}l} {A}(\mu) & = e^{\tilde{A}(\mu)}. \end{array} $$
A simplification, which we also used for our simulations, would be to assume the model parameters except the coupling factors to be identical for all frequencies. In this case, the delay and the decay constants can be computed directly from the energy decay curve (EDC) defined as
$$\begin{array}{*{20}l} \text{EDC}(i) = \frac{\sum\limits_{i'=i}^{N_{\text{LM}}-1}\lvert {h}_{\text{LM},i'}\rvert^{2}}{\sum\limits_{i'=0}^{N_{\text{LM}}-1}\lvert {h}_{\text{LM},i'}\rvert^{2}}. \end{array} $$
The EDC describes the remaining energy in the system at time instance i normalized to 0 dB. An example of an EDC measured in a car cabin can be seen in Fig. 5. The delay TD is the time instant where the EDC begins to drop. Using this value, the delay in frames can be derived as
$$ P(\mu) \approx P = \left\lfloor T_{\mathrm{D}}\cdot\frac{f_{\mathrm{S}}}{L}\right\rfloor \,\, \forall \mu, $$
Energy decay curve and coupling factors of an impulse response measured in a car cabin
with ⌊...⌋ denoting rounding towards the next smaller integer. The reverberation time T60 is the time instant when the energy that remains in the tail of the room impulse response reaches − 60 dB. Often, this cannot be found in the EDC, because there is measurement noise dominating when using a measured impulse response. In this case, the EDC has to be extrapolated linearly, as it was done in the example shown in Fig. 5. Using this value the decay instant can be derived:
$$\begin{array}{*{20}l} \gamma(\mu) \approx \gamma = \frac{2 \cdot 3 \,\ln (10) \, L}{T_{\textrm{60}}\,f_{\text{S}}} \,\,\forall \mu. \end{array} $$
The remaining parameters which need to be estimated are the coupling factors. These are the absolute squared values of the maximum of the subband impulse response, which can be found at frame index k=P according to Eq.(30). Therefore, the smoothed version of Eq. (29) should be used to reduce the variance along the frequency axis to yield
$$\begin{array}{*{20}l} A(\mu)= \big\lvert{\bar{H}}{_{\text{LM}}(\mu,P)}\big\rvert^{2}. \end{array} $$
In Fig. 6, a time-frequency analysis of the measured impulse response and the corresponding model can be seen.
Measured and modeled time-frequency representation of the energy envelope of hLM
The proposed schemes were tested in an ICC application. For this, an impulse response measured in a van was used. It is the same one as shown in Fig. 5. The complete setup is shown in Fig. 7.
Structure of a single-channel system used for the simulation
For our simulation we used clean speech signals from different male and female speakers sampled at 44.1 kHz. The DFT order was set to N=512 and we used a frameshift of L=256. The NLMS-based adaptive filter
$$\begin{array}{*{20}l} \notag \hat{{\boldsymbol {H}}}(\mu,k)=& \hat{{\boldsymbol {H}}}(\mu,k-1) \\ &+ \alpha ~\frac{E^{*}(\mu,k-1){\boldsymbol {X}}(\mu,k-1)}{\Vert{\boldsymbol {X}}(\mu,k -1)\Vert^{2}} \end{array} $$
with a fixed step-size α and E(μ,k) and X(μ,k) being the error signal or the excitation signal vector, respectively, was adapted until a specific system distance for all subbands was obtained. Afterwards the adaption was stopped by setting α to zero. As the system distance for this particular simulation was known, we used this value also as a parameter in model B and C to avoid the influence of estimation errors. M was set to four frames, corresponding to a filter length of 46.4 ms. Afterwards the step-size was set to zero for the simulation. This was done since the models would affect the convergence behavior of the adaptive filter, so the results would also be affected. The aim of the postfilter is to reduce the residual feedback in the microphone signal as much as possible without affecting the desired speech signal. To prove this, the ratio of the mean logarithmic speech power and the mean weighted logarithmic speech power was calculated as
$$\begin{array}{*{20}l} P_{s} &=\frac{\sum\limits_{k= 0}^{N_{\text{Frames}}-1}\sum\limits_{\mu= \mu_{\text{Start}}}^{{ \mu_{\text{Stop}}}}\tilde{P}_{s}(\mu,k)}{\sum\limits_{k= 0}^{N_{\text{Frames}}-1}\sum\limits_{\mu= \mu_{\text{Start}}}^{{ \mu_{\text{Stop}}}}\Phi_{s}(\mu,k)} \end{array} $$
$$\begin{array}{*{20}l} \!\!\!\!\!\!\!\!\tilde{P}_{s}(\mu,k) &=10\,\log_{10}\left(\frac{S_{{ss}}(\mu,k)\,\Phi_{s}(\mu,k)}{S_{{ss}}(\mu,k)\,\Phi_{s}(\mu,k)\,G_{\mathrm{X}}^{2}(\mu,k)}\right) \notag \\ &=\left. 10\,\log_{10}\left(\frac{1}{G_{\mathrm{X}}^{2}(\mu,k)}\right)\right|_{\Phi_{s}(\mu,k) \ne 0}. \end{array} $$
The same can be done for the reverberation parts of the signal
$$\begin{array}{*{20}l} {P}_{r} &=\frac{\sum\limits_{k= 0}^{N_{\text{Frames}}-1}\sum\limits_{\mu= \mu_{\text{Start}}}^{{ \mu_{\text{Stop}}}}\tilde{P}_{r}(\mu,k)}{\sum\limits_{k= 0}^{N_{\text{Frames}}-1}\sum\limits_{\mu= \mu_{\text{Start}}}^{{ \mu_{\text{Stop}}}}\Phi_{r}(\mu,k)} \end{array} $$
$$\begin{array}{*{20}l} \!\!\!\!\!\!\!\!\tilde{P}_{r}(\mu,k) &=10\,\log_{10}\left(\frac{S_{{ss}}(\mu,k)\,\Phi_{r}(\mu,k)}{S_{{ss}}(\mu,k)\,\Phi_{r}(\mu,k)\,G_{\mathrm{X}}^{2}(\mu,k)}\right) \notag \\ &=\left. 10\,\log_{10}\left(\frac{1}{G_{\mathrm{X}}^{2}(\mu,k)}\right)\right|_{\Phi_{r}(\mu,k) \ne 0}, \end{array} $$
where NFrames is the total number of processed frames in this simulation and μz∈[μStart,μStart+1,⋯,μStop] are the investigated subbands representing frequencies between about 90 Hz and 8000 Hz, where most of the speech power is located. Sss(μ,k) is the short-term power spectrum of the input frame of the clean speech signal. Φs(μ,k) is a binary mask based on a subband voice activity detection which is one for subbands where voice is detected and zero for those without voice. Φr(μ,k) is the equivalent for the reverberation signal with the additional condition that all time-frequency bins where Φs(μ,k)=1 are set to zero. An example of the described masks can be seen in Fig. 8, where the speech signal of the investigated subbands is at the top, the mask for the clean speech is in the middle, and the mask for the reverberation is at the bottom. Using these masks, the impairment is only evaluated where the respective signals are present. The reason for using this method is that we wanted to observe the ratio between speech impairment and feedback suppression as well as both quantities alone. This is important because a large speech impairment could lead to audible artifacts. More established methods like segmental SNR would only show the feedback reduction, which is not sufficient for our purpose.
Spectrogram of the speech signal at the top, speech active mask in the middle and reverberation active mask at the bottom
Furthermore, we used a short-time objective intelligibility (STOI) measure, which was proposed in [34]. This measurement shows good performance in evaluating degradation caused by time-frequency based algorithms e.g. noise reduction. As a reference signal we used the clean speech s(n). The loudspeaker signal x(n) was the signal to be evaluated.
In Table 1, the simulation results for different system distances are shown. Ps is the unwanted impairment of the clean speech signal, Pr is the equivalent for the reverberation signal. Pr−Ps is the distance between both. Since one impairment is wanted and the other one is not, it describes the attenuation of the unwanted signal. Consequently Ps has to be treated as an offset and needs to be compensated.
Table 1 Simulation results for the four different models at different system distances
When the system distance is at − 40 dB, the results in terms of the different approaches (B–D) for Pr−Ps as well as STOI are similar. However, it can also be seen that the impairment of the clean speech is significantly higher in case of model A, which is the original unadapted model. Model B, which is assumed to be correct for this simulation because the filter is well adapted and has a constant system mismatch, is very similar to models C and D with respect to Ps and Pr. This happens due to the fact that in case of a small system distance models (B–D) are nearly the same. When the system distance is increased, the impairment of the clean speech caused by models A and D are nearly constant whereas it increases in case of models B and C. This is exactly what one would expect because of the short-time stationary nature of speech leading to a large correlation when reducing the lag between an input frame and the models response. The system distance is used as a weight. By increasing it while decreasing the response time the correlation between the wanted and the unwanted signal increases.
In case of model B, there is an immediate response to an input signal. Model C also produces a response in the region of early feedback, resulting in a higher value of Ps when increasing the system distance. However, Pr−Ps is slightly better than in case of all other models. The best compromise regarding the impairment of the clean speech is reached with models C and D. Evaluating with STOI shows similar results. It can be seen that the results improve when the system distance decreases. For large system distances the scores are low even if there is no postfilter applied (-). This is due to the existing feedback in the processed signal. For a system distance of −10 dB it can also be seen that the score of model D is slightly higher, although Pr−Ps of model C is slightly larger. This is due to the fact that model D causes less speech impairment in this particular setup. Furthermore, a relation between STOI and Ps can be observed. A large value for Ps leads to a low STOI rating, which is due to the fact that STOI evaluates the degradation of the speech signal, which is mainly influenced by Ps. The best results in terms of STOI are reached when there is no postfilter at all. However, this does not mean that it makes no sense to use a postfilter at all, because one of its main purpose is to increase stability while saving computing power.
The impulse responses used for the model-based approach were measured under certain conditions. For example, this could be an empty vehicle at a certain temperature. In reality, however, these are subject to permanent fluctuations due to room changes. For example, a car could be fully loaded and fully occupied or empty. In addition, objects directly in front of sound sources or microphones could cause large attenuation.
Even changes in the distance between loudspeaker and microphone are conceivable. In the following, we will investigate such a situation where the model parameters are determined based on an impulse response of an empty van, but in fact there is a fully occupied interior. This variation of the acoustic path results in a reduced T60=80.3 ms compared to the original 119.9 ms as well as different coupling factors, which can be seen in Fig. 9.
Coupling factors of the empty car used as model parameters for simulation and true coupling factors of the fully loaded car
The results (see Table 2) show that both STOI and (Pr−Ps) are only slightly worse than before, but still very good.
Table 2 Simulation results for the four different models at different system distances with incorrect system parameters after a room change
In order to have a more robust evaluation, we simulated a second acoustic path. This time it is one that was recorded in a lecture hall and has a significantly longer decay time T60 of 777.8 ms. The delay TD is 18.7 ms. The results can be seen in Table 3. The higher reverberation time results in a slightly higher influence on the desired signal than in the simulation before. However, a clear attenuation of the feedback can still be seen. The previously discussed effects of the different models apply here without restriction.
Table 3 Simulation results for the four different models at different system distances with impulse response of a lecture room
In order to evaluate the subjective impairment of the desired signal we conducted a listening test with 26 untrained participants aged between 21 and 46 years. We used the same setup as shown before with a system distance of −30 dB.
Overall the procedure was a degradation category rating (DCR) according to ITU-T Rec. P.800 [35], which was modified for our purpose. We always played the unprocessed clean speech signal as reference and then the simulated versions with either one of the models (A–D) or the cancelation only (–) in random order. We used three female and two male speakers saying German sentences according to ITU-T Rec. P.501 [36]. In sum, every participant had to rate 25 signals. One of the female speakers was used for a trial run which we did not take into account to give the participants the opportunity to get used to the test procedure. The signals are provided on a web page [37].
The rating was defined as following:
5. Excellent – Speech sounds like the unprocessed signal
4. Good – Speech is slightly impaired, but sounds natural
3. Fair – Speech is impaired, but the artifacts are not disturbing
2. Poor – Speech quality degrades, interfering artifacts are clearly audible
1. Bad – Speech is heavily impaired
The results in terms of a mean opinion score (MOS) can be seen in Fig. 10.
Results (mean opinion score and variance) of subjective evaluation for the adaptive filter only (−), the "standard" model (A), the model with const. system mismatch (B), the model with an exponentially decaying shape of the system mismatch (C), and the model based on perfect adaptive filter (D)
The unadapted model and the version without any postfilter were rated with a mean opinion score below 2.5. This shows that artifacts caused by the postfilter are as bad as the residual feedback when there is no postfilter at all. All of the adapted models are rated with a MOS between 3.2 and 3.6 which means that the impairment is less compared to the unadapted model. Here model C shows the best results compared to models B and D. However, a Tukey honest significant difference (HSD) test with α=0.05 as suggested in [35] shows that there is no significant difference between models B, C, and D as well as between model A and no postfilter at all. However, the approaches can be grouped in model (B, C, and D) and (A and −).
The results in Table 4 are consistent with the objective results, with the exception of STOI. In the hearing test, the subjects rated a larger feedback as disturbing as a strong degradation of the speech signal. In contrast, the results according to STOI must be interpreted in such a way that a stronger feedback has less influence than the degradation of speech.
Table 4 Results of Tukey's HSD test with α=0.05
In a next step, we want to evaluate the model influence on the convergence behavior of an adaptive filter. For this, we used an NLMS-based adaptive filter based on pseudo-optimal step-size
$$\begin{array}{*{20}l} \alpha_{{opt}}(\mu,k) \approx \frac{\mathrm{E}\left\{| E_{u}(\mu,k)|^{2}\right\}}{\mathrm{E}\left\{| E(\mu,k)|^{2}\right\}} \end{array} $$
according to [26], where the expected value E{·} was approximated by first order IIR smoothing. The so-called undisturbed error Eu(μ,k), which is the error signal E(μ,k) without local signals, must be estimated as well. For this, it is replaced by
$$\begin{array}{*{20}l} E_{u}(\mu,k)&= E(\mu,k)-S(\mu,k)-B(\mu,k) \end{array} $$
$$\begin{array}{*{20}l} &= {\boldsymbol{H}}_{\Delta}^{H}(\mu,k){\boldsymbol{X}}(\mu,k), \end{array} $$
with HΔ(μ,k) being still not known. First, we replace it with an estimate yielding
$$\begin{array}{*{20}l} \left| \bar{\hat{E}}_{u}(\mu,k)\right|^{2} &= | \bar{X}(\mu,k)|^{2}\cdot \beta_{x}^{2}(\mu,k). \end{array} $$
In acoustic echo cancelation, \(\beta _{x}^{2}(\mu,k)\) could be estimated by minimum tracking the power of the noise-reduced error signal, which is then divided by the smoothed power of X(μ,k). However, this is not possible in feedback cancelation due to the permanent presence of local speech. Here, an approach [38] is to split \(\beta _{x}^{2}(\mu,k)\) into
$$\begin{array}{*{20}l} \beta_{x}^{2}(\mu,k) &= \beta_{\text{\tiny LEM}}^{2}(\mu,k)\cdot\frac{\lvert E(\mu,k)\rvert^{2}-\hat{S}_{{bb}}(\mu,k)}{\lvert Y(\mu,k)\rvert^{2}-\hat{S}_{{bb}}(\mu,k)} \end{array} $$
$$\begin{array}{*{20}l} &= \beta_{\text{\tiny LEM}}^{2}(\mu,k)\cdot \beta_{y}^{2}(\mu,k), \end{array} $$
with \(\beta _{\text {\tiny LEM}}^{2}(\mu,k)\) being a pre-measured quantity based on A(μ) and \(\beta _{y}^{2}(\mu,k)\) being the smoothed power ratio of the microphone and the error signal.
For the simulation, we used speech signals recorded in a car at 100 km/h. Now we replaced the fixed values of ||HΔ(μ,k)||2 with its estimate which we get from the step-size control \(\beta _{x}^{2}(\mu,k)\). Q(μ,k) in model C was replaced with \(M A(\mu)\beta _{x}^{2}(\mu,k)\). The loop gain initially was at 5 dB and was increasd with 0.8 dB/second until it reached the final value of 28 dB. The results in terms of the system distance over time with the same adaptive filter and the different models for the postfilter are shown in Fig. 11.
System distance of adaptive filter over time with different models
It can be seen that the best performance is reached when there is no postfilter at all. This is due to the fact that the achievable system distance at a fixed step-size depends only on the power ratio between feedback signal and local signal [26]. Even with an adaptive step-size control, as in this case, it does not always work well enough to compensate for this. As mentioned before, this is mainly due to Ps. This value attenuates the desired signal, which reduces the power of the loudspeaker signal by the same amount. To adjust the achievable filter performance, the filtered signal must be amplified by the value Ps, which is shown in Fig. 12.
System distance of adaptive filter over time with different models with gain adjustment
Here, we adjusted the gain by an offset of 5 dB in case of models C and D and 10 dB in case of model A. For these three models this nearly matches the individual values of Ps. However, this did not work for model B, where we had to add 17 dB to the loop gain. The difference is that Ps of models A,C, and D has only a small or no dependency on the system distance, whereas it increases with decreasing system difference in case of model B.
In this work, we investigated existing and proposed slightly extended postfilter schemes which are capable of suppressing residual feedback in closed-loop systems where adaptive feedback cancelers are used. We showed that there are different ways to adapt the reverberation model with respect to the feedback canceler. We were able to show by means of subjective and objective evaluation that all of our adapted models provide a better performance then using the standard reverberation model (model A) as a postfilter in a system with acoustic feedback canceler. However, there is a drawback. In the model, it was assumed that knowledge about the current system distance is available. This is, however, a quantity which is not available in real systems. But there are several step-size control methods available where a robust estimation of this quantity is included. In this case, it can also be used for the model-based postfilter. In all other cases, when no estimation of the system distance exists we propose to use the other adapted model, which assumes the system distance to be zero (model D). Beside models A to D several other models could be thought of and some of them were also tested during this research work, but at the end, we decided to continue only with these four approaches to keep this publication at a reasonable length.
The sample files which we used for the subjective evaluation can be found online [37].
STOI:
Short-time objective intelligibility
DCR:
Degradation category rating
PA:
HSD:
Honest significant difference
ICC:
In-car communication
MOS:
Mean-opinion score
Room-impulse response
EDC:
Energy-decay curve
FIR:
Finite impulse response
NLMS:
Normalized least mean square
PSD:
Power spectral density
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The authors would like to thank all participants of the hearing test.
Open Access funding enabled and organized by Projekt DEAL.
Marco Gimm and Philipp Bulling contributed equally to this work.
Digital Signal Processing and System Theory, Kiel University, Kaiserstr. 2, Kiel, 24143, Germany
Marco Gimm & Gerhard Schmidt
Cerence, Soeflinger Strasse 100, Ulm, 89077, Germany
Philipp Bulling
Marco Gimm
Gerhard Schmidt
MG and PB have conducted the research and analyzed the data. MG, PB, and GS authored the paper. All authors read and approved the final manuscript.
Correspondence to Gerhard Schmidt.
Gimm, M., Bulling, P. & Schmidt, G. Residual feedback suppression with extended model-based postfilters. J AUDIO SPEECH MUSIC PROC. 2021, 21 (2021). https://doi.org/10.1186/s13636-021-00205-8
Residual feedback suppression
In-car communication system
Feedback cancelation
Postfilter | CommonCrawl |
\begin{document}
\title{f The Delone Peak ootnote{The author is very thankful to his friend, professor Marjorie Senechal for spending much trouble on polishing an original text written poor author's English. He also thanks A.B. Delone and V.S.Makarov for presenting photos.}
\renewcommand{Abstract}{Abstract} \renewcommand{References}{References}
\begin{abstract} One of beautifull mountains in the neighborhood of the Belukha, the tallest summit in the Altai (in Siberia), is called the Delone Peak in the name of Boris Nikolaevich Delone (Delaunay), prominent mathematician. Here we are going to say about the life and science of this extremely suprising man \footnote{The paper is a seriously remade and supplemented version of the author's paper "The Delone Peak" published in "Kvant", 1987, No (Russian)}.
\end{abstract}
\textbf{Childhood}
Boris Nikolaevich Delone (Delaunay) (March 15, 1890, Saint-Peterbourg - July 17, 1980, Moscow) \footnote{In childhood and in the first part of his life, publishing papers in French,
Boris Delone used the spelling \textit{Delaunay}. Beginning in the 1930s, he published his papers mostly in Russian with the Russian spelling of his name. When, later, his work was translated into foreign languages his name was transliterated as \textit{Delone}. Through out the paper we will use the "Russian-English" spelling \textit{Delone} for Boris Nikolaevich and French spelling \textit{Delaunay} for his ancestors. } was, above all, an outstanding mathematician, and mathematics was certainly the main theme of his life. However, in a polyphonic musical piece it is hard to separate different themes. For people who knew Delone well, it is impossible to consider the mathematical part of his life separately from his other also very strong, interests and activities.
Boris Delone's father, Nikolay Delaunay\footnote{The Delaunay family originated in France: Boris' great grandfather Pierre Delaunay was a medical doctor in Napoleon's army when it invaded Russia in 1812. He was sent from burnt-down Moscow to Simbirsk. After he release he returned to France. But, being in love with a young beautifull gentlewoman from a noble Russian family, Pierre returned to Russia, married and remained there for the rest of his life. Reports that the Commandant of the Bastille in Paris, Marquis De Launay, an early victim of the French Revolution killed by rebels on July 14, 1789, was Boris Nikolaevich's ancester turned out to be wrong. According to later investigations by professor of astronomy Alexander Sharov (B. Delone's son-in-law), the Marquis belonged to another family.}, was a well-known professor of mechanics, an author of university textbooks. For Boris' biography, it turned out to be important that in the early 1900's, while his father was a professor of Warsaw university\footnote{At that time the territory of Poland was a part of the Russian empire.} he was a close friend of Georgy Voronoi. Later Boris Delone used to recall that Voronoi visited his home and the young teenager Boris often attended evening conversations between his dad and Voronoi.
Boris Delone had an excellent, many-facet education. His studies in music were quite solid: he played many pieces of Bach and Mozart, all the sonatas of Beethoven, and composed many himself. A music teacher recommended that the musically gifted boy enter a conservatory to study composition. However, his drawing teacher insisted that he continue his education in an art academy!
While parents and teachers tried to determine ``the future'' of Boris, the boy painted landscapes and played soccer, reproduced (in pencil) "The Last Supper" by Leonardo, and climbed trees with a younger sister on his shoulders (``for physical load'', Delone explained later). At the same time Boris turned his room into a real physics laboratory in which he made many of the devices himself. Later he often recalled proudly small technical tricks which allowed him, for example, to obtain, in a Leyden jar he made, a ``so-o-o huge spark''. Being keen on astronomy, Boris made a telescope for which he also polished a mirror. Later, each time he mentioned this episode, he said that "polishing a mirror from bronze was a stupid idea: it was very labor-consuming and grew dim fast".
It is impossible not to mention one more event in Boris Delone's youth. His father was a friend of a famous Russian scientist Nikolay Egorovich Zhukovski, the "grandfather of Russian aviation''. Under his influence in 1907 Nikolay Delaunay organized the first gliding circle in Russia in Kiev, after the family moved from Warsaw. The 17 year old Boris took an active part in this circle. During the next two years he constructed 5 gliders and flew on them. Needless to say we are talking here about flights of several tens of meters. Nevertheless, this circle played a notable role in development of aviation in Russia and in the world. In particular, among the participants was Boris's friend Igor Sikorski, also 17. Later, in 1913, Igor Sikorski constructed a huge (for that time) four-engine aircraft. After the Bolshevik revolution in 1917, Sikorski emigrated to the United States, where very soon he organized a famous helicopter company. One of his helicopters was the first to cross the Atlantic.
Before discussing the place of mathematics in Boris's childhood, one should mention one more passion that appeared at age of 12-15 and continued throughout his life: a passion for mountaineering and hiking. This passion was began and grew in early 1900's in the Swiss Alps, where Boris' family used to go for summer vacations. Later Delone talked about his climbing and hiking around Mont Ros and the Matterhorn above Zermat.
Boris Delone's mathematical talent was evident very early. By age 12 Boris was familiar with the elements of Calculus. At 13 he started his own investigations in Algebra and Number Theory. His father encouraged Boris in his mathematics studies and in 1904 took him to the International Congress of Mathematicians in Heidelberg. There Delone attended talks by Hilbert and Minkowski. Voronoi too took an active part in the congress, and got to know Minkowski. His conversation with Minkowski likely inspired Voronoi's tempest activity in the geometry of numbers in the last years of his life.
\textbf{Univerity years}
Voronoi's work strongly influenced Delone's, though Boris had no direct scientific contacts with him. Voronoi died suddenly at the age of 40 in 1908, exactly the same year that Delone entered the physics and mathematics department of Kiev university.
Several extremely talented young people entered the department at about that time, among them Otto Yu.Schmidt (1891-1956) and Nikolay Chebotarev (1894-1947). Chebotarev became an outstanding algebraist. Schmidt also became a prominent algebraist. He wrote a well-known book ``Abstract theory of groups'', founded a famous Moscow algebraic school, and became a full member of the USSR Academy of Sciences.
Moreover, the name of Schmidt became legendary in the Soviet Union as an explorer of the Arctic Ocean and a leader of a scientific expedition on the ice-breaker "Chelyuskin" in 1934.
Boris Delone, Schmidt, Chebotarev, and several others formed the core of the seminar led by Dmitrii Grave. This seminar on algebra and the algebraic theory of numbers, legendary in the history of Russian mathematics, determined Delone's field of studies for many years. Delone's first student work, ``Link between the theory of ideals and the Galois theory'', was awarded the ``Great golden medal'' of Kiev university. Delone's first published paper, ``On determining algebraic domain by means of congruence'', was dedicated to a new proof of a famous theorem of Kronecker on absolutely abelian fields.
\textbf{Cubic diophantine equations}
At about this time Delone began his investigation of the theory of Diophantine equations of degree 3 with two variables. This turned out to be the summit of his entire mathematical career.
Diophantine equations are
of the form
$$
p(x_1,\ldots, x_n )=0, \eqno(1)
$$
where $p(x_1,\ldots , x_n)=0$ is a polynomial with integer coefficients, for which one needs to find integer (or sometimes rational) solutions. For example, the equation $x^2+y^2=z^2$ has infinitely many integer solutions, the well-known Pythagorean triples (3,4,5), (5,12,13), and so on. These triples correspond to right-angled triangles with integer sides.
In Hilbert's famous list 23 problems the 10th problem asked: is there an algorithm to determine, from the coefficients of equation (1), whether or not solutions exist? Today, after the work by Yu. Matiyasevich, it is well-known that there is no such algorithm. In fact, the negative answer was expected by all experts on diophantine equations. They were very well aware of how hard each new result in this area was won.
The simplest form of diophantine equation is a linear equation with two variables:
$$
ax+by=1,
$$
where $a$ and $b$ are integers. This equation was investigated by the Indian mathematician Ariabkhata in 5th - 6th centuries.
Diophantine equations of degree 2 turned out to be more difficult. They were studied by Euler, Lagrange (a full theory of Pell's equation $ax^2+y^2=1$, where $a$ is a non-square number), Gauss and other outstanding mathematicians of 19th century.
As for diophantine equations of degree 3 or greater, a very deep result was obtained by the Norwegian mathematician A. Thue in 1908. Thue proved that diophantine equations of the following type
$$ f(x,y)=c,
$$
where $f(x,y)$ is a homogenous polynomial of degree 3 or greater, has at most a finite number of (integer) solutions, if the polynomial $f(x,y)$ cannot be decomposed into a product of polynomials of degree 2. This fact contrasts with quadratic Pell's equations which have an infinite number of solutions. One should note that Thue's theorem does not give any tools for finding these solutions.
It may be fortunate that the student Delone did not know Thue's work. Delone decided to investigate a longstanding problem of cubic diophantine equations equations of the form
$$ f(x,y)=1,
$$
where $f(x,y)$ is a homogenous form of degree 3 with negative discriminant. (Delone understood that if any Diophantine equations of degree could have infinitely many solutions, it would be this class.) First, he studied a cubic analogue of Pell's equation: $$ x^3q +y^3=1, \eqno(2) $$ where $q$ is an integer but not a cube. (Otherwise, a form $x^3q +y^3$ would be decomposable and the problem would reduce to solving diophantine equations of degree 1 and 2.)
Equation (1) has always a trivial solution (0; 1). The problem is to find other solutions, if any.
Delone introduced into consideration a ring $\sigma$ of algebraic numbers $z\sqrt[3]{q^2} + x\sqrt[3]{q} + y$, where $(z,x,y)$ are integer triples. It turns out that if $(x,y)$ is a solution (in integers) of equation (2) then
a number $\varepsilon= x\sqrt[3]{q} + y$ belongs to the ring $\sigma$ together with its reciprocal $\varepsilon ^{-1}$ . Such elements of the ring $\sigma$ are called \textit{units}. Thus, any solution of (2) corresponds to a unit of $\sigma$. Moreover, this unit $\varepsilon$ is a bi-term unit since one term, $z\sqrt[3]{q^2}$, is equal to $0$. It follows from a theorem of Dirichelt that the bi-term unit $\varepsilon\in \sigma$ is equal to a power $m$ of some so-called \textit{basic unit} $\varepsilon_0$ of $\sigma$: $\varepsilon=\varepsilon_0^m$. Indeed, it was also known that the degree $m>0$ if $0< \varepsilon<1 $.
Further, Delone proved that if the basic unit $\varepsilon_0$ is bi-term itself then no positive power $m$ of $\varepsilon_0$, except $m=1$, is equal to a bi-term unit. After that he investigated the case $\varepsilon=\varepsilon_0^m$, where $\varepsilon_0$ ia a three-term basic unit. By means of sparkling wit Delone proved that a bi-term unit of sort $\varepsilon_0^m$ can be just a bi-term basic unit $\varepsilon_0$ itself.
Thus, Delone obtained a final result: the cubic analogue of Pell's equation (2), besides a trivial solution (0; 1), has at most one more non-trivial solution. In order to get this solution one needs to find a basic unit. Here one needs to emphasize that about 20 years earlier, Voronoi had constructed an algorithm for computing basic units in the algebraic ring $\sigma$. If this unit is bi-term of the sort $x\sqrt[3]{q} + y$ then $(x,y)$ is the only non trivial solution of equation (2). If this is not the case then there are no non-trivial solutions at all.
After this strong success Delone proceeded to a more general problem: $$f(x,y)=1, \eqno(3)$$ where $f(x,y)=x^3 + ax^2y +bxy^2+cy^3$ is a cubic form in two variables with a negative discriminant.
In this more general case Delone again reduced the problem to an investigation of bi-term units of sort $x+y\rho$ in a ring $\sigma$ of algebraic numbers $x+y\rho +z\rho^2$. By means of his so-called "algorithm of ascent" Delone proved the following fundamental theorem:
\textit{In general case equation (3) has at most 3 integer solutions; in two concrete cases it has exactly 4 solutions; one concrete equation has sharply 5 solutions. No equations of sort (2) have more than 5 integer solutions.}
Moreover, by means of the algorithm of ascent, Delone was able to find all solutions for each concrete equation of form (3). For example, the equation $x^3 - xy^2+y^3=1$ has the 5 solutions (1; 1), (1; 0), (0; 1), (-1; 1), (4; -3). The fact that the equation has no other solutions follows from the theorem by Delone mentioned above.
However, Delone failed to rigorously prove that this method of obtaining solutions is workable for any equation. The matter was that then there was no upper bound expressed in terms of coefficients of the equation (3). Therefore, it was impossible to indicate the point when the algorithm of ascent had already given all solutions or not in a general case. The existence of such a upper bound was obtained in 1960's by A. Baker. This work of Baker has been awarded of the Fields medal.
Finally, one should emphasize that after classical results by Euler, Lagrange, Gauss et al on quadratic diophantine equations, works by Delone represented a serious breakthrough in the theory of cubic equations and remained unsurpassed until the late 1960's (A.Baker). Boris Delone himself evaluated his investigations on cubic equations with two unknowns as the best in all his scientific work.
\textbf{Life in Kiev, move to Saint-Peterbourg (Leningrad)}
The success was a result, as Delone used to say, of thousands of hours of very intensive work. One should say that Delone managed to win a very complicate problem regardless extremely hard political and living conditions which existed in Kiev. During WW I Kiev was captured by German troops. Therefore, in 1915-1916 the Delone family (together with other university professors) was forced to leave Kiev for Saratov. Nikolai Chebotarev went to Saratov too. Over there Boris Delone strongly influenced Nikolai, who was 4 years younger.
When the Delones returned to Kiev, in Russian territory, World War I was giving way to a more cruel and bloody civil war (1918-1922). Kiev was at the core of this war too. One political and military power was superceeded by another permanently. German troops left the city to the Red army (bol'shevics); in their turn the reds left Kiev to the White army (protectors of the Russian tsarist monarchy). The whites were pushed out again by the reds. The reds were replaced by Petlura's squadrons (a sort of ukranian national movement), which were replaced by Polish troops, then by the greens, the greens by the yellow-blues and so on.
Later B.N. Delone laughed about one dramatic and illustrative story from his family's life. B.N. had a younger brother Alexander (nicknamed Alik) who was an officer in Denikin's army\footnote{General Denikin was the White army comander, one of the most prominent activists of the White movement in Russia.}. Once, during a sudden capture of Kiev by red troops, Alik was forced to escape from Kiev. For this he exchanged his military uniform for civil clothes. The military form was left in a two door wardrobe in his father's apartment, where Boris also lived. The next day a red marine patrol came to them for inspection. During a search of the apartment a commander of the patrol approached the wardrobe and abruptly opened the right door. Suddenly a big dol fell from an upper shelf onto the floor and started crying loudly. This doll had been brought from Paris for someone in the family a few years before. At that time a crying doll was a real wonder. It surprised all the patrols so much that the commandant forgot to open the other door of the wardrobe behind which the white officer uniform hung... The time was very cruel and ruthless, shooting was an ordinary matter.
In this period Delone worked successively as a math teacher in a gymnasium, an educator at Kiev university, and an associate professor at Kiev polytechnical university. In 1920 he presented his work on cubic diophantine equations as a doctoral dissertation to Saint-Peterbourg (at that time named Petrograd) university. Andrei Markov (of the Markov processes) was a leader of a commission which considered the dissertation and highly evaluated the thesis. In 1922 Delone was invited to be a professor of Saint-Peterbourg university. The university hosted a famous Saint-Peterbourg school of number theory which had been begun by Euler and flourished in the late XIX century under P. Chebyshev's. A. Markov, G. Voronoi, A. Korkin, E. Zolotarev, A. Lyapunov and others formed the core of this school.
In due time, according to the traditions of this famous number theoretical school, the outstanding member G. Voronoi clothed a very geometrical work on parallelohedra in analytical clothes. In contrast, Boris Delone, possessing sure geometrical gift, geometrized numerous algebraic works, including the Voronoi algorithm for computing the basic unit in a ring which belongs to a field of degree 3 with a negative discriminant. After that he constructed a very nice geometric theory of cubic binary forms. In these and other works Delone interpreted the ring of cubic irrationalities as an integer lattice with a natural multiplication rule.
\textbf{The empty sphere method}
Delone's study of algebraic problems with geometric tools continued into the late 1950s. Here we are not going to focus any longer on this part of his activity. It is interesting that already in 1920s he wrote a paper on the empty sphere. A tendency to clarity and transparency was always peculiar to Delone. He gave an extremely clear description of the geometry of a tiling dual to the well-known Dirichlet tiling. Delone called this dual tiling a $L$-\textit{tiling}, one can only guess why. Likely, it was in honor of the first part of Dirichlet's surname (his full name was Johan Peter Gustav \textit{Lejeuene Dirichlet}) \footnote{Dirichlet, (1805-1859, used such 2- and 3-dimensional tessellations in his studies on quadratic forms.}
For this,
Delone introduced an \textit{(r,R)-set} $X$, where $r,R$ are positive numbers. Today such sets are also called \textit{Delone sets}, or \textit{separated nets}. By definition, such a set fulfils: \newline (1) any open ball with radius $r$ contains at most one point of $x$; \newline (2) any closed ball with radius $R$ contains at least one point of $x$.
Next, Delone displaced into space a small ball free of points of $X$. Such an empty ball exists due to the property (1). Then, leaving its center fixed, he expands the ball until it touches at least one point $x$ of $X$. This event must happen due to point (2). Moreover, it has to occur while the radius still remains smaller $R$.
Next, keeping $x$ on its boundary, Delone expands ball (still empty inside) away from $x$. Sooner or later a new point $x'$ of $X$ appears on the boundary of the growing ball. At the next stage Delone increases the ball further, keeping the points $x$ and $y$ on its boundary. This continues until the boundary contains a full set of independent points. If it is to remain empty, the ball cannot be increased further. The radius of this locally maximal ball $B$ cannot exceed $R$ (due to condition (2)).
By taking the convex hull of the points of $X$ on the boundary of $B$ Delone gets a convex inscribed polyhedron he called an $L$-\textit{solid}. The last crucial point is that the set of all $L$-solids forms a face-to-face tiling of space. Delone called it an $L$-\textit{tiling}.
Needless to say, in the 1960-70s Delone and we, his students, working with these tilings, did not suspect that very soon they would be called \textit{Delaunay triangulations}. I remember how proud Delone was of the crystallographic terminology \textit{Delone symbol} and \textit{Delone sort}. However, the use of these terms, named in honor of their inventor, cannot compare in use and dissemination with Delone triangulations.
Unfortunately this term came to Russia from the West after Delone's death. Here one should note a contribution Delone and Coxeter made together for this term to enter the mathematical language.
It was rather curious story. One day, sometime in the late 1950s, Delone saw a paper by H. S. M. Coxeter in which Coxeter had introduced and used a Delone tiling with no reference to Delone's work. Delone wrote Coxeter a letter in which, as Delone told it, he informed Coxeter that he had studied these tilings in the early 1920s. Delone asked Professor Steklov \footnote{Vladimir Andreevich Steklov (1864 - 1926) was a prominent Russian mathematician, academician and, in the 1920's, organizer and the first director of the Institute of Mathematics and Physics of Russian Academy of sciences. Later, in 1934, this institute was divided into two institutes; one of them is Steklov Mathematical institute of Russian academy of sciences.}, who went in 1924 to an international congress of mathematicians held in Toronto, to deliver
Delone's work. This paper was published in the proceedings of the congress\footnote{Sur la sph\'ere vide. Proceedings of the International mathematical congress held in Toronto, August 11 - 16, 1924, V. 1, Toronto, Univ. of Toronto press, 1928, p. 695 - 700.}. In addition, in his letter Delone mentioned that the congress was held in the "Coxeter's native city," Toronto. Coxeter's reply was, Delone said, very polite. He found, in the materials of Toronto congress, some mention on Delone's work. Coxeter asked Delone to forgive him because in 1924 he, Coxeter, was so young that he ``walked under table in short pants''. At the same time Coxeter wrote a letter about this to Prof. C.A.Rogers in Cambridge who was completing his book, ``Packings and Coverings''. Rogers inserted a mention of the Delaunay tiling. Very likely the term of Delaunay tilings (in Computational Geometry - triangulations) first appeared in C. A. Rogers' book.
In 1993, I had a chance to meet Professor Coxeter in person and I asked him about this story. On the whole Professor Coxeter confirmed it, except for ``walking under table in short pants''. In 1924 Coxeter (1907 - 2003) was already 17 years old. The phrase ``walking under the table'' was typical of Delone's expressive and humorous style.
\textbf{Parallelohedra and the Geometry of Positive Quadratic Forms}
The work on the empty sphere began a new line in scientific work for Delone. If earlier his geometric gift showed in the geometrization of algebraic and number theory, in the late 1920s geometry itself became the main object of his interest. In 1929 Delone completed a large paper in which he derived all combinatorial types of 4-dimensional parallelohedra\footnote{Indeed, Delone found 51 parallelohedra and missed one which later, in 1969, was discovered by his student M.I.Stogrin.} In this paper Delone continued the classical work of Fedorov (all 5 types of 3-dimensional parallelohedra) and Voronoi (general theory of Voronoi parallelohedra for arbitrary dimension, all 3 primitive 4-dimensional parallelohedra). A parallelohedron of dimension $d$ is a convex $d$-polyhedron which tiles euclidean $d$-space, i.e. paves space without gaps, in face-to-face way. This concept was introduced by the crystallographer Evgraph Stepanovich Fedorov.
A very important special case of parallelohedra is the Voronoi domains for points of integer lattices in space. These parallelohedra are called now Voronoi parallelohedra as Voronoi, in his two last outstanding memoirs of 1908, invented and developed a general theory of such parallelohedra. By the way, these memoirs contain certain very deep ideas which still remain unknown to modern mathematicians. In particular, a well-known idea on the lift of a Voronoi tiling to a paraboloid of revolution, discovered by computational geometers about 30 years ago, was introduced and actively used in Voronoi's memoirs. Voronoi had a strategic plan to investigate first of all Voronoi parallelohedra and constructed an exhaustive theory. In parallel, he posed a conjecture: any parallelohedron is affinely equivalent to some Voronoi parallelohedron. This conjecture was proved by Voronoi for the most generic class of parallelohedra - primitive parallelohedra. (Recall that a parallelohedron is \textit{primitive} if at any vertex of a tiling by such parallelohedron the minimum number of cells (i.e. $d+1$) meet. Later O.Zhitomirski proved affine equivalence of some Voronoi parallelohedron of any parallelohedron which is just \textit{primitive at $(d-2)$-faces}, i.e. the minimum number (three) of cells meet at any $(d-2)$-face \footnote{O.K. Zhitomirski was a student of Delone}.
Since this result the Voronoi conjecture has remained unsolved for over 70 years.
In his work on 4-parallelohedra, Delone proved that \textit{any 4-dimensional parallelohedron is affinely equivalent to some Voronoi parallelohedron}. For dimension 5 and higher the answer to the Voronoi conjecture remains unknown.
At this time Delone started the further development of a theory which appeared in works by Gauss and was further developed by Voronoi. Delone introduced a term for this field: \textit{geometry of positive quadratic forms}. The main idea of the geometry of positive quadratic forms (PQF geometry) is as follows.
Given an integer lattice $\Lambda \subset \mathbb{E}^d$ of rank $d$, i.e. the set of all points $ \textbf{x}=(x_1,\ldots , x_d)$ all of whose co-ordinates $x_i$ are integral with respect to some vector basis ${\cal E}=\{\textbf{e}_1,\ldots , \textbf{e}_d\}$. A $(d\times d)$-matrix $A=(a_{ij})$, where $a_{ij}=(\textbf{e}_i, \textbf{e}_j)$, is symmetric matrix called a Gram matrix. A quadratic form $\textbf{x}A \textbf{x}^{\small T}= \sum_{ij}a_{ij}x_ix_j$ is a positively definite form. Each quadratic form $f=\sum_{ij}a_{ij}x_ix_j$ with $d$ variables is assigned to a point $f\in \mathbb{E}^N$, where $N=\frac{d(d+1)}{2}$. Positive forms occupy in $\mathbb{E}^N$ an open convex cone $K$. Since a lattice $\Lambda$ has infinitely many bases ${\cal E}_1, {\cal E}_2, \ldots $, each lattice $\Lambda$ is associated with infinitely many equivalent points $f_1, f_2,\ldots \in {\cal E}$. The cone $K$ is divided into classes of equivalent forms, i.e. forms corresponding to the same lattice $\Lambda$. Therefore, a primary task of PQF geometry is the reduction theory which chooses from each class an appropriate representative, the so-called \textit{reduced form}.
The merit of Delone's PQF geometry is that he looked at the basic problems of the PQF theory from a unified geometric viewpoint. The empty sphere method lies at the core of PQF geometry, as does the interpretation of a $d$-dimensional lattice as a point in a convex cone in a high dimensional space $\mathbb{E}^{\frac{d(d+1}{2}}$. On this geometric foundation, Delone was able to systematize and enrich the results of his predecessors.
We mention here just one result relevant to the optimal covering of space by balls. Given a lattice $\Lambda\subset \mathbb{E}^d$, take the set of all equal balls
$$\{B_{\textbf{x}}(\rho)\, |\, \textbf{x}\in \Lambda\} \eqno(4)$$ centered at points $\textbf{x}$ of $\Lambda$ with radius $\rho$. One supposes here that $\rho$ is a minimal covering radius, i.e. the set (4) of balls covers $d$-space, but balls of smaller radius do not. The main goal of the theory is to find a lattice (with a unit volume of the fundamental parallelepiped) giving the thinnest covering of space by equal balls. Due to efforts of Voronoi, Delone and others, the problem of finding of the thinnest lattice covering is now reduced to the following. Voronoi found that the reduction domain in the cone $K$ with apex $O$ is divided into finite number of so-called \textit{$V$-type (Voronoi) domains} $V_1, V_2\ldots $. Each $V$-type domain $V_i$ is a polyhedral cone with the apex $O$. Each inner point $f\in V_i$ in the domain $V_i$ represents a lattice $\Lambda$ such that a Voronoi parallelohedron in the tiling $V(\Lambda )$ is primitive and the type of Voronoi tiling $V(\Lambda )$ remains unchangeable whi
le $\Lambda$ stays inside of $V_i$.
The Delone tiling $D(\Lambda )$ for this lattice is dual to the $V(\Lambda )$. Therefore, the $L$-type (or Delone type) domain coincides with the $V$-type domain. From the definition of a Delone cell it immediately follows that the covering radius for a lattice $\Lambda$ is equal to the biggest radius of an empty sphere circumscribed around simplices
in the Delone tiling. Take a simplex $S({\cal E})$, whose all vertices have integer coordinates with respect to a certain basis $\cal E$, and let its circumradius be $R(S, {\cal E})$. It turns out that a $R_0$-level surface $F(R_0)\subset \mathbb{E}^{\frac{d(d+1)}{2}}$ consisting of all bases $\cal E$ such that all circumradii $R(S, {\cal E})=R_0$ is \textit{convex}\footnote{B.N.~Delone, N.P.~Dolbilin, S.S.~Ryshkov and M.I.~Stogrin. A new construction in the theory of lattice coverings of an $n$-dimensional space by equal spheres. {\em Math.\ USSR Izv.} {\bf 4} (1970), 293--302.}. Due to this important fact, in each $L$-domain there exists at most one locally thinnest lattice. Therefore, the problem of the optimal lattice is reduced to a concrete optimization problem for any given $L$-type domain. After that by comparing the maxima over all finitely many $L$-type domains one can get the thinnest lattice covering of space by equal balls.
Among Delone's other results geometrizing the algebraic theory of numbers we mention a brilliant geometric description of the Voronoi algorithm for computing a basic unit of a ring contained in the field of the third degree with a negative discriminant\footnote{B.N.Delone. Saint-Peterbourg school of the number theory, 2005.}. This was not by chance. As we remember, a basic unit of an algebraic integer ring and its computation played an important role in Delone's research on diophantine cubic equations.
\textbf{Moscow period}
From 1922 to 1934 Delone stayed in Leningrad (now Saint-Peterbourg)and taught in Leningrad university. Due to his outstanding pedagogical and artistic talents Delone became famous as an outstanding, remarkable professor. In 1930-1934 he held the algebra and number theory chair of Leningrad university. In 1929 Delone was elected to the USSR Academy of Sciences as a corresponding member. He joined Institute of Mathematics and Physics just founded by Steklov. In 1934 the Institute was divided into the Steklov Mathematical Institute and the Lebedev Physics Institute. Since that time Delone's career has been tied to the Steklov institute of the Academy of Sciences. That same year, 1934, the USSR Academy of Sciences' headquarter moved to Moscow. The Steklov Institute was subdivided into two parts and one of them moved to Moscow too.
Delone joined Moscow branch of the Steklov Institute where he was the head of the algebra department (1945 - 1960) and later (1960 - 1980) the head of the geometry department. Besides his research at the institute, Delone was doing intensive pedagogical work, which continued into the early 1960s,
teaching at Moscow university (professor, 1935-1958, and the head of the higher geometry chair, 1935 - 1943). His long teaching experience resulted in a two-volume textbook on analytical geometry\footnote{B.N. Delone, D.E. Raikov, Analytical Geometry, V. I, (1948), pp. 456, V. II, (1949), pp. 516 (in Russian).}. Indeed, this enormous book can be thought of as a geometric encyclopaedia. The famous topologist Paul Alexandrov said that due to the beauty of its geometric ideas, Delone's book has no equal. Delone was also an organizer and the first head of the mathematics chair of the newly inaugurated Moscow Institute of Physics and Technology (1947 - 1961).
During his Moscow period, Delone did research mainly on PQF geometry, the geometrization of diophantine cubic equations, and application of mathematics to crystallography. In 1937-38 he published a large memoir (in two parts) on PQF\footnote{B.N.Delone. Geometry of positive quadratic forms, P.I, Russian surveys, \textbf{3}, (1937), p.16-62; P.II, Russian surveys, \textbf{4}, (1938), p.102-164, (in Russian)}. This paper gave a name to the whole field of geometry of numbers, as we had already said, the name of \textit{PQF geometry}. Geometric methods of studying cubic diophantine equations and cubic irrationalities were summarized in a monograph written jointly with his closest student, the outstanding algebraist D.K. Faddeev\footnote{Delone B.N., Faddeev D.K.. The theory of irrationalities of the third degree, M. - L., (1940), pp. 340 (in Russian), English translation: Translations of Mathematical Monographs V. 10, Providence, Rhode Island, AMS, (1964), pp. 513.}.
Among his crystallographic works one must mention Delone's discovery of the so-called \textit{Delone sorts of lattices}. Delone defines two lattices $\Lambda$ and $\Lambda '$ to be of the same sort if their Voronoi tilings $V(\Lambda)$ and $V(\Lambda ')$ are combinatorially isomorphic, the symmetry groups $Sym (V(\Lambda ))$ and $Sym (V(\Lambda '))$ are isomorphic, and the isomorphisms commute. Delone classified all 3-lattices into 24 Delone sorts. This classification involves not only the coincidence of lattice group properties, but also the geometry of the regular tilings corresponding to a given lattice. Therefore Delone's classification was a subsequent, more detailed development of Bravais classification of lattices, and proved itself a very natural and helpful for crystallography,
\textbf{Isohedral Tilings and Local Theory}
In the late 1950s B.Delone showed a strong interest in the theory of isohedral tilings. A tiling is called \textit{isohedral} \footnote{Delone himself called such tilings \textit{regular}.} if its symmetry
group operates on the set of tiles in transitive way, i.e. for any pair of tiles $P$ and $P'$ of a tiling there a symmetry of the tiling which maps $P$ to $P'$. An isohedral tiling is assumed to be face-to-face and the tiles convex. The cells of such a tiling are called \textit{stereohedra}\footnote{The term a stereohedron is due to E. Fedorov. A parallelohedron is a special case of stereohedron.}. Due to the Schoenflies-Bieberbach theorem\footnote{This principal theorem on the structure of crystallographic groups was an answer to Hilbert's 18th problem. In fact this problem consisted of two problems. One was to prove that for any dimension $d$ any crystallographic group operating in Euclidean $d$-space has a translational subgroup of finite index.}, any isohedral tiling with stereohedra is partitioned into several "lattices" of stereohedra. The set of stereohedra in one lattice is invariant and transitive with respect to the translation subgroup.
In contrast to the theory of parallelohedra, well-developed by H.Minkowski, G.Voronoi, B.Delone, et al, the theory of stereohedra remained undeveloped. Jointly with N.Sandakova Delone found the upper bound for the number of facets of a $d$-dimensional stereohedron in the given tiling: $$ f_{d-1}\leq 2(2^d-1)+(h-1)2^d, \eqno(5) $$ where $h$ denotes the number of lattices of stereohedra in the tiling. Since $h$ is upper bounded for the given dimension $d$ (for example, for all $d\geq 10$ $h\leq 2^dd!$) (5) implies a universal upper bound for the number of facets in any $d$-dimensional stereohedra. Thus, Delone generalized a celebrated upper bound by Minkowski for the number of facets in a parallelohedron, $$ f_{d-1}\leq 2(2^d-1). \eqno(6) $$ Though the upper bound (6) is obtained from (5) by putting $h=1$, it is true that, in contrast to (5) which is non-refinable, the upper bound (5) is very rough for $h>1$.
Nevertheless, from this upper bound the upper bound for the number of combinatorial types of isohedral tilings of euclidean space follows for any given dimension. This was a key idea in constructing a general theory of euclidean stereohedra.
One should mention also the so-called \textit{local theory} of isohedral tilings and regular point sets. An isohedral tiling (or regular point set = crystallographic orbit) is an appropriate mathematical model of a crystalline structure. The problem is that these mathematical concepts are defined by the notion of symmetry group yet a real crystalline structure is formed without any idea of a group: it is just a result of the interaction of nearby atoms. Assume we are dealing with a solution consisting of atoms of one sort. Under appropriate physical conditions (concentration, temperature, pressure) the crystallization process starts. Each atom tries to surround itself with other atoms in exactly the same way. So, it is likely that the source of global order in a crystalline structure is caused by the local identity of the structure in the neighborhood of each atom. Something like that can be watched in an army. When soldiers are stepping in rank each soldier aligns himself not with all rest but just with the fourth one on the right. Thus a \textit{local rule} provides a \textit{global order} in a rank.
Early on, Delone and his students suggested searching for mathematical theorems explaining the link between local identity and global order in crystals\footnote{B.N.Delone, N.P.Dolbilin, M.I.Stogrin, R.V.Galiulin. A local criterion for regularity of a system of points. Soviet Math. Dokl. 17 (1976), 319-322}. Later a more complete local theory, whose main goal was searching for and describing local conditions in discrete structures that provide global crystallographic order, was developed in a series of works by M.I.Stogrin, N.P. Dolbilin, and others.
With his research, Delone actively attracted young people and influenced their progress. Among his students there are several outstanding scholars who themselves founded their own schools. They include, most notably, the geometer academician Alexander Alexandrov, and the algebraists academician Igor Shafarevich and corresponding member of USSR Academy of sciences Dmitrii Faddeev. Among others one should mention the names of Tartakovskii V.A., Zhitomirskii O.K., Ryshkov S.S..
\textbf{Not only science}.
The Delone's scientific work is varied, valuable, and held in very high respect. However, the extreme brilliance of Delone's personality consists in that, besides outstanding scientific talent, he possessed other talents also and, what is most amazing, he was able to demonstrate explicitly these talents in his life. Likely, the ability to live as independently as possible, according to his own principles and goals, among which service to mathematics stood in first place, was Delone's main talent. I remember well how one gray day in March 1970 the great Andrei Kolmogorov came to Delone's small study in the Steklov to congratulate him on the occasion of his 80th birthday. Kolmogorov presented a gift (it was a rolled up gravure showing, in the old Dutch style, a marine landscape with war sailing vessels). During their talk Kolmogorov mentioned how much he envied Delone that he (Delone) had been able to avoid the temptations of seductive offers of this or that high position in scientific administration or something like that, and to organize his life by his own principles and enjoy daily life. It was really true to fact...
Delone was a fantastic lecturer. His popular lectures on mathematics and science designed for a wide audience sometimes attracted hundreds listeners. He had a lot of tricks to make his lectures very lively, attractive, unforgettable. For example, in 1940s he loved to give a talk "The Zhukovskii theorem on the supporting force of a plane's wing". The presentation, which was aimed at a very wide audience, started as follows:
\noindent \textit{Delone}: Do you know the famous mathematician, academician Paul Alexandrov? \newline \textit{Someone in the audience}: Yes, we do. \newline \textit{Delone}: Then, do you know that the academician Paul Alexandrov never flies in a plane? \newline \textit{Audience} is puzzled. \newline \textit{Delone}: Do you have any idea why academician Alexandrov is afraid of flying? \newline \textit{Audience} keeps puzzling \newline \textit{Delone}: Academician Alexandrov never flies on plane because he does not know the Zhukovski theorem. But I do and I fly very much. Today I will make you acquainted with this wonderful theorem and you also will never fear flying.
At this point the audience was ``warmed-up'' and ready to be acquainted with a ``wonderful theorem''.
We mention one more example, Delone's \textit{affine cat} which has firmly entered Russian mathematical folk lore. During his lectures on analytic geometry Delone drew with a chalk on a blackboard a cat and several its affine images. The affine cats looked funny and gave additional representation to the geometry of affine transformations.
Mathematics students of Moscow university write jokes about their most popular professors on the theme: Who and in what does a soup cook? A version of the joke relevant to Delone contained a bit of irony: "Delone used to cook a soup in an integer point lattice. True, the soup used to spill, but the \textit{obviousness} remained."
In 1934 Delone initiated and organized the first mathematical olympiad in Russia. Participants in this and subsequent olympiads have left very warm and amazing reminiscences of Delone. Talking to pupils, Delone evaluated the olympiads highly and loved to repeat that the only difference between an olympiad problem and a real research problem is that an olympiad problem takes few hours while the research problem takes few thousand hours. Later for many years B.N. was often the head of olympiad organizing committee.
Enthusiasm for mountain climbing, as has been already said, started in his childhood and continued throughout his life; it was, likely, his strongest passion after mathematics. His admiration for the beauty of mountain peaks and ridges was the stronger for its geometrical associations. For Delone, one of the most attractive components of mountaineering was the psychological fine-tuning needed for overcoming difficulties faced in mountains, for gaining a final victory over a summit and over oneself. Delone possessed these special psychological features and, there can be no doubt, these alpinist qualities helped him gain victory over serious mathematics problems.
Delone's mountain climbing was quite professional: he held a title, \textit{ master of Soviet mountaineering},
organized several alpinist camps, and served as a climbing coach. He was one of the best experts on the Western Caucasus. In 1937 he published a guide-book for alpinists "Summits of the Western Caucasus". This book contains detailed descriptions of paths on all the main mountains of this region. All of the numerous sketch-maps of routes of ascent on the tens of summits were drawn by Delone himself. But the main feature of the guide-book is a very detailed panorama of the ridge chain of the whole Western Caucasus (over 200 kilometers long). This panorama of the ridge chain was drawn by Delone himself sequentially from several outlying mountains of the region.
He also knew the Altai very well, especially the region of the Belukha. One of the paths up to the Belukha goes along an edge in which there is an intermediate mountain named the Delone Peak in the honor of one of its first subjugators. However, his most beloved place in the Altai was not Delone Peak but the lake Shavlo under the Skazka mountain (``skazka'' is the Russian word for ``tale''). He visited this beautiful place many times, the last time in 1970, in his jubilee year. Delone was convinced that the beauty of this place exceeded any place in the Alps (he knew the Swiss Alps very well) and admired it all his life.
\textbf{``An Evening of Life''}
In 1970 the German academy Leopoldina sent its congratulations on the occasion of the 80th anniversary of its member Boris Delone and wished him "a nice and quiet evening of life". The evening continued for 10 more years and turned out to be very saturated, but not quiet at all.
Between his 80th and 90th years, Delone continued his very active life: research, popularization of science, care of family, hiking, etc..
Delone was a very strong man from the birth and maintained his physical strength throughout his life by weekly hiking, climbing, etc.. Each Saturday and Sunday he went on a 30-kilometer walking or skiing tour, usually a very remote path going through wild but beautiful places. These weekly tours took place in any weather, even very bad and ugly. Usually a group of 5 or 6 people joined him in these tours and enjoyed the beauty of nature. Delone's intercourse with nature was sacramental. His close colleagues, who knew him well, recognized those instants when one should keep silence in order not to violate the voice of nature.
On Summer holidays Delone went to real mountains: the Caucasus, the Altai, Tien Shan, the Pamirs. These tours were filled with varied, incredible events. Stories of his climbing or mountain hiking tours told by Delone himself were of extraordinary interest and could constitute the whole book. Here is just one small typical episode which occurred to Delone in his 86th year.
July 6, 1975, Delone spends a cold night ( -25C) in a tent on a glacier under the famous peak of Khan-Tengri (7000 m, Tien-Shan mountain system, Central Asia) at a height about 4200 m. In the morning he comes down by helicopter to Frundze (now Bishkek, the capital of Kirgizstan) where heat exceeds +40C. After standing few hours in line he succeeds in purchasing an air ticket to Moscow and arrived that night in a sub-Moscow airport. Taking the last local train he arrives at the very small station of Abramtsevo at 2 am and walks in deep night through a dark dense forest to his "dacha" (country house). He loses the way and leaves his heavy rucksack in a secluded place. Only in the morning Delone succeeds in getting home safely.
It is not surprising that Delone was held in respect and liked by all his family. In his turn he remained a real head and support for his family and sought to resolve many of their problems. In particular, he cared very much for his beloved grandson Vadim Delone. The young Vadim was a talented poet and known dissident. On August 25, 1968, he took part in a celebrated political demonstration organized by 6 dissidents on the Red Square in Moscow against the Soviet Army's intervention in Czechoslovakia. For "one minute of freedom" with an unfolded slogan "For your and our freedom," the 20 year old Vadim was sentenced to 3 years imprisonment in a Siberian concentration camp. Delone did his best to ease his grandson's fate. He wrote letters to the KGB and asked many outstanding people to show support for Vadim, but nothing helped. All he was able to do was to visit the grandson in the camp as many times as he was allowed. One should add that Delone was not alone: Delone's former student Alexander Alexandrov\footnote{Academician A.D. Alexandrov (1912 - 1999) is an outstanding geometer of the XX century.} was also able to get permission from the authorities and joined Delone in visiting Vadim.
There was one more recipient of his direct care. This was his wife Mariya Henrikhovna. She originated from a Denmark immigrant family. Since 1924 when they married, Mariya was B.N.'s closest and hopeful friend.
They were born on the same day, March 15, though she was one year older. That Delone had a long, successful, highly creative life was in great part due to her. In particular She made sure that he would not miss his walking, hiking, and climbing tours. One day she told me how happy she was that B. N. had young colleagues and friends who were ready to join his weekly tours. When, at age about 85, she became seriously ill, Delone took care of her to the end. This care was very touching. For two or three years, these duties seriously limited his possibilities for walking. Nevertheless, even at this time he had at least one serious walking tour (30-40 kilometers long) a week.
After his wife's death in February, 1976, Delone's "one hundred days" began. He through himself into a storm of activity: daily research, lecturing, and on his two days off, 30 kilometers of hiking. Sometimes, on a working day that did not go well, B.N. made a suggestion: "Let us have some tea". This meant that he briefly stopped at home to get his well-known campfire teapot. After that we, two or three, went to a rail-station (1 hour), took a train (over 1 hour), then walked in deep snow (half an hour) until we reached a beautiful glade surrounded by high splendid pine trees. We cooked tea made of snow, enjoyed a very nice sunny day for about an hour and started back, in order to be home in the metropolis by the late evening. On the train B.N. became an inexhaustible storyteller, another of his many talents. "Do you know that my cousin Liza Pilenko is holy?" he might suddenly ask. "What
do you mean?" ask his puzzled companions (fellow-travelers.
"Yes, yes, true, really holy", B. N. insists with a sly smile. After a theatrical pause he starts telling a long, amazing, story about his cousin. After the revolution in 1917, a young poetess Liza Pilenko (married name Kuz'mina-Karavaeva) emigrated to Paris, where she changed professions several times, took the veil, and took the name Mother Mariya. In the early 1940's she entered the French Resistance, was imprisoned, and died in a concentration camp. Later the Catholic church officially canonized her.
One week in May turned out too stormy and ended with a hemorrhage of the brain. The "one hundred days" were over, "the Waterloo battle" had begun. Very limited mobility and speech, loss of memory, age 86: this promised nothing good. The doctors refused to give any consolatory prognosis. Nevertheless, B.N. started fighting for his health, for a life of full value. Certainly, he was not alone in this battle. His relatives, students, colleagues did their best in help. And B.N. won his Waterloo battle: indeed, a few months later he returned to his normal life. Certainly, his schedule became less hectic: instead of 30 kilometer walking tours now he did just 10. Wild, impassable, remote forests gave way to more "industrial" landscapes close to Moscow. However, B.N. had a life of full value. Though this victory over such serious illness at such an age did not bring him additional fame, it was likely the most valuable victory in his life.
Nevertheless, his life approached its natural end. I will never forget, how in March 1980, three or four months before he died, he took a book and read one remarkable statement by Poincar\'e. Overcome by emotion, his voice was trembling: "Life is just a short episode between two eternities of death and ... even in this episode conscious thought has lasted and will last only an instant. A thought is just a lightning flash in the middle of the infinite night. But this flash is everything".
\end{document} | arXiv |
Andrei Okounkov
Andrei Yuryevich Okounkov (Russian: Андре́й Ю́рьевич Окунько́в, Andrej Okun'kov) (born July 26, 1969) is a Russian mathematician who works on representation theory and its applications to algebraic geometry, mathematical physics, probability theory and special functions. He is currently a professor at the University of California, Berkeley and the academic supervisor of HSE International Laboratory of Representation Theory and Mathematical Physics.[1] In 2006, he received the Fields Medal "for his contributions to bridging probability, representation theory and algebraic geometry."[2]
Andrei Okounkov
Born
Andrei Yuryevich Okounkov
(1969-07-26) July 26, 1969
Moscow, Soviet Union
NationalityRussian
Alma materMoscow State University
AwardsFields Medal (2006)
EMS Prize (2004)
Scientific career
FieldsMathematics
InstitutionsColumbia University
National Research University – Higher School of Economics
Princeton University
University of California, Berkeley
University of Chicago
Doctoral advisorAlexandre Kirillov
Education and career
He graduated with a B.S. in mathematics, summa cum laude, from Moscow State University in 1993 and received his doctorate, also at Moscow State, in 1995 under Alexandre Kirillov and Grigori Olshanski.[3] He was a professor at Columbia University between 2010 and 2022. He previously was a professor at Princeton University, where he was awarded a Packard Fellowship (2001), the European Mathematical Society Prize (2004), and the Fields Medal (2006); an assistant and associate professor at Berkeley, where he was awarded a Sloan Research Fellowship; and an instructor at the University of Chicago. He rejoined the faculty at Berkeley in the summer of 2022.
Work
He has worked on the representation theory of infinite symmetric groups, the statistics of plane partitions, and the quantum cohomology of the Hilbert scheme of points in the complex plane. Much of his work on Hilbert schemes was joint with Rahul Pandharipande.
Okounkov, along with Pandharipande, Nikita Nekrasov, and Davesh Maulik, has formulated well-known conjectures relating the Gromov–Witten invariants and Donaldson–Thomas invariants of threefolds.
In 2006, at the 25th International Congress of Mathematicians in Madrid, Spain, he received the Fields Medal "for his contributions to bridging probability, representation theory and algebraic geometry."[2] In 2016, he became a fellow of the American Academy of Arts and Sciences.[4]
See also
• Newton–Okounkov body
References
1. "International Laboratory of Representation Theory and Mathematical Physics". mf.hse.ru.
2. "Information about Andrei Okounkov, Fields Medal winner" Archived 2016-10-20 at the Wayback Machine, ICM Press Release
3. "G. Olshanski - Grigori Olshanski". www.iitp.ru.
4. Newly Elected Members, American Academy of Arts and Sciences, April 2016, retrieved 2016-04-20
External links
• Andrei Okounkov home page at Columbia
• Andrei Okounkov home page at Princeton
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| Wikipedia |
Curve-shortening flow
In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a geometric flow, and is the one-dimensional case of the mean curvature flow. Other names for the same process include the Euclidean shortening flow, geometric heat flow,[1] and arc length evolution.
As the points of any smooth simple closed curve move in this way, the curve remains simple and smooth. It loses area at a constant rate, and its perimeter decreases as quickly as possible for any continuous curve evolution. If the curve is non-convex, its total absolute curvature decreases monotonically, until it becomes convex. Once convex, the isoperimetric ratio of the curve decreases as the curve converges to a circular shape, before collapsing to a single point of singularity. If two disjoint simple smooth closed curves evolve, they remain disjoint until one of them collapses to a point. The circle is the only simple closed curve that maintains its shape under the curve-shortening flow, but some curves that cross themselves or have infinite length keep their shape, including the grim reaper curve, an infinite curve that translates upwards, and spirals that rotate while remaining the same size and shape.
An approximation to the curve-shortening flow can be computed numerically, by approximating the curve as a polygon and using the finite difference method to calculate the motion of each polygon vertex. Alternative methods include computing a convolution of polygon vertices and then resampling vertices on the resulting curve, or repeatedly applying a median filter to a digital image whose black and white pixels represent the inside and outside of the curve.
The curve-shortening flow was originally studied as a model for annealing of metal sheets. Later, it was applied in image analysis to give a multi-scale representation of shapes. It can also model reaction–diffusion systems, and the behavior of cellular automata. The curve-shortening flow can be used to find closed geodesics on Riemannian manifolds, and as a model for the behavior of higher-dimensional flows.
Definitions
A flow is a process in which the points of a space continuously change their locations or properties over time. More specifically, in a one-dimensional geometric flow such as the curve-shortening flow, the points undergoing the flow belong to a curve, and what changes is the shape of the curve, its embedding into the Euclidean plane determined by the locations of each of its points.[2] In the curve-shortening flow, each point of a curve moves in the direction of a normal vector to the curve, at a rate proportional to the curvature. For an evolving curve represented by a two-parameter function C(s,t) where s parameterizes the arc length along the curve and t parameterizes a time in the evolution of the curve, the curve-shortening flow can be described by the parabolic partial differential equation
${\frac {\partial C}{\partial t}}={\frac {\partial ^{2}C}{\partial s^{2}}}=\kappa n,$
a form of the heat equation, where κ is the curvature and n is the unit normal vector.[3]
Because the ingredients of this equation, the arc length, curvature, and time, are all unaffected by translations and rotations of the Euclidean plane, it follows that the flow defined by this equation is invariant under translations and rotations (or more precisely, equivariant). If the plane is scaled by a constant dilation factor, the flow remains essentially unchanged, but is slowed down or sped up by the same factor.[4]
Non-smooth curves
In order for the flow to be well defined, the given curve must be sufficiently smooth that it has a continuous curvature. However, once the flow starts, the curve becomes analytic, and remains so until reaching a singularity at which the curvature blows up. For a smooth curve without crossings, the only possible singularity happens when the curve collapses to a point, but immersed curves can have other types of singularity.[5] In such cases, with some care it is possible to continue the flow past these singularities until the whole curve shrinks to a single point.[6]
For a simple closed curve, using an extension of the flow to non-smooth curves based on the level-set method, there are only two possibilities. Curves with zero Lebesgue measure (including all polygons and piecewise-smooth curves) instantly evolve into smooth curves, after which they evolve as any smooth curve would. However, Osgood curves with nonzero measure instead immediately evolve into a topological annulus with nonzero area and smooth boundaries.[7] The topologist's sine curve is an example that instantly becomes smooth, despite not even being locally connected; examples such as this show that the reverse evolution of the curve-shortening flow can take well-behaved curves to complicated singularities in a finite amount of time.[8]
Non-Euclidean surfaces
The curve-shortening flow, and many of the results about the curve-shortening flow, can be generalized from the Euclidean plane to any two-dimensional Riemannian manifold. In order to avoid additional types of singularity, it is important for the manifold to be convex at infinity; this is defined to mean that every compact set has a compact convex hull, as defined using geodesic convexity. The curve-shortening flow cannot cause a curve to depart from its convex hull, so this condition prevents parts of the curve from reaching the boundary of the manifold.[9]
Space curves
The curve-shortening flow has also been studied for curves in three-dimensional Euclidean space. The normal vector in this case can be defined (as in the plane) as the derivative of the tangent vector with respect to arc length, normalized to be a unit vector; it is one of the components of the Frenet–Serret frame. It is not well defined at points of zero curvature, but the product of the curvature and the normal vector remains well defined at those points, allowing the curve-shortening flow to be defined. Curves in space may cross each other or themselves according to this flow, and the flow may lead to singularities in the curves; every singularity is asymptotic to a plane.[10] However, spherical curves and curves which can be orthogonally projected into a regular convex planar curve are known to remain simple.[11] The curve shortening flow for space curves has been used as a way to define flow past singularities in plane curves.[12]
Beyond curves
It is possible to extend the definition of the flow to more general inputs than curves, for instance by using rectifiable varifolds or the level-set method. However, these extended definitions may allow parts of curves to vanish instantaneously or fatten into sets of nonzero area.[13]
A commonly studied variation of the problem involves networks of interior-disjoint smooth curves, with junctions where three or more of the curves meet. When the junctions all have exactly three curves meeting at angles of 2π/3 (the same conditions seen in an optimal Steiner tree or two-dimensional foam of soap bubbles) the flow is well-defined for the short term. However, it may eventually reach a singular state with four or more curves meeting at a junction, and there may be more than one way to continue the flow past such a singularity.[14]
Behavior
Avoidance principle, radius, and stretch factor
If two disjoint smooth simple closed curves undergo the curve-shortening flow simultaneously, they remain disjoint as the flow progresses. The reason is that, if two smooth curves move in a way that creates a crossing, then at the time of first crossing the curves would necessarily be tangent to each other, without crossing. But, in such a situation, the two curves' curvatures at the point of tangency would necessarily pull them apart rather than pushing them together into a crossing. For the same reason, a single simple closed curve can never evolve to cross itself. This phenomenon is known as the avoidance principle.[15]
The avoidance principle implies that any smooth closed curve must eventually reach a singularity, such as a point of infinite curvature. For, if a given smooth curve C is surrounded by a circle, both will remain disjoint for as long as they both exist. But the enclosing circle shrinks under the curvature flow, remaining circular, until it collapses, and by the avoidance principle C must remain contained within it. So, if C were to never reach a singularity, it would be trapped at a single point at the time when the circle collapses, which is impossible for a smooth curve. This can be quantified by observing that the radius of the smallest circle that encloses C must decrease at a rate that is at least as fast as the decrease in radius of a circle undergoing the same flow.[16]
Huisken (1998) quantifies the avoidance principle for a single curve in terms of the ratio between the arc length (of the shorter of two arcs) and Euclidean distance between pairs of points, sometimes called the stretch factor. He shows that the stretch factor is strictly decreasing at each of its local maxima, except for the case of the two ends of a diameter of a circle in which case the stretch factor is constant at π. This monotonicity property implies the avoidance principle, for if the curve would ever touch itself the stretch factor would become infinite at the two touching points.[17]
Length
As a curve undergoes the curve-shortening flow, its length L decreases at a rate given by the formula
${\frac {dL}{dt}}=-\int \kappa ^{2}\,ds,$
where the integral is taken over the curve, κ is the curvature, and s is arc length along the curve. The integrand is always non-negative, and for any smooth closed curve there exist arcs within which it is strictly positive, so the length decreases monotonically. More generally, for any evolution of curves whose normal speed is f, the rate of change in length is
${\frac {dL}{dt}}=-\int f\kappa \,ds,$
which can be interpreted as a negated inner product between the given evolution and the curve-shortening flow. Thus, the curve-shortening flow can be described as the gradient flow for length, the flow that (locally) decreases the length of the curve as quickly as possible relative to the L2 norm of the flow. This property is the one that gives the curve-shortening flow its name.[18]
Area
For a simple closed curve, the area enclosed by the curve shrinks, at the constant rate of 2π units of area per unit of time, independent of the curve. Therefore, the total time for a curve to shrink to a point is proportional to its area, regardless of its initial shape.[19] Because the area of a curve is reduced at a constant rate, and (by the isoperimetric inequality) a circle has the greatest possible area among simple closed curves of a given length, it follows that circles are the slowest curves to collapse to a point under the curve-shortening flow. All other curves take less time to collapse than a circle of the same length.[20]
The constant rate of area reduction is the only conservation law satisfied by the curve-shortening flow. This implies that it is not possible to express the "vanishing point" where the curve eventually collapses as an integral over the curve of any function of its points and their derivatives, because such an expression would lead to a forbidden second conservation law.[21] However, by combining the constant rate of area loss with the avoidance principle, it is possible to prove that the vanishing point always lies within a circle, concentric with the minimum enclosing circle, whose area is the difference in areas between the enclosing circle and the given curve.[22]
Total absolute curvature
The total absolute curvature of a smooth curve is the integral of the absolute value of the curvature along the arc length of the curve,
$K=\int |\kappa |\,ds.$
It can also be expressed as a sum of the angles between the normal vectors at consecutive pairs of inflection points. It is 2π for convex curves and larger for non-convex curves, serving as a measure of non-convexity of a curve.[23]
New inflection points cannot be created by the curve-shortening flow.[24] Each of the angles in the representation of the total absolute curvature as a sum decreases monotonically, except at the instants when two consecutive inflection points reach the same angle or position as each other and are both eliminated. Therefore, the total absolute curvature can never increase as the curve evolves. For convex curves it is constant at 2π and for non-convex curves it decreases monotonically.[25]
Gage–Hamilton–Grayson theorem
If a smooth simple closed curve undergoes the curve-shortening flow, it remains smoothly embedded without self-intersections. It will eventually become convex, and once it does so it will remain convex. After this time, all points of the curve will move inwards, and the shape of the curve will converge to a circle as the whole curve shrinks to a single point. This behavior is sometimes summarized by saying that every simple closed curve shrinks to a "round point".[26]
This result is due to Michael Gage, Richard S. Hamilton, and Matthew Grayson. Gage (1983, 1984) proved convergence to a circle for convex curves that contract to a point. More specifically Gage showed that the isoperimetric ratio (the ratio of squared curve length to area, a number that is 4π for a circle and larger for any other convex curve) decreases monotonically and quickly. Gage & Hamilton (1986) proved that all smooth convex curves eventually contract to a point without forming any other singularities, and Grayson (1987) proved that every non-convex curve will eventually become convex.[27] Andrews & Bryan (2011) provide a simpler proof of Grayson's result, based on the monotonicity of the stretch factor.
Similar results can be extended from closed curves to unbounded curves satisfying a local Lipschitz condition. For such curves, if both sides of the curve have infinite area, then the evolved curve remains smooth and singularity-free for all time. However, if one side of an unbounded curve has finite area, and the curve has finite total absolute curvature, then its evolution reaches a singularity in time proportional to the area on the finite-area side of the curve, with unbounded curvature near the singularity.[28] For curves that are graphs of sufficiently well-behaved functions, asymptotic to a ray in each direction, the solution converges in shape to a unique shape that is asymptotic to the same rays.[29] For networks formed by two disjoint rays on the same line, together with two smooth curves connecting the endpoints of the two rays, an analogue of the Gage–Hamilton–Grayson theorem holds, under which the region between the two curves becomes convex and then converges to a vesica piscis shape.[30]
Singularities of self-crossing curves
Curves that have self-crossings may reach singularities before contracting to a point. For instance, if a lemniscate (any smooth immersed curve with a single crossing, resembling a figure 8 or infinity symbol) has unequal areas in its two lobes, then eventually the smaller lobe will collapse to a point. However, if the two lobes have equal areas, then they will remain equal throughout the evolution of the curve, and the isoperimetric ratio will diverge as the curve collapses to a singularity.[4]
When a locally convex self-crossing curve approaches a singularity as one of its loops shrinks, it either shrinks in a self-similar way or asymptotically approaches the grim reaper curve (described below) as it shrinks. When a loop collapses to a singularity, the amount of total absolute curvature that is lost is either at least 2π or exactly π.[31]
On Riemannian manifolds
On a Riemannian manifold, any smooth simple closed curve will remain smooth and simple as it evolves, just as in the Euclidean case. It will either collapse to a point in a finite amount of time, or remain smooth and simple forever. In the latter case, the curve necessarily converges to a closed geodesic of the surface.[32]
Immersed curves on Riemannian manifolds, with finitely many self-crossings, become self-tangent only at a discrete set of times, at each of which they lose a crossing. As a consequence the number of self-crossing points is non-increasing.[33]
Curve shortening on a sphere can be used as part of a proof of the tennis ball theorem. This theorem states that every smooth simple closed curve on the sphere that divides the sphere's surface into two equal areas (like the seam of a tennis ball) must have at least four inflection points. The proof comes from the observation that curve shortening preserves the smoothness and area-bisection properties of the curve, and does not increase its number of inflection points. Therefore, it allows the problem to be reduced to the problem for curves near the limiting shape of curve shortening, a great circle.[34]
Huisken's monotonicity formula
Main article: Huisken's monotonicity formula
According to Huisken's monotonicity formula, the convolution of an evolving curve with a time-reversed heat kernel is non-increasing. This result can be used to analyze the singularities of the evolution.[35]
Specific curves
Curves with self-similar evolution
Because every other simple closed curve converges to a circle, the circle is the only simple closed curve that keeps its shape under the curve-shortening flow. However, there are many other examples of curves that are either non-simple (they include self-crossings) or non-closed (they extend to infinity) and keep their shape. In particular,[36]
• Every line stays unchanged by the curve-shortening flow. Lines are the only curves that are unaffected by the curve-shortening flow,[36] although there exist more complex stable networks of curves, such as the hexagonal tiling of the plane.
• The grim reaper curve y = − log cos x moves upwards without changing its shape. In the same way, any curve similar to the grim reaper is translated by the curve-shortening flow, shifted in the direction of the symmetry axis of the curve without changing its shape or orientation. The grim reaper is the only curve with this property.[36] It is also called the hairpin model in the physics literature.[37]
• A family of self-crossing closed curves, derived from projections of torus knots, shrink homothetically but remain self-similar under the curve-shortening flow.[36] These have come to be known as the Abresch–Langer curves, after the work of Abresch & Langer (1986),[38] although they were mentioned earlier by Mullins (1956) and rediscovered independently by Epstein & Weinstein (1987). These curves are locally convex, and therefore can be described by their support functions. Suitably scaled versions of these support functions obey the differential equation
$h''+h={\frac {1}{h}},$
which has positive periodic solutions (corresponding to curves with self-similar evolution) for any period that is strictly between π and $\pi {\sqrt {2}}$.[38]
• Other curves, including some infinite spirals, remain self-similar with more complicated motions including rotation or combinations of rotation, shrinking or expansion, and translation.[36]
• For networks of smooth curves, meeting in threes at junctions with angles of 2π/3, the self-similar shrinking solutions include a double bubble surrounding two equal areas, a lens shape (vesica piscis) bounded by two congruent arcs of circles together with two collinear rays having their apexes at the corners of the lens, and a "fish-shaped" network bounded by a line segment, two rays, and a convex curve. Any other self-similar shrinking networks involve a larger number of curves.[39] Another family of networks grows homothetically and remains self-similar; these are tree-like networks of curves, meeting at angles of 2π/3 at triple junctions, asymptotic to a fan of two or more rays that meet at a common endpoint. The two-ray case of these shapes is an unbounded smooth curve; for three or more rays the evolution of these shapes may be defined using generalized variants of the curve-shortening flow such as the one for varifolds. A given fan of four or more rays may be asymptotic to more than one different solution of this type, so these solutions do not provide a unique definition for the curve-shortening flow starting from a fan of rays.[40]
Ancient solutions
An ancient solution to a flow problem is a curve whose evolution can be extrapolated backwards for all time, without singularities. All of the self-similar solutions that shrink or stay the same size rather than growing are ancient solutions in this sense; they can be extrapolated backwards by reversing the self-similarity transformation that they would undergo by the forwards curve-shortening flow. Thus, for instance, the circle, grim reaper, and Abresch–Langer curves are all ancient solutions.[41]
There are also examples which are not self-similar. An explicit example is the Angenent oval solution after the work of Angenent (1992). This family of curves may be parameterized by specifying the curvature as a function of the tangent angle using the formula
$k(\theta ,t)={\sqrt {\cos 2\theta -\operatorname {coth} 2t}}$
and have as their limiting shape under reverse evolution a pair of grim reaper curves approaching each other from opposite directions.[42] In the Cartesian coordinate system, they may be given by the implicit curve equation[43]
$\cosh y-e^{-t}\cos x=0.$
In the physics literature, the same shapes are known as the paperclip model.[37]
The Angenent oval and shrinking circle solutions are the only ancient solutions whose timeslices bound bounded convex sets.[41] The Grim Reaper, stationary halfspace and stationary strip solutions are the only examples whose timeslices bound unbounded convex sets.[44] There exist many further (nonembedded) locally convex examples as well as many further (nonconvex) embedded examples.[45][46]
Numerical approximations
In order to compute the curve-shortening flow efficiently, both a continuous curve and the continuous evolution of the curve need to be replaced by a discrete approximation.
Front tracking
Front tracking methods have long been used in fluid dynamics to model and track the motion of boundaries between different materials, of steep gradients in material properties such as weather fronts, or of shock waves within a single material. These methods involve deriving the equations of motion of the boundary, and using them to directly simulate the motion of the boundary, rather than simulating the underlying fluid and treating the boundary as an emergent property of the fluid.[47] The same methods can also be used to simulate the curve-shortening flow, even when the curve undergoing the flow is not a boundary or shock.
In front tracking methods for curve shortening, the curve undergoing the evolution is discretized as a polygon. The finite difference method is used to derive formulas for the approximate normal vector and curvature at each vertex of the polygon, and these values are used to determine how to move each vertex in each time step.[48] Although the curve-shortening flow is defined by the motion of a curve perpendicularly to itself, some parameterizations of the curve-shortening flow may allow the vertices that approximate the curve to move non-perpendicularly. In effect, this allows the vertices to move along the curve, as the curve evolves. Choosing a careful reparameterization can help redistribute the vertices more evenly along the curve in situations where perpendicular motion would cause them to bunch up.[49] Merriman, Bence & Osher (1992) write that these methods are fast and accurate but that it is much more complicated to extend them to versions of the curve-shortening flow that apply to more complicated inputs than simple closed curves, where it is necessary to deal with singularities and changes of topology.
For most such methods, Cao (2003) warns that "The conditions of stability cannot be determined easily and the time step must be chosen ad hoc."[50] Another finite differencing method by Crandall & Lions (1996) modifies the formula for the curvature at each vertex by adding to it a small term based on the Laplace operator. This modification is called elliptic regularization, and it can be used to help prove the existence of generalized flows as well as in their numerical simulation.[51] Using it, the method of Crandall and Lions can be proven to converge and is the only numerical method listed by Cao that is equipped with bounds on its convergence rate.[52] For an empirical comparison of the forward Euler, backward Euler, and more accurate Crank–Nicolson finite difference methods, see Balažovjech & Mikula (2009).
Resampled convolution
Mokhtarian & Mackworth (1992) suggest a numerical method for computing an approximation to the curve-shortening flow that maintains a discrete approximation to the curve and alternates between two steps:
• Resample the current curve by placing new sample points at a uniform spacing, as measured by normalized arc length.
• Convolve the locations of the points with a Gaussian function with small standard deviation, in effect replacing each point's location with a weighted average of the locations of nearby points along the curve, with Gaussian weights. The standard deviation of the Gaussian should be chosen to be small enough that, after this step, the sample points still have nearly-uniform spacing.
As they show, this method converges to the curve-shortening distribution in the limit as the number of sample points grows and the normalized arc length of the convolution radius shrinks.[53]
Median filtering
Merriman, Bence & Osher (1992) describe a scheme operating on a two-dimensional square grid – effectively an array of pixels. The curve to be evolved is represented by assigning the value 0 (black) to pixels exterior to the curve, and 1 (white) to pixels interior to the curve, giving the indicator function for the interior of the curve. This representation is updated by alternating two steps:
• Convolve the pixelated image with a heat kernel to simulate its evolution under the heat equation for a short time step. The result is a Gaussian blur of the image, or equivalently the Weierstrass transform of the indicator function, with radius proportional to the square root of the time step.
• Set every pixel with numerical value less than 1/2 to 0, and every pixel with numerical value greater than 1/2 to 1, thresholding the image back to its original values in new positions.
In order for this scheme to be accurate, the time step must be large enough to cause the curve to move by at least one pixel even at points of low curvature, but small enough to cause the radius of blurring to be less than the minimum radius of curvature. Therefore, the size of a pixel must be O(min κ/max κ2), small enough to allow a suitable intermediate time step to be chosen.[54]
The method can be generalized to the evolution of networks of curves, meeting at junctions and dividing the plane into more than three regions, by applying the same method simultaneously to each region.[54] Instead of blurring and thresholding, this method can alternatively be described as applying a median filter with Gaussian weights to each pixel. It is possible to use kernels other than the heat kernel, or to adaptively refine the grid so that it has high resolution near the curve but does not waste time and memory on pixels far from the curve that do not contribute to the outcome.[55] Instead of using only the two values in the pixelated image, a version of this method that uses an image whose pixel values represent the signed distance to the curve can achieve subpixel accuracy and require lower resolution.[56]
Applications
Annealing metal sheets
An early reference to the curve-shortening flow by William W. Mullins (1956) motivates it as a model for the physical process of annealing, in which heat treatment causes the boundaries between grains of crystallized metal to shift. Unlike soap films, which are forced by differences in air pressure to become surfaces of constant mean curvature, the grain boundaries in annealing are subject only to local effects, which cause them to move according to the mean curvature flow. The one-dimensional case of this flow, the curve-shortening flow, corresponds to annealing sheets of metal that are thin enough for the grains to become effectively two-dimensional and their boundaries to become one-dimensional.[57]
Shape analysis
In image processing and computer vision, Mokhtarian & Mackworth (1992) suggest applying the curve-shortening flow to the outline of a shape derived from a digital image, in order to remove noise from the shape and provide a scale space that provides a simplified description of the shape at different levels of resolution. The method of Mokhtarian and Mackworth involves computing the curve-shortening flow, tracking the inflection points of the curve as they progress through the flow, and drawing a graph that plots the positions of the inflection points around the curve against the time parameter. The inflection points will typically be removed from the curve in pairs as the curve becomes convex (according to the Gage–Hamilton–Grayson theorem) and the lifetime of a pair of points corresponds to the salience of a feature of the shape. Because of the resampled convolution method that they describe for computing a numerical approximation of the curve-shortening flow, they call their method the resampled curvature scale space. They observe that this scale space is invariant under Euclidean transformations of the given shape, and assert that it uniquely determines the shape and is robust against small variations in the shape. They compare it experimentally against several related alternative definitions of a scale space for shapes, and find that the resampled curvature scale space is less computationally intensive, more robust against nonuniform noise, and less strongly influenced by small-scale shape differences.[58]
Reaction–diffusion
In reaction–diffusion systems modeled by the Allen–Cahn equation, the limiting behavior for fast reaction, slow diffusion, and two or more local minima of energy with the same energy level as each other is for the system to settle into regions of different local minima, with the fronts delimiting boundaries between these regions evolving according to the curve-shortening flow.[59]
Cellular automata
In a cellular automaton, each cell in an infinite grid of cells may have one of a finite set of states, and all cells update their states simultaneously based only on the configuration of a small set of neighboring cells. A Life-like cellular automaton rule is one in which the grid is the infinite square lattice, there are exactly two cell states, the set of neighbors of each cell are the eight neighbors of the Moore neighborhood, and the update rule depends only on the number of neighbors with each of the two states rather than on any more complicated function of those states. In one particular life-like rule, introduced by Gerard Vichniac and called the twisted majority rule or annealing rule, the update rule sets the new value for each cell to be the majority among the nine cells given by it and its eight neighbors, except when these cells are split among four with one state and five with the other state, in which case the new value of the cell is the minority rather than the majority. The detailed dynamics of this rule are complicated, including the existence of small stable structures.[60] However, in the aggregate (when started with all cells in random states) it tends to form large regions of cells that are all in the same state as each other, with the boundaries between these regions evolving according to the curve-shortening flow.[61]
Construction of closed geodesics
The curve-shortening flow can be used to prove an isoperimetric inequality for surfaces whose Gaussian curvature is a non-increasing function of the distance from the origin, such as the paraboloid. On such a surface, the smooth compact set that has any given area and minimum perimeter for that area is necessarily a circle centered at the origin. The proof applies the curve-shortening flow to two curves, a metric circle and the boundary of any other compact set, and compares the change in perimeter of the two curves as they are both reduced to a point by the flow.[62] The curve-shortening flow can also be used to prove the theorem of the three geodesics, that every smooth Riemannian manifold topologically equivalent to a sphere has three geodesics that form simple closed curves.[63]
Related flows
Other geometric flows related to the curve-shortening flow include the following ones.
• For simulating the behavior of crystals or other anisotropic materials, it is important to have variants of the curve-shortening flow for which the speed of flow depends on the orientation of a curve as well as on its curvature. One way of doing this is to define the energy of a curve to be the integral of a smooth function γ of its normal vectors, and form the gradient flow of this energy, according to which the normal speed at which the curve flows is proportional to an anisotropic analog of the curvature. This flow can be simulated by discretizing the curve as a polygon. In numerical experiments, initial curves appear to converge to the Wulff shape for γ before shrinking to a point.[64] Alternatively, one can let the curve flow with speed a(θ)κ + b(θ) where κ is the (usual) curvature and a and b are smooth functions of the orientation θ. When a(θ + π) = a(θ) and b(θ + π) = −b(θ) (so that the flow is invariant under point reflection), the resulting flow can be shown to obey the avoidance principle and an analog of the Gage–Hamilton–Grayson theorem.[65]
• The affine curve-shortening flow was first investigated by Alvarez et al. (1993) and Sapiro & Tannenbaum (1993). In this flow, the normal speed of the curve is proportional to the cube root of the curvature.[66] The resulting flow is invariant (with a corresponding time scaling) under the affine transformations of the Euclidean plane, a larger symmetry group than the similarity transformations under which the curve-shortening flow is invariant. Under this flow, an analogue of the Gage–Hamilton–Grayson theorem applies, under which any simple closed curve eventually becomes convex and then converges to an ellipse as it collapses to a point.[67]
• Transforming a curve with equal normal speeds at all points has been called the grassfire transform. Curves evolved in this way will in general develop sharp corners, the trace of which forms the medial axis of the curve.[68] A closely related curve evolution which moves straight segments of a polygonal curve at equal speeds but allows concave corners to move more quickly than unit speed instead forms a different type of topological skeleton of the given curve, its straight skeleton.[69]
• For surfaces in higher dimensions, there is more than one definition of curvature, including extrinsic (embedding-dependent) measures such as the mean curvature and intrinsic measures such as the scalar curvature and Ricci curvature. Correspondingly, there are several ways of defining geometric flows based on curvature, including the mean curvature flow (in which the normal speed of an embedded surface is its mean curvature), the Ricci flow (an intrinsic flow on the metric of a space based on its Ricci curvature), the Gauss curvature flow, and the Willmore flow (the gradient flow for an energy functional combining the mean curvature and Gaussian curvature). The curve-shortening flow is a special case of the mean curvature flow and of the Gauss curvature flow for one-dimensional curves.[20]
• In real-time path planning for mobile robots, a modified version of the curve-shortening flow with additional forces has been used to find paths that strike a balance between being short and staying clear of obstacles.[70]
• Inspired by the curve-shortening flow on smooth curves, researchers have studied methods for flowing polygons so that they stay polygonal, with applications including pattern formation and synchronization in distributed systems of robots.[71] Length-preserving polygonal flows can be used to solve the carpenter's rule problem.[72]
• In computer vision, the active contour model for edge detection and image segmentation is based on curve shortening, and evolves curves based on a combination of their curvature and the features of an image.[73]
Notes
1. The phrase "geometric heat flow" has also been used for flows on other kinds of object than curves, such as differential forms.
2. Devadoss & O'Rourke (2011), p.140: "a geometric flow [is] an evolution of the geometry of C over time t."
3. Devadoss & O'Rourke (2011), p. 140.
4. Grayson (1989a).
5. Grayson (1989a); White (2002).
6. Angenent (1991a); Altschuler & Grayson (1992).
7. Lauer (2013).
8. Lam & Lauer (2016).
9. Ritoré & Sinestrari (2010), p. 72.
10. Altschuler (1991).
11. Minarčík & Beneš (2020).
12. Altschuler & Grayson (1992).
13. Brakke (1978); White (1989); Cao (2003), "4.7.1 Brakke's varifold solution", p. 100. Lauer (2013).
14. Ilmanen, Neves & Schulze (2014).
15. White (2002), p. 526.
16. White (2002), p. 527.
17. Huisken (1998).
18. Chou & Zhu (2001), p. vii; White (2002), p. 526.
19. Brakke (1978), Appendix B, Proposition 1, p. 230; Chou & Zhu (2001), p. vii; White (2002), Theorem 1, p. 527.
20. White (1989).
21. Bryant & Griffiths (1995).
22. Kimmel (2004), pp. 182–183.
23. Brook, Bruckstein & Kimmel (2005).
24. Cao (2003), p. 143.
25. Brakke (1978), Appendix B, Proposition 2, p. 230; Chou & Zhu (2001), Lemma 5.5, p. 130; "6.1 The decrease in total absolute curvature", pp. 144–147.
26. Chou & Zhu (2001), p. vii; White (2002), Theorems 2 and 3, pp. 527–528; Cao (2003), Theorem 3.26, p. 47; Devadoss & O'Rourke (2011), p. 141.
27. Chou & Zhu (2001), p. vii; Cao (2003), p. 47; Devadoss & O'Rourke (2011), p. 141.
28. Chou & Zhu (1998).
29. Ishimura (1995).
30. Schnürer et al. (2011); Bellettini & Novaga (2011).
31. Angenent (1991b).
32. Grayson (1989b); White (2002), p. 528; Ritoré & Sinestrari (2010), Theorem 2.2.1, p. 73. This result was already stated as a conjecture by Gage & Hamilton (1986).
33. Angenent (1991a).
34. Angenent (1999).
35. Huisken (1990).
36. Mullins (1956); Abresch & Langer (1986); Epstein & Weinstein (1987); Chou & Zhu (2001), "2. Invariant solutions for the curve-shortening flow", pp. 27–44; Halldórsson (2012); Altschuler et al. (2013).
37. Lukyanov, Vitchev & Zamolodchikov (2004); Huisken & Sinestrari (2015).
38. Au (2010).
39. Schnürer et al. (2011).
40. The two-ray case was already described by Mullins (1956). For the generalization to two or more rays and issues of non-uniqueness see Brakke (1978), Appendix C, pp. 235–237 and Ilmanen, Neves & Schulze (2014).
41. Daskalopoulos, Hamilton & Sesum (2010).
42. Angenent (1992).
43. Broadbridge & Vassiliou (2011).
44. Bourni, Langford & Tinaglia (2020).
45. Angenent & You (2021).
46. You (2014).
47. See, e.g., Scriven (1960); Holden & Risebro (2015).
48. Merriman, Bence & Osher (1992); Mikula & Ševčovič (1999); Cao (2003), "5.1.1 Finite difference methods", pp. 107–108.
49. Kimura (1994); Deckelnick & Dziuk (1995); Mikula & Ševčovič (2001); Barrett, Garcke & Nürnberg (2011); Elliott & Fritz (2017).
50. Cao (2003), "5.1.1 Finite difference methods", pp. 107–108.
51. Ilmanen (1994), p. 1.
52. Crandall & Lions (1996); Deckelnick (2000); Cao (2003), "5.2.3 A monotone and convergent finite difference schemes", p. 109.
53. Mokhtarian & Mackworth (1992), pp. 796–797; Cao (2003), pp. 10–11.
54. Merriman, Bence & Osher (1992).
55. Cao (2003), "5.2.4 Bence, Merriman and Osher scheme for mean curvature motion", pp. 109–110. For the correctness of median filtering with other isotropic kernels, see section 4.4.1, pp. 90–92.
56. Esedoḡlu, Ruuth & Tsai (2010).
57. Mullins (1956); Rhines, Craig & DeHoff (1974); Brakke (1978), Appendix A, pp. 224–228.
58. Mokhtarian & Mackworth (1992).
59. Rubinstein, Sternberg & Keller (1989).
60. Pickover (1993).
61. Vichniac (1986); Chopard & Droz (1998).
62. Benjamini & Cao (1996); Ritoré & Sinestrari (2010), Theorem 2.3.1, p. 75.
63. Grayson (1989b).
64. Dziuk (1999); Haußer & Voigt (2006).
65. Chou & Zhu (2001), Chapter 6: A Class of Non-convex Anisotropic Flows, pp. 143–177.
66. Cao (2003), "3.2.3 The affine invariant flow: the simplest affine invariant curve flow", pp. 42–46.
67. Angenent, Sapiro & Tannenbaum (1998); Cao (2003), Theorem 3.28, p. 47.
68. Sapiro & Tannenbaum (1993).
69. Aichholzer et al. (1995).
70. Huptych & Röck (2021).
71. Smith, Broucke & Francis (2007).
72. Cantarella et al. (2004).
73. Kichenassamy et al. (1995).
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Xue Wang1 &
Zhisheng Yin1
For the compressed sensing of multiband signals, modulated wideband converter (MWC) is used as the sampling system, and the signal is reconstructed by the simultaneous orthogonal matching pursuit algorithm (SOMP) and its derivative algorithms. In order to find matching atoms, we need to obtain the inner product between atoms in sensing matrix and columns in residual matrix. Next, several inner products corresponding to the same atom constitute an inner product vector. By calculating its 2-norm, we can find the maximum value, whose corresponding atom is the matched atom. However, the inner product actually cannot reflect the relevancy between atoms and residual matrix very accurately, which may eventually lead to wrong results for a certain probability. The main idea of this paper is to change the inner product into the correlation coefficient, so that we can measure the relevancy between atoms and the residual better. Simulation results show that the improved algorithms can get higher probability of the signal reconstruction compared with the original algorithms in the condition of high signal-noise ratio (SNR). It also means that less samples were needed to reconstruct signals than traditional algorithms when the number of bands is unchanged. Since calculating correlation coefficient at each iteration will cost a lot of time, we also proposed a simplified algorithm, which can also improve reconstruction probability and reconstruction time is about the same as corresponding traditional algorithms.
Traditional compressed sensing theory is mainly used to process discrete and finite-dimensional digital signal. However, it is expected to develop a technology which can sample continuous and infinite dimensional analog signal at sub-Nyquist rate, so as to truly break the bandwidth limitation of existing ADC equipment and reconstruct original signal from the samples of baseband signal after low-pass filters and finally ease the pressure of hardware sampling. To address this problem, a variety of solutions have been proposed. Analog to information convertor (AIC) and Xampling for multiband analog signal [1–5] are two relatively mature technologies among them. Xampling uses modulated wideband converter [6] (MWC) to sample, whose results are infinite measurement vectors (IMV), and it cannot directly reconstruct signal using the traditional reconstruction algorithms. To solve this problem, we can reconstruct the original signal by turning IMV problem into multiple measurement vector (MMV) problem using continuous to finite (CTF) modular [7] under the premise of joint sparse [8].
The simultaneous orthogonal matching pursuit (SOMP) [9, 10], treated as a derivative algorithm of matching pursuit simultaneous (OMP) algorithm to solve MMV problem, is what used for reconstruction in CTF at present. Based on this method, many novel MMV algorithms have been proposed according to the derivative algorithms of OMP such as regularized orthogonal matching pursuit (ROMP) [11], stagewise orthogonal matching pursuit (StOMP) [12], compressive sampling matching pursuit (CoSaMP) [13], and subspace pursuit (SP) [14]. The main idea of these algorithms is firstly calculating the inner product between atoms in sensing matrix and columns in residual matrix. Then, several inner products corresponding to the same atom constitute an inner product vector. By calculating its 2-norm. We can find the maximum value whose corresponding atom is the matched atom. In fact, the inner product cannot measure the relevancy between atoms and residual matrix very well and sometimes will eventually lead to errors.
The idea of this paper is to change the inner product into the correlation coefficient, so that we can measure the matching degree between atoms and the residual better. In order to check the performance of the changement, the SOMP algorithm and its derivative algorithms such as MMV-ROMP, MMV-StOMP, MMV-CoSaMP, and MMV-SP, which can solve MMV problems, are chosen for comparison. The proposed algorithms are improved by changing the inner product into correlation coefficient as screening criterion of atoms, then compare the signal reconstruction probability of these methods with original algorithms. The results show that the improved algorithms have better reconstruction probability and there are less samples required.
Signal model of multiband
Assuming x(t) is a continuous signal in L 2 space which satisfies the square integration as follows.
$$ \int_{- \infty }^{+ \infty} {{\left| {x(t)} \right|}^{2}} dt < + \infty $$
And its Fourier transformation can be expressed as
$$ X(f) = \int_{- \infty }^{+ \infty} {x(t){e^{- j2\pi ft}}dt} $$
If X(f) is finite banded and its frequency spectrum is in [−1/2T,1/2T], then its Nyquist sampling frequency should be f NYQ=1/T. Besides, if X(f) satisfies the structure shown in Fig. 1, namely, it contains N disjoint sub-frequency bands whose bandwidths are all less than B, then x(t) is a multiband signal.
Model of multiband signal
Modulated wideband converter
The model of MWC for multiband signal is shown as Fig. 2.
Model of MWC
Multiband modulated wideband signal converter is shown in Fig. 2. There are m sampling channels, and the mixing function p i (t) is a pseudo-random sequence with T p -period. Its value is {+1,−1}, and it has M pulses in each period whose interval is T s . The values of p i (t) in the first k intervals are denoted as α ik , h(t) is a low-pass filter whose cutoff frequency is 1/2T s , and the sampling frequency is 1/T s . The signal x(t) is transmitted through m channels at the same time, and it is multiplied with different pseudo-random sequences in each branch whose cycle is abiding and obey the same distribution. The samples y i [n],n=1,2,…,m can be obtained after low-pass filters and low-speed sampling.
And the fourier transform can be expressed as
$$ {}{Y_{i}}({e^{j2\pi f{T_{s}}}}) = \sum\limits_{n = -\infty}^{\infty} {{y_{i}}[n]{e^{-j2\pi fn{T_{s}}}} = \sum\limits_{l = -{L_{0}}}^{{L_{0}}} {{c_{il}} X\left(\,f - l{f_{p}}\right)}} $$
where f p =1/T p , F s =[−f s /2,f s /2], \({L_{0}} = \left [ {\frac {{f_{\textit {NYQ}}} + {f_{s}}}{2{f_{p}}}} \right ]- 1\), and \({c_{\textit {il}}} = \frac {1}{{T_{p}}} \int _{0}^{T_{p}} {{p_{i}}(t){e^{-j \frac {2\pi }{T_{p}} lt}}} dt\).
For convenience, the formula can be expressed in matrix form
$$ \textbf{y}(f) = {\textbf{A}\alpha}(f)f \in {\textbf{F}_{s}} $$
From Eq. (4), it can be seen that y(f) is a m×1 dimension vector which is made of \({Y_{i}}\left ({e^{j2\pi f{T_{s}}}}\right)\). A is a M×L dimensional matrix which is composed of \({A_{\textit {il}}} = {c_{i, -l}} = c_{\textit {il}}^{*}, -{L_{0}} \leq l \leq {L_{0}}\), L=2L 0+1, and L≈f NYQ/B in common. Moreover, α(f) is a L×1 dimensional vector which is composed of α i (f)=X(f+(i−L 0−1)f p ),f∈F s . To recover a signal, we need to firstly solve the sparse spectrum α(f) from Eq. (4). Then, the estimated value \(\hat {\mathbf {x}}(t)\) of x(t) is obtained by using the inverse Fourier transform. However, f is defined over a continuous interval whose samples are infinite dimensional vector. Thus, in fact, this is a problem about how to gain the sparse solution from one more of the numerous less demand equations. Moreover, it cannot be calculated by the conventional compressed sensing reconstruction algorithms.Therefore, we should get the support set S of signal by using CTF module and MMV problem can be turned into SMV problem by joint sparsity; then, SOMP algorithm is adopted. In CTF, we first use y[n] to construct a matrix as follows
$$ \textbf{Q} = \int_{f \in {\textbf{F}_{s}}} {\textbf{y}(f)} {\textbf{y}^{H}}(f)df = \sum\limits_{n = -\infty}^{+\infty} {\textbf{y}[n]{\textbf{y}^{T}}[n]} $$
where y[n]=[y 1[n],y 2[n],…,y m [n]]T, then matrix Q is divided into Q=V V H, then we can get framework matrix V. Then, the sparsest support set \(\bar {\mathbf {U}}\) of V=AU is obtained. According to the support set \(\bar {\mathbf {U}}\) and y[n], we can solve the signal's support set S, and finally, compose the matrix A S by the columns in the matrix A which correspond to the elements of S, then reconstruct sparse spectrum α(f) by using Eq. (6), so as to reconstruct the traditional signal.
$$ \begin{aligned} {\mathbf{\alpha}_{\textbf{s}}}(f) = \textbf{A}_{\textbf{s}}^{\mathbf{\dag}}\textbf{y}(f) \\ {\alpha_{i}}(f) = 0,i \notin \textbf{S} \end{aligned} $$
Improved simultaneous orthogonal matching pursuit algorithm
The reconstruction algorithm for the MWC system is usually simultaneous orthogonal matching pursuit algorithm, whose steps are as follows:
Input: M×L dimensional sensing matrix A, the number of sub frequency bands K, m×2K dimensional frame vector V, and residual threshold θ.
Output: 2K×1 dimensional support set S.
Initialization: support set S=∅, residual matrix R=V.
Iteration: Repeat the following steps until the residual is less than the threshold or the number of iteration reach K.
(1) P=A T R.
(2) d k =∥P k ∥2,k=1,2,…,L, P k is kth column of matrix P.
(3) \({z_{\text {k}}} = \frac {d_{k}}{{\left \| {\textbf {A}_{k}} \right \|}_{2}}\), k=1,2,…,L, z={z 1,z 2…,z L }, A k is the kth column of matrix A.
(4) Find the largest item in the vector z and add its corresponding index k into the support set S, and the symmetric index value L+1−k is also added to the support set V. It is generally considered that the traditional signal is real signal, and its support set is symmetric.
(5) Construct the matrix A S corresponding to the columns in matrix A and support set S.
(6) Evaluate \(\hat {\mathbf {U}}\), \(\hat {\mathbf {U}} = \mathbf {A}_{\textbf {S}}^{\mathbf {\dag }} \textbf {V} = {\left (\textbf {A}_{\textbf {S}}^{T}{\textbf {A}_{\textbf {S}}}\right)^{-1}} \textbf {A}_{\textbf {S}}^{T}\textbf {V}\).
(7) Update signal residuals \(\textbf {R} = \textbf {V} - {\textbf {A}_{\textbf {S}}}\hat {\mathbf {U}}\).
It can be found that the inner product between atoms in sensing matrix and columns in residual matrix can be obtained from steps (1) to (4) firstly. Then, several inner products corresponding to the same atom constitute an inner product vector. By calculating its 2-norm, we can find the maximum value, whose corresponding atom is matched atom. The relevancy between residual matrix and sensing matrix actually depends on the relevancy between residual matrix columns and atoms in the sensing matrix. However, because of the randomness of sensing matrix and sampling, the inner product cannot accurately reflect the relevancy between vectors. Wrong atoms can be selected when the number of sampling points is relatively small. However, if the correlation coefficient is used as the criterion, it can reflect the correlation between the random variables better. The reason for the difference can be traced to the calculation method of the inner product and correlation coefficient. For an n dimensional vector \(\vec {a}\) and \(\vec {b}\), their inner product formula is
$$ \vec{a} \bullet \vec{b} = \sum\limits_{i = 1}^{n} {{a_{i}}{b_{i}}} $$
In two dimensions, the inner product can be expressed as
$$ \vec{a} \bullet \vec{b} = \left| {\vec{a}} \right|\left| {\vec{b}} \right|\cos \theta $$
The inner product itself cannot reflect the relevancy between vectors because it is influenced by the norm of vectors. But if \(\vec {a}\) and \(\vec {b}\) are the unit vectors, the inner product can reflect the relevancy between \(\vec {a}\) and \(\vec {b}\). If \(\vec {b}\) is regarded as the column vector of the residual matrix, while the vector \(\vec {a}\) is regarded as the atom in the matrix A, then, the inner product can reflect the relevancy between the residuals and the atoms.With this relative, the correlation coefficient can also reflect the relevancy, and its formula is
$$ {r_{XY}} = \frac{\sum\limits_{i = 1}^{N} {\left({X_{i}} - \bar{X}\right)\left({Y_{i}} - \bar{Y}\right)}} {\sqrt{\sum\limits_{i = 1}^{N} {{\left({X_{i}} - \bar{X}\right)}^{2}}} \sqrt{\sum\limits_{i = 1}^{N} {{\left({Y_{i}} - \bar{Y}\right)}^{2}}}} $$
It also excludes the effect of the norm of atoms, and the mean value of atom is zero. Although both the inner product and the correlation coefficient can reflect the relevancy, but in some cases, for the same signal and sensing matrix, the atoms they selected are different, just as the following example: There are two atoms X=(1,2,3), Y=(2,2,3), signal Z=(4,5,6), after normalizing them, we can get X∗,Y∗, and Z∗. If we use the inner product as screening criterion of atoms, the inner product of X∗ and Z∗ is 0.975 and the inner product of Y∗ and Z∗ is 0.995.The most matched atoms should be Y∗. However, when the correlation coefficient is used as screening criterion of atoms, the correlation coefficient of X∗ and Z∗ is 1 and the correlation coefficient of Y∗ and Z∗ is 0.866. The most matched atoms should be X∗. The results are different using two criterions. Then which one is better? To solve this problem, we need to analyze the process of signal reconstruction.
In CTF, the signal support set S is calculated indirectly by calculating the most sparse solution \(\bar {\mathbf {U}}\) of V=AU, since the support set of \(\bar {\mathbf {U}}\) and the support set of signal y[n] are the same. Then, construct the matrix A S corresponding to the columns of matrix A and support set S. Finally, reconstruct the sparse spectrum α(f) and the original signal by using Eq. (6). If
$$ \textbf{A} = \left[ { \begin{array}{ccccccc} {a_{11}} & \cdots & {a_{1q}} & \cdots & {a_{1i}} & \cdots & {a_{1L}} \\ 0\vdots & & \vdots & & \vdots & & \vdots \\ {a_{r1}} & \cdots & {a_{rq}} & \cdots & {a_{ri}} & \cdots & {a_{rL}} \\ \vdots & & \vdots & & \vdots & & \vdots \\ {a_{m1}} & \cdots & {a_{mq}} & \cdots & {a_{mi}} & \cdots & {a_{mL}} \end{array}} \right] $$
In order to simplify the analysis, V=AU is expressed as
$$ \begin{aligned} &\left[ {\begin{array}{ccccccc} {a_{11}} & \cdots & {a_{1q}} & \cdots & {a_{1i}} & \cdots & {a_{1L}} \\ \vdots & & \vdots & & \vdots & & \vdots \\ {a_{r1}} & \cdots & {a_{rq}} & \cdots & {a_{ri}} & \cdots & {a_{rL}} \\ \vdots & & \vdots & & \vdots & & \vdots \\ {a_{m1}} & \cdots & {a_{mq}} & \cdots & {a_{mi}} & \cdots & {a_{mL}} \end{array}} \right] \times \\ &\left[ {\begin{array}{ccccc} 0 & \cdots & 0 & \cdots & 0 \\ \vdots & & \vdots & & \vdots \\ {u_{q1}} & \cdots & {u_{qt}} & \cdots & {u_{qp}} \\ \vdots & & \vdots & & \vdots \\ {u_{i1}} & \cdots & {u_{it}} & \cdots & {u_{ip}} \\ \vdots & & \vdots & & \vdots \\ 0 & \cdots & 0 & \cdots & 0 \end{array}} \right] = \left[ \begin{array}{c} {a_{1q}{u_{q1}} + {a_{1i}}{u_{i1}}} \\ \vdots \\ {{a_{rq}}{u_{q1}} + {a_{ri}}{u_{i1}}} \\ \vdots \\ {{a_{mq}}{u_{q1}} + {a_{mi}}{u_{i1}}} \end{array} \right. \\ &\left. {\begin{array}{ccc} {\cdots \ {a_{1q}}{u_{qt}} + {a_{1i}}{u_{it}}} & \cdots & {{a_{iq}}{u_{qp}} + {a_{1i}}{u_{ip}}} \\ \vdots & & \vdots \\ {\cdots \ {a_{rq}}{u_{qt}} + {a_{ri}}{u_{it}}} & \cdots & {{a_{rq}}{u_{qp}} + {a_{ri}}{u_{ip}}} \\ \vdots & & \vdots \\ {\cdots \ {a_{mq}}{u_{qt}} + {a_{mi}}{u_{it}}} & \cdots & {{a_{mq}}{u_{qp}} + {a_{mi}}{u_{ip}}} \end{array}} \right] = \\ &\left[{{A_{q}}{u_{q1}} + {A_{i}}{u_{i1}} \cdots {A_{q}}{u_{qt}} + {A_{i}}{u_{it}} \cdots {A_{q}}{u_{qp}} + {A_{i}}{u_{ip}}} \right] \end{aligned} $$
In which, A q and A i are the qth column and ith column of matrix A, respectively. According to SOMP algorithm, the method of using inner product as screening criterion of atoms can be expressed as
$$ \begin{aligned} &\lambda = {\left\| {{A_{s}}\textbf{R}} \right\|_{2}} = {\left\| {{A_{s}}\textbf{V}} \right\|_{2}} = \\ &{\left(\begin{array}{l} {\left({A_{s}^{T}}{A_{q}}{u_{q1}} + {A_{s}^{T}}{A_{i}}{u_{i1}}\right)^{2}} + \cdots \\ + {\left({A_{s}^{T}}{A_{q}}{u_{qp}} + {A_{s}^{T}}{A_{i}}{u_{ip}}\right)^{2}} \end{array} \right)^{1/2}} \end{aligned} $$
We hope that when s=q or s=i, λ has the maximum value. Since the column vectors of matrix A are approximately orthogonal, one of \({A_{s}^{T}}{A_{q}}{u_{\textit {qt}}}\) and \({A_{s}^{T}}{A_{i}}{u_{\textit {it}}}\) is relatively large, while the other one is quite small. Assuming
$$ \begin{aligned} \lambda &\approx \sqrt{{{\left({A_{s}^{T}}{A_{q}}{u_{q1}}\right)}^{2}} + \cdots + {{\left({A_{s}^{T}}{A_{q}}{u_{qp}}\right)}^{2}}} \\ &= \sqrt{{{\left({A_{s}^{T}}{A_{q}}\right)}^{2}}\left(u_{q1}^{2} + u_{q2}^{2} + \cdots u_{qp}^{2}\right)} \end{aligned} $$
where the value of λ is independent of the matrix U, only related to A s . Most of the time,
$$ {\left({A_{q}^{T}}{A_{q}}\right)^{2}} > {\left({A_{s}^{T}}{A_{q}}\right)^{2}},s \neq q $$
But sometimes, the result is opposite (even if the atoms are normalized). Then, the SOMP algorithm will select the wrong atoms. And if the correlation coefficient is used as screening criterion of atoms,
$$ {r_{{A_{s}}{A_{q}}}} \leq 1 $$
if and only if when s=q, the equality holds. Thus, using correlation coefficient as screening criterion of atoms is more accurate.
Since the sensing matrix A and the residual matrix R are multidimensional, we cannot calculate the correlation coefficient between them directly, so we need to calculate the correlation coefficient between atoms in matrix A and columns in residual matrix R, and then squaring the correlation coefficients and sum the correlation coefficients corresponding to the same atom; finally, extract the root of the sum and find the largest one, whose corresponding atom is matched atom. This improvement will only change the first step of the SOMP algorithm into \({P_{k}} = \sum \limits _{i = 1}^{m} {r_{{\textbf {A}_{k}}{\textbf {R}_{i}}}}\), other steps remain unchanged.
To test whether the improved algorithm can improve the reconstruction probability, we compare the performance of MMV-ROMP, MMV-StOMP, MMV-CoSaMP, MMV-SP, and SOMP and their improved algorithms. We perform 1000 Monte-Carlo simulation for each algorithm. The signal expression is
$$ x(t) = \sum\limits_{n = 1}^{N/2} {\sqrt{B{E_{i}}} {\text{sinc}}\left(B(t - {\tau_{i}})\cos (2\pi {f_{i}}(t - {\tau_{i}})\right)} $$
The signal contains N frequency bands which are origin symmetric. The maximum bandwidth of sub-frequency bands is 50 MHz. The carrier frequency f i is a uniform distribution random number among [−5 GHz,5 GHz]. The Nyquist sampling rate is f NYQ=10 GHz. The energy E i of each frequency band is random, time shift τ i is a uniform distribution random number among [0,1]μ s. L 0=97,L=195,M=195,SNR=0, the sampling period, and the period of the pseudo random sequence p i (t) are f p =f s =f NYQ/L=51.28 MHz, respectively. The value of p i (t) is either 1 or −1 with equal probability and obey Bernoulli distribution. The numbers of sampling channels in Figs. 3 and 7 are 100, 100, 50, 50, and 120. The simulation results are as follows.
Reconstruction probability comparison between MMV-CoSaMP and improved MMV-CoSaMP
As can be seen from the simulation, new algorithms are improved in reconstruction probability compared with the traditional algorithms, especially for MMV-ICoSaMP algorithm and MMV-ISP algorithm, the improvement is very significant. When traditional algorithms are completely unable to reconstruct the signal, as is shown in Figs. 3 and 4, the improved algorithm still can reconstruct the signal with 100 % probability. It also means that less samples were needed to reconstruct signals than traditional algorithms when the number of bands is unchanged. The other three algorithms can also improve the reconstruction probability to a limited extent, as is shown in Figs. 5, 6 and 7. Because of the improvement about correlation coefficient, the complexity of the algorithm is increased. In addition, relatively small SNR has an adverse effect on reconstruction probability of improved algorithm, as is shown in Figs. 8 and 9, where the number of sampling channels is 100 and the number of bands is 20.
Reconstruction probability comparison between MMV-SP and improved MMV-SP
Reconstruction probability comparison between MMV-ROMP and improved MMV-ROMP
Reconstruction probability comparison between SOMP and improved SOMP
Reconstruction probability comparison between MMV-StOMP and improved MMV-StOMP
Reconstruction probability comparison between MMV-CoSaMP and improved MMV-CoSaMP in different SNRs
Reconstruction probability comparison between MMV-SP and improved MMV-SP in different SNRs
Although changing the inner product into correlation coefficient as screening criterion of atoms can improve the performance of the algorithm, calculating correlation coefficient at each iteration will cost a lot of time, so we can consider the possibility of simplified this process. A significant difference between the inner product and correlation coefficient is that correlation coefficient excludes the impact of data mean, so we can consider whether it is feasible if the column vectors of sensing matrix are centered or normalized firstly and still use inner product as screening criterion of matching atoms. Simulation result shows that compared to traditional MMV-SP algorithms, normalization cannot improve the reconstruction probability, but centralization can. If the column vectors of sensing matrix are centered firstly and still use inner product as screening criterion of matching atoms, the reconstruction probability is silently worse than MMV-ISP algorithm, just as shown in Fig. 10, where MMV-SISP means simplified MMV-ISP. The advantage of this simplification is that it can reduce the complexity of the proposed algorithm so that the reconstruction time is about the same as MMV-SP algorithm.
Comparison of the reconstructed probability of the simplified and unsimplified algorithms
In order to improve the reconstruction probability of MWC sampling system, we improve the SOMP algorithm and its derivative algorithms in this paper by changing the inner product into correlation coefficient as screening criterion of atoms. The simulation experiments show that the improved algorithm can increase the reconstruction probability, especially for MMV-ICoSaMP algorithm and MMV-ISP algorithm, the improvement is very significant. For SOMP, MMV-ROMP, and MMV-StOMP algorithms, the improvement can also increase the reconstruction probability to a limited extent. It also means that less samples were needed to reconstruct signals than traditional algorithms when the number of bands is unchanged. Because of the improvement of correlation coefficient, the complexity of the algorithms increase, so the reconstruction time correspondingly increase and the algorithm is not suitable for the system with high real-time requirements. For systems that require high accuracy, the improved algorithms can be used. Since calculating correlation coefficient at each iteration will cost a lot of time, we also proposed a simplified algorithm, which can also improve reconstruction probability and reconstruction time is about the same as traditional algorithms. In addition, relatively small SNR has an adverse effect on reconstruction probability of improved algorithms, so the improved algorithms are more suitable for low noise channels.
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Communication Research Center, Harbin Institute of Technology, Yikuang Street, Harbin, 150080, China
Min Jia
, Yao Shi
, Xuemai Gu
, Xue Wang
& Zhisheng Yin
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Correspondence to Min Jia.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Jia, M., Shi, Y., Gu, X. et al. Improved algorithm based on modulated wideband converter for multiband signal reconstruction. J Wireless Com Network 2016, 50 (2016) doi:10.1186/s13638-016-0547-y
DOI: https://doi.org/10.1186/s13638-016-0547-y
Compressed sensing
Greedy algorithm
Radar and Sonar Networks | CommonCrawl |
\begin{document}
\title{Computing the $k$ Nearest-Neighbors for all Vertices via
Dijkstra}
\author{Sariel Har-Peled\SarielThanks{Work on this paper
was partially supported by a NSF AF awards
CCF-1421231, and
CCF-1217462.
}}
\date{\today}
\maketitle
\begin{abstract}
We are given a directed graph ${G} = ({V},{E})$ with
$n$ vertices and $m$ edges, with positive weights on the edges,
and a parameter $k >0$. We show how to compute, for every vertex
$v \in {V}$, its $k$ nearest-neighbors. The algorithm runs
in $O( k ( n \log n + m ) )$ time, and follows by a somewhat
careful modification of Dijkstra's shortest path algorithm.
This result is probably folklore, but we were unable to find a
reference to it -- thus, this note. \end{abstract}
\section{The problem}
Let ${G} = ({V}, {E})$ be an undirected graph with $n$ vertices, $m$ edges, and with positive weights on the edges. For $u,v \in {V}$, let $\distG{u}{v}$ denote the shortest path distances in ${G}$, and assume the sake of the simplicity of exposition that all the non-trivial shortest path distances in the graph are distinct.
A vertex $u = \nnY{v}{i}$ is the \emphi{$i$\th nearest-neighbor} to $v$, if one partition ${V}$ into three disjoint sets $C, \{u_i\}, F$, such that \begin{inparaenum}[(i)]
\item $\cardin{C} = i-1$, and
\item $\forall c \in C$, $\forall f \in F$ we have
$\distG{c}{v} < \distG{u}{v} < \distG{f}{v}$. \end{inparaenum} Observe that $\nnY{v}{0} = v$. For any integer $k$, the \emphi{$k$
nearest-neighbors} to $v$, are the members of the set \begin{align*}
\NNY{v}{k} = \brc{ \Bigl. \nnY{v}{0}, \nnY{v}{1}, \ldots,
\nnY{v}{k-1}}; \end{align*} that is, they are the $k$ vertices in ${G}$ that are closest to $v$.
Our purpose here is to compute for every vertex $v \in {V}$, the set $\NNY{v}{k}$; that is, to compute for $v$ the $k$ distinct vertices closest to it.
\section{A simple (but slower) randomized algorithm}
\subsection{Algorithm} Let $t = O( k \log n)$, and let $R_1, \ldots, R_t$ be random samples from ${V}$, where every vertex is picked into the $i$\th random sample with probability $p=1/k$. Now, compute for every vertex $v \in {V}$ its distance from its nearest-neighbor in $R_i$, for all $i$. For a specific $i$, this can be done by performing Dijkstra in ${G}$ starting from all the vertices of $R_i$ simultaneously (i.e., we create a fake source vertex $s$, add it to ${G}$, and add edges of weight $0$ from $s$ to all the vertices of $R_i$, and perform regular Dijkstra from $s$). For a vertex $v$, let $D'(v)$ be its $t$ candidate nearest-neighbors computed by these $t$ executions of Dijkstra. Using hashing, remove duplicates in $D'(v)$ (i.e., a vertex $s$ might be the nearest-neighbor for $v$ in several of these executions -- note however that in such a case it is always the same distance). Now, compute the $k$ vertices with the smallest numbers associated with them in $D'(v)$, and let $D(v)$ be the resulting set of nearest-neighbors. We claim that, for all $v$, the sets $D(v)$ are the desired $k$ nearest-neighbors.
\newcommand{\mathcal{E}}{\mathcal{E}}
\subsection{Analysis}
For the running time, observe that performing Dijkstra shortest path algorithm $t$ times takes $O( t (n + m \log n) )$ time. All the other work is dominated by this.
As for correctness, we need to argue that if $u$ is the $j$\th nearest-neighbor to $v$, for $j \leq k$, then $u \in D(v)$. To this end, consider the event $\mathcal{E}_i$ that $u \in R_i$, and none of the vertices of $\NNY{v}{j-1}$ are in $R_i$. We have that \begin{align*}
\alpha
=
\Prob{\mathcal{E}_i}
=
p (1-p)^{j-1}
\geq
(1-1/k)^{k} /k
\geq
\exp( -1/2k)^k /k
\geq
1/10k. \end{align*} If this happens, then $u \in D'(v)$, which in turn implies that $u \in D(v)$, as desired. The probability that none of the events $\mathcal{E}_1, \ldots, \mathcal{E}_t$ happens is $(1-\alpha)^t \leq (1-1/10k)^t < 1/n^{c}$ by making $t$ sufficiently large, where $c$ is an arbitrary constant. Since there are $n k$ pairs of $(u,v)$ such that $u$ is one of the $k$ nearest neighbors to $v$, it follows that the probability this algorithm fails is at most $1/n^{c-2}$.
\subsection{The result}
\begin{lemma}
Given a directed graph ${G} = ({V},{E})$ with $n$
vertices and $m$ edges, with positive weights on the edges, and a
parameter $k >0$, one can compute, in
$O\pth{ \pth{ n \log n + m} k \log n}$ time, for every vertex
$v \in {V}$, its $k$ nearest-neighbors in ${G}$. The
algorithm succeeds with high probability. \end{lemma}
The above randomized algorithm is inspired by the Clarkson-Shor technique \cite{cs-arscg-89} and this trick is useful in many other scenarios.
\section{A faster algorithm}
\subsection{The algorithm -- a first try}
We are going to run (conceptually) $n$ copies of the shortest-path algorithm simultaneously. In particular, let $A_v$ be the shortest-path algorithm starting from the vertex $v$, for all $v \in {V}$. We use global heap for the events for all these algorithms together. Here, every event of $A_v$ would be indexed by the source vertex $v$ associated with this algorithm. The algorithm is going to maintain for each vertex the set of $D'(v)$ of nearest-neighbors found so far, and a count $c_v = \cardin{D'(v)}$.
Now, when the algorithm extract the next vertex to be visited (i.e., the one of the with lowest candidate distance), we get a triple $(v, s, d)$, where $v$ is the vertex to be handled, $s$ is the source vertex, and $d$ is the proposed distance. Using a hash table, we check in constant time whether $v$ has $s$ as one of its computed nearest neighbors (i.e., check if $s \in D'(v)$), and if so the algorithm continues to the next iteration. Otherwise, the algorithm \begin{compactenum}[\qquad(i)]
\item adds $s$ to $D'(v)$,
\item increase $c_v$, and
\item perform the standard relax operation from $v$ for all the
outgoing edges of $v$ (these operations are ``marked'' by
the source vertex $s$ they are being done for).
Specifically, consider an edge $\dirEdge{v}{z}$ being relaxed,
during the handling of the event $(v,s,d)$. The new associated
event is $\pth{z, s, d+ w(\dirEdge{v}{z})}$, where
$w(\dirEdge{v}{z})$ is the weight of the edge $\dirEdge{v}{z}$. If
$s \notin D'(z)$ the algorithm ``schedule'' this event by
inserting it to the heap, otherwise it ignores it. \end{compactenum}
The basic observation is that once a vertex was visited by $k$ of these parallel executions, it is no longer needed, and it can be ``disabled'' blocking it from being visited by any other Dijkstra (i.e., as soon as $c_v = k$). From this point on, the algorithm ignores update operations for triplets of the form $(v, \cdot, \cdot)$.
\subsubsection{Analysis}
\begin{lemma}
The algorithm computes correctly, for each vertex $v \in
{V}$, its $k$ nearest-neighbors in ${G}$. \end{lemma} \begin{proof}
By induction on the distance being computed. For $\ell=0$, for
each vertex $v \in {V}$, the vertex $v$ is its own $0$\th
nearest-neighbor, of distance $0$, and it was computed
correctly.
Assume all the relevant distances $< \ell$ were computed correctly
(they are $0,1,\ldots, k$ nearest-neighbor distances for some
pairs of vertices in the graph).
So consider a vertex $v$, where its $i$\th nearest-neighbor is
$s$, let its (real) shortest path from $s$ to $v$ be $\pi = s =
v_0 v_1 v_2 \ldots v_t = v$, and assume the length of $\pi$ is
$\ell$.
The key observation is that for all the vertices on $\pi$, the
vertex $s$ must be one of their first $i$
nearest-neighbors. Indeed, if not, then there exists $i$ vertices
in ${G}$ that are closer to $v$ than $s$, which is a
contradiction.
Thus, by induction all the relevant distances for $v_0, v_1,
\ldots, v_{t-1}$ were computed correctly by the algorithm, and as
such, this path would be considered by the algorithm. Namely, the
$i$\th distance to $v$ would be set correctly. \end{proof}
\paragraph{Running time.} Since for a vertex $v$, the counter $c_v$ can be increased only $k$ times, it follows that every edge participates in $k$ relax operations. As such, the total number of relax operations handled by this algorithm is $O(k m)$, and this bounds the maximum size of the global heap. The global heap might perform $O(k m)$ extract-min operations on the global heap. As such, the overall running time of this algorithm is $O( k m \log n)$.
\subsection{Speeding up the algorithm}
Now, our purpose is to improve the running time -- this requires some cleverness with the data-structures being used.
Our first task is to avoid having extract-min operations in the global heap involving a vertex $v$ such that $c_v = k$. To this end, for each vertex $v \in {V}$, the algorithm maintains a separate queue $Q_u$ that handles all the events for $u$. Every source vertex $s \in {V}$ might maintain at most one value at $Q_u$. Every such queue would maintain its current minimum, and would update it in the global queue $Q$ as necessary (all of these operations are either \texttt{insert} or \texttt{decrease\_key}, and both operations can be done in constant time using the standard Fibonacci heap). That is, the global queue contains at most $n$ values, potentially one from each vertex queue. In particular, once $c_v \geq k$, we disable the queue $Q_v$, and it not longer participates in the global queue, or takes updates to values in its queue.
\subsection{Analysis}
\begin{lemma}
The algorithm running time is $O\pth{ k \pth{ n \log n + m}}$. \end{lemma}
\begin{proof}
Every vertex distance is going to be set by at most $k$ of these
``parallel'' executions of Dijkstra. Whenever it happens, the
algorithm performs a relax operation on all the adjacent edges. It
follows, that an edge would be relaxed $O( k)$ times. The cost of
a relax operation is $O(1)$, and thus the overall cost of these is
$O( k m)$. Setting one of the $k$ shortest distance values of a
vertex takes $O(\log n)$ time, as it involves
\texttt{extract\_min} from the global heap. As such, each vertex
would require overall $O( k \log n)$ time for its
\texttt{extract\_min} operations from the global heap and the
local heap of this vertex. The bound on the running time now
follows readily. \end{proof}
\subsection{The result}
\begin{theorem}
\thmlab{main}
Given a directed graph ${G} = ({V},{E})$ with $n$
vertices and $m$ edges, with positive weights on the edges, and a
parameter $k >0$, one can compute, in
$O\pth{ k \pth{ n \log n + m}}$ time, for every vertex
$v \in {V}$, its $k$ nearest-neighbors in ${G}$.
The algorithm uses hashing (i.e., randomization), a deterministic
version of the algorithm runs in time
$O\pth{ k n \log n + k m \log k }$. \end{theorem}
\begin{proof}
For the deterministic running time, replace the hash table
maintained by each vertex $v$ (which stores all the
nearest-neighbors to $v$ discovered so far) by a balanced binary
tree. The key observation is that such a tree, for a node $v$,
needs to maintain only the $k$ smallest candidate distances
offered to this vertex. In particular, if a new candidate distance
is larger than all these $k$ values, we can immediately reject
it. Similarly, after insertion we reject any value larger than $k$
smallest values in this data-structure.
As such, each binary tree stores at most $O(k)$ elements, so every
basic operation takes $O( \log k)$ time. The algorithm performs
$O( k(n+m))$ operations on these lookup data structures, which
implies the claimed running time. \end{proof}
\begin{remark}
Consider the settings of \thmref{main}, but in addition there is a
set of terminals $T \subseteq {V}$. One can compute for each
vertex of ${G}$ the $k$ closest \emph{terminals} to it in
${G}$ using the algorithm of \thmref{main}. The only
modification being that we start the Dijkstra process only from
the vertices of $T$. The running time of the modified algorithm is
the same. \end{remark}
\paragraph*{Acknowledgments.}
The author thanks Chandra Chekuri and David Eppstein for useful comments on this note. In addition, Vivek Madan came up independently with a similar solution to the one described in this note.
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\end{document} | arXiv |
Jordan curve theorem
In topology, the Jordan curve theorem asserts that every Jordan curve (a plane simple closed curve) divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points. Every continuous path connecting a point of one region to a point of the other intersects with the curve somewhere. While the theorem seems intuitively obvious, it takes some ingenuity to prove it by elementary means. "Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it." (Tverberg (1980, Introduction)). More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces.
The Jordan curve theorem is named after the mathematician Camille Jordan (1838–1922), who found its first proof. For decades, mathematicians generally thought that this proof was flawed and that the first rigorous proof was carried out by Oswald Veblen. However, this notion has been overturned by Thomas C. Hales and others.[1]
Definitions and the statement of the Jordan theorem
A Jordan curve or a simple closed curve in the plane R2 is the image C of an injective continuous map of a circle into the plane, φ: S1 → R2. A Jordan arc in the plane is the image of an injective continuous map of a closed and bounded interval [a, b] into the plane. It is a plane curve that is not necessarily smooth nor algebraic.
Alternatively, a Jordan curve is the image of a continuous map φ: [0,1] → R2 such that φ(0) = φ(1) and the restriction of φ to [0,1) is injective. The first two conditions say that C is a continuous loop, whereas the last condition stipulates that C has no self-intersection points.
With these definitions, the Jordan curve theorem can be stated as follows:
Theorem — Let C be a Jordan curve in the plane R2. Then its complement, R2 \ C, consists of exactly two connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior), and the curve C is the boundary of each component.
In contrast, the complement of a Jordan arc in the plane is connected.
Proof and generalizations
The Jordan curve theorem was independently generalized to higher dimensions by H. Lebesgue and L. E. J. Brouwer in 1911, resulting in the Jordan–Brouwer separation theorem.
Theorem — Let X be an n-dimensional topological sphere in the (n+1)-dimensional Euclidean space Rn+1 (n > 0), i.e. the image of an injective continuous mapping of the n-sphere Sn into Rn+1. Then the complement Y of X in Rn+1 consists of exactly two connected components. One of these components is bounded (the interior) and the other is unbounded (the exterior). The set X is their common boundary.
The proof uses homology theory. It is first established that, more generally, if X is homeomorphic to the k-sphere, then the reduced integral homology groups of Y = Rn+1 \ X are as follows:
${\tilde {H}}_{q}(Y)={\begin{cases}\mathbb {Z} ,&q=n-k{\text{ or }}q=n,\\\{0\},&{\text{otherwise}}.\end{cases}}$
This is proved by induction in k using the Mayer–Vietoris sequence. When n = k, the zeroth reduced homology of Y has rank 1, which means that Y has 2 connected components (which are, moreover, path connected), and with a bit of extra work, one shows that their common boundary is X. A further generalization was found by J. W. Alexander, who established the Alexander duality between the reduced homology of a compact subset X of Rn+1 and the reduced cohomology of its complement. If X is an n-dimensional compact connected submanifold of Rn+1 (or Sn+1) without boundary, its complement has 2 connected components.
There is a strengthening of the Jordan curve theorem, called the Jordan–Schönflies theorem, which states that the interior and the exterior planar regions determined by a Jordan curve in R2 are homeomorphic to the interior and exterior of the unit disk. In particular, for any point P in the interior region and a point A on the Jordan curve, there exists a Jordan arc connecting P with A and, with the exception of the endpoint A, completely lying in the interior region. An alternative and equivalent formulation of the Jordan–Schönflies theorem asserts that any Jordan curve φ: S1 → R2, where S1 is viewed as the unit circle in the plane, can be extended to a homeomorphism ψ: R2 → R2 of the plane. Unlike Lebesgue's and Brouwer's generalization of the Jordan curve theorem, this statement becomes false in higher dimensions: while the exterior of the unit ball in R3 is simply connected, because it retracts onto the unit sphere, the Alexander horned sphere is a subset of R3 homeomorphic to a sphere, but so twisted in space that the unbounded component of its complement in R3 is not simply connected, and hence not homeomorphic to the exterior of the unit ball.
Discrete version
The Jordan curve theorem can be proved from the Brouwer fixed point theorem (in 2 dimensions),[2] and the Brouwer fixed point theorem can be proved from the Hex theorem: "every game of Hex has at least one winner", from which we obtain a logical implication: Hex theorem implies Brouwer fixed point theorem, which implies Jordan curve theorem.[3]
It is clear that Jordan curve theorem implies the "strong Hex theorem": "every game of Hex ends with exactly one winner, with no possibility of both sides losing or both sides winning", thus the Jordan curve theorem is equivalent to the strong Hex theorem, which is a purely discrete theorem.
The Brouwer fixed point theorem, by being sandwiched between the two equivalent theorems, is also equivalent to both.[4]
In reverse mathematics, and computer-formalized mathematics, the Jordan curve theorem is commonly proved by first converting it to an equivalent discrete version similar to the strong Hex theorem, then proving the discrete version.[5]
Application to image processing
In image processing, a binary picture is a discrete square grid of 0 and 1, or equivalently, a compact subset of $\mathbb {Z} ^{2}$. Topological invariants on $\mathbb {R} ^{2}$, such as number of components, might fail to be well-defined for $\mathbb {Z} ^{2}$ if $\mathbb {Z} ^{2}$ does not have an appropriately defined graph structure.
There are two obvious graph structures on $\mathbb {Z} ^{2}$:
• the "4-neighbor square grid", where each vertex $(x,y)$ is connected with $(x+1,y),(x-1,y),(x,y+1),(x,y-1)$.
• the "8-neighbor square grid", where each vertex $(x,y)$ is connected with $(x',y')$ iff $|x-x'|\leq 1,|y-y'|\leq 1$, and $(x,y)\neq (x',y')$.
Both graph structures fail to satisfy the strong Hex theorem. The 4-neighbor square grid allows a no-winner situation, and the 8-neighbor square grid allows a two-winner situation. Consequently, connectedness properties in $\mathbb {R} ^{2}$, such as the Jordan curve theorem, do not generalize to $\mathbb {Z} ^{2}$ under either graph structure.
If the "6-neighbor square grid" structure is imposed on $\mathbb {Z} ^{2}$, then it is the hexagonal grid, and thus satisfies the strong Hex theorem, allowing the Jordan curve theorem to generalize. For this reason, when computing connected components in a binary image, the 6-neighbor square grid is generally used.[6]
Steinhaus chessboard theorem
The Steinhaus chessboard theorem in some sense shows that the 4-neighbor grid and the 8-neighbor grid "together" implies the Jordan curve theorem, and the 6-neighbor grid is a precise interpolation between them.[7][8]
The theorem states that: suppose you put bombs on some squares on a $n\times n$ chessboard, so that a king cannot move from the bottom side to the top side without stepping on a bomb, then a rook can move from the left side to the right side stepping only on bombs.
History and further proofs
The statement of the Jordan curve theorem may seem obvious at first, but it is a rather difficult theorem to prove. Bernard Bolzano was the first to formulate a precise conjecture, observing that it was not a self-evident statement, but that it required a proof. It is easy to establish this result for polygons, but the problem came in generalizing it to all kinds of badly behaved curves, which include nowhere differentiable curves, such as the Koch snowflake and other fractal curves, or even a Jordan curve of positive area constructed by Osgood (1903).
The first proof of this theorem was given by Camille Jordan in his lectures on real analysis, and was published in his book Cours d'analyse de l'École Polytechnique.[9] There is some controversy about whether Jordan's proof was complete: the majority of commenters on it have claimed that the first complete proof was given later by Oswald Veblen, who said the following about Jordan's proof:
His proof, however, is unsatisfactory to many mathematicians. It assumes the theorem without proof in the important special case of a simple polygon, and of the argument from that point on, one must admit at least that all details are not given.[10]
However, Thomas C. Hales wrote:
Nearly every modern citation that I have found agrees that the first correct proof is due to Veblen... In view of the heavy criticism of Jordan’s proof, I was surprised when I sat down to read his proof to find nothing objectionable about it. Since then, I have contacted a number of the authors who have criticized Jordan, and each case the author has admitted to having no direct knowledge of an error in Jordan’s proof.[11]
Hales also pointed out that the special case of simple polygons is not only an easy exercise, but was not really used by Jordan anyway, and quoted Michael Reeken as saying:
Jordan’s proof is essentially correct... Jordan’s proof does not present the details in a satisfactory way. But the idea is right, and with some polishing the proof would be impeccable.[12]
Earlier, Jordan's proof and another early proof by Charles Jean de la Vallée Poussin had already been critically analyzed and completed by Schoenflies (1924).[13]
Due to the importance of the Jordan curve theorem in low-dimensional topology and complex analysis, it received much attention from prominent mathematicians of the first half of the 20th century. Various proofs of the theorem and its generalizations were constructed by J. W. Alexander, Louis Antoine, Ludwig Bieberbach, Luitzen Brouwer, Arnaud Denjoy, Friedrich Hartogs, Béla Kerékjártó, Alfred Pringsheim, and Arthur Moritz Schoenflies.
New elementary proofs of the Jordan curve theorem, as well as simplifications of the earlier proofs, continue to be carried out.
• Elementary proofs were presented by Filippov (1950) and Tverberg (1980).
• A proof using non-standard analysis by Narens (1971).
• A proof using constructive mathematics by Gordon O. Berg, W. Julian, and R. Mines et al. (1975).
• A proof using the Brouwer fixed point theorem by Maehara (1984).
• A proof using non-planarity of the complete bipartite graph K3,3 was given by Thomassen (1992).
The root of the difficulty is explained in Tverberg (1980) as follows. It is relatively simple to prove that the Jordan curve theorem holds for every Jordan polygon (Lemma 1), and every Jordan curve can be approximated arbitrarily well by a Jordan polygon (Lemma 2). A Jordan polygon is a polygonal chain, the boundary of a bounded connected open set, call it the open polygon, and its closure, the closed polygon. Consider the diameter $\delta $ of the largest disk contained in the closed polygon. Evidently, $\delta $ is positive. Using a sequence of Jordan polygons (that converge to the given Jordan curve) we have a sequence $\delta _{1},\delta _{2},\dots $ presumably converging to a positive number, the diameter $\delta $ of the largest disk contained in the closed region bounded by the Jordan curve. However, we have to prove that the sequence $\delta _{1},\delta _{2},\dots $ does not converge to zero, using only the given Jordan curve, not the region presumably bounded by the curve. This is the point of Tverberg's Lemma 3. Roughly, the closed polygons should not thin to zero everywhere. Moreover, they should not thin to zero somewhere, which is the point of Tverberg's Lemma 4.
The first formal proof of the Jordan curve theorem was created by Hales (2007a) in the HOL Light system, in January 2005, and contained about 60,000 lines. Another rigorous 6,500-line formal proof was produced in 2005 by an international team of mathematicians using the Mizar system. Both the Mizar and the HOL Light proof rely on libraries of previously proved theorems, so these two sizes are not comparable. Nobuyuki Sakamoto and Keita Yokoyama (2007) showed that in reverse mathematics the Jordan curve theorem is equivalent to weak Kőnig's lemma over the system ${\mathsf {RCA}}_{0}$.
Application
Main article: Point in polygon § Ray casting algorithm
In computational geometry, the Jordan curve theorem can be used for testing whether a point lies inside or outside a simple polygon.[14][15][16]
From a given point, trace a ray that does not pass through any vertex of the polygon (all rays but a finite number are convenient). Then, compute the number n of intersections of the ray with an edge of the polygon. Jordan curve theorem proof implies that the point is inside the polygon if and only if n is odd.
See also
• Denjoy–Riesz theorem, a description of certain sets of points in the plane that can be subsets of Jordan curves
• Lakes of Wada
• Quasi-Fuchsian group, a mathematical group that preserves a Jordan curve
Notes
1. Hales, Thomas C. (2007). "Jordan's proof of the Jordan curve theorem" (PDF). From Insight to Proof: Festschrift in Honour of Andrzej Trybulec. Studies in Logic, Grammar and Rhetoric. University of Białystok. 10 (23).
2. Maehara (1984), p. 641.
3. Gale, David (December 1979). "The Game of Hex and the Brouwer Fixed-Point Theorem". The American Mathematical Monthly. 86 (10): 818–827. doi:10.2307/2320146. ISSN 0002-9890. JSTOR 2320146.
4. Nguyen, Phuong; Cook, Stephen A. (2007). "The Complexity of Proving the Discrete Jordan Curve Theorem". 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007). IEEE. pp. 245–256. arXiv:1002.2954. doi:10.1109/lics.2007.48. ISBN 978-0-7695-2908-0.
5. Hales, Thomas C. (December 2007). "The Jordan Curve Theorem, Formally and Informally". The American Mathematical Monthly. 114 (10): 882–894. doi:10.1080/00029890.2007.11920481. ISSN 0002-9890. S2CID 887392.
6. Nayar, Shree (Mar 1, 2021). "First Principles of Computer Vision: Segmenting Binary Images | Binary Images". YouTube.
7. Šlapal, J (April 2004). "A digital analogue of the Jordan curve theorem". Discrete Applied Mathematics. 139 (1–3): 231–251. doi:10.1016/j.dam.2002.11.003. ISSN 0166-218X.
8. Surówka, Wojciech (1993). "A discrete form of Jordan curve theorem". ISSN 0860-2107. {{cite journal}}: Cite journal requires |journal= (help)
9. Camille Jordan (1887)
10. Oswald Veblen (1905)
11. Hales (2007b)
12. Hales (2007b)
13. A. Schoenflies (1924). "Bemerkungen zu den Beweisen von C. Jordan und Ch. J. de la Vallée Poussin". Jahresber. Deutsch. Math.-Verein. 33: 157–160.
14. Richard Courant (1978)
15. "V. Topology". 1. Jordan curve theorem (PDF). Edinburg: University of Edinburgh. 1978. p. 267.
16. "PNPOLY - Point Inclusion in Polygon Test - WR Franklin (WRF)". wrf.ecse.rpi.edu. Retrieved 2021-07-18.
References
• Berg, Gordon O.; Julian, W.; Mines, R.; Richman, Fred (1975), "The constructive Jordan curve theorem", Rocky Mountain Journal of Mathematics, 5 (2): 225–236, doi:10.1216/RMJ-1975-5-2-225, ISSN 0035-7596, MR 0410701
• Courant, Richard (1978). "V. Topology". Written at Oxford. What is mathematics? : an elementary approach to ideas and methods. Herbert Robbins ([4th ed.] ed.). United Kingdom: Oxford University Press. p. 267. ISBN 978-0-19-502517-0. OCLC 6450129.
• Filippov, A. F. (1950), "An elementary proof of Jordan's theorem" (PDF), Uspekhi Mat. Nauk (in Russian), 5 (5): 173–176
• Hales, Thomas C. (2007a), "The Jordan curve theorem, formally and informally", The American Mathematical Monthly, 114 (10): 882–894, doi:10.1080/00029890.2007.11920481, ISSN 0002-9890, MR 2363054, S2CID 887392
• Hales, Thomas (2007b), "Jordan's proof of the Jordan Curve theorem" (PDF), Studies in Logic, Grammar and Rhetoric, 10 (23)
• Jordan, Camille (1887), Cours d'analyse (PDF), pp. 587–594
• Maehara, Ryuji (1984), "The Jordan Curve Theorem Via the Brouwer Fixed Point Theorem", The American Mathematical Monthly, 91 (10): 641–643, doi:10.2307/2323369, ISSN 0002-9890, JSTOR 2323369, MR 0769530
• Narens, Louis (1971), "A nonstandard proof of the Jordan curve theorem", Pacific Journal of Mathematics, 36: 219–229, doi:10.2140/pjm.1971.36.219, ISSN 0030-8730, MR 0276940
• Osgood, William F. (1903), "A Jordan Curve of Positive Area", Transactions of the American Mathematical Society, 4 (1): 107–112, doi:10.2307/1986455, ISSN 0002-9947, JFM 34.0533.02, JSTOR 1986455
• Ross, Fiona; Ross, William T. (2011), "The Jordan curve theorem is non-trivial", Journal of Mathematics and the Arts, 5 (4): 213–219, doi:10.1080/17513472.2011.634320, S2CID 3257011. author's site
• Sakamoto, Nobuyuki; Yokoyama, Keita (2007), "The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic", Archive for Mathematical Logic, 46 (5): 465–480, doi:10.1007/s00153-007-0050-6, ISSN 0933-5846, MR 2321588, S2CID 33627222
• Thomassen, Carsten (1992), "The Jordan–Schönflies theorem and the classification of surfaces", American Mathematical Monthly, 99 (2): 116–130, doi:10.2307/2324180, JSTOR 2324180
• Tverberg, Helge (1980), "A proof of the Jordan curve theorem" (PDF), Bulletin of the London Mathematical Society, 12 (1): 34–38, CiteSeerX 10.1.1.374.2903, doi:10.1112/blms/12.1.34
• Veblen, Oswald (1905), "Theory on Plane Curves in Non-Metrical Analysis Situs", Transactions of the American Mathematical Society, 6 (1): 83–98, doi:10.2307/1986378, JSTOR 1986378, MR 1500697
External links
• M.I. Voitsekhovskii (2001) [1994], "Jordan theorem", Encyclopedia of Mathematics, EMS Press
• The full 6,500 line formal proof of Jordan's curve theorem in Mizar.
• Collection of proofs of the Jordan curve theorem at Andrew Ranicki's homepage
• A simple proof of Jordan curve theorem (PDF) by David B. Gauld
• Brown, R.; Antolino-Camarena, O. (2014). "Corrigendum to "Groupoids, the Phragmen-Brouwer Property, and the Jordan Curve Theorem", J. Homotopy and Related Structures 1 (2006) 175-183". arXiv:1404.0556 [math.AT].
| Wikipedia |
Neutrons, energy distribution, neutron interactions.
Neutrons have roughly the same mass as a proton, but no charge.
Neutrons have a larger range than charged particles, because they don't interact with the electrons in the atoms of matter, as they have no charge.
Free neutrons
Free neutrons are unstable, with a halflife of about 10.4 minutes. They undergo \(\beta^-\) decay, yielding a proton, an electron and an antineutrino: $${}^1_0n \to {}^1_1p + {}^{\ \ \,0}_{-1}e + \overline{v}$$
Free neutrons can be classified according to their kinetic energy. They can be roughly divided into three energy ranges:
Thermal neutrons (up to 1 eV) are in a thermal equilibrium with their surrounding, with the most probable energy at 20 degrees C being 0.025 eV (~ 2km/s). They often have a much larger neutron absorption cross-section than fast neutrons.
Resonance neutrons (1 eV - 100 keV) have energies where the neutron capture cross-sections peak, and the probability of capture exceeds the probability of fusion.
Fast neutrons (1-10 MeV) are produced by nuclear fission. They have a Maxwell-Boltzmann distribution of energy with a mean energy of 2 MeV in the case of uranium-235 fission.
Thermal reactors use thermal neutrons to sustain the fission chain reaction. They thus slow down the produced fast neutrons using a neutron moderator, until their energy is in thermal equilibrium with the atoms in the system. Fast reactors, on the other hand, use fast neutrons to sustain the fission chain reaction, not using any neutron moderator.
The energy of a neutron has a big impact on its cross-sections. An important property of a reactor core thus is its neutron energy distribution.
Neutron interactions
Since neutrons have no charge, they pass easily through electron clouds and directly interact with nuclei. Those interactions can be thought of as two steps: first the neutron is absorbed by the nucleus, forming a compound nucleus, which then decays.
There are two main kinds of interactions: scattering and absorption.
Elastic scattering (n,n): The neutron and nucleus are left unchanged, no energy is transferred. That is, Q=0. It can be understood as a collision where the neutron bounces off the nucleus, and is the only interaction that is not considered to form a compound nucleus.
Inelastic scattering (n,n'): The neutron penetrates the nucleus, forming a compound nucleus in an excited state. This compound nucleus returns to its ground state by gamma decay, emitting a neutron (on a different energy level than the incoming one, so energy was transferred).
In general, scattering refers to nuclear reactions in which the outgoing particle is the same as the incident particle.
Radioactive capture (n, \(\gamma\) ): The neutron is absorbed by the nucleus, and gamma rays are emitted. Other neutron absorption reactions are (n, \(\alpha\) ) and (n,p), which can be either exothermic or endothermic.
Neutron production (n,2n) and (n,3n): The nucleus emits one or more neutrons. For example, \({}^2H\) and \({}^9Be\) have loosely bound neutrons that can easily be extracted.
Fission: The neutron causes the nucleus to split into two fission products, emitting 2 to 4 neutrons and releasing energy. $$X \to Y + Z + {}^1_0n$$ This can also happen spontaneously, but its probability is low. (Californium has an exceedingly high probability for spontaneous fission, ~3%, which is why it is sometimes used to jumpstart nuclear reactors).
Example: Uranium fission
$${}^1_0n + {}^{235}_{92}U \,\to\, {}^{236}_{92}U \,\to\, {}^{92}_{36}Kr + {}^{141}_{56}Ba + 2.44{}^1_0n$$
The energy of the resulting neutrons is a probability distribution, with the most likely energy ~700 keV and the average energy ~2 MeV. The are, however, too high-energy to engage in further fission events. In a reactor, they thus need to be slowed down by a moderator like water, where neutrons successively lose energy through elastic scattering.
Cross-sections
Cross-sections characterize the probability of a neutron interaction, quantified in terms of characteristic target areas (where a larger area means a larger probability of interaction).
The microscopic cross-section \(\sigma\) specifies the probability of a particular neutron interaction with a target nucleus. It depends on the target nucleus, the type of reaction, the neutron energy, and the target energy.
The macroscopic cross-section \(\Sigma\) specifies this probability of a neutron interaction for the whole medium. It is considered per unit path length that the neutron travels. The average distance a neutron travels before interacting with a nucleus is its mean free path \(\frac{1}{\Sigma}\) .
With \(N\) the nuclear density:
$$\sigma \times N = \Sigma$$ $$\Sigma \times \text{neutron flux} = \text{reaction rate}$$
Resonance region:
When the sum of the kinetic energy of the neutron and the binding energy correspond to an energy level of the compound nucleus, the neutron cross section exhibits a spike in its probability of interaction which are called resonances.
Fission cross-section for U-235:
Neutrons generated by fission are in the higher energy range, the red circle marks the thermal energy range. U-235 fissions with neutrons at all energy levels (while U-238 requires neutrons of at least 1 MeV), however in the range of 1-100 eV there are also strong resonance regions for neutron capture. So in order to keep a fission chain reaction going with U-235, neutrons need to be slowed down into the thermal region (~0.025 eV), quickly in order to avoid spending too much time in the resonance regions for capture.
In calculation, neutron energy is often simplified by splitting the energy range into two groups (thermal and fast) and taking the average in each group.
2022 (c) Christina Unger
Rendered with Hugo. Theme based on Kube. | CommonCrawl |
Memory for own actions in parrots
Mental representation and episodic-like memory of own actions in dogs
Claudia Fugazza, Péter Pongrácz, … Ádám Miklósi
Dogs distinguish human intentional and unintentional action
Britta Schünemann, Judith Keller, … Juliane Bräuer
Innovation in wild Barbary macaques (Macaca sylvanus)
Federica Amici, Alvaro L. Caicoya, … Anja Widdig
An Orangutan Hangs Up a Tool for Future Use
Nicholas J. Mulcahy
Visuospatial information foraging describes search behavior in learning latent environmental features
David L. Barack, Akram Bakkour, … C. Daniel Salzman
Using an Innovation Arena to compare wild-caught and laboratory Goffin's cockatoos
Theresa Rössler, Berenika Mioduszewska, … Alice M. I. Auersperg
Out-of-reach rewards elicit human-oriented referential communicative behaviours in family dogs but not in family pigs
Paula Pérez Fraga, Boglárka Morvai, … Attila Andics
Selection for specific behavioural traits does not influence preference of chasing motion and visual strategy in dogs
Judit Abdai & Ádám Miklósi
The effect of age on discrimination learning and self-control in a marshmallow test for pigs
Annika Krause, Maren Kreiser, … Sandra Düpjan
Sara Torres Ortiz1,2,
Simeon Q. Smeele3,4,5,
Juliette Champenois1,2 &
Auguste M. P. von Bayern1,2
Scientific Reports volume 12, Article number: 20561 (2022) Cite this article
The ability to recall one's past actions is a crucial prerequisite for mental self-representation and episodic memory. We studied whether blue-throated macaws, a social macaw species, can remember their previous actions. The parrots were trained to repeat four previously learned actions upon command. Test sessions included repeat trials, double repeat trials and trials without repeat intermixed to test if the parrots repeated correctly, only when requested and not relying on a representation of the last behavioral command. Following their success, the parrots also received sessions with increasing time delays preceding the repeat command and successfully mastered 12–15 s delays. The parrots successfully transferred the repeat command spontaneously at first trial to three newly trained behaviors they had never repeated before, and also succeeded in a second trial intermixed with already trained actions (untrained repeat tests). This corroborates that successful repeating is not just an artifact of intense training but that blue-throated macaws can transfer the abstract "repeat rule" to untrained action. It also implies that an important aspect of self-representation has evolved in this avian group and might be adaptive, which is consistent with the complex socio-ecological environment of parrots and previous demonstrations of their complex cognition.
Most of human everyday memories involve past actions, i.e., memories of what oneself or others did in the past1. Such memories of past events involving past actions shape current behavior, as well as guide future behavior2. Yet, most memory studies on humans and non-human animals are based on remembering passively presented stimuli, without overtly involving and often even restricting actions1,3,4. Besides involving overt motor behavior, actions are also different from the traditional passive stimulus presentation approach because they are self-performed1,3,4,5. Given that the definition of human episodic memory6,7 is the "memory for personally experienced events" and that self-performed actions are per se "personally experienced", memory of self-performed actions should be considered episodic accordingly. Being able to recall and represent one's own past actions is at the same time considered as of the main prerequisites or main "building blocks" for the ability to represent the self8,9,10. It has been speculated that it may have evolved early in mammalian evolution and could be widespread phylogenetically, since it has recently been documented in domestic dogs3,8,11 (Canis lupus familiaris), as well as dolphins, primates and pinnipeds12,13,14. Given that self-representation in turn is a prerequisite for human episodic memory15, this applies to remembering one's own past actions inevitably. As self-awareness of one's own personal experiences is not clearly proven in animals, the analogous memory processes in animals are often referred to as episodic-like memory15,16. To date, several studies have reported evidence for episodic-like memory in a few animal species although most of them remain debated. For example Clayton and colleagues17 claimed the first evidence for episodic-like memory showing that scrub jays (Aphelocoma californica) are able to remember where and when they have cached a particular food item, and claimed it, could imply the ability of episodic memory for the species. However, it was argued that food caching and retrieval behavior is very likely genetically fixed, and the paradigm used failed to provide evidence of conscious ('autonoetic') components of episodic memory possible in a scrub jay18, thus, the ability was called episodic-like memory19,20,21.
Testing an animal's ability to repeat its own previous behavior, i.e., the "repeat paradigm", has been put forward as a method to determine animals' memory for their own actions12,22 and recently has been proposed as a method of testing episodic-like memory in two studies on dogs8,11. The methodology has great potential as a comparative test for episodic-like memory content in animals, but the prerequisites for truly episodic-like memory retrieval rely on the assumption that the animals must not expect to be asked about the past event23. The unexpectedness of the test rules out a prepared behavioral response (i.e., the animal expects the trial and gets ready to offer the correct behavioral response rather than retrieving the past event from their memory) from the animal, instead of an episodic content retrieval8,24.
To date, the "repeat paradigm" has been employed in pinnipeds, pigtail macaques (Macaca nemestrina), dogs and dolphins (Tursiops truncatus)8,11,12,13,14, i.e., four mammalian taxa. Even though all tested species learned to repeat their previous behavior on a range of behaviors they could perform on command, the extent to which the different species relied on the underlying repeat concept, i.e., "repeat your last action" remains unclear. Little detail was given for the monkeys, dogs and dolphins regarding the training procedure and previous experience, making comparisons between species difficult. In any case, dolphins and dogs were reported to generalize the repeat command to novel behaviors8,11,12 and macaques were found to fail13 while pinnipeds were not tested14. Concerning the duration of memory retention, the species differed substantially, but given the methodological differences between the studies, one cannot directly compare these results and draw reliable conclusions from them.
In order to gain deeper insights into the phylogenetic distribution of the ability to recall self-performed actions, it is necessary to broaden the comparison to species outside of the mammalian clade. Parrots, a diverse avian order, appear an interesting first model group for investigating memory for own actions because together with dolphins, primates and corvids they stand out among vertebrates in terms of both their relative brain size25 and neuron density26,27 and are considered on a par with great apes in terms of complex cognition25. Parrots exhibit high social complexity and long-term monogamy28 and share several socio-ecological and cognitive traits with dolphins and primates29,30,31. Various memory studies have been conducted in different avian species, ranging from corvids to pigeons16,32,33,34 but to date, memory for own actions has not been investigated in birds. In parrots, no study has focused on memory specifically, although it seems equally adaptive for them than it has been suggested to be for corvids35,36,37. Albeit, many cognitive abilities that have been studied in parrots involve short-term and possibly other types of memory. For example, previous work on African grey parrots (Psittacus erithacus) has shown that they can categorize objects, recognize similarity and point out missing information, all of which requires short-term memory38. For instance, in order to show appropriate social behavior, individuals may have to remember specific social events (in terms of what happened where and when and who was involved) that could lead to, for example, sudden changes in the hierarchy. According to Clayton et al.16 monitoring the interactions of conspecifics and remembering specific social events is crucial for an updated knowledge of the relationships and social dynamics between one's group members. The type of memory necessary to perform these functions fits the behavioral criteria of episodic memory. Such a memory of social interactions would of course also include events of one's own interactions with conspecifics. A good example is a study by Emery and Clayton39 conducted with scrub-jays where one group of birds experienced stealing other birds' caches whereas a second group lacked this experience. Only the group that had experienced being a thief re-cached their food after having been observed by other individuals35,39. If convergent trends in the evolution of sociality and cognition hold, we should therefore see robust memory for one's own past actions in parrots, another avian taxon.
The objective of the current study was to extend our understanding of memory for own actions in mammalian taxa to a distantly related avian model, and to examine the parrots' concept learning ability. We tested blue-throated macaws (Ara glaucogularis), a social, large-brained species, endemic to Bolivian Savannah type habitats40 that had been trained to perform four different arbitrary behaviors upon specific commands (hand signals). After the training of the specific actions, they were trained to respond to the repeat command reliably, i.e., repeating one out of the four actions they had previously performed upon command when requested to "repeat" (details of their training procedure are provided in Supplementary materials). Depending on the experimental condition, they had to repeat straight after the performed action, or after an increasing time delay. Extending previous studies, we also tested the parrot's ability to generalize the repeat rule to novel behaviors. Considering the results from the mammalian studies8,11,13,14,22 and given that parrots share similar socio-ecological environments30,41 and comparable cognitive abilities41,42,43,44,45, we hypothesized that the macaws will be able to learn the abstract rule of repeating their own previous action, to remember self-performed behavior for delays comparable to those of the mammals tested, and to transfer the repeat command to novel actions.
Repeat test
Single repeat trials
All parrots showed overall performance well above chance level across all four behaviors, about 75% with a chance level of 25% (see Fig. 1; see Table SP1) in the single repeat trials (Fig. 1). There were no differences between the individuals (sigma: 0.46, 89% PI: 0.04–1.31) and the behavior to be repeated had little effect on the performance except for 'Head shake', which all parrots were less likely to successfully repeat (sigma: 0.45, 89% PI: 0.09–0.95). The time between the command for the action and the repeat command was five seconds (Fig. 2A).
Performance in the single repeat trials. Gray density plots show the prior centered around chance-level (gray dashed line). Colored density plots show the posterior distributions for the average performance. Dashed colored lines are the means. Dots are outcomes per trial.
(A) Illustration for the double-repeat trial procedure and the delays between the different commands. (B) Delayed repeat test procedure. If the parrot repeated correctly, the delay increased three seconds in the next trial, otherwise, the delay decreased three seconds (artwork by Sara Torres Ortiz).
Double repeat trials
In the double repeat trials, all three birds also performed well above chance level across all four behaviors with an overall performance of 60% correct, with a chance level of 6.25% (Fig. 3). There were no significant differences between individuals (sigma: 0.47, 89% PI: 0.03–1.4) and behaviors (sigma: 0.89, 89% PI: 0.19–1.7). The time between the action command and the second repeat was 10 s (Fig. 2B).
Performance in the double repeat trials. Gray density plots show the prior centered around chance-level (gray dashed line). Colored density plots show the posterior distributions for the average performance. Dashed colored lines are the means. Dots are outcomes per trial.
Delayed repeat test
The parrots were able to perform above chance level for delays up to 12–15 s (see Fig. 4). There were no significant differences between the three individuals (sigma: 0.48, 89% PI: 0.03–1.37) and behaviors (sigma: 0.55, 89% PI: 0.04–1.30). The performance in the delayed (single) repeat test with a corresponding delay of 10 s was 41% (0.89 PI: 26–57%; see Fig. 4).
Performance in the delayed repeat test. Colored dots are average performance per delay and individual (green—Mr Huang, yellow—Charlie, red—Gargamel). Dashed line is the predicted performance and gray shaded area the 89% PI.
Novel behavior repeat test
All three parrots were able to repeat newly learned behaviors spontaneously in the first and in a second test trial, which were intermixed with already repeated behaviors. Table 1 summarizes the animals' performance in repeating novel behaviors. "Success" refers to successful repeating in both test trials. All three parrots succeeded repeating the behavior "vocalization". Concerning the second novel behavior, "lift left leg", only Charlie lifted the correct same leg when asked to repeat, while Mr Huang and Gargamel lifted the right leg. The last tested novel behavior, "Move ring into lid", was mastered spontaneously by Mr Huang and Gargamel, they both moved the ring into the lid. Charlie instead lifted the left leg during the session, when asked to repeat the "Move ring into lid", so performed the previous learned behavior. Thus, all individuals repeated two out of three novel behaviors in their first and second trial without training.
Table 1 Summary of the results of the novel behavior repeat test.
The blue-throated macaws tested in this study could recall their own previous actions well above chance levels, providing the first evidence for mental representation and memorization of their own actions in an avian taxon. The overall (single) repeat performance of parrots was similar to those of mammals. Fugazza et al.8 hypothesized that one of the building blocks of the supposedly complex ability to represent the self might be the capacity to mentally represent one's own behavior. Our study shows that this ability is also present in parrots and therefore suggests it has evolved in birds too. This is an interesting finding considering the evolutionary distance between mammals and birds. Birds also have differently structured brains compared to mammals although their forebrain (nido- and mesopallium) represents a homologous structure to the mammalian neocortex28,46,47,48. Like the neocortex, this telencephalic structure fulfills higher cognitive functions even if it lacks the cortex-typical lamination and is structured differently49. Even though parrots' brains may be small in absolute size, their relative brain sizes stand out amongst all vertebrates like otherwise only those of corvids, primates and dolphins31. Additionally, parrots exhibit an astonishing neuronal density in their pallium with neuron numbers exceeding those of primate species27. It may be the large absolute numbers of telencephalic neurons in their pallium that explains their advanced behavioral and cognitive complexity27,28,50. Together, the architectural differences in the brains of mammals and birds and the great evolutionary distance indicate that the ability to remember their own actions is likely a result of convergent evolution.
The finding that the birds performed above chance level and similarly well in single and double repeat trials suggests that the birds relied on their memory of their last own previous action rather than the last command (i.e., the previous hand signal) they received. The interval between the command for a behavioral action and the second repeat command was on average 10 s. If the parrots were remembering the hand gesture, their performance of the second repeat should have been similar to the performance of the delayed condition at 10 s (Fig. 2A,B). However, the parrots' performance in the delayed (single) repeat test with a corresponding delay of 10 s was 20% lower compared to their performance in the double repeat. It is therefore unlikely that the parrots remembered the last command rather than responding to the repeat command. The better performance at double repeating compared to remembering following a delay of 10 s suggests that the parrots remembered their last action, which was performed only 5 s earlier, rather than the last command.
Our other objective was to examine whether the parrots had actually abstracted the underlying rule of repeating their last performed action, which can be considered as an abstract concept11. The parrots had been trained with four behaviors only before it was tested whether they had learned the "repeat" rule and could transfer it to novel actions they had not repeated before (novel behavior repeat test). Their spontaneous success shows that indeed they had generalized the repeat rule. The fact that parrots learned this abstract concept after only being trained for four actions is remarkable. By comparison, pigeons, monkeys, parrots and corvids needed thousands of trials to learn the same-different concept required to succeed in match-to-sample tests51,52. Similarly, sea lions need over two thousand trials to form equivalence relations, where the animal needs to spontaneously transferred the relationship on a set that is reflexive (A ~ A), symmetric (if A ~ B, then B ~ A) and transitive (if A ~ B and B ~ C, then A ~ C)53,54. In general, a conceptual ability provides great efficiency to learning55. Fast concept learning and generalization may be adaptive under many circumstances as it allows the individual to apply prior learning to a new environment avoiding the costs and risks associated with new trial-and-error learning55. Why the macaws tested in this study showed such a rapid concept learning ability, needs to be solidified by further direct comparisons of different taxa in the same tasks and remains speculation until it is investigated further. Maybe parrots need to adapt fast to novel environments or social constellations but the same is likely true for primates and pinnipeds14,56.
The other important conclusion one can draw from the parrots' spontaneous transfer ability is that they could recall their own previous behavior without being trained to do so. Given that the repeat command was never associated with novel actions, we can assume that they did not expect the repeat command after the novel action in the experimental test. According to Fugazza et al.8, the unexpectedness of the test rules out a "prepared behavioral response" explanation and suggests accidental encoding of the memory8. In our experiment, it can be assumed that repeating the novel behavior is unexpected by the parrots. Firstly, because the novel behaviors were never asked to be repeated, and secondly, because of the long break between the delayed repeat experiment and the novel behavior repeat test (~ 45 days). Accordingly, the parrot's success in retrieving the memory of the novel behavior would have to be explained by episodic-like memory. Of the four previously tested mammalian taxa, only dogs have been tested for accidental encoding (albeit with substantially longer time scales)8 and only dolphins and dogs were shown to generalize the repeat command to new behaviors8,11,12. The pinniped species were not tested in this respect and the macaques were not able to transfer the repeat rule having been trained with three behaviors only13,14. The dolphins12,22 transferred the repeat rule to new behaviors but already knew the repeat command for a long time and there is no information on how many behaviors they required to learn the repeat rule before they were capable of generalizing. Both dog studies8,11 reported that the subjects were able to transfer the repeat rule after they had trained the repeat command on six to seven behaviors. Although the data on the different species are not directly comparable due to the differences in methodology, particularly concerning the previous training exposure, our results suggest that parrots have a capacity to generalize abstract rules comparable to big-brained mammals as dolphins in the same paradigm. Domestic dogs which also generalized well, may represent a special case because of domestication and enculturation57,58. For instance, studies on wolves raised by humans and dogs with little human contact showed that dogs were still better at reading human communicative gestures, suggesting that artificial selection for a set of social-cognitive abilities enhanced their responsiveness to human communicative signals59. Similar findings have been reported in birds as well, e.g., artificial selection negatively affecting spatial learning ability in white leghorn chicken compared to their ancestor the red jungle fowl60. In any case, such effects of artificial selection and enculturation should be considered in phylogenetic comparisons of cognitive abilities and may explain why dogs performed so well in this task.
Concerning the duration of memory retention, our results show that parrots could remember their own behavior for up to 12–15 s. The retention interval is similar to those previously reported in the wide variety of species tested in Delayed-Match-to-Sample, a common paradigm to test animals' working memory4,61. Our results are also quite similar to those reported in pinnipeds, macaques and dolphins tested previously in this paradigm13,14,22 but dogs exhibited longer retention intervals. These differences can be explained by methodological differences. In one of the dog studies, the delay intervals were fixed with 10 and 30 s, in our case, if the parrots made a mistake, the interval would decrease 3 s so that the parrots were never exposed to such long intervals11. In the second dog study, the trial for the delay condition only contained a single action that had to be repeated after 1 h8. For parrots, each experimental session had 16 trials, which adds noise into their memory system increasing the attention needed to perform correctly8. As parrots paralleled other mammal species in performance for single and double repeats, it is likely that the reported differences are a result of different methodology. Unfortunately, details of the training methodology of the previous studies are lacking. Replicating the previous pinniped study in this respect, the parrots were not trained for expecting the delays before entering the delayed repeat test, so it was tested how well they remembered when not actively trying to remember. Concerning the dogs, in contrast, no details are given as to whether the delay was learned with training steps gradually increasing in duration, or whether the animals were directly tested at their longest retention durations in an unexpected manner. The same challenge is present when comparing results of Delayed-Match-to-Sample tasks, as the amount of training is not always reported in sufficient detail, or at all, making species comparisons in memory performance very challenging4.
The incredibly complex brains of humans and varied faculties are likely to have evolved from simpler prototypes of our ancestors62. Paul Cisek developed the theory of "phylogenetic refinement" where behaviors and brain structures are the consequence of evolutionary refinement from more basic building blocks63. One may see the parrots' ability to represent and remember their own actions as an important building block of the ability to represent "the self"8. In order to experience one's self in the past one would need an awareness of self also in the present time1. Thus, the parrots could be considered have evolved a prerequisite of episodic-like memory. Fugazza and colleagues8,24 state that previous studies can be explained by a "prepared behavioral response" by the subjects as they expected the repeat command to be given. We argue that the repetition of a novel action can be considered as accidental encoding in our study given that the parrots had never been asked to repeat those actions before. Additionally, the test of the first untrained action for the novel repeat test took place following a ~ 45-day break after the delayed repeat test had been completed, so that the parrots were not in a repeat testing routine anymore.
Like most parrots, blue-throated macaws are very social, show fission–fusion dynamics with temporally changing flock composition (e.g., for foraging or during certain developmental phases40) and live in long-term monogamous pairs throughout their lives64, all of which likely has selected for socio-cognitive adaptations and even larger relative brain sizes26,50. Being able to remember one's own behavior may be equally adaptive for species living in complex societies31,65 than remembering the behavior and interactions of third-parties50,66,67. Concerning parrots, remembering one's own past action might be particularly important for coordinating cooperative behaviors with one's mate (e.g., when jointly rearing offspring28) or group members (e.g., when foraging individuals of a flock leave to join other groups29) for which parrots exhibit the cognitive and motivational basis42,44,68,69,70. Parrots also have been shown to display capacity for reciprocity, which may require long-term episodic memories of own and others' behavior44,66,69 for the reciprocal sharing of resources.
In summary, blue-throated macaws were proficient in learning and generalizing the abstract concept of "repeat my most recent action". Our results show that parrots are capable of reporting their own previous actions upon command. They therefore exhibit a pre-requisite for self-representation and for episodic-like memory previously only shown in social, large-brained, domesticated or enculturated mammals. Given the evolutionary distance between mammals and birds31 with a common ancestor around 297 mya31, this ability is likely to constitute an example of convergent or independent evolution, due to similar selection pressures28,30,31,71. Our study provides new insights into the independent evolution of functionally equivalent building blocks of self-representation and episodic-like memory in distantly related taxa10,20.
Subjects and housing
Three adult male blue-throated macaws (Ara glaucogularis) (between 10 and 12 years old) were trained and subsequently tested. Details on age and sex of the individuals are specified in Table SP2. The training of the behavioral commands and repeat commands are described in the Supplementary materials.
The birds were housed in the Comparative Cognition Research Station, inside Loro Parque zoo, in Tenerife (Spain). Loro Parque Foundation staff hand-raised and group-reared all the parrots used in this study (more details on housing and diet are described in Supplementary materials). Water was provided at libitum and the parrots were fed twice a day. All parrots participated voluntarily both in the training sessions and the experimental sessions. During training, sunflower seeds were used as rewards, and during testing small pieces of walnut were used.
Experimental setup and general procedures
The birds were trained and tested individually in separate testing rooms (1.5 × 1.5 × 1.5 m) artificially lit with daylight lamps (Arcadia Zoo Bars ©), which the birds were well habituated to. During the experiment, the subject sat on a perch facing the experimenter who stood inside the test room on the opposite side wearing mirror-glass and blinded sunglasses. A second person, the assistant, also wore mirrored but see-through sunglasses and observed the experiment through a window from the neighboring room. The experimenter gave the experimental commands (Table SP1) to the parrot, but could not see its response. If the parrot responded correctly to the command, the assistant in the neighboring room gave the parrot a whistle blow as immediate conditioned reinforcer, followed by a food reward (a piece of walnut) as positive reinforcement. If the parrot did not respond correctly, the whistle was not blown, no reward was given and the next trial started following a three second pause. The experimenter gave an equal piece of walnut every time the parrot performed the correct behavior in control trials. The order of the behavioral commands and repeat commands given in each session, was pseudo-randomized and counterbalanced across birds and determined before the session. The assistant also signaled to the experimenter which command to give next by showing the behavior label on the screen of an iPad visible only for the experimenter (see Video 1).
Testing criterion
After all animals had been trained for the four actions and the repeat command, the reliability of their response to the repeat command was tested in a 20-trial session. The command for one of the four trained behaviors was given either followed by a "repeat" command in 60% of the trials or by a command for one of the four behaviors (40%). This was implemented to prevent the animal from learning to simply repeat the first command in every new trial. The list of commands was randomized and counter-balanced. To reach criterion, the animal had to perform at least seven repeats out of 12 (58%) correctly during the session.
Experimental conditions
To test the animal's ability to repeat their own previous behavior on command, eight experimental sessions consisting of 26 trials each were completed. A "single repeat" trial was composed of the command for one of the four trained actions followed by the repeat command. A "double repeat" trial started with the command for one of the four trained actions followed by a repeat and a second repeat command. There was no training for the double repeat trials. The "control" trials consisted of requesting one of the four trained actions, followed by the command for another one of the four trained actions. For each session four of the trials (15%) were "single repeats", eight (30%) were "double repeats" and ten trials (38%) were "controls". The first "repeat" of a "double repeat" trial was analyzed as a "single repeat trial" for the results.
To test how long the animals could remember their own previous behavior, gradually increasing delays were introduced between the behavior and the repeat command. Only single repeat trials (16 trials per session, and 6 sessions) were performed. A staircase paradigm in which a delay increased 3 s after a correct response and decreased 3 s after an incorrect response was implemented.
For the delay test, the assistant stood next to the experimenter inside the room together with the parrot. The assistant held a computer and communicated the next command to be given verbally to the experimenter (see Video 2). The list of behaviors was imported into an R session (RStudio, version 1.1.38345). After having entered the parrot's response (correct or not) and the delay had passed, the computer displayed the next behavior to be requested to the assistant. The assistant then communicated it to the experimenter, The computer automatically updated the delay duration for the following trial.
To test if the parrots could generalize the repeat rule and apply it to novel behaviors, i.e., trained behaviors that the parrots display upon a specific gestural command, but that they had never been asked to repeat before15,46. We tested if the parrots were capable of repeating three newly trained behaviors spontaneously from the first repeat trial. The parrots were therefore trained to perform three novel behaviors upon specific commands (see Supplementary Material for description). The training of the new actions occurred after the end of the delayed repeat test and took ~ 45 days. Once the parrot performed the novel behavior associated with the hand signal reliably, we started the test. The first test session started with a trial in which the subject was requested to perform the new behavior followed by the repeat command. Subsequently, the experimenter continued with 2–4 control trials with previously known behaviors and then requested the new behavior again, followed by the repeat command. If the new behavior was repeated correctly in the first trial and second critical trial spontaneously, the parrot was considered successful. And the training for the next novel behavior started the following day.
Performance in the repeat test
To test if individuals could remember their own previous behavior, we estimated the probability of responding correctly to the repeat command and compared this to chance-level. We assumed the chance-level to be 1/4 if they chose one of the four behaviors randomly or if they showed a preference for one of these behavior on all trials. To estimate the probability of success we used a Bayesian multilevel model with the following structure:
$$\begin{array}{ll}{\text{response}}_{i}& \sim {\text{binomial}}(1,\hspace{0.25em}{p}_{i})\\ {\text{logit}}({p}_{i})& ={\alpha }_{{\text{individual}}[i]}+{\gamma }_{{\text{behaviour}}[i]}\\ \alpha & \sim {\text{normal}}(\overline{\alpha },\hspace{0.25em}{\sigma }_{\alpha })\\ \overline{\alpha }& \sim {\text{normal}}(-0.5,\hspace{0.25em}1)\\ \gamma & \sim {\text{normal}}(0,\hspace{0.25em}{\sigma }_{\gamma })\\ {\sigma }_{\alpha }& \sim {\text{exponential}}(1)\\ {\sigma }_{\gamma }& \sim {\text{exponential}}(2)\end{array}$$
The prior for the average performance was chosen to center most mass around the chance level of 1/4.
To test performance on the double repeat task the same model was used, with the only modification that the prior for \(\overline{\alpha }\) was centered around − 1. Models were fitted using ulam from the rethinking package72 which runs the Stan sampler using the cmstanrinterface. We ran four chains with 8000 iterations and a 500-iteration warm-up. Rhat and divergence were monitored.
Performance in the delayed "repeat" test
To test if and how performance declined with increased delay between the initial behavior and the repeat command a similar Bayesian model was used with a multilevel slope added:
$$\begin{array}{ll}{\text{response}}_{i}& \sim {\text{binomial}}(1,\hspace{0.25em}{p}_{i})\\ {\text{logit}}({p}_{i})& ={\alpha }_{{\text{individual}}[i]}+{\gamma }_{{\text{behaviour}}[i]}+{\beta }_{[i]}*\text{log(time)}\\ \alpha & \sim {\text{normal}}(\overline{\alpha },\hspace{0.25em}{\sigma }_{\alpha })\\ \overline{\alpha }& \sim {\text{normal}}(-1,\hspace{0.25em}1)\\ \gamma & \sim {\text{normal}}(0,\hspace{0.25em}{\sigma }_{\gamma })\\ {\beta }_{[i]}& ={\zeta }_{{\text{individual}}[i]}+{\iota }_{{\text{behaviour}}[i]}\\ \zeta & \sim {\text{normal}}(0,\hspace{0.25em}{\sigma }_{\zeta })\\ \iota & \sim {\text{normal}}(0,\hspace{0.25em}{\sigma }_{\iota })\\ {\sigma }_{\alpha },{\sigma }_{\gamma },{\sigma }_{\zeta },{\sigma }_{\iota }& \sim {\text{exponential}}(2)\end{array}$$
All applicable international, national, and institutional guidelines for the care and use of experimental animals were followed. In accordance with the German Animal Welfare Act of 25th May 1998, Section V, Article 7 and the Spanish Animal Welfare Act 32/2007 of 7th November 2007, Preliminary Title, Article 3, the study was classified as non-animal experiment and did not require any approval from a relevant body. The experiments did not require an application to the Animal Ethics Committee of neither Germany nor Spain, as animals participated voluntarily in the experiments and were not affected by them in any way. This article does not contain any studies with human participants performed by any of the authors. The ARRIVE guidelines for the reporting of animal experiments were followed. The informed consent to publish was obtained from all subjects and/or their legal guardian(s) for the publication of identifying information/images.
All data and code is publicly available on GitHub (https://github.com/simeonqs/Memory_for_own_actions_in_parrots) and permanently stored on Edmond (https://doi.org/10.17617/3.XYSJ66—link will work upon publication). For review data can be accessed on Edmond using the following link: https://edmond.mpdl.mpg.de/privateurl.xhtml?token=2a7cf6ab-7b5f-4732-91ee-d9e1360c9299.
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We thank the Loro Parque and its president, Mr. Wolfgang Kiessling for their generous support, the access to the birds and the research facilities. We thank the Loro Parque Fundación and its president Mr. Christoph Kiessling for their collaboration and the staff of the Loro Parque Fundación, the animal caretakers and the veterinary department for their constant support. We are grateful to Ariana Hernandez and Anouk Verstraeten for assisting in part of the data collection and to Mathias Osvath, Fernando Colmenares, Tobias Staudigl, Manfred Gahr and Jason Bruck for comments to improve the manuscript. Lastly, we thank Caterina Bonizzato and Janneke Vos for their artwork. Simeon Quirinus Smeele received funding from the International Max Planck Research School of Quantitative Behaviour Ecology and Evolution and Sara Torres Ortiz from the Animal Minds Project e.V. Simeon and Sara are members of the International Max Planck Research School for Organismal Biology.
Open Access funding enabled and organized by Projekt DEAL.
Max Planck Institute for Biological Intelligence, in Foundation, Seewiesen Eberhard-Gwinner-Strasse, 82319, Starnberg, Germany
Sara Torres Ortiz, Juliette Champenois & Auguste M. P. von Bayern
Max-Planck Comparative Cognition Research Station, Loro Parque Fundación, Av. Loro Parque, 38400, Puerto de la Cruz, Tenerife, Spain
Max Planck Institute of Animal Behavior, Am Obstberg 1, 78315, Radolfzell Am Bodensee, Germany
Simeon Q. Smeele
Max Planck Institute for Evolutionary Anthropology, Deutscher Platz 6, 04103, Leipzig, Germany
Department of Biology, University of Konstanz, Konstanz, Germany
Sara Torres Ortiz
Juliette Champenois
Auguste M. P. von Bayern
Conceptualization, S.T.O. and A.V.B.; methodology, S.T.O., S.Q.S. and A.V.B.; investigation, S.T.O., J.C. and A.V.B.; formal analysis, S.T.O. and S.Q.S.; software, S.Q.S.; writing—original draft, S.T.O., S.Q.S., T.J.S.B. and O.N.L.; writing—review and editing, S.T.O., S.Q.S., J.C. and A.V.B.; resources, S.T.O. and A.V.B.; supervision, A.V.B.
Correspondence to Sara Torres Ortiz or Auguste M. P. von Bayern.
The authors declare no competing interests.
Supplementary Information 1.
Supplementary Video 1.
Torres Ortiz, S., Smeele, S.Q., Champenois, J. et al. Memory for own actions in parrots. Sci Rep 12, 20561 (2022). https://doi.org/10.1038/s41598-022-25199-x | CommonCrawl |
\begin{document}
\author{Jean B\'{e}rard, Didier Piau}
\address[Jean B\'{e}rard]{\noindent Universit\'e de Lyon ; Universit\'e Lyon 1 ; Institut Camille Jordan CNRS UMR 5208 ; 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex; France.
\newline e-mail: \rm
\texttt{[email protected]}}
\address[Didier Piau]{\noindent Institut Fourier - UMR 5582,
Universit\'e Joseph Fourier Grenoble 1, 100 rue des Maths, BP 74,
38402 Saint Martin d'H\`eres, France.
\newline e-mail: \rm \texttt{[email protected]}}
\date{\today}
\title[CFTP times with ambiguities for particle systems]{Coupling from the past times with ambiguities and perturbations of interacting particle systems}
\keywords{Interacting particle systems, Coupling, Perturbations, Stochastic models of nucleotide substitutions}
\subjclass[2000]{60J25, 60K35, 92D20}
\begin{abstract} We discuss coupling from the past techniques (CFTP) for perturbations of interacting particle systems on $\Z^d$ with a finite set of states, within the framework of the graphical construction of the dynamics based on Poisson processes. We first develop general results for what we call CFTP times with ambiguities. These are analogous to classical coupling (from the past) times, except that the coupling property holds only provided that some ambiguities concerning the stochastic evolution of the system are resolved. If these ambiguities are rare enough on average, CFTP times with ambiguities can be used to build actual CFTP times, whose properties can be controlled in terms of those of the original CFTP time with ambiguities. We then prove a general perturbation result, which can be stated informally as follows. Start with an interacting particle system possessing a CFTP time whose definition involves the exploration of an exponentially integrable number of points in the graphical construction, and which satisfies the positive rates property. Then consider a perturbation obtained by adding new transitions to the original dynamics. Our result states that, provided that the perturbation is small enough (in the sense of small enough rates), the perturbed interacting particle system too possesses a CFTP time (with nice properties such as an exponentially decaying tail). The proof consists in defining a CFTP time with ambiguities for the perturbed dynamics, from the CFTP time for the unperturbed dynamics. Finally, we discuss examples of particle systems to which this result can be applied. Concrete examples include a class of neighbor-dependent nucleotide substitution model, and variations of the classical voter model, illustrating the ability of our approach to go beyond the case of weakly interacting particle systems. \end{abstract}
\maketitle
\section{Introduction}
The present paper discusses coupling from the past (CFTP) techniques for interacting particle systems. The key idea of CFTP, as described in the seminal paper \cite{ProWil} by Propp and Wilson, consists in simulating coupled trajectories of a finite state-space Markov chain from further and further in the past, until eventually the present state of the Markov chain is the same for all trajectories, regardless of their starting point. One thus obtains an exact realization of the stationary distribution of the corresponding Markov chain, and, under a certain monotonicity condition on the transitions of the chain, CFTP leads to a practical algorithm for sampling from the stationary distribution. Many extensions of this scheme have been developed since, notably to include processes on more general state-spaces, and situations where the monotonicity condition is not met (see the online bibliography \cite{Wil}).
Here, we consider interacting particle systems in the sense of \cite{Lig}, that is, continuous-time Markov processes describing the evolution of a system of states attached to the sites of $\Z^d$, the evolution at a site being governed by local transition rates involving the states of the neighboring sites. Our discussion is limited to particle systems with a finite state space whose dynamics can be prescribed by a finite family or transition rules (see below for a precise definition). For an ergodic particle system, one is interested in using CFTP to sample from the stationary distribution of the system. In general, it is not feasible to sample from the full stationary distribution, if only because a full configuration of the particle system is an infinite-dimensional object, comprising one definite state for each site of $\Z^d$. A more reasonable goal is the following: given a finite set of sites in $\Z^d$, use CFTP to sample from the marginal of the stationary distribution on this set of sites. This turns out to be possible when the interacting particle system possesses what we call in this paper a CFTP time, a precise definition being given below.
In the rest of this introduction, we give a quick overview of our results, some formal definitions and statements being postponed to later sections. A discussion of the existing literature and how the present work fits into it, is given at the end of the introduction.
\subsection{Interacting particle system dynamics and the graphical construction}
We consider interacting particle systems with a finite state space $\S$, whose set of sites is $\Z^d$ for some $d \geq 1$. To specify the dynamics of the system, we use the notion of a transition rule. Such a rule is a triple $\mathfrak{R}} \def\ZZ{\mathfrak{Z} = (f, A, r)$, where $A$ is a finite subset of $\Z^d$, $f \ : \ \S^A \to S$ is a map, and $r \geq 0$ is a non-negative real number. Given a configuration of the system $\eta = (\eta(z))_{z \in \Z^d} \in \S^{\Z^d}$, and $x \in \Z^d$, we denote by $\mathfrak{R}} \def\ZZ{\mathfrak{Z}^x \eta$ the configuration defined by $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}^x \eta)(x) = f( ( \eta(x+y))_{y \in A} )$ and $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}^x \eta)(z) = \eta(z)$ for $z \neq x$. (Our convention when $A=\emptyset$ is that the set $\S^A$ is a singleton on which $f$ takes a single well-defined value.)
Now, given a finite list of such transition rules $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}_i)_{i \in \I}=(f_i, A_i, r_i)_{i \in \I}$, we consider the interacting particle system dynamics specified by the infinitesimal generator $\LL$ satisfying \begin{equation}\label{e:inf-generator}\LL \phi (\eta) = \sum_{i \in \I} \sum_{x \in \Z^d} r_i \left( \phi( \mathfrak{R}} \def\ZZ{\mathfrak{Z}^x \eta ) - \phi(\eta) \right),\end{equation} for all functions $\phi \ : \ \S^{\Z^d} \to \R$ that depend only on a finite number of coordinates. Informally, \eqref{e:inf-generator} means that, during an infinitesimal time-interval of length $dt$, independently at each site $x \in \Z^d$, the transformation $\mathfrak{R}} \def\ZZ{\mathfrak{Z}_i^x$ is applied to the current system configuration with probability $r_i dt$. It is standard to check (see \cite{Lig}) that \eqref{e:inf-generator} uniquely characterizes a c\`adl\`ag continuous-time Markov process $(\eta_t)_t$ on $\S^{\Z^d}$ equipped with the product topology.
In the sequel, we assume that the dynamics is in fact built through the graphical construction associated with the list of rules $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}_i)_{i \in \I}$ (see \cite{Lig2} for examples of this construction). Specifically, we consider a Poisson point process $\Nsp$ on $\Z^d \times \I \times \R$ with intensity $J$ defined by $dJ(x,i,t) := r_i d (c_{\Z^d} \otimes c_{\I} \otimes \ell_{\R})(x,i,t)$, where $c_{\Z^d}$ and $c_{\I}$ denote the counting measure respectively on $\Z^d$ and $\I$, while $\ell_{R}$ denotes the Lebesgue measure on $\R$. The realization of the point process $\Nsp$ prescribes the dynamics of the particle system through the fact that, for every $x$, $(\eta_t(x))_t$ is a jump process whose state may change only at times $t$ for which there exists an (a.s. unique) $i$ such that $(x,i,t) \in \Nsp$, and that, for such a $t$, one has \begin{equation}\label{e:graph-construct} \eta_t = \mathfrak{R}} \def\ZZ{\mathfrak{Z}_i^x ( \eta_{t-}) .\end{equation} Given $t_1 \leq t_2$ and $\xi \in \S^{\Z^d}$, define $\Phi_{t_1}^{t_2}(\xi)$ to be the configuration of the system at time $t_2$ obtained by starting in configuration $\xi$ at time $t_1-$, and using the transitions specified by $\Nsp$ through \eqref{e:graph-construct}. We refer to $\Phi$ as the stochastic flow on $\S^{\Z^d}$ induced by $\Nsp$. Given $t \leq 0$, we use the notation $\Nsp_t = \Nsp \cap (\Z^d \times [t,0[ \times \I )$, and let $\F_t = \sigma(\Nsp_t)$.
\subsection{CFTP times with and without ambiguities}
We now consider the coupling properties of the flow $\Phi$. We say that a negative and a.s. finite random variable $T$ is a CFTP time (for site zero) if the following property holds on the event\footnote{Throughout the paper, we adopt the convention that all possible pathologies of CFTP times and their variants are concentrated on the event that the corresponding time takes the value $-\infty$, so that one does not have to bother excluding undesirable zero probability events when the corresponding time takes a finite value.} $\{ T > -\infty \}$, \begin{equation}\label{e:coupling} \mbox{for all }\xi_1, \xi_2 \in \S^{\Z^d}, \ [ \Phi_{T}^{0-}(\xi_1) ](0) = [ \Phi_{T}^{0-}(\xi_2) ](0).\end{equation}
One checks that the existence of a CFTP time implies ergodicity of the particle system. Moreover, starting from an arbitrary configuration $\xi \in \S^{\Z^d}$, the distribution of $[\Phi_T^{0-}(\xi)](0)$ is exactly the marginal at site $0$ of the corresponding stationary distribution. To obtain a sample from the marginal of this distribution on an arbitrary finite set of sites, one then has to repeat (after suitable translation) the procedure leading to $[\Phi_T^{0-}(\xi)](0)$ to each site in the set of interest.
The notion of CFTP time with ambiguities is a weakening of the notion of CFTP time, in which property \eqref{e:coupling} holds only when the "ambiguities" associated with the rules attached to a specific random subset $H$ of $\Nsp_T$, are resolved. To give a precise definition, let us consider, for each $\alpha=(x,i,t) \in \Nsp$, and $s<t$, the random variable $e(\alpha,\xi,s)$ denoting the value at site $x$ produced by the application of the rule attached to $\alpha$ when starting in state $\xi$ at time $s-$, more formally: \begin{equation} e(\alpha,s,\xi) = [\Phi_s^t(\xi)](x).\end{equation} When there exist two distinct $\xi_1, \xi_2$ such that $e(\alpha, s, \xi_1) \neq e(\alpha, s, \xi_2)$, we say that there is an ambiguity as to the result of the application of the rule attached to $\alpha$, when we start at time $s-$. A coupling time with ambiguities consists of a negative a.s. finite random variable $T$, together with a random subset $H$ of $\Nsp_T$, which is assumed to be finite on the event $\{ T > -\infty \}$, and such that the following modification of \eqref{e:coupling} holds: \begin{gather}\label{e:coupl-ambig} \mbox{ for all }\xi_1, \xi_2 \in \S^{\Z^d}, \ [ \Phi_{T}^{0-}(\xi_1) ](0) = [ \Phi_{T}^{0-}(\xi_2) ](0) \mbox{ provided that } \\
e(\alpha, T, \xi_1) = e(\alpha, T, \xi_2) \mbox{ for all }\alpha \in H. \nonumber \end{gather} Note that, when $H= \emptyset$, \eqref{e:coupl-ambig} is exactly \eqref{e:coupling}. In addition, we require that $H$ has the stopping property, i.e. $H \cap \Nsp_t$ is $\F_t$-measurable for all $t$.
\subsection{Description of the main results}
Our first main result is that, starting from a CFTP time with ambiguities $(T,H)$, one can build an actual CFTP time $T^*$, provided that $H$ contains few enough points on average. To give a precise statement, introduce the quantity
\begin{equation}\label{d:growth} \mathfrak{g} := \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{(x,t,i) \in H} |A_i| \right). \end{equation} \begin{theorem}\label{t:finitude} If $\mathfrak{g}<1$, one can construct a CFTP time $T^*$ for the interacting particle system. \end{theorem} The construction of $T^*$ is explained in Section \ref{s:definitions}. Here is an informal description. Starting with $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_0 := \{ (0,0) \}$, we recursively define a sequence $(\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n)_{n \geq 0}$ of random subsets of $\Z^d \times \R$ in the following way. First, we apply the coupling time with ambiguities $(T,H)$ at each space-time point in $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n$. This generates a set of elements of $\Nsp$, with respect to which ambiguities have to be resolved. Then $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{n+1}$ is defined as the set of space-time points upon which the resolution of these ambiguities directly depends, i.e. for $\alpha=(x,i,t)$, the set $\{ (x+y, t); \ y \in A_i \}$. The overall set of points generated by this process is $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty} := \bigcup_{n \geq 0} \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n,$ and $T^*$ is defined as the lowest value of $T$ obtained when applying the coupling time with ambiguities $(T,H)$ to the space-time points in $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}$. The idea underlying this construction is that, if $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}$ is finite, one can resolve ambiguities in a step-by-step manner, starting from the points in $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}$ that are furthest in the past and thus associated with an empty set of ambiguities, down to the origin where we can determine the value of $[\Phi_{T^*}^{0-}(\xi)](0)$ (a precise formulation is given in Proposition \ref{p:base-ambig} in Section \ref{s:preuve-1}).
Without giving precise statements (see Theorems \ref{t:temps-exp} and \ref{t:espace-exp} in Section \ref{s:complete-results}), let us mention that, in addition to Theorem \ref{t:finitude}, it is possible to obtain estimates on the tail of $T^*$ and on the range of its space-dependence (in terms of bounds on exponential moments), from analogous properties for $T$ and $H$.
Our second main result deals with perturbations of interacting particle systems. To formalize this notion, consider a particle system whose dynamics is defined by a list of rules $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}_i)_{i \in \I^u}$. This corresponds to the original, unperturbed, particle system. Then consider the dynamics defined by a list of rules of the form $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}_i)_{i \in \I^u \cup \I^p}$, where $\I^p$ is disjoint from $\I^u$. This corresponds to the perturbed particle system. Our result gives general conditions under which the existence of a CFTP time $T^u$ for the unperturbed particle system leads to the existence of a CFTP time for the perturbed particle system, provided that the perturbation is small enough. Our first condition is that the unperturbed dynamics possesses the positive rates property, which means that, for every $v \in \S$, there exists a rule with index in $\I^u$ whose application inconditionally leads to the value $v$. Our second condition requires that the definition of $T^u$ involves the exploration of an exponentially integrable\footnote{For a non-negative random variable $X$, we say that $X$ is exponentially integrable if there exists $\mu > 0$ such that $E(e^{\mu X})<+\infty$.} number of points in $\Nsp$, a notion whose precise formulation is given in Section \ref{s:definitions}, and involves what we call the exploration process associated with $T^u$. Finally, the smallness of the perturbation is measured through two parameters $\epsilon$ and $\kappa$, that admit explicit definitions in terms of $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}_i)_{i \in \I^u}$ and $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}_i)_{i \in \I^p}$ (see Section \ref{ss:pos-rates-prop}).
\begin{theorem}\label{t:theoreme-principal} Assume that the unperturbed dynamics has the positive rates property, and possesses a CFTP time associated with an exploration process whose total number of points is exponentially integrable. Then, for any perturbation with small enough $\epsilon$ and $\kappa$, we can construct a CFTP time with ambiguities $(T,H)$ for the perturbed dynamics, satisfying condition \eqref{d:growth}, so that the corresponding $T^*$ is a CFTP time for the perturbed dynamics. \end{theorem}
The construction of $(T,H)$ is explained in Section \ref{s:definitions}. Note that, in addition to satisfying \eqref{d:growth}, $(T,H)$ also satisfies the assumptions of Theorems \ref{t:temps-exp} and \ref{t:espace-exp} for sufficiently small $\epsilon$ and $\kappa$, leading to exponential moment bounds on the tail of $T^*$ and on the range of its space-dependence. This extension of Theorem \ref{t:theoreme-principal} is stated as Theorem \ref{t:theoreme-principal-extension} in Section \ref{s:complete-results}.
We illustrate Theorem \ref{t:theoreme-principal} with applications to several kinds of interacting particle systems. A first class of examples is given by systems that satisfy what we call the finite factor property (see Section \ref{s:applications}). Informally, this means that the state of a site at a certain time depends on the points in the graphical construction and on the initial condition only through a window of fixed size around $x$. The simplest example is provided by dynamics for which distinct sites do not interact, whose perturbations correspond to weakly interacting interacting particle systems. A more elaborate concrete example is a family of nucleotide substitution models called RN+YpR (see \cite{BerGouPia}), which allows for arbitrarily strong interactions between neighboring sites and yet satisfies the finite factor property. Another class of systems to which we apply Theorem \ref{t:theoreme-principal} (and for which the finite factor property is not satisfied), is given by noisy voter models on $\Z^d$. Specifically, we consider the case of a classical linear voter model with an arbitrary finite alphabet, and a variation we call the voter model with asymmetric polling, that uses the classical $\{ +, - \}$ alphabet (see Section \ref{s:applications}). Note that these examples too extend beyond the weakly interacting case.
Although we do not enter into the details here, let us note that the existence of a CFTP algorithm is not only interesting for simulation purposes, but can also provide interesting theoretical results on the particle system. Indeed, the existence of a CFTP time automatically implies that the interacting particle system is ergodic, and estimates on the tail of the CFTP time such as those provided by Theorem \ref{t:temps-exp} immediately lead to bounds on the speed of convergence to the stationary distribution. Similarly, estimates on the range of the space-dependence such as those provided by Theorem \ref{t:espace-exp} yield bounds on the decay of spatial correlations. As a consequence, our results can be readily used to derive interesting conclusions about the behavior of the perturbed particle systems to which Theorem \ref{t:theoreme-principal} applies.
Also, note that, in this paper, we do not explicitly address issues related to the practical implementation of CFTP. However, from the definition of $T^*$ in terms of $T$ and $H$ it should be clear that, if $T$ and $H$ lend themselves to an actual algorithmic implementation, so is the case for $T^*$. Similarly, Theorem \ref{t:theoreme-principal} yields an actual CFTP algorithm for the perturbed particle system provided that $T^u$ and the associated exploration process are compatible with an actual algorithmic implementation.
Finally, let us point out that a key role in the proof of our results is played by first-moment arguments, that allow us to essentially bypass the quite intricate analysis of dependencies that would otherwise be required to study the combinatorial objects we have to deal with (e.g. the sequence of sets $(\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n)_{n \geq 0}$, or the tree-indexed exploration process $\hat{\X}$ used to define a coupling time with ambiguities from $T^u$). In fact, first-moment arguments allow us to largely ignore these dependencies and obtain results in very much the same way as for classical branching processes.
\subsection{Discussion}
For ergodic particle systems satisfying a monotonicity condition similar to that of \cite{ProWil}, CFTP is always possible, as shown by van den Berg and Steif in \cite{vdBSte}. For systems lacking monotonicity, CFTP algorithms have been developed under "high-noise" or "weak interaction" type assumptions, meaning that the strength of the interaction between neighboring sites has to be sufficiently small. In other words, the particle system under consideration has to be a sufficiently small perturbation of a system in which distinct sites do not interact. One example is given by Haggstr\"om and Steif \cite{HagSte}, who use a bounding set approach to control the coalescence of trajectories (see also \cite{dSaPic} for some refinements). Another example is given by Galves, Garcia, L\"ocherbach \cite{GalGarLoc} (see also \cite{GalGarLocOrl, GalLocOrl}), whose approach is based on a branching construction\footnote{A very similar construction was already used in \cite{FerFerGar} to devise CFTP algorithms in a different framework. In fact, various constructions of this kind appear in the literature, though not explicitly in the context of CFTP, see e.g. \cite{Fer, DobKryToo}.} of which the one we use in the present paper can be seen as a generalization. One interest of the present paper is that it provides a general criterion under which small perturbations of an interacting particle system retain some of the CFTP properties of the original unperturbed system, allowing one to go beyond the weakly interacting case. Let us mention that some of our applications overlap with the recent paper \cite{MohNewRav}, where a specific kind of perturbation of noisy voter models is considered, and ergodicity is proved for sufficiently small perturbations. Finally, note that the present paper is a revised and extended version of an earlier manuscript \cite{BerPia}, where the results were limited to perturbations of RN+YpR nucleotide substitution models.
\subsection{Organization of the paper}
The rest of the paper is organized as follows. Section \ref{s:definitions} contains the definitions of the various notions and objects that were encountered in the introduction but not formally defined. Section \ref{s:complete-results} contains additional results that were not stated in the introduction. Section \ref{s:applications} contains the examples of application of Theorem \ref{t:theoreme-principal}. Section \ref{s:preuve-1} describes the proofs of Theorems \ref{t:finitude}, \ref{t:temps-exp} and \ref{t:espace-exp}. Section \ref{s:preuve-1} describes the proof of Theorem \ref{t:theoreme-principal} (and its extension Theorem \ref{t:theoreme-principal-extension}).
\section{Some formal definitions}\label{s:definitions}
\subsection{Canonical probability space}
We assume throughout the paper that $\Nsp$ is defined on a canonical probability space $(\Omega, \F, \P)$ that we now describe. First, $\Omega$ is the set of locally finite subsets $\omega$ of $\Z^d \times \I \times \R $ satisfying the additional requirements that (i) no two points in $\omega$ share the same $\R-$coordinate, (ii) for every $(x,i) \in \Z^d \times \I$, both sets $ \omega \cap (\{ (x,i) \} \times \R_+)$
and $ \omega \cap (\{ (x,i) \} \times \R_-)$ are infinite, (iii) for every $(x,i,t) \in \omega$, and any sequence $(y_n,j_n,s_n)_{n \geq 0}$ starting at $( y_0,j_0,s_0 ) = (x,i,t)$ and satisfying $y_{n+1} \in y_n+A_{j_n}$ and $s_{n+1} < s_n$ for all $n \geq 0$, one has $\lim_{n \to +\infty} s_n = -\infty$. Then $\F$ is defined as the $\sigma-$algebra generated by all the maps of the form $\omega \mapsto | \omega \cap B |$, where $B$ is a Borel subset of $\Z^d \times \I \times \R$. Finally, we set $\Nsp(\omega):=\omega$, and $\P$ is uniquely defined on $(\Omega, \F)$ by the requirement that $\Nsp$ is a Poisson process with intensity $J$ (this definition makes sense since (i), (ii), (iii) are almost sure properties of a Poisson process with intensity $J$). Note that thanks to (i), (ii), (iii), $\Phi_{t_1}^{t_2}(\xi)$ is well-defined for any $t_1<t_2$, $\xi \in \S^{\Z^d}$ and $\omega \in \Omega$.
Given $(x,t) \in \Z^d \times \R$, we define the space-time shift $\tau_{x,t}$ on $\Omega$ by $\tau_{x,t}(\omega) := \bigcup_{ (z,i,s) \in \omega } \{ (z-x,i,s-t) \}.$ (This definition is possible since (i), (ii), (iii) all are shift-invariant properties.) Given $a \in \N \cup \{ \pm \infty \}$, we define $\Nsp^{-a,a} := \Nsp \cap ([-a, a]^d \times \I \times \R) $, and let $\F^{-a,a} = \sigma(\Nsp^{-a,a})$. On occasions, we use the notation $\Nsp^{-a,a}_t := \Nsp \cap ([-a,a]^d \times [t,0[ \times \I )$.
Finally, to properly define the notion of a random subset of $\Nsp$, we introduce the space $\tilde{\Omega}$ formed by the subsets of elements of $\Omega$, equipped with the $\sigma-$algebra $\tilde{\F}$ defined just as $\F$. Note that, as a rule, in the rest of the paper, we mention measurability issues only when they involve a non-trivial argument.
In the sequel, we have to consider two distinct probability spaces, one associated with the unperturbed dynamics, and one with the perturbed dynamics. We denote by $(\Omega^u, \F^u, \P^u)$ the probability space associated with the unperturbed dynamics specified by $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}_i)_{i \in \I^u}$, while $(\Omega, \F, \P)$ refers to the perturbed dynamics specified by the full list $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}_i)_{i \in \I}$. The corresponding Poisson processes are denoted respectively $\Nsp^u$ and $\Nsp$.
\subsection{Construction of $T^*$}
Given a coupling time with ambiguities $(T,H)$, define by induction the following random sequence of subsets of $\Z^d \times ]-\infty, 0]$: \begin{equation}\label{e:rec-amb} \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_0 := \{ (0,0) \}, \ \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{n+1} := \bigcup_{(x,t) \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n} \bigcup_{(z,i,s) \in \tau_{x,t}^{-1} \circ H \circ \tau_{x,t}} \bigcup_{y \in A_i} \{ (z+y,s) \}.\end{equation} Then let $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty} := \bigcup_{n \geq 0} \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n,$
and $T^* := \inf_{(x,t) \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}} t+T \circ \tau_{x,t}$ in the case where $|\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}| < +\infty$, while $T^* := -\infty$ otherwise.
\subsection{Exploration process}
Here, we define the notion of an exploration process attached to an interacting particle system.
Given a non-empty finite subset $B \subset \Z^d$ and $t \leq 0$, define $\mathfrak{n}(B,t)$ to be the element of $\Nsp \cap (B \times \I \times ] - \infty , t [)$ with the highest time coordinate\footnote{Note that we are dealing with negative numbers here, so that the highest time coordinate here corresponds to the time coordinate with the least absolute value.} (this is always well-defined with our choice of $\Omega$). Let $\Omega_f$ denote the set of all finite subsets of sets in $\Omega$, and let $\theta$ denote a measurable map from $\Omega_f$ to the set of finite subsets of $\Z^d$ (where $\Omega_f$ is equipped with a $\sigma-$algebra defined as $\F$).
The exploration process associated to $\theta$ is defined as follows. Start with $\mathfrak{X}_0:=\emptyset$, $\gamma_0:=0$. Then, for all $n \geq 0$, let $B_{n} := \theta(\mathfrak{X}_n)$. If $B_n \neq \emptyset$, denote $\mathfrak{n}(B_n,\gamma_n)=:(x_n, i_n, t_n)$, and let $\mathfrak{X}_{n+1}:= \mathfrak{X}_n \cup \{ (x_n, i_n, t_n) \}$ and $\gamma_{n+1}:=t_n$. If $B_n = \emptyset$, then $\mathfrak{X}_{n+1} := \mathfrak{X}_n, \gamma_{n+1} := \gamma_n$.
Let $\mathfrak{X}_{\infty} := \bigcup_{n \geq 0} \mathfrak{X}_n$ and $\gamma_{\infty} := \lim_{n \to +\infty} \gamma_n$. The total number of points in the exploration process is then defined as $|\mathfrak{X}_{\infty}|$.
We say that a CFTP time $T$ defined on $(\Omega, \F, \P)$ is associated with such an exploration process if, on $\{ |\mathfrak{X}_{\infty}|<+\infty \}$, one has $T = \gamma_{\infty}$, or equivalently, $T = \inf \{ t ; \ (x,i,t) \in \mathfrak{X}_{\infty} \}$, while $T=-\infty$ when $|\mathfrak{X}_{\infty}|=+\infty $, and if, on $\{ T > -\infty \}$, the value of $[\Phi^{0-}_{T}(\xi)](0)$, which is the same for every $\xi \in \S^{\Z^d}$, is measurable with respect to $\mathfrak{X}_{\infty}$.
We shall always assume that there exists a deterministic function $\beta \ : \ \N \to \N$ such that for every $\ell \geq 0$, \begin{equation}\label{e:borne-taille} \theta(\mathfrak{X}_{\ell}) \subset \{ -\beta (\ell), \ldots, \beta (\ell) \}^d ,\end{equation} and such that $\beta(\ell) = O(\ell)$ as $\ell$ goes to infinity.
In the sequel, we assume that a CFTP time $T^u$ for the unperturbed dynamics is defined on $(\Omega^u, \F^u, \P^u)$, and that $T^u$ is associated with an exploration process of the type we have just described. We denote the corresponding process by $\mathfrak{X}^u$ to emphasize the fact that this process is defined for the unperturbed dynamics, but, for the sake of readability, we use $\theta$, $x_n$, $i_n$, $t_n$, $\gamma_n,$ etc. instead of the awkward $\theta^u$, $x_n^u$, $i_n^u$, $t_n^u$, $\gamma_n^u$, etc.
\begin{remark} Given a CFTP time $T$ and an exploration process $(\mathfrak{X}_n)_{n}$, the fact that $T = \gamma_{\infty}$ does not in general imply that $[\Phi^{0-}_{T}(\xi)](0)$ is measurable with respect to $\mathfrak{X}_{\infty}$, so this last condition has to be added to the definition of an exploration process associated to a CFTP time. \end{remark}
\subsection{Positive rates property, $\epsilon$ and $\kappa$}\label{ss:pos-rates-prop}
The positive rates property for the set of non-perturbative rules $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}_i)_{i \in \I^u}$ means that, for every $v \in \S$, there exists a rule with index in $\I^u$ which is of the form $(A,f,r)$ with $r>0$, $A=\emptyset$ and $f \equiv v$. We denote by $\iota_v$ the\footnote{One may assume without loss of generality that, for any $v$, there is a single such rule, since identical rules with distinct indices may always be merged into a single rule.} index of such a rule. We then control the smallness of the perturbation of $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}_i)_{i \in \I^u}$ by $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}_i)_{i \in \I^p}$ through the following two parameters \begin{equation}\label{e:def-epsilon}\epsilon := \sup_{v \in \S} \left( \sum_{j \in \I^p ; \ v \in f_j(A_j)} r_j \right) (r_{\iota_v})^{-1},\end{equation}
\begin{equation}\label{e:def-kappa} \kappa:= \left( \sum_{i \in \I^p} | A_i | r_i \right) \left( \sum_{i \in \I} r_i \right)^{-1}. \end{equation}
\subsection{Construction of a coupling time with ambiguities $(T,H)$ from $T^u$}
We now define what we call the exploration process with locking of perturbative ambiguities attached to the perturbed dynamics, and associated to the map $\theta$ used to define the exploration process $\mathfrak{X}^u$ of the unperturbed dynamics. This is the process we use to define a coupling time with ambiguities for the perturbed dynamics.
Informally, the construction can be described as follows: run the exploration process associated with the unperturbed dynamics on $\Nsp$ (instead of $\Nsp^u$). When an $\alpha=(x,i,t)$ corresponding to a perturbative rule, i.e. $i \in \I^p$ is encountered, split the exploration process into $|f(A_i)|$ exploration processes evolving in parallel, one for each $v \in f(A_i)$, in which $(x,i,t)$ is replaced by $(x, \iota_v, t)$.
The formal construction uses a recursively constructed tree $\mathbb{T}$ to label the process. Let us start with the root of $\mathbb{T}$, denoted $r$, for which we set $\hat{\X}_r := \emptyset $ and $\gamma_r:=0$. Then, for any vertex $a \in \mathbb{T}$, we set $B_a := \theta(\hat{\X}_a)$. Then, if $B_a \neq \emptyset$, denote $\mathfrak{n}(B_a,\gamma_a)=(x_a, i_a, t_a)$. If $i_a \in \I^u$, we attach a single son $b$ to $a$, and let $\hat{\X}_{b}:= \hat{\X}_a \cup \{ (x_a, i_a, t_a) \}$ and $\gamma_{b}:=t_a$. On the other hand, if $i_a \in \I^p$, we attach to $a$ a list of sons $\left(b_v, \ v \in f_{i_a}(\S^{A_{i_a}}) \right)$, and let
$\hat{\X}_{b_v}:= \hat{\X}_a \cup \{ (x_a, \iota_v,t_a) \}$ and $\gamma_{b_v}:=t_a$. If $B_a = \emptyset$, then $a$ has no son. Finally, we let $\hat{\X}_{\infty} := \bigcup_{a \in \mathbb{T}} \hat{\X}_a$.
We now define $T$ by \begin{equation}\label{d:def-T}T := \inf \{ t ; \ (x,i,t) \in \hat{\X}_{\infty} \} \mbox{ if } | \hat{\X}_{\infty} | < +\infty,\end{equation}
while $T:=-\infty$ if $| \hat{\X}_{\infty} | = +\infty$, and $H$ is defined by \begin{equation}\label{d:def-H}H := \{ (x_a, i_a, t_a); \ a \in \mathbb{T}', \ i_a \in \I^p \},\end{equation} where $\mathbb{T}'$ denotes the subtree of $\mathbb{T}$ obtained by removing the leaves of $\mathbb{T}$. Note that one may view $\hat{\X}_{\infty}$ as the result of the exploration process associated to a certain map $\hat{\theta}$. However, the representation with a labelled tree turns out to be more convenient for our purposes.
\begin{remark}
The definition of $\hat{\X}_{\infty}$ makes sense whether or not $| \mathfrak{X}^u_{\infty} |$ has an exponentially decaying tail, as assumed in the statement of Theorem \ref{t:theoreme-principal}: provided that $|\hat{\X}_{\infty}|$ is a.s. finite, $(T,H)$ is indeed a CFTP time with ambiguities (see Proposition \ref{p:base-perturb}), and the role of the tail condition on $| \mathfrak{X}^u_{\infty} |$ in Theorem \ref{t:theoreme-principal} is indeed to ensure that $|\hat{\X}_{\infty}|$ is a.s. finite. \end{remark}
\section{Additional results}\label{s:complete-results}
The first result shows that the exponential moments of $T^*$ can be controlled in terms of similar moments for $T$ and $H$.
For $\lambda \in \R$, define
\begin{equation}\label{e:def-Laplace-temps}\Lambda_T(\lambda) := \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(\exp(\lambda T)), \ \Lambda_{H,time}(\lambda) := \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{(x,i,t) \in H} |A_i| \exp(\lambda t) \right).\end{equation} \begin{theorem}\label{t:temps-exp} Assume that $\lambda \le 0$ is such that $ \Lambda_T(\lambda) < +\infty$ and $\Lambda_{H,time}(\lambda) < 1$. Then $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(\exp(\lambda T^*)) \leq \Lambda_T(\lambda) (1 - \Lambda_{H,time}(\lambda))^{-1}.$$ \end{theorem}
Our next result deals with the range of space-dependence of $T^*$. To formalize this notion, say that a $\N \cup \{ + \infty \}$-valued random variable $L$ defines a stopping box in $\Z^d$ if, for any $a \in \N$, one has $\{L = a \} \in \F^{-a,a}$. We say that an a.s. finite such random variable bounds the width of a CFTP time $T$ if, on $\{ T > -\infty \}$, the value of $[\Phi_{T}^{0-}(\xi)](0)$ (which by definition does not depend on the choice of $\xi \in \S^{\Z^d}$) is measurable with respect to $\F^{-L,L}$. We say that $L$ bounds the width of a CFTP time with ambiguities if $H$ is measurable with respect to $\F^{-L, L}$
and if there exists a measurable map\footnote{To be more specific about measurability assumptions concerning $(e(\alpha, T, \xi))_{\alpha \in H}$, we assume that it is encoded as the random subset of $\Z^d \times \I \times \R \times \S$ defined by $\bigcup _{\alpha \in H} (\alpha, e(\alpha, T, \xi))$, where the $\sigma-$algebra on the set of locally finite subsets of $\Z^d \times \I \times \R \times \S$ is generated by maps of the form $\varpi \to | \varpi \cap B|$, where $B$ is a Borel subset of $\Z^d \times \I \times \R \times \S$.} $\Theta$ such that, on $\{ T > -\infty \}$, for all $\xi \in \S^{\Z^d}$, $[\Phi_{T}^{0-}(\xi)](0) = \Theta\left(L, \Nsp^{-L, L} , (e(\alpha, T, \xi))_{\alpha \in H} \right) $.
For $\lambda \in \R$, and $1 \leq q \leq d$, define \begin{equation}\label{e:def-Laplace-esp}\Lambda_{L}(\lambda) := \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(e^{\lambda L}), \ \Lambda_{H,space}(\lambda,q) := \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{(x,t,i) \in H} \sum_{z \in A_i} e^{\lambda (x_q+z_q)} \right).\end{equation} Then define $L^*_+$ by $L^*_+ := \sup_{1 \leq q \leq d} \sup_{ (x,t) \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty} } x_q + L \circ \tau_{x,t}$, and $L^*_-$ by $L^*_- := \inf_{1 \leq q \leq d} \inf_{ (x,t) \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty} } x_q - L \circ \tau_{x,t}$. Finally, let $L^* := \max(L^+, -L^-)$. \begin{theorem}\label{t:espace-exp} If $\mathfrak{g}<1$, and $L$ bounds the width of $(T,H)$ then $L^*$ bounds the width of $T^*$. Moreover, if $\lambda > 0$ is such that $ \Lambda_{L}(\lambda) < +\infty$ and $\Lambda_{H,space}( \pm \lambda,q) < 1$ for all $q$, then $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(\exp(\lambda L_+^*)) \leq \Lambda_{L}(\lambda) \sup_{1 \leq q \leq d} (1 - \Lambda_{H,space}(\lambda,q))^{-1},$$ $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(\exp(-\lambda L_-^*)) \leq \Lambda_{L}(\lambda) \sup_{1 \leq q \leq d} (1 - \Lambda_{H,space}(-\lambda,q))^{-1}.$$ \end{theorem}
Finally, we have the following extension of Theorem \ref{t:theoreme-principal}.
\begin{theorem}[Extension of Theorem \ref{t:theoreme-principal}]\label{t:theoreme-principal-extension}
Under the assumptions of Theorem \ref{t:theoreme-principal}, for any list of perturbative rules with small enough $\epsilon$ and $\kappa$, the pair $(T,H)$ defined by \eqref{d:def-T} and \eqref{d:def-H} defines a CFTP time with ambiguities that satisfies the assumptions of Theorem \ref{t:finitude}, together with the assumptions of Theorems \ref{t:temps-exp} and \ref{t:espace-exp} for small enough $|\lambda|$. \end{theorem}
For the sake of readability, we did not include explicit estimates in the statement of Theorem \ref{t:theoreme-principal-extension}. However, looking at the proofs given in Section \ref{s:preuve-2}, it is easy to obtain explicit control upon the characteristics of $(T,H)$ (namely, $\mathfrak{g}$, $\Lambda_T$, $\Lambda_{H, time}$, $\Lambda_L$, $\Lambda_{H, space}$) in terms of $\epsilon, \kappa$, and the parameters $(\mathfrak{R}} \def\ZZ{\mathfrak{Z}_i)_{i \in \I^u}$ of the unperturbed model. Combined with Theorems \ref{t:finitude}, \ref{t:temps-exp} and \ref{t:espace-exp}, this leads to an explicit control upon the characteristics of the resulting CFTP time $T^*$.
\section{Applications}\label{s:applications}
In this section, we give some examples of dynamics which satisfy the properties required for the unperturbed dynamics in Theorem \ref{t:theoreme-principal}. Since we discuss unperturbed dynamics only, it is unnecessary to use $^u$ superscripts to distinguish between perturbed and unperturbed dynamics, and consequently such superscripts are not used in this section.
\subsection{Perturbations of finite factor models}
We say that the dynamics possess the finite factor property if there exists $b \in \N$ such that, for all $t < 0$, $\left[\Phi^{0-}_{t}(\xi)\right] $ is measurable with respect to $\Nsp^{-b,b}_t$ and $\left(\xi(x); \ x \in \{-b, \ldots, b\}^d \right)$.
\begin{prop}\label{p:finite-factor} Any dynamics with the finite factor property and the positive rates property satisfies the assumptions of Theorem \ref{t:theoreme-principal}, i.e. there exists a CFTP time associated with an exploration process whose total size has some finite exponential moment. \end{prop}
\begin{proof}
Given $X \in \Omega_f$, let $q:=|X|$, and denote by $(y_k, j_k, s_k)_{0 \leq k \leq q-1}$ the list of elements of $X$, indexed by decreasing order of time, so that $s_0>\cdots > s_{q-1}$. Let also $h:= (2b+1)^d$. Now consider the exploration process associated with the map $\theta$ defined as follows. Set $\theta(X) := \emptyset$ when the following three conditions are met \begin{itemize}
\item[a)] $|X| \geq h$, \item[b)] $\{ y_{q-1},\ldots, y_{q-h} \} = \{ -b, \ldots, b \}^d $, \item[c)] for all $q-h \leq k \leq q-1$, $A_{j_k} = \emptyset$, \end{itemize} Otherwise, let $\theta(X) := \{-b, \ldots, b\}^d$. Denote by $(\mathfrak{X}_n)_n$ the corresponding exploration process, and observe that condition \eqref{e:borne-taille} is satisfied with $\beta(\ell) := b$ for all $\ell$. One checks that given $\mathfrak{X}_n$, the probability that $\mathfrak{X}_{n+h}$ satisfies $\theta(\mathfrak{X}_{n+h}) = \emptyset$ is bounded below by $h ! h^{-h} \rho^h$, where $\rho := (\sum_{i \in \I} r_i \mathbf{1}} \def\dd{\mathrm d(A_i = \emptyset) )(\sum_{i \in \I} r_i)^{-1}$. This proves the fact that there exists $\mu>0$ such that
$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}\left( e^{\mu | \mathfrak{X}_{\infty}} | \right) < +\infty$. Let us now check that $T:=\gamma_{\infty}$ is a CFTP time for the dynamics, associated with the exploration process defined by $\theta$.
Define $U$ to be the a.s. finite smallest index $k$ such that $\theta(\mathfrak{X}_k) = \emptyset$. From conditions a) b) c), one has that, on $\{ U < +\infty \}$, for all $x \in \{ -b, \ldots, b \}^d$,
$[\Phi_{T}^{\gamma_{U-h}-}(\xi)](x)$ takes the same value for every $\xi$, and this value is measurable with respect to $(y_{U}, j_{U}), \ldots, (y_{U-h+1}, j_{U-h+1})$. On the other hand, the fact that $\theta(\mathfrak{X}_k) = \{ -b, \ldots, b \}^d$ for all $k \leq U-h$ shows that $\mathfrak{X}_{U-h} = \Nsp^{-b,b}_{\gamma_{U-h}} $. Now by the definition of the flow, one has that
$$[\Phi_T^{0-}(\xi)](0) = \left[\Phi^{0-}_{\gamma_{U-h}} \left(\Phi_T^{\gamma_{U-h}-}(\xi)\right) \right](0).$$ The finite factor property then yields that $[\Phi_T^{0-}(\xi)](0)$ is the same whatever the value of $\xi$, and that this value is measurable with respect to $\mathfrak{X}_{U}$. \end{proof}
The simplest example of dynamics with the finite factor property is the case where distinct sites do not interact, i.e. $A_i \subset \{ 0 \}$ for every $i \in \I$. In this case, the factor property holds with $b=0$, and each site evolves independently according to a continuous-time Markov chain on $\S$.
A more sophisticated example, whose study was our original motivation for this work, is the so-called class of RN+YpR nucleotide substitution models, see \cite{BerGouPia}, whose goal is to provide tractable models that include neighbor-dependent effects such as the well-known hypermutability of CpG dinucleotides. These models use the nucleotidic alphabet $\S := \{ A, C, G, T \}$ as their state space, and $\Z$ as their set of sites, with $\S^{\Z}$ being an idealized representation of a DNA sequence. Additionally, $\S$ is divided into the set of pyrimidines $Y:= \{ C, T \}$, and purines $R := \{ A, G \}$, and we say that $Y$ is the type of $C$ and $T$, while $R$ is the type of $A$ and $G$.
The RN+YpR dynamics is specified through the following list of rules (each rule is of the form $(f,A,r)$): \begin{itemize} \item unconditional rules: for each $v \in \S$, a rule with $A := \emptyset$, $r>0$ and $f \equiv v$; \item transversion rules: for each $v \in \S$, a rule with $A: = \{ 0 \}$ and $f(w) := v$ if $v$ and $w$ are not of the same type, $f(w):=w$ otherwise; \item transition rules: for each $v \in \S$, a rule with $A := \{ 0 \}$ and $f(w) := v$ if $v$ and $w$ are of the same type, $f(w):=w$ otherwise; \item left-dependent rules: for each $u \in Y$, $v \in R$, $v' \in R$, a rule with $A := \{ -1, 0 \}$, $f(w_{-1},w_0):=v'$ if $(w_{-1},w_0)=(u,v)$, $f(w_{-1},w_0):=w_0$ otherwise; \item right-dependent rules: for each $u \in Y$, $v \in R$, $u' \in Y$, a rule with $A := \{ 0, 1 \}$, $f(w_0,w_1):=u'$ if $(w_0,w_1)=(u,v)$, $f(w_0,w_1):=w_0$ otherwise. \end{itemize}
It turns out (see \cite{BerGouPia}) that RN+YpR models have the finite factor property with $b:=1$. Let us insist that the rates of left- and right-dependent rules, whence the strength of the interaction between sites, may be arbitrarily large, so that the RN+YpR class contains models that are not weakly dependent. Note that one can generalize this class of models to produce interacting particle systems with an arbitrarily long range of dependence, where the minimal $b$ for which the finite factor property holds can be made arbitrarily large, although these seem less biologically motivated. Note also that, in the case of the RN+YpR model, one can define alternative coupling times which, as opposed to the one defined in the proof of Proposition \ref{p:finite-factor}, do not get larger and larger when the interaction strength (given by the rates of the rules involving interactions between neighboring sites) gets large, see \cite{BerPia}.
\subsection{Perturbations of voter-like models}
We now describe how Theorem \ref{t:theoreme-principal} can be applied to variants of classical interacting particle systems such as the voter model on $\Z^d$ (see \cite{Lig2}).
\subsubsection{Classical linear voter model.}
Let $p(\cdot)$ denote a probability measure on $\Z^d$ with finite support. The dynamics of the classical voter model can be defined thanks to the following set of rules: \begin{itemize} \item state-copying rules: for each $x$ in the support of $p(\cdot)$, a rule with $A := \{ 0,x \}$ and $r:=p(x)$, with $f(w_0, w_x):=w_x$. \end{itemize} One might interpret this model as describing the evolution of opinions of individuals attached to the sites of $\Z^d$, with $\S$ representing the set of possible opinions. The individual at $x$ waits for a unit exponential time, then chooses a random location $y \in \Z^d$ with probability $p(y-x)$, and adopts the opinion of the individual attached to site $y$. As such, the voter model does not satisfy the assumptions of Theorem \ref{t:theoreme-principal}, since it does not enjoy the positive rates property. As a consequence, we add to this model a list of unconditional rules so as to enforce this property: \begin{itemize} \item unconditional rules: for each $v \in \S$, a rule with $A := \emptyset$, $r>0$ and $f \equiv v$. \end{itemize} We call the resulting model "noisy voter model". Note that this addition dramatically changes the dynamics of the voter model, since it automatically turns it into an ergodic interacting particle system. Note that we may consider this addition as part of the perturbation of the original voter model we want to study, but this part of the perturbation has to be included in the dynamics prior to the application of Theorem \ref{t:theoreme-principal}.
\begin{prop} The noisy voter model satisfies the assumptions of Theorem \ref{t:theoreme-principal}. \end{prop}
\begin{proof} The corresponding exploration process is defined as follows. First $\theta(\emptyset) := \{ 0 \}$.
Then, given a non-empty $X \in \Omega_f$, denote by $(y,j,s)$ the element of $X$ with the lowest time-coordinate. Then let $\theta(X):=\emptyset$ if $A_j = \emptyset$. Otherwise, $A_j$ is of the form $\{ 0, x \}$, and we let $\theta(X):= \{ y+x \}$. We denote by $(\mathfrak{X}_n)_{n}$ the corresponding exploration process. Note that condition \eqref{e:borne-taille} is satisfied with $\beta(\ell) := \sup \{ |z|; \ p(z) \neq 0 \} \times \ell$. To prove that the $|\mathfrak{X}_{\infty}|$ has some finite exponential moment, note that, conditional upon $\mathfrak{X}_{n}$, if $\theta(\mathfrak{X}_n) \neq \emptyset$, the probability that the next point to be included in $\mathfrak{X}_{n+1}$ corresponds to a rule of the form $f \equiv v$ for some $v$, is bounded below by the ratio $(\sum_{v \in \S} r_{\iota_v}) (\sum_{i \in \I} r_i)^{-1}$. Since in this case $\theta(\mathfrak{X}_{n+1})=\emptyset$, a geometric upper bound holds for the tail of $|\mathfrak{X}_{\infty}|$. Let us now check that $T$ defined as the least time-coordinate of an element in $\mathfrak{X}_{\infty}$ is a CFTP time for the dynamics, associated with the exploration process defined by $\theta$. Indeed, it is clear from the definition of the dynamics that if the element of $X_{\infty}$ with the least time-coordinate is associated with the rule $\iota_v$, then $\left[\Phi_{T}^{0-}(\xi)\right](0)=v$ for all $\xi$. \end{proof}
\subsubsection{Voter model with asymmetric polling}
We now consider a variation upon the classical voter model. Let $A^{(1)},\ldots, A^{(m)}$ denote a list of finite non-empty subsets of $\Z^d$, $r^{(1)},\ldots, r^{(m)}$ denote a list of non-negative real numbers, and take as a state space $S := \{ +, - \}$. The set of rules characterizing our model is the following: \begin{itemize} \item polling rules: for each $1 \leq i \leq m$, a rule with $A:=A^{(i)}$ and $f(w):=+$ if $w_x = +$ for at least one $x \in A^{(i)}$, $f(w):=-$ otherwise. \end{itemize} We call this model the voter model with asymmetric polling. Here, an individual performs a poll within a randomly chosen finite subset of individuals, and adopts an opinion that depends on the results of the poll in an asymmetric way: indeed, the individual will adopt the opinion denoted $+$ if any of the individuals in the poll expresses the opinion $+$, while, to adopt the opinion denoted $-$, consensus within the poll is required.
As in the case of the classical linear voter model, we add to the above set of rules a list of unconditional rules ensuring the positive rates property: \begin{itemize} \item unconditional rules: for $v=+,-$, a rule with $A := \emptyset$, $r>0$ and $f \equiv v$. \end{itemize} The resulting model is called noisy voter model with asymmetric polling.
\begin{prop} The noisy voter model with asymmetric polling satisfies the assumptions of Theorem \ref{t:theoreme-principal}. \end{prop}
\begin{proof} The corresponding exploration process is defined as follows. First $\theta(\emptyset) := \{ 0 \}$. Then, given a non-empty set $X \in \Omega_f$, denote by $(y,j,s)$ the element of $X$ with the least time-coordinate. Then let $\theta(X):=\emptyset$ if $A_j = \emptyset$ and $f_j \equiv +$. If $A_j = \emptyset$ and $f_j \equiv -$, then let $\theta(X) := \theta(X \setminus \{ (y,j,s) \}) \setminus \{ j \}$. Otherwise, $A_j$ is of the form $A^{(k)}$ for some $1 \leq k \leq m$, and we let
$\theta(X) := \theta(X \setminus \{ (y,j,s) \}) \cup (y+A^{(k)})$.
We denote by $(\mathfrak{X}_n)_{n}$ the corresponding exploration process. Note that condition \eqref{e:borne-taille} is satisfied with $\beta(\ell) := \sup \{ |z|; z \in \cup_{1 \leq k \leq m} A^{(k)} \} \times \ell$.
To prove that the number of points in $|\mathfrak{X}_{\infty}|$ has some finite exponential moment, note that, conditional upon $\mathfrak{X}_{n}$, if $\theta(\mathfrak{X}_n) \neq \emptyset$, the probability that the next point to be included in $\mathfrak{X}_{n+1}$ corresponds to the rule with $f \equiv +$, is bounded below by the ratio $r_{\iota_+} (\sum_{i \in \I} r_i)^{-1}$. Since in this case
$\theta(X_{n+1})=\emptyset$, a geometric upper bound holds for the tail of $|\mathfrak{X}_{\infty}|$. Let us now check that $T$ defined as the least time-coordinate of an element in $\mathfrak{X}_{\infty}$ is a CFTP time for the dynamics, associated with the exploration process defined by $\theta$. Indeed, it is clear from the definition of the dynamics that if the element of $\mathfrak{X}_{\infty}$ with the least time coordinate is associated with the rule $\iota_+$, $\left[\Phi_{T}^{0-}(\xi)\right](0)=+$ for all $\xi$, while, if this element is associated with
the rule $\iota_-$, $\left[\Phi_{T}^{0-}(\xi)\right](0)=-$ for all $\xi$. \end{proof}
\section{Proofs of Theorems \ref{t:finitude}, \ref{t:temps-exp} and \ref{t:espace-exp}}\label{s:preuve-1}
We start with a proposition showing that, if $T^*$ is finite with probability one, then $T^*$ is indeed a CFTP time.
\begin{prop}\label{p:base-ambig} If $\P(T^* > - \infty)=1$, then $T^*$ is a CFTP time. \end{prop}
\begin{proof}
Note that, with our definitions, $T^*>-\infty$ implies that $|\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}|<+\infty$. The proof is by induction on $|\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}|$. Specifically, we shall show for all $n \geq 0$ that the following property $(P_n)$ is true: on $\{ |\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}|=n , \ T^* > - \infty\}$, for all $\xi_1, \xi_2 \in \S^{\Z^d}$, $[\Phi_{T^*}^{0-}(\xi_1)](0) = [\Phi_{T^*}^{0-}(\xi_2)](0)$.
Assume throughout that $T^* > -\infty$, and let us start with $n=1$. If $|\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}|=1$, a first possibility is that
$H = \emptyset$. In this case, the definition of a CFTP time with ambiguities shows that $[\Phi_{T}^{0-}(\xi_1)](0) = [\Phi_{T}^{0-}(\xi_2)](0)$ for all $\xi_1, \xi_2 \in \S^{\Z^d}$, whence, since by definition $T^* \leq T$, the fact that $[\Phi_{T^*}^{0-}(\xi_1)](0) = [\Phi_{T^*}^{0-}(\xi_2)](0)$ for all $\xi_1, \xi_2 \in \S^{\Z^d}$. If $H \neq \emptyset$, the fact that $|\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}|=1$ shows that every $\alpha=(x,i,t) \in H$ is such that
$A_i = \emptyset$. In this case, for any $s \leq t$, $e(\alpha,\xi,s)$ depends neither on $\xi$ nor $s$, so that again $[\Phi_{T^*}^{0-}(\xi_1)](0) = [\Phi_{T^*}^{0-}(\xi_2)](0)$ for all $\xi_1, \xi_2 \in \S^{\Z^d}$. We now show that $(P_{n+1})$ is valid provided that $(P_k)$ is valid for all $1 \leq k \leq n$. Assume that $|\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}|=n+1$. It is enough to prove that, for any $\alpha = (z,i,t) \in H$ such that $A_i \neq \emptyset$, $e(\alpha, T^*, \xi)$ admits the same value for every $\xi \in \S^{\Z^d}$. Consider such an $\alpha=(z,i,t)$, let $y \in A_i$, and $x:=z+y$. Then observe that, by definition, \begin{equation}\label{e:inclusion-translation}\tau_{x,t}^{-1} \circ \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty} \circ \tau_{x,t} \subset \bigcup_{k \geq 1} \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_k.\end{equation}
Since $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{0} = \{ (0,0) \}$ while $(0,0) \notin \bigcup_{k \geq 1} \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_k$, our assumption that $|\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}|=n+1$ implies that $| \bigcup_{k \geq 1} \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_k | = n$. We thus deduce from \eqref{e:inclusion-translation} that $| \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty} \circ \tau_{x,t} | \leq n$. Moreover, \eqref{e:inclusion-translation} shows that $t+T^* \circ \tau_{x,t} \geq T^*$, so that our assumption that $T^*>-\infty$ implies that $T^* \circ \tau_{x,t} > -\infty$. Our induction hypothesis then implies that for all $\xi_1, \xi_2 \in \S^{\Z^d}$, $[\Phi_{T^*}^{0-}(\xi_1)](0) \circ \tau_{x,t}= [\Phi_{T^*}^{0-}(\xi_2)](0) \circ \tau_{x,t}$, which rewrites as $[\Phi_{t+T^* \circ \tau_{x,t}}^{t-}(\xi_1)](x) = [\Phi_{t+T^* \circ \tau_{x,t}}^{t-}(\xi_2)](x)$. We have seen that $T^* \leq t+T^* \circ \tau_{x,t}$, so we can deduce that $e(\alpha, T^*, \xi)$ does not depend on $\xi \in \S^{\Z^d}$. \end{proof}
Let $M$ denote the intensity measure of the set $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_1$, i.e. the positive measure on $\Z^d \times \R$ defined for all Borel set $C$ by
$$M(C) := \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} ( |\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_1 \cap C| ) = \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{(z,i,s) \in H } \sum_{y \in A_i} \mathbf{1}} \def\dd{\mathrm d( (z+y,s) \in C ) \right).$$ We use the notation $\star$ for the convolution product of measures on $\Z^d \times \R$. For all $n \geq 0$, $M^{\star n}$ denotes the convolution product $M \star \cdots \star M$ with $n$ factors, with the convention $M^{\star 0} := \delta_{(0,0)}$.
Our key first-moment estimates are given in the next proposition and its corollary.
\begin{prop}\label{p:branch-ambig} For any measurable $f \ : \ \Z^d \times \R \to \R_+$, and any $n \geq 0$, one has $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}\left( \sum_{\zeta \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n} f(\zeta) \right) \leq \int f(\zeta) d M^{\star n}(\zeta),$$ with the convention $M^{\star 0} := \delta_{(0,0)}$. \end{prop}
\begin{coroll}\label{c:branch-ambig} For measurable $f \ : \ \Z^d \times \R \to \R_+$, any non-negative $\F_0-$measurable random variable $D$, and any $n \geq 0$, one has $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}\left( \sum_{\zeta \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n} f(\zeta) \cdot D \circ \tau_{\zeta} \right) \leq \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(D) \cdot \int f(\zeta) d M^{\star n}(\zeta),$$ with the convention $M^{\star 0} := \delta_{(0,0)}$. \end{coroll}
The proof makes use of the so-called refined Campbell theorem (see \cite{StoKenMec}), which we quote here in the special form we need: \begin{theorem}\label{t:Campbell} For any measurable map $\Psi \ : \ (\Z^d \times \I \times \R) \times \Omega \to \R_+$, one has the following identity:
$$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{\alpha \in \Nsp} \Psi(\alpha, \Nsp) \right) = \int \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( \Psi(\alpha, \Nsp \cup \{ \alpha \}) ) dJ(\alpha),$$
where $J$ denotes the intensity measure of $\Nsp$. \end{theorem}
A crucial property of the sets $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n$ is that they possess the stopping property, as stated in the following lemma. \begin{lemma}\label{l:mesurabil-Amb} For all $n \geq 0$, $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n$ has the stopping property, i.e. for all $t<0$, $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n \cap \Nsp_t$ is $\F_t-$measurable. \end{lemma}
Before we prove Lemma \ref{l:mesurabil-Amb}, we need the following lemma. \begin{lemma}\label{l:progres} There exists a measurable map $\Xi$ from $\tilde{\Omega} \times ]-\infty,0[$ to $\tilde{\Omega}$ such that, for all $t <0$, $$H \cap \Nsp_t = \Xi(t, \Nsp_t).$$ \end{lemma}
\begin{proof} We first prove that there exists a measurable map $V$ from $\Omega \times ]-\infty,0[$ to $\tilde{\Omega}$ such that, for all $t >0$, \begin{equation}\label{e:mesurabilite-infini}H \cap \Nsp_t = V(t, \Nsp).\end{equation} For each $(x,i) \in \Z^d \times \I$, let $(\psi(x,i,k))_{k \geq 1}$ denote the successive points $ \Nsp$ whose coordinate on $\Z^d \times \I$ is $(x,i)$, in decreasing order of the $\R-$coordinate. We let $\psi(x,i,k) =: (x,i, s(x,i,k))$. Given a Borel set $B$ of $ \Z^d \times \I \times \R$, one has that, for all $t<0$,
$$| H \cap \Nsp_t \cap B| = \sum_{x,i,k} \mathbf{1}} \def\dd{\mathrm d(\psi(x,i,k) \in H) \mathbf{1}} \def\dd{\mathrm d(s(x,i,k) \geq t) \mathbf{1}} \def\dd{\mathrm d( \psi(x,i,k) \in B).$$ Since $H$ is a measurable map from $(\Omega,\F)$ to $(\tilde{\Omega}, \tilde{F})$, one has that $\omega \mapsto \mathbf{1}} \def\dd{\mathrm d(\psi(x,i,k) \in H)(\omega)$ is a measurable map from $(\Omega, \F)$ to $\R$. This is also the case for $\omega \mapsto \mathbf{1}} \def\dd{\mathrm d(\psi(x,i,k) \in B)(\omega)$. Finally, $(t, \omega) \mapsto (s(x,i,k) - t)$ is measurable from $\Omega \times ]-\infty,0[$ to $\R$, so this is also the case for $ \mathbf{1}} \def\dd{\mathrm d(s(x,i,k) \geq t)$. We conclude that $(\omega, t) \mapsto H \cap \Nsp_t $ is measurable from $\Omega \times ]-\infty,0[$ to $\tilde{\Omega}$, whence the existence of $V$. Consider now an arbitrarily fixed element $\omega_0 \in \Omega$ that contains no point with $0$ $\R-$coordinate, and define the map $\mathfrak{a}$ from $\tilde{\Omega} \times ]-\infty,0[$ to $\Omega$ by $$\mathfrak{a}(\tilde{\omega},t) := \left( \tilde{\omega} \cap (\Z^d \times \I \times [t,0[) \right) \cup \tau_{0,t}(\omega_0).$$
One checks that $\mathfrak{a}$ is measurable by writing$$| \mathfrak{a}(\omega,t) \cap B | = \sum_{x,i,k} \mathbf{1}} \def\dd{\mathrm d(s(x,i,k)(\tilde{\omega}) \geq t) \mathbf{1}} \def\dd{\mathrm d( \psi(x,i,k)(\tilde{\omega}) \in B)+|\tau_{0,t}(\omega_0) \cap B|,$$ where we have extended the definition of $\psi(x,i,k)$ to $\tilde{\Omega}$ in the obvious way, with the convention that $s(x,i,k)$ takes the value $-\infty$ when the value of $k$ excesses the number of points to be indexed. Now, since, for any given $t<0$, $H \cap \Nsp_t $ is $\F_t-$measurable by assumption, one has that, in view of \eqref{e:mesurabilite-infini}, for any $t<0$, $$ V( t, \Nsp ) = V(t, \mathfrak{a}( \Nsp_t, t )).$$ As a consequence, the conclusion of the proposition is achieved by defining $$\Xi(t, \tilde{\omega}) := V(t, \mathfrak{a}( \tilde{\omega}, t )).$$
\end{proof}
\begin{proof}[Proof of Lemma \ref{l:mesurabil-Amb}] We re-use the notations introduced in the proof of Lemma \ref{l:progres}. The proof is by induction. For $n=0$, the result is obvious since $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_0 := \{ (0,0) \}$. Now assume the result to be true for a given $n \geq 0$. Define $D_{n}:= \{ (w,i,k) ; \ \pi(\psi(w,i,k)) \cap \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{n} \neq \emptyset \}$. By definition, one has \begin{equation}\label{e:redef-Amb}\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{n+1} := \bigcup_{(x,i,k) \in D_n} \pi \left( \tau_{x,s(x,i,k)}^{-1} \circ H \circ \tau_{x,s(x,i,k)} \right),\end{equation} with the slight abuse of notation that, given a subset $C$ of $\Z^d \times \I \times \R$, $\pi(C) := \bigcup_{c \in C} \pi(c)$.
Now consider $t<0$. From Lemma \ref{l:progres}, we deduce that the map on $\Omega \times [t,0]$ defined by $(\omega, s) \mapsto \left( \tau_{x,s}^{-1} \circ H \circ \tau_{x,s} (\omega) \right) \cap \Nsp_t(\omega)$ is $\F_t \otimes \mathcal{B}([t,0])-$measurable. On the other hand, our induction hypothesis shows that for any $(x,i,k)$, the event $\{ (x,i,k) \in D_n , \ s(x,i,k) \geq t \}$ is $\F_t-$measurable. We can then deduce from \eqref{e:redef-Amb} that $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{n+1} \cap \Nsp_t$ is $\F_t-$measurable. \end{proof}
\begin{proof}[Proof of Proposition \ref{p:branch-ambig}] We proceed by induction. For $n=0$, the result is immediate since by definition $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_0 := \{ (0,0) \}$, while, for $n=1$, the result is a direct consequence of $M$ being the intensity measure of $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_1$. Now consider $n \geq 1$, and note that, by definition, \begin{equation}\label{e:contrib-parents}\sum_{\zeta \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{n+1} } f(\zeta) \leq \sum_{(x,t) \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n} g_{x,t},\end{equation} with $$g_{x,t} := \sum_{(z,i,s) \in \tau_{x,t}^{-1} \circ H \circ \tau_{x,t}} \sum_{y \in A_i} f(z+y,i,s).$$
For $(x,i,t) \in \Z^d \times \I \times \R$, define $\pi(x,i,t) := \bigcup_{z \in A_i} \{ (x+z,t) \}$. Now we rewrite $$ \sum_{(x,t) \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n} g_{x,t} = \sum_{\alpha \in \Nsp} \Psi( \alpha, \Nsp ), $$ with $$\Psi(\alpha, \omega) := \mathbf{1}} \def\dd{\mathrm d (\pi(\alpha)\subset \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n(\omega)) \sum_{(x,t) \in \pi(\alpha)} g_{x,t}(\omega).$$ Applying Campbell's theorem (Theorem \ref{t:Campbell}), we deduce that \begin{equation}\label{e:Campbell-1} \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{\zeta \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{n+1} } f(\zeta) \right) \leq \int \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( \Psi(\alpha, \Nsp \cup \{ \alpha \}) ) dJ(\alpha).\end{equation} No we deduce from Lemma \ref{l:mesurabil-Amb} that, for all $\alpha=(w,j,t) \in \Z^d \times \I \times \R$ such that $t<0$, the event $\{ \pi(\alpha) \subset \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n(\Nsp \cup \{ \alpha \}) \}$ is $\F_t-$measurable. On the other hand, for all $(x,t) \in \pi(\alpha)$, the random variable $g_{x,t}$ is measurable with respect to $\sigma(\Nsp_{<t})$, where $\Nsp_{<t} := \Nsp \cap (\Z^d \times \I \times ]-\infty, t [)$. As a consequence, $ \mathbf{1}} \def\dd{\mathrm d (\pi(\alpha)\subset \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n(\Nsp \cup \{ \alpha \}) $ and $ \sum_{(x,t) \in \pi(\alpha)} g_{x,t}(\Nsp \cup \{ \alpha \})$ are independent. Moreover, one has that $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{(x,t) \in \pi(\alpha)} g_{x,t}(\Nsp \cup \{ \alpha \} ) \right) = \sum_{(x,t) \in \pi(\alpha)} \varphi(x,t),$$ with $$\varphi(x,t) := \int f(\zeta) d (\delta_{(x,t)} \star M)(\zeta).$$ Thus \begin{equation} \label{e:indep-temps} \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( \Psi(\alpha, \Nsp \cup \{ \alpha \}) ) = \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \mathbf{1}} \def\dd{\mathrm d( \pi(\alpha) \subset \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n(\Nsp \cup \{ \alpha \}) ) \times \sum_{(x,t) \in \pi(\alpha)} \varphi(x,t) \right) ,\end{equation} Applying again Campbell's theorem, we deduce from \eqref{e:indep-temps} that \begin{eqnarray*} \int \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( \Psi(\alpha, \Nsp \cup \{ \alpha \}) ) dJ(\alpha) &=& \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{\alpha \in \Nsp} \mathbf{1}} \def\dd{\mathrm d( \pi(\alpha) \subset \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n) \sum_{(x,t) \in \pi(\alpha)} \varphi(x,t) \right) \\ &=& \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{\zeta \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n} \varphi(\zeta) \right).\end{eqnarray*} Assuming the conclusion of the proposition to be true for $n$, we deduce that $$ \int \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( \Psi(\alpha, \Nsp \cup \{ \alpha \}) ) dJ(\alpha) = \int \varphi(\zeta) d M^{\star n}(\zeta) = \int f(\zeta) d M^{\star (n+1)}(\zeta).$$ In view of \eqref{e:Campbell-1}, this establishes the conclusion of the proposition for $n+1$. \end{proof}
\begin{proof}[Proof of Corollary \ref{c:branch-ambig}] For $n=0$ the result is immediate. For $n \geq 1$, $$\sum_{\zeta \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n} f(\zeta) \cdot D \circ \tau_{\zeta} = \sum_{\alpha \in \Nsp} \Psi'(\alpha, \Nsp),$$ with $$ \Psi'(\alpha,\omega) := \mathbf{1}} \def\dd{\mathrm d(\pi(\alpha) \subset \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n(\omega)) \sum_{(x,t) \in \pi(\alpha)} g'_{x,t}(\omega),$$ $$g'_{x,t} := f(x,t) \cdot D \circ \tau_{x,t}(\omega).$$ Then Campbell's theorem shows that $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}\left( \sum_{\zeta \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n} f(\zeta) \cdot D \circ \tau_{\zeta} \right) = \int \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( \Psi'(\alpha, \Nsp \cup \{ \alpha \}) ) dJ(\alpha).$$ As in the proof of Proposition \ref{p:branch-ambig}, given $\alpha=(w,j,t) \in \Z^d \times \I \times \R$ such that $t<0$, the event $\{ \pi(\alpha) \subset \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n(\Nsp \cup \{ \alpha \}) \}$ is $\F_t-$measurable while, for all $(x,t) \in \pi(\alpha)$, the random variable $g'_{x,t}$ is measurable with respect to $\sigma(\Nsp_{<t})$. Moreover, $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( g'_{x,t} (\Nsp \cup \{ \alpha \} \right)) = f(x,t) \cdot \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(D),$$ so that $$ \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( \Psi'(\alpha, \Nsp \cup \{ \alpha \}) ) = \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \mathbf{1}} \def\dd{\mathrm d(\pi(\alpha) \subset \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n(\omega)) \sum_{(x,t) \in \pi(\alpha)} f(x,t) \cdot \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(D) \right).$$ Another application of Campbell's theorem yields that \begin{eqnarray*} \int \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( \Psi'(\alpha, \Nsp \cup \{ \alpha \}) ) dJ(\alpha) &=& \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(D) \cdot \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{\alpha \in \Nsp} \mathbf{1}} \def\dd{\mathrm d( \pi(\alpha) \subset \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n) \sum_{(x,t) \in \pi(\alpha)} f(x,t) \right) \\ &=& \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(D) \cdot \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{\zeta \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n} f(\zeta) \right).\end{eqnarray*}
Proposition \ref{p:branch-ambig} then yields the conclusion. \end{proof}
\begin{proof}[Proof of Theorem \ref{t:finitude}] Assume that $\mathfrak{g} < 1$, and note that, by definition, one has
$\int d M(\zeta) = \mathfrak{g}$. We now use Proposition \ref{p:branch-ambig} with $f \equiv 1$, and obtain that, for all $n \geq 0$, $\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(|\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n|) \leq \mathfrak{g}^n$. As a consequence,
$$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(|\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}|) \leq \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{n \geq 0} |\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n| \right) = \sum_{n \geq 0} \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( |\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n| \right) \leq \sum_{n \geq 0} \mathfrak{g}^n < +\infty.$$
It is now clear that $\P (| \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}|<+\infty)=1$. Similarly, applying Corollary \ref{c:branch-ambig} with $f \equiv 1$ and $D := \mathbf{1}} \def\dd{\mathrm d(T=-\infty)$ yields that, for all $n \geq 0$, $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{(x,t) \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n } \mathbf{1}} \def\dd{\mathrm d(T \circ \tau_{x,t} = -\infty) \right) = 0.$$ As a consequence, with probability one,
$T \circ \tau_{x,t} > -\infty$ for all $(x,t) \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}$. We have thus proved that $\P(T^* > -\infty) =1$. The conclusion of the theorem is now a consequence of Proposition \ref{p:base-ambig}. \end{proof}
\begin{proof}[Proof of Theorem \ref{t:temps-exp}] We apply Corollary \ref{c:branch-ambig} with $f(x,t) \equiv \exp(\lambda t)$ and $D := \exp(\lambda T)$. As a result, for all $n \geq 0$, $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{ (x,t) \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_n } e^{\lambda (t+ T \circ \tau_{x,t})} \right) \leq \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}\left( e^{\lambda T} \right) \left( \int e^{\lambda t} dM^{\star n}(x,t) \right).$$ One has $$\int e^{\lambda t} dM^{\star n}(x,t) = \left( \int e^{\lambda t} dM(x,t) \right)^n = (\Lambda_{H,time}(\lambda))^n.$$ Summing over $n \geq 0$, we obtain that $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{ (x,t) \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty} } e^{\lambda (t+ T \circ \tau_{x,t})} \right) \leq \sum_{n \geq 0} \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}\left( e^{\lambda T} \right) (\Lambda_{H,time}(\lambda))^n = \frac{ \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}\left( e^{\lambda T} \right) }{1 - \Lambda_{H,time}(\lambda)}.$$ Now by definition of $T^*$, using the fact that $\lambda < 0$, $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(\exp(\lambda T^*)) \leq \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{ (x,t) \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty} } e^{\lambda (t+ T \circ \tau_{x,t})} \right).$$ The conclusion follows. \end{proof}
\begin{proof}[Proof of Theorem \ref{t:espace-exp} (sketch)] First note that, when $\mathfrak{g}<1$, $L^*_+$ and $L^*_-$ are a.s. finite, using an argument similar to the one establishing that $T^*$ is a.s. finite in the proof of Theorem \ref{t:finitude}. Moreover, the proof of the estimates on $\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(\exp(\lambda L^*_+))$ and $\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(\exp(-\lambda L^*_-))$ is completely similar to the proof of Theorem \ref{t:temps-exp}. It remains to prove that $L^*$ indeed bounds the width of $T^*$. This is done by adapting the proof of Proposition \ref{p:base-ambig} as follows. We work on the event $T^*>-\infty$. First note that, thanks to the fact that $L$ defines a stopping box, $L$ is $\F^{-L', L'}-$measurable for any random variable such that $L \leq L'$. Then observe that $\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}$ is measurable with respect to $\F^{-L^*,L^*}$, since, for each $n \geq 0$ and $(x,t) \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{n}$, one has that
$L^*_- \leq x - L \circ \tau_{x,t} \leq x + L \circ \tau_{x,t} \leq L^*_+$. We now start the induction with the case $|\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}|=1$. Then the values of the $e(\alpha, T, \xi)$ for $\alpha \in H$ are completely determined by $H$ itself, and $H$ is measurable with respect to $\F^{-L, L}$, so we are done. If $|\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty}|=n+1$, we apply the induction hypothesis to every $|\mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_{\infty} \circ \tau_{x,t}|$ such that $x=z+y$ for some $\alpha=(z,i,t) \in \mathrm{Amb}} \def\pred{\mathrm{Preced}} \def\perf{\mathrm{Perf}_1$ and $y \in A_i$, then deduce that $[\Phi_{T^*}^{0-}(\xi)](0)$ has the required measurability properties. \end{proof}
\section{Proof of Theorem \ref{t:theoreme-principal}}\label{s:preuve-2}
We start with a proposition showing that $(T,H)$ is a CFTP time with ambiguities for the perturbed dynamics if $\hat{\X}_{\infty}$ is a finite set $\P-$a.s.
\begin{prop}\label{p:base-perturb}
If $\P(|\hat{\X}_{\infty}|<+\infty)=1$, then $(T, H)$ is a CFTP time with ambiguities. \end{prop}
\begin{proof}
First note that the stopping property of $\hat{\X}_{\infty}$ is a direct consequence of the way the process is constructed. We now work on the event that $|\hat{\X}_{\infty}|$ is finite. Consider $\xi \in \S^{\Z^d}$, and let $\mathcal{Q}} \def\Nsp{\mathcal{P}$ be the element of $\Omega^u$ obtained from $\Nsp$ by replacing any $\alpha =(x,i,s) \in \Nsp$ such that $s \geq T$ and $i \in \I^p$ by $(x,\iota_{e(\alpha,T,\xi)},s)$, and suppressing any $\alpha =(x,i,s) \in \Nsp$ such that $s < T$ and $i \in \I^p$. Let $h_t(\xi)$ denote the random variable defined on $\Omega^u$ by $$h_t(\xi) := [(\Phi^u)_{t}^{0-}(\xi)](0),$$ where $\Phi^u$ denotes the stochastic flow defined by $\Nsp^u$ on $\Omega^u$. From the definition of the dynamics, we see that \begin{equation}\label{e:premiere-substitution}[\Phi_T^{0-}(\xi)](0) = [h_T(\xi)](\mathcal{Q}} \def\Nsp{\mathcal{P}).\end{equation} Now define a path $c_0,\ldots, c_m$ in $\mathbb{T}$ as follows. Start with $c_0:=r$. Then assume that $c_0,\ldots, c_k$ have been defined. If $c_k$ has no son in $\mathbb{T}$, the path ends at $c_k$, so that $m:=k$. If $c_k$ has a single son in $\mathbb{T}$, then $c_{k+1}$ is defined to be this single son. Finally, if $c_k$ has several sons in $\mathbb{T}$, $c_{k+1}$ is defined to be the son associated with the value $v:=e((x_{c_k}, i_{c_k}, t_{c_k}),T,\xi)$.
By definition of the exploration processes $\mathfrak{X}^u$ and $\hat{\X}$, one has that $\mathfrak{X}^u_{\infty}(\mathcal{Q}} \def\Nsp{\mathcal{P}) = \hat{\X}_{c_m}$. Then by definition $T^u(\mathcal{Q}} \def\Nsp{\mathcal{P}) = \inf \{ t; \ (x,i,t) \in \mathfrak{X}^u_{\infty}(\mathcal{Q}} \def\Nsp{\mathcal{P}) \}$, so that the identity $\mathfrak{X}^u_{\infty}(\mathcal{Q}} \def\Nsp{\mathcal{P}) = \hat{\X}_{c_m}$ implies that $T^u(\mathcal{Q}} \def\Nsp{\mathcal{P}) \geq T$. From the fact that $T^u$ is associated with the exploration process specified by $\theta$, there exists a measurable map $G$ defined on the set $\Omega^u_f$ such that, on the event $\{ T^u > -\infty \} = \{ |\mathfrak{X}^u_{\infty}|<+\infty \}$, one has, for every $\chi \in \S^{\Z^d}$, $h_{T^u}(\chi) = G(\mathfrak{X}^u_{\infty})$. As a consequence,
\begin{equation}\label{e:seconde-substitution}[h_T(\xi)](\mathcal{Q}} \def\Nsp{\mathcal{P}) = G( \hat{\X}_{c_m} ).\end{equation} It is now immediate from \eqref{e:premiere-substitution} and \eqref{e:seconde-substitution} that if $e(\alpha, \xi_1, T) = e(\alpha, \xi_2, T)$ for every $\alpha \in H$, then $[\Phi_T^{0-}(\xi_1)](0) = [\Phi_T^{0-}(\xi_2)](0)$, since both values of $ \hat{\X}_{c_m}$ obtained starting from $\xi_1$ or $\xi_2$ are identical. \end{proof}
The next two propositions are the key first-moment estimates needed to control $\hat{\X}_{\infty}$.
Define a kernel $K$ on $\Omega^u_f$ as follows. If $\theta(X) = \emptyset$, then $K(X, \cdot) = \delta_{X}(\cdot)$. If $\theta(X) \neq \emptyset$, let $s := \inf \{ t; \ (x,i,t) \in X \}$, and
let $$d K(X, X \cup \{ (x,i,t) \}) = r_i \exp(|\theta(X) | \mathfrak{r}_u (t-s)) \mathbf{1}} \def\dd{\mathrm d(t < s) \mathbf{1}} \def\dd{\mathrm d(x \in \theta(X)) dJ^u(x,i,t),$$ with $\mathfrak{r}_u := \sum_{j \in \I^u} r_j$. From the definition, one has the following. \begin{prop}\label{p:Markov} The sequence $(\mathfrak{X}^u_{\ell})_{\ell \geq 0}$ is a Markov chain on $\Omega^u_f$ with initial state $\emptyset$ and transition kernel $K$. \end{prop}
Now define a kernel $\hat{K}$ on the set of elements $X \in \Omega^u_f$ such that $\theta(X) \neq \emptyset$ as follows. Let $s := \inf \{ t; \ (x,i,t) \in X \}$, and
let $$d \hat{K}(X, X \cup \{ (x,i,t) \}) = \hat{r}_i \exp(|\theta(X) | \mathfrak{r} (t-s)) \mathbf{1}} \def\dd{\mathrm d(t < s) \mathbf{1}} \def\dd{\mathrm d(x \in \theta(X)) dJ^u(x,i,t),$$ with, for $i \in \I^u$, $\hat{r}_i := r_i + \sum_{j \in \I^p } \sum_{ v \in f_j(A_j) } r_j \mathbf{1}} \def\dd{\mathrm d(\iota_v = i)$, and $\mathfrak{r} := \sum_{j \in \I} r_j$. Define also the kernel $L$ by
$$dL(X, (x,i,t)) = r_i \exp(|\theta(X) | \mathfrak{r} (s-t)) \mathbf{1}} \def\dd{\mathrm d(t < s) \mathbf{1}} \def\dd{\mathrm d(x \in \theta(X)) dJ(x,i,t).$$
For $\ell \geq 0$, let $\mathbb{T}_{\ell}$ (resp. $\mathbb{T}'_{\ell}$) denote the set of vertices at distance $\ell$ from the root in $\mathbb{T}$ (resp. $\mathbb{T}'$); let also $$\Gamma_{\ell} := \left\{ (X_0,\ldots, X_{\ell}) \in (\Omega^u_f)^{\ell+1}; \ \theta(X_0) \neq \emptyset, \ldots, \theta(X_{\ell-1}) \neq \emptyset \right\},$$ $$\Delta_{\ell} := \left\{ (X_0,\ldots, X_{\ell}) \in (\Omega^u_f)^{\ell+1}; \ \theta(X_0) \neq \emptyset, \ldots, \theta(X_{\ell}) \neq \emptyset \right\}.$$
\begin{prop}\label{p:calcul-branchant} For every $\ell \geq 0$, and any measurable map $F \ : \ \Omega^u_f \to \R_+$, one has the identity $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}\left( \sum_{a \in \mathbb{T}_{\ell}} F(\hat{\X}_a)\right) = \int_{\Gamma_{\ell}} F(X_{\ell}) d \delta_{\emptyset}(X_0) d\hat{K}(X_0,X_1) \cdots d\hat{K}(X_{\ell-1},X_{\ell}).$$ For every $\ell \geq 0$, and any measurable map $f \ : \ \Z^d \times \I \times \R \to \R_+$, one has the identity $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}\left( \sum_{a \in \mathbb{T}'_{\ell}} f(x_a,i_a,t_a)\right) = \int f(\alpha) d \delta_{\emptyset}(X_0) d\hat{K}(X_0,X_1) \cdots d\hat{K}(X_{\ell-1},X_{\ell}) dL(X_{\ell}, \alpha),$$ where the integral is over $(X_0,\ldots, X_{\ell}, \alpha) \in \Delta_{\ell} \times (\Z^d \times \I \times \R)$. \end{prop}
\begin{proof}[Proof] The proof is similar to that of Proposition \ref{p:branch-ambig}. We start with the first identity, whose proof is by induction. For $\ell=0,1$, the result is a direct consequence of the definition. Now for $\ell \geq 1$, write \begin{equation} \sum_{a \in \mathbb{T}_{\ell+1}} F(\hat{\X}_a) = \sum_{\alpha \in \Nsp} Z_1(\alpha, \Nsp) ,\end{equation} where $$Z_1(\alpha, \cdot) := \sum_{c \in \mathbb{T}'_{\ell-1}} \mathbf{1}} \def\dd{\mathrm d(\alpha = (x_c,i_c,t_c)) \sum_{b \in \mathbb{T}_{\ell}, \ b \leftarrow c} \mathbf{1}} \def\dd{\mathrm d(\theta(\hat{\X}_b) \neq \emptyset) Z_2(\hat{\X}_b, \cdot),$$ with $b \leftarrow c$ meaning that $b$ is a son of $c$ in $\mathbb{T}$, and with $$ Z_2(X, \cdot):= \mathbf{1}} \def\dd{\mathrm d(j \in \I^u) F(X \cup \{ (y, j, s) \} ) + \mathbf{1}} \def\dd{\mathrm d(j \in \I^p) \sum_{ v \in f_{j}(A_{j}) } F(X \cup \{ (y, \iota_v, s) \} ),$$ with $(y,j,s) := \mathfrak{n}(\theta(X), t),$ and $t:= \inf \{ \nu; \ (w,k,\nu) \in X \}$. (We write $Z_1(\alpha, \cdot)$ and $Z(X,\cdot)$ to make the dependence on $\omega$ explicit.)
By Campbell's theorem (Theorem \ref{t:Campbell}), one has that $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{a \in \mathbb{T}_{\ell+1}} F(\hat{\X}_a) \right) = \int \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( Z_1(\alpha, \Nsp \cup \{ \alpha \})) d J(\alpha).$$ Now given $\alpha = (x,i,t)$, define the random finite counting measure\footnote{We equip the set of finite counting measures on $\Omega^u_f$ with the $\sigma-$algebra generated by all the maps of the form $Y \mapsto Y(B)$, where $B$ belongs to the $\sigma-$algebra defined on $\Omega^u_f$.} $\mathcal{M}$ on $\Omega^u_f$ by $$ \mathcal{M} := \sum_{c \in \mathbb{T}'_{\ell-1}} \mathbf{1}} \def\dd{\mathrm d(\alpha = (x_c,i_c,t_c)) \sum_{b \in \mathbb{T}_{\ell}, \ b \leftarrow c} \mathbf{1}} \def\dd{\mathrm d(\theta(\hat{\X}_b) \neq \emptyset) \delta_{\hat{\X}_b},$$ so that $$Z_1(\alpha, \cdot) = \int Z_2(X, \cdot) d \mathcal{M}(X, \cdot).$$
Note that $\mathcal{M}$ is $\F_t-$measurable, while, for any $X \in \Omega^u_f$ such that $\theta(X) \neq \emptyset$ and $t:= \inf \{ \nu; \ (w,k,\nu) \in X \}$, $Z_2(X ,\cdot)$ is measurable with respect to $\sigma(\Nsp_{<t})$ and satisfies $\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(Z_2(X,\cdot)) = \hat{K}F(X)$.
We deduce that $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( Z_1(\alpha, \Nsp \cup \{ \alpha \}) ) = \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \int \hat{K}F(X) d \mathcal{M}(X, \Nsp \cup \{ \alpha \}) \right) = \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(Z_3(\alpha, \Nsp \cup \{ \alpha \})),$$ where $$Z_3(\alpha, \cdot) := \sum_{c \in \mathbb{T}'_{\ell-1}} \mathbf{1}} \def\dd{\mathrm d(\alpha = (x_c,i_c,t_c)) \sum_{b \in \mathbb{T}_{\ell}, \ b \leftarrow c} \hat{K} F( \hat{\X}_b ).$$ Using Campbell's theorem again shows that $$\int \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(Z_3(\alpha, \Nsp \cup \{ \alpha \}))dJ(\alpha) = \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{b \in \mathbb{T}_{\ell}} \hat{K} F( \hat{\X}_b ) \right).$$ This computation allows induction over $\ell$ to be used to prove the desired identity for all $\ell \geq 1$. The second identity of the Proposition can be deduced from the first one, using an argument similar to the derivation of Corollary \ref{c:branch-ambig} from Proposition \ref{p:branch-ambig}. \end{proof}
Combined with Propositions \ref{p:Markov} and \ref{p:calcul-branchant}, the following remark is the key to obtaining estimates on $\hat{\X}$ from the properties of $\mathfrak{X}^u$. Consider $X \in \Omega^u_f$ such that $\theta(X) \neq \emptyset$. From the definition of $\epsilon$ given in \eqref{e:def-epsilon} and the fact that $\mathfrak{r}_u \leq \mathfrak{r}$, one has that \begin{equation}\label{e:compare-densite}d\hat{K}(X, X \cup \{ (x,i,t) \}) \leq (1+\epsilon) d K (X, X \cup \{ (x,i,t) \}).\end{equation}
We can now prove the various estimates that are needed in the proof of Theorem \ref{t:theoreme-principal}.
\begin{lemma}\label{l:estimation-1} For every $\ell \geq 1$,
$$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(|\mathbb{T}_{\ell}|) \leq (1+\epsilon)^{\ell} \P^u( |\mathfrak{X}^u_{\infty}| \geq \ell ).$$ \end{lemma}
\begin{proof} By Proposition \ref{p:calcul-branchant}, one has that
$$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(|\mathbb{T}_{\ell}| ) = \int_{(X_0,\ldots, X_{\ell}) \in \Gamma_{\ell}} d \delta_{\emptyset}(X_0) d\hat{K}(X_0,X_1) \cdots d\hat{K}(X_{\ell-1},X_{\ell}),$$ From \eqref{e:compare-densite}, one deduces that
$$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(|\mathbb{T}_{\ell}|) \leq (1+\epsilon)^{\ell} \int_{(X_0,\ldots, X_{\ell}) \in \Gamma_{\ell}} d \delta_{\emptyset}(X_0) dK(X_0,X_1) \cdots d K(X_{\ell-1},X_{\ell}).$$ But by Proposition \ref{p:Markov},
$$\int_{(X_0,\ldots, X_{\ell}) \in \Gamma_{\ell}} d \delta_{\emptyset}(X_0) dK(X_0,X_1) \cdots d K(X_{\ell-1},X_{\ell}) = \P^u( |\mathfrak{X}^u_{\infty}| \geq \ell ),$$ so that
$$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(|\mathbb{T}_{\ell}|) \leq (1+\epsilon)^{\ell} \P^u( |\mathfrak{X}^u_{\infty}| \geq \ell ).$$ \end{proof}
\begin{lemma}\label{l:estimation-2} For every $\ell \geq 0$,
$$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{a \in \mathbb{T}'_{\ell}} |A_{i_a}| \mathbf{1}} \def\dd{\mathrm d(i_a \in \I^p) \right) \leq \kappa (1+\epsilon)^{\ell} \P^u( |\mathfrak{X}^u_{\infty}| > \ell ).$$ \end{lemma}
\begin{proof}
Let $f(x,i,t):= |A_{i}| \mathbf{1}} \def\dd{\mathrm d(i \in \I^p)$. Note that, given $X \in \Omega^u_f$ such that $\theta(X) \neq \emptyset$, one has $\int_{(x,i,t) \in \Z^d \times \I \times \R} f(x,i,t) dL(X,(x,i,t)) = \kappa$. Using Proposition \ref{p:calcul-branchant} and \eqref{e:compare-densite}, one obtains that
$$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{a \in \mathbb{T}'_{\ell}} |A_{i_a}| \mathbf{1}} \def\dd{\mathrm d(i_a \in \I^p) \right) \leq \kappa (1+\epsilon)^{\ell} I_2,$$ where
$$I_2=\int_{(X_0,\cdots, X_{\ell}) \in \Delta_{\ell}} d \delta_{\emptyset}(X_0) d K(X_0,X_1) \cdots d K(X_{\ell-1},X_{\ell})= \P^u( |\mathfrak{X}^u_{\infty}| > \ell).$$
\end{proof}
\begin{lemma}\label{l:estimation-3} For every $\ell \geq 0$, and $\lambda < \mathfrak{r}$,
$$ \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{a \in \mathbb{T}'_{\ell}} \exp(\lambda t_a) \right) \leq \frac{\mathfrak{r}}{\mathfrak{r} - \lambda} (1+\epsilon)^{\ell} \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}^u(\exp(\lambda t_{\ell-1}) \mathbf{1}} \def\dd{\mathrm d(|\mathfrak{X}^u_{\infty}| > \ell )),$$ with the convention $t_{-1} := 0$ (remember that $(x_n, i_n, t_n)_{n \geq 0}$ corresponds to the exploration process $\mathfrak{X}^u$). \end{lemma}
\begin{lemma}\label{l:estimation-4} For every $\ell \geq 0$, and $\lambda < \mathfrak{r}$,
$$ \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{a \in \mathbb{T}'_{\ell}} |A_i| \exp(\lambda t_a) \mathbf{1}} \def\dd{\mathrm d(i_a \in \I^p) \right) \leq \frac{\kappa \mathfrak{r}}{\mathfrak{r} - \lambda} (1+\epsilon)^{\ell} \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}^u(\exp(\lambda t_{\ell-1}) \mathbf{1}} \def\dd{\mathrm d(|\mathfrak{X}^u_{\infty}| > \ell )),$$ with the convention $t_{-1} := 0$. \end{lemma}
\begin{proof}
We prove Lemma \ref{l:estimation-4}, the proof of Lemma \ref{l:estimation-3} being quite similar. Let $f(x,i,t):= |A_i| \exp(\lambda t) \mathbf{1}} \def\dd{\mathrm d(i \in \I^p)$. Consider $X \in \Omega^u_f$ such that $\theta(X) \neq \emptyset$, and let $s := \inf \{ t; \ (x,i,t) \in X \}$. One has
$\int_{(x,i,t) \in \Z^d \times \I \times \R} f(x,i,t) dL(X,(x,i,t)) = \kappa \mathfrak{r} | \theta(X)| ( \mathfrak{r} |\theta(X)| - \lambda )^{-1} \exp( \lambda s) \leq \kappa \mathfrak{r} (\mathfrak{r} - \lambda)^{-1} \exp(\lambda s),$
since $|\theta(X)| \geq 1$. Using Proposition \ref{p:calcul-branchant} and \eqref{e:compare-densite}, we deduce that $$ \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{a \in \mathbb{T}'_{\ell}} \exp(\lambda t_a) \mathbf{1}} \def\dd{\mathrm d(i_a \in \I^p) \right) \leq \kappa\mathfrak{r} (\mathfrak{r} - \lambda)^{-1} (1+\epsilon)^{\ell} I_3,$$ with $$I_3=\int_{(X_0,\cdots, X_{\ell}) \in \Delta_{\ell}} \exp(\lambda s_{\ell-1}) d \delta_{\emptyset}(X_0) d K(X_0,X_1) \cdots d K(X_{\ell-1},X_{\ell})$$
and $s_{\ell-1} := \inf \{ t; \ (x,i,t) \in X_{\ell} \}$, with the convention $s_{\ell-1}:=0$. Then note that $I_3 = \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}^u(\exp(\lambda t_{\ell-1}) \mathbf{1}} \def\dd{\mathrm d(|\mathfrak{X}^u_{\infty}| > \ell ))$. \end{proof}
\begin{lemma}\label{l:estimation-6} For every $\ell \geq 1$, $\lambda \in \R$ and $1 \leq q \leq d$,
$$ \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{a \in \mathbb{T}'_{\ell}} \sum_{z \in A_{i_a}} e^{\lambda (x_a+z)_q} \mathbf{1}} \def\dd{\mathrm d\mbox{\tiny $(i_a \in \I^p)$} \right) \leq Q (1+\epsilon)^{\ell} \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}^u \left( \frac{1}{|\theta(\mathfrak{X}^u_{\ell})|} \sum_{x \in \theta(\mathfrak{X}^u_{\ell})} e^{\lambda x_q} \mathbf{1}} \def\dd{\mathrm d\mbox{\tiny $(|\mathfrak{X}^u_{\infty}| > \ell )$} \right),$$ where \begin{equation}Q := \left( \sum_{i \in \I^p} (r_i/\mathfrak{r}) \sum_{z \in A_i} \exp(\lambda z_q) \right).\end{equation}
\end{lemma}
We are now ready to prove Theorem \ref{t:theoreme-principal}. From now on, we assume that there exists $\mu>0$ such that
$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}^u\left( e^{\mu | \mathfrak{X}^u_{\infty}} | \right) < +\infty$. As a consequence, there exists a finite constant $C$ such that, for all $\ell \geq 1$,
\begin{equation}\label{e:queue-exp} \P^u(|\mathfrak{X}^u_{\infty}| \geq \ell) \leq C \exp(-\mu \ell).\end{equation}
We first prove that, as soon as $\epsilon < \mu$, $\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( |\hat{\X}_{\infty}|)<+\infty$. Indeed, one has that $|\hat{\X}_{\infty}| \leq |\mathbb{T}|$, so that
\begin{equation}\label{e:somme-tranches}\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(|\hat{\X}_{\infty}|) \leq \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(|\mathbb{T}|) = \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{\ell \geq 0} |\mathbb{T}_{\ell}| \right)= \sum_{\ell \geq 0} \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(|\mathbb{T}_{\ell}|).\end{equation}
By Lemma \ref{l:estimation-1}, for all $\ell \geq 1$, one has $\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(|\mathbb{T}_{\ell}|) \leq (1+\epsilon)^{\ell} \P^u( |\mathfrak{X}^u_{\infty}| \geq \ell )$.
Combining \eqref{e:queue-exp} and \eqref{e:somme-tranches}, we see that $\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( |\hat{\X}_{\infty}|)<+\infty$ when $\epsilon < \mu$. By Proposition \ref{p:base-perturb}, this proves that the pair $(T,H)$ defined by \eqref{d:def-T} and \eqref{d:def-H} is indeed a CFTP time with ambiguities.
We now prove that, for small enough $\epsilon$ and $\kappa$, the pair $(T,H)$ satisfies $\mathfrak{g} < 1$. Using the definition, then Lemma \ref{l:estimation-2} , we have that
$$\mathfrak{g} = \sum_{\ell \geq 0} \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{a \in \mathbb{T}'_{\ell}} |A_{i_a}| \mathbf{1}} \def\dd{\mathrm d(i_a \in \I^p) \right) \leq \kappa \sum_{\ell \geq 0} (1+\epsilon)^{\ell} \P^u( |\mathfrak{X}^u_{\infty}| > \ell ).$$ From \eqref{e:queue-exp}, we see that $\mathfrak{g} < 1$ for all $\epsilon < \mu$ and small enough $\kappa$.
We now prove that $\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(\exp(\lambda T))<+\infty$ for all small enough $\epsilon$ and $\lambda$. We start with the observation that $$\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( \exp(\lambda T)) \leq \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P} \left( \sum_{a \in \mathbb{T}'} \exp(\lambda t_a) \right) = \sum_{\ell \geq 0} \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}\left( \sum_{a \in \mathbb{T}'_{\ell}} \exp(\lambda t_a) \right).$$ From Lemma \ref{l:estimation-3}, we deduce that, for all $\lambda < \mathfrak{r}$,
\begin{equation}\label{e:somme-tranches-2}\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}( \exp(\lambda T)) \leq \frac{ \mathfrak{r}} {\mathfrak{r} - \lambda} \sum_{\ell \geq 0} (1+\epsilon)^{\ell} \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}^u(\exp(\lambda t_{\ell-1}) \mathbf{1}} \def\dd{\mathrm d(|\mathfrak{X}^u_{\infty}| > \ell )).\end{equation} By Schwarz's inequality,
$$ \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}^u(\exp(\lambda t_{\ell-1}) \mathbf{1}} \def\dd{\mathrm d(|\mathfrak{X}^u_{\infty}| > \ell )) \leq \left( \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}^u(\exp( 2\lambda t_{\ell-1}) \right)^{1/2} \P^u(|\mathfrak{X}^u_{\infty}| > \ell )^{1/2}.$$ Bounding above $t_{\ell-1}$ by the sum of $\ell$ independent exponential random variables with parameter $\mathfrak{r}_u$ on one hand, and using \eqref{e:queue-exp} on the other hand, one obtains that, when $\lambda < \mathfrak{r}_u/2$,
\begin{equation}\label{e:borne-sauvage} \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}^u(\exp(\lambda t_{\ell}) \mathbf{1}} \def\dd{\mathrm d(|\mathfrak{X}^u_{\infty}| > \ell )) \leq C^{1/2} \left(\frac{\mathfrak{r}_u}{\mathfrak{r}_u-2 \lambda}\right)^{\ell} \exp(-\mu(\ell+1)/2). \end{equation} Combining \eqref{e:somme-tranches-2} and \eqref{e:borne-sauvage}, we have that $\mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}(\exp(\lambda T))<+\infty$ for all small enough $\epsilon$ and $\lambda$.
We now prove that $\Lambda_{H,time}(\lambda)<1$ for all small enough $\epsilon, \kappa, \lambda$. Using Lemma \ref{l:estimation-4}, we obtain that
\begin{equation}\label{e:somme-tranches-3}\Lambda_{H,time}(\lambda) \leq \frac{\kappa \mathfrak{r}}{\mathfrak{r} - \lambda} \sum_{\ell \geq 0} (1+\epsilon)^{\ell} \mathbb{E}} \def\N{\mathbb{N}} \def\P{\mathbb{P}^u(\exp(\lambda t_{\ell}) \mathbf{1}} \def\dd{\mathrm d(|\mathfrak{X}^u_{\infty}| > \ell )).\end{equation} Using again \eqref{e:borne-sauvage}, one concludes that $\Lambda_{H,time}(\lambda)<1$ for all small enough $\epsilon, \kappa, \lambda$.
Now let $R$ denote the depth of $\mathbb{T}$, and define $L := \beta_u(R)$ (remember that $\beta$ is defined in \eqref{e:borne-taille}). By definition of the exploration process with locking of ambiguities, one checks that $L$ defines a stopping box and that $H$ is measurable with respect to $\F^{-L, L}$. Now, \eqref{e:premiere-substitution} and \eqref{e:seconde-substitution} show that $\left[\Phi_{T}^{0-}(\xi) \right](0)$ satisfies the required measurability properties.
Using the obvious inequality $R \leq |\mathbb{T}|$, Lemma \ref{l:estimation-1} shows that $\Lambda_{L}(\lambda)$ is finite for small enough $\epsilon, \lambda$. Finally, Lemma \ref{l:estimation-6} shows that $\Lambda_{H,space}(\lambda,q)<1$ for all $q$, when $\epsilon, \kappa, \lambda$ are small enough.
\end{document} | arXiv |
\begin{document}
\RUNAUTHOR{Henry Lam}
\RUNTITLE{Statistical Guarantees via the Empirical DRO}
\TITLE{Recovering Best Statistical Guarantees via the Empirical Divergence-based Distributionally Robust Optimization}
\ARTICLEAUTHORS{ \AUTHOR{Henry Lam} \AFF{Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, \EMAIL{[email protected]}}
}
\ABSTRACT{ We investigate the use of distributionally robust optimization (DRO) as a tractable tool to recover the asymptotic statistical guarantees provided by the Central Limit Theorem, for maintaining the feasibility of an expected value constraint under ambiguous probability distributions. We show that using empirically defined Burg-entropy divergence balls to construct the DRO can attain such guarantees. These balls, however, are not reasoned from the standard data-driven DRO framework since by themselves they can have low or even zero probability of covering the true distribution. Rather, their superior statistical performances are endowed by linking the resulting DRO with empirical likelihood and empirical processes. We show that the sizes of these balls can be optimally calibrated using $\chi^2$-process excursion. We conduct numerical experiments to support our theoretical findings. }
\KEYWORDS{distributionally robust optimization, empirical likelihood, empirical process, chi-square process, central limit theorem}
\maketitle
\section{Statistical Motivation of Distributionally Robust Optimization}
We consider an expected value constraint in the form \begin{equation} Z_0(x):=E_0[h(x;\xi)]\leq0\label{constraint} \end{equation} where $\xi\in\Xi$ is a random object under the probability measure $P_0$, $E_0[\cdot]$ denotes the corresponding expectation, $x\in\Theta\subset\mathbb R^m$ is the decision variable, and $h$ is a known function. The generic constraint \eqref{constraint} has appeared in various applications such as resource allocation (\cite{atlason2004call}), risk management (\cite{palmquist1999portfolio,fabian2008handling}), among others.
In practice, the probability measure $P_0$ is often unknown, but rather is observed via a finite collection of data. Such uncertainty has been considered in the stochastic and the robust optimization literature. Our main goal in this paper is to investigate, in a statistical sense, the \emph{best} data-driven reformulation of \eqref{constraint} in terms of feasibility guarantees.
\subsection{Initial Attempt: Sample Average Approximation} To define what ``best" means, we start by discussing arguably the most natural attempt for handling \eqref{constraint}, namely the sample average approximation (SAA) (\cite{shapiro2014lectures,wang2008sample,kleywegt2002sample}). Suppose we have i.i.d. data $\xi_1,\ldots,\xi_n$. SAA entails replacing the unknown expectation $Z_0(x)$ with the sample average $(1/n)\sum_{i=1}^nh(x;\xi_i)$, leading to \begin{equation} \hat h(x):=\frac{1}{n}\sum_{i=1}^nh(x;\xi_i)\leq0\label{SAA} \end{equation} The issue with naively using SAA in this setting is that a solution feasible according to \eqref{SAA} may be mistakeably infeasible for \eqref{constraint}. Since for any $x$ the true mean $Z_0(x)$ can lie above or below its sample average, both with substantial probabilities, the $x$'s close to the boundary of the feasible region according to \eqref{SAA} could, with overwhelming probabilities, be infeasible for the original constraint \eqref{constraint}. Consequently, the probability $$P\left(\hat h(x)\leq0\ \Rightarrow\ Z_0(x)\leq0\right)$$ where $P$ is with respect to the generation of data, can be much lower than an acceptable level.
One way to boost the confidence of SAA is to insert a margin, namely by using the constraint \begin{equation} \hat h(x)+\epsilon_n\leq0\label{SAA relaxation} \end{equation} This idea has appeared in various contexts (e.g., \cite{wang2008sample,nagaraj2014stochastically}). Choosing $\epsilon_n>0$ suitably can guarantee that \begin{equation} P\left(\hat h(x)+\epsilon_n\leq0\ \Rightarrow\ Z_0(x)\leq0\right)\geq1-\alpha\label{conf guarantee} \end{equation} where $1-\alpha$ is a prescribed confidence level chosen by the modeler (a typical choice is $\alpha=0.05$). This is achieved by finding $\epsilon_n$ such that \begin{equation} P\left(Z_0(x)\leq\hat h(x)+\epsilon_n\text{\ \ for all\ \ }x\in\Theta\right)\geq1-\alpha\label{conf guarantee simplify} \end{equation} Such a choice of $\epsilon_n$ can be obtained in terms of the maximal variance of $h(x;\xi)$ over all $x\in\Theta$, and other information such as the diameter of the space $\Theta$ (e.g., \cite{wang2008sample} provides one such choice).
\subsection{The Statistician's Approach: Confidence Bounds from the Central Limit Theorem} Though \eqref{conf guarantee simplify} could provide a good feasibility guarantee, the use of one single number $\epsilon_n$ as the margin adjustment may unnecessarily penalize $x$ whose $h(x;\xi)$ bears only a small variation. From a ``classical statistician"'s viewpoint, we adopt a margin adjustment that takes into account the variability of $h(x;\xi)$ at each point of $x$, and at the same time provides a $1-\alpha$ confidence guarantee, by formulating the constraint as \begin{equation} \hat h(x)+z\frac{\hat\sigma(x)}{\sqrt n}\leq0\label{CB} \end{equation} where $z$ is the critical value of a suitable sampling distribution, and $\hat\sigma(x)$ is an estimate of $\sqrt{Var_0(h(x;\xi))}$ ($Var_0(\cdot)$ denotes the variance under $P_0$), i.e., $\hat\sigma(x)/\sqrt n$ is the standard error. A judicious choice of $z$ can lead to the \emph{asymptotically exact} guarantee \begin{equation} \lim_{n\to\infty}P\left(Z_0(x)\leq\hat h(x)+z\frac{\hat\sigma(x)}{\sqrt n}\text{\ \ for all\ \ }x\in\Theta\right)=1-\alpha\label{guarantee exact} \end{equation} Without making further assumption on the optimization objective, we set the reformulation \eqref{CB} and the guarantee \eqref{guarantee exact} as our benchmark in this paper, since they stem from the central limit theorem (CLT) widely used in statistics.
The problem with directly using \eqref{CB} is that (sample) standard deviation is not a tractability-preserving operation, e.g., $\hat\sigma(x)$ may not be convex in $x$ even though the function $h(x;\xi)$ is. Thus the constraint \eqref{CB} can be intractable despite that \eqref{constraint} is tractable. This motivates the investigation of a distributionally robust optimization (DRO) approach, namely, by using \begin{equation} \max_{P\in\mathcal U}E_P[h(x;\xi)]\leq0\label{DRO constraint} \end{equation} where $E_P[\cdot]$ denotes the expectation under $P$, and $\mathcal U:=\mathcal U(\xi_1,\ldots,\xi_n)$ is an uncertainty set (also known as ambiguity set), calibrated from data, that contains a collection of distributions. As documented in many previous work (e.g., \cite{delage2010distributionally,ben2013robust}), \eqref{DRO constraint} can be made tractable by suitably choosing $\mathcal U$. One central question in this paper is to ask: \\
\noindent\emph{Is there a tractable choice of $\mathcal U$ that can recover the statistician's asymptotically exact guarantee, namely \begin{equation} \lim_{n\to\infty}P\left(Z_0(x)\leq\max_{P\in\mathcal U}E_P[h(x;\xi)]\text{\ \ for all\ \ }x\in\Theta\right)=1-\alpha\label{main guarantee} \end{equation} and that \begin{equation} \max_{P\in\mathcal U}E_P[h(x;\xi)]\approx\hat h(x)+z\frac{\hat\sigma(x)}{\sqrt n}\ ?\label{mimic CI} \end{equation} }
\subsection{Data-driven Distributionally Robust Optimization and Statistically ``Good" Uncertainty Sets}\label{sec:good set} To answer the above question, let us first revisit the common argument in the literature of data-driven DRO. To facilitate discussion, we call an uncertainty set $\mathcal U$ statistically ``good" if it allows \begin{equation} \liminf_{n\to\infty}P\left(Z_0(x)\leq\max_{P\in\mathcal U}E_P[h(x;\xi)]\text{\ \ for all\ \ }x\in\Theta\right)\geq1-\alpha\label{good guarantee} \end{equation} In contrast, a statistically ``best" uncertainty set in the sense of \eqref{main guarantee} sharpens the inequality in \eqref{good guarantee} to equality.
The data-driven DRO framework provides a general methodology in guaranteeing \eqref{good guarantee}. First, one calibrates an uncertainty set $\mathcal U$ from data so that it contains the true distribution with probability $1-\alpha$, namely $P(P_0\in\mathcal U)\geq1-\alpha$. Note that since $P_0\in\mathcal U$ implies that $Z_0(x)=E_0[h(x;\xi)]\leq\max_{P\in\mathcal U}E_P[h(x;\xi)]$ for all $x$, we have \begin{equation} P\left(Z_0(x)\leq\max_{P\in\mathcal U}E_P[h(x;\xi)]\text{\ \ for all\ \ }x\in\Theta\right)\geq P(P_0\in\mathcal U)\geq1-\alpha\label{good guarantee argument} \end{equation} Similarly, a set $\mathcal U$ constructed with the asymptotic property $\liminf_{n\to\infty}P(P_0\in\mathcal U)\geq1-\alpha$ guarantees that \eqref{good guarantee} holds and, in fact, so is the stronger guarantee $$\liminf_{n\to\infty}P\left(\min_{P\in\mathcal U}E_P[h(x;\xi)]\leq Z_0(x)\leq\max_{P\in\mathcal U}E_P[h(x;\xi)]\text{\ \ for all\ \ }x\in\Theta\right)\geq1-\alpha$$
Thus, good uncertainty sets can be readily created as confidence regions for $P_0$. Constructing these confidence regions and their tractability have been substantially investigated. A non-exhaustive list includes moment and deviation-type constraints (\cite{delage2010distributionally,goh2010distributionally,wiesemann2014distributionally}), Wasserstein balls (\cite{esfahani2015data,gao2016distributionally}), $\phi$-divergence balls (\cite{ben2013robust}), likelihood-based (\cite{Wang2015}) and goodness-of-fit-based regions (\cite{bertsimas2014robust}). Recently, \cite{gupta2015near} further investigates the smallest of such confidence regions as a baseline to measure the degree of conservativeness of a given uncertainty set.
\subsection{Our Contributions} Despite the availability of all the good uncertainty sets, finding the statistically best one in the sense of \eqref{main guarantee} has not been addressed in the literature. In this paper, we construct an uncertainty set that is close to the best (the meaning of ``close to" will be apparent in our later exposition) by leveraging one of the good sets, namely the Burg-entropy divergence ball.
Intriguingly, the way we construct these balls, and the associated statistical explanation, is completely orthogonal to the standard data-driven DRO framework discussed above. These balls are empirically defined (as we will explain in detail) and do not have any interpretation as confidence regions by themselves. In fact, they have low, or even zero, probability of covering the true distribution. Yet the resulting DRO has the best statistical performances among all DRO formulations. This disentanglement between set coverage and ultimate performance can be explained by a duality relation between our resulting DRO and the empirical likelihood theory, a connection that has been briefly discussed in a few previous work (e.g., \cite{Wang2015,lam2015quantifying}) but not been fully exploited as far as we know.
Importantly, through setting up such a connection, we study optimal calibration of the sizes of these sets by using a generalization of $\chi^2$-quantiles that involves the excursion of so-called $\chi^2$-processes. As a by-product, our proposed method also resolves some technical challenges reported in the previous literature in calibrating divergence balls (e.g., \cite{jiang2012data,esfahani2015data}). More precisely, since divergence is only properly defined between absolutely continuous distributions, it has been suggested, in the case of continuous distributions, that one needs to construct the ball using kernel estimation of density and the divergence, which is statistically challenging, or resorting to a parametric framework. The approach we take here, on the other hand, bypasses these issues.
To summarize, our main contributions of this paper are:\emph{ \begin{enumerate} \item We systematically build an uncertainty set that, in a precise sense, is close to recovering the guarantees \eqref{main guarantee} and \eqref{mimic CI} provided by the CLT. \item In doing so, we expand the view on the meaning of uncertainty sets beyond the notion of confidence regions, by showing that our empirical Burg-entropy divergence ball recovers the best guarantees despite being a low or zero-coverage set. This is achieved through connecting the dual of the resulting DRO with the empirical likelihood theory. \item To achieve our claimed guarantees, we study an approach to optimally calibrate the sizes of these balls using quantiles of $\chi^2$-process excursion. \item As a by-product, our approach resolves the technical difficulties in enforcing absolute continuity when calibrating divergence balls that are raised in previous works in data-driven DRO. \end{enumerate} }
Finally, while the viewpoint taken by this paper is primarily statistical, we mention that there are other valuable perspectives in the DRO literature motivated from risk or tractability considerations (see, e.g., the survey \cite{gabrel2014recent}); these are, however, beyond the scope of this work.
The rest of this paper is organized as follows. Section \ref{sec:empircal uncertainty set} motivates our proposed uncertainty sets. Section \ref{sec:theory} presents methods to calibrate their sizes and the theoretical explanation of their statistical performances. Section \ref{sec:numerics} shows results of our numerical experiments. Section \ref{sec:conlusion} concludes and discusses future directions. Section \ref{sec:proofs} provides all the proofs. Appendices \ref{sec:EP} and \ref{sec:thms} list some auxiliary concepts and theorems.
\section{Towards the Empirical DRO}\label{sec:empircal uncertainty set} We first review some background in divergence-based inference and how to use it to create confidence regions for probability distributions in Section \ref{sec:DI}. Through a preliminary numerical investigation in Section \ref{sec:initial}, we motivate and present, in Section \ref{sec:empirical divergence ball}, the \emph{empirical divergence ball} as our main tool.
\subsection{Divergence-based Inference and Confidence Regions}\label{sec:DI} A $\phi$-divergence ball is in the form \begin{equation} \mathcal U=\{P\in\mathcal P_Q:D_\phi(P,Q)\leq\eta\}\label{divergence} \end{equation} where $$D_\phi(P,Q)=\int\phi\left(\frac{dP}{dQ}\right)dQ$$ for some baseline distribution $Q$ and suitable function $\phi(\cdot)$, and $dP/dQ$ is the likelihood ratio given by the Radon-Nikodym derivative between $P$ and $Q$. The latter is well-defined only for $P$ within $\mathcal P_Q$, the set of all distributions absolutely continuous with respect to $Q$. The function $\phi:\mathbb R^+\to\mathbb R$ is convex and satisfies $\phi(1)=0$.
Suppose the random variable $\xi$ lies on a finite discrete support $\{s_1,\ldots,s_k\}$. One way to construct a statistically good divergence ball is as follows (\cite{ben2013robust}). Set the baseline distribution as the histogram of the i.i.d. data given by $\hat{\mathbf p}=(\hat p_i)_{i=1,\ldots,k}$, where $\hat p_i=n_i/n$, $n_i$ is the counts on support $s_i$, and $n$ is the total sample size. The divergence ball \eqref{divergence} can be written as \begin{align} \mathcal U&=\{\mathbf p\in\mathcal P_{\hat{\mathbf p}}:D_\phi(\mathbf p,\hat{\mathbf p})\leq\eta\}\notag\\ &=\left\{(p_1,\ldots,p_k):\sum_{i=1}^k\hat p_i\phi\left(\frac{p_i}{\hat p_i}\right)\leq\eta,\ \sum_{i=1}^kp_i=1,\ p_i\geq0\text{\ for all\ }i=1,\ldots,k\right\}\label{divergence discrete} \end{align} Under twice continuous differentiability condition on $\phi$, the theory of divergence-based inference (\cite{pardo2005statistical}) stipulates that $$\frac{2n}{\phi''(1)}D_\phi(\mathbf p,\hat{\mathbf p})\Rightarrow\chi^2_{k-1}\text{\ \ as $n\to\infty$}$$ where $\chi^2_{k-1}$ is the $\chi^2$-distribution with degree of freedom $k-1$, and ``$\Rightarrow$" denotes convergence in distribution. This implies that taking $\eta=\frac{\phi''(1)}{2n}\chi^2_{k-1,1-\alpha}$ in \eqref{divergence discrete}, where $\chi^2_{k-1,1-\alpha}$ is the $1-\alpha$ quantile of $\chi^2_{k-1}$, forms an uncertainty set $\mathcal U$ that contains the true distribution with probability asymptotically $1-\alpha$. This in turn implies that $\mathcal U$ is a good uncertainty set satisfying \eqref{good guarantee}.
For instance, $\phi(x)=(x-1)^2$ yields the $\chi^2$-distance, and setting $\eta$ at $\chi^2_{k-1,1-\alpha}/n$ results in the confidence region associated with the standard $\chi^2$ goodness-of-fit test for categorical data (\cite{agresti2011categorical}). On the other hand, $\phi(x)=-\log x+x-1$ yields the Burg-entropy (or the Kullback-Leibler) divergence (\cite{kullback1951information}), and $\eta$ in this case should be set at $\chi^2_{k-1,1-\alpha}/(2n)$. Since the Burg-entropy divergence is important in our subsequent discussion, for convenience, we denote its divergence ball as \begin{equation} \mathcal U_{Burg}=\left\{(p_1,\ldots,p_k):-\sum_{i=1}^k\hat p_i\log\frac{p_i}{\hat p_i}\leq\frac{\chi^2_{k-1,1-\alpha}}{2n},\ \sum_{i=1}^kp_i=1,\ p_i\geq0\text{\ for all\ }i=1,\ldots,k\right\}\label{Burg} \end{equation} From the discussion above, $\mathcal U_{Burg}$ is a good uncertainty set and moreover satisfies \begin{equation} \lim_{n\to\infty}P\left(P_0\in\mathcal U_{Burg}\right)=1-\alpha\label{Burg guarantee} \end{equation} for a finite discrete true distribution $P_0$.
The computational tractability of divergence balls has been studied in depth in \cite{ben2013robust}, who reformulate $\max_{P\in\mathcal U}E_P[h(x;\xi)]$ in terms of the conjugate function of $\phi$ and propose efficient optimization algorithms. Because of this we will not drill further on tractability and instead refer interested readers therein.
\subsection{An Initial Numerical Investigation on Coverage Accuracy}\label{sec:initial} To get a sense of the coverage performance provided by $\mathcal U_{Burg}$, we run an experiment on estimating $Z_0(x)=E_0[h(x;\xi)]$, where we set $h$ as \begin{equation} h(x;\xi)=-v\min(x,\xi)-s(x-\xi)^++l(\xi-x)^++cx+\rho\label{h example} \end{equation} with $v=10$, $s=5$, $l=4$, $c=3$, and $\rho=40$. This function $h$ is adapted from the example in Section 6.3 in \cite{ben2013robust}. As an application, \eqref{h example} can represent the loss amount in excess of the threshold $\rho$ for a newsvendor. In this case, $v$ is the selling price per unit, $s$ the salvage value per unit, $l$ the shortage cost per unit, $c$ the cost per unit, $\xi$ a random demand, and $x$ the quantity to order.
For now, let us fix the solution at $x=30$ (so it is purely about estimating $Z_0(30)$). We set the random variable $\xi$ as an exponential random variable with mean $20$ that is discretized uniformly over a $k$-grid on the interval $[0,50]$, or more precisely, \begin{equation} \begin{array}{l} P\left(\xi=\frac{50j}{k}\right)=P\left(\frac{50(j-1)}{k}<Exp\left(\frac{1}{20}\right)<\frac{50j}{k}\right)\text{\ \ for\ }j=1,\ldots,k-1\\ P\left(\xi=50\right)=P\left(Exp\left(\frac{1}{20}\right)>\frac{50(k-1)}{k}\right)\end{array}\label{exp specification} \end{equation} We repeat $1,000$ times: \begin{enumerate} \item Simulate $n$ i.i.d. data $\xi_1,\ldots,\xi_n$ from the $k$-discretized $Exp(1/20)$. \item Construct $\mathcal U_{Burg}$, and compute $\min_{\mathbf p\in\mathcal U_{Burg}}E_{\mathbf p}[h(x;\xi)]$ and $\max_{\mathbf p\in\mathcal U_{Burg}}E_{\mathbf p}[h(x;\xi)]$
with $\alpha=0.05$. \item Output $I\left(\min_{\mathbf p\in\mathcal U_{Burg}}E_{\mathbf p}[h(x;\xi)]\leq Z_0(x)\leq\max_{\mathbf p\in\mathcal U_{Burg}}E_{\mathbf p}[h(x;\xi)]\right)$, where $Z_0(x)$ is the true quantity calculable in closed-form, and $I(\cdot)$ is the indicator function. \end{enumerate} We then output the point estimate and the $95\%$ confidence interval (CI) of the coverage probability from the $1,000$ replications.
Step 2 above is carried out by using duality and numerically solving \begin{align*} \min_{\mathbf p\in\mathcal U_{Burg}}E_{\mathbf p}[h(x;\xi)]&=\max_{\lambda\geq0,\gamma}\sum_{i=1}^n\frac{\lambda}{n}\log\left(1-\frac{-h(\xi_i)+\gamma}{\lambda}\right)-\lambda\eta+\gamma\\ \max_{\mathbf p\in\mathcal U_{Burg}}E_{\mathbf p}[h(x;\xi)]&=\min_{\lambda\geq0,\gamma}-\sum_{i=1}^n\frac{\lambda}{n}\log\left(1-\frac{h(\xi_i)+\gamma}{\lambda}\right)+\lambda\eta-\gamma \end{align*} where $-0\log(1-t/0):=0$ for $t\leq0$ and $-0\log(1-t/0):=\infty$ for $t>0$ (see \cite{ben2013robust}).
Table \ref{table:DRO} shows the estimates of coverage probabilities for different support size $k$. The sample size for $\xi$ is $n=30$. The coverage probabilities are all greater than $95\%$, showing correct statistical guarantees. However, more noticeable is that they are all higher than $99\%$, and are consistently close to $100\%$ for $k=10$ or above, thus leading to severe over-coverage. Note that this phenomenon occurs despite that $\mathcal U_{Burg}$ has asymptotically exactly $1-\alpha$ probability of covering the true distribution as guaranteed in \eqref{Burg guarantee}.
\begin{table}[!htb]
\begin{subtable}{0.33\textwidth}
\centering
{\footnotesize\begin{tabular}{c|c|c}
$k$&Cover.&$95\%$ C.I. of\\
&Prob.&Cover. Prob.\\
\hline
5&99.6\%&(99.3\%, 99.9\%)\\ 10&100.0\%&(100.0\%, 100.0\%)\\ 15&100.0\%&(100.0\%, 100.0\%)\\ 20&100.0\%&(100.0\%, 100.0\%)
\end{tabular}}
\subcaption{DRO with Burg-ball of size $\chi^2_{k-1,0.95}/(2n)$}
\label{table:DRO}
\end{subtable}
\begin{subtable}{0.33\textwidth}
\centering
{\footnotesize\begin{tabular}{c|c|c}
$k$&Cover.&$95\%$ C.I. of\\
&Prob.&Cover. Prob.\\
\hline
5&94.5\%&(93.3\%, 95.7\%)\\ 10&95.0\%&(94.0\%, 96.2\%)\\ 15&95.3\%&(94.2\%, 96.4\%)\\ 20&94.8\%&(93.7\%, 96.0\%)
\end{tabular}}
\subcaption{Standard CLT}
\label{table:CLT}
\end{subtable}
\begin{subtable}{0.33\textwidth}
\centering
{\footnotesize\begin{tabular}{c|c|c}
$k$&Cover.&$95\%$ C.I. of\\
&Prob.&Cover. Prob.\\
\hline
5&94.1\%&(92.9\%, 95.3\%)\\ 10&94.4\%&(93.2\%, 95.6\%)\\ 15&95.4\%&(94.3\%, 96.5\%)\\ 20&95.3\%&(94.2\%, 96.4\%)
\end{tabular}}
\subcaption{DRO with Burg-ball of size $\chi^2_{1,0.95}/(2n)$}
\label{table:empirical}
\end{subtable}
\caption{Coverage probabilities for different methods and support sizes for discrete distributions}
\end{table}
As a comparison, we repeat the experiment, but this time checking the coverage of the standard $95\%$ CI generated from the CLT $$\left[\hat h(30)-z_{1-\alpha/2}\frac{\hat\sigma(30)}{\sqrt n},\hat h(30)+z_{1-\alpha/2}\frac{\hat\sigma(30)}{\sqrt n}\right]$$ where $\hat h(x)=\frac{1}{n}\sum_{i=1}^nh(x;\xi_i)$, $\hat\sigma^2(x)=\frac{1}{n-1}\sum_{i=1}^n(h(x;\xi_i)-\bar h)^2$, and $z_{1-\alpha/2}$ is the $(1-\alpha/2)$-quantile of standard normal distribution. Table \ref{table:CLT} shows that, unlike data-driven DRO, the coverage probabilities are now very close to $95\%$, regardless of the values of $k$. This result is, of course, as predicted by the CLT.
To investigate the source of inferiority in the data-driven DRO approach, we shall interpret the degree of freedom in the $\chi^2$-distribution from another angle: In maximum likelihood theory (\cite{cox1979theoretical}), the degree of freedom in the limiting $\chi^2$-distribution of the so-called log-likelihood ratio is equal to the number of effective parameters to be estimated. In our experiment, this number is \emph{one}, because we are only interested in estimating a single quantity $Z_0(30)$. Indeed, Table \ref{table:empirical} shows that the coverage probabilities of DRO, using the quantile of $\chi^2_1$ instead of $\chi^2_{k-1}$, are equally competitive as the CLT approach. This motivates us to propose our key definition of uncertainty set next.
\subsection{The Empirical Divergence Ball}\label{sec:empirical divergence ball} Given i.i.d. data $\xi_1,\ldots,\xi_n$, we define the \emph{empirical Burg-entropy divergence ball} as \begin{equation} \mathcal U_n(\eta)=\left\{\mathbf w=(w_1,\ldots,w_n):-\frac{1}{n}\sum_{i=1}^n\log(nw_i)\leq\eta,\ \sum_{i=1}^nw_i=1,\ w_i\geq0\text{\ for all\ }i=1,\ldots,n\right\}\label{empirical uncertainty} \end{equation} where $\mathbf w$ is the probability weight vector on the $n$ support points from data (some possibly with the same values). The set \eqref{empirical uncertainty} is well-defined whether the distribution of $\xi$ is discrete or continuous. It is a Burg-entropy divergence ball centered at the empirical distribution, with radius $\eta>0$, pretending that the support of the distribution is solely on the data. For convenience, we call the corresponding DRO over the empirical divergence ball as the \emph{empirical DRO}.
The discussion in Section \ref{sec:initial} suggests to put $\eta=\chi^2_{1,1-\alpha}/(2n)$. One intriguing observation is that $\mathcal U_n(\chi^2_{1,1-\alpha}/(2n))$ under-covers the true probability distribution. This can be seen by noting that, in the discrete case, $\mathcal U_n(\chi^2_{1,1-\alpha}/(2n))$ is equivalent to $\mathcal U_{Burg}$ except that $\chi^2_{k-1,1-\alpha}$ in its definition is replaced by $\chi^2_{1,1-\alpha}$ (see Proposition \ref{EL discrete} later). Since $\mathcal U_{Burg}$ is asymptotically exact in providing $1-\alpha$ coverage for the true distribution, and that $\chi^2_{1,1-\alpha}<\chi^2_{k-1,1-\alpha}$, $\mathcal U_n(\chi^2_{1,1-\alpha}/(2n))$ must be asymptotically under-covering. What is more, in the continuous case, the empirical distribution is singular with respect to the true distribution. Thus $\mathcal U_n(\chi^2_{1,1-\alpha}/(2n))$ has as low as \emph{zero} coverage. Clearly, the performance of the empirical uncertainty set cannot be reasoned using the standard data-driven DRO framework discussed in Section \ref{sec:good set}.
We close this section with Table \ref{table:cont}, which shows additional experimental results for the same example as above, this time the data being generated from the \emph{continuous} distribution $Exp(1/20)$. As we can see in Table \ref{table:empirical cont}, the coverages using $\mathcal U_n(\chi^2_{1,1-\alpha}/(2n))$ are maintained at close to $95\%$ when $n=40$ or above. As a comparison, Table \ref{table:CLT cont} shows that the standard CLT performs similarly as the empirical DRO (except that it tends to over-cover instead of under-cover when $n$ is small). Note that, unlike the discrete case, there is no well-defined choice of $k$ in this setting.
\begin{table}[!htb]
\begin{subtable}{0.5\textwidth}
\centering
\begin{tabular}{c|c|c}
$n$&Cover.&$95\%$ C.I. of\\
&Prob.&Cover. Prob.\\
\hline
20&91.9\%&(90.5\%, 93.3\%)\\ 30&92.8\%&(91.5\%, 94.1\%)\\
50&94.5\%&(93.3\%, 95.7\%)\\
80&94.4\%&(93.2\%, 95.6\%)
\end{tabular}
\subcaption{Empirical DRO with ball size $\chi^2_{1,0.95}/(2n)$}
\label{table:empirical cont}
\end{subtable}
\begin{subtable}{0.5\textwidth}
\centering
\begin{tabular}{c|c|c}
$n$&Cover.&$95\%$ C.I. of\\
&Prob.&Cover. Prob.\\
\hline
20&96.1\%&(95.1\%, 97.1\%)\\ 30&96.4\%&(95.4\%, 97.4\%)\\
50&94.3\%&(93.1\%, 95.5\%)\\
80&96.4\%&(95.4\%, 97.4\%)
\end{tabular}
\subcaption{Standard CLT}
\label{table:CLT cont}
\end{subtable}
\caption{Coverage probabilities for different methods and sample sizes for continuous distributions}
\label{table:cont}
\end{table}
\section{Statistical Guarantees}\label{sec:theory} We present our theoretical justification in two subsections. Section \ref{sec:EL} first connects the dual of the empirical DRO with the empirical likelihood (EL) method. Sections \ref{sec:excursion}, \ref{sec:PNLRP} and \ref{sec:Euler} elaborate this connection to develop the calibration method for the radius $\eta$ in the empirical divergence ball, via estimating the excursion of $\chi^2$-processes. We defer all proofs to Appendix \ref{sec:proofs}.
Throughout our exposition, ``$\Rightarrow$" denotes weak convergence (or convergence in distribution), ``a.s." abbreviates ``almost surely", and ``ev." abbreviates ``eventually".
\subsection{The Empirical Likelihood Method}\label{sec:EL} The EL method, first proposed by Owens (\cite{owen1988empirical,owen2001empirical}), can be viewed as a nonparametric counterpart of maximum likelihood theory. Given a set of i.i.d. data $\xi_1,\ldots,\xi_n$, one can view the empirical distribution, formed by putting probability weight $1/n$ on each data point, as a nonparametric maximum likelihood in the following sense. We define the nonparametric likelihood of any distributions supported on the data as \begin{equation} \prod_{i=1}^nw_i\label{nonparametric likelihood} \end{equation} where $\mathbf w=(w_1,\ldots,w_n)\in\mathcal P_n$ is any probability vector on $\{\xi_1,\ldots,\xi_n\}$. Then the likelihood of the empirical distribution, given by \begin{equation} \prod_{i=1}^n\frac{1}{n}\label{nonparametric MLE} \end{equation} maximizes \eqref{nonparametric likelihood}. This observation can be easily verified by a simple convexity argument. Moreover, \eqref{nonparametric MLE} still maximizes even if one considers other distributions that are not only supported on the data, since these distributions would have $\sum_{i=1}^nw_i<1$, making \eqref{nonparametric likelihood} even smaller.
The key of EL is a nonparametric analog of the celebrated Wilks' Theorem (\cite{cox1979theoretical}), stating the convergence of the so-called logarithmic likelihood ratio to $\chi^2$-distribution. In the EL framework, the nonparametric likelihood ratio is defined as the ratio between any nonparametric likelihood and the maximum likelihood, given by $$\prod_{i=1}^n\frac{w_i}{1/n}=\prod_{i=1}^n(nw_i)$$
To carry out inference we need to specify a quantity of interest to be estimated. Suppose we are interested in estimating $\mu_0=E_0[g(\xi)]$ for some function $g(\cdot)$, where $E_0[\cdot]$ is the expectation with respect to the true distribution generating the data (and similarly, $Var_0(\cdot)$ denotes its variance). The EL method utilizes the \emph{profile nonparametric likelihood ratio} \begin{equation} R(\mu)=\max\left\{\prod_{i=1}^n(nw_i):\sum_{i=1}^ng(\xi_i)w_i=\mu,\ \sum_{i=1}^nw_i=1,\ w_i\geq0\text{\ for all\ }i=1,\ldots,n\right\}\label{profile LR} \end{equation} where the likelihood ratios are ``profiled" according to the value of $\sum_{i=1}^ng(\xi_i)w_i$. With this definition, we have:
\begin{theorem}[The Empirical Likelihood Theorem; \cite{owen1988empirical}] Let $\xi_1,\ldots,\xi_n\in\Xi$ be i.i.d. data under $P_0$. Let $\mu_0=E_0[g(\xi)]<\infty$, and assume that $0<Var_0(g(\xi))<\infty$. Then \begin{equation} -2\log R(\mu_0)\Rightarrow\chi^2_1\text{\ \ as $n\to\infty$}\label{Wilks EL} \end{equation} where $-2\log R(\mu_0)$ is defined as $\infty$ if there is no feasible solution in defining $R(\mu_0)$ in \eqref{profile LR}.\label{ELT} \end{theorem} The degree of freedom 1 in the limiting $\chi^2$-distribution in \eqref{Wilks EL} counts the number of effective parameters, which is only $\mu_0$ in this case.
Phrasing in terms of our problem setup, we define \begin{equation} R(x;Z)=\max\left\{\prod_{i=1}^n(nw_i):\sum_{i=1}^nh(x;\xi_i)w_i=Z(x),\ \sum_{i=1}^nw_i=1,\ w_i\geq0\text{\ for\ }i=1,\ldots,n\right\}\label{EL process1} \end{equation} and hence \begin{eqnarray*} &&-2\log R(x;Z)\\ &=&\min\left\{-2\sum_{i=1}^n\log(nw_i):\sum_{i=1}^nh(x;\xi_i)w_i=Z(x),\ \sum_{i=1}^nw_i=1,\ w_i\geq0\text{\ for all\ }i=1,\ldots,n\right\} \end{eqnarray*} From Theorem \ref{ELT}, we conclude $P(-2\log R(x;Z_0)\leq\chi^2_{1,1-\alpha})\to1-\alpha$ as $n\to\infty$ for a fixed $x$. The important implication of Theorem \ref{ELT} arises from a duality relation between $-2\log R(x;Z_0)$ and the optimal values of the empirical DRO, in the sense that $-2\log R(x;Z_0)\leq\kappa$ if and only if $$\min_{\mathbf w\in\mathcal U_n(\kappa/(2n))}\sum_{i=1}^nh(x;\xi_i)w_i\leq Z_0(x)\leq\max_{\mathbf w\in\mathcal U_n(\kappa/(2n))}\sum_{i=1}^nh(x;\xi_i)w_i$$ where $\mathcal U_n(\eta)$ is the empirical divergence ball defined in \eqref{empirical uncertainty}. This implies: \begin{theorem} Fix $x\in\Theta$, and let $\xi_1,\ldots,\xi_n\in\Xi$ be i.i.d. data under $P_0$. Assume that $0<Var_0(h(x;\xi))<\infty$, and $Z_0(x)=E_0[h(x;\xi)]<\infty$. We have \begin{equation} \lim_{n\to\infty}P\left(\underline Z_n(x)\leq Z_0(x)\leq\overline Z_n(x)\right)=1-\alpha\label{exact asymptotic} \end{equation} where \begin{align} \underline Z_n(x)&=\min_{\mathbf w\in\mathcal U_n\left(\chi^2_{1,1-\alpha}/(2n)\right)}\sum_{i=1}^nh(x;\xi_i)w_i\label{min single}\\ \overline Z_n(x)&=\max_{\mathbf w\in\mathcal U_n\left(\chi^2_{1,1-\alpha}/(2n)\right)}\sum_{i=1}^nh(x;\xi_i)w_i\label{max single} \end{align}
\label{EL basic} \end{theorem}
Next, we argue that, in the discrete case, the empirical DRO given by $\underline Z_n(x)$ and $\overline Z_n(x)$ reduces to the standard divergence-based DRO given by $\max/\min_{\mathbf p\in\mathcal U_{Burg}}E_{\mathbf p}[h(x;\xi)]$, except that the degree of freedom in the $\chi^2$-quantile is replaced by 1. This explains the experimental results in Section \ref{sec:initial}. \begin{proposition} Fix $x\in\Theta$. When $\xi$ is discrete on the support set $\{s_1,\ldots,s_k\}$, $\underline Z_n(x)$ and $\overline Z_n(x)$ defined in \eqref{min single} and \eqref{max single} are equal to $\min_{\mathbf p\in\mathcal U_{Burg}'}E_{\mathbf p}[h(x;\xi)]$ and $\max_{\mathbf p\in\mathcal U_{Burg}'}E_{\mathbf p}[h(x;\xi)]$ respectively, where \begin{equation} \mathcal U_{Burg}'=\left\{(p_1,\ldots,p_k):-\sum_{i=1}^k\hat p_i\log\frac{p_i}{\hat p_i}\leq\frac{\chi^2_{1,1-\alpha}}{2n},\ \sum_{i=1}^kp_i=1,\ p_i\geq0\text{\ for all\ }i=1,\ldots,k\right\}\label{Burg EL} \end{equation} and $\hat p_i=n_i/n$, the proportion of data falling onto $s_i$.\label{EL discrete} \end{proposition}
We complement Theorem \ref{EL basic} with a consistency result: \begin{theorem} Under the same conditions in Theorem \ref{EL basic}, for any fixed $x\in\Theta$, both $\underline Z_n(x)\stackrel{a.s.}{\to}Z_0(x)$ and $\overline Z_n(x)\stackrel{a.s.}{\to}Z_0(x)$ as $n\to\infty$.\label{consistency simple} \end{theorem}
We note that, in the data-driven DRO framework, if $\xi$ is continuous, the absolute continuity condition requires a divergence ball to center at a continuous distribution to have any chance of containing the true distribution. This observation has been pointed out by several authors (e.g., \cite{jiang2012data,esfahani2015data}) and forces the use of kernel density estimators to set the baseline. Unless one assumes a parametric framework, calibrating the ball radius requires nonparametric divergence estimation, which involves challenging statistical analyses on bandwidth tuning and loss of estimation efficiency (e.g., \cite{moon2014multivariate,nguyen2007estimating,pal2010estimation}). The empirical DRO based on the EL framework cleanly bypasses these issues.
Our discussion in this subsection is also related to likelihood robust optimization studied in \cite{Wang2015}, which also discusses EL as well as other connections such as Bayesian statistics. \cite{Wang2015} focuses on finite discrete distributions. The work \cite{lam2015quantifying} also investigates EL, among other techniques like the bootstrap, in constructing confidence bounds for the optimal values of stochastic programs. However, none of these formalizes the connection, or more precisely, the \emph{disconnection} between set coverage and the statistical performance of DRO. As our next subsection shows, this formalization is important in capturing a statistical price to attain our best guarantee in \eqref{main guarantee}. This will be our focus next.
\subsection{Asymptotically Exact Coverage via $\chi^2$-Process Excursion}\label{sec:excursion} The discussion so far presumes a fixed $x\in\Theta$. Recall in Section \ref{sec:good set} that, in data-driven DRO, a confidence region given by $\mathcal U$ guarantees $Z_0(x)\leq\max_{P\in\mathcal U}E_P[h(x;\xi)]$ with at least the same confidence level thanks to \eqref{good guarantee argument}. This guarantee holds regardless of a fixed $x$ or uniformly over all $x\in\Theta$. This is because the construction of such confidence regions is completely segregated from the expected value constraint of interest.
In contrast, the statistical performance of our empirical divergence ball is highly coupled with $h$, since $E_0[h(x;\xi)]$ can be viewed as the parameter we want to estimate in the EL method. Consequently, the reasoning for Theorem \ref{EL basic} only applies to situations where $x$ is fixed, and the empirical divergence ball constructed there is not big enough to guarantee \eqref{main guarantee}, which requires a bound simultaneous for all $x\in\Theta$.
The main result in this section is to explain and to show how, depending on the ``complexity" of $h$, one can suitably inflate the size of the ball to match a statistical performance close to \eqref{main guarantee}.
We begin our discussion by imposing the following assumptions: \begin{assumption}[Finite mean] $Z_0(x)=E_0[h(x;\xi)]<\infty$ for all $x\in\Theta$.\label{finite mean} \end{assumption}
\begin{assumption}[Non-degeneracy]
$\inf_{x\in\Theta}E_0|h(x;\xi)-Z_0(x)|>0$.\label{nondegeneracy} \end{assumption}
\begin{assumption}[$L_2$-boundedness]
$E_0\sup_{x\in\Theta}|h(x;\xi)-Z_0(x)|^2<\infty$\label{L2} \end{assumption}
\begin{assumption}[Function complexity] The collection of functions
\begin{equation}
\mathcal H_\Theta=\{h(x;\cdot):\Xi\to\mathbb R|x\in\Theta\}\label{function class}
\end{equation}
is a $P_0$-Donsker class.\label{complexity} \end{assumption}
The first three assumptions are mild moment conditions on the quantity $h(x;\xi)$. The last assumption, the so-called Donsker condition, means that the function class $\mathcal H_\Theta$ is ``simple" enough to allow the associated empirical process indexed by $\mathcal H_\Theta$ to converge weakly to a Brownian bridge (see Definition \ref{def:Donsker} in Appendix \ref{sec:EP}).
The following theorem precisely describes the radius of the empirical divergence ball needed to attain the best guarantee in \eqref{main guarantee}: \begin{theorem}[Optimal Calibration of Empirical Divergence Ball] Let $\xi_1,\ldots,\xi_n\in\Xi$ be i.i.d. data under $P_0$. Suppose Assumptions \ref{finite mean}, \ref{nondegeneracy}, \ref{L2} and \ref{complexity} hold. Let $q_n$ be the $(1-\alpha)$-quantile of $\sup_{x\in\Theta}J_n(x)$, i.e. \begin{equation} P_{\bm\xi}\left(\sup_{x\in\Theta}J_n(x)\geq q_n\right)=\alpha\label{calibration} \end{equation} where $J_n(x)=G_n(x)^2$ and $G_n(\cdot)$ is a Gaussian process indexed by $\Theta$ that is centered, i.e. mean zero, with covariance \begin{equation} Cov(G_n(x_1),G_n(x_2))=\frac{\sum_{i=1}^n(h(x_1;\xi_i)-\hat h(x_1))(h(x_2;\xi_i)-\hat h(x_2))}{\sqrt{\sum_{i=1}^n(h(x_1;\xi_i)-\hat h(x_1))^2\sum_{i=1}^n(h(x_2;\xi_i)-\hat h(x_2))^2}}\label{cov} \end{equation} for any $x_1,x_2\in\Theta$, and $\hat h(x)=(1/n)\sum_{i=1}^nh(x;\xi_i)$ is the sample mean of $h(x;\xi_i)$'s. $P_{\bm\xi}$ denotes the probability conditional on the data $\xi_1,\ldots,\xi_n$.
We have \begin{equation} \lim_{n\to\infty}P(\underline Z_n^*(x)\leq Z_0(x)\leq\overline Z_n^*(x)\text{\ \ for all\ \ }x\in\Theta)=1-\alpha\label{asymptotic guarantee new} \end{equation}
where \begin{align*} \underline Z_n^*(x)&=\min_{\mathbf w\in\mathcal U_n(q_n/(2n))}\sum_{i=1}^nh(x;\xi_i)w_i\\ \overline Z_n^*(x)&=\max_{\mathbf w\in\mathcal U_n(q_n/(2n))}\sum_{i=1}^nh(x;\xi_i)w_i \end{align*} \label{main} \end{theorem}
Note that, other than being a two-sided bound instead of one-sided, the guarantee \eqref{asymptotic guarantee new} is precisely \eqref{main guarantee}.
The process $J_n(\cdot)$, as the square of a Gaussian process, is known as a $\chi^2$-process (or $\chi^2$ random field; e.g., \cite{adler2009random}). Its covariance structure can be expressed explicitly in terms of the function $h$ and the data. The quantity $P_{\bm\xi}\left(\sup_{x\in\Theta}J_n(x)\geq u\right)$ is the excursion probability of $J_n(\cdot)$ above $u$. Note that we have ignored some subtle measurability issues in stating our result. To avoid unnecessary diversion, we will stay silent on measurability throughout the paper and refer the reader to \cite{van1996weak} for detailed treatments.
We observe some immediate connection of $\sup_xJ_n(x)$ to the $\chi^2_1$-distribution used in Theorem \ref{EL basic}. In addition to the fact that the marginal distribution of $J_n(x)$ at any $x$ is a $\chi^2_1$-distribution, we also have, by the Borell-TIS inequality (\cite{adler1990introduction}), that the asymptotic tail probability of $\sup_xJ_n(x)$ has the same exponential decay rate as that of $\chi^2_1$, i.e. $$\frac{\log P\left(\sup_{x\in\Theta}G^2(x)\geq\nu\right)}{\log P(Y\geq\nu)}\to1$$ as $\nu\to\infty$, where $Y$ is a $\chi^2_1$ random variable. This suggests a relatively small overhead in using $q_n$ instead of $\chi^2_{1,1-\alpha}$ in calibrating the empirical ball when $\alpha$ is small.
Nevertheless, Theorem \ref{main} offers some insights beyond Theorem \ref{EL basic}. First, it requires the Donsker condition on the class $\mathcal H_\Theta$. One sufficient condition of $P_0$-Donsker is: \begin{lemma} Suppose that $Z_0(x)=E_0[h(x;\xi)]<\infty$ and $Var_0(h(x;\xi))<\infty$ for all $x\in\Theta$. Also assume that there exists a random variable $M$ with $E_0M^2<\infty$ such that
$$|h(x_1;\xi)-h(x_2;\xi)|\leq M\|x_1-x_2\|_2$$ a.s. for all $x_1,x_2\in\Theta$. Then $\mathcal H_\Theta$ as defined in \eqref{function class} is $P_0$-Donsker.\label{Donsker sufficiency} \end{lemma} Lemma \ref{Donsker sufficiency} is a consequence of the Jain-Marcus Theorem (e.g., \cite{van1996weak}, Example 2.11.13). It is worth noting that the condition in Lemma \ref{Donsker sufficiency} is also a standard sufficient condition in guaranteeing the central limit convergence for SAA (\cite{shapiro2014lectures}, Theorem 5.7). This is not a coincidence, as the machinery behind Theorem \ref{main} involves an underpinning CLT, much like in the convergence analysis of SAA.
Secondly, even though $q_n\approx\chi^2_{1,1-\alpha}$ when $\alpha\approx0$, $q_n$ is strictly larger than $\chi^2_{1,1-\alpha}$ since $\sup_{x\in\Theta}J_n(x)$ stochastically dominates $\chi^2_1$ (unless degeneracy occurs). Thus the ball constructed in Theorem \ref{main} is always bigger than that in Theorem \ref{EL basic}. One way to estimate this inflation is by approximating the excursion probability of $\chi^2$-process using the theory of random geometry. We delegate this discussion to Section \ref{sec:Euler}. For now, we will delve into more details underlying Theorem \ref{main} and other properties of the empirical DRO.
\subsection{The Profile Nonparametric Likelihood Ratio Process and Other Properties of the Empirical DRO}\label{sec:PNLRP} We explain briefly the machinery leading to Theorem \ref{main}, leaving the details to Appendix \ref{sec:proofs}. Our starting point is to define the profile nonparametric likelihood ratio in \eqref{EL process1} at the process level \begin{equation} \{R(x;Z):x\in\Theta\}\label{EL process} \end{equation} We call \eqref{EL process} the \emph{profile nonparametric likelihood ratio process} indexed by $x\in\Theta$.
Denote the space \begin{equation}
\ell^\infty(\Theta)=\left\{y:\Theta\to\mathbb R\Bigg|\|y\|_\Theta<\infty\right\}\label{topology} \end{equation}
where we define $\|y\|_\Theta=\sup_{x\in\Theta}|y(x)|$ for any function $y:\Theta\to\mathbb R$. We have a convergence theorem for $R(x;Z)$ uniformly over $x\in\Theta$, in the following sense:
\begin{theorem}[Limit Theorem of the Profile Nonparametric Likelihood Ratio Process]
Under Assumptions \ref{finite mean}, \ref{nondegeneracy}, \ref{L2} and \ref{complexity}, the profile likelihood ratio process defined in \eqref{EL process}
satisfies $$-2\log R(\cdot;Z_0)\Rightarrow J(\cdot)\text{\ \ in\ \ }\ell^\infty(\Theta)$$ where $J(x)=G(x)^2$ and $G(\cdot)$ is a Gaussian process indexed by $x\in\Theta$ that has mean zero and covariance $$Cov(G(x_1),G(x_2))=\frac{Cov_0(h(x_1;\xi),h(x_2;\xi))}{\sqrt{Var_0(h(x_1;\xi))Var_0(h(x_2;\xi))}}$$ for any $x_1,x_2\in\Theta$. \label{EL process thm} \end{theorem}
Theorem \ref{EL process thm} is the empirical-process generalization of Theorem \ref{EL basic}. It implies that $P\left(\sup_{x\in\Theta}\{-2\log R(x;Z_0)\}\leq q^*\right)\to1-\alpha$ for $q^*$ selected such that $P\left(\sup_{x\in\Theta}J(x)\leq q^*\right)=1-\alpha$. By a duality-type argument similar to that in Section \ref{sec:EL}, we have $-2\log R(x;Z_0)\leq q^*$ for all $x\in\Theta$, if and only if $\min_{\mathbf w\in\mathcal U_n(q^*/(2n))}\sum_{i=1}^nh(x;\xi_i)w_i\leq Z_0(x)\leq\max_{\mathbf w\in\mathcal U_n(q^*/(2n))}\sum_{i=1}^nh(x;\xi_i)w_i$ for all $x\in\Theta$, which implies $\lim_{n\to\infty}P\left(\min_{\mathbf w\in\mathcal U_n(q^*/(2n))}\sum_{i=1}^nh(x;\xi_i)w_i\leq Z_0(x)\leq\max_{\mathbf w\in\mathcal U_n(q^*/(2n))}\sum_{i=1}^nh(x;\xi_i)w_i\text{\ for all\ }x\in\Theta\right)=1-\alpha$ for the same choice of $q^*$. However, since $J$ relies on information about the unknown true distribution $P_0$, $q^*$ is unknown. The following result closes the gap by arguing that $J$ can be ``plugged-in" by $J_n$, and consequently $q^*$ by $q_n$ as depicted in Theorem \ref{main}: \begin{lemma} Under Assumptions \ref{finite mean}, \ref{nondegeneracy}, \ref{L2} and \ref{complexity}, conditional on almost every data realization $(P_n:n\geq1)$, $$G_n(\cdot)\Rightarrow G(\cdot)\text{\ \ in\ \ }\ell^\infty(\Theta)$$ where $G_n(\cdot)$ and $G(\cdot)$ are defined in Theorems \ref{main} and \ref{EL process thm} respectively.\label{sample Gaussian} \end{lemma}
Theorem \ref{main} can then be proved by combining Theorem \ref{EL process thm} and Lemma \ref{sample Gaussian}. Moreover, consistency of the empirical DRO also holds uniformly over $x\in\Theta$: \begin{theorem}[Uniform Strong Consistency] Under Assumptions \ref{finite mean}, \ref{nondegeneracy}, \ref{L2} and \ref{complexity}, \begin{align*}
\sup_{x\in\Theta}|\underline Z_n^*(x)-Z_0(x)|\stackrel{a.s.}{\to}0\\
\sup_{x\in\Theta}|\overline Z_n^*(x)-Z_0(x)|\stackrel{a.s.}{\to}0 \end{align*} as $n\to\infty$. \label{consistency} \end{theorem}
Lastly, the following theorem highlights that the width of the confidence band $[\underline Z_n^*(x),\overline Z_n^*(x)]$ varies with the standard deviation at each $x$: \begin{theorem}[Pertaining to the Variability at Each Decision Point] Suppose Assumptions \ref{finite mean}, \ref{nondegeneracy}, \ref{L2} and \ref{complexity} hold. Additionally, suppose that $h(\cdot;\cdot)$ is bounded. Then $$\underline Z_n^*(x)=\hat h(x)-\sqrt{q_n}\frac{\hat\sigma(x)}{\sqrt n}+O\left(\frac{1}{n}\right)$$ $$\overline Z_n^*(x)=\hat h(x)+\sqrt{q_n}\frac{\hat\sigma(x)}{\sqrt n}+O\left(\frac{1}{n}\right)$$ uniformly over $x\in\Theta$ a.s.. Here $\hat h(x)=\frac{1}{n}\sum_{i=1}^nh(x;\xi_i)$ is the sample mean, $\hat\sigma^2(x)=\frac{1}{n}\sum_{i=1}^n(h(x;\xi_i)-\hat h(x))^2$ is the sample variance at each $x$, and $q_n$ is defined in Theorem \ref{main}.\label{asymptotic equivalence} \end{theorem} Theorem \ref{asymptotic equivalence} gives rise to \eqref{mimic CI}. In particular, $\sqrt{q_n}$ is analogous to the critical value in a confidence band. In summary, Theorems \ref{main} and \ref{asymptotic equivalence} show that our empirical divergence ball $\mathcal U_n(q_n/(2n))$, calibrated via the quantile of $\chi^2$-process excursion $q_n$, satisfies our benchmark guarantees \eqref{main guarantee} and \eqref{mimic CI}, except that it provides a two-sided bound instead of one-sided. The difference of two- versus one-sided bound is the reason we have claimed ``close to" the best in Section \ref{sec:good set}.
\subsection{Approximating the Quantile of $\chi^2$-process Excursion}\label{sec:Euler} We discuss how to estimate $q_n$ in Theorem \ref{main}. One approach is to approximate the excursion probability of $\chi^2$-process by the mean Euler characteristic approximation (e.g., \cite{adler2009random}, Theorem 13.4.1 and Section 15.10.2, and \cite{adler2011topological}, Theorem 4.8.1):
\begin{equation} P\left(\sup_{x\in\Theta}J_n(x)\geq u\right)\approx\sum_{j=0}^m(2\pi)^{-j/2}\mathcal L_j(\Theta)\mathcal M_j(u)\label{Euler} \end{equation}
Here $m$ is the dimension of the decision space $\Theta\subset\mathbb R^m$. The coefficients $\mathcal L_j(\Theta)$ on the RHS of \eqref{Euler} are known as the Lipschitz-Killing curvatures of the domain $\Theta$, which measure the ``intrinsic volumes" of the domain $\Theta$ using the Riemannian metric induced by the Gaussian process $G_n$ (\cite{adler2009random}, equation (12.2.2)). In particular, the highest-dimensional coefficient is given by $$\mathcal L_m(\Theta)=\int_\Theta det(\Lambda(x))^{1/2}dx$$ (\cite{adler2009random}, equation (12.2.22), and \cite{adler2011topological}, equation (5.4.1)) where $\Lambda(x)=(\Lambda_{ij}(x))_{i,j=1,\ldots,m}\in\mathbb R^{m\times m}$, and \begin{equation}
\Lambda_{ij}(x)=Cov\left(\frac{\partial G_n(x)}{\partial x_i},\frac{\partial G_n(x)}{\partial x_j}\right)=\frac{\partial^2}{\partial y_i\partial z_j}Cov(G_n(y),G_n(z))\Big|_{y=x,z=x}\label{general moment} \end{equation} for differentiable $G_n$ (in the $L^2$ sense), with $x_i$ and $x_j$ the $i$ and $j$-th components of $x$.
Thus \eqref{general moment} can be evaluated by differentiating \eqref{cov}. Lower-dimensional coefficients can be evaluated by integration over lower-dimensional surfaces of $\Theta$, and $\mathcal L_0(\Theta)=1$.
On the other hand, the quantities $\mathcal M_j(u)$'s are the Gaussian Minkowski functionals for the excursion set, independent of $\Theta$ and $h$, and are given by
$$\mathcal M_j(u)=(-1)^j\frac{d^j}{dy^j}P(Y\geq y)\Big|_{y=\sqrt u}$$ where $Y$ is the square root of a $\chi^2_1$ random variable. Thus, for instance, $\mathcal M_0(u)=P(\chi^2_1\geq u)$, and $\mathcal M_1(u)=2\phi(\sqrt u)$ where $\phi(\cdot)$ is the standard normal density.
\eqref{Euler} is a very accurate approximation for $P(\sup_{x\in\Theta}J_n(x)\geq u)$ in the sense
$$\left|P\left(\sup_{x\in\Theta}J_n(x)\geq u\right)-\sum_{j=0}^m(2\pi)^{-j/2}\mathcal L_j(\Theta)\mathcal M_j(u)\right|\leq Ce^{-\beta u/2}$$ where $C>0$ and $\beta>1$ (\cite{atw16}, Section 5.3.2). In other words, the approximation error is exponentially smaller than all the terms in \eqref{Euler} as $u$ increases. In practice, however, the formula for $\mathcal L_j(\Theta)$ could get increasingly complex as $j$ decreases, in which case only the first and the second highest order coefficients of $\mathcal L_j(\Theta)$ are used.
In light of the above, an accurate approximation of $q_n$ can be found by solving the root of $$\sum_{j=0}^m(2\pi)^{-j/2}\mathcal L_j(\Theta)\mathcal M_j(u)=\alpha$$
As an explicit illustration, when $m=1$, and $h$ is twice differentiable almost everywhere, we have
$$P\left(\sup_{x\in\Theta}J_n(x)\geq u\right)=P(\chi^2_1\geq u)+\int_\Theta \sqrt{\frac{\partial^2}{\partial y\partial z}Cov(G_n(y),G_n(z))\Big|_{y=x,z=x}}dx\ \frac{e^{-u/2}}{\pi}+O(e^{-\beta u/2})$$ for some $\beta>1$. An approximate $q_n$ can then be found by solving the root of \begin{equation}
P(\chi^2_1\geq u)+\int_\Theta \sqrt{\frac{\partial^2}{\partial y\partial z}Cov(G_n(y),G_n(z))\Big|_{y=x,z=x}}dx\ \frac{e^{-u/2}}{\pi}=\alpha\label{calibrate inflated} \end{equation}
\section{Numerical Illustrations for the Empirical DRO}\label{sec:numerics} This section shows some numerical results on the statistical performance of empirical DRO. We use the newsvendor loss function in \eqref{h example} as our $h$. We repeat $1,000$ times: \begin{enumerate} \item Simulate $n$ i.i.d. data $\xi_1,\ldots,\xi_n$ from the $k$-discretized $Exp(1/20)$. \item Estimate $q_n$ using \eqref{calibrate inflated}, and compute $\underline Z_n^*(x)=\min_{\mathbf w\in\mathcal U_n(q_n/(2n))}\sum_{i=1}^nh(x;\xi_i)w_i$ and $\overline Z_n^*(x)=\max_{\mathbf w\in\mathcal U_n(q_n/(2n))}\sum_{i=1}^nh(x;\xi_i)w_i$, with $\alpha$ set to be $0.05$. \item Output $$I\left(\underline Z_n^*(x)\leq Z_0(x)\leq\overline Z_n^*(x)\text{\ \ for\ }x=\frac{50j}{20},\ j=1,\ldots,20\right)$$ and $$I\left(Z_0(x)\leq\overline Z_n^*(x)\text{\ \ for\ }x=\frac{50j}{20},\ j=1,\ldots,20\right)$$ where $Z_0(x)$ is the true function of interest that is calculable in closed-form. \end{enumerate} We then output the point estimates and the $95\%$ CIs of the two- and one-sided coverage probabilities, i.e. $P\left(\underline Z_n^*(x)\leq Z_0(x)\leq\overline Z_n^*(x)\text{\ \ for\ }x=\frac{50j}{20},\ j=1,\ldots,20\right)$ and $P\left(Z_0(x)\leq\overline Z_n^*(x)\text{\ \ for\ }x=\frac{50j}{20},\ j=1,\ldots,20\right)$, from the $1,000$ replications. These probabilities serve as proxies for the probabilities $P\left(\underline Z_n^*(x)\leq Z_0(x)\leq\overline Z_n^*(x)\text{\ \ for\ }x\in\Theta\right)$ and $P\left(Z_0(x)\leq\overline Z_n^*(x)\text{\ \ for\ }x\in\Theta\right)$ respectively.
We first set $\xi$ as a $k$-discretized $Exp(1/20)$ as in \eqref{exp specification}. For comparison, we also repeat the above experiment but using $\chi^2_{k-1,0.95}$ and $\chi^2_{1,0.95}$ in place of $q_n$. Table \ref{table:comparison discrete n} shows the results of two-sided coverage probabilities as we vary the sample size from $n=20$ to $80$. The coverage probabilities appear to be stable already starting at $n=20$. As we can see, the coverages using the $\chi^2_{k-1,0.95}$ calibration (Table \ref{table:DRO simultaneous n}) are around $99\%$, much higher than $95\%$, as $k-1$ is over-determining the number of parameters we want to estimate from the EL perspective. The coverage probabilities using the $\chi^2_{1,0.95}$ calibration (Table \ref{table:CLT simultaneous n}), on the other hand, are in the range $86\%$ to $87\%$, significantly lower than $95\%$, since it does not account for simultaneous estimation errors. Lastly, the coverage probabilities using the $\chi^2$-process excursion (Table \ref{table:empirical simultaneous n}) are very close to $95\%$ in all cases, thus confirming the superiority of our approach.
\begin{table}[!htb]
\begin{subtable}{0.33\textwidth}
\centering
{\footnotesize\begin{tabular}{c|c|c}
$n$&2-sided&$95\%$ C.I. of\\
&Cover. Prob.&Cover. Prob.\\
\hline
20&98.3\%&(98.0\%, 98.6\%)\\ 30&98.8\%&(98.7\%, 100.0\%)\\ 40&98.9\%&(98.8\%, 100.0\%)\\ 50&98.8\%&(98.7\%, 100.0\%)\\ 60&98.9\%&(98.8\%, 100.0\%)\\ 80&98.8\%&(98.6\%, 98.9\%)
\end{tabular}}
\subcaption{$\chi^2_{k-1,0.95}/(2n)$}
\label{table:DRO simultaneous n}
\end{subtable}
\begin{subtable}{0.33\textwidth}
\centering
{\footnotesize\begin{tabular}{c|c|c}
$n$&2-sided&$95\%$ C.I. of\\
&Cover. Prob.&Cover. Prob.\\
\hline 20&85.6\%&(85.0\%, 86.2\%)\\ 30&85.9\%&(85.4\%, 86.5\%)\\ 40&86.6\%&(86.1\%, 87.1\%)\\ 50&86.6\%&(86.1\%, 87.1\%)\\ 60&86.1\%&(85.5\%, 86.7\%)\\ 80&86.8\%&(86.3\%, 87.3\%)
\end{tabular}}
\subcaption{$\chi^2_{1,0.95}/(2n)$}
\label{table:CLT simultaneous n}
\end{subtable}
\begin{subtable}{0.33\textwidth}
\centering
{\footnotesize\begin{tabular}{c|c|c}
$n$&2-sided&$95\%$ C.I. of\\
&Cover. Prob.&Cover. Prob.\\
\hline 20&94.4\%&(93.8\%, 95.0\%)\\ 30&94.6\%&(94.0\%, 95.2\%)\\ 40&94.7\%&(94.2\%, 95.3\%)\\ 50&94.7\%&(94.2\%, 95.3\%)\\ 60&94.4\%&(93.8\%, 94.9\%)\\ 80&95.0\%&(94.5\%, 95.5\%)
\end{tabular}}
\subcaption{Approximate $95\%$-quantile of $\sup_xJ_n(x)$}
\label{table:empirical simultaneous n}
\end{subtable}
\caption{Two-sided coverage probabilities for different Burg-divergence ball sizes and sample sizes for a discrete distribution with $k=5$} \label{table:comparison discrete n}
\end{table}
Next, Table \ref{table:comparison discrete n 1-sided} shows the results for one-sided coverage instead of two-sided. These one-sided coverage probabilities are slightly higher than the two-sided counterparts as the coverage condition is now more relaxed. Nonetheless, the magnitudes of these changes are very small compared to the effects brought by the choice of calibration methods. In particular, using $\chi^2_{k-1,0.95}$ appears to be severely over-covering at about $99\%$ to $100\%$, while using $\chi^2_{1,0.95}$ gives under-coverage at about $89\%$ to $91\%$. Using the $\chi^2$-process excursion shows $95\%$ to $96\%$ coverage performances, thus significantly better than the other two methods. These show that, even though our statistical guarantees in Theorem \ref{main} are two-sided, the loss of inaccuracy for one-sided coverage is very minor compared to the improvement in the calibration method used. This experimentally justifies our claim of ``close to" the best at the end of Section \ref{sec:PNLRP}.
\begin{table}[!htb]
\begin{subtable}{0.33\textwidth}
\centering
{\footnotesize\begin{tabular}{c|c|c}
$n$&1-sided&$95\%$ C.I. of\\
&Cover. Prob.&Cover. Prob.\\
\hline
20&98.6\%&(98.1\%, 99.2\%)\\ 30&99.6\%&(99.4\%, 100.0\%)\\ 40&99.7\%&(99.6\%, 100.0\%)\\ 50&99.6\%&(99.4\%, 100.0\%)\\ 60&99.7\%&(99.6\%, 100.0\%)\\ 80&99.5\%&(99.3\%, 99.8\%)
\end{tabular}}
\subcaption{$\chi^2_{k-1,0.95}/(2n)$}
\label{table:DRO simultaneous n 1-sided}
\end{subtable}
\begin{subtable}{0.33\textwidth}
\centering
{\footnotesize\begin{tabular}{c|c|c}
$n$&1-sided&$95\%$ C.I. of\\
&Cover. Prob.&Cover. Prob.\\
\hline
20&88.5\%&(87.2\%, 89.8\%)\\ 30&89.2\%&(88.0\%, 90.4\%)\\ 40&90.5\%&(89.4\%, 91.6\%)\\ 50&90.5\%&(89.4\%, 91.6\%)\\ 60&89.5\%&(88.3\%, 90.7\%)\\ 80&90.9\%&(89.9\%, 91.9\%)
\end{tabular}}
\subcaption{$\chi^2_{1,0.95}/(2n)$}
\label{table:CLT simultaneous n 1-sided}
\end{subtable}
\begin{subtable}{0.33\textwidth}
\centering
{\footnotesize\begin{tabular}{c|c|c}
$n$&1-sided&$95\%$ C.I. of\\
&Cover. Prob.&Cover. Prob.\\
\hline
20&94.8\%&(93.7\%, 96.0\%)\\ 30&95.1\%&(94.0\%, 96.3\%)\\ 40&95.4\%&(94.3\%, 96.5\%)\\ 50&95.4\%&(94.3\%, 96.5\%)\\ 60&94.7\%&(93.5\%, 95.8\%)\\ 80&96.0\%&(95.0\%, 97.0\%)
\end{tabular}}
\subcaption{Approximate $95\%$-quantile of $\sup_xJ_n(x)$}
\label{table:empirical simultaneous n 1-sided}
\end{subtable}
\caption{One-sided coverage probabilities for different Burg-divergence ball sizes and sample sizes for a discrete distribution with $k=5$} \label{table:comparison discrete n 1-sided}
\end{table}
Finally, we repeat the experiments using the continuous distribution $Exp(1/20)$. We compare the use of $\chi^2_{1,0.95}$ with the $\chi^2$-process excursion (there is no notion of $k$ in this case). Table \ref{table:comparison cont} shows that the two-sided coverages using $\chi^2_{1,0.95}$ are under-covering at between $82\%$ and $85\%$. The $\chi^2$-process excursion gives about $93\%$ at $n=20$ and converges to close to $95\%$ at $n=80$. Thus, similar to the discrete case, the calibration using $\chi^2$-process excursion gives significantly more accurate two-sided coverages than using $\chi^2_{1,0.95}$. Table \ref{table:comparison cont 1-sided} draws similar conclusion for one-sided coverages. For $\chi^2_{1,0.95}$, the coverage probability is about $84\%$ at $n=20$ and $90\%$ at $n=80$, therefore severely under-covering. On the other hand, $\chi^2$-process excursion gives $94\%$ to $96\%$ coverages among all the $n$'s. This once again shows the insignificance of one- versus two-sided coverage compared to the improvement in the choice of calibration method. In overall, our proposed scheme of using $\chi^2$-process excursion gives much more accurate coverages than using $\chi^2_{1,0.95}$.
\begin{table}[!htb]
\begin{subtable}{0.5\textwidth}
\centering
{\footnotesize\begin{tabular}{c|c|c}
$n$&2-sided&$95\%$ C.I. of\\
&Cover. Prob.&Cover. Prob.\\
\hline
20&82.4\%&(81.3\%, 83.5\%)\\ 30&82.8\%&(81.7\%, 83.8\%)\\ 40&83.5\%&(82.5\%, 84.5\%)\\ 50&84.3\%&(83.3\%, 85.2\%)\\ 60&84.2\%&(83.2\%, 85.2\%)\\ 80&85.1\%&(84.2\%, 86.1\%)
\end{tabular}}
\subcaption{$\chi^2_{1,0.95}/(2n)$}
\label{table:DRO simultaneous cont}
\end{subtable}
\begin{subtable}{0.5\textwidth}
\centering
{\footnotesize\begin{tabular}{c|c|c}
$n$&2-sided&$95\%$ C.I. of\\
&Cover. Prob.&Cover. Prob.\\
\hline
20&92.8\%&(91.1\%, 94.6\%)\\ 30&93.6\%&(92.3\%, 94.9\%)\\ 40&94.4\%&(93.1\%, 95.8\%)\\ 50&94.4\%&(93.1\%, 95.8\%)\\ 60&95.7\%&(94.6\%, 96.9\%)\\ 80&95.3\%&(94.1\%, 96.5\%)
\end{tabular}}
\subcaption{Approximate $95\%$-quantile of $\sup_xJ_n(x)$}
\label{table:empirical simultaneous cont}
\end{subtable}
\caption{Two-sided coverage probabilities for different Burg-divergence ball sizes and sample sizes for a continuous distribution} \label{table:comparison cont}
\end{table}
\begin{table}[!htb]
\begin{subtable}{0.5\textwidth}
\centering
{\footnotesize\begin{tabular}{c|c|c}
$n$&1-sided&$95\%$ C.I. of\\
&Cover. Prob.&Cover. Prob.\\
\hline
20&83.9\%&(81.7\%, 86.1\%)\\ 30&84.6\%&(82.4\%, 86.8\%)\\ 40&86.1\%&(84.0\%, 88.2\%)\\ 50&87.7\%&(85.7\%, 89.7\%)\\ 60&87.5\%&(85.4\%, 89.6\%)\\ 80&89.5\%&(87.6\%, 91.4\%)
\end{tabular}}
\subcaption{$\chi^2_{1,0.95}/(2n)$}
\label{table:DRO simultaneous cont 1-sided}
\end{subtable}
\begin{subtable}{0.5\textwidth}
\centering
{\footnotesize\begin{tabular}{c|c|c}
$n$&1-sided&$95\%$ C.I. of\\
&Cover. Prob.&Cover. Prob.\\
\hline
20&93.6\%&(91.9\%, 95.3\%)\\ 30&94.4\%&(93.1\%, 95.7\%)\\ 40&95.2\%&(93.9\%, 96.5\%)\\ 50&95.2\%&(93.9\%, 96.5\%)\\ 60&96.5\%&(95.4\%, 97.6\%)\\ 80&96.1\%&(94.9\%, 97.3\%)
\end{tabular}}
\subcaption{Approximate $95\%$-quantile of $\sup_xJ_n(x)$}
\label{table:empirical simultaneous cont 1-sided}
\end{subtable}
\caption{One-sided coverage probabilities for different Burg-divergence ball sizes and sample sizes for a continuous distribution} \label{table:comparison cont 1-sided}
\end{table}
\section{Conclusion}\label{sec:conlusion} We have motivated and investigated the construction of tractable uncertainty sets that can recover the feasibility guarantees on par with the implications of CLT. We have shown that the empirical Burg-entropy divergence balls are capable of achieving such guarantees. We have also shown, intriguingly, that these balls are invalid confidence regions in the standard framework of data-driven DRO, and can have low or zero coverages on the true underlying distributions. Rather, we have explained their statistical performances via linking the resulting DRO with empirical likelihood. This link allows us to derive the optimal sizes of these balls, using the quantiles of $\chi^2$-process excursion. Such a calibration approach also bypasses some documented difficulties in using divergence balls in the data-driven DRO literature. Future work includes further developments of the theory and calibration methods to incorporate optimization objectives and more general constraints.
\ACKNOWLEDGMENT{The author gratefully acknowledges support from the National Science Foundation under grants CMMI-1400391/1542020 and CMMI-1436247/1523453. He also thanks Zhiyuan Huang and Yanzhe Jin for assisting with the numerical experiments.}
\begin{APPENDICES} \label{sec:appendix} \section{Technical Proofs}\label{sec:proofs} Theorem \ref{EL basic} is a simple consequence of the following proposition: \begin{proposition} Under the same conditions as Theorem \ref{EL basic}, $\underline Z_n(x)\leq Z_0(x)\leq\overline Z_n(x)$ if and only if $-2\log R(Z_0(x))\leq\chi^2_{1,1-\alpha}$.\label{duality} \end{proposition}
\proof{Proof of Proposition \ref{duality}.} We first argue that the optimization defining \eqref{profile1} must have an optimal solution, if it is feasible. Since $-2\sum_{i=1}^n\log(nw_i)\to\infty$ as $w_i\to0$ for any $i$, it suffices to consider only $w_i$ such that $w_i\geq\epsilon$ for some small $\epsilon>0$. Since the set $\left\{\sum_{i=1}^nh(x;\xi_i)w_i=Z_0(x),\ \sum_{i=1}^nw_i=1,\ w_i\geq\epsilon\text{\ for all\ }i=1,\ldots,n\right\}$ is compact, by Weierstrass Theorem, there exists an optimal solution for \eqref{profile1}.
Suppose $-2\log R(Z_0(x))\leq\chi^2_{1,1-\alpha}$. Then the optimization in $-2\log R(Z_0(x))$ is feasible, and there must exist a probability vector $\mathbf w=(w_1,\ldots,w_n)$ such that $-2\sum_{i=1}^n\log(nw_i)\leq\chi^2_{1,1-\alpha}$ and $\sum_{i=1}^nh(x;\xi_i)w_i=Z_0(x)$. This implies $\underline Z_n(x)\leq Z_0(x)\leq\overline Z_n(x)$.
To show the reverse direction, note first that the set $$\left\{\sum_{i=1}^nh(x;\xi_i)w_i:-2\sum_{i=1}^n\log(nw_i)\leq\chi^2_{1,1-\alpha},\ \sum_{i=1}^nw_i=1,\ w_i\geq0\text{\ for all\ }i=1,\ldots,n\right\}$$ is an interval, since $\sum_{i=1}^nh(x;\xi_i)w_i$ is a linear function of the convex set $$\left\{(w_1,\ldots,w_n):-2\sum_{i=1}^n\log(nw_i)\leq\chi^2_{1,1-\alpha},\ \sum_{i=1}^nw_i=1,\ w_i\geq0\text{\ for all\ }i=1,\ldots,n\right\}$$ Moreover, since the latter set is compact, by Weierstrass Theorem again, there must exist optimal solutions in the optimization pair $$\max/\min\left\{\sum_{i=1}^nh(x;\xi_i)w_i:-2\sum_{i=1}^n\log(nw_i)\leq\chi^2_{1,1-\alpha},\ \sum_{i=1}^nw_i=1,\ w_i\geq0\text{\ for all\ }i=1,\ldots,n\right\}$$ Therefore, $\underline Z_n(x)\leq Z_0(x)\leq\overline Z_n(x)$ implies that there exists a probability vector $\mathbf w$ such that $\sum_{i=1}^nh(x;\xi_i)w_i=Z_0(x)$ and $-2\sum_{i=1}^n\log(nw_i)\leq\chi^2_{1,1-\alpha}$, leading to $-2\log R(Z_0(x))\leq\chi^2_{1,1-\alpha}$.\Halmos \endproof
\proof{Proof of Theorem \ref{EL basic}.} By Theorem \ref{ELT}, we have $\lim_{n\to\infty}P(-2\log R(Z_0(x))\leq\chi^2_{1,1-\alpha})=1-\alpha$ for a fixed $x\in\Theta$, where \begin{eqnarray} &&-2\log R(Z_0(x))\notag\\ &=&\min\left\{-2\sum_{i=1}^n\log(nw_i):\sum_{i=1}^nh(x;\xi_i)w_i=Z_0(x),\ \sum_{i=1}^nw_i=1,\ w_i\geq0\text{\ for all\ }i=1,\ldots,n\right\}\label{profile1} \end{eqnarray} Thus, to show \eqref{exact asymptotic}, it suffices to prove that $\underline Z_n(x)\leq Z_0(x)\leq\overline Z_n(x)$ if and only if $-2\log R(Z_0(x))\leq\chi^2_{1,1-\alpha}$. Proposition \ref{duality} finishes the proof.\Halmos \endproof
\proof{Proof of Proposition \ref{EL discrete}.} By relabeling the weights under membership of the support points, we rewrite \begin{eqnarray} \underline Z_n(x)&=&\min\Bigg\{\sum_{i=1}^kh(x;s_i)\sum_{j=1}^{n_i}w_{ij}:-\frac{1}{n}\sum_{i=1}^k\sum_{j=1}^{n_i}\log(nw_{ij})\leq\frac{\chi^2_{1,1-\alpha}}{2n},{}\notag\\ &&{}\ \sum_{i=1}^k\sum_{j=1}^{n_i}w_{ij}=1,\ w_{ij}\geq0\text{\ for\ }i=1,\ldots,k,\ j=1,\ldots,n_i\Bigg\}\label{opt EL dual min2} \end{eqnarray} and \begin{eqnarray} \overline Z_n(x)&=&\max\Bigg\{\sum_{i=1}^kh(x;s_i)\sum_{j=1}^{n_i}w_{ij}:-\frac{1}{n}\sum_{i=1}^k\sum_{j=1}^{n_i}\log(nw_{ij})\leq\frac{\chi^2_{1,1-\alpha}}{2n},{}\notag\\ &&{}\ \sum_{i=1}^k\sum_{j=1}^{n_i}w_{ij}=1,\ w_{ij}\geq0\text{\ for\ }i=1,\ldots,k,\ j=1,\ldots,n_i\Bigg\}\label{opt EL dual max2} \end{eqnarray}
To avoid repetition, we focus on the maximization formulation. We show that, for any feasible $\mathbf p$ in $\max_{\mathbf p\in\mathcal U_{Burg}'}E_{\mathbf p}[h(x;\xi)]$, we can construct a feasible $\mathbf w$ for $\overline Z_n(x)$ that attains the same objective value, and vice versa.
To this end, for any $\mathbf p=(p_1,\ldots,p_k)\in\mathcal U_{Burg}'$, we define $w_{ij}=p_i/n_i$ for all $j=1,\ldots,n_i$. Then $$-\frac{1}{n}\sum_{i=1}^k\sum_{j=1}^{n_i}\log(nw_{ij})=-\sum_{i=1}^k\frac{n_i}{n}\log\frac{np_i}{n_i}=-\sum_{i=1}^k\hat p_i\log\frac{p_i}{\hat p_i}\leq\frac{\chi^2_{1,1-\alpha}}{2n}$$ as well as $\sum_{i=1}^k\sum_{j=1}^{n_i}w_{ij}=\sum_{i=1}^kp_i=1$, and $w_{ij}\geq0$ for all $i$ and $j$. Hence $w_{ij}$ is feasible for \eqref{opt EL dual max2}. Moreover, $\sum_{i=1}^kh(x;s_i)\sum_{j=1}^{n_i}w_{ij}=\sum_{i=1}^kh(x;s_i)p_i$, thus the same objective value is attained.
On the other hand, suppose $\mathbf w=(w_{ij})$ is a feasible solution for \eqref{opt EL dual max2}. We then define $p_i=\sum_{j=1}^{n_i}w_{ij}$. By Jensen's inequality we have $-\log(p_i/n_i)\leq-(1/n_i)\sum_{j=1}^{n_i}\log w_{ij}$, and so $$-\sum_{i=1}^k\frac{n_i}{n}\log\frac{np_i}{n_i}\leq-\frac{1}{n}\sum_{i=1}^k\sum_{j=1}^{n_i}\log(nw_{ij})\leq\frac{\chi^2_{1,1-\alpha}}{2n}$$ Together with the simple observation that $\sum_{i=1}^kp_i=\sum_{i=1}^k\sum_{j=1}^{n_i}w_{ij}=1$ and $p_i\geq0$ for all $i$, we get that $\mathbf p=(p_i)$ is feasible for $\max_{\mathbf p\in\mathcal U_{Burg}'}E_{\mathbf p}[h(x;\xi)]$. Moreover, $\sum_{i=1}^kh(x;s_i)p_i=\sum_{i=1}^kh(x;s_i)\sum_{j=1}^{n_i}w_{ij}$, thus the same objective value is attained in this case as well.
Similar arguments apply to the minimization formulation, and we conclude the proof. \Halmos \endproof
\proof{Proof of Theorem \ref{EL process thm}.} First, Assumption \ref{finite mean} allows us to define $\tilde h(x;\xi)=h(x;\xi)-Z_0(x)$. Also, we denote the classes of functions $\Xi\to\mathbb R$
$$\mathcal H_\Theta^1=\{|\tilde h(x;\cdot)|:x\in\Theta\}$$ $$\mathcal H_\Theta^2=\{\tilde h(x;\cdot)^2:x\in\Theta\}$$ $$\mathcal H_\Theta^+=\{\tilde h(x;\cdot)^+:x\in\Theta\}$$ $$\mathcal H_\Theta^-=\{\tilde h(x;\cdot)^-:x\in\Theta\}$$ where $$y^+=\left\{\begin{array}{ll}y&\text{\ \ if\ }y\geq0\\0&\text{\ \ if\ }y<0\end{array}\right.\text{\ \ \ \ and\ \ \ \ }y^-=\left\{\begin{array}{ll}0&\text{\ \ if\ }y>0\\-y&\text{\ \ if\ }y\leq0\end{array}\right.$$
Since $\mathcal H_\Theta$ is a $P_0$-Donsker class, it is $P_0$-Glivenko-Cantelli (GC) (e.g., the discussion before Example 2.1.3 in \cite{van1996weak}). By the preservation theorem (Theorem \ref{preservation} in Appendix \ref{sec:EP}), since $E_0\|\tilde h(\cdot;\xi)^2\|_\Theta=E\|\tilde h(\cdot;\xi)\|_\Theta^2<\infty$ by Assumption \ref{L2}, $\mathcal H_\Theta^2$ is also $P_0$-GC. Moreover, since $E\|\tilde h(\cdot;\xi)^\pm\|_\Theta\leq E\|\tilde h(\cdot;\xi)\|_\Theta\leq\sqrt{E\|\tilde h(\cdot;\xi)\|_\Theta^2}<\infty$, $\mathcal H_\Theta^+$, $\mathcal H_\Theta^-$ and $\mathcal H_\Theta^1$ are all $P_0$-GC as well. Letting $P_n$ be the empirical measure generated from $\xi_1,\ldots,\xi_n$, the above imply \begin{eqnarray}
\|P_n-P_0\|_{\mathcal H_\Theta^+}\stackrel{a.s.}{\to}0\label{GC+}\\
\|P_n-P_0\|_{\mathcal H_\Theta^-}\stackrel{a.s.}{\to}0\label{GC-}\\
\|P_n-P_0\|_{\mathcal H_\Theta^2}\stackrel{a.s.}{\to}0\label{GC var} \end{eqnarray}
where $\|P_n-P_0\|_{\mathcal F}=\sup_{f\in\mathcal F}|P_n(f)-P_0(f)|$ for $P_n$ indexed by $\mathcal F$, and similarly defined for $P_0$ (see Appendix \ref{sec:EP}).
Note that \eqref{GC var} in particular implies the uniform convergence of the empirical variance \begin{equation}
\left\|\frac{1}{n}\sum_{i=1}^n(h(\cdot;\xi_i)-Z_0(\cdot))^2-\sigma_0(\cdot)\right\|_\Theta\stackrel{a.s.}{\to}0\label{convergence var} \end{equation} where $\sigma_0(x)=Var_0(h(x;\xi))$.
Now, for each $x$,
$$E_0\tilde h(x;\xi)^++E_0\tilde h(x;\xi)^-=E_0|\tilde h(x;\xi)|$$ $$E_0\tilde h(x;\xi)^+-E_0\tilde h(x;\xi)^-=E_0\tilde h(x;\xi)=0$$
which gives $E_0\tilde h(x;\xi)^+=E_0\tilde h(x;\xi)^-=E_0|\tilde h(x;\xi)|/2$. Hence \begin{equation}
\inf_{x\in\Theta}E_0\tilde h(x;\xi)^\pm=\inf_{x\in\Theta}\frac{E_0|\tilde h(x;\xi)|}{2}\geq\frac{c}{2}\label{lower bdd pm} \end{equation}
where we define $c$ as a constant such that $\inf_{x\in\Theta}E_0|h(x;\xi)-Z_0(x)|\geq c$, which exists by Assumption \ref{nondegeneracy}. By Jensen's inequality, \begin{equation}
\inf_{x\in\Theta}E_0\tilde h(x;\xi)^2\geq\left(\inf_{x\in\Theta}E_0|\tilde h(x;\xi)|\right)^2\geq c^2\label{lower bdd var} \end{equation} by using Assumption \ref{nondegeneracy} again. From \eqref{GC+}, \eqref{GC-} and \eqref{lower bdd pm}, we have $\inf_{x\in\Theta}(1/n)\sum_{i=1}^n\tilde h(x;\xi_i)^+$ and $\inf_{x\in\Theta}(1/n)\sum_{i=1}^n\tilde h(x;\xi_i)^->0$ for large enough $n$ a.s.. When this occurs, $\min_{1\leq i\leq n}h(x;\xi_i)<Z_0(x)<\max_{1\leq i\leq n}h(x;\xi_i)$ for every $x$, and the optimization defining $-\log R(x;Z_0)$, namely \begin{equation}
\min\left\{-\sum_{i=1}^n\log(nw_i)\Bigg|\sum_{i=1}^nh(x;\xi_i)w_i=Z_0(x),\ \sum_{i=1}^nw_i=1,\ w_i\geq0\text{\ for\ }i=1,\ldots,n\right\}\label{EL opt transformed} \end{equation} has a unique optimal solution $\mathbf w(x)=(w_1(x),\ldots,w_n(x))$ with $w_i(x)>0$ for all $i$, for any $x$. This is because setting any $w_i(x)=0$ would render $-2\sum_{i=1}^n\log(nw_i)=\infty$ which is clearly suboptimal. Hence it suffices to replace $w_i\geq0$ with $w_i\geq\epsilon$ for all $i$ for some small enough $\epsilon>0$. In this modified region, the optimum exists and is unique since $-\sum_{i=1}^n\log(nw_i)$ is strictly convex.
Now consider the optimization \eqref{EL opt transformed} when $\min_{1\leq i\leq n}h(x;\xi_i)<Z_0(x)<\max_{1\leq i\leq n}h(x;\xi_i)$. We adopt the proof technique in Section 11.2 of \cite{owen2001empirical}, but generalize at the empirical process level. For convenience, we write $\tilde h_i=\tilde h(x;\xi_i)=h(x;\xi_i)-Z_0(x)$, and also suppress the $x$ in $w_i=w_i(x)$ and $Z_0=Z_0(x)$. The Lagrangian is written as $$-\sum_{i=1}^n\log(nw_i)+\lambda\left(\sum_{i=1}^n\tilde h_iw_i-Z_0\right)+\gamma\left(\sum_{i=1}^nw_i-1\right)$$ where $\lambda=\lambda(x)$ and $\gamma=\gamma(x)$ are the Lagrange multipliers. Differentiating with respect to $w_i$ and setting it to zero, we have \begin{equation} -\frac{1}{w_i}+\lambda\tilde h_i+\gamma=0\label{interim} \end{equation} Setting $\sum_{i=1}^n\tilde h_iw_i=0$ and $\sum_{i=1}^nw_i=1$, multiplying both sides of \eqref{interim} by $w_i$ and summing up over $i$, we get $\gamma=n$. Using \eqref{interim} again, we have \begin{equation} w_i=\frac{1}{n}\frac{1}{1+\lambda\tilde h_i}\label{w} \end{equation} where the $\lambda$ in \eqref{w} is rescaled by a factor of $n$. Note that we can find $\lambda$ such that \begin{equation} \sum_{i=1}^n\frac{1}{n}\frac{\tilde h_i}{1+\lambda\tilde h_i}=0\label{interim3} \end{equation} and $\frac{1}{n}\frac{1}{1+\lambda\tilde h_i}>0$ for all $i$, upon which the Karush-Kuhn-Tucker (KKT) condition can be seen to hold and conclude that $w_i$ in \eqref{w} is the optimal solution. Indeed, let $\tilde h^*=\max_i\tilde h_i>0$ and $\tilde h_*=\min_i\tilde h_i<0$. Note that $\sum_{i=1}^n\frac{1}{n}\frac{\tilde h_i}{1+\lambda\tilde h_i}\to\infty$ as $\lambda\to-1/\tilde h^*$, and $\to-\infty$ as $\lambda\to-1/\tilde h_*$. Since $\sum_{i=1}^n\frac{1}{n}\frac{\tilde h_i}{1+\lambda\tilde h_i}$ is a continuous function in $\lambda$ between $-1/\tilde h^*$ and $-1/\tilde h_*$, there must exist a $\lambda$ that solves \eqref{interim3}. Moreover, $\frac{1}{n}\frac{1}{1+\lambda\tilde h_i}>0$ for all $i$ for this $\lambda$.
Given this characterization of the optimal solution, the rest of the proof is to derive the asymptotic behavior of $-2\log R(x;Z_0)$ as $n\to\infty$. First, we write $$\frac{1}{1+\lambda\tilde h_i}=1-\frac{\lambda\tilde h_i}{1+\lambda\tilde h_i}$$ Multiplying both sides by $\tilde h_i/n$ and summing up over $i$, we get $$\sum_{i=1}^n\frac{1}{n}\frac{\tilde h_i}{1+\lambda\tilde h_i}=\bar h-\lambda\sum_{i=1}^n\frac{1}{n}\frac{\tilde h_i^2}{1+\lambda\tilde h_i}$$ where $\bar h:=\bar h(x)=(1/n)\sum_{i=1}^n\tilde h(x;\xi_i)$, and hence \begin{equation} \bar h=\lambda\sum_{i=1}^n\frac{1}{n}\frac{\tilde h_i^2}{1+\lambda\tilde h_i}\label{interim2} \end{equation} by \eqref{interim3}. Now let $$s:=s(x)=\frac{1}{n}\sum_{i=1}^n\tilde h(x;\xi_i)^2$$ be the empirical variance. Note that, from \eqref{w}, $w_i>0$ implies $1+\lambda\tilde h_i>0$. Together with $s\geq0$, we get \begin{align*}
|\lambda|s&\leq\left|\lambda\sum_{i=1}^n\frac{1}{n}\frac{\tilde h_i^2}{1+\lambda\tilde h_i}\right|\left(1+|\lambda|\max_{1\leq i\leq n}|\tilde h_i|\right)\\
&=|\bar h|\left(1+|\lambda|\max_{1\leq i\leq n}|\tilde h_i|\right) \end{align*} by using \eqref{interim2}, and hence \begin{equation}
|\lambda|\left(s-|\bar h|\max_{1\leq i\leq n}|\tilde h_i|\right)\leq|\bar h|\label{interim new} \end{equation}
By Lemma \ref{lemma max} in Appendix \ref{sec:thms}, $E\|\tilde h(x;\xi)\|_\Theta^2<\infty$ in Assumption \ref{L2} implies that \begin{equation}
\max_{1\leq i\leq n}\|\tilde h_i\|_\Theta=o(n^{1/2})\text{\ a.s.}\label{interim new1} \end{equation}
Moreover, since $\mathcal H_\Theta$ is $P_0$-Donsker, we have $\sqrt n\bar h=\sqrt n(P_n(h(\cdot;\cdot))-P_0(h(\cdot;\cdot)))\Rightarrow\tilde G$ in $\ell^\infty(\mathcal H_\Theta)$, where $\tilde G(\cdot)$ is a tight Gaussian process indexed by $h(x;\cdot)\in\mathcal H_\Theta$ that is centered and has covariance $Cov(\tilde G(h(x_1;\cdot)),\tilde G(h(x_2;\cdot)))=Cov_0(h(x_1;\xi),h(x_2;\xi))$ for any $h(x_1;\cdot),h(x_2;\cdot)\in\mathcal H_\Theta$. Noting that the map $\ell^\infty(\mathcal H_\Theta)\to\ell^\infty(\Theta)$ defined by $y(\cdot)\mapsto y(h(\cdot;\cdot))$ is continuous, by the continuous mapping theorem (Theorem \ref{continuous mapping} in Appendix \ref{sec:thms}), we have $\sqrt n\bar h\Rightarrow\tilde G$ in $\ell^\infty(\Theta)$ where $\tilde G$ is now indexed by $x\in\Theta$. As the norm in $\ell^\infty(\Theta)$, $\|\cdot\|_\Theta$ is a continuous map. By the continuous mapping theorem again, $\sqrt n\|\bar h\|_\Theta=\|\sqrt n\bar h\|_\Theta\Rightarrow\|\tilde G\|_\Theta$,
so that $\|\bar h\|_\Theta=O_p(n^{-1/2})$. Moreover, $\||\bar h|\max_{1\leq i\leq n}|\tilde h_i|\|_\Theta\leq\|\bar h\|_\Theta\max_{1\leq i\leq n}\|\tilde h_i\|_\Theta=O_p(n^{-1/2})o(n^{1/2})=o_p(1)$.
Next, from \eqref{convergence var} and \eqref{lower bdd var}, we have $\inf_{x\in\Theta}s(x)\geq c_1$ for some $c_1>0$ ev.. Pick any constant $\varepsilon<c_1$. We have \begin{equation}
P\left(\inf_{x\in\Theta}\left\{s(x)-|\bar h(x)|\max_{1\leq i\leq n}|\tilde h_i(x)|\right\}\geq c_1-\varepsilon\right)\geq P\left(\inf_{x\in\Theta}s(x)\geq c_1,\left\||\bar h(x)|\max_{1\leq i\leq n}|\bar h_i(x)|\right\|_\Theta\leq\varepsilon\right)\to1\label{interim11} \end{equation}
Over the set $\{\inf_{x\in\Theta}\{s(x)-|\bar h(x)|\max_{1\leq i\leq n}|\bar h_i(x)|\}>c_1-\varepsilon\}$, \eqref{interim new} implies
$$|\lambda(x)|\leq\frac{|\bar h(x)|}{s-|\bar h(x)|\max_{1\leq i\leq n}|\bar h_i(x)|}$$ for all $x\in\Theta$, so that \begin{equation}
\|\lambda\|_\Theta\leq\frac{\|\bar h\|_\Theta}{c_1-\varepsilon}\label{interim updated1} \end{equation}
We argue that $\|\lambda\|_\Theta=O_p(n^{-1/2})$. This is because, for any given $\delta>0$, we can find a large enough $B>0$ such that \begin{eqnarray*}
&&\limsup_{n\to\infty}P(\|\lambda\|_\Theta>Bn^{-1/2})\\
&\leq&\limsup_{n\to\infty}\Bigg(P\left(\|\lambda\|_\Theta>Bn^{-1/2},\ \inf_{x\in\Theta}\left\{s(x)-|\bar h(x)|\max_{1\leq i\leq n}|\tilde h_i(x)|\right\}>c_1-\varepsilon\right){}\\
&&{}+P\left(\inf_{x\in\Theta}\left\{s(x)-|\bar h(x)|\max_{1\leq i\leq n}|\tilde h_i(x)|\right\}\leq c_1-\varepsilon\right)\Bigg)\\
&\leq&\limsup_{n\to\infty}\left(P(\|\bar h\|_\Theta>(c_1-\varepsilon)Bn^{-1/2})+P\left(\inf_{x\in\Theta}\left\{s(x)-|\bar h(x)|\max_{1\leq i\leq n}|\tilde h_i(x)|\right\}\leq c_1-\varepsilon\right)\right){}\\ &&{}\text{\ \ by \eqref{interim updated1}}\\ &<&\delta \end{eqnarray*}
by \eqref{interim11} and that $\|\bar h\|_\Theta=O_p(n^{-1/2})$ as shown above. This and \eqref{interim new1} together gives \begin{equation}
\max_{1\leq i\leq n}\sup_{x\in\Theta}|\lambda(x)\tilde h_i(x)|\leq\|\lambda\|_\Theta\max_{1\leq i\leq n}\|\tilde h_i\|_\Theta=O_p(n^{-1/2})o(n^{1/2})=o_p(1)\label{interim6} \end{equation}
Now \eqref{interim3} can be rewritten as \begin{align} 0&=\sum_{i=1}^n\frac{1}{n}\tilde h_i\left(1-\lambda\tilde h_i+\frac{\lambda^2\tilde h_i^2}{1+\lambda\tilde h_i}\right)\notag\\ &=\bar h-\lambda s+\sum_{i=1}^n\frac{1}{n}\frac{\lambda^2\tilde h_i^3}{1+\lambda\tilde h_i}\label{interim4} \end{align} The last term in \eqref{interim4} satisfies
$$\left|\sum_{i=1}^n\frac{1}{n}\frac{\lambda^2\tilde h_i^3}{1+\lambda\tilde h_i}\right|\leq\frac{1}{n}\sum_{i=1}^n\tilde h_i^2\lambda^2\max_{1\leq i\leq n}|\tilde h_i|\max_{1\leq i\leq n}(1+\lambda\tilde h_i)^{-1}$$ Taking $\sup_{\theta\in\Theta}$ on both sides, we get \begin{equation}
\left\|\sum_{i=1}^n\frac{1}{n}\frac{\lambda^2\tilde h_i^3}{1+\lambda\tilde h_i}\right\|_\Theta\leq\|s\|_\Theta\|\lambda\|_\Theta^2\max_{1\leq i\leq n}\|\tilde h_i\|_\Theta\max_{1\leq i\leq n}\sup_{x\in\Theta}(1+\lambda\tilde h(x,\xi_i))^{-1}\label{interim new2} \end{equation}
Now, $\|s\|_\Theta\to\|\sigma_0\|_\Theta$ by \eqref{convergence var}. Moreover, for any small $\varepsilon>0$, \begin{align*} P\left(\max_{1\leq i\leq n}\sup_{x\in\Theta}(1+\lambda\tilde h(x,\xi_i))^{-1}>\frac{1}{1-\varepsilon}\right)&=P\left(\frac{1}{1+\lambda\tilde h(x,\xi_i)}>\frac{1}{1-\varepsilon}\text{\ \ for some $1\leq i\leq n$ and $\theta\in\Theta$}\right)\\ &\leq P(\lambda\tilde h(x,\xi_i)<-\varepsilon\text{\ \ for some $1\leq i\leq n$ and $x\in\Theta$})\\
&\leq P\left(\max_{1\leq i\leq n}\sup_{x\in\Theta}|\lambda\tilde h(x,\xi_i)|>\varepsilon\right)\to0 \end{align*} by \eqref{interim6}. Thus \eqref{interim new2} is bounded from above by \begin{equation} O(1)O_p(n^{-1})o(n^{1/2})O_p(1)=o_p(n^{-1/2})\label{interim5} \end{equation}
From \eqref{interim4} and \eqref{interim5}, we have $$0=\bar h-\lambda s+\epsilon$$
where $\|\epsilon\|_\Theta=o_p(n^{-1/2})$. Since \eqref{convergence var} and \eqref{lower bdd var} implies $\inf_{x\in\Theta}s(x)\geq c_1$ for some $c_1>0$ ev., we further get \begin{equation} \lambda=s^{-1}(\bar h+\epsilon)\label{interim new4} \end{equation}
Now consider \begin{align} -2\log R(x;Z_0)&=-2\sum_{i=1}^n\log(nw_i)\notag\\ &=2\sum_{i=1}^n\log(1+\lambda\tilde h_i)\notag\\ &=2\sum_{i=1}^n\left(\lambda\tilde h_i-\frac{1}{2}\lambda^2\tilde h_i^2+\nu_i\right)\label{interim7} \end{align} where $$\nu_i=\frac{1}{3}\frac{1}{(1+\zeta_i)^3}(\lambda\tilde h_i)^3$$ with $\zeta_i=\zeta_i(x)$ between 0 and $\lambda\tilde h_i$ by Taylor's expansion. So \begin{equation}
|\nu_i|\leq\frac{1}{3}\frac{1}{|1+\zeta_i|^3}|\lambda\tilde h_i|^3\label{interim new3} \end{equation} For any large enough $B_1>0$, we have \begin{eqnarray}
&&P(|\nu_i(x)|>B_1|\lambda(x)\tilde h(x,\xi_i)|^3\text{\ \ for all $x\in\Theta$ and $1\leq i\leq n$})\notag\\
&\leq&P\left((3B_1)^{1/3}\max_{1\leq i\leq n}\|1+\zeta_i\|_\Theta<1\right)\text{\ \ from \eqref{interim new3}}\notag\\
&=&P\left((3B_1)^{1/3}\max_{1\leq i\leq n}\|1+\zeta_i\|_\Theta<1,\max_{1\leq i\leq n}\|\zeta_i\|_\Theta<\varepsilon\right){}\notag\\
&&{}+P\left((3B_1)^{1/3}\max_{1\leq i\leq n}\|1+\zeta_i\|_\Theta<1,\max_{1\leq i\leq n}\|\zeta_i\|_\Theta>\varepsilon\right)\text{\ \ for some sufficiently large $0<\varepsilon<1$}\notag\\
&\leq&P\left((3B_1)^{1/3}\max_{1\leq i\leq n}(1-\varepsilon)<1\right)+P\left(\max_{1\leq i\leq n}\|\zeta_i\|_\Theta>\varepsilon\right)\notag\\ &\to&0\label{interim13} \end{eqnarray}
since $\max_{1\leq i\leq n}\|\zeta_i\|_\Theta\leq\max_{1\leq i\leq n}\|\lambda\tilde h_i\|_\Theta=o_p(1)$ by \eqref{interim6}.
Now \eqref{interim7} gives \begin{eqnarray} &&2n\lambda\bar h-\lambda^2ns+2\sum_{i=1}^n\nu_i\notag\\ &=&2ns^{-1}(\bar h+\epsilon)\bar h-nss^{-2}(\bar h^2+2\epsilon\bar h+\epsilon^2)+2\sum_{i=1}^n\nu_i\text{\ \ by \eqref{interim new4}}\notag\\ &=&ns^{-1}\bar h^2-ns^{-1}\epsilon^2+2\sum_{i=1}^n\nu_i\label{interim10} \end{eqnarray} Note that \begin{equation}
\|ns^{-1}\epsilon^2\|_\Theta\leq nO(1)o_p(n^{-1})=o_p(1)\label{interim8} \end{equation}
since $\inf_{x\in\Theta}s(x)\geq c_1>0$ ev., and $\|\epsilon\|_\Theta=o_p(n^{-1/2})$. Moreover, over the set $\{|\nu_i(x)|\leq B_1|\lambda(x)\tilde h_i(x)|^3$ for all $x\in\Theta$ and $1\leq i\leq n\}$, we have
$$\left|\sum_{i=1}^n\nu_i\right|\leq B_1|\lambda|^3\sum_{i=1}^n|\tilde h_i|^2\max_{1\leq i\leq n}|\tilde h_i|$$ for all $x\in\Theta$. Note that \begin{equation}
\left\|B_1|\lambda|^3\sum_{i=1}^n|\tilde h_i|^2\max_{1\leq i\leq n}|\tilde h_i|\right\|_\Theta\leq B_1O_p(n^{-3/2})nO(1)o(n^{1/2})=o_p(1)\label{interim12} \end{equation}
since $\|\lambda\|_\Theta=O_p(n^{-1/2})$, $\|s\|_\Theta=O(1)$, and $\max_{1\leq i\leq n}\|\tilde h_i\|_\Theta=o(n^{1/2})$. Now, for any $\varepsilon>0$, \begin{eqnarray*}
&&P\left(\left\|\sum_{i=1}^n\nu_i\right\|_\Theta>\varepsilon\right)\\
&\leq&P\left(\left\|\sum_{i=1}^n\nu_i\right\|_\Theta>\varepsilon,\ |\nu_i(x)|\leq B_1|\lambda(x)\tilde h_i(x)|^3\text{\ \ for some $x\in\Theta$ and $1\leq i\leq n$}\right){}\\
&&{}+P(|\nu_i(x)|>B_1|\lambda(x)\tilde h_i(x)|^3\text{\ \ for all $x\in\Theta$ and $1\leq i\leq n$})\\
&\leq&P\left(\left\|B_1|\lambda|^3\sum_{i=1}^n|\tilde h_i|^2\max_{1\leq i\leq n}|\tilde h_i|\right\|_\Theta>\varepsilon\right)+P(|\nu_i|>B_1|\lambda\tilde h_i|^3\text{\ \ for all $x\in\Theta$ and $1\leq i\leq n$})\\ &\to&0 \end{eqnarray*} by \eqref{interim13} and \eqref{interim12}. Hence we have \begin{equation}
\left\|\sum_{i=1}^n\nu_i\right\|_\Theta=o_p(1)\label{interim9} \end{equation} Using \eqref{interim8} and \eqref{interim9}, \eqref{interim10} implies that \begin{equation} -2\log R(x;Z_0)=ns^{-1}\bar h^2+\epsilon_1\label{interim new5} \end{equation}
where $\|\epsilon_1\|_\Theta=o_p(1)$.
Note that $\sqrt n\bar h\Rightarrow\tilde G$ in $\ell^\infty(\Theta)$ where $\tilde G(\cdot)$ is defined previously as the centered Gaussian process with covariance $Cov(\tilde G(x_1),\tilde G(x_2))=Cov(h(x_1;\xi),h(x_2;\xi))$ for any $x_1,x_2\in\Theta$. By Slutsky's Theorem (Theorem \ref{Slutsky} in Appendix \ref{sec:thms}) and \eqref{convergence var}, $(\sqrt n\bar h,s)\Rightarrow(\tilde G,\sigma_0)$ in $(\ell^\infty\times\ell^\infty)(\Theta)$ defined as
$$(\ell^\infty\times\ell^\infty)(\Theta)=\left\{(y_1,y_2):\Theta\to\mathbb R^2\Bigg|\|y_1\|_\Theta+\|y_2\|_\Theta<\infty\right\}$$ Note that pointwise division and $(\cdot)^2$ are continuous maps on $(\ell^\infty\times\ell^\infty)(\Theta)\to\ell^\infty(\Theta)$ and $\ell^\infty(\Theta)\to\ell^\infty(\Theta)$ respectively. Also, $\inf_{x\in\Theta}\sigma_0(x)>0$ by Assumption \ref{nondegeneracy} and Jensen's inequality. By continuous mapping theorem, we have $ns^{-1}\bar h^2\Rightarrow J$ in $\ell^\infty(\Theta)$ where $J(\cdot)$ is as defined in the theorem.
Finally, from \eqref{interim new5} and $\|\epsilon_1\|_\Theta=o_p(1)$, we get further that $-2\log R(\cdot;Z_0)\Rightarrow J(\cdot)$ in $\ell^\infty(\Theta)$. This concludes the proof.\Halmos \endproof
\proof{Proof of Lemma \ref{sample Gaussian}.} First define $\tilde G(\cdot)$ as a centered Gaussian process indexed by $x\in\Theta$ with covariance $Cov(\tilde G(x_1),\tilde G(x_2))=Cov_0(h(x_1;\xi),h(x_2;\xi))$. Conditional on almost every data realization $(P_n:n\geq1)$, define $\tilde G_n(\cdot)$ as a centered Gaussian process indexed by $x\in\Theta$ with covariance $Cov(\tilde G_n(x_1),\tilde G_n(x_2))=(1/n)\sum_{i=1}^n(h(x_1;\xi)-\hat h(x_1))(h(x_2;\xi)-\hat h(x_2))$, and $\hat h(x)=(1/n)\sum_{i=1}^nh(x;\xi_i)$.
We first show that $\tilde G_n(\cdot)\Rightarrow\tilde G(\cdot)$ in $\ell^\infty(\Theta)$. Note that, by the property of Gaussian processes, any finite-dimensional vector $(\tilde G_n(x_1),\ldots,\tilde G_n(x_d))$ is distributed as $N(\mathbf 0,\Sigma_n)$, where $\mathbf 0$ is the zero vector and $$\Sigma_n=\left(\frac{1}{n}\sum_{k=1}^n(h(x_i;\xi_k)-\hat h(x_i))(h(x_j;\xi_k)-\hat h(x_j))\right)_{i,j=1,\ldots,d}$$
On the other hand, $(\tilde G(x_1),\ldots,\tilde G(x_d))$ is distributed as $N(\mathbf 0,\Sigma)$, where $\Sigma=(Cov_0(h(x_i;\xi),h(x_j;\xi))_{i,j=1,\ldots,d}$. Note that $\Sigma_n\to\Sigma$ a.s. in each entry, and hence $(\tilde G_n(x_1),\ldots,\tilde G_n(x_d))\Rightarrow(\tilde G(x_1),\ldots,\tilde G(x_d))$ (by using for example convergence of the characteristic function).
Next, note that by Assumption \ref{complexity}, $\mathcal H_\Theta$ is $P_0$-Donsker and hence is totally bounded equipped with the semi-metric $\rho_0(h(x_1;\cdot),h(x_2;\cdot)):=(Var_0(h(x_1;\xi)-h(x_2;\xi)))^{1/2}$ (Section 2.1.2 in \cite{van1996weak}). Equivalently, $\Theta$ is totally bounded under the semi-metric $\rho_0(x_1,x_2):=(Var_0((h(x_1;\xi)-h(x_2;\xi)))^{1/2}$.
We shall also show that $\tilde G_n(\cdot)$ is uniformly equicontinuous in probability under the same semi-metric. To this end, we want to show that \begin{equation}
\lim_{\delta\to0}\limsup_{n\to\infty}P_{\bm\xi}\left(\sup_{\rho_0(x_1x_2)<\delta}|\tilde G_n(x_1)-\tilde G_n(x_2)|>\epsilon\right)=0\label{asymptotic equicontinuity} \end{equation} where $P_{\bm\xi}(\cdot)$ is the probability conditional on the data $\bm\xi$. First, by using the covariance structure of the Gaussian process $\tilde G_n(\cdot)$, \begin{eqnarray*} &&E(\tilde G_n(x_1)-\tilde G_n(x_2))^2=\widehat{Var}_n(h(x_1;\xi)-h(x_2;\xi))\\ &=&\frac{1}{n}\sum_{i=1}^n(h(x_1;\xi)-\hat h(x_1))^2+\frac{1}{n}\sum_{i=1}^n(h(x_2;\xi)-\hat h(x_2))^2-\frac{2}{n}\sum_{i=1}^n(h(x_1;\xi)-\hat h(x_1))(h(x_2;\xi)-\hat h(x_2)) \end{eqnarray*}
Now define $\tilde h(x;\xi)=h(x;\xi)-Z_0(x)$ under Assumption \ref{finite mean}. Note that \begin{align*}
E\left[\sup_{x,y\in\Theta}(\tilde h(x;\xi)-\tilde h(y;\xi))^2\right]&\leq E\left[\sup_{x\in\Theta}\tilde h(x;\xi)^2+\sup_{y\in\Theta}\tilde h(y;\xi)^2+2\sup_{x\in\Theta}|\tilde h(x;\xi)|\sup_{y\in\Theta}|\tilde h(x;\xi)|\right]\\
&=4E\|\tilde h(\cdot;\xi)\|_\Theta^2<\infty \end{align*} by Assumption \ref{L2}. Viewing $\tilde h(x;\cdot)$ and $\tilde h(y;\cdot)$ each as a function $(x,y)\in\Theta^2\to\mathbb R$, we can apply the preservation theorem to conclude that the class of functions $$\mathcal H_\Theta^\Pi=\{(\tilde h(x;\cdot)-\tilde h(y;\cdot))^2:(x,y)\in\Theta^2\}$$ is a $P_0$-GC class. Therefore, \begin{equation}
\sup_{x,y\in\Theta}\left|\frac{1}{n}\sum_{i=1}^n(\tilde h(x;\xi_i)-\tilde h(y;\xi_i))^2-E_0(\tilde h(x;\xi_i)-\tilde h(y;\xi_i))^2\right|\to0\text{\ \ a.s.}\label{interim new22} \end{equation} Now, note that \begin{eqnarray} &&\widehat{Var}_n(h(x_1;\xi)-h(x_2;\xi))\notag\\ &=&\frac{1}{n}\sum_{i=1}^n\left(\left(h(x_1;\xi_i)-\hat h(x_1)\right)-\left(h(x_2;\xi_i)-\hat h(x_2)\right)\right)^2\notag\\ &=&\frac{1}{n}\sum_{i=1}^n\left(\left(\tilde h(x_1;\xi_i)-\tilde h(x_2;\xi_i)\right)-\left((\hat h(x_1)-Z_0(x_1))-(\hat h(x_2)-Z_0(x_2))\right)\right)^2\notag\\ &=&\frac{1}{n}\sum_{i=1}^n\left(\tilde h(x_1;\xi_i)-\tilde h(x_2;\xi_i)\right)^2-\left(\left(\hat h(x_1)-Z_0(x_1)\right)-\left(\hat h(x_2)-Z_0(x_2)\right)\right)^2\label{interim updated2} \end{eqnarray}
Since $\mathcal H_\Theta$ is $P_0$-GC, $\|\hat h(\cdot)-Z_0(\cdot)\|_\Theta\to0$ a.s.. Hence $\sup_{x_1,x_2\in\Theta}\left(\left(\hat h(x_1)-Z_0(x_1)\right)-\left(\hat h(x_2)-Z_0(x_2)\right)\right)^2\to0$ a.s.. Combining with \eqref{interim new22}, we have, from \eqref{interim updated2}, \begin{equation}
\sup_{x_1,x_2\in\Theta}|\widehat{Var}_n(h(x_1;\xi)-h(x_2;\xi))-Var_0(h(x_1,\xi)-h(x_2,\xi))|\to0\text{\ \ a.s.}\label{interim new23} \end{equation} by noting that $Var_0(h(x_1,\xi)-h(x_2,\xi))=E_0(\tilde h(x;\xi)-\tilde h(y;\xi))^2$.
Therefore, by \eqref{interim new23}, for any $\gamma>1$, we have $$E_{\bm\xi}(\tilde G_n(x_1)-\tilde G_n(x_2))^2=\widehat{Var}_n(h(x_1;\xi)-h(x_2;\xi))\leq\gamma Var_0(h(x_1;\xi)-h(x_2;\xi))=E(\tilde G_\gamma(x_1)-\tilde G_\gamma(x_2))^2$$ a.s. for any $x_1,x_2\in\Theta$, when $n$ is sufficiently large, where $\tilde G_\gamma(\cdot):=\gamma G(\cdot)$ and $E_{\bm\xi}[\cdot]$ denotes the expectation conditional on $\bm\xi$. Thus, by the argument for the Sudakov-Fernique inequality (the first equation in the proof of Theorem 2.9 in \cite{adler1990introduction}), we have
$$E_{\bm\xi}\left[\sup_{\rho_0(x_1,x_2)<\delta}|\tilde G_n(x_1)-\tilde G_n(x_2)|\right]\leq E\left[\sup_{\rho_0(x_1,x_2)<\delta}|\tilde G_\gamma(x_1)-\tilde G_\gamma(x_2)|\right]=\gamma E\left[\sup_{\rho_0(x_1,x_2)<\delta}|\tilde G(x_1)-\tilde G(x_2)|\right]$$ when $n$ is large. Note that
$$\lim_{\delta\to0}E\left[\sup_{\rho_0(x_1,x_2)<\delta}|\tilde G(x_1)-\tilde G(x_2)|\right]=0$$ since $\tilde G(\cdot)$ is tight by the $P_0$-Donsker property of $\mathcal H_\Theta$. Thus \begin{eqnarray*}
&&\limsup_{n\to\infty}P_{\bm\xi}\left(\sup_{\rho_0(x_1x_2)<\delta}|\tilde G_n(x_1)-\tilde G_n(x_2)|>\epsilon\right)\\
&\leq&\limsup_{n\to\infty}\frac{E_{\bm\xi}\left[\sup_{\rho_0(x_1x_2)<\delta}|\tilde G_n(x_1)-\tilde G_n(x_2)|\right]}{\epsilon}\text{\ \ by Chebyshev's inequality}\\
&\leq&\frac{\gamma E_{\bm\xi}\left[\sup_{\rho_0(x_1x_2)<\delta}|\tilde G(x_1)-\tilde G(x_2)|\right]}{\epsilon}\\ &\to&0 \end{eqnarray*} as $\delta\to0$. We have therefore proved \eqref{asymptotic equicontinuity}.
Together with total boundedness, we have $\tilde G_n(\cdot)\Rightarrow\tilde G(\cdot)$ in $\ell^\infty(\Theta)$ (Section 2.1.2 in \cite{van1996weak}).
Finally, note that $\inf_{x\in\Theta}\sigma_0(x)>0$ by Assumption \ref{nondegeneracy} and Jensen's inequality. Using \eqref{convergence var} and that pointwise division is a continuous map $(\ell^\infty\times\ell^\infty)(\Theta)\to\ell^\infty(\Theta)$, Slutsky's Theorem and the continuous mapping theorem conclude that $G_n(\cdot)\Rightarrow G(\cdot)$ in $\ell^\infty(\Theta)$. \Halmos
\endproof
\proof{Proof of Theorem \ref{main}.} By Theorem \ref{EL process thm} and Lemma \ref{sample Gaussian}, and using the fact that $(\cdot)^2$ and $\sup_{x\in\Theta}\cdot$ are continuous maps $\ell^\infty(\Theta)\to\ell^\infty(\Theta)$ and $\ell^\infty(\Theta)\to\mathbb R$ respectively, we have $\sup_{x\in\Theta}\{-2\log R(x;Z_0)\}\Rightarrow\sup_{x\in\Theta}J(x)$ and $\sup_{x\in\Theta}J_n(x)\Rightarrow\sup_{x\in\Theta}J(x)$, where $J_n(\cdot)$ and $J(\cdot)$ are defined in Theorems \ref{main} and \ref{EL process thm}. Moreover, since $\sup_{x\in\Theta}J(x)$ has a continuous distribution function, pointwise convergence of distribution functions to that of $\sup_{x\in\Theta}J(x)$ implies uniform convergence. Hence we have \begin{equation}
\sup_{q\in\mathbb R^+}\left|P_{\bm\xi}\left(\sup_{x\in\Theta}J_n(x)\leq q\right)-P\left(\sup_{x\in\Theta}J(x)\leq q\right)\right|\to0\label{interim new24} \end{equation} and \begin{equation}
\sup_{q\in\mathbb R^+}\left|P\left(\sup_{x\in\Theta}\{-2\log R(x;Z_0)\}\leq q\right)-P\left(\sup_{x\in\Theta}J(x)\leq q\right)\right|\to0\label{interim new25} \end{equation} Selecting $q_n$ such that $P_{\bm\xi}\left(\sup_{x\in\Theta}J_n(x)\leq q_n\right)=1-\alpha$, \eqref{interim new24} and \eqref{interim new25} implies that $$P\left(\sup_{x\in\Theta}\{-2\log R(x;Z_0)\}\leq q_n\right)\to1-\alpha$$ By applying Proposition \ref{duality} to every point $x\in\Theta$ and with $\chi^2_{1,1-\alpha}$ with $q_n$, we have $-2\log R(x;Z_0)\leq q_n$ if and only if $\underline Z_n^*(x)\leq Z_0(x)\leq\overline Z_n^*(x)$, for each $x\in\Theta$.
Hence \begin{align*} P\left(\sup_{x\in\Theta}\{-2\log R(x;Z_0)\}\leq q_n\right)&=P\left(-2\log R(x;Z_0)\leq q_n\text{\ for all\ }x\in\Theta\right)\\%P\left(\{Z(x)\}_{x\in\Theta}\in\left\{\{\tilde Z(x)\}_{x\in\Theta}:\sup_{x\in\Theta}\{-2\log R(\tilde Z(x))\}\leq\eta\right\}\right)= &=P(\underline Z_n^*(x)\leq Z_0(x)\leq\overline Z_n^*(x)\text{\ for all\ }x\in\Theta)\to1-\alpha \end{align*}\Halmos \endproof
\proof{Proof of Theorem \ref{consistency}.}
To avoid repetition, we focus on $\overline Z_n(x)$. Consider \begin{equation} \overline Z_n(x)-Z_0(x)=\max_{\mathbf w\in\mathcal U_n(q_n/(2n))}\sum_{i=1}^n\tilde h(x;\xi_i)w_i\label{opt consistency1} \end{equation} where $\tilde h(x;\xi)=h(x;\xi)-Z_0(x)$. With Lagrangian relaxation, the program \eqref{opt consistency1} can be written as \begin{eqnarray} &&\min_{\lambda\geq0,\gamma}\max_{\mathbf w\geq\mathbf 0}\sum_{i=1}^n\tilde h(x;\xi_i)w_i-\lambda\left(-\frac{1}{n}\sum_{i=1}^n\log(nw_i)-\frac{q_n}{2n}\right)+\gamma\left(\sum_{i=1}^nw_i-1\right)\notag\\ &=&\min_{\lambda\geq0,\gamma}\sum_{i=1}^n\frac{\lambda}{n}\max_{w_i\geq0}\left\{\frac{\tilde h(x;\xi_i)+\gamma}{\lambda}nw_i+\log(nw_i)-nw_i+1\right\}+\lambda\frac{q_n}{2n}-\gamma\notag\\ &=&\min_{\lambda\geq0,\gamma}-\sum_{i=1}^n\frac{\lambda}{n}\log\left(1-\frac{\tilde h(x;\xi_i)+\gamma}{\lambda}\right)+\lambda\frac{q_n}{2n}-\gamma\label{interim new26} \end{eqnarray} where $-0\log(1-t/0):=0$ for $t\leq0$ and $-0\log(1-t/0):=\infty$ for $t>0$, by using the conjugate function of $-\log r+r-1$ as $\sup_{r\geq0}\{tr+\log r-r+1\}=-\log(1-t)$ for $t<1$, and $\infty$ for $t\geq1$ (e.g., \cite{ben2013robust}).
Now, to get an upper bound for \eqref{interim new26}, pick $\gamma=0$, and $\lambda$ as $\lambda_n=\Theta(n^\varepsilon)$ where $1/2<\varepsilon<1$. Then, by using \eqref{interim new1}, we have
$$\max_{1\leq i\leq n}\|\tilde h(\cdot;\xi_i)\|_\Theta\leq\frac{\lambda_n}{2}\text{\ \ ev.}$$
Using the fact that $-\log(1-t)\leq t+2t^2$ for any $|t|\leq1/2$, we have \eqref{interim new26} bounded from above by \begin{eqnarray} &&\sum_{i=1}^n\frac{\lambda_n}{n}\left(\frac{\tilde h(x;\xi_i)}{\lambda_n}+2\frac{\tilde h(x;\xi_i)^2}{\lambda_n^2}\right)+\lambda_n\frac{q_n}{2n}\notag\\ &=&\frac{1}{n}\sum_{i=1}^n\tilde h(x;\xi_i)+2\frac{(1/n)\sum_{i=1}^n\tilde h(x;\xi_i)^2}{\lambda_n}+\lambda_n\frac{q_n}{2n}\label{interim new27} \end{eqnarray} where $\frac{1}{n}\sum_{i=1}^n\tilde h(x;\xi_i)\to0$ and $\frac{1}{n}\sum_{i=1}^n\tilde h(x;\xi_i)^2$ satisfy
$$\left\|\frac{1}{n}\sum_{i=1}^n\tilde h(\cdot;\xi_i)\right\|_\Theta\to0\text{\ \ a.s.}$$
$$\left\|\frac{1}{n}\sum_{i=1}^n\tilde h(\cdot;\xi_i)^2-\sigma_0^2(\cdot)\right\|_\Theta\to0\text{\ \ a.s.}$$ by the $P_0$-GC property of $\mathcal H_\Theta$ and \eqref{convergence var}. Moreover, by \eqref{interim new24} in the proof of Theorem \ref{main}, we have $P(\sup_{x\in\Theta}J(x)\leq q_n)\to1-\alpha$ a.s.. By the continuity of $\sup_{x\in\Theta}J(x)$, we get $q_n\to q^*$ a.s. where $q^*$ satisfies $P(\sup_{x\in\Theta}J(x)\leq q^*)=1-\alpha$. Hence $q_n/(2n)=\Theta(1/n)$. These imply that \eqref{interim new27} converges to 0 uniformly over $\Theta$.
On the other hand, plugging in $\mathbf w=(1/n)_{1\leq i\leq n}$, $\overline Z_n^*(x)-Z_0(x)$ in \eqref{opt consistency1} is bounded from below by $(1/n)\sum_{i=1}^n\tilde h(x;\xi_i)$, which converges to 0 uniformly over $\Theta$. Combining with above, we get \begin{equation}
\|\overline Z_n^*(\cdot)-Z_0(\cdot)\|_\Theta\to0\text{\ \ a.s.}\label{interim new28} \end{equation}
\Halmos \endproof
\proof{Proof of Theorem \ref{consistency simple}.} For any fixed $x$, $\underline Z_n(x)\to Z_0(x)$ and $\overline Z_n(x)\to Z_0(x)$ follows as a special case of \eqref{interim new28}.\Halmos \endproof
\proof{Proof of Theorem \ref{asymptotic equivalence}.} To avoid redundancy, we focus only on the upper bound $\overline Z_n^*(x)$. Consider $\overline Z_n^*(x)-\hat h(x)$, which can be written as \begin{equation} \max\left\{\sum_{i=1}^nw_i\hat h(x;\xi_i):-\frac{1}{n}\sum_{i=1}^n\log(nw_i)\leq\frac{q_n}{2n},\ \sum_{i=1}^nw_i=1,\ w_i\geq0\text{\ for all\ }i=1,\ldots,n\right\}\label{interim asymptotic} \end{equation} where $\hat h(x;\xi_i)=h(x;\xi_i)-\hat h_n(x)$. Similar to the proof of Theorem \ref{consistency}, a Lagrangian relaxation of \eqref{interim asymptotic} gives \begin{eqnarray} &&\min_{\lambda\geq0,\gamma}\max_{\mathbf w\geq\mathbf 0}\sum_{i=1}^nw_i\hat h(x;\xi_i)-\lambda\left(-\frac{1}{n}\sum_{i=1}^n\log(nw_i)-\frac{q_n}{2n}\right)+\gamma\left(\sum_{i=1}^nw_i-1\right)\notag\\ &\leq&\min_{\lambda>0,\gamma}\sum_{i=1}^n\frac{\lambda}{n}\max_{w_i\geq0}\left\{nw_i\frac{\hat h(x;\xi_i)+\gamma}{\lambda}+\log(nw_i)-nw_i+1\right\}+\frac{\lambda q_n}{2n}-\gamma\notag\\ &\leq&\min_{\lambda>0,\gamma,\frac{\hat h(x;\xi_i)+\gamma}{\lambda}<1\text{\ for all\ }i=1,\ldots,n}-\sum_{i=1}^n\frac{\lambda}{n}\log\left(1-\frac{\hat h(x;\xi_i)+\gamma}{\lambda}\right)+\frac{\lambda q_n}{2n}-\gamma\label{interim asymptotic1} \end{eqnarray} by using the fact that $-\log(1-t)=\sup_{r\geq0}\{tr+\log r-r+1\}$ is the conjugate function of $-\log r+r-1$, defined for $t<1$.
Now, given $x$, we choose $\gamma=0$, and $\lambda=\frac{\sqrt n\hat\sigma(x)}{\sqrt{q_n}}$. Similar to the proof of Theorem \ref{consistency}, we have
$$\left\|\frac{1}{n}\sum_{i=1}^n\hat h(\cdot;\xi_i)^2-\sigma_0^2(\cdot)\right\|_\Theta\to0\text{\ \ a.s.}$$ by the $P_0$-GC property of $\mathcal H_\Theta$ and \eqref{convergence var}, and $q_n\to q^*$ a.s. where $q^*$ satisfies $P(\sup_{x\in\Theta}J(x)\leq q^*)=1-\alpha$. Moreover, $$\hat\sigma^2(x)=\frac{1}{n}\sum_{i=1}^n(h(x;\xi_i)-Z_0(x))^2-(\hat h(x)-Z_0(x))^2$$ By Assumptions \ref{finite mean}, \ref{L2} and \ref{complexity}, $\{(h(x;\cdot)-Z_0(x))^2:x\in\Theta\}$ is $P_0$-Donsker, and we have
$$\sup_{x\in\Theta}|\hat\sigma^2(x)-\sigma_0^2(x)|\to0\text{\ \ as $n\to\infty$ a.s.}$$ Together with the assumption that $h$ is bounded, we have \begin{equation} \frac{\hat h(x;\xi_i)+\gamma}{\lambda}=\frac{\hat h(x;\xi_i)\sqrt{q_n}}{\sqrt n\hat\sigma(x)}\to0\label{interim asymptotic2} \end{equation} a.s. uniformly for all $i=1,\ldots,n$ as $n\to\infty$.
Therefore, \eqref{interim asymptotic1} is bounded from above ev. by \begin{eqnarray} &&-\sum_{i=1}^n\frac{\hat\sigma(x)}{\sqrt{nq_n}}\log\left(1-\frac{\hat h(x;\xi_i)\sqrt q_n}{\sqrt n\hat\sigma(x)}\right)+\frac{\sqrt q_n\hat\sigma(x)}{2\sqrt n}\notag\\ &=&\sum_{i=1}^n\frac{\hat\sigma(x)}{\sqrt{nq_n}}\left(\frac{\tilde h(x;\xi_i)\sqrt q_n}{\sqrt n\hat\sigma(x)}+\frac{1}{2}\left(\frac{\tilde h(x;\xi_i)\sqrt q_n}{\sqrt n\hat\sigma(x)}\right)^2+O\left(\left(\frac{\hat h(x;\xi_i)\sqrt q_n}{\sqrt n\hat\sigma(x)}\right)^3\right)\right)+\frac{\sqrt q_n\hat\sigma(x)}{2\sqrt n}{}\notag\\ &&{}\text{\ \ where $O(\cdot)$ is uniform over $x\in\Theta$}\notag\\ &=&\frac{\sqrt{q_n}\hat\sigma(x)}{2\sqrt n}+O\left(\frac{\hat\mu_3(x)q_n}{n\hat\sigma(x)^2}\right)+\frac{\sqrt q_n\hat\sigma(x)}{2\sqrt n}{}\notag\\ &&{}\text{\ \ since $\sum_{i=1}^n\hat h(x;\xi_i)=0$ and $\frac{1}{n}\sum_{i=1}^n\hat h(x;\xi_i)^2=\hat\sigma^2(x)$,}{}\notag\\
&&{}\text{\ \ where $\hat\mu_3(x)=\frac{1}{n}\sum_{i=1}^n|\hat h(x;\xi_i)|^3$, which is uniformly bounded over $x\in\Theta$ a.s. since $h$ is bounded}\notag\\ &=&\frac{\sqrt{q_n}\hat\sigma(x)}{\sqrt n}+O\left(\frac{1}{n}\right)\label{interim asymptotic3} \end{eqnarray}
On the other hand, choose $$w_i=\frac{1}{n}\left(1+\frac{\hat h(x;\xi_i)\sqrt{q_n}}{\sqrt n\hat\sigma(x)}\left(1-\frac{C}{\sqrt n}\right)\right)$$ for some large enough $C>0$. When $n$ is large enough, we have $w_i>0$ a.s. for all $i=1,\ldots,n$ and $x\in\Theta$ since $\frac{\hat h(x;\xi_i)\sqrt{q_n}}{\sqrt n\hat\sigma(x)}<1$ ev. by the same argument in \eqref{interim asymptotic2}. Note that $\sum_{i=1}^nw_i=1$ since $\sum_{i=1}^n\hat h(x;\xi_i)=0$ by definition. Moreover, \begin{eqnarray*} &&-\frac{1}{n}\sum_{i=1}^n\log(nw_i)\\ &=&-\frac{1}{n}\sum_{i=1}^n\log\left(1+\frac{\hat h(x;\xi_i)\sqrt{q_n}}{\sqrt n\hat\sigma(x)}\left(1-\frac{C}{\sqrt n}\right)\right)\\ &=&-\frac{1}{n}\sum_{i=1}^n\frac{\hat h(x;\xi_i)\sqrt{q_n}}{\sqrt n\hat\sigma(x)}\left(1-\frac{C}{\sqrt n}\right)+\frac{1}{n}\frac{1}{2}\sum_{i=1}^n\frac{\hat h(x;\xi_i)^2q_n}{n\hat\sigma(x)^2}\left(1-\frac{C}{\sqrt n}\right)^2+O\left(\frac{\hat\mu_3(x)q_n^{3/2}}{n^{3/2}\hat\sigma(x)^3}\left(1-\frac{C}{\sqrt n}\right)^3\right){}\\ &&{}\text{\ \ where $O(\cdot)$ is uniform over $x\in\Theta$}\\ &=&\frac{q_n}{2n}\left(1-\frac{2C}{\sqrt n}\right)+O\left(\frac{1}{n^2}\right)+O\left(\frac{\hat\mu_3(x)q_n^{3/2}}{n^{3/2}\hat\sigma(x)^3}\right){}\\ &&{}\text{\ \ where the last $O(\cdot)$ has leading term that is independent of $C$}\\ &\leq&\frac{q_n}{2n} \end{eqnarray*} when $n$ is large enough, by choosing a large enough $C$. Therefore, the chosen $w_i$'s form a feasible solution in $\mathcal U_n(q_n/(2n))$. We have \begin{align} \sum_{i=1}^nw_i\hat h(x;\xi_i)&=\sum_{i=1}^n\hat h(x;\xi_i)\frac{1}{n}\left(1+\frac{\hat h(x;\xi_i)\sqrt{q_n}}{\sqrt n\hat\sigma(x)}\left(1-\frac{C}{\sqrt n}\right)\right)\notag\\ &=\sqrt{q_n}\frac{\hat\sigma(x)}{\sqrt n}\left(1-\frac{C}{\sqrt n}\right)\notag\\ &=\sqrt{q_n}\frac{\hat\sigma(x)}{\sqrt n}+O\left(\frac{1}{n}\right)\label{interim asymptotic4} \end{align}
Combining the bound for the dual and the primal bounds \eqref{interim asymptotic3} and \eqref{interim asymptotic4}, we conclude that $\overline Z_n^*(x)=\sqrt{q_n}\frac{\hat\sigma(x)}{\sqrt n}+O\left(\frac{1}{n}\right)$ uniformly over $x\in\Theta$. The proof for $\underline Z_n^*(x)$ follows by merely replacing $h$ with $-h$. This concludes the theorem.\Halmos \endproof
\section{Review of Empirical Processes}\label{sec:EP}
We review some terminologies and results in the empirical process theory that are related to our developments.
Given a class of functions $\mathcal F=\{f:\Xi\to\mathbb R\}$, we define the empirical measure $\mathbb P_n$, generated from i.i.d. $\xi_1,\ldots,\xi_n$ each under $P$, as a map from $\mathcal F$ to $\mathbb R$ such that $$\mathbb P_n(f)=\frac{1}{n}\sum_{i=1}^nf(\xi_i)$$
We also define $P(f)=\int f(\xi)dP(\xi)=E_P[f(\xi)]$ where $E_P[\cdot]$ is the expectation under $P$. The empirical process indexed by $f\in\mathcal F$ is defined as $$\sqrt n(\mathbb P_n-P)$$
For any functions $y:\mathcal F\to\mathbb R$, we define $\|y\|_{\mathcal F}=\sup_{f\in\mathcal F}|y(f)|$. We also define the envelope of $\mathcal F$ as a function that maps from $\Xi$ to $\mathbb R$ given by
$$\sup_{f\in\mathcal F}|f(\xi)|$$
\begin{definition} We call $\mathcal F$ a $P$-Glivenko-Cantelli (GC) class if the empirical measure under $P$ satisfies
$$\|\mathbb P_n-P\|_{\mathcal F}:=\sup_{f\in\mathcal F}|\mathbb P_n(f)-P(f)|\stackrel{a.s.}{\to}0\text{\ \ as\ }n\to\infty$$ \end{definition} \begin{definition} We call $\mathcal F$ a $P$-Donsker class if the empirical process under $P$ satisfies \begin{equation} \sqrt n(\mathbb P_n-P)\Rightarrow\mathbb G\text{\ \ in\ }\ell^\infty(\mathcal F)\label{Donsker thm} \end{equation} where $\mathbb G$ is a Gaussian process indexed by $\mathcal F$, centered, with covariance function $$Cov(\mathbb G(f_1),\mathbb G(f_2))=Cov_P(f_1(\xi),f_2(\xi))=P(f_1f_2)-P(f_1)P(f_2)$$ where $Cov_P(\cdot,\cdot)$ denotes the covariance under $P$, and
$$\ell^\infty(\mathcal F)=\left\{y:\mathcal F\to\mathbb R\Bigg|\|y\|_{\mathcal F}<\infty\right\}$$ Moreover, the process $\mathbb G$ has uniformly continuous sample paths with respect to the canonical semi-metric $\rho_P(f_1,f_2)=Var_P(f_1(\xi)-f_2(\xi))$, where $Var_P(\cdot)$ denotes the variance under $P$.\label{def:Donsker} \end{definition}
We have ignored the measurability issues, in particular the use of outer and inner probability measures, in the definitions (see \cite{van1996weak}).
\begin{theorem}[Preservation of GC classes; \cite{van2000preservation}, Theorem 3] Suppose that $\mathcal F_1,\ldots,\mathcal F_k$ are $P$-GC classes of functions, and that $\varphi:\mathbb R^k\to\mathbb R$ is continuous. Then $\mathcal H=\varphi(\mathcal F_1,\ldots,\mathcal F_k)$ is $P$-GC provided that it has an integrable envelope function.\label{preservation} \end{theorem}
\section{Other Useful Theorems}\label{sec:thms} \begin{lemma}[\cite{owen2001empirical}, Lemma 11.2]
Let $Y_i$ be i.i.d. random variables on $\mathbb R$ with $EY_i^2<\infty$. Then $\max_{1\leq i\leq n}|Y_i|=o(n^{1/2})$ a.s..\label{lemma max} \end{lemma}
\begin{theorem}[Continuous Mapping Theorem;\cite{van1996weak},Theorem 1.3.6] Let $g:\mathbb D\to\mathbb E$ be continuous at every point $\mathbb D_0\subset\mathbb D$. If $X_n\Rightarrow X$ and $X$ takes its values in $\mathbb D_0$, then $g(X_n)\Rightarrow g(X)$.\label{continuous mapping} \end{theorem}
\begin{theorem}[Slutsky's Theorem; \cite{van1996weak}, Example 1.4.7] If $X_n\Rightarrow X$ and $Y_n\Rightarrow c$ where $X$ is separable and $c$ is a constant, then $(X_n,Y_n)\Rightarrow(X,c)$ under the product topology.\label{Slutsky} \end{theorem}
\end{APPENDICES}
\end{document} | arXiv |
Random variable
Random variable explained
In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is described informally as a variable whose values depend on outcomes of a random phenomenon.[1] The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a probability space that maps from the sample space to the real numbers.[2] A random variable's possible values might represent the possible outcomes of a yet-to-be-performed experiment, or the possible outcomes of a past experiment whose already-existing value is uncertain (for example, because of imprecise measurements or quantum uncertainty). They may also conceptually represent either the results of an "objectively" random process (such as rolling a die) or the "subjective" randomness that results from incomplete knowledge of a quantity. The meaning of the probabilities assigned to the potential values of a random variable is not part of probability theory itself, but is instead related to philosophical arguments over the interpretation of probability. The mathematics works the same regardless of the particular interpretation in use.
As a function, a random variable is required to be measurable, which allows for probabilities to be assigned to sets of its potential values. It is common that the outcomes depend on some physical variables that are not predictable. For example, when tossing a fair coin, the final outcome of heads or tails depends on the uncertain physical conditions, so the outcome being observed is uncertain. The coin could get caught in a crack in the floor, but such a possibility is excluded from consideration.
The domain of a random variable is called a sample space, defined as the set of possible outcomes of a non-deterministic event. For example, in the event of a coin toss, only two possible outcomes are possible: heads or tails.
A random variable has a probability distribution, which specifies the probability of Borel subsets of its range. Random variables can be discrete, that is, taking any of a specified finite or countable list of values (having a countable range), endowed with a probability mass function that is characteristic of the random variable's probability distribution; or continuous, taking any numerical value in an interval or collection of intervals (having an uncountable range), via a probability density function that is characteristic of the random variable's probability distribution; or a mixture of both.
Two random variables with the same probability distribution can still differ in terms of their associations with, or independence from, other random variables. The realizations of a random variable, that is, the results of randomly choosing values according to the variable's probability distribution function, are called random variates.
Although the idea was originally introduced by Christiaan Huygens, the first person to think systematically in terms of random variables was Pafnuty Chebyshev.[3] [4]
A random variable
is a measurable function
X\colon\Omega\toE
from a set of possible outcomes
\Omega
to a measurable space
. The technical axiomatic definition requires
to be a sample space of a probability triple
(\Omega,l{F},\operatorname{P})
(see the measure-theoretic definition). A random variable is often denoted by capital roman letters such as
.[5] [6]
The probability that
takes on a value in a measurable set
S\subseteqE
is written as
\operatorname{P}(X\inS)=\operatorname{P}(\{\omega\in\Omega\midX(\omega)\inS\})
Standard case
In many cases,
is real-valued, i.e.
E=R
. In some contexts, the term random element (see extensions) is used to denote a random variable not of this form.
When the image (or range) of
is countable, the random variable is called a discrete random variable[7] and its distribution is a discrete probability distribution, i.e. can be described by a probability mass function that assigns a probability to each value in the image of
. If the image is uncountably infinite (usually an interval) then
is called a continuous random variable.[8] [9] In the special case that it is absolutely continuous, its distribution can be described by a probability density function, which assigns probabilities to intervals; in particular, each individual point must necessarily have probability zero for an absolutely continuous random variable. Not all continuous random variables are absolutely continuous,[10] a mixture distribution is one such counterexample; such random variables cannot be described by a probability density or a probability mass function.
Any random variable can be described by its cumulative distribution function, which describes the probability that the random variable will be less than or equal to a certain value.
The term "random variable" in statistics is traditionally limited to the real-valued case (
). In this case, the structure of the real numbers makes it possible to define quantities such as the expected value and variance of a random variable, its cumulative distribution function, and the moments of its distribution.
of values. Thus one can consider random elements of other sets
, such as random boolean values, categorical values, complex numbers, vectors, matrices, sequences, trees, sets, shapes, manifolds, and functions. One may then specifically refer to a random variable of type
, or an
-valued random variable.
This more general concept of a random element is particularly useful in disciplines such as graph theory, machine learning, natural language processing, and other fields in discrete mathematics and computer science, where one is often interested in modeling the random variation of non-numerical data structures. In some cases, it is nonetheless convenient to represent each element of
, using one or more real numbers. In this case, a random element may optionally be represented as a vector of real-valued random variables (all defined on the same underlying probability space
, which allows the different random variables to covary). For example:
A random word may be represented as a random integer that serves as an index into the vocabulary of possible words. Alternatively, it can be represented as a random indicator vector, whose length equals the size of the vocabulary, where the only values of positive probability are
(1 0 0 0 … )
and the position of the 1 indicates the word.
A random sentence of given length
may be represented as a vector of
random words.
A random graph on
given vertices may be represented as a
N x N
matrix of random variables, whose values specify the adjacency matrix of the random graph.
may be represented as a collection of random variables
, giving the function's values at the various points
in the function's domain. The
are ordinary real-valued random variables provided that the function is real-valued. For example, a stochastic process is a random function of time, a random vector is a random function of some index set such as
1,2,\ldots,n
, and random field is a random function on any set (typically time, space, or a discrete set).
Distribution functions
If a random variable
X\colon\Omega\toR
defined on the probability space
is given, we can ask questions like "How likely is it that the value of
is equal to 2?". This is the same as the probability of the event
\{\omega:X(\omega)=2\}
which is often written as
P(X=2)
pX(2)
for short.
Recording all these probabilities of output ranges of a real-valued random variable
yields the probability distribution of
. The probability distribution "forgets" about the particular probability space used to define
and only records the probabilities of various values of
. Such a probability distribution can always be captured by its cumulative distribution function
FX(x)=\operatorname{P}(X\lex)
and sometimes also using a probability density function,
. In measure-theoretic terms, we use the random variable
to "push-forward" the measure
to a measure
.The underlying probability space
is a technical device used to guarantee the existence of random variables, sometimes to construct them, and to define notions such as correlation and dependence or independence based on a joint distribution of two or more random variables on the same probability space. In practice, one often disposes of the space
altogether and just puts a measure on
that assigns measure 1 to the whole real line, i.e., one works with probability distributions instead of random variables. See the article on quantile functions for fuller development.
Discrete random variable
In an experiment a person may be chosen at random, and one random variable may be the person's height. Mathematically, the random variable is interpreted as a function which maps the person to the person's height. Associated with the random variable is a probability distribution that allows the computation of the probability that the height is in any subset of possible values, such as the probability that the height is between 180 and 190 cm, or the probability that the height is either less than 150 or more than 200 cm.
Another random variable may be the person's number of children; this is a discrete random variable with non-negative integer values. It allows the computation of probabilities for individual integer values – the probability mass function (PMF) – or for sets of values, including infinite sets. For example, the event of interest may be "an even number of children". For both finite and infinite event sets, their probabilities can be found by adding up the PMFs of the elements; that is, the probability of an even number of children is the infinite sum
\operatorname{PMF}(0)+\operatorname{PMF}(2)+\operatorname{PMF}(4)+ …
In examples such as these, the sample space is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space. But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from the same random person, for example so that questions of whether such random variables are correlated or not can be posed.
If \, \ are countable sets of real numbers, b_n >0 and
\sumnbn=1
, then
F=\sumnbn
\delta
is a discrete distribution function. Here
\deltat(x)=0
x<t
x\get
. Taking for instance an enumeration of all rational numbers as
\{an\}
, one gets a discrete distribution function that is not a step function or piecewise constant.
The possible outcomes for one coin toss can be described by the sample space
\Omega=\{heads,tails\}
. We can introduce a real-valued random variable
that models a $1 payoff for a successful bet on heads as follows:
Y(\omega)= \begin{cases} 1,&if\omega=heads,\\[6pt] 0,&if\omega=tails. \end{cases}
given by:
fY(y)= \begin{cases} \tfrac12,&ify=1,\\[6pt] \tfrac12,&ify=0, \end{cases}
Dice roll
A random variable can also be used to describe the process of rolling dice and the possible outcomes. The most obvious representation for the two-dice case is to take the set of pairs of numbers n1 and n2 from (representing the numbers on the two dice) as the sample space. The total number rolled (the sum of the numbers in each pair) is then a random variable X given by the function that maps the pair to the sum:
X((n1,n2))=n1+n2
and (if the dice are fair) has a probability mass function ƒX given by:
fX(S)=
min(S-1,13-S)
,forS\in\{2,3,4,5,6,7,8,9,10,11,12\}
Continuous random variable
Formally, a continuous random variable is a random variable whose cumulative distribution function is continuous everywhere.[11] There are no "gaps", which would correspond to numbers which have a finite probability of occurring. Instead, continuous random variables almost never take an exact prescribed value c (formally, \forall c \in \mathbb:\; \Pr(X = c) = 0) but there is a positive probability that its value will lie in particular intervals which can be arbitrarily small. Continuous random variables usually admit probability density functions (PDF), which characterize their CDF and probability measures; such distributions are also called absolutely continuous; but some continuous distributions are singular, or mixes of an absolutely continuous part and a singular part.
An example of a continuous random variable would be one based on a spinner that can choose a horizontal direction. Then the values taken by the random variable are directions. We could represent these directions by North, West, East, South, Southeast, etc. However, it is commonly more convenient to map the sample space to a random variable which takes values which are real numbers. This can be done, for example, by mapping a direction to a bearing in degrees clockwise from North. The random variable then takes values which are real numbers from the interval [0, 360), with all parts of the range being "equally likely". In this case, '''''X''''' = the angle spun. Any real number has probability zero of being selected, but a positive probability can be assigned to any ''range'' of values. For example, the probability of choosing a number in [0, 180] is . Instead of speaking of a probability mass function, we say that the probability density of X is 1/360. The probability of a subset of [0, 360) can be calculated by multiplying the measure of the set by 1/360. In general, the probability of a set for a given continuous random variable can be calculated by integrating the density over the given set. More formally, given any [[Interval (mathematics)|interval]] I = [a, b] = \, a random variable
XI\sim\operatorname{U}(I)=\operatorname{U}[a,b]
is called a "continuous uniform random variable" (CURV) if the probability that it takes a value in a subinterval depends only on the length of the subinterval. This implies that the probability of
falling in any subinterval
[c,d]\sube[a,b]
is proportional to the length of the subinterval, that is, if, one has
\Pr\left(X_I \in [c,d]\right) = \frac
where the last equality results from the unitarity axiom of probability. The probability density function of a CURV
X\sim\operatorname{U}[a,b]
is given by the indicator function of its interval of support normalized by the interval's length: f_X(x) = \begin \displaystyle, & a \le x \le b \\ 0, & \text. \endOf particular interest is the uniform distribution on the unit interval
[0,1]
. Samples of any desired probability distribution
\operatorname{D}
can be generated by calculating the quantile function of
on a randomly-generated number distributed uniformly on the unit interval. This exploits properties of cumulative distribution functions, which are a unifying framework for all random variables.
Mixed type
A mixed random variable is a random variable whose cumulative distribution function is neither piecewise-constant (a discrete random variable) nor everywhere-continuous. It can be realized as the sum of a discrete random variable and a continuous random variable; in which case the will be the weighted average of the CDFs of the component variables.
An example of a random variable of mixed type would be based on an experiment where a coin is flipped and the spinner is spun only if the result of the coin toss is heads. If the result is tails, X = −1; otherwise X = the value of the spinner as in the preceding example. There is a probability of that this random variable will have the value −1. Other ranges of values would have half the probabilities of the last example.
Most generally, every probability distribution on the real line is a mixture of discrete part, singular part, and an absolutely continuous part; see . The discrete part is concentrated on a countable set, but this set may be dense (like the set of all rational numbers).
Measure-theoretic definition
The most formal, axiomatic definition of a random variable involves measure theory. Continuous random variables are defined in terms of sets of numbers, along with functions that map such sets to probabilities. Because of various difficulties (e.g. the Banach–Tarski paradox) that arise if such sets are insufficiently constrained, it is necessary to introduce what is termed a sigma-algebra to constrain the possible sets over which probabilities can be defined. Normally, a particular such sigma-algebra is used, the Borel σ-algebra, which allows for probabilities to be defined over any sets that can be derived either directly from continuous intervals of numbers or by a finite or countably infinite number of unions and/or intersections of such intervals.
The measure-theoretic definition is as follows.
(\Omega,l{F},P)
be a probability space and
(E,l{E})
a measurable space. Then an
-valued random variable is a measurable function
, which means that, for every subset
B\inl{E}
, its preimage is
l{F}
-measurable;
X-1(B)\inl{F}
, where
X-1(B)=\{\omega:X(\omega)\inB\}
. This definition enables us to measure any subset
in the target space by looking at its preimage, which by assumption is measurable.
In more intuitive terms, a member of
is a possible outcome, a member of
is a measurable subset of possible outcomes, the function
gives the probability of each such measurable subset,
represents the set of values that the random variable can take (such as the set of real numbers), and a member of
l{E}
is a "well-behaved" (measurable) subset of
(those for which the probability may be determined). The random variable is then a function from any outcome to a quantity, such that the outcomes leading to any useful subset of quantities for the random variable have a well-defined probability.
is a topological space, then the most common choice for the σ-algebra
is the Borel σ-algebra
l{B}(E)
, which is the σ-algebra generated by the collection of all open sets in
. In such case the
-valued random variable is called an
-valued random variable. Moreover, when the space
is the real line
, then such a real-valued random variable is called simply a random variable.
Real-valued random variables
In this case the observation space is the set of real numbers. Recall,
is the probability space. For a real observation space, the function
X\colon\Omega → R
is a real-valued random variable if
\{\omega:X(\omega)\ler\}\inl{F} \forallr\inR.
This definition is a special case of the above because the set
\{(-infty,r]:r\in\R\}
generates the Borel σ-algebra on the set of real numbers, and it suffices to check measurability on any generating set. Here we can prove measurability on this generating set by using the fact that
\{\omega:X(\omega)\ler\}=X-1((-infty,r])
The probability distribution of a random variable is often characterised by a small number of parameters, which also have a practical interpretation. For example, it is often enough to know what its "average value" is. This is captured by the mathematical concept of expected value of a random variable, denoted
\operatorname{E}[X]
, and also called the first moment. In general,
\operatorname{E}[f(X)]
is not equal to
f(\operatorname{E}[X])
. Once the "average value" is known, one could then ask how far from this average value the values of
typically are, a question that is answered by the variance and standard deviation of a random variable.
can be viewed intuitively as an average obtained from an infinite population, the members of which are particular evaluations of
Mathematically, this is known as the (generalised) problem of moments: for a given class of random variables
, find a collection
\{fi\}
of functions such that the expectation values
\operatorname{E}[fi(X)]
fully characterise the distribution of the random variable
Moments can only be defined for real-valued functions of random variables (or complex-valued, etc.). If the random variable is itself real-valued, then moments of the variable itself can be taken, which are equivalent to moments of the identity function
f(X)=X
of the random variable. However, even for non-real-valued random variables, moments can be taken of real-valued functions of those variables. For example, for a categorical random variable X that can take on the nominal values "red", "blue" or "green", the real-valued function
[X=green]
can be constructed; this uses the Iverson bracket, and has the value 1 if
has the value "green", 0 otherwise. Then, the expected value and other moments of this function can be determined.
Functions of random variables
g\colonR → R
to the outcomes of a real-valued random variable
. That is,
Y=g(X)
. The cumulative distribution function of
is then
FY(y)=\operatorname{P}(g(X)\ley).
If function
is invertible (i.e.,
h=g-1
exists, where
's inverse function) and is either increasing or decreasing, then the previous relation can be extended to obtain
FY(y)=\operatorname{P}(g(X)\ley)= \begin{cases} \operatorname{P}(X\leh(y))=FX(h(y)),&ifh=g-1increasing,\\ \\ \operatorname{P}(X\geh(y))=1-FX(h(y)),&ifh=g-1decreasing. \end{cases}
With the same hypotheses of invertibility of
, assuming also differentiability, the relation between the probability density functions can be found by differentiating both sides of the above expression with respect to
, in order to obtain
fY(y)=fXl(h(y)r)\left|
dh(y)
\right|.
If there is no invertibility of
but each
admits at most a countable number of roots (i.e., a finite, or countably infinite, number of
such that
y=g(xi)
) then the previous relation between the probability density functions can be generalized with
fY(y)=\sumifX(g
(y))\left|
\right|
xi=
, according to the inverse function theorem. The formulas for densities do not demand
to be increasing.
In the measure-theoretic, axiomatic approach to probability, if a random variable
and a Borel measurable function
is also a random variable on
, since the composition of measurable functions is also measurable. (However, this is not necessarily true if
is Lebesgue measurable.) The same procedure that allowed one to go from a probability space
(\Omega,P)
(R,dFX)
can be used to obtain the distribution of
be a real-valued, continuous random variable and let
Y=X2
FY(y)=\operatorname{P}(X2\ley).
y<0
P(X2\leqy)=0
, so
FY(y)=0 \hbox{if} y<0.
y\geq0
\operatorname{P}(X2\ley)=\operatorname{P}(|X|\le\sqrt{y}) =\operatorname{P}(-\sqrt{y}\leX\le\sqrt{y}),
FY(y)=FX(\sqrt{y})-FX(-\sqrt{y}) \hbox{if} y\ge0.
Suppose
is a random variable with a cumulative distribution
FX(x)=P(X\leqx)=
(1+e-x)\theta
\theta>0
is a fixed parameter. Consider the random variable
Y=log(1+e-X).
FY(y)=P(Y\leqy)=P(log(1+e-X)\leqy)=P(X\geq-log(ey-1)).
The last expression can be calculated in terms of the cumulative distribution of
\begin{align} FY(y)&=1-
X(-log(e
-1))\\[5pt] &=1-
(1 +
log(ey-1)
)\theta
\\[5pt] &=1-
(1+ey-1)\theta
\\[5pt] &=1-e-y. \end{align}
which is the cumulative distribution function (CDF) of an exponential distribution.
is a random variable with a standard normal distribution, whose density is
fX(x)=
\sqrt{2\pi
}e^.
Consider the random variable
Y=X2.
We can find the density using the above formula for a change of variables:
In this case the change is not monotonic, because every value of
has two corresponding values of
(one positive and negative). However, because of symmetry, both halves will transform identically, i.e.,
fY(y)=
X(g
dg-1(y)
The inverse transformation is
x=g-1(y)=\sqrt{y}
and its derivative is
2\sqrt{y
} .
fY(y)=2
}e^ \frac = \frace^.
This is a chi-squared distribution with one degree of freedom.
is a random variable with a normal distribution, whose density is
\sqrt{2\pi\sigma2
(one positive and negative). Differently from the previous example, in this case however, there is no symmetry and we have to compute the two distinct terms:
fY(y)=fX(g
\right|+fX(g
x=
(y)=\pm\sqrt{y}
=\pm
} \frac (e^+e^) .
This is a noncentral chi-squared distribution with one degree of freedom.
Some properties
The probability distribution of the sum of two independent random variables is the convolution of each of their distributions.
Probability distributions are not a vector space—they are not closed under linear combinations, as these do not preserve non-negativity or total integral 1—but they are closed under convex combination, thus forming a convex subset of the space of functions (or measures).
Equivalence of random variables
There are several different senses in which random variables can be considered to be equivalent. Two random variables can be equal, equal almost surely, or equal in distribution.
In increasing order of strength, the precise definition of these notions of equivalence is given below.
Equality in distribution
If the sample space is a subset of the real line, random variables X and Y are equal in distribution (denoted
X\stackrel{d}{=}Y
) if they have the same distribution functions:
\operatorname{P}(X\lex)=\operatorname{P}(Y\lex) forallx.
To be equal in distribution, random variables need not be defined on the same probability space. Two random variables having equal moment generating functions have the same distribution. This provides, for example, a useful method of checking equality of certain functions of independent, identically distributed (IID) random variables. However, the moment generating function exists only for distributions that have a defined Laplace transform.
Almost sure equality
Two random variables X and Y are equal almost surely (denoted
X \stackrel{a.s.
} \; Y) if, and only if, the probability that they are different is zero:
\operatorname{P}(X ≠ Y)=0.
For all practical purposes in probability theory, this notion of equivalence is as strong as actual equality. It is associated to the following distance:
dinfty(X,Y)=\operatorname{ess}\sup\omega|X(\omega)-Y(\omega)|,
where "ess sup" represents the essential supremum in the sense of measure theory.
Finally, the two random variables X and Y are equal if they are equal as functions on their measurable space:
X(\omega)=Y(\omega) \hbox{forall}\omega.
This notion is typically the least useful in probability theory because in practice and in theory, the underlying measure space of the experiment is rarely explicitly characterized or even characterizable.
See main article: Convergence of random variables. A significant theme in mathematical statistics consists of obtaining convergence results for certain sequences of random variables; for instance the law of large numbers and the central limit theorem.
There are various senses in which a sequence
of random variables can converge to a random variable
. These are explained in the article on convergence of random variables.
Aleatoricism
Algebra of random variables
Event (probability theory)
Multivariate random variable
Pairwise independent random variables
Observable variable
Random element
Random measure
Random number generator produces a random value
Random vector
Stochastic process
Relationships among probability distributions
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Book: Kallenberg, Olav . 2001 . Foundations of Modern Probability . 2nd . . Berlin . 0-387-95313-2 .
Book: Papoulis, Athanasios . Athanasios Papoulis . 1965 . Probability, Random Variables, and Stochastic Processes . . Tokyo . 9th . 0-07-119981-0 .
Book: Blitzstein. Joe. Hwang. Jessica. Introduction to Probability. 2014. CRC Press. 9781466575592.
Web site: Economics 245A – Introduction to Measure Theory . Steigerwald . Douglas G. . University of California, Santa Barbara . April 26, 2013.
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Bulletin of the American Mathematical Society . New Series. 3. 1. July 1980. HARMONIC ANALYSIS AS THE EXPLOITATION OF SYMMETRY - A HISTORICAL SURVEY. George Mackey.
Web site: 2020-04-26. List of Probability and Statistics Symbols. 2020-08-21. Math Vault. en-US.
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Book: Yates . Daniel S. . Moore . David S . Starnes . Daren S. . 2003 . The Practice of Statistics . 2nd . . New York . 978-0-7167-4773-4 . dead . https://web.archive.org/web/20050209001108/http://bcs.whfreeman.com/yates2e/ . 2005-02-09 .
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Random variable". | CommonCrawl |
SPIKE algorithm
The SPIKE algorithm is a hybrid parallel solver for banded linear systems developed by Eric Polizzi and Ahmed Sameh^
Overview
The SPIKE algorithm deals with a linear system AX = F, where A is a banded $n\times n$ matrix of bandwidth much less than $n$, and F is an $n\times s$ matrix containing $s$ right-hand sides. It is divided into a preprocessing stage and a postprocessing stage.
Preprocessing stage
In the preprocessing stage, the linear system AX = F is partitioned into a block tridiagonal form
${\begin{bmatrix}{\boldsymbol {A}}_{1}&{\boldsymbol {B}}_{1}\\{\boldsymbol {C}}_{2}&{\boldsymbol {A}}_{2}&{\boldsymbol {B}}_{2}\\&\ddots &\ddots &\ddots \\&&{\boldsymbol {C}}_{p-1}&{\boldsymbol {A}}_{p-1}&{\boldsymbol {B}}_{p-1}\\&&&{\boldsymbol {C}}_{p}&{\boldsymbol {A}}_{p}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {X}}_{1}\\{\boldsymbol {X}}_{2}\\\vdots \\{\boldsymbol {X}}_{p-1}\\{\boldsymbol {X}}_{p}\end{bmatrix}}={\begin{bmatrix}{\boldsymbol {F}}_{1}\\{\boldsymbol {F}}_{2}\\\vdots \\{\boldsymbol {F}}_{p-1}\\{\boldsymbol {F}}_{p}\end{bmatrix}}.$
Assume, for the time being, that the diagonal blocks Aj (j = 1,...,p with p ≥ 2) are nonsingular. Define a block diagonal matrix
D = diag(A1,...,Ap),
then D is also nonsingular. Left-multiplying D−1 to both sides of the system gives
${\begin{bmatrix}{\boldsymbol {I}}&{\boldsymbol {V}}_{1}\\{\boldsymbol {W}}_{2}&{\boldsymbol {I}}&{\boldsymbol {V}}_{2}\\&\ddots &\ddots &\ddots \\&&{\boldsymbol {W}}_{p-1}&{\boldsymbol {I}}&{\boldsymbol {V}}_{p-1}\\&&&{\boldsymbol {W}}_{p}&{\boldsymbol {I}}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {X}}_{1}\\{\boldsymbol {X}}_{2}\\\vdots \\{\boldsymbol {X}}_{p-1}\\{\boldsymbol {X}}_{p}\end{bmatrix}}={\begin{bmatrix}{\boldsymbol {G}}_{1}\\{\boldsymbol {G}}_{2}\\\vdots \\{\boldsymbol {G}}_{p-1}\\{\boldsymbol {G}}_{p}\end{bmatrix}},$
which is to be solved in the postprocessing stage. Left-multiplication by D−1 is equivalent to solving $p$ systems of the form
Aj[Vj Wj Gj] = [Bj Cj Fj]
(omitting W1 and C1 for $j=1$, and Vp and Bp for $j=p$), which can be carried out in parallel.
Due to the banded nature of A, only a few leftmost columns of each Vj and a few rightmost columns of each Wj can be nonzero. These columns are called the spikes.
Postprocessing stage
Without loss of generality, assume that each spike contains exactly $m$ columns ($m$ is much less than $n$) (pad the spike with columns of zeroes if necessary). Partition the spikes in all Vj and Wj into
${\begin{bmatrix}{\boldsymbol {V}}_{j}^{(t)}\\{\boldsymbol {V}}_{j}'\\{\boldsymbol {V}}_{j}^{(b)}\end{bmatrix}}$ and ${\begin{bmatrix}{\boldsymbol {W}}_{j}^{(t)}\\{\boldsymbol {W}}_{j}'\\{\boldsymbol {W}}_{j}^{(b)}\\\end{bmatrix}}$
where V (t)
j
, V (b)
j
, W (t)
j
and W (b)
j
are of dimensions $m\times m$. Partition similarly all Xj and Gj into
${\begin{bmatrix}{\boldsymbol {X}}_{j}^{(t)}\\{\boldsymbol {X}}_{j}'\\{\boldsymbol {X}}_{j}^{(b)}\end{bmatrix}}$ and ${\begin{bmatrix}{\boldsymbol {G}}_{j}^{(t)}\\{\boldsymbol {G}}_{j}'\\{\boldsymbol {G}}_{j}^{(b)}\\\end{bmatrix}}.$
Notice that the system produced by the preprocessing stage can be reduced to a block pentadiagonal system of much smaller size (recall that $m$ is much less than $n$)
${\begin{bmatrix}{\boldsymbol {I}}_{m}&{\boldsymbol {0}}&{\boldsymbol {V}}_{1}^{(t)}\\{\boldsymbol {0}}&{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{1}^{(b)}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {W}}_{2}^{(t)}&{\boldsymbol {I}}_{m}&{\boldsymbol {0}}&{\boldsymbol {V}}_{2}^{(t)}\\&{\boldsymbol {W}}_{2}^{(b)}&{\boldsymbol {0}}&{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{2}^{(b)}&{\boldsymbol {0}}\\&&\ddots &\ddots &\ddots &\ddots &\ddots \\&&&{\boldsymbol {0}}&{\boldsymbol {W}}_{p-1}^{(t)}&{\boldsymbol {I}}_{m}&{\boldsymbol {0}}&{\boldsymbol {V}}_{p-1}^{(t)}\\&&&&{\boldsymbol {W}}_{p-1}^{(b)}&{\boldsymbol {0}}&{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{p-1}^{(b)}&{\boldsymbol {0}}\\&&&&&{\boldsymbol {0}}&{\boldsymbol {W}}_{p}^{(t)}&{\boldsymbol {I}}_{m}&{\boldsymbol {0}}\\&&&&&&{\boldsymbol {W}}_{p}^{(b)}&{\boldsymbol {0}}&{\boldsymbol {I}}_{m}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {X}}_{1}^{(t)}\\{\boldsymbol {X}}_{1}^{(b)}\\{\boldsymbol {X}}_{2}^{(t)}\\{\boldsymbol {X}}_{2}^{(b)}\\\vdots \\{\boldsymbol {X}}_{p-1}^{(t)}\\{\boldsymbol {X}}_{p-1}^{(b)}\\{\boldsymbol {X}}_{p}^{(t)}\\{\boldsymbol {X}}_{p}^{(b)}\end{bmatrix}}={\begin{bmatrix}{\boldsymbol {G}}_{1}^{(t)}\\{\boldsymbol {G}}_{1}^{(b)}\\{\boldsymbol {G}}_{2}^{(t)}\\{\boldsymbol {G}}_{2}^{(b)}\\\vdots \\{\boldsymbol {G}}_{p-1}^{(t)}\\{\boldsymbol {G}}_{p-1}^{(b)}\\{\boldsymbol {G}}_{p}^{(t)}\\{\boldsymbol {G}}_{p}^{(b)}\end{bmatrix}}{\text{,}}$
which we call the reduced system and denote by S̃X̃ = G̃.
Once all X (t)
j
and X (b)
j
are found, all X′j can be recovered with perfect parallelism via
${\begin{cases}{\boldsymbol {X}}_{1}'={\boldsymbol {G}}_{1}'-{\boldsymbol {V}}_{1}'{\boldsymbol {X}}_{2}^{(t)}{\text{,}}\\{\boldsymbol {X}}_{j}'={\boldsymbol {G}}_{j}'-{\boldsymbol {V}}_{j}'{\boldsymbol {X}}_{j+1}^{(t)}-{\boldsymbol {W}}_{j}'{\boldsymbol {X}}_{j-1}^{(b)}{\text{,}}&j=2,\ldots ,p-1{\text{,}}\\{\boldsymbol {X}}_{p}'={\boldsymbol {G}}_{p}'-{\boldsymbol {W}}_{p}{\boldsymbol {X}}_{p-1}^{(b)}{\text{.}}\end{cases}}$
SPIKE as a polyalgorithmic banded linear system solver
Despite being logically divided into two stages, computationally, the SPIKE algorithm comprises three stages:
1. factorizing the diagonal blocks,
2. computing the spikes,
3. solving the reduced system.
Each of these stages can be accomplished in several ways, allowing a multitude of variants. Two notable variants are the recursive SPIKE algorithm for non-diagonally-dominant cases and the truncated SPIKE algorithm for diagonally-dominant cases. Depending on the variant, a system can be solved either exactly or approximately. In the latter case, SPIKE is used as a preconditioner for iterative schemes like Krylov subspace methods and iterative refinement.
Preprocessing stage
The first step of the preprocessing stage is to factorize the diagonal blocks Aj. For numerical stability, one can use LAPACK's XGBTRF routines to LU factorize them with partial pivoting. Alternatively, one can also factorize them without partial pivoting but with a "diagonal boosting" strategy. The latter method tackles the issue of singular diagonal blocks.
In concrete terms, the diagonal boosting strategy is as follows. Let 0ε denote a configurable "machine zero". In each step of LU factorization, we require that the pivot satisfy the condition
|pivot| > 0ε‖A‖1.
If the pivot does not satisfy the condition, it is then boosted by
$\mathrm {pivot} ={\begin{cases}\mathrm {pivot} +\epsilon \lVert {\boldsymbol {A}}_{j}\rVert _{1}&{\text{if }}\mathrm {pivot} \geq 0{\text{,}}\\\mathrm {pivot} -\epsilon \lVert {\boldsymbol {A}}_{j}\rVert _{1}&{\text{if }}\mathrm {pivot} <0\end{cases}}$
where ε is a positive parameter depending on the machine's unit roundoff, and the factorization continues with the boosted pivot. This can be achieved by modified versions of ScaLAPACK's XDBTRF routines. After the diagonal blocks are factorized, the spikes are computed and passed on to the postprocessing stage.
The two-partition case
In the two-partition case, i.e., when p = 2, the reduced system S̃X̃ = G̃ has the form
${\begin{bmatrix}{\boldsymbol {I}}_{m}&{\boldsymbol {0}}&{\boldsymbol {V}}_{1}^{(t)}\\{\boldsymbol {0}}&{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{1}^{(b)}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {W}}_{2}^{(t)}&{\boldsymbol {I}}_{m}&{\boldsymbol {0}}\\&{\boldsymbol {W}}_{2}^{(b)}&{\boldsymbol {0}}&{\boldsymbol {I}}_{m}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {X}}_{1}^{(t)}\\{\boldsymbol {X}}_{1}^{(b)}\\{\boldsymbol {X}}_{2}^{(t)}\\{\boldsymbol {X}}_{2}^{(b)}\end{bmatrix}}={\begin{bmatrix}{\boldsymbol {G}}_{1}^{(t)}\\{\boldsymbol {G}}_{1}^{(b)}\\{\boldsymbol {G}}_{2}^{(t)}\\{\boldsymbol {G}}_{2}^{(b)}\end{bmatrix}}{\text{.}}$
An even smaller system can be extracted from the center:
${\begin{bmatrix}{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{1}^{(b)}\\{\boldsymbol {W}}_{2}^{(t)}&{\boldsymbol {I}}_{m}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {X}}_{1}^{(b)}\\{\boldsymbol {X}}_{2}^{(t)}\end{bmatrix}}={\begin{bmatrix}{\boldsymbol {G}}_{1}^{(b)}\\{\boldsymbol {G}}_{2}^{(t)}\end{bmatrix}}{\text{,}}$
which can be solved using the block LU factorization
${\begin{bmatrix}{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{1}^{(b)}\\{\boldsymbol {W}}_{2}^{(t)}&{\boldsymbol {I}}_{m}\end{bmatrix}}={\begin{bmatrix}{\boldsymbol {I}}_{m}\\{\boldsymbol {W}}_{2}^{(t)}&{\boldsymbol {I}}_{m}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{1}^{(b)}\\&{\boldsymbol {I}}_{m}-{\boldsymbol {W}}_{2}^{(t)}{\boldsymbol {V}}_{1}^{(b)}\end{bmatrix}}{\text{.}}$
Once X (b)
1
and X (t)
2
are found, X (t)
1
and X (b)
2
can be computed via
X (t)
1
= G (t)
1
− V (t)
1
X (t)
2
,
X (b)
2
= G (b)
2
− W (b)
2
X (b)
1
.
The multiple-partition case
Assume that p is a power of two, i.e., p = 2d. Consider a block diagonal matrix
D̃1 = diag(D̃ [1]
1
,...,D̃ [1]
p/2
)
where
${\boldsymbol {\tilde {D}}}_{k}^{[1]}={\begin{bmatrix}{\boldsymbol {I}}_{m}&{\boldsymbol {0}}&{\boldsymbol {V}}_{2k-1}^{(t)}\\{\boldsymbol {0}}&{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{2k-1}^{(b)}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {W}}_{2k}^{(t)}&{\boldsymbol {I}}_{m}&{\boldsymbol {0}}\\&{\boldsymbol {W}}_{2k}^{(b)}&{\boldsymbol {0}}&{\boldsymbol {I}}_{m}\end{bmatrix}}$
for k = 1,...,p/2. Notice that D̃1 essentially consists of diagonal blocks of order 4m extracted from S̃. Now we factorize S̃ as
S̃ = D̃1S̃2.
The new matrix S̃2 has the form
${\begin{bmatrix}{\boldsymbol {I}}_{3m}&{\boldsymbol {0}}&{\boldsymbol {V}}_{1}^{[2](t)}\\{\boldsymbol {0}}&{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{1}^{[2](b)}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {W}}_{2}^{[2](t)}&{\boldsymbol {I}}_{m}&{\boldsymbol {0}}&{\boldsymbol {V}}_{2}^{[2](t)}\\&{\boldsymbol {W}}_{2}^{[2](b)}&{\boldsymbol {0}}&{\boldsymbol {I}}_{3m}&{\boldsymbol {V}}_{2}^{[2](b)}&{\boldsymbol {0}}\\&&\ddots &\ddots &\ddots &\ddots &\ddots \\&&&{\boldsymbol {0}}&{\boldsymbol {W}}_{p/2-1}^{[2](t)}&{\boldsymbol {I}}_{3m}&{\boldsymbol {0}}&{\boldsymbol {V}}_{p/2-1}^{[2](t)}\\&&&&{\boldsymbol {W}}_{p/2-1}^{[2](b)}&{\boldsymbol {0}}&{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{p/2-1}^{[2](b)}&{\boldsymbol {0}}\\&&&&&{\boldsymbol {0}}&{\boldsymbol {W}}_{p/2}^{[2](t)}&{\boldsymbol {I}}_{m}&{\boldsymbol {0}}\\&&&&&&{\boldsymbol {W}}_{p/2}^{[2](b)}&{\boldsymbol {0}}&{\boldsymbol {I}}_{3m}\end{bmatrix}}{\text{.}}$
Its structure is very similar to that of S̃2, only differing in the number of spikes and their height (their width stays the same at m). Thus, a similar factorization step can be performed on S̃2 to produce
S̃2 = D̃2S̃3
and
S̃ = D̃1D̃2S̃3.
Such factorization steps can be performed recursively. After d − 1 steps, we obtain the factorization
S̃ = D̃1⋯D̃d−1S̃d,
where S̃d has only two spikes. The reduced system will then be solved via
X̃ = S̃ −1
d
D̃ −1
d−1
⋯D̃ −1
1
G̃
.
The block LU factorization technique in the two-partition case can be used to handle the solving steps involving D̃1, ..., D̃d−1 and S̃d for they essentially solve multiple independent systems of generalized two-partition forms.
Generalization to cases where p is not a power of two is almost trivial.
Truncated SPIKE
When A is diagonally-dominant, in the reduced system
${\begin{bmatrix}{\boldsymbol {I}}_{m}&{\boldsymbol {0}}&{\boldsymbol {V}}_{1}^{(t)}\\{\boldsymbol {0}}&{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{1}^{(b)}&{\boldsymbol {0}}\\{\boldsymbol {0}}&{\boldsymbol {W}}_{2}^{(t)}&{\boldsymbol {I}}_{m}&{\boldsymbol {0}}&{\boldsymbol {V}}_{2}^{(t)}\\&{\boldsymbol {W}}_{2}^{(b)}&{\boldsymbol {0}}&{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{2}^{(b)}&{\boldsymbol {0}}\\&&\ddots &\ddots &\ddots &\ddots &\ddots \\&&&{\boldsymbol {0}}&{\boldsymbol {W}}_{p-1}^{(t)}&{\boldsymbol {I}}_{m}&{\boldsymbol {0}}&{\boldsymbol {V}}_{p-1}^{(t)}\\&&&&{\boldsymbol {W}}_{p-1}^{(b)}&{\boldsymbol {0}}&{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{p-1}^{(b)}&{\boldsymbol {0}}\\&&&&&{\boldsymbol {0}}&{\boldsymbol {W}}_{p}^{(t)}&{\boldsymbol {I}}_{m}&{\boldsymbol {0}}\\&&&&&&{\boldsymbol {W}}_{p}^{(b)}&{\boldsymbol {0}}&{\boldsymbol {I}}_{m}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {X}}_{1}^{(t)}\\{\boldsymbol {X}}_{1}^{(b)}\\{\boldsymbol {X}}_{2}^{(t)}\\{\boldsymbol {X}}_{2}^{(b)}\\\vdots \\{\boldsymbol {X}}_{p-1}^{(t)}\\{\boldsymbol {X}}_{p-1}^{(b)}\\{\boldsymbol {X}}_{p}^{(t)}\\{\boldsymbol {X}}_{p}^{(b)}\end{bmatrix}}={\begin{bmatrix}{\boldsymbol {G}}_{1}^{(t)}\\{\boldsymbol {G}}_{1}^{(b)}\\{\boldsymbol {G}}_{2}^{(t)}\\{\boldsymbol {G}}_{2}^{(b)}\\\vdots \\{\boldsymbol {G}}_{p-1}^{(t)}\\{\boldsymbol {G}}_{p-1}^{(b)}\\{\boldsymbol {G}}_{p}^{(t)}\\{\boldsymbol {G}}_{p}^{(b)}\end{bmatrix}}{\text{,}}$
the blocks V (t)
j
and W (b)
j
are often negligible. With them omitted, the reduced system becomes block diagonal
${\begin{bmatrix}{\boldsymbol {I}}_{m}\\&{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{1}^{(b)}\\&{\boldsymbol {W}}_{2}^{(t)}&{\boldsymbol {I}}_{m}\\&&&{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{2}^{(b)}\\&&&\ddots &\ddots &\ddots \\&&&&{\boldsymbol {W}}_{p-1}^{(t)}&{\boldsymbol {I}}_{m}\\&&&&&&{\boldsymbol {I}}_{m}&{\boldsymbol {V}}_{p-1}^{(b)}\\&&&&&&{\boldsymbol {W}}_{p}^{(t)}&{\boldsymbol {I}}_{m}\\&&&&&&&&{\boldsymbol {I}}_{m}\end{bmatrix}}{\begin{bmatrix}{\boldsymbol {X}}_{1}^{(t)}\\{\boldsymbol {X}}_{1}^{(b)}\\{\boldsymbol {X}}_{2}^{(t)}\\{\boldsymbol {X}}_{2}^{(b)}\\\vdots \\{\boldsymbol {X}}_{p-1}^{(t)}\\{\boldsymbol {X}}_{p-1}^{(b)}\\{\boldsymbol {X}}_{p}^{(t)}\\{\boldsymbol {X}}_{p}^{(b)}\end{bmatrix}}={\begin{bmatrix}{\boldsymbol {G}}_{1}^{(t)}\\{\boldsymbol {G}}_{1}^{(b)}\\{\boldsymbol {G}}_{2}^{(t)}\\{\boldsymbol {G}}_{2}^{(b)}\\\vdots \\{\boldsymbol {G}}_{p-1}^{(t)}\\{\boldsymbol {G}}_{p-1}^{(b)}\\{\boldsymbol {G}}_{p}^{(t)}\\{\boldsymbol {G}}_{p}^{(b)}\end{bmatrix}}$
and can be easily solved in parallel .
The truncated SPIKE algorithm can be wrapped inside some outer iterative scheme (e.g., BiCGSTAB or iterative refinement) to improve the accuracy of the solution.
SPIKE for tridiagonal systems
The first SPIKE partitioning and algorithm was presented in and was designed as the means to improve the stability properties of a parallel Givens rotations-based solver for tridiagonal systems. A version of the algorithm, termed g-Spike, that is based on serial Givens rotations applied independently on each block was designed for the NVIDIA GPU . A SPIKE-based algorithm for the GPU that is based on a special block diagonal pivoting strategy is described in .
SPIKE as a preconditioner
The SPIKE algorithm can also function as a preconditioner for iterative methods for solving linear systems. To solve a linear system Ax = b using a SPIKE-preconditioned iterative solver, one extracts center bands from A to form a banded preconditioner M and solves linear systems involving M in each iteration with the SPIKE algorithm.
In order for the preconditioner to be effective, row and/or column permutation is usually necessary to move "heavy" elements of A close to the diagonal so that they are covered by the preconditioner. This can be accomplished by computing the weighted spectral reordering of A.
The SPIKE algorithm can be generalized by not restricting the preconditioner to be strictly banded. In particular, the diagonal block in each partition can be a general matrix and thus handled by a direct general linear system solver rather than a banded solver. This enhances the preconditioner, and hence allows better chance of convergence and reduces the number of iterations.
Implementations
Intel offers an implementation of the SPIKE algorithm under the name Intel Adaptive Spike-Based Solver . Tridiagonal solvers have also been developed for the NVIDIA GPU and the Xeon Phi co-processors. The method in is the basis for a tridiagonal solver in the cuSPARSE library.[1] The Givens rotations based solver was also implemented for the GPU and the Intel Xeon Phi.[2]
References
1. NVIDIA, Accessed October 28, 2014. CUDA Toolkit Documentation v. 6.5: cuSPARSE, http://docs.nvidia.com/cuda/cusparse.
2. Venetis, Ioannis; Sobczyk, Aleksandros; Kouris, Alexandros; Nakos, Alexandros; Nikoloutsakos, Nikolaos; Gallopoulos, Efstratios (2015-09-03). "A general tridiagonal solver for coprocessors: Adapting g-Spike for the Intel Xeon Phi" – via ResearchGate.
1. ^ Polizzi, E.; Sameh, A. H. (2006). "A parallel hybrid banded system solver: the SPIKE algorithm". Parallel Computing. 32 (2): 177–194. doi:10.1016/j.parco.2005.07.005.
2. ^ Polizzi, E.; Sameh, A. H. (2007). "SPIKE: A parallel environment for solving banded linear systems". Computers & Fluids. 36: 113–141. doi:10.1016/j.compfluid.2005.07.005.
3. ^ Mikkelsen, C. C. K.; Manguoglu, M. (2008). "Analysis of the Truncated SPIKE Algorithm". SIAM J. Matrix Anal. Appl. 30 (4): 1500–1519. CiteSeerX 10.1.1.514.8748. doi:10.1137/080719571.
4. ^ Manguoglu, M.; Sameh, A. H.; Schenk, O. (2009). "PSPIKE: A Parallel Hybrid Sparse Linear System Solver". Euro-Par 2009 Parallel Processing. Lecture Notes in Computer Science. Vol. 5704. pp. 797–808. Bibcode:2009LNCS.5704..797M. doi:10.1007/978-3-642-03869-3_74. ISBN 978-3-642-03868-6.
5. ^ "Intel Adaptive Spike-Based Solver - Intel Software Network". Retrieved 2009-03-23.
6. ^ Sameh, A. H.; Kuck, D. J. (1978). "On Stable Parallel Linear System Solvers". Journal of the ACM. 25: 81–91. doi:10.1145/322047.322054. S2CID 17109524.
7. ^ Venetis, I.E.; Kouris, A.; Sobczyk, A.; Gallopoulos, E.; Sameh, A. H. (2015). "A direct tridiagonal solver based on Givens rotations for GPU architectures". Parallel Computing. 25: 101–116. doi:10.1016/j.parco.2015.03.008.
8. ^ Chang, L.-W.; Stratton, J.; Kim, H.; Hwu, W.-M. (2012). "A scalable, numerically stable, high-performance tridiagonal solver using GPUs". Proc. Int'l. Conf. High Performance Computing, Networking Storage and Analysis (SC'12). Los Alamitos, CA, USA: IEEE Computer Soc. Press: 27:1–27:11. ISBN 978-1-4673-0804-5.
Further reading
• Gallopoulos, E.; Philippe, B.; Sameh, A.H. (2015). Parallelism in Matrix Computations. Springer. ISBN 978-94-017-7188-7.
Numerical linear algebra
Key concepts
• Floating point
• Numerical stability
Problems
• System of linear equations
• Matrix decompositions
• Matrix multiplication (algorithms)
• Matrix splitting
• Sparse problems
Hardware
• CPU cache
• TLB
• Cache-oblivious algorithm
• SIMD
• Multiprocessing
Software
• MATLAB
• Basic Linear Algebra Subprograms (BLAS)
• LAPACK
• Specialized libraries
• General purpose software
| Wikipedia |
Why don't we talk about angular momentum at all in fluid mechanics?
People usually talk about similar (or maybe not?) things like vorticity or enstrophy in fluid mechanics, but no one talks about angular momentum, why?
fluid-dynamics angular-momentum rotational-dynamics
physixfanphysixfan
$\begingroup$ Symmetry of the stress tensor takes care of it, most of the time $\endgroup$ – Hydro Guy Jul 22 '14 at 21:32
$\begingroup$ A quick search for 'fluid angular momentum' suggests that people do talk about the angular momentum of fluid (well, discrete parcels of fluid at least). $\endgroup$ – Kyle Oman Jul 22 '14 at 21:34
$\begingroup$ @Kyle Well, that's true but when you look at fluid textbooks no one talks about angular momentum... $\endgroup$ – physixfan Jul 22 '14 at 21:39
$\begingroup$ Batchelor discusses angular momentum in fluids right on the first chapter, have you read it? $\endgroup$ – Hydro Guy Jul 22 '14 at 21:45
$\begingroup$ @user23873 I just looked it up and it also didn't talk about angular momentum in detail.. $\endgroup$ – physixfan Jul 22 '14 at 21:52
We don't need to talk about angular momentum because the conservation law is summed up by vorticity. Consider the vorticity equation (in the context of a rotating frame as well): $$ \frac{D\boldsymbol\omega}{Dt}=\boldsymbol\omega\cdot\nabla\mathbf u $$ (ignoring all other terms that are normally contained in this term). If we take the coordinate system where $s$ is along the vortex line, then the component of this gives $$ \frac{D\omega_s}{Dt}=\omega\frac{\partial u_s}{\partial s} $$ This shows that the vorticity along $s$ changes due to the stretching of the vortex lines, which is principle of angular momentum conservation.
Kyle KanosKyle Kanos
A fluid is modelled as a vector field and therefore we use vorticity to describe its spinning motion. Angular momentum is more often used for a single object or particle, but not so often for a vector field (even though it is still applicable in principle). For a fluid in general, vorticity is twice the mean angular velocity and this fact to me makes it less useful as a quantity when modelling fluids.
Constandinos DamalasConstandinos Damalas
$\begingroup$ But I can calculate the angular momentum of any subset of the vector field by integrating over it. If I do it for the whole field, I get its total angular momentum. It's not like it's undefined... $\endgroup$ – Kyle Oman Jul 22 '14 at 21:33
$\begingroup$ @Kyle that is true but it seems to me it wouldn't be as useful as vorticity. Where would you use angular momentum for example? $\endgroup$ – Constandinos Damalas Jul 22 '14 at 21:43
$\begingroup$ @Kyle I have made an edit which hopefully answers (at least partially?) your question. $\endgroup$ – Constandinos Damalas Jul 22 '14 at 21:55
$\begingroup$ I wouldn't, I would use vorticity ;) I was more just pointing out that angular momentum is well defined for a vector field (well, you'd also need an associated scalar (density) field). If you desired, you could formulate a constraint on the mechanics of a dissipationless flow based on the conservation of angular momentum. Really I'm just quibbling that your answer reads as "we can't use angular momentum for vector fields" instead of "there is a more useful alternative in this case". $\endgroup$ – Kyle Oman Jul 22 '14 at 22:19
$\begingroup$ To paraphrase something the mods keep saying: comments aren't meant to be permanent, can be and often are deleted, etc. -> make an edit! $\endgroup$ – Kyle Oman Jul 22 '14 at 22:26
There is a paper that discusses the delicate (and often superficially brushed off) issue of diffusion of angular momentum in the Naiver-Stokes equation, the stress tensor, and its symmetries by Berdahl and Strang titled: The Behavior of a Vorticity-Influenced Asymmetric Stress Tensor in Fluid Flow. Here is a link: http://www.dtic.mil/dtic/tr/fulltext/u2/a181244.pdf .
Consider an incompressible, isotropic, Newtonian, viscous fluid that is truly continuous (i.e. it is not composed of discrete particles that have some non-zero rotational inertia) in a large cylindrical drum (with full slip boundary conditions in the fluid state) and initially frozen solid. Spin the drum at some angular speed along its center z-axis, then, at time t=0, blast the solid with a burst of radiation that instantaneously melts it into its fluid state and raises it to an arbitrarily high temperature that is spatially uniform. The initial condition for this flow has zero shear strain rate – but everywhere has a constant vorticity equal to twice the rotation rate.
Rotational motion does not magically shut off diffusion along radial lines. Since the speed in the theta direction is larger at r + dr than it is at r, this velocity in the theta direction will initially diffuse radially inward and "the band at r + dr" will exert a viscous torque on "the band at r" and cause it to experience an angular acceleration in the theta direction. Angular momentum is conserved as it initially diffuses radially inward – not because the stress tensor is symmetric.
In fact, if the stress tensor were symmetric, this would imply that e.g. a shear flow v = (Cz,0,0) would not only have a shear stress exerted on the x-y plane in the positive x direction, but it would also have a shear stress on the z-y plane in the positive z direction - without any physical mechanism to account for this.
Since this is a truly continuous fluid, the inwards diffusion of angular momentum leads to an infinite angular speed at r = 0 for t > 0. Roughly, the viscous torque on a cylindrical volume element initially scales as r-squared and is equal to the rotational inertia of the axially centered fluid element (which scales as the density times r to the 4th power) times the angular acceleration which must then blow up as 1/r-squared. If such a fluid could be created in the physical world, angular speed would in fact blow up at the origin and this model would then be correct.
In the real world however, fluids are composed of particles that have non-zero radii and non-zero rotational inertias that prevent viscous torques from creating infinite angular speeds (there is an additional length scale that is relevant as well – the average inter-particle distance…). Consider replacing the above fluid with radon atoms. The same initial velocity gradient would cause the radon spheres to acquire an average spin. As these atoms collide, these spins would then average back to zero and indirectly mediate a transition of angular momentum of the fluid inward. If the relaxation rate (from intrinsic to extrinsic angular momentum) is sufficiently high, one can get away with symmetrizing the extrinsic stress tensor, avoiding the troublesome infinities, and allowing the intrinsic stress tensor to indirectly approximate the evolution of the fluid flow.
The continual relaxation of the spins in the case of the pure shear flow also indirectly accounts for the additional perpendicular shear stress - which resolves the paradox mentioned above.
Spin2He2Spin2He2
The total angular momentum of a continuum is the vector sum of net angular momenta of the particles it is comprised of. The net angular momentum of a particle accounts for its spin as well as its translational motion about the point about which angular momentum is to be calculated (moment of its momentum).
Vorticity is a kinematic quantity measuring (half of) the angular velocity of a point relative to a different point in the same continuum. As a result, vorticity equation is like a statement of conservation of moment of momentum.
However, the angular momentum corresponding to the spin of each particle is assumed to be 0 in most of the continuum treatment. This assumption follows the random distribution of molecular angular momenta (rather all vector quantities) inside the continuum. Very closely related to the assumption of kinetic theory of gases.
Prateek GuptaPrateek Gupta
Angular momentum is about a fixed point (you decide it) that is the same for every lump of fluid in the domain. Vorticity is (1/2) the angular momentum of the fluid about EACH fluid lump's OWN center of gravity.
This is why a potential vortex has zero vorticity (except at the origin). But lots of angular momentum at each point in the flow (especially about the obvious axis of the center of the vortex, but also about any other axis).
They are related, but definitely not the same (or just off by a factor of 1/2). Vorticity is more useful. In physics - utility is always the final answer. Engineers use angular momentum control volume equations (when they are useful, like for turbine problems). An undergrad engineering fluids book (any) will have a section in the control volume chapter on the angular momentum equations.
BlairBlair
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Simple Coatings to Render Polystyrene Protein Resistant
Marcelle Hecker
Full Text Available Non-specific protein adsorption is detrimental to the performance of many biomedical devices. Polystyrene is a commonly used material in devices and thin films. Simple reliable surface modification of polystyrene to render it protein resistant is desired in particular for device fabrication and orthogonal functionalisation schemes. This report details modifications carried out on a polystyrene surface to prevent protein adsorption. The trialed surfaces included Pluronic F127 and PLL-g-PEG, adsorbed on polystyrene, using a polydopamine-assisted approach. Quartz crystal microbalance with dissipation (QCM-D results showed only short-term anti-fouling success of the polystyrene surface modified with F127, and the subsequent failure of the polydopamine intermediary layer in improving its stability. In stark contrast, QCM-D analysis proved the success of the polydopamine assisted PLL-g-PEG coating in preventing bovine serum albumin adsorption. This modified surface is equally as protein-rejecting after 24 h in buffer, and thus a promising simple coating for long term protein rejection of polystyrene.
Simple sequence proteins in prokaryotic proteomes
Ramachandran Srinivasan
Full Text Available Abstract Background The structural and functional features associated with Simple Sequence Proteins (SSPs are non-globularity, disease states, signaling and post-translational modification. SSPs are also an important source of genetic and possibly phenotypic variation. Analysis of 249 prokaryotic proteomes offers a new opportunity to examine the genomic properties of SSPs. Results SSPs are a minority but they grow with proteome size. This relationship is exhibited across species varying in genomic GC, mutational bias, life style, and pathogenicity. Their proportion in each proteome is strongly influenced by genomic base compositional bias. In most species simple duplications is favoured, but in a few cases such as Mycobacteria, large families of duplications occur. Amino acid preference in SSPs exhibits a trend towards low cost of biosynthesis. In SSPs and in non-SSPs, Alanine, Glycine, Leucine, and Valine are abundant in species widely varying in genomic GC whereas Isoleucine and Lysine are rich only in organisms with low genomic GC. Arginine is abundant in SSPs of two species and in the non-SSPs of Xanthomonas oryzae. Asparagine is abundant only in SSPs of low GC species. Aspartic acid is abundant only in the non-SSPs of Halobacterium sp NRC1. The abundance of Serine in SSPs of 62 species extends over a broader range compared to that of non-SSPs. Threonine(T is abundant only in SSPs of a couple of species. SSPs exhibit preferential association with Cell surface, Cell membrane and Transport functions and a negative association with Metabolism. Mesophiles and Thermophiles display similar ranges in the content of SSPs. Conclusion Although SSPs are a minority, the genomic forces of base compositional bias and duplications influence their growth and pattern in each species. The preferences and abundance of amino acids are governed by low biosynthetic cost, evolutionary age and base composition of codons. Abundance of charged amino acids Arginine
A simple theory of motor protein kinetics and energetics. II.
Qian, H
A three-state stochastic model of motor protein [Qian, Biophys. Chem. 67 (1997) pp. 263-267] is further developed to illustrate the relationship between the external load on an individual motor protein in aqueous solution with various ATP concentrations and its steady-state velocity. A wide variety of dynamic motor behavior are obtained from this simple model. For the particular case of free-load translocation being the most unfavorable step within the hydrolysis cycle, the load-velocity curve is quasi-linear, V/Vmax = (cF/Fmax-c)/(1-c), in contrast to the hyperbolic relationship proposed by A.V. Hill for macroscopic muscle. Significant deviation from the linearity is expected when the velocity is less than 10% of its maximal (free-load) value--a situation under which the processivity of motor diminishes and experimental observations are less certain. We then investigate the dependence of load-velocity curve on ATP (ADP) concentration. It is shown that the free load Vmax exhibits a Michaelis-Menten like behavior, and the isometric Fmax increases linearly with ln([ATP]/[ADP]). However, the quasi-linear region is independent of the ATP concentration, yielding an apparently ATP-independent maximal force below the true isometric force. Finally, the heat production as a function of ATP concentration and external load are calculated. In simple terms and solved with elementary algebra, the present model provides an integrated picture of biochemical kinetics and mechanical energetics of motor proteins.
A Simple Combinatorial Codon Mutagenesis Method for Targeted Protein Engineering.
Belsare, Ketaki D; Andorfer, Mary C; Cardenas, Frida S; Chael, Julia R; Park, Hyun June; Lewis, Jared C
Directed evolution is a powerful tool for optimizing enzymes, and mutagenesis methods that improve enzyme library quality can significantly expedite the evolution process. Here, we report a simple method for targeted combinatorial codon mutagenesis (CCM). To demonstrate the utility of this method for protein engineering, CCM libraries were constructed for cytochrome P450 BM3 , pfu prolyl oligopeptidase, and the flavin-dependent halogenase RebH; 10-26 sites were targeted for codon mutagenesis in each of these enzymes, and libraries with a tunable average of 1-7 codon mutations per gene were generated. Each of these libraries provided improved enzymes for their respective transformations, which highlights the generality, simplicity, and tunability of CCM for targeted protein engineering.
A SIMPLE FLUORESCENT LABELING METHOD FOR STUDIES OF PROTEIN OXIDATION, PROTEIN MODIFICATION, AND PROTEOLYSIS
Pickering, Andrew. M.; Davies, Kelvin. J. A.
Proteins are sensitive to oxidation, and oxidized proteins are excellent substrates for degradation by proteolytic enzymes such as the Proteasome and the mitochondrial Lon protease. Protein labeling is required for studies of protein turnover. Unfortunately, most labeling techniques involve 3H or 14C methylation which is expensive, exposes researchers to radioactivity, generates large amounts of radioactive waste, and allows only single-point assays because samples require acid-precipitation. Alternative labeling methods, have largely proven unsuitable, either because the probe itself is modified by the oxidant(s) being studied, or because the alternative labeling techniques are too complex or too costly for routine use. What is needed is a simple, quick, and cheap labeling technique that uses a non-radioactive marker, that binds strongly to proteins, is resistant to oxidative modification, and emits a strong signal. We have devised a new reductive method for labeling free carboxyl groups of proteins with the small fluorophore 7-amino-4-methycoumarin (AMC). When bound to target proteins, AMC fluoresces very weakly but when AMC is released by proteinases, proteases, or peptidases, it fluoresces strongly. Thus, without acid-precipitation, the proteolysis of any target protein can be studied continuously, in multiwell plates. In direct comparisons, 3H-labeled proteins and AMC-labeled proteins exhibited essentially identical degradation patterns during incubation with trypsin, cell extracts, and purified proteasome. AMC-labeled proteins are well-suited to study increased proteolytic susceptibility following protein modification, since the AMC-protein bond is resistant to oxidizing agents such as hydrogen peroxide and peroxynitrite, and is stable over time and to extremes of pH, temperature (even boiling), freeze-thawing, mercaptoethanol, and methanol. PMID:21988844
Prediction of thermodynamic instabilities of protein solutions from simple protein–protein interactions
D'Agostino, Tommaso; Solana, José Ramón; Emanuele, Antonio
Highlights: ► We propose a model of effective protein–protein interaction embedding solvent effects. ► A previous square-well model is enhanced by giving to the interaction a free energy character. ► The temperature dependence of the interaction is due to entropic effects of the solvent. ► The validity of the original SW model is extended to entropy driven phase transitions. ► We get good fits for lysozyme and haemoglobin spinodal data taken from literature. - Abstract: Statistical thermodynamics of protein solutions is often studied in terms of simple, microscopic models of particles interacting via pairwise potentials. Such modelling can reproduce the short range structure of protein solutions at equilibrium and predict thermodynamics instabilities of these systems. We introduce a square well model of effective protein–protein interaction that embeds the solvent's action. We modify an existing model [45] by considering a well depth having an explicit dependence on temperature, i.e. an explicit free energy character, thus encompassing the statistically relevant configurations of solvent molecules around proteins. We choose protein solutions exhibiting demixing upon temperature decrease (lysozyme, enthalpy driven) and upon temperature increase (haemoglobin, entropy driven). We obtain satisfactory fits of spinodal curves for both the two proteins without adding any mean field term, thus extending the validity of the original model. Our results underline the solvent role in modulating or stretching the interaction potential
Simple method for identification of plasmid-coded proteins
Sancar, A.; Hack, A.M.; Rupp, W.D.
Proteins encoded by plasmid DNA are specifically labeled in uv-irradiated cells of Escherichia coli carrying recA and uvrA mutations because extensive degradation of the chromosome DNA occurs concurrently with amplification of plasmid DNA
Feasibilty of zein proteins, simple sequence repeats and phenotypic ...
Widespread adoption of quality protein maize (QPM), especially among tropical farming systems has been slow mainly due to the slow process of generating varieties with acceptable kernel quality and adaptability to different agroecological contexts. A molecular based foreground selection system for opaque 2 (o2), the ...
A simple method for labelling proteins with 211At via diazotized aromatic diamine
Wunderlich, G.; Franke, W.-G.; Fischer, S.; Dreyer, R.
A simple and rapid method for labelling proteins with 211 At by means of a 1,4-diaminobenzene link is described. This link is transformed into the diazonium salt and subsequently reactions of both 211 At and proteins with the diazonium salt take place simultaneously. For possibly high yields of astatized protein an appropriate temperature of 273 K was found. The results demonstrate the difference between the reaction mechanisms of iodine and astatine with proteins. (author)
A simple immunoblotting method after separation of proteins in agarose gel
Koch, C; Skjødt, K; Laursen, I
A simple and sensitive method for immunoblotting of proteins after separation in agarose gels is described. It involves transfer of proteins onto nitrocellulose paper simply by diffusion through pressure, a transfer which only takes about 10 min. By this method we have demonstrated the existence ...
A simple and effective method for detecting precipitated proteins in MALDI-TOF MS.
Oshikane, Hiroyuki; Watabe, Masahiko; Nakaki, Toshio
MALDI-TOF MS has developed rapidly into an essential analytical tool for the life sciences. Cinnamic acid derivatives are generally employed in routine molecular weight determinations of intact proteins using MALDI-TOF MS. However, a protein of interest may precipitate when mixed with matrix solution, perhaps preventing MS detection. We herein provide a simple approach to enable the MS detection of such precipitated protein species by means of a "direct deposition method" -- loading the precipitant directly onto the sample plate. It is thus expected to improve routine MS analysis of intact proteins. Copyright © 2018. Published by Elsevier Inc.
PERMutation Using Transposase Engineering (PERMUTE): A Simple Approach for Constructing Circularly Permuted Protein Libraries.
Jones, Alicia M; Atkinson, Joshua T; Silberg, Jonathan J
Rearrangements that alter the order of a protein's sequence are used in the lab to study protein folding, improve activity, and build molecular switches. One of the simplest ways to rearrange a protein sequence is through random circular permutation, where native protein termini are linked together and new termini are created elsewhere through random backbone fission. Transposase mutagenesis has emerged as a simple way to generate libraries encoding different circularly permuted variants of proteins. With this approach, a synthetic transposon (called a permuteposon) is randomly inserted throughout a circularized gene to generate vectors that express different permuted variants of a protein. In this chapter, we outline the protocol for constructing combinatorial libraries of circularly permuted proteins using transposase mutagenesis, and we describe the different permuteposons that have been developed to facilitate library construction.
Simple and Efficient Purification of Recombinant Proteins Using the Heparin-Binding Affinity Tag.
Jayanthi, Srinivas; Gundampati, Ravi Kumar; Kumar, Thallapuranam Krishnaswamy Suresh
Heparin, a member of the glycosaminoglycan family, is known to interact with more than 400 different types of proteins. For the past few decades, significant progress has been made to understand the molecular details involved in heparin-protein interactions. Based on the structural knowledge available from the FGF1-heparin interaction studies, we have designed a novel heparin-binding peptide (HBP) affinity tag that can be used for the simple, efficient, and cost-effective purification of recombinant proteins of interest. HBP-tagged fusion proteins can be purified by heparin Sepharose affinity chromatography using a simple sodium chloride gradient to elute the bound fusion protein. In addition, owing to the high density of positive charges on the HBP tag, recombinant target proteins are preferably expressed in their soluble forms. The purification of HBP-fusion proteins can also be achieved in the presence of chemical denaturants, including urea. Additionally, polyclonal antibodies raised against the affinity tag can be used to detect HBP-fused target proteins with high sensitivity. © 2017 by John Wiley & Sons, Inc. Copyright © 2017 John Wiley & Sons, Inc.
Electrostatics of cysteine residues in proteins: Parameterization and validation of a simple model
Salsbury, Freddie R.; Poole, Leslie B.; Fetrow, Jacquelyn S.
One of the most popular and simple models for the calculation of pKas from a protein structure is the semi-macroscopic electrostatic model MEAD. This model requires empirical parameters for each residue to calculate pKas. Analysis of current, widely used empirical parameters for cysteine residues showed that they did not reproduce expected cysteine pKas; thus, we set out to identify parameters consistent with the CHARMM27 force field that capture both the behavior of typical cysteines in proteins and the behavior of cysteines which have perturbed pKas. The new parameters were validated in three ways: (1) calculation across a large set of typical cysteines in proteins (where the calculations are expected to reproduce expected ensemble behavior); (2) calculation across a set of perturbed cysteines in proteins (where the calculations are expected to reproduce the shifted ensemble behavior); and (3) comparison to experimentally determined pKa values (where the calculation should reproduce the pKa within experimental error). Both the general behavior of cysteines in proteins and the perturbed pKa in some proteins can be predicted reasonably well using the newly determined empirical parameters within the MEAD model for protein electrostatics. This study provides the first general analysis of the electrostatics of cysteines in proteins, with specific attention paid to capturing both the behavior of typical cysteines in a protein and the behavior of cysteines whose pKa should be shifted, and validation of force field parameters for cysteine residues. PMID:22777874
Simple Protein Modification Using Zwitterionic Polymer to Mitigate the Bioactivity Loss of Conjugated Insulin.
Xie, Jinbing; Lu, Yang; Wang, Wei; Zhu, Hui; Wang, Zhigang; Cao, Zhiqiang
Polymer-protein conjugation has been extensively explored toward a better protein drug with improved pharmacokinetics. However, a major problem with polymer-protein conjugation is that the polymers drastically reduce the bioactivity of the modified protein. There is no perfect solution to prevent the bioactivity loss, no matter the polymer is conjugated in a non-site specific way, or a more complex site-specific procedure. Here the authors report for the first time that when zwitterionic carboxybetaine polymer (PCB) is conjugated to insulin through simple conventional coupling chemistry. The resulting PCB-insulin does not show a significant reduction of in vitro bioactivity. The obtained PCB-insulin shows two significant advantages as a novel pharmaceutical agent. First, its therapeutic performance is remarkable. For PCB-insulin, there is a 24% increase of in vivo pharmacological activity of lowering blood glucose compared with native insulin. Such uncommonly seen increase has rarely been reported and is expected to be due to both the improved pharmacokinetics and retained bioactivity of PCB-insulin. Second, the production is simple from manufacturing standpoints. Conjugation procedure involves only one-step coupling reaction without complex site-specific linkage technique. The synthesized PCB-insulin conjugates do not require chromatographic separation to purify and obtain particular isoforms. © 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.
Exploring the role of internal friction in the dynamics of unfolded proteins using simple polymer models
Cheng, Ryan R.; Hawk, Alexander T.; Makarov, Dmitrii E.
Recent experiments showed that the reconfiguration dynamics of unfolded proteins are often adequately described by simple polymer models. In particular, the Rouse model with internal friction (RIF) captures internal friction effects as observed in single-molecule fluorescence correlation spectroscopy (FCS) studies of a number of proteins. Here we use RIF, and its non-free draining analog, Zimm model with internal friction, to explore the effect of internal friction on the rate with which intramolecular contacts can be formed within the unfolded chain. Unlike the reconfiguration times inferred from FCS experiments, which depend linearly on the solvent viscosity, the first passage times to form intramolecular contacts are shown to display a more complex viscosity dependence. We further describe scaling relationships obeyed by contact formation times in the limits of high and low internal friction. Our findings provide experimentally testable predictions that can serve as a framework for the analysis of future studies of contact formation in proteins.
Simultaneous pre-concentration and separation on simple paper-based analytical device for protein analysis.
Niu, Ji-Cheng; Zhou, Ting; Niu, Li-Li; Xie, Zhen-Sheng; Fang, Fang; Yang, Fu-Quan; Wu, Zhi-Yong
In this work, fast isoelectric focusing (IEF) was successfully implemented on an open paper fluidic channel for simultaneous concentration and separation of proteins from complex matrix. With this simple device, IEF can be finished in 10Â min with a resolution of 0.03 pH units and concentration factor of 10, as estimated by color model proteins by smartphone-based colorimetric detection. Fast detection of albumin from human serum and glycated hemoglobin (HBA1c) from blood cell was demonstrated. In addition, off-line identification of the model proteins from the IEF fractions with matrix-assisted laser desorption ionization time-of-flight mass spectrometry (MALDI-TOF-MS) was also shown. This PAD IEF is potentially useful either for point of care test (POCT) or biomarker analysis as a cost-effective sample pretreatment method.
Charcot-Marie-Tooth disease-linked protein SIMPLE functions with the ESCRT machinery in endosomal trafficking
Lee, Samuel M.; Chin, Lih-Shen; Li, Lian
Mutations in small integral membrane protein of lysosome/late endosome (SIMPLE) cause autosomal dominant, Charcot-Marie-Tooth disease (CMT) type 1C. The cellular function of SIMPLE is unknown and the pathogenic mechanism of SIMPLE mutations remains elusive. Here, we report that SIMPLE interacted and colocalized with endosomal sorting complex required for transport (ESCRT) components STAM1, Hrs, and TSG101 on early endosomes and functioned with the ESCRT machinery in the control of endosome-to...
Electrostatics of cysteine residues in proteins: parameterization and validation of a simple model.
Salsbury, Freddie R; Poole, Leslie B; Fetrow, Jacquelyn S
One of the most popular and simple models for the calculation of pK(a) s from a protein structure is the semi-macroscopic electrostatic model MEAD. This model requires empirical parameters for each residue to calculate pK(a) s. Analysis of current, widely used empirical parameters for cysteine residues showed that they did not reproduce expected cysteine pK(a) s; thus, we set out to identify parameters consistent with the CHARMM27 force field that capture both the behavior of typical cysteines in proteins and the behavior of cysteines which have perturbed pK(a) s. The new parameters were validated in three ways: (1) calculation across a large set of typical cysteines in proteins (where the calculations are expected to reproduce expected ensemble behavior); (2) calculation across a set of perturbed cysteines in proteins (where the calculations are expected to reproduce the shifted ensemble behavior); and (3) comparison to experimentally determined pK(a) values (where the calculation should reproduce the pK(a) within experimental error). Both the general behavior of cysteines in proteins and the perturbed pK(a) in some proteins can be predicted reasonably well using the newly determined empirical parameters within the MEAD model for protein electrostatics. This study provides the first general analysis of the electrostatics of cysteines in proteins, with specific attention paid to capturing both the behavior of typical cysteines in a protein and the behavior of cysteines whose pK(a) should be shifted, and validation of force field parameters for cysteine residues. Copyright © 2012 Wiley Periodicals, Inc.
A Simple and Rapid Method for Preparing a Cell-Free Bacterial Lysate for Protein Synthesis.
Nitzan Krinsky
Full Text Available Cell-free protein synthesis (CFPS systems are important laboratory tools that are used for various synthetic biology applications. Here, we present a simple and inexpensive laboratory-scale method for preparing a CFPS system from E. coli. The procedure uses basic lab equipment, a minimal set of reagents, and requires less than one hour to process the bacterial cell mass into a functional S30-T7 extract. BL21(DE3 and MRE600 E. coli strains were used to prepare the S30-T7 extract. The CFPS system was used to produce a set of fluorescent and therapeutic proteins of different molecular weights (up to 66 kDa. This system was able to produce 40-150 μg-protein/ml, with variations depending on the plasmid type, expressed protein and E. coli strain. Interestingly, the BL21-based CFPS exhibited stability and increased activity at 40 and 45°C. To the best of our knowledge, this is the most rapid and affordable lab-scale protocol for preparing a cell-free protein synthesis system, with high thermal stability and efficacy in producing therapeutic proteins.
A Simple and Rapid Method for Preparing a Cell-Free Bacterial Lysate for Protein Synthesis
Kaduri, Maya; Shainsky-Roitman, Janna; Goldfeder, Mor; Ivanir, Eran; Benhar, Itai; Shoham, Yuval; Schroeder, Avi
Cell-free protein synthesis (CFPS) systems are important laboratory tools that are used for various synthetic biology applications. Here, we present a simple and inexpensive laboratory-scale method for preparing a CFPS system from E. coli. The procedure uses basic lab equipment, a minimal set of reagents, and requires less than one hour to process the bacterial cell mass into a functional S30-T7 extract. BL21(DE3) and MRE600 E. coli strains were used to prepare the S30-T7 extract. The CFPS system was used to produce a set of fluorescent and therapeutic proteins of different molecular weights (up to 66 kDa). This system was able to produce 40–150 μg-protein/ml, with variations depending on the plasmid type, expressed protein and E. coli strain. Interestingly, the BL21-based CFPS exhibited stability and increased activity at 40 and 45°C. To the best of our knowledge, this is the most rapid and affordable lab-scale protocol for preparing a cell-free protein synthesis system, with high thermal stability and efficacy in producing therapeutic proteins. PMID:27768741
A simple purification and activity assay of the coagulant protein from Moringa oleifera seed.
Ghebremichael, Kebreab A; Gunaratna, K R; Henriksson, Hongbin; Brumer, Harry; Dalhammar, Gunnel
Use of extracts from Moringa oleifera (MO) is of great interest for low-cost water treatment. This paper discusses water and salt extraction of a coagulant protein from the seed, purification using ion exchange, its chemical characteristics, coagulation and antimicrobial properties. The coagulant from both extracts is a cationic protein with pI greater than 9.6 and molecular mass less than 6.5 kDa. Mass spectrometric analysis of the purified water extract indicated that it contained at least four homologous proteins, based on MS/MS peptide sequence data. The protein is thermoresistant and remained active after 5h heat treatment at 95 degrees C. The coagulant protein showed both flocculating and antibacterial effects of 1.1--4 log reduction. With samples of high turbidity, the MO extract showed similar coagulation activity as alum. Cecropin A and MO extract were found to have similar flocculation effects for clay and microorganisms. Simple methods for both the purification and assay of MO coagulating proteins are presented, which are necessary for large-scale water treatment applications.
Structure, Function, Self-Assembly and Origin of Simple Membrane Proteins
Pohorille, Andrew
Integral membrane proteins perform such essential cellular functions as transport of ions, nutrients and waste products across cell walls, transduction of environmental signals, regulation of cell fusion, recognition of other cells, energy capture and its conversion into high-energy compounds. In fact, 30-40% of genes in modem organisms codes for membrane proteins. Although contemporary membrane proteins or their functional assemblies can be quite complex, their transmembrane fragments are usually remarkably simple. The most common structural motif for these fragments is a bundle of alpha-helices, but occasionally it could be a beta-barrel. In a series of molecular dynamics computer simulations we investigated self-organizing properties of simple membrane proteins based on these structural motifs. Specifically, we studied folding and insertion into membranes of short, nonpolar or amphiphatic peptides. We also investigated glycophorin A, a peptide that forms sequence-specific dimers, and a transmembrane aggregate of four identical alpha-helices that forms an efficient and selective voltage-gated proton channel was investigated. Many peptides are attracted to water-membrane interfaces. Once at the interface, nonpolar peptides spontaneously fold to a-helices. Whenever the sequence permits, peptides that contain both polar and nonpolar amino also adopt helical structures, in which polar and nonpolar amino acid side chains are immersed in water and membrane, respectively. Specific identity of side chains is less important. Helical peptides at the interface could insert into the membrane and adopt a transmembrane conformation. However, insertion of a single helix is unfavorable because polar groups in the peptide become completely dehydrated upon insertion. The unfavorable free energy of insertion can be regained by spontaneous association of peptides in the membrane. The first step in this process is the formation of dimers, although the most common are aggregates of 4
A simple and reliable approach to docking protein-protein complexes from very sparse NOE-derived intermolecular distance restraints
Tang, Chun; Clore, G. Marius
A simple and reliable approach for docking protein-protein complexes from very sparse NOE-derived intermolecular distance restraints (as few as three from a single point) in combination with a novel representation for an attractive potential between mapped interaction surfaces is described. Unambiguous assignments of very sparse intermolecular NOEs are obtained using a reverse labeling strategy in which one the components is fully deuterated with the exception of selective protonation of the δ-methyl groups of isoleucine, while the other component is uniformly 13 C-labeled. This labeling strategy can be readily extended to selective protonation of Ala, Leu, Val or Met. The attractive potential is described by a 'reduced' radius of gyration potential applied specifically to a subset of interfacial residues (those with an accessible surface area ≥ 50% in the free proteins) that have been delineated by chemical shift perturbation. Docking is achieved by rigid body minimization on the basis of a target function comprising the sparse NOE distance restraints, a van der Waals repulsion potential and the 'reduced' radius of gyration potential. The method is demonstrated for two protein-protein complexes (EIN-HPr and IIA Glc -HPr) from the bacterial phosphotransferase system. In both cases, starting from 100 different random orientations of the X-ray structures of the free proteins, 100% convergence is achieved to a single cluster (with near identical atomic positions) with an overall backbone accuracy of ∼2 A. The approach described is not limited to NMR, since interfaces can also be mapped by alanine scanning mutagenesis, and sparse intermolecular distance restraints can be derived from double cycle mutagenesis, cross-linking combined with mass spectrometry, or fluorescence energy transfer
Tang, Chun; Clore, G. Marius [National Institutes of Health, Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases (United States)], E-mail: [email protected]
A simple and reliable approach for docking protein-protein complexes from very sparse NOE-derived intermolecular distance restraints (as few as three from a single point) in combination with a novel representation for an attractive potential between mapped interaction surfaces is described. Unambiguous assignments of very sparse intermolecular NOEs are obtained using a reverse labeling strategy in which one the components is fully deuterated with the exception of selective protonation of the {delta}-methyl groups of isoleucine, while the other component is uniformly {sup 13}C-labeled. This labeling strategy can be readily extended to selective protonation of Ala, Leu, Val or Met. The attractive potential is described by a 'reduced' radius of gyration potential applied specifically to a subset of interfacial residues (those with an accessible surface area {>=} 50% in the free proteins) that have been delineated by chemical shift perturbation. Docking is achieved by rigid body minimization on the basis of a target function comprising the sparse NOE distance restraints, a van der Waals repulsion potential and the 'reduced' radius of gyration potential. The method is demonstrated for two protein-protein complexes (EIN-HPr and IIA{sup Glc}-HPr) from the bacterial phosphotransferase system. In both cases, starting from 100 different random orientations of the X-ray structures of the free proteins, 100% convergence is achieved to a single cluster (with near identical atomic positions) with an overall backbone accuracy of {approx}2 A. The approach described is not limited to NMR, since interfaces can also be mapped by alanine scanning mutagenesis, and sparse intermolecular distance restraints can be derived from double cycle mutagenesis, cross-linking combined with mass spectrometry, or fluorescence energy transfer.
Simple knowledge-based descriptors to predict protein-ligand interactions. Methodology and validation
Nissink, J. Willem M.; Verdonk, Marcel L.; Klebe, Gerhard
A new type of shape descriptor is proposed to describe the spatial orientation for non-covalent interactions. It is built from simple, anisotropic Gaussian contributions that are parameterised by 10 adjustable values. The descriptors have been used to fit propensity distributions derived from scatter data stored in the IsoStar database. This database holds composite pictures of possible interaction geometries between a common central group and various interacting moieties, as extracted from small-molecule crystal structures. These distributions can be related to probabilities for the occurrence of certain interaction geometries among different functional groups. A fitting procedure is described that generates the descriptors in a fully automated way. For this purpose, we apply a similarity index that is tailored to the problem, the Split Hodgkin Index. It accounts for the similarity in regions of either high or low propensity in a separate way. Although dependent on the division into these two subregions, the index is robust and performs better than the regular Hodgkin index. The reliability and coverage of the fitted descriptors was assessed using SuperStar. SuperStar usually operates on the raw IsoStar data to calculate propensity distributions, e.g., for a binding site in a protein. For our purpose we modified the code to have it operate on our descriptors instead. This resulted in a substantial reduction in calculation time (factor of five to eight) compared to the original implementation. A validation procedure was performed on a set of 130 protein-ligand complexes, using four representative interacting probes to map the properties of the various binding sites: ammonium nitrogen, alcohol oxygen, carbonyl oxygen, and methyl carbon. The predicted `hot spots' for the binding of these probes were compared to the actual arrangement of ligand atoms in experimentally determined protein-ligand complexes. Results indicate that the version of SuperStar that applies to
Virtual Protein Purification: A Simple Exercise to Introduce pH as A Parameter That Effects Ion Exchange Chromatography
Clark, Daniel D.; Edwards, Daniel J.
This article describes a simple exercise using a free, easy-to-use, established online program. The exercise helps to reinforce protein purification concepts and introduces undergraduates to pH as a parameter that affects anion-exchange chromatography. The exercise was tested with biochemistry majors at California State University-Chico. Given the…
A Simple Spreadsheet Program to Simulate and Analyze the Far-UV Circular Dichroism Spectra of Proteins
Abriata, Luciano A.
A simple algorithm was implemented in a spreadsheet program to simulate the circular dichroism spectra of proteins from their secondary structure content and to fit [alpha]-helix, [beta]-sheet, and random coil contents from experimental far-UV circular dichroism spectra. The physical basis of the method is briefly reviewed within the context of…
Birth and death of protein domains: A simple model of evolution explains power law behavior
Berezovskaya Faina S
Full Text Available Abstract Background Power distributions appear in numerous biological, physical and other contexts, which appear to be fundamentally different. In biology, power laws have been claimed to describe the distributions of the connections of enzymes and metabolites in metabolic networks, the number of interactions partners of a given protein, the number of members in paralogous families, and other quantities. In network analysis, power laws imply evolution of the network with preferential attachment, i.e. a greater likelihood of nodes being added to pre-existing hubs. Exploration of different types of evolutionary models in an attempt to determine which of them lead to power law distributions has the potential of revealing non-trivial aspects of genome evolution. Results A simple model of evolution of the domain composition of proteomes was developed, with the following elementary processes: i domain birth (duplication with divergence, ii death (inactivation and/or deletion, and iii innovation (emergence from non-coding or non-globular sequences or acquisition via horizontal gene transfer. This formalism can be described as a birth, death and innovation model (BDIM. The formulas for equilibrium frequencies of domain families of different size and the total number of families at equilibrium are derived for a general BDIM. All asymptotics of equilibrium frequencies of domain families possible for the given type of models are found and their appearance depending on model parameters is investigated. It is proved that the power law asymptotics appears if, and only if, the model is balanced, i.e. domain duplication and deletion rates are asymptotically equal up to the second order. It is further proved that any power asymptotic with the degree not equal to -1 can appear only if the hypothesis of independence of the duplication/deletion rates on the size of a domain family is rejected. Specific cases of BDIMs, namely simple, linear, polynomial and rational
A simple quantitative model of macromolecular crowding effects on protein folding: Application to the murine prion protein(121-231)
Bergasa-Caceres, Fernando; Rabitz, Herschel A.
A model of protein folding kinetics is applied to study the effects of macromolecular crowding on protein folding rate and stability. Macromolecular crowding is found to promote a decrease of the entropic cost of folding of proteins that produces an increase of both the stability and the folding rate. The acceleration of the folding rate due to macromolecular crowding is shown to be a topology-dependent effect. The model is applied to the folding dynamics of the murine prion protein (121-231). The differential effect of macromolecular crowding as a function of protein topology suffices to make non-native configurations relatively more accessible.
Research Techniques Made Simple: Emerging Methods to Elucidate Protein Interactions through Spatial Proximity.
Che, Yonglu; Khavari, Paul A
Interactions between proteins are essential for fundamental cellular processes, and the diversity of such interactions enables the vast variety of functions essential for life. A persistent goal in biological research is to develop assays that can faithfully capture different types of protein interactions to allow their study. A major step forward in this direction came with a family of methods that delineates spatial proximity of proteins as an indirect measure of protein-protein interaction. A variety of enzyme- and DNA ligation-based methods measure protein co-localization in space, capturing novel interactions that were previously too transient or low affinity to be identified. Here we review some of the methods that have been successfully used to measure spatially proximal protein-protein interactions. Copyright © 2017 The Authors. Published by Elsevier Inc. All rights reserved.
Simple fluorescence-based detection of protein kinase A activity using a molecular beacon probe.
Ma, Changbei; Lv, Xiaoyuan; Wang, Kemin; Jin, Shunxin; Liu, Haisheng; Wu, Kefeng; Zeng, Weimin
Protein kinase A was detected by quantifying the amount of ATP used after a protein kinase reaction. The ATP assay was performed using the T4 DNA ligase and a molecular beacon (MB). In the presence of ATP, DNA ligase catalyzed the ligation of short DNA. The ligation product then hybridized to MB, resulting in a fluorescence enhancement of the MB. This assay was capable of determining protein kinase A in the range of 12.5∼150 nM, with a detection limit of 1.25 nM. Furthermore, this assay could also be used to investigate the effect of genistein on protein kinase A. It was a universal, non-radioisotopic, and homogeneous method for assaying protein kinase A.
MyPMFs: a simple tool for creating statistical potentials to assess protein structural models.
Postic, Guillaume; Hamelryck, Thomas; Chomilier, Jacques; Stratmann, Dirk
Evaluating the model quality of protein structures that evolve in environments with particular physicochemical properties requires scoring functions that are adapted to their specific residue compositions and/or structural characteristics. Thus, computational methods developed for structures from the cytosol cannot work properly on membrane or secreted proteins. Here, we present MyPMFs, an easy-to-use tool that allows users to train statistical potentials of mean force (PMFs) on the protein structures of their choice, with all parameters being adjustable. We demonstrate its use by creating an accurate statistical potential for transmembrane protein domains. We also show its usefulness to study the influence of the physical environment on residue interactions within protein structures. Our open-source software is freely available for download at https://github.com/bibip-impmc/mypmfs. Copyright © 2018. Published by Elsevier B.V.
JACOP: A simple and robust method for the automated classification of protein sequences with modular architecture
Pagni Marco
Full Text Available Abstract Background Whole-genome sequencing projects are rapidly producing an enormous number of new sequences. Consequently almost every family of proteins now contains hundreds of members. It has thus become necessary to develop tools, which classify protein sequences automatically and also quickly and reliably. The difficulty of this task is intimately linked to the mechanism by which protein sequences diverge, i.e. by simultaneous residue substitutions, insertions and/or deletions and whole domain reorganisations (duplications/swapping/fusion. Results Here we present a novel approach, which is based on random sampling of sub-sequences (probes out of a set of input sequences. The probes are compared to the input sequences, after a normalisation step; the results are used to partition the input sequences into homogeneous groups of proteins. In addition, this method provides information on diagnostic parts of the proteins. The performance of this method is challenged by two data sets. The first one contains the sequences of prokaryotic lyases that could be arranged as a multiple sequence alignment. The second one contains all proteins from Swiss-Prot Release 36 with at least one Src homology 2 (SH2 domain – a classical example for proteins with modular architecture. Conclusion The outcome of our method is robust, highly reproducible as shown using bootstrap and resampling validation procedures. The results are essentially coherent with the biology. This method depends solely on well-established publicly available software and algorithms.
A simple detection method for low-affinity membrane protein interactions by baculoviral display.
Toshiko Sakihama
Full Text Available BACKGROUND: Membrane protein interactions play an important role in cell-to-cell recognition in various biological activities such as in the immune or neural system. Nevertheless, there has remained the major obstacle of expression of the membrane proteins in their active form. Recently, we and other investigators found that functional membrane proteins express on baculovirus particles (budded virus, BV. In this study, we applied this BV display system to detect interaction between membrane proteins important for cell-to-cell interaction in immune system. METHODOLOGY/PRINCIPAL FINDINGS: We infected Sf9 cells with recombinant baculovirus encoding the T cell membrane protein CD2 or its ligand CD58 and recovered the BV. We detected specific interaction between CD2-displaying BV and CD58-displaying BV by an enzyme-linked immunosorbent assay (ELISA. Using this system, we also detected specific interaction between two other membrane receptor-ligand pairs, CD40-CD40 ligand (CD40L, and glucocorticoid-induced TNFR family-related protein (GITR-GITR ligand (GITRL. Furthermore, we observed specific binding of BV displaying CD58, CD40L, or GITRL to cells naturally expressing their respective receptors by flowcytometric analysis using anti-baculoviral gp64 antibody. Finally we isolated CD2 cDNA from a cDNA expression library by magnetic separation using CD58-displaying BV and anti-gp64 antibody. CONCLUSIONS: We found the BV display system worked effectively in the detection of the interaction of membrane proteins. Since various membrane proteins and their oligomeric complexes can be displayed on BV in the native form, this BV display system should prove highly useful in the search for natural ligands or to develop screening systems for therapeutic antibodies and/or compounds.
Effect of simple cook and defatten processing method on the protein ...
The cooked-defatted diet (CDD) and casilan diet (CAD) supported growth while Nitrogen-free diet (NFD), raw diet (RD) and cooked diet (CD) did not support growth. The protein efficiency ratio (PER), net protein utilization (NPU), apparent and true digestibility of the raw diets were very low (-4.01 ± 1.66, 48.35 ± 7.64, 44.90 ...
A simple strategy for the purification of native recombinant full-length human RPL10 protein from inclusion bodies.
Pereira, Larissa M; Silva, Luana R; Alves, Joseane F; Marin, Nélida; Silva, Flavio Sousa; Morganti, Ligia; Silva, Ismael D C G; Affonso, Regina
The L10 ribosomal protein (RPL10) plays a role in the binding of the 60 S and 40 S ribosomal subunits and in mRNA translation. The evidence indicates that RPL10 also has multiple extra-ribosomal functions, including tumor suppression. Recently, the presence of RPL10 in prostate and ovarian cancers was evaluated, and it was demonstrated to be associated with autistic disorders and premature ovarian failure. In the present work, we successfully cloned and expressed full-length human RPL10 (hRPL10) protein and isolated inclusion bodies containing this protein that had formed under mild growth conditions. The culture produced 376mg of hRPL10 protein per liter of induced bacterial culture, of which 102.4mg was present in the soluble fraction, and 25.6mg was recovered at approximately 94% purity. These results were obtained using a two-step process of non-denaturing protein extraction from pelleted inclusion bodies. We studied the characteristics of this protein using circular dichroism spectroscopy and by monitoring the changes induced by the presence or absence of zinc ions using fluorescence spectrometry. The results demonstrated that the protein obtained using these non-conventional methods retained its secondary and tertiary structure. The conformational changes induced by the incorporation of zinc suggested that this protein could interact with Jun or the SH3 domain of c-yes. The results suggested that the strategy used to obtain hRPL10 is simple and could be applied to obtaining other proteins that are susceptible to degradation. Copyright © 2014 Elsevier Inc. All rights reserved.
A simple biosynthetic method for stereospecific resonance assignment of prochiral methyl groups in proteins
Plevin, Michael J.; Hamelin, Olivier; Boisbouvier, Jérôme; Gans, Pierre
A new method for stereospecific assignment of prochiral methyl groups in proteins is presented in which protein samples are produced using U-[ 13 C]glucose and subsaturating amounts of 2-[ 13 C]methyl-acetolactate. The resulting non-uniform labeling pattern allows proR and proS methyl groups to be easily distinguished by their different phases in a constant-time two-dimensional 1 H- 13 C correlation spectra. Protein samples are conveniently prepared using the same media composition as the main uniformly-labeled sample and contain higher levels of isotope-enrichment than fractional labeling approaches. This new strategy thus represents an economically-attractive, robust alternative for obtaining isotopically-encoded stereospecific NMR assignments of prochiral methyl groups.
A simple tool for tubing modification to improve spiral high-speed counter-current chromatography for protein purification.
Ito, Yoichiro; Ma, Xiaofeng; Clary, Robert
A simple tool is introduced which can modify the shape of tubing to enhance the partition efficiency in high-speed countercurrent chromatography. It consists of a pair of interlocking identical gears, each coaxially holding a pressing wheel to intermittently compress plastic tubing in 0 - 10 mm length at every 1 cm interval. The performance of the processed tubing is examined in protein separation with 1.6 mm ID PTFE tubing intermittently pressed in 3 mm and 10 mm width both at 10 mm intervals at various flow rates and revolution speeds. A series of experiments was performed with a polymer phase system composed of polyethylene glycol and dibasic potassium phosphate each at 12.5% (w/w) in deionized water using three protein samples. Overall results clearly demonstrate that the compressed tubing can yield substantially higher peak resolution than the non-processed tubing. The simple tubing modifier is very useful for separation of proteins with high-speed countercurrent chromatography.
Using Simple Manipulatives to Improve Student Comprehension of a Complex Biological Process: Protein Synthesis
Guzman, Karen; Bartlett, John
Biological systems and living processes involve a complex interplay of biochemicals and macromolecular structures that can be challenging for undergraduate students to comprehend and, thus, misconceptions abound. Protein synthesis, or translation, is an example of a biological process for which students often hold many misconceptions. This article…
Introducing Proteomics in the Undergraduate Curriculum: A Simple 2D Gel Electrophoresis Exercise with Serum Proteins
Kim, Thomas D.; Craig, Paul A.
Two-dimensional gel electrophoresis (2DGE) remains an important tool in the study of biological systems by proteomics. While the use of 2DGE is commonplace in research publications, there are few instructional laboratories that address the use of 2DGE for analyzing complex protein samples. One reason for this lack is the fact that the preparation…
A simple Pichia pastoris fermentation and downstream processing strategy for making recombinant pandemic Swine Origin Influenza a virus Hemagglutinin protein.
Athmaram, T N; Singh, Anil Kumar; Saraswat, Shweta; Srivastava, Saurabh; Misra, Princi; Kameswara Rao, M; Gopalan, N; Rao, P V L
The present Influenza vaccine manufacturing process has posed a clear impediment to initiation of rapid mass vaccination against spreading pandemic influenza. New vaccine strategies are therefore needed that can accelerate the vaccine production. Pichia offers several advantages for rapid and economical bulk production of recombinant proteins and, hence, can be attractive alternative for producing an effective influenza HA based subunit vaccine. The recombinant Pichia harboring the transgene was subjected to fed-batch fermentation at 10Â L scale. A simple fermentation and downstream processing strategy is developed for high-yield secretory expression of the recombinant Hemagglutinin protein of pandemic Swine Origin Influenza A virus using Pichia pastoris via fed-batch fermentation. Expression and purification were optimized and the expressed recombinant Hemagglutinin protein was verified by sodium dodecyl sulfate polyacrylamide gel electrophoresis, Western blot and MALDI-TOF analysis. In this paper, we describe a fed-batch fermentation protocol for the secreted production of Swine Influenza A Hemagglutinin protein in the P. pastoris GS115 strain. We have shown that there is a clear relationship between product yield and specific growth rate. The fed-batch fermentation and downstream processing methods optimized in the present study have immense practical application for high-level production of the recombinant H1N1 HA protein in a cost effective way using P. pastoris.
Determination of olanzapine in whole blood using simple protein precipitation and liquid chromatography-tandem mass spectrometry
Nielsen, Marie Katrine Klose; Johansen, Sys Stybe
A simple, sensitive, and reproducible liquid chromatography-tandem mass spectrometry method has been developed and validated for the quantification of the antipsychotic drug olanzapine in whole blood using dibenzepine as internal standard (IS). After acidic methanol-induced protein precipitation......, and stability. The absolute recovery obtained was 103% for olanzapine and 68% for IS. An LOQ of 0.005 mg/kg olanzapine in whole blood was achieved. Inter- and intraday precision were less than 11% within concentrations from 0.01 to 0.50 mg/kg, and the accuracy ranged from 85 to 115%. The method was subsequently...
Simple protein structure-sensitive chronopotentiometric analysis with dithiothreitol-modified Hg electrodes
Ostatná, Veronika; Černocká, Hana; Pale�ek, Emil
Ro�. 87, SI (2012), s. 84-88 ISSN 1567-5394 R&D Projects: GA AV ČR(CZ) KJB100040901; GA ČR(CZ) GAP301/11/2055; GA MŠk(CZ) LC06035 Institutional research plan: CEZ:AV0Z50040507; CEZ:AV0Z50040702 Keywords : protein electroanalysis * DTT-modified electrodes * electrocatalysis Subject RIV: BO - Biophysics Impact factor: 3.947, year: 2012
A simple coated-tube assay for alpha-foeto protein for clinical use
Dakubu, S.; Ahene, I.S.; Foli, A.K.
A standard method for coating plastic tubes with antiserum has been applied to coat tubes with rabbit antiserum to human alpha-foeto protein. The coated plastic tubes have been used to set up a radioimmunoassay system which is sensitive and convenient for use on the occasional clinical sample. For a successful coated-tube assay, it was found necessary to modify the final incubation mixture from what was suitable in a standard double antibody assay system. (orig.) [de
Fast and simple protein-alignment-guided assembly of orthologous gene families from microbiome sequencing reads.
Huson, Daniel H; Tappu, Rewati; Bazinet, Adam L; Xie, Chao; Cummings, Michael P; Nieselt, Kay; Williams, Rohan
Microbiome sequencing projects typically collect tens of millions of short reads per sample. Depending on the goals of the project, the short reads can either be subjected to direct sequence analysis or be assembled into longer contigs. The assembly of whole genomes from metagenomic sequencing reads is a very difficult problem. However, for some questions, only specific genes of interest need to be assembled. This is then a gene-centric assembly where the goal is to assemble reads into contigs for a family of orthologous genes. We present a new method for performing gene-centric assembly, called protein-alignment-guided assembly, and provide an implementation in our metagenome analysis tool MEGAN. Genes are assembled on the fly, based on the alignment of all reads against a protein reference database such as NCBI-nr. Specifically, the user selects a gene family based on a classification such as KEGG and all reads binned to that gene family are assembled. Using published synthetic community metagenome sequencing reads and a set of 41 gene families, we show that the performance of this approach compares favorably with that of full-featured assemblers and that of a recently published HMM-based gene-centric assembler, both in terms of the number of reference genes detected and of the percentage of reference sequence covered. Protein-alignment-guided assembly of orthologous gene families complements whole-metagenome assembly in a new and very useful way.
A Simple Sonication Improves Protein Signal in Matrix-Assisted Laser Desorption Ionization Imaging
Lin, Li-En; Su, Pin-Rui; Wu, Hsin-Yi; Hsu, Cheng-Chih
Proper matrix application is crucial in obtaining high quality matrix-assisted laser desorption ionization (MALDI) mass spectrometry imaging (MSI). Solvent-free sublimation was essentially introduced as an approach of homogeneous coating that gives small crystal size of the organic matrix. However, sublimation has lower extraction efficiency of analytes. Here, we present that a simple sonication step after the hydration in standard sublimation protocol significantly enhances the sensitivity of MALDI MSI. This modified procedure uses a common laboratory ultrasonicator to immobilize the analytes from tissue sections without noticeable delocalization. Improved imaging quality with additional peaks above 10 kDa in the spectra was thus obtained upon sonication treatment. [Figure not available: see fulltext.
A simple and efficient method for assembling TALE protein based on plasmid library.
Zhang, Zhiqiang; Li, Duo; Xu, Huarong; Xin, Ying; Zhang, Tingting; Ma, Lixia; Wang, Xin; Chen, Zhilong; Zhang, Zhiying
DNA binding domain of the transcription activator-like effectors (TALEs) from Xanthomonas sp. consists of tandem repeats that can be rearranged according to a simple cipher to target new DNA sequences with high DNA-binding specificity. This technology has been successfully applied in varieties of species for genome engineering. However, assembling long TALE tandem repeats remains a big challenge precluding wide use of this technology. Although several new methodologies for efficiently assembling TALE repeats have been recently reported, all of them require either sophisticated facilities or skilled technicians to carry them out. Here, we described a simple and efficient method for generating customized TALE nucleases (TALENs) and TALE transcription factors (TALE-TFs) based on TALE repeat tetramer library. A tetramer library consisting of 256 tetramers covers all possible combinations of 4 base pairs. A set of unique primers was designed for amplification of these tetramers. PCR products were assembled by one step of digestion/ligation reaction. 12 TALE constructs including 4 TALEN pairs targeted to mouse Gt(ROSA)26Sor gene and mouse Mstn gene sequences as well as 4 TALE-TF constructs targeted to mouse Oct4, c-Myc, Klf4 and Sox2 gene promoter sequences were generated by using our method. The construction routines took 3 days and parallel constructions were available. The rate of positive clones during colony PCR verification was 64% on average. Sequencing results suggested that all TALE constructs were performed with high successful rate. This is a rapid and cost-efficient method using the most common enzymes and facilities with a high success rate.
Simple clinical means of documenting increased pulmonary endothelial permeability to protein
Mishkin, F.S.; Niden, A.; Kumar, A.; Thomas, A.; Reese, I.C.; Vasinrapee, P.
The authors investigated a simple method that can be used at the bedside for documenting the net accumulation of albumin in the lung. The technique employs measurement with a computer-linked gamma camera of the activity ratio in an area of the right lung compared with the same-sized area in the heart at 20 minutes and three hours following intravenous injection of technetium Tc 99m albumin. They applied this measurement to three groups of patients: a control group and patients with roentgenographic evidence of edema classified according to clinically available criteria as either hydrostatic edema or permeability edema to see if they could document differences among these groups. In control patients this ratio did not increase by more than seven units between the 20-minute and three-hour measurements. Of 18 patients classified by other routine clinical means as having hydrostatic pulmonary edema, 89% showed no increase in lung albumin accumulation. In 29 patients with permeability edema associated with the so-called adult respiratory distress syndrome, 31% showed evidence of net pulmonary albumin accumulation. These findings suggest that some patients otherwise classified as having hydrostatic edema have concomitant permeability changes in the microvasculature and that permeability edema represents a spectrum of endothelial damage
The authors investigated a simple method that can be used at the bedside for documenting the net accumulation of albumin in the lung. The technique employs measurement with a computer-linked gamma camera of the activity ratio in an area of the right lung compared with the same-sized area in the heart at 20 minutes and three hours following intravenous injection of technetium Tc 99m albumin. They applied this measurement to three groups of patients: a control group and patients with roentgenographic evidence of edema classified according to clinically available criteria as either hydrostatic edema or permeability edema to see if they could document differences among these groups. In control patients this ratio did not increase by more than seven units between the 20-minute and three-hour measurements. Of 18 patients classified by other routine clinical means as having hydrostatic pulmonary edema, 89% showed no increase in lung albumin accumulation. In 29 patients with permeability edema associated with the so-called adult respiratory distress syndrome, 31% showed evidence of net pulmonary albumin accumulation. These findings suggest that some patients otherwise classified as having hydrostatic edema have concomitant permeability changes in the microvasculature and that permeability edema represents a spectrum of endothelial damage.
Unification of Cas protein families and a simple scenario for the origin and evolution of CRISPR-Cas systems
Wolf Yuri I
Full Text Available Abstract Background The CRISPR-Cas adaptive immunity systems that are present in most Archaea and many Bacteria function by incorporating fragments of alien genomes into specific genomic loci, transcribing the inserts and using the transcripts as guide RNAs to destroy the genome of the cognate virus or plasmid. This RNA interference-like immune response is mediated by numerous, diverse and rapidly evolving Cas (CRISPR-associated proteins, several of which form the Cascade complex involved in the processing of CRISPR transcripts and cleavage of the target DNA. Comparative analysis of the Cas protein sequences and structures led to the classification of the CRISPR-Cas systems into three Types (I, II and III. Results A detailed comparison of the available sequences and structures of Cas proteins revealed several unnoticed homologous relationships. The Repeat-Associated Mysterious Proteins (RAMPs containing a distinct form of the RNA Recognition Motif (RRM domain, which are major components of the CRISPR-Cas systems, were classified into three large groups, Cas5, Cas6 and Cas7. Each of these groups includes many previously uncharacterized proteins now shown to adopt the RAMP structure. Evidence is presented that large subunits contained in most of the CRISPR-Cas systems could be homologous to Cas10 proteins which contain a polymerase-like Palm domain and are predicted to be enzymatically active in Type III CRISPR-Cas systems but inactivated in Type I systems. These findings, the fact that the CRISPR polymerases, RAMPs and Cas2 all contain core RRM domains, and distinct gene arrangements in the three types of CRISPR-Cas systems together provide for a simple scenario for origin and evolution of the CRISPR-Cas machinery. Under this scenario, the CRISPR-Cas system originated in thermophilic Archaea and subsequently spread horizontally among prokaryotes. Conclusions Because of the extreme diversity of CRISPR-Cas systems, in-depth sequence and structure
Simple assay method for proteins carrying sexual hormones (PTHS); values in men, women, and during pregnancy
Tafurt, C.A.; Estrada, R. de.
Starting from the fact that the binding forces between steroid hormones and their carrier proteins are similar to those between antigens and antibodies, the paper describes PTHS determination by a dilution method analogous to antiserum labelling for radioimmunoassay. The method consists of the following steps: 1) Plasma dilution, 2) incubation of the solutions with 20,000 dpm 1,2 3 H testosterone, 3) separation of the tracer fraction bound to PTHS by precipitation with ammonium sulfate, 4) centrifugation and measurement of the supernatant, 5) presentation of the findings in a graphical system with the bound steroid fraction, referred to the free steroid (U/L) as the ordinate and the plasma dilutions as the abscissa. The values represent the label in 50% of the sites. The method offers the highest sensitivity, i.e. the steepest parts of the dilution curves where 50% of the binding sites are located. The method also dispenses with tedious processes such as dialysis. The following PTHS values were obtained: 1/5 in men, 1/93 in women, and 1/360 in pregnant women. There were no cross-reactions. (AJ) [de
A simple electroelution method for rapid protein purification: isolation and antibody production of alpha toxin from Clostridium septicum
Lorena Vázquez-Iglesias
Full Text Available Clostridium septicum produces a number of diseases in human and farm animals which, in most of the cases, are fatal without clinical intervention. Alpha toxin is an important agent and the unique lethal virulent factor produced by Clostridium septicum. This toxin is haemolytic, highly lethal and necrotizing activities but is being used as an antigen to develop animal vaccines. The aim of this study was to isolate the alpha toxin of Clostridium septicum and produce highly specific antibodies against it. In this work, we have developed a simple and efficient method for alpha toxin purification, based on electroelution that can be used as a time-saving method for purifying proteins. This technique avoids contamination by other proteins that could appear during other protein purification techniques such chromatography. The highly purified toxin was used to produce polyclonal antibodies. The specificity of the antibodies was tested by western blot and these antibodies can be applied to the quantitative determination of alpha toxin by slot blot.
Simple and efficient machine learning frameworks for identifying protein-protein interaction relevant articles and experimental methods used to study the interactions.
Agarwal, Shashank; Liu, Feifan; Yu, Hong
Protein-protein interaction (PPI) is an important biomedical phenomenon. Automatically detecting PPI-relevant articles and identifying methods that are used to study PPI are important text mining tasks. In this study, we have explored domain independent features to develop two open source machine learning frameworks. One performs binary classification to determine whether the given article is PPI relevant or not, named "Simple Classifier", and the other one maps the PPI relevant articles with corresponding interaction method nodes in a standardized PSI-MI (Proteomics Standards Initiative-Molecular Interactions) ontology, named "OntoNorm". We evaluated our system in the context of BioCreative challenge competition using the standardized data set. Our systems are amongst the top systems reported by the organizers, attaining 60.8% F1-score for identifying relevant documents, and 52.3% F1-score for mapping articles to interaction method ontology. Our results show that domain-independent machine learning frameworks can perform competitively well at the tasks of detecting PPI relevant articles and identifying the methods that were used to study the interaction in such articles. Simple Classifier is available at http://sourceforge.net/p/simpleclassify/home/ and OntoNorm at http://sourceforge.net/p/ontonorm/home/.
A simple three-dimensional-focusing, continuous-flow mixer for the study of fast protein dynamics.
Burke, Kelly S; Parul, Dzmitry; Reddish, Michael J; Dyer, R Brian
We present a simple, yet flexible microfluidic mixer with a demonstrated mixing time as short as 80 μs that is widely accessible because it is made of commercially available parts. To simplify the study of fast protein dynamics, we have developed an inexpensive continuous-flow microfluidic mixer, requiring no specialized equipment or techniques. The mixer uses three-dimensional, hydrodynamic focusing of a protein sample stream by a surrounding sheath solution to achieve rapid diffusional mixing between the sample and sheath. Mixing initiates the reaction of interest. Reactions can be spatially observed by fluorescence or absorbance spectroscopy. We characterized the pixel-to-time calibration and diffusional mixing experimentally. We achieved a mixing time as short as 80 μs. We studied the kinetics of horse apomyoglobin (apoMb) unfolding from the intermediate (I) state to its completely unfolded (U) state, induced by a pH jump from the initial pH of 4.5 in the sample stream to a final pH of 2.0 in the sheath solution. The reaction time was probed using the fluorescence of 1-anilinonaphthalene-8-sulfonate (1,8-ANS) bound to the folded protein. We observed unfolding of apoMb within 760 μs, without populating additional intermediate states under these conditions. We also studied the reaction kinetics of the conversion of pyruvate to lactate catalyzed by lactate dehydrogenase using the intrinsic tryptophan emission of the enzyme. We observe sub-millisecond kinetics that we attribute to Michaelis complex formation and loop domain closure. These results demonstrate the utility of the three-dimensional focusing mixer for biophysical studies of protein dynamics.
A Simple Fractionated Extraction Method for the Comprehensive Analysis of Metabolites, Lipids, and Proteins from a Single Sample.
Salem, Mohamed; Bernach, Michal; Bajdzienko, Krzysztof; Giavalisco, Patrick
Understanding of complex biological systems requires the measurement, analysis and integration of multiple compound classes of the living cell, usually determined by transcriptomic, proteomic, metabolomics and lipidomic measurements. In this protocol, we introduce a simple method for the reproducible extraction of metabolites, lipids and proteins from biological tissues using a single aliquot per sample. The extraction method is based on a methyl tert-butyl ether: methanol: water system for liquid: liquid partitioning of hydrophobic and polar metabolites into two immiscible phases along with the precipitation of proteins and other macromolecules as a solid pellet. This method, therefore, provides three different fractions of specific molecular composition, which are fully compatible with common high throughput 'omics' technologies such as liquid chromatography (LC) or gas chromatography (GC) coupled to mass spectrometers. Even though the method was initially developed for the analysis of different plant tissue samples, it has proved to be fully compatible for the extraction and analysis of biological samples from systems as diverse as algae, insects, and mammalian tissues and cell cultures.
A simple microfluidic platform to study age-dependent protein abundance and localization changes in Saccharomyces cerevisiae
Margarita Cabrera
Full Text Available The budding yeast Saccharomyces cerevisiae divides asymmetrically, with a smaller daughter cell emerging from its larger mother cell. While the daughter lineage is immortal, mother cells age with each cell division and have a finite lifespan. The replicative ageing of the yeast mother cell has been used as a model to study the ageing of mitotically active human cells. Several microfluidic platforms, which use fluid flow to selectively remove daughter cells, have recently been developed that can monitor cell physiology as mother cells age. However, these platforms are not trivial to set up and users often require many hours of training. In this study, we have developed a simple system, which combines a commercially available microfluidic platform (the CellASIC ONIX Microfluidic Platform and a genetic tool to prevent the proliferation of daughter cells (the Mother Enrichment Program, to monitor protein abundance and localization changes during approximately the first half of the yeast replicative lifespan. We validated our system by observing known age-dependent changes, such as decreased Sir2 abundance, and have identified a protein with a previously unknown age-dependent change in localization.
A Simple, Fast, Low Cost, HPLC/UV Validated Method for Determination of Flutamide: Application to Protein Binding Studies.
Esmaeilzadeh, Sara; Valizadeh, Hadi; Zakeri-Milani, Parvin
The main goal of this study was development of a reverse phase high performance liquid chromatography (RP-HPLC) method for flutamide quantitation which is applicable to protein binding studies. Ultrafilteration method was used for protein binding study of flutamide. For sample analysis, flutamide was extracted by a simple and low cost extraction method using diethyl ether and then was determined by HPLC/UV. Acetanilide was used as an internal standard. The chromatographic system consisted of a reversed-phase C8 column with C8 pre-column, and the mobile phase of a mixture of 29% (v/v) methanol, 38% (v/v) acetonitrile and 33% (v/v) potassium dihydrogen phosphate buffer (50 mM) with pH adjusted to 3.2. Acetanilide and flutamide were eluted at 1.8 and 2.9 min, respectively. The linearity of method was confirmed in the range of 62.5-16000 ng/ml (r(2) > 0.99). The limit of quantification was shown to be 62.5 ng/ml. Precision and accuracy ranges found to be (0.2-1.4%, 90-105%) and (0.2-5.3 %, 86.7-98.5 %) respectively. Acetanilide and flutamide capacity factor values of 1.35 and 2.87, tailing factor values of 1.24 and 1.07 and resolution values of 1.8 and 3.22 were obtained in accordance with ICH guidelines. Based on the obtained results a rapid, precise, accurate, sensitive and cost-effective analysis procedure was proposed for quantitative determination of flutamide.
Effects of protein versus simple sugar intake on weight loss in polycystic ovary syndrome (according to the National Institutes of Health criteria).
Kasim-Karakas, Sidika E; Almario, Rogelio U; Cunningham, Wendy
To compare the effects of protein vs. simple sugars on weight loss, body composition, and metabolic and endocrine parameters in polycystic ovary syndrome (PCOS). A 2-month, free-living, randomized, single-blinded study. University PCOS clinic. Thirty-three patients with PCOS. To achieve a final energy reduction of 450 kcal/day, first the daily energy intake was reduced by 700 kcal; then a 240-kcal supplement containing either whey protein or simple sugars was added. Changes in weight, fat mass, fasting glucose and insulin, plasma lipoproteins, and sex steroids. Twenty-four subjects (13 in the simple sugars group and 11 in the protein group) completed the study. The protein group lost more weight (-3.3 +/- 0.8 kg vs. -1.1 +/- 0.6 kg) and more fat mass (-3.1 +/- 0.9 kg vs. -0.5 +/- 0.6 kg) and had larger decreases in serum cholesterol (-33.0 +/- 8.4 mg/dL vs. -2.3 +/- 6.8 mg/dL), high-density lipoprotein cholesterol (-4.5 +/- 1.3 mg/dL vs. -0.4 +/- 1.3 mg/dL), and apoprotein B (-20 +/- 5 mg/dL vs. 3 +/- 5 mg/dL). In patients with PCOS, a hypocaloric diet supplemented with protein reduced body weight, fat mass, serum cholesterol, and apoprotein B more than the diet supplemented with simple sugars.
Direct, simple derivatization of disulfide bonds in proteins with organic mercury in alkaline medium without any chemical pre-reducing agents
Campanella, Beatrice; Onor, Massimo [National Research Council of Italy, C.N.R., Istituto di Chimica dei Composti Organo Metallici-ICCOM- UOS Pisa, Area di Ricerca, Via G. Moruzzi 1, 56124 Pisa (Italy); Ferrari, Carlo [National Research Council of Italy, C.N.R., Istituto Nazionale di Ottica, INO-UOS Pisa, Area di Ricerca, Via G. Moruzzi 1, 56124 Pisa (Italy); D'Ulivo, Alessandro [National Research Council of Italy, C.N.R., Istituto di Chimica dei Composti Organo Metallici-ICCOM- UOS Pisa, Area di Ricerca, Via G. Moruzzi 1, 56124 Pisa (Italy); Bramanti, Emilia, E-mail: [email protected] [National Research Council of Italy, C.N.R., Istituto di Chimica dei Composti Organo Metallici-ICCOM- UOS Pisa, Area di Ricerca, Via G. Moruzzi 1, 56124 Pisa (Italy)
Highlights: • A simple procedure for the derivatization of proteins disulfide bonds. • Cysteine groups in several proteins derivatised with pHMB in alkaline media. • 75–100% labelling of cysteines in proteins with pHMB. - Abstract: In this work we have studied the derivatization of protein disulfide bonds with p-Hydroxymercurybenzoate (pHMB) in strong alkaline medium without any preliminary reduction. The reaction has been followed by the determination of the protein–pHMB complex using size exclusion chromatography coupled to a microwave/UV mercury oxidation system for the on-line oxidation of free and protein-complexed pHMB and atomic fluorescence spectrometry (SEC–CVG–AFS) detection. The reaction has been optimized by an experimental design using lysozyme as a model protein and applied to several thiolic proteins. The proposed method reports, for the first time, that it is possible to label 75–100% cysteines of proteins and, thus, to determine thiolic proteins without the need of any reducing step to obtain reduced -SH groups before mercury labelling. We obtained a detection limit of 100 nmol L{sup −1} based on a signal-to-noise ratio of 3 for unbound and complexed pHMB, corresponding to a detection limit of proteins ranged between 3 and 360 nmol L{sup −1}, depending on the number of cysteines in the protein sequence.
Campanella, Beatrice; Onor, Massimo; Ferrari, Carlo; D'Ulivo, Alessandro; Bramanti, Emilia
Highlights: • A simple procedure for the derivatization of proteins disulfide bonds. • Cysteine groups in several proteins derivatised with pHMB in alkaline media. • 75–100% labelling of cysteines in proteins with pHMB. - Abstract: In this work we have studied the derivatization of protein disulfide bonds with p-Hydroxymercurybenzoate (pHMB) in strong alkaline medium without any preliminary reduction. The reaction has been followed by the determination of the protein–pHMB complex using size exclusion chromatography coupled to a microwave/UV mercury oxidation system for the on-line oxidation of free and protein-complexed pHMB and atomic fluorescence spectrometry (SEC–CVG–AFS) detection. The reaction has been optimized by an experimental design using lysozyme as a model protein and applied to several thiolic proteins. The proposed method reports, for the first time, that it is possible to label 75–100% cysteines of proteins and, thus, to determine thiolic proteins without the need of any reducing step to obtain reduced -SH groups before mercury labelling. We obtained a detection limit of 100 nmol L −1 based on a signal-to-noise ratio of 3 for unbound and complexed pHMB, corresponding to a detection limit of proteins ranged between 3 and 360 nmol L −1 , depending on the number of cysteines in the protein sequence
A simple and robust protocol for high-yield expression of perdeuterated proteins in Escherichia coli grown in shaker flasks
Cai, Mengli; Huang, Ying; Yang, Renbin; Craigie, Robert; Clore, G. M.
We present a simple, convenient and robust protocol for expressing perdeuterated proteins in E. coli BL21(DE3) cells in shaker flasks that reduces D_2O usage tenfold and d_7-glucose usage by 30 %. Using a modified M9 medium and optimized growth conditions, we were able to grow cells in linear log phase to an OD_6_0_0 of up to 10. Inducing the cells with isopropyl β-d-1-thiogalactopyranoside at an OD_6_0_0 of 10, instead of less than 1, enabled us to increase the cell mass tenfold per unit volume of cell culture. We show that protein expression levels per cell are the same when induced at an OD_6_0_0 between 1 and 10 under these growth conditions. Thus, our new protocol can increase protein yield per unit volume of cell culture tenfold. Adaptation of E. coli from H_2O-based to D_2O-based medium is also key for ensuring high levels of protein expression in D_2O. We find that a simple three-step adaptation approach—Luria–Bertani (LB) medium in H_2O to LB in D_2O to modified-M9 medium in D_2O is both simple and reliable. The method increases the yield of perdeuterated proteins by up to tenfold using commonly available air shakers without any requirement for specialized fermentation equipment.
Sorting protein lists with nwCompare: a simple and fast algorithm for n-way comparison of proteomic data files.
Pont, Frédéric; Fournié, Jean Jacques
MS, the reference technology for proteomics, routinely produces large numbers of protein lists whose fast comparison would prove very useful. Unfortunately, most softwares only allow comparisons of two to three lists at once. We introduce here nwCompare, a simple tool for n-way comparison of several protein lists without any query language, and exemplify its use with differential and shared cancer cell proteomes. As the software compares character strings, it can be applied to any type of data mining, such as genomic or metabolomic datalists.
A simple elution strategy for biotinylated proteins bound to streptavidin conjugated beads using excess biotin and heat.
Cheah, Joleen S; Yamada, Soichiro
Protein-protein interactions are the molecular basis of cell signaling. Recently, proximity based biotin identification (BioID) has emerged as an alternative approach to traditional co-immunoprecipitation. In this protocol, a mutant biotin ligase promiscuously labels proximal binding partners with biotin, and resulting biotinylated proteins are purified using streptavidin conjugated beads. This approach does not require preservation of protein complexes in vitro, making it an ideal approach to identify transient or weak protein complexes. However, due to the high affinity bond between streptavidin and biotin, elution of biotinylated proteins from streptavidin conjugated beads requires harsh denaturing conditions, which are often incompatible with downstream processing. To effectively release biotinylated proteins bound to streptavidin conjugated beads, we designed a series of experiments to determine optimal binding and elution conditions. Interestingly, the concentrations of SDS and IGEPAL-CA630 during the incubation with streptavidin conjugated beads were the key to effective elution of biotinylated proteins using excess biotin and heating. This protocol provides an alternative method to isolate biotinylated proteins from streptavidin conjugated beads that is suitable for further downstream analysis. Copyright © 2017 Elsevier Inc. All rights reserved.
TA-GC cloning: A new simple and versatile technique for the directional cloning of PCR products for recombinant protein expression.
Athanasios Niarchos
Full Text Available During the last few decades, the recombinant protein expression finds more and more applications. The cloning of protein-coding genes into expression vectors is required to be directional for proper expression, and versatile in order to facilitate gene insertion in multiple different vectors for expression tests. In this study, the TA-GC cloning method is proposed, as a new, simple and efficient method for the directional cloning of protein-coding genes in expression vectors. The presented method features several advantages over existing methods, which tend to be relatively more labour intensive, inflexible or expensive. The proposed method relies on the complementarity between single A- and G-overhangs of the protein-coding gene, obtained after a short incubation with T4 DNA polymerase, and T and C overhangs of the novel vector pET-BccI, created after digestion with the restriction endonuclease BccI. The novel protein-expression vector pET-BccI also facilitates the screening of transformed colonies for recombinant transformants. Evaluation experiments of the proposed TA-GC cloning method showed that 81% of the transformed colonies contained recombinant pET-BccI plasmids, and 98% of the recombinant colonies expressed the desired protein. This demonstrates that TA-GC cloning could be a valuable method for cloning protein-coding genes in expression vectors.
Niarchos, Athanasios; Siora, Anastasia; Konstantinou, Evangelia; Kalampoki, Vasiliki; Lagoumintzis, George; Poulas, Konstantinos
During the last few decades, the recombinant protein expression finds more and more applications. The cloning of protein-coding genes into expression vectors is required to be directional for proper expression, and versatile in order to facilitate gene insertion in multiple different vectors for expression tests. In this study, the TA-GC cloning method is proposed, as a new, simple and efficient method for the directional cloning of protein-coding genes in expression vectors. The presented method features several advantages over existing methods, which tend to be relatively more labour intensive, inflexible or expensive. The proposed method relies on the complementarity between single A- and G-overhangs of the protein-coding gene, obtained after a short incubation with T4 DNA polymerase, and T and C overhangs of the novel vector pET-BccI, created after digestion with the restriction endonuclease BccI. The novel protein-expression vector pET-BccI also facilitates the screening of transformed colonies for recombinant transformants. Evaluation experiments of the proposed TA-GC cloning method showed that 81% of the transformed colonies contained recombinant pET-BccI plasmids, and 98% of the recombinant colonies expressed the desired protein. This demonstrates that TA-GC cloning could be a valuable method for cloning protein-coding genes in expression vectors.
Cai, Mengli [National Institutes of Health, Laboratories of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases (United States); Huang, Ying; Yang, Renbin; Craigie, Robert, E-mail: [email protected] [National Institutes of Health, Laboratories of Molecular Biology, National Institute of Diabetes and Digestive and Kidney Diseases (United States); Clore, G. M., E-mail: [email protected] [National Institutes of Health, Laboratories of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases (United States)
We present a simple, convenient and robust protocol for expressing perdeuterated proteins in E. coli BL21(DE3) cells in shaker flasks that reduces D{sub 2}O usage tenfold and d{sub 7}-glucose usage by 30 %. Using a modified M9 medium and optimized growth conditions, we were able to grow cells in linear log phase to an OD{sub 600} of up to 10. Inducing the cells with isopropyl β-d-1-thiogalactopyranoside at an OD{sub 600} of 10, instead of less than 1, enabled us to increase the cell mass tenfold per unit volume of cell culture. We show that protein expression levels per cell are the same when induced at an OD{sub 600} between 1 and 10 under these growth conditions. Thus, our new protocol can increase protein yield per unit volume of cell culture tenfold. Adaptation of E. coli from H{sub 2}O-based to D{sub 2}O-based medium is also key for ensuring high levels of protein expression in D{sub 2}O. We find that a simple three-step adaptation approach—Luria–Bertani (LB) medium in H{sub 2}O to LB in D{sub 2}O to modified-M9 medium in D{sub 2}O is both simple and reliable. The method increases the yield of perdeuterated proteins by up to tenfold using commonly available air shakers without any requirement for specialized fermentation equipment.
Development of a simple measurement method for GluR2 protein expression as an index of neuronal vulnerability
Chihiro Sugiyama
Full Text Available In vitro estimating strategies for potential neurotoxicity are required to screen multiple substances. In a previous study, we showed that exposure to low-concentrations of some chemicals, such as organotin, decreased the expression of GluR2 protein, which is a subunit of alpha-amino-3-hydroxy-5-methyl-4-isoxazole propionic acid (AMPA-type glutamate receptors, and led to neuronal vulnerability. This result suggested that GluR2 decreases as an index of neuronal cell sensitivity and vulnerability to various toxic insults. Accordingly, we developed a versatile method that is a large scale determination of GluR2 protein expression in the presence of environmental chemicals by means of AlphaLISA technology. Various analytical conditions were optimized, and then GluR2 protein amount was measured by the method using AlphaLISA. The GluR2 amounts were strongly correlated with that of measured by western blotting, which is currently used to determine GluR2 expression. An ideal standard curve could be written with the authentic GluR2 protein from 0Â ng to 100Â ng. Subsequently, twenty environmental chemicals were screened and nitenpyram was identified as a chemical which lead to decrease in GluR2 protein expression. This assay may provide a tool for detecting neurotoxic chemicals according to decreases in GluR2 protein expression.
Resolution-by-proxy: a simple measure for assessing and comparing the overall quality of NMR protein structures
Berjanskii, Mark; Zhou Jianjun; Liang Yongjie; Lin Guohui; Wishart, David S.
In protein X-ray crystallography, resolution is often used as a good indicator of structural quality. Diffraction resolution of protein crystals correlates well with the number of X-ray observables that are used in structure generation and, therefore, with protein coordinate errors. In protein NMR, there is no parameter identical to X-ray resolution. Instead, resolution is often used as a synonym of NMR model quality. Resolution of NMR structures is often deduced from ensemble precision, torsion angle normality and number of distance restraints per residue. The lack of common techniques to assess the resolution of X-ray and NMR structures complicates the comparison of structures solved by these two methods. This problem is sometimes approached by calculating "equivalent resolution� from structure quality metrics. However, existing protocols do not offer a comprehensive assessment of protein structure as they calculate equivalent resolution from a relatively small number (<5) of protein parameters. Here, we report a development of a protocol that calculates equivalent resolution from 25 measurable protein features. This new method offers better performance (correlation coefficient of 0.92, mean absolute error of 0.28 Å) than existing predictors of equivalent resolution. Because the method uses coordinate data as a proxy for X-ray diffraction data, we call this measure "Resolution-by-Proxy� or ResProx. We demonstrate that ResProx can be used to identify under-restrained, poorly refined or inaccurate NMR structures, and can discover structural defects that the other equivalent resolution methods cannot detect. The ResProx web server is available at http://www.resprox.cahttp://www.resprox.ca.
DBAC: A simple prediction method for protein binding hot spots based on burial levels and deeply buried atomic contacts
Background A protein binding hot spot is a cluster of residues in the interface that are energetically important for the binding of the protein with its interaction partner. Identifying protein binding hot spots can give useful information to protein engineering and drug design, and can also deepen our understanding of protein-protein interaction. These residues are usually buried inside the interface with very low solvent accessible surface area (SASA). Thus SASA is widely used as an outstanding feature in hot spot prediction by many computational methods. However, SASA is not capable of distinguishing slightly buried residues, of which most are non hot spots, and deeply buried ones that are usually inside a hot spot. Results We propose a new descriptor called "burial level� for characterizing residues, atoms and atomic contacts. Specifically, burial level captures the depth the residues are buried. We identify different kinds of deeply buried atomic contacts (DBAC) at different burial levels that are directly broken in alanine substitution. We use their numbers as input for SVM to classify between hot spot or non hot spot residues. We achieve F measure of 0.6237 under the leave-one-out cross-validation on a data set containing 258 mutations. This performance is better than other computational methods. Conclusions Our results show that hot spot residues tend to be deeply buried in the interface, not just having a low SASA value. This indicates that a high burial level is not only a necessary but also a more sufficient condition than a low SASA for a residue to be a hot spot residue. We find that those deeply buried atoms become increasingly more important when their burial levels rise up. This work also confirms the contribution of deeply buried interfacial atomic contacts to the energy of protein binding hot spot. PMID:21689480
A FREQUENCY-BASED LINGUISTIC APPROACH TO PROTEIN DECODING AND DESIGN: SIMPLE CONCEPTS, DIVERSE APPLICATIONS, AND THE SCS PACKAGE
Kenta Motomura
Full Text Available Protein structure and function information is coded in amino acid sequences. However, the relationship between primary sequences and three-dimensional structures and functions remains enigmatic. Our approach to this fundamental biochemistry problem is based on the frequencies of short constituent sequences (SCSs or words. A protein amino acid sequence is considered analogous to an English sentence, where SCSs are equivalent to words. Availability scores, which are defined as real SCS frequencies in the non-redundant amino acid database relative to their probabilistically expected frequencies, demonstrate the biological usage bias of SCSs. As a result, this frequency-based linguistic approach is expected to have diverse applications, such as secondary structure specifications by structure-specific SCSs and immunological adjuvants with rare or non-existent SCSs. Linguistic similarities (e.g., wide ranges of scale-free distributions and dissimilarities (e.g., behaviors of low-rank samples between proteins and the natural English language have been revealed in the rank-frequency relationships of SCSs or words. We have developed a web server, the SCS Package, which contains five applications for analyzing protein sequences based on the linguistic concept. These tools have the potential to assist researchers in deciphering structurally and functionally important protein sites, species-specific sequences, and functional relationships between SCSs. The SCS Package also provides researchers with a tool to construct amino acid sequences de novo based on the idiomatic usage of SCSs.
Motomura, Kenta; Nakamura, Morikazu; Otaki, Joji M.
Protein structure and function information is coded in amino acid sequences. However, the relationship between primary sequences and three-dimensional structures and functions remains enigmatic. Our approach to this fundamental biochemistry problem is based on the frequencies of short constituent sequences (SCSs) or words. A protein amino acid sequence is considered analogous to an English sentence, where SCSs are equivalent to words. Availability scores, which are defined as real SCS frequencies in the non-redundant amino acid database relative to their probabilistically expected frequencies, demonstrate the biological usage bias of SCSs. As a result, this frequency-based linguistic approach is expected to have diverse applications, such as secondary structure specifications by structure-specific SCSs and immunological adjuvants with rare or non-existent SCSs. Linguistic similarities (e.g., wide ranges of scale-free distributions) and dissimilarities (e.g., behaviors of low-rank samples) between proteins and the natural English language have been revealed in the rank-frequency relationships of SCSs or words. We have developed a web server, the SCS Package, which contains five applications for analyzing protein sequences based on the linguistic concept. These tools have the potential to assist researchers in deciphering structurally and functionally important protein sites, species-specific sequences, and functional relationships between SCSs. The SCS Package also provides researchers with a tool to construct amino acid sequences de novo based on the idiomatic usage of SCSs. PMID:24688703
The expression of a xylanase targeted to ER-protein bodies provides a simple strategy to produce active insoluble enzyme polymers in tobacco plants.
Immaculada Llop-Tous
Full Text Available BACKGROUND: Xylanases deserve particular attention due to their potential application in the feed, pulp bleaching and paper industries. We have developed here an efficient system for the production of an active xylanase in tobacco plants fused to a proline-rich domain (Zera of the maize storage protein γ-zein. Zera is a self-assembling domain able to form protein aggregates in vivo packed in newly formed endoplasmic reticulum-derived organelles known as protein bodies (PBs. METHODOLOGY/PRINCIPAL FINDINGS: Tobacco leaves were transiently transformed with a binary vector containing the Zera-xylanase coding region, which was optimized for plant expression, under the control of the 35S CaMV promoter. The fusion protein was efficiently expressed and stored in dense PBs, resulting in yields of up to 9% of total protein. Zera-xylanase was post-translationally modified with high-mannose-type glycans. Xylanase fused to Zera was biologically active not only when solubilized from PBs but also in its insoluble form. The resistance of insoluble Zera-xylanase to trypsin digestion demonstrated that the correct folding of xylanase in PBs was not impaired by Zera oligomerization. The activity of insoluble Zera-xylanase was enhanced when substrate accessibility was facilitated by physical treatments such as ultrasound. Moreover, we found that the thermostability of the enzyme was improved when Zera was fused to the C-terminus of xylanase. CONCLUSION/SIGNIFICANCE: In the present work we have successfully produced an active insoluble aggregate of xylanase fused to Zera in plants. Zera-xylanase chimeric protein accumulates within ER-derived protein bodies as active aggregates that can easily be recovered by a simple density-based downstream process. The production of insoluble active Zera-xylanase protein in tobacco outlines the potential of Zera as a fusion partner for producing enzymes of biotechnological relevance. Zera-PBs could thus become efficient and low
relaxGUI: a new software for fast and simple NMR relaxation data analysis and calculation of ps-ns and μs motion of proteins
Bieri, Michael; D'Auvergne, Edward J.; Gooley, Paul R.
Investigation of protein dynamics on the ps-ns and μs-ms timeframes provides detailed insight into the mechanisms of enzymes and the binding properties of proteins. Nuclear magnetic resonance (NMR) is an excellent tool for studying protein dynamics at atomic resolution. Analysis of relaxation data using model-free analysis can be a tedious and time consuming process, which requires good knowledge of scripting procedures. The software relaxGUI was developed for fast and simple model-free analysis and is fully integrated into the software package relax. It is written in Python and uses wxPython to build the graphical user interface (GUI) for maximum performance and multi-platform use. This software allows the analysis of NMR relaxation data with ease and the generation of publication quality graphs as well as color coded images of molecular structures. The interface is designed for simple data analysis and management. The software was tested and validated against the command line version of relax.
relaxGUI: a new software for fast and simple NMR relaxation data analysis and calculation of ps-ns and μs motion of proteins.
Bieri, Michael; d'Auvergne, Edward J; Gooley, Paul R
Entropic potential field formed for a linear-motor protein near a filament: Statistical-mechanical analyses using simple models.
Amano, Ken-Ichi; Yoshidome, Takashi; Iwaki, Mitsuhiro; Suzuki, Makoto; Kinoshita, Masahiro
We report a new progress in elucidating the mechanism of the unidirectional movement of a linear-motor protein (e.g., myosin) along a filament (e.g., F-actin). The basic concept emphasized here is that a potential field is entropically formed for the protein on the filament immersed in solvent due to the effect of the translational displacement of solvent molecules. The entropic potential field is strongly dependent on geometric features of the protein and the filament, their overall shapes as well as details of the polyatomic structures. The features and the corresponding field are judiciously adjusted by the binding of adenosine triphosphate (ATP) to the protein, hydrolysis of ATP into adenosine diphosphate (ADP)+Pi, and release of Pi and ADP. As the first step, we propose the following physical picture: The potential field formed along the filament for the protein without the binding of ATP or ADP+Pi to it is largely different from that for the protein with the binding, and the directed movement is realized by repeated switches from one of the fields to the other. To illustrate the picture, we analyze the spatial distribution of the entropic potential between a large solute and a large body using the three-dimensional integral equation theory. The solute is modeled as a large hard sphere. Two model filaments are considered as the body: model 1 is a set of one-dimensionally connected large hard spheres and model 2 is a double helical structure formed by two sets of connected large hard spheres. The solute and the filament are immersed in small hard spheres forming the solvent. The major findings are as follows. The solute is strongly confined within a narrow space in contact with the filament. Within the space there are locations with sharply deep local potential minima along the filament, and the distance between two adjacent locations is equal to the diameter of the large spheres constituting the filament. The potential minima form a ringlike domain in model 1
A simple electrostatic switch important in the activation of type I protein kinase A by cyclic AMP.
Vigil, Dominico; Lin, Jung-Hsin; Sotriffer, Christoph A; Pennypacker, Juniper K; McCammon, J Andrew; Taylor, Susan S
Cyclic AMP activates protein kinase A by binding to an inhibitory regulatory (R) subunit and releasing inhibition of the catalytic (C) subunit. Even though crystal structures of regulatory and catalytic subunits have been solved, the precise molecular mechanism by which cyclic AMP activates the kinase remains unknown. The dynamic properties of the cAMP binding domain in the absence of cAMP or C-subunit are also unknown. Here we report molecular-dynamics simulations and mutational studies of the RIalpha R-subunit that identify the C-helix as a highly dynamic switch which relays cAMP binding to the helical C-subunit binding regions. Furthermore, we identify an important salt bridge which links cAMP binding directly to the C-helix that is necessary for normal activation. Additional mutations show that a hydrophobic "hinge" region is not as critical for the cross-talk in PKA as it is in the homologous EPAC protein, illustrating how cAMP can control diverse functions using the evolutionarily conserved cAMP-binding domains.
Graybill, George
Just how simple are simple machines? With our ready-to-use resource, they are simple to teach and easy to learn! Chocked full of information and activities, we begin with a look at force, motion and work, and examples of simple machines in daily life are given. With this background, we move on to different kinds of simple machines including: Levers, Inclined Planes, Wedges, Screws, Pulleys, and Wheels and Axles. An exploration of some compound machines follows, such as the can opener. Our resource is a real time-saver as all the reading passages, student activities are provided. Presented in s
Concentrated Solutions of Single-Chain Nanoparticles: A Simple Model for Intrinsically Disordered Proteins under Crowding Conditions.
Moreno, Angel J; Lo Verso, Federica; Arbe, Arantxa; Pomposo, José A; Colmenero, Juan
By means of large-scale computer simulations and small-angle neutron scattering (SANS), we investigate solutions of single-chain nanoparticles (SCNPs), covering the whole concentration range from infinite dilution to melt density. The analysis of the conformational properties of the SCNPs reveals that these synthetic nano-objects share basic ingredients with intrinsically disordered proteins (IDPs), as topological polydispersity, generally sparse conformations, and locally compact domains. We investigate the role of the architecture of the SCNPs in their collapse behavior under macromolecular crowding. Unlike in the case of linear macromolecules, which experience the usual transition from self-avoiding to Gaussian random-walk conformations, crowding leads to collapsed conformations of SCNPs resembling those of crumpled globules. This behavior is already found at volume fractions (about 30%) that are characteristic of crowding in cellular environments. The simulation results are confirmed by the SANS experiments. Our results for SCNPs--a model system free of specific interactions--propose a general scenario for the effect of steric crowding on IDPs: collapse from sparse conformations at high dilution to crumpled globular conformations in cell environments.
A simple model-based control for Pichia pastoris allows a more efficient heterologous protein production bioprocess.
Cos, Oriol; Ramon, Ramon; Montesinos, José Luis; Valero, Francisco
A predictive control algorithm coupled with a PI feedback controller has been satisfactorily implemented in the heterologous Rhizopus oryzae lipase production by Pichia pastoris methanol utilization slow (Mut(s)) phenotype. This control algorithm has allowed the study of the effect of methanol concentration, ranging from 0.5 to 1.75 g/L, on heterologous protein production. The maximal lipolytic activity (490 UA/mL), specific yield (11,236 UA/g(biomass)), productivity (4,901 UA/L . h), and specific productivity (112 UA/g(biomass)h were reached for a methanol concentration of 1 g/L. These parameters are almost double than those obtained with a manual control at a similar methanol set-point. The study of the specific growth, consumption, and production rates showed different patterns for these rates depending on the methanol concentration set-point. Results obtained have shown the need of implementing a robust control scheme when reproducible quality and productivity are sought. It has been demonstrated that the model-based control proposed here is a very efficient, robust, and easy-to-implement strategy from an industrial application point of view. (c) 2006 Wiley Periodicals, Inc.
Simple prostatectomy
... Han M, Partin AW. Simple prostatectomy: open and robot-assisted laparoscopic approaches. In: Wein AJ, Kavoussi LR, ... M. is also a founding member of Hi-Ethics and subscribes to the principles of the Health ...
Simple unification
Ponce, W.A.; Zepeda, A.
We present the results obtained from our systematic search of a simple Lie group that unifies weak and electromagnetic interactions in a single truly unified theory. We work with fractionally charged quarks, and allow for particles and antiparticles to belong to the same irreducible representation. We found that models based on SU(6), SU(7), SU(8) and SU(10) are viable candidates for simple unification. (author). 23 refs
A simple method for measuring signs of {sup 1}H{sup N} chemical shift differences between ground and excited protein states
Bouvignies, Guillaume; Korzhnev, Dmitry M.; Neudecker, Philipp; Hansen, D. Flemming [University of Toronto, Departments of Molecular Genetics, Biochemistry and Chemistry (Canada); Cordes, Matthew H. J. [University of Arizona, Department of Chemistry and Biochemistry (United States); Kay, Lewis E., E-mail: [email protected] [University of Toronto, Departments of Molecular Genetics, Biochemistry and Chemistry (Canada)
NMR relaxation dispersion spectroscopy is a powerful method for studying protein conformational dynamics whereby visible, ground and invisible, excited conformers interconvert on the millisecond time-scale. In addition to providing kinetics and thermodynamics parameters of the exchange process, the CPMG dispersion experiment also allows extraction of the absolute values of the chemical shift differences between interconverting states, |{Delta}{omega}-tilde|, opening the way for structure determination of excited state conformers. Central to the goal of structural analysis is the availability of the chemical shifts of the excited state that can only be obtained once the signs of {Delta}{omega}-tilde are known. Herein we describe a very simple method for determining the signs of {sup 1}H{sup N} {Delta}{omega}-tilde values based on a comparison of peak positions in the directly detected dimensions of a pair of {sup 1}H{sup N}-{sup 15}N correlation maps recorded at different static magnetic fields. The utility of the approach is demonstrated for three proteins that undergo millisecond time-scale conformational rearrangements. Although the method provides fewer signs than previously published techniques it does have a number of strengths: (1) Data sets needed for analysis are typically available from other experiments, such as those required for measuring signs of {sup 15}N {Delta}{omega}-tilde values, thus requiring no additional experimental time, (2) acquisition times in the critical detection dimension can be as long as necessary and (3) the signs obtained can be used to cross-validate those from other approaches.
Simple protein precipitation extraction technique followed by validated chromatographic method for linezolid analysis in real human plasma samples to study its pharmacokinetics.
Mohammed, Samah A; Eissa, Maya S; Ahmed, Hytham M
Fast and sensitive HPLC method was developed, optimized and validated for quantification of linezolid (LNZ) in human plasma using guaifenesin as an internal standard (IS). Analyte and IS were extracted from plasma by simple protein precipitation extraction technique using methanol as the precipitating solvent. The pretreated samples were injected in a mobile phase formed of acetonitrile:water:methanol (20:70:10v/v/v) in an isocratic mode at a flow rate of 1.5mL/min with UV detection at 251nm. Separation was done using Aglient ODS C 18 . The method showed linearity in the range of 0.75-50μg/mL with correlation coefficients equals to 0.9991. Precision and accuracy were in conformity with the criteria normally accepted in bio-analytical method validation. The RSDs for intra- and inter-day assays were <3.56 and 4.63%, respectively. The intra- and inter-day accuracies were 94.67-98.28% and 91.25-96.18%, respectively. The mean absolute recoveries ranged from 92.56±1.78 to 95.24±2.84. According to stability results, LNZ was stable in human plasma during the storage and analysis. LNZ a pharmacokinetic behavior was studied by applying the proposed analytical method. Copyright © 2016 Elsevier B.V. All rights reserved.
Simple synthesis of carbon-11-labeled chromen-4-one derivatives as new potential PET agents for imaging of DNA-dependent protein kinase (DNA-PK) in cancer
Gao, Mingzhang; Wang, Min; Miller, Kathy D.; Zheng, Qi-Huang
Carbon-11-labeled chromen-4-one derivatives were synthesized as new potential PET agents for imaging of DNA repair enzyme DNA-dependent protein kinase (DNA-PK) in cancer. The target tracers, X-[ 11 C]methoxy-2-morpholino-4H-chromen-4-ones (X=8, 7, 6, 5; [ 11 C]4a–d), were prepared from their corresponding precursors, X-hydroxy-2-morpholino-4H-chromen-4-ones (X=8, 7, 6, 5; 5a–d), with [ 11 C]CH 3 OTf through O-[ 11 C]methylation and isolated by a simplified solid-phase extraction (SPE) method using a C-18 Sep-Pak Plus cartridge. The radiochemical yields decay corrected to end of bombardment (EOB), from [ 11 C]CO 2 , were 40–60%. The specific activity at end of synthesis (EOS) was 185–370 GBq/μmol. - Highlights: ► New chromen-4-one derivatives were synthesized. ► New carbon-11-labeled chromen-4-one derivatives were synthesized. ► Simple solid-phase extraction (SPE) method was employed in radiosynthesis.
A simple recipe for the non-expert bioinformaticist for building experimentally-testable hypotheses for proteins with no known homologs
Zawaira, A
Full Text Available The study of the protein-protein interactions (PPIs) of unique ORFs is a strategy for deciphering the biological roles of unique ORFs of interest. For uniform reference, we define unique ORFs as those for which no matching protein is found after PDB...
Migration of the guinea pig sperm membrane protein PH-20 from one localized surface domain to another does not occur by a simple diffusion-trapping mechanism.
Cowan, A E; Myles, D G; Koppel, D E
The redistribution of membrane proteins on the surface of cells is a prevalent feature of differentiation in a variety of cells. In most cases the mechanism responsible for such redistribution is poorly understood. Two potential mechanisms for the redistribution of surface proteins are: (1) passive diffusion coupled with trapping, and (2) active translocation. We have studied the process of membrane protein redistribution for the PH-20 protein of guinea pig sperm, a surface protein required for sperm binding to the egg zona pellucida (P. Primakoff, H. Hyatt, and D. G. Myles (1985). J. Cell Biol. 101, 2239-2244). PH-20 protein is localized to the posterior head plasma menbrane of the mature sperm cell. Following the exocytotic acrosome reaction, PH-20 protein moves into the newly incorporated inner acrosomal membrane (IAM), placing it in a position favorable for a role in binding sperm to the egg zona pellucida (D. G. Myles, and P. Primakoff (1984), J. Cell Biol. 99, 1634-1641). To analyze the mechanistic basis for this protein migration, we have used fluorescence microscopy and digital image processing to characterize PH-20 protein migration in individual cells. PH-20 protein was observed to move against a concentration gradient in the posterior head plasma membrane. This result argues strongly against a model of passive diffusion followed by trapping in the IAM, and instead suggests that an active process serves to concentrate PH-20 protein toward the boundary separating the posterior head and IAM regions. A transient gradient of PH-20 concentration observed in the IAM suggests that once PH-20 protein reaches the IAM, it is freely diffusing. Additionally, we observed that migration of PH-20 protein was calcium dependent.
A simple sheathless CE-MS interface with a sub-micrometer electrical contact fracture for sensitive analysis of peptide and protein samples
Nguyen, Tam T. T. N.; Petersen, Nickolaj J.; Rand, Kasper Dyrberg
Online coupling of capillary electrophoresis (CE) to electrospray ionization mass spectrometry (MS) has shown considerable potential, however, technical challenges have limited its use. In this study, we have developed a simple and sensitive sheathless CE-MS interface based on the novel concept o...
Ion-Exchange Sample Displacement Chromatography as a Method for Fast and Simple Isolation of Low- and High-Abundance Proteins from Complex Biological Mixtures
Martina Srajer Gajdosik
Full Text Available Sample displacement chromatography (SDC in reversed phase and ion-exchange modes was introduced at the end of 1980s. This chromatographic method was first used for preparative purification of synthetic peptides, and subsequently adapted for protein fractionation, mainly in anion-exchange mode. In the past few years, SDC has been successfully used for enrichment of low- and medium-abundance proteins from complex biological fluids on both monolithic and bulk chromatographic supports. If aqueous mobile phase is used with the application of mild chromatographic conditions, isolated proteins are not denatured and can also keep their biological activity. In this paper, the use of SDC in anion-exchange mode on a high-capacity chromatographic resin for separation of proteins from complex biological mixtures such as human plasma is demonstrated. By use of three and more columns coupled in series during sample application, and subsequent parallel elution of detached columns, additional separation of bound proteins was achieved. Highly enriched human serum albumin fraction and a number of physiologically active medium- and low-abundance proteins could be fractionated and detected by electrospray ionization tandem mass spectrometry (ESI-MS/MS and matrix assisted laser desorption/ionization time-of-flight tandem mass spectrometry (MALDI-TOF/TOF-MS. The use of the aforementioned columns that can be sanitized with 1 M sodium hydroxide for further application of SDC in biotechnology and food technology was discussed.
DOCLASP - Docking ligands to target proteins using spatial and electrostatic congruence extracted from a known holoenzyme and applying simple geometrical transformations.
Chakraborty, Sandeep
The ability to accurately and effectively predict the interaction between proteins and small drug-like compounds has long intrigued researchers for pedagogic, humanitarian and economic reasons. Protein docking methods (AutoDock, GOLD, DOCK, FlexX and Glide to name a few) rank a large number of possible conformations of protein-ligand complexes using fast algorithms. Previously, it has been shown that structural congruence leading to the same enzymatic function necessitates the congruence of electrostatic properties (CLASP). The current work presents a methodology for docking a ligand into a target protein, provided that there is at least one known holoenzyme with ligand bound - DOCLASP (Docking using CLASP). The contact points of the ligand in the holoenzyme defines a motif, which is used to query the target enzyme using CLASP. If there are significant matches, the holoenzyme and the target protein are superimposed based on congruent atoms. The same linear and rotational transformations are also applied to the ligand, thus creating a unified coordinate framework having the holoenzyme, the ligand and the target enzyme. In the current work, the dipeptidyl peptidase-IV inhibitor vildagliptin was docked to the PI-PLC structure complexed with myo-inositol using DOCLASP. Also, corroboration of the docking of phenylthiourea to the modelled structure of polyphenol oxidase (JrPPO1) from walnut is provided based on the subsequently solved structure of JrPPO1 (PDBid:5CE9). Analysis of the binding of the antitrypanosomial drug suramin to nine non-homologous proteins in the PDB database shows a diverse set of binding motifs, and multiple binding sites in the phospholipase A2-likeproteins from the Bothrops genus of pitvipers. The conformational changes in the suramin molecule on binding highlights the challenges in docking flexible ligands into an already 'plastic' binding site. Thus, DOCLASP presents a method for 'soft docking' ligands to proteins with low computational
Reference distributions for complement proteins C3 and C4: a practical, simple and clinically relevant approach in a large cohort.
Ritchie, Robert F; Palomaki, Glenn E; Neveux, Louis M; Navolotskaia, Olga; Ledue, Thomas B; Craig, Wendy Y
The two serum proteins of the complement cascade in the highest concentrations, C3 and C4, respond to various conditions in much the same manner as do other positive acute-phase proteins. A major difference is that they are relatively sluggish in response to cytokine drive, requiring several days rather than hours to be detectably elevated by serial measurements. As with other acute-phase proteins, there are many processes that up- or down-regulate synthesis, including infection or inflammation, hepatic failure, and immune-complex formation. Clinicians may find it difficult to distinguish among these processes, because they often occur simultaneously. The situation is further complicated by genetic polymorphism, with rare instances of markedly reduced synthesis and circulating levels, and consequent vulnerability to infection. C3 and C4 are measured for clinical purposes to help define certain rheumatic and immunologically mediated renal diseases. Interpreting the measured blood levels of these two components requires one to consider the intensity of the inflammatory drive, the timing of the suspected clinical process, the production of complement-consuming immune complexes, and the possible existence of benign circumstances. In this fifth article in a series, reference ranges for serum levels of two complement proteins (C3 and C4) are examined. The study is based on a cohort of over 55,000 Caucasian individuals from northern New England, who were tested in our laboratory in 1994-1999. Measurements were standardized against certified reference material (CRM) 470/reference preparation for proteins in human serum (RPPHS), and analyzed using a previously described statistical approach. Individuals with unequivocal laboratory evidence of inflammation (C-reactive protein of 10 mg/L or higher) were excluded. Our results show that the levels of C3 and C4 change little during life and between the sexes, except that they increase slightly and then fall after age 20 in males
Evaluation of a simple protein extraction method for species identification of clinically relevant staphylococci by matrix-assisted laser desorption ionization-time of flight mass spectrometry.
Matsuda, Naoto; Matsuda, Mari; Notake, Shigeyuki; Yokokawa, Hirohide; Kawamura, Yoshiaki; Hiramatsu, Keiichi; Kikuchi, Ken
In clinical microbiology, bacterial identification is labor-intensive and time-consuming. A solution for this problem is the use of matrix-assisted laser desorption ionization-time of flight mass spectrometry (MALDI-TOF MS). In this study, we evaluated a modified protein extraction method of identification performed on target plates (on-plate extraction method) with MALDI-TOF (Bruker Microflex LT with Biotyper version 3.0) and compared it to 2 previously described methods: the direct colony method and a standard protein extraction method (standard extraction method). We evaluated the species of 273 clinical strains and 14 reference strains of staphylococci. All isolates were characterized using the superoxide dismutase A sequence as a reference. For the species identification, the on-plate, standard extraction, and direct colony methods identified 257 isolates (89.5%), 232 isolates (80.8%), and 173 isolates (60.2%), respectively, with statistically significant differences among the three methods (P extraction method is at least as good as standard extraction in identification rate and has the advantage of a shorter processing time.
Investigation on the Aggregation Behaviors and Filament Morphology of Tau Protein by a Simple 90° Angle Light-Scattering Assay
Hai-Lin Liao
Full Text Available The in vitro aggregation of tau constructs was monitored by a simple 90° angle light-scattering (LS approach which was conducted directly on fluorescence instrument. At the optimum incident wavelength (550 nm, unpolarized, the sensitivity of LS was high enough to detect tau aggregation at micromolar range. The nucleation and elongation, different events in the aggregation process of 4RMBD construct (corresponding with the four repeated units of tau Microtubule Binding Domain could be observed by this approach, as compared with ThS fluorescence assay. The validity of this technique was demonstrated over a range of tau concentrations with different tau filaments. Linear regression of scattering light against concentration yielded the x-intercept, the critical concentrations of tau constructs. The critical concentrations of 4RMBD and its S305N mutant are 5.26 μM and 4.04 μM respectively, indicating point mutation S305N, which is associated with FTDP-17, appear to enhance the heparin-induced tau aggregation in vitro. Furthermore, the slopes of concentration dependence curves, as well as the angle dependence, were discussed based on the filaments morphology examined by electron microscopy and ultrasonication experiment.
Simple and green synthesis of protein-conjugated CdS nanoparticles and spectroscopic study on the interaction between CdS and zein
Qin, Dezhi, E-mail: [email protected]; Zhang, Li; Du, Xian; Wang, Yabo; Zhang, Qiuxia [Pingdingshan University, College of Chemistry and Environmental Engineering (China)
The present study demonstrates the role of zein molecules in synthesizing CdS nanoassemblies through protein-directed, green synthetic approach. Zein molecules can as capping ligand and stabilizing agent to regulate the nucleation and growth of CdS nanocrystals, and the obtained products are organic–inorganic nanocomposites. The analysis of surface charge and conductivity indicates that strong electrostatic force restricts mobility of ions, which creates a local supersaturation surrounding the binding sites of zein and reduces the activated energy of nucleation. The interaction between Cd{sup 2+}/CdS and zein molecules was systematically investigated through spectroscopy techniques. Fourier transform infrared (FT-IR) spectra were used to envisage the binding of the functional groups of zein with the surface of CdS nanoparticles. Ultraviolet visible (UV–Vis) and photoluminescence (PL) spectra results show that Cd{sup 2+}/CdS might interact with the aromatic amino acids of protein molecules and change its chemical microenvironment. The quantum-confined effect of nanocrystals is confirmed by optical absorption spectrum due to the small size (3–5 nm) of CdS particles. The data of circular dichroism (CD) spectra indicate that the formation of CdS nanocrystals could lead to the conformational change of zein molecules. Moreover, the possible mechanism of CdS nanocrystals growth in zein solution was also discussed. The weak interactions such as Van der Waals, hydrophobic forces and hydrogen bonds in zein molecules should play a crucial factor in the self-assembly of small nanoparticles.
Simple Kidney Cysts
... Solitary Kidney Your Kidneys & How They Work Simple Kidney Cysts What are simple kidney cysts? Simple kidney cysts are abnormal, fluid-filled ... that form in the kidneys. What are the kidneys and what do they do? The kidneys are ...
A simple method for measuring sex-hormone binding protein (SHBP) - typical values in men and women and in pregnant women
Tafurt, C.A.; Estrada, R. de
Assuming that the binding forces between steroid hormones and their binding proteins are similar to those between antigens and their antibodies, the authors describe how to determine SHBP activity by a dilution method analogous to that used for titration of antisera in radioimmunoassay. The method consists of the following stages: (1) plasma dilution; (2) incubation of the dilution with 20,000dis/min of 1,2- 3 H-testosterone; (3) separation of the fraction of tracer bound to the SHBP by precipitation with ammonium sulphate; (4) centrifugation and measurement of the supernatant; and (5) plotting of the results on a graph where the axis of ordinates represents the quotient given by bound steroid over free steroid (U/L) and the abscissa represents the plasma dilutions. The values are expressed as the 50% bound titre. An advantage of the method is the higher sensitivity of the dilution curves in the steepest part where the 50% bound is encountered; it is thus not necessary to use the saturation part of the curves where sensitivity is lost owing to the steeper slope. A further advantage of the method is that there is no need for costly processes such as dialysis. The SHBP values obtained for healthy subjects were as follows: 1/5 for men, 1/93 for women, and 1/360 in pregnant women. These physiological values showed no overlapping. (author)
Serum C-reactive protein (CRP) as a simple and independent prognostic factor in extranodal natural killer/T-cell lymphoma, nasal type.
Li, Ya-Jun; Li, Zhi-Ming; Xia, Yi; Huang, Jia-Jia; Huang, Hui-Qiang; Xia, Zhong-Jun; Lin, Tong-Yu; Li, Su; Cai, Xiu-Yu; Wu-Xiao, Zhi-Jun; Jiang, Wen-Qi
C-reactive protein (CRP) is a biomarker of the inflammatory response, and it shows significant prognostic value for several types of solid tumors. The prognostic significance of CRP for lymphoma has not been fully examined. We evaluated the prognostic role of baseline serum CRP levels in patients with extranodal natural killer (NK)/T-cell lymphoma (ENKTL). We retrospectively analyzed 185 patients with newly diagnosed ENKTL. The prognostic value of the serum CRP level was evaluated for the low-CRP group (CRP≤10 mg/L) versus the high-CRP group (CRP>10 mg/L). The prognostic value of the International Prognostic Index (IPI) and the Korean Prognostic Index (KPI) were evaluated and compared with the newly developed prognostic model. Patients in the high-CRP group tended to display increased adverse clinical characteristics, lower rates of complete remission (P60 years, hypoalbuminemia, and elevated lactate dehydrogenase levels were independent adverse predictors of OS. Based on these four independent predictors, we constructed a new prognostic model that identified 4 groups with varying OS: group 1, no adverse factors; group 2, 1 factor; group 3, 2 factors; and group 4, 3 or 4 factors (PKPI in distinguishing between the low- and intermediate-low-risk groups, the intermediate-low- and high-intermediate-risk groups, and the high-intermediate- and high-risk groups. Our results suggest that pretreatment serum CRP levels represent an independent predictor of clinical outcome for patients with ENKTL. The prognostic value of the new prognostic model is superior to both IPI and KPI.
Crossing simple resonances
Collins, T.
A simple criterion governs the beam distortion and/or loss of protons on a fast resonance crossing. Results from numerical integrations are illustrated for simple sextupole, octupole, and 10-pole resonances
A simple criterion governs the beam distortion and/or loss of protons on a fast resonance crossing. Results from numerical integrations are illustrated for simple sextupole, octupole, and 10-pole resonances.
Simple WZW currents
Fuchs, J.
A complete classification of simple currents of WZW theory is obtained. The proof is based on an analysis of the quantum dimensions of the primary fields. Simple currents are precisely the primaries with unit quantum dimension; for WZW theories explicit formulae for the quantum dimensions can be derived so that an identification of the fields with unit quantum dimension is possible. (author). 19 refs.; 2 tabs
Strategy as simple rules.
Eisenhardt, K M; Sull, D N
The success of Yahoo!, eBay, Enron, and other companies that have become adept at morphing to meet the demands of changing markets can't be explained using traditional thinking about competitive strategy. These companies have succeeded by pursuing constantly evolving strategies in market spaces that were considered unattractive according to traditional measures. In this article--the third in an HBR series by Kathleen Eisenhardt and Donald Sull on strategy in the new economy--the authors ask, what are the sources of competitive advantage in high-velocity markets? The secret, they say, is strategy as simple rules. The companies know that the greatest opportunities for competitive advantage lie in market confusion, but they recognize the need for a few crucial strategic processes and a few simple rules. In traditional strategy, advantage comes from exploiting resources or stable market positions. In strategy as simple rules, advantage comes from successfully seizing fleeting opportunities. Key strategic processes, such as product innovation, partnering, or spinout creation, place the company where the flow of opportunities is greatest. Simple rules then provide the guidelines within which managers can pursue such opportunities. Simple rules, which grow out of experience, fall into five broad categories: how- to rules, boundary conditions, priority rules, timing rules, and exit rules. Companies with simple-rules strategies must follow the rules religiously and avoid the temptation to change them too frequently. A consistent strategy helps managers sort through opportunities and gain short-term advantage by exploiting the attractive ones. In stable markets, managers rely on complicated strategies built on detailed predictions of the future. But when business is complicated, strategy should be simple.
Simple Finite Sums
Alabdulmohsin, Ibrahim M.
We will begin our treatment of summability calculus by analyzing what will be referred to, throughout this book, as simple finite sums. Even though the results of this chapter are particular cases of the more general results presented in later chapters, they are important to start with for a few reasons. First, this chapter serves as an excellent introduction to what summability calculus can markedly accomplish. Second, simple finite sums are encountered more often and, hence, they deserve special treatment. Third, the results presented in this chapter for simple finite sums will, themselves, be used as building blocks for deriving the most general results in subsequent chapters. Among others, we establish that fractional finite sums are well-defined mathematical objects and show how various identities related to the Euler constant as well as the Riemann zeta function can actually be derived in an elementary manner using fractional finite sums.
Excel 2010 Made Simple
Katz, Abbott
Get the most out of Excel 2010 with Excel 2010 Made Simple - learn the key features, understand what's new, and utilize dozens of time-saving tips and tricks to get your job done. Over 500 screen visuals and clear-cut instructions guide you through the features of Excel 2010, from formulas and charts to navigating around a worksheet and understanding Visual Basic for Applications (VBA) and macros. Excel 2010 Made Simple takes a practical and highly effective approach to using Excel 2010, showing you the best way to complete your most common spreadsheet tasks. You'll learn how to input, format,
Droids Made Simple
Mazo, Gary
If you have a Droid series smartphone - Droid, Droid X, Droid 2, or Droid 2 Global - and are eager to get the most out of your device, Droids Made Simple is perfect for you. Authors Martin Trautschold, Gary Mazo and Marziah Karch guide you through all of the features, tips, and tricks using their proven combination of clear instructions and detailed visuals. With hundreds of annotated screenshots and step-by-step directions, Droids Made Simple will transform you into a Droid expert, improving your productivity, and most importantly, helping you take advantage of all of the cool features that c
Clusters in simple fluids
Sator, N.
This article concerns the correspondence between thermodynamics and the morphology of simple fluids in terms of clusters. Definitions of clusters providing a geometric interpretation of the liquid-gas phase transition are reviewed with an eye to establishing their physical relevance. The author emphasizes their main features and basic hypotheses, and shows how these definitions lead to a recent approach based on self-bound clusters. Although theoretical, this tutorial review is also addressed to readers interested in experimental aspects of clustering in simple fluids
Association "Les Simples"
Thouzery, Michel
Fondée par les producteurs du Syndicat Inter-Massifs pour la Production et l'Économie des Simples (S.I.M.P.L.E.S), l'association base son action sur la recherche et le maintien d'une production de qualité (herboristerie et préparations à base de plantes) qui prend en compte le respect de l'environnement et la pérennité des petits producteurs en zone de montagne. Actions de formation Stages de découverte de la flore médicinale sauvage, Stages de culture et transformation des plantes médicinale...
A simple electron multiplexer
Dobrzynski, L; Akjouj, A; Djafari-Rouhani, B; Al-Wahsh, H; Zielinski, P
We present a simple multiplexing device made of two atomic chains coupled by two other transition metal atoms. We show that this simple atomic device can transfer electrons at a given energy from one wire to the other, leaving all other electron states unaffected. Closed-form relations between the transmission coefficients and the inter-atomic distances are given to optimize the desired directional electron ejection. Such devices can be adsorbed on insulating substrates and characterized by current surface technologies. (letter to the editor)
Simple Driving Techniques
Rosendahl, Mads
-like language. Our aim is to extract a simple notion of driving and show that even in this tamed form it has much of the power of more general notions of driving. Our driving technique may be used to simplify functional programs which use function composition and will often be able to remove intermediate data...
A Simple Tiltmeter
Dix, M. G.; Harrison, D. R.; Edwards, T. M.
Bubble vial with external aluminum-foil electrodes is sensing element for simple indicating tiltmeter. To measure bubble displacement, bridge circuit detects difference in capacitance between two sensing electrodes and reference electrode. Tiltmeter was developed for experiment on forecasting seismic events by changes in Earth's magnetic field.
A Simple Hydrogen Electrode
Eggen, Per-Odd
This article describes the construction of an inexpensive, robust, and simple hydrogen electrode, as well as the use of this electrode to measure "standard" potentials. In the experiment described here the students can measure the reduction potentials of metal-metal ion pairs directly, without using a secondary reference electrode. Measurements…
Structure of simple liquids
Blain, J.F.
The results obtained by application to argon and sodium of the two important methods of studying the structure of liquids: scattering of X-rays and neutrons, are presented on one hand. On the other hand the principal models employed for reconstituting the structure of simple liquids are exposed: mathematical models, lattice models and their derived models, experimental models. (author) [fr
Simple mathematical fireworks
De Luca, R; Faella, O
Mathematical fireworks are reproduced in two dimensions by means of simple notions in kinematics and Newtonian mechanics. Extension of the analysis in three dimensions is proposed and the geometric figures the falling tiny particles make on the ground after explosion are determined. (paper)
simple sequence repeat (SSR)
In the present study, 78 mapped simple sequence repeat (SSR) markers representing 11 linkage groups of adzuki bean were evaluated for transferability to mungbean and related Vigna spp. 41 markers amplified characteristic bands in at least one Vigna species. The transferability percentage across the genotypes ranged ...
A Simple Wave Driver
Temiz, Burak Kagan; Yavuz, Ahmet
This study was done to develop a simple and inexpensive wave driver that can be used in experiments on string waves. The wave driver was made using a battery-operated toy car, and the apparatus can be used to produce string waves at a fixed frequency. The working principle of the apparatus is as follows: shortly after the car is turned on, the…
Complexity is simple!
Cottrell, William; Montero, Miguel
In this note we investigate the role of Lloyd's computational bound in holographic complexity. Our goal is to translate the assumptions behind Lloyd's proof into the bulk language. In particular, we discuss the distinction between orthogonalizing and `simple' gates and argue that these notions are useful for diagnosing holographic complexity. We show that large black holes constructed from series circuits necessarily employ simple gates, and thus do not satisfy Lloyd's assumptions. We also estimate the degree of parallel processing required in this case for elementary gates to orthogonalize. Finally, we show that for small black holes at fixed chemical potential, the orthogonalization condition is satisfied near the phase transition, supporting a possible argument for the Weak Gravity Conjecture first advocated in [1].
Unicameral (simple) bone cysts.
Baig, Rafath; Eady, John L
Since their original description by Virchow, simple bone cysts have been studied repeatedly. Although these defects are not true neoplasms, simple bone cysts may create major structural defects of the humerus, femur, and os calcis. They are commonly discovered incidentally when x-rays are taken for other reasons or on presentation due to a pathologic fracture. Various treatment strategies have been employed, but the only reliable predictor of success of any treatment strategy is the age of the patient; those being older than 10 years of age heal their cysts at a higher rate than those under age 10. The goal of management is the formation of a bone that can withstand the stresses of use by the patient without evidence of continued bone destruction as determined by serial radiographic follow-up. The goal is not a normal-appearing x-ray, but a functionally stable bone.
Information technology made simple
Carter, Roger
Information Technology: Made Simple covers the full range of information technology topics, including more traditional subjects such as programming languages, data processing, and systems analysis. The book discusses information revolution, including topics about microchips, information processing operations, analog and digital systems, information processing system, and systems analysis. The text also describes computers, computer hardware, microprocessors, and microcomputers. The peripheral devices connected to the central processing unit; the main types of system software; application soft
Modern mathematics made simple
Murphy, Patrick
Modern Mathematics: Made Simple presents topics in modern mathematics, from elementary mathematical logic and switching circuits to multibase arithmetic and finite systems. Sets and relations, vectors and matrices, tesselations, and linear programming are also discussed.Comprised of 12 chapters, this book begins with an introduction to sets and basic operations on sets, as well as solving problems with Venn diagrams. The discussion then turns to elementary mathematical logic, with emphasis on inductive and deductive reasoning; conjunctions and disjunctions; compound statements and conditional
Dimensional analysis made simple
Lira, Ignacio
An inductive strategy is proposed for teaching dimensional analysis to second- or third-year students of physics, chemistry, or engineering. In this strategy, Buckingham's theorem is seen as a consequence and not as the starting point. In order to concentrate on the basics, the mathematics is kept as elementary as possible. Simple examples are suggested for classroom demonstrations of the power of the technique and others are put forward for homework or experimentation, but instructors are encouraged to produce examples of their own. (paper)
Applied mathematics made simple
Applied Mathematics: Made Simple provides an elementary study of the three main branches of classical applied mathematics: statics, hydrostatics, and dynamics. The book begins with discussion of the concepts of mechanics, parallel forces and rigid bodies, kinematics, motion with uniform acceleration in a straight line, and Newton's law of motion. Separate chapters cover vector algebra and coplanar motion, relative motion, projectiles, friction, and rigid bodies in equilibrium under the action of coplanar forces. The final chapters deal with machines and hydrostatics. The standard and conte
Data processing made simple
Data Processing: Made Simple, Second Edition presents discussions of a number of trends and developments in the world of commercial data processing. The book covers the rapid growth of micro- and mini-computers for both home and office use; word processing and the 'automated office'; the advent of distributed data processing; and the continued growth of database-oriented systems. The text also discusses modern digital computers; fundamental computer concepts; information and data processing requirements of commercial organizations; and the historical perspective of the computer industry. The
ASP made simple
Deane, Sharon
ASP Made Simple provides a brief introduction to ASP for the person who favours self teaching and/or does not have expensive computing facilities to learn on. The book will demonstrate how the principles of ASP can be learned with an ordinary PC running Personal Web Server, MS Access and a general text editor like Notepad.After working through the material readers should be able to:* Write ASP scripts that can display changing information on a web browser* Request records from a remote database or add records to it* Check user names & passwords and take this knowledge forward, either for their
Theory of simple liquids
Hansen, Jean-Pierre
This book gives a comprehensive and up-to-date treatment of the theory of ""simple"" liquids. The new second edition has been rearranged and considerably expanded to give a balanced account both of basic theory and of the advances of the past decade. It presents the main ideas of modern liquid state theory in a way that is both pedagogical and self-contained. The book should be accessible to graduate students and research workers, both experimentalists and theorists, who have a good background in elementary mechanics.Key Features* Compares theoretical deductions with experimental r
Probabilistic simple sticker systems
Selvarajoo, Mathuri; Heng, Fong Wan; Sarmin, Nor Haniza; Turaev, Sherzod
A model for DNA computing using the recombination behavior of DNA molecules, known as a sticker system, was introduced by by L. Kari, G. Paun, G. Rozenberg, A. Salomaa, and S. Yu in the paper entitled DNA computing, sticker systems and universality from the journal of Acta Informatica vol. 35, pp. 401-420 in the year 1998. A sticker system uses the Watson-Crick complementary feature of DNA molecules: starting from the incomplete double stranded sequences, and iteratively using sticking operations until a complete double stranded sequence is obtained. It is known that sticker systems with finite sets of axioms and sticker rules generate only regular languages. Hence, different types of restrictions have been considered to increase the computational power of sticker systems. Recently, a variant of restricted sticker systems, called probabilistic sticker systems, has been introduced [4]. In this variant, the probabilities are initially associated with the axioms, and the probability of a generated string is computed by multiplying the probabilities of all occurrences of the initial strings in the computation of the string. Strings for the language are selected according to some probabilistic requirements. In this paper, we study fundamental properties of probabilistic simple sticker systems. We prove that the probabilistic enhancement increases the computational power of simple sticker systems.
Simple stochastic simulation.
Schilstra, Maria J; Martin, Stephen R
Stochastic simulations may be used to describe changes with time of a reaction system in a way that explicitly accounts for the fact that molecules show a significant degree of randomness in their dynamic behavior. The stochastic approach is almost invariably used when small numbers of molecules or molecular assemblies are involved because this randomness leads to significant deviations from the predictions of the conventional deterministic (or continuous) approach to the simulation of biochemical kinetics. Advances in computational methods over the three decades that have elapsed since the publication of Daniel Gillespie's seminal paper in 1977 (J. Phys. Chem. 81, 2340-2361) have allowed researchers to produce highly sophisticated models of complex biological systems. However, these models are frequently highly specific for the particular application and their description often involves mathematical treatments inaccessible to the nonspecialist. For anyone completely new to the field to apply such techniques in their own work might seem at first sight to be a rather intimidating prospect. However, the fundamental principles underlying the approach are in essence rather simple, and the aim of this article is to provide an entry point to the field for a newcomer. It focuses mainly on these general principles, both kinetic and computational, which tend to be not particularly well covered in specialist literature, and shows that interesting information may even be obtained using very simple operations in a conventional spreadsheet.
Crystal nucleation in simple and complex fluids.
Oxtoby, David W
The application of density-functional methods from statistical mechanics to the nucleation of crystals from the melt is described. Simple fluids such as metals, with sizes comparable with the range of their attractive forces, are compared with complex fluids such as colloidal suspensions and proteins dissolved in solution. A different mechanism for crystal nucleation is proposed in the latter case, in which density (concentration) changes before periodic crystalline order appears. This leads to a theoretical foundation for empirical observations on the 'crystallization window' in protein crystallization. Comparisons are made with the results of computer simulation via molecular dynamics.
Beyond Simple Headquarters Configurations
Dellestrand, Henrik; Kappen, Philip; Nell, Phillip Christopher
We investigate "dual headquarters involvement�, i.e. corporate and divisional headquarters' simultaneous involvement in subsidiaries' innovation development projects. Analyses draw on 85 innovation projects in 23 multibusiness firms and reveal that cross-divisional innovation importance, i.......e., an innovation that is important for the firm beyond the divisional boundaries, drives dual headquarters involvement in innovation development. Contrary to expectations, on average, a non-significant effect of cross-divisional embeddedness on dual headquarters involvement is found. Yet, both cross......-divisional importance and embeddedness effects are contingent on the overall complexity of the innovation project as signified by the size of the development network. The results lend support for the notion that parenting in complex structures entails complex headquarters structures and that we need to go beyond simple...
Simple relation algebras
Givant, Steven
This monograph details several different methods for constructing simple relation algebras, many of which are new with this book. By drawing these seemingly different methods together, all are shown to be aspects of one general approach, for which several applications are given. These tools for constructing and analyzing relation algebras are of particular interest to mathematicians working in logic, algebraic logic, or universal algebra, but will also appeal to philosophers and theoretical computer scientists working in fields that use mathematics. The book is written with a broad audience in mind and features a careful, pedagogical approach; an appendix contains the requisite background material in relation algebras. Over 400 exercises provide ample opportunities to engage with the material, making this a monograph equally appropriate for use in a special topics course or for independent study. Readers interested in pursuing an extended background study of relation algebras will find a comprehensive treatme...
A Simple Harmonic Universe
Graham, Peter W.; /Stanford U., ITP; Horn, Bart; Kachru, Shamit; /Stanford U., ITP /SLAC; Rajendran, Surjeet; /Johns Hopkins U. /Stanford U., ITP; Torroba, Gonzalo; /Stanford U., ITP /SLAC
We explore simple but novel bouncing solutions of general relativity that avoid singularities. These solutions require curvature k = +1, and are supported by a negative cosmological term and matter with -1 < w < -1 = 3. In the case of moderate bounces (where the ratio of the maximal scale factor a{sub +} to the minimal scale factor a{sub -} is {Omicron}(1)), the solutions are shown to be classically stable and cycle through an infinite set of bounces. For more extreme cases with large a{sub +} = a{sub -}, the solutions can still oscillate many times before classical instabilities take them out of the regime of validity of our approximations. In this regime, quantum particle production also leads eventually to a departure from the realm of validity of semiclassical general relativity, likely yielding a singular crunch. We briefly discuss possible applications of these models to realistic cosmology.
SIMPLE for industrial radiography
Azhar Azmi; Abd Nassir Ibrahim; Siti Madiha Muhammad Amir; Glam Hadzir Patai Mohamad; Saidi Rajab
The first thing industrial radiographers have to do before commencing radiography works is to determine manually the amount of correct exposure that the film need to be exposed in order to obtain the right density. The amount of exposure depends on many variables such as type of radioisotope, type of film, nature of test-object and its orientation, and specific arrangement related to object location and configuration. In many cases radiography works are rejected because of radiographs fail to meet certain reference criteria as defined in the applicable standard. One of the main reasons of radiograph rejection is due to inadequate exposure received by the films. SIMPLE is a software specially developed to facilitate the calculation of gamma-radiography exposure. By using this software and knowing radiographic parameters to be encountered during the work, it is expected that human error will be minimized, thus enhancing the quality and productivity of NDT jobs. (Author)
Molecular genetics made simple
Heba Sh. Kassem
Full Text Available Genetics have undoubtedly become an integral part of biomedical science and clinical practice, with important implications in deciphering disease pathogenesis and progression, identifying diagnostic and prognostic markers, as well as designing better targeted treatments. The exponential growth of our understanding of different genetic concepts is paralleled by a growing list of genetic terminology that can easily intimidate the unfamiliar reader. Rendering genetics incomprehensible to the clinician however, defeats the very essence of genetic research: its utilization for combating disease and improving quality of life. Herein we attempt to correct this notion by presenting the basic genetic concepts along with their usefulness in the cardiology clinic. Bringing genetics closer to the clinician will enable its harmonious incorporation into clinical care, thus not only restoring our perception of its simple and elegant nature, but importantly ensuring the maximal benefit for our patients.
Kassem, Heba Sh.; Girolami, Francesca; Sanoudou, Despina
Abstract Genetics have undoubtedly become an integral part of biomedical science and clinical practice, with important implications in deciphering disease pathogenesis and progression, identifying diagnostic and prognostic markers, as well as designing better targeted treatments. The exponential growth of our understanding of different genetic concepts is paralleled by a growing list of genetic terminology that can easily intimidate the unfamiliar reader. Rendering genetics incomprehensible to the clinician however, defeats the very essence of genetic research: its utilization for combating disease and improving quality of life. Herein we attempt to correct this notion by presenting the basic genetic concepts along with their usefulness in the cardiology clinic. Bringing genetics closer to the clinician will enable its harmonious incorporation into clinical care, thus not only restoring our perception of its simple and elegant nature, but importantly ensuring the maximal benefit for our patients. PMID:25610837
SIMPLE LIFE AND RELIGION
Ahmet YILDIRIM
Full Text Available Individuals in terms of the economy in which we live is one of the most important phenomenon of the century. This phenomenon present itself as the only determinant of people's lives by entering almost makes itself felt. The mo st obvious objective needs of the economy by triggering motive is to induce people to consume . Consumer culture pervades all aspects of the situation are people . Therefore, these people have the blessing of culture , beauty and value all in the name of w hatever is consumed. This is way out of the siege of moral and religious values we have is to go back again . Referred by local cultural and religious values, based on today increasingly come to the fore and the Muslim way of life appears to be close to th e plain / lean preferred by many people life has been a way of life. Even the simple life , a way of life in the Western world , a conception of life , a philosophy, a movement as it has become widely accepted. Here in determining the Muslim way of life Pr ophet. Prophet (sa lived the kind of life a very important model, sample, and determining which direction is known. Religious values, which is the carrier of the prophets, sent to the society they have always been examples and models. Because every aspect of human life, his life style and the surrounding area has a feature. We also value his life that he has unknowingly and without learning and skills and to understand it is not possible to live our religion . We also our presentation, we mainly of Islam o utlook on life and predicted life - style, including the Prophet of Islam 's (sa simple life to scrutinize and lifestyle issues related to reveal , in short Islam's how life has embraced and the Prophet. Prophet's will try to find answers to questions reg arding how to live.
Quasispecies made simple.
Full Text Available Quasispecies are clouds of genotypes that appear in a population at mutation-selection balance. This concept has recently attracted the attention of virologists, because many RNA viruses appear to generate high levels of genetic variation that may enhance the evolution of drug resistance and immune escape. The literature on these important evolutionary processes is, however, quite challenging. Here we use simple models to link mutation-selection balance theory to the most novel property of quasispecies: the error threshold-a mutation rate below which populations equilibrate in a traditional mutation-selection balance and above which the population experiences an error catastrophe, that is, the loss of the favored genotype through frequent deleterious mutations. These models show that a single fitness landscape may contain multiple, hierarchically organized error thresholds and that an error threshold is affected by the extent of back mutation and redundancy in the genotype-to-phenotype map. Importantly, an error threshold is distinct from an extinction threshold, which is the complete loss of the population through lethal mutations. Based on this framework, we argue that the lethal mutagenesis of a viral infection by mutation-inducing drugs is not a true error catastophe, but is an extinction catastrophe.
A simple scaler timer
Narayanan, R.; Kalavathy, K.R.
In any nuclear reactor, the start-up channels monitor the neutron flux during the start-up operation and give the alarm signals for safety purposes. Normally, a fission chamber is used as a detector to detect the low level neutron fluxes. The output of the detector after amplification and discrimination is shaped in a pulse shaper to provide constant width, constant height pulses for further processing in rate meters. The shaped pulses also go to a scaler timer, where they are counted for fixed time intervals and the accumulated counts displayed. The scaler timer described in this paper uses LSIs to get at a simple, compact and reliable unit. The design is centered around two LSIs. MOS Counter Timebase LSI type MK 5009P (U1) is used to generate the gating pulses. A 1 MHz crystal is used to generate the system clock. A 4 bit address selects the desired gating intervals of 1 or 10 or 100 seconds. In fact, MK 5009 is a very versatile LSI in a 16 pin DIP package, consisting of a MOS oscillator and divider chain. It is binary encoded for frequency division selection ranging from 1 to 36 x 10. With an input frequency of 1 MHz, MK 5009 provides the time periods of 1 μs to 100 seconds, one minute, ten minute and one hour periods. (author)
A Simple Accelerometer Calibrator
Salam, R A; Islamy, M R F; Khairurrijal; Munir, M M; Latief, H; Irsyam, M
High possibility of earthquake could lead to the high number of victims caused by it. It also can cause other hazards such as tsunami, landslide, etc. In that case it requires a system that can examine the earthquake occurrence. Some possible system to detect earthquake is by creating a vibration sensor system using accelerometer. However, the output of the system is usually put in the form of acceleration data. Therefore, a calibrator system for accelerometer to sense the vibration is needed. In this study, a simple accelerometer calibrator has been developed using 12 V DC motor, optocoupler, Liquid Crystal Display (LCD) and AVR 328 microcontroller as controller system. The system uses the Pulse Wave Modulation (PWM) form microcontroller to control the motor rotational speed as response to vibration frequency. The frequency of vibration was read by optocoupler and then those data was used as feedback to the system. The results show that the systems could control the rotational speed and the vibration frequencies in accordance with the defined PWM. (paper)
A Simple Spectral Observer
Lizeth Torres
Full Text Available The principal aim of a spectral observer is twofold: the reconstruction of a signal of time via state estimation and the decomposition of such a signal into the frequencies that make it up. A spectral observer can be catalogued as an online algorithm for time-frequency analysis because is a method that can compute on the fly the Fourier transform (FT of a signal, without having the entire signal available from the start. In this regard, this paper presents a novel spectral observer with an adjustable constant gain for reconstructing a given signal by means of the recursive identification of the coefficients of a Fourier series. The reconstruction or estimation of a signal in the context of this work means to find the coefficients of a linear combination of sines a cosines that fits a signal such that it can be reproduced. The design procedure of the spectral observer is presented along with the following applications: (1 the reconstruction of a simple periodical signal, (2 the approximation of both a square and a triangular signal, (3 the edge detection in signals by using the Fourier coefficients, (4 the fitting of the historical Bitcoin market data from 1 December 2014 to 8 January 2018 and (5 the estimation of a input force acting upon a Duffing oscillator. To round out this paper, we present a detailed discussion about the results of the applications as well as a comparative analysis of the proposed spectral observer vis-Ã -vis the Short Time Fourier Transform (STFT, which is a well-known method for time-frequency analysis.
A simple and highly sensitive on-line column extraction liquid chromatography-tandem mass spectrometry method for the determination of protein-unbound tacrolimus in human plasma samples.
Bittersohl, Heike; Schniedewind, Björn; Christians, Uwe; Luppa, Peter B
Therapeutic drug monitoring (TDM) of the immunosuppressive drug tacrolimus is essential to avoid side effects and rejection of the allograft after transplantation. In the blood circulation, tacrolimus is largely located inside erythrocytes or bound to plasma proteins and less than 0.1% is protein-unbound (free). One basic principle of clinical pharmacology is that only free drug is pharmacologically active and monitoring this portion has the potential to better reflect the drug effect than conventional measurements of total tacrolimus in whole blood. To address this, a highly sensitive and straightforward on-line liquid chromatography-tandem mass spectrometry (LC-MS/MS) method was developed, validated and applied to patient plasma samples. The sample preparation included ultracentrifugation and addition of the stable isotope labeled drug analogue D2,13C-tacrolimus, followed by on-line sample extraction and measurement using a Sciex QTRAP ® 6500 in the multiple reaction monitoring mode. Due to very low concentrations of protein-unbound tacrolimus, it was important to develop a highly sensitive, precise and accurate assay. Here, we first report the efficient formation of tacrolimus lithium adduct ions, which greatly increased assay sensitivity. A lower limit of quantification (LLOQ) of 1 pg/mL (10 fg on column) was achieved and the assay was linear between 1 and 200 pg/mL. There was no carry-over detected. The inaccuracy ranged from -9.8 to 7.4% and the greatest imprecision was 7.5%. The matrix factor was found to be smaller than 1.1%. In summary, this method represents a suitable tool to investigate the potential clinical value of free tacrolimus monitoring in organ transplant recipients. Copyright © 2018 Elsevier B.V. All rights reserved.
a simple a simple excitation control excitation control excitation
eobe
field voltages determined follow a simple quadratic relationship that offer a very simple control scheme, dependent on only the stator current. Keywords: saturated reactances, no-load field voltage, excitation control, synchronous generators. 1. Introduction. Introduction. Introduction. The commonest generator in use today is ...
Is simple nephrectomy truly simple? Comparison with the radical alternative.
Connolly, S S; O'Brien, M Frank; Kunni, I M; Phelan, E; Conroy, R; Thornhill, J A; Grainger, R
The Oxford English dictionary defines the term "simple" as "easily done" and "uncomplicated". We tested the validity of this terminology in relation to open nephrectomy surgery. Retrospective review of 215 patients undergoing open, simple (n = 89) or radical (n = 126) nephrectomy in a single university-affiliated institution between 1998 and 2002. Operative time (OT), estimated blood loss (EBL), operative complications (OC) and length of stay in hospital (LOS) were analysed. Statistical analysis employed Fisher's exact test and Stata Release 8.2. Simple nephrectomy was associated with shorter OT (mean 126 vs. 144 min; p = 0.002), reduced EBL (mean 729 vs. 859 cc; p = 0.472), lower OC (9 vs. 17%; 0.087), and more brief LOS (mean 6 vs. 8 days; p < 0.001). All parameters suggest favourable outcome for the simple nephrectomy group, supporting the use of this terminology. This implies "simple" nephrectomies are truly easier to perform with less complication than their radical counterpart.
Simple-MSSM: a simple and efficient method for simultaneous multi-site saturation mutagenesis.
Cheng, Feng; Xu, Jian-Miao; Xiang, Chao; Liu, Zhi-Qiang; Zhao, Li-Qing; Zheng, Yu-Guo
To develop a practically simple and robust multi-site saturation mutagenesis (MSSM) method that enables simultaneously recombination of amino acid positions for focused mutant library generation. A general restriction enzyme-free and ligase-free MSSM method (Simple-MSSM) based on prolonged overlap extension PCR (POE-PCR) and Simple Cloning techniques. As a proof of principle of Simple-MSSM, the gene of eGFP (enhanced green fluorescent protein) was used as a template gene for simultaneous mutagenesis of five codons. Forty-eight randomly selected clones were sequenced. Sequencing revealed that all the 48 clones showed at least one mutant codon (mutation efficiency = 100%), and 46 out of the 48 clones had mutations at all the five codons. The obtained diversities at these five codons are 27, 24, 26, 26 and 22, respectively, which correspond to 84, 75, 81, 81, 69% of the theoretical diversity offered by NNK-degeneration (32 codons; NNK, K = T or G). The enzyme-free Simple-MSSM method can simultaneously and efficiently saturate five codons within one day, and therefore avoid missing interactions between residues in interacting amino acid networks.
Simple Solutions for Dry Eye
Patient Education Sheet Simple Solutions for Dry Eye The SSF thanks J. Daniel Nelson, MD, Associate Medical Director, Specialty Care HealthPartners Medical Group & Clinics, and Professor of Ophthalmology, University of ...
Simple Tidal Prism Models Revisited
Luketina, D.
Simple tidal prism models for well-mixed estuaries have been in use for some time and are discussed in most text books on estuaries. The appeal of this model is its simplicity. However, there are several flaws in the logic behind the model. These flaws are pointed out and a more theoretically correct simple tidal prism model is derived. In doing so, it is made clear which effects can, in theory, be neglected and which can not.
Protein Structure Prediction by Protein Threading
Xu, Ying; Liu, Zhijie; Cai, Liming; Xu, Dong
The seminal work of Bowie, Lüthy, and Eisenberg (Bowie et al., 1991) on "the inverse protein folding problem" laid the foundation of protein structure prediction by protein threading. By using simple measures for fitness of different amino acid types to local structural environments defined in terms of solvent accessibility and protein secondary structure, the authors derived a simple and yet profoundly novel approach to assessing if a protein sequence fits well with a given protein structural fold. Their follow-up work (Elofsson et al., 1996; Fischer and Eisenberg, 1996; Fischer et al., 1996a,b) and the work by Jones, Taylor, and Thornton (Jones et al., 1992) on protein fold recognition led to the development of a new brand of powerful tools for protein structure prediction, which we now term "protein threading." These computational tools have played a key role in extending the utility of all the experimentally solved structures by X-ray crystallography and nuclear magnetic resonance (NMR), providing structural models and functional predictions for many of the proteins encoded in the hundreds of genomes that have been sequenced up to now.
Simple arithmetic: not so simple for highly math anxious individuals.
Chang, Hyesang; Sprute, Lisa; Maloney, Erin A; Beilock, Sian L; Berman, Marc G
Fluency with simple arithmetic, typically achieved in early elementary school, is thought to be one of the building blocks of mathematical competence. Behavioral studies with adults indicate that math anxiety (feelings of tension or apprehension about math) is associated with poor performance on cognitively demanding math problems. However, it remains unclear whether there are fundamental differences in how high and low math anxious individuals approach overlearned simple arithmetic problems that are less reliant on cognitive control. The current study used functional magnetic resonance imaging to examine the neural correlates of simple arithmetic performance across high and low math anxious individuals. We implemented a partial least squares analysis, a data-driven, multivariate analysis method to measure distributed patterns of whole-brain activity associated with performance. Despite overall high simple arithmetic performance across high and low math anxious individuals, performance was differentially dependent on the fronto-parietal attentional network as a function of math anxiety. Specifically, low-compared to high-math anxious individuals perform better when they activate this network less-a potential indication of more automatic problem-solving. These findings suggest that low and high math anxious individuals approach even the most fundamental math problems differently. © The Author (2017). Published by Oxford University Press.
Generalized Gradient Approximation Made Simple
Perdew, J.P.; Burke, K.; Ernzerhof, M.
Generalized gradient approximations (GGA close-quote s) for the exchange-correlation energy improve upon the local spin density (LSD) description of atoms, molecules, and solids. We present a simple derivation of a simple GGA, in which all parameters (other than those in LSD) are fundamental constants. Only general features of the detailed construction underlying the Perdew-Wang 1991 (PW91) GGA are invoked. Improvements over PW91 include an accurate description of the linear response of the uniform electron gas, correct behavior under uniform scaling, and a smoother potential. copyright 1996 The American Physical Society
Emergence Issues - not so simple
Anaesthetics Supplement: Emergence Issues - not so simple. S Afr Fam Pract 2014. Vol 56 No 2 Supplement 1. Introduction. Emergence from anaesthesia is by definition the process of return to baseline physiological function of all organ systems after cessation of administration of general anaesthesia and is the stage from ...
On framed simple Lie groups
MINAMI, Haruo
For a compact simple Lie group $G$, we show that the element $[G, \\mathcal{L}] \\in \\pi^S_*(S^0)$ represented by the pair $(G, \\mathcal{L})$ is zero, where $\\mathcal{L}$ denotes the left invariant framing of $G$. The proof relies on the method of E. Ossa [Topology, 21 (1982), 315–323].
The simple ethers of glycerin
Kimsanov, B.Kh.; Karimov, M.B.
From glycerin derivatives the considerable interest is present simple ethers because many of them are biological active and found wide practical using as an effect drugs, inters for thin organic synthesis, vehicle for injections, regulators of plants growth, reagents, components for perfumery-cosmetic goods and etc
Solving Simple Kinetics without Integrals
de la Pen~a, Lisandro Herna´ndez
The solution of simple kinetic equations is analyzed without referencing any topic from differential equations or integral calculus. Guided by the physical meaning of the rate equation, a systematic procedure is used to generate an approximate solution that converges uniformly to the exact solution in the case of zero, first, and second order…
Grief: Difficult Times, Simple Steps.
Waszak, Emily Lane
This guide presents techniques to assist others in coping with the loss of a loved one. Using the language of 9 layperson, the book contains more than 100 tips for caregivers or loved ones. A simple step is presented on each page, followed by reasons and instructions for each step. Chapters include: "What to Say"; "Helpful Things to Do"; "Dealing…
Simple stålrammebygninger
Ellum, J.C.
Anvisningen gennemgår dimensioneringen og bringer detaljerede konstruktionstegninger til simple stålrammebygninger, dvs. lukkede, fritliggende bygninger i én etage, hvor tagkonstruktionen ud over egenlast kun er påvirket af naturlaster, dvs. sne og vind. Dimensioneringen sker ved at udfylde et di...
Simple models with ALICE fluxes
Striet, J
We introduce two simple models which feature an Alice electrodynamics phase. In a well defined sense the Alice flux solutions we obtain in these models obey first order equations similar to those of the Nielsen-Olesen fluxtube in the abelian higgs model in the Bogomol'nyi limit. Some numerical solutions are presented as well.
Structure of simple liquids; Structure des liquides simples
Blain, J F [Commissariat a l' Energie Atomique, Fontenay-aux-Roses (France). Centre d' Etudes Nucleaires
The results obtained by application to argon and sodium of the two important methods of studying the structure of liquids: scattering of X-rays and neutrons, are presented on one hand. On the other hand the principal models employed for reconstituting the structure of simple liquids are exposed: mathematical models, lattice models and their derived models, experimental models. (author) [French] On presente d'une part les resultats obtenus par application a l'argon et au sodium des deux principales methodes d'etude de la structure des liquides: la diffusion des rayons X et la diffusion des neutrons; d'autre part, les principaux modeles employes pour reconstituer la structure des liquides simples sont exposes: modeles mathematiques, modeles des reseaux et modeles derives, modeles experimentaux. (auteur)
Simple apparatus for polarization sensing of analytes
Gryczynski, Zygmunt; Gryczynski, Ignacy; Lakowicz, Joseph R.
We describe a simple device for fluorescence sensing based on an unexpansive light source, a dual photocell and a Watson bridge. The emission is detected from two fluorescent samples, one of which changes intensity in response to the analyte. The emission from these two samples is observed through two orthogonally oriented polarizers and an analyzer polarizer. The latter polarizer is rotated to yield equal intensities from both sides of the dual photocell, as determined by a zero voltage from the Watson bridge. Using this device, we are able to measure fluorescein concentration to an accuracy near 2% at 1 (mu) M fluorescein, and pH values accurate to +/- 0.02 pH units. We also use this approach with a UV hand lamp and a glucose-sensitive protein to measure glucose concentrations near 2 (mu) M to an accuracy of +/- 0.1 (mu) M. This approach requires only simple electronics, which can be battery powered. Additionally, the method is generic, and can be applied with any fluorescent sample that displays a change in intensity. One can imagine this approach being used to develop portable point-of-care clinical devices.
A simple radioimmunoassay for plasma cortisol
Seth, J.; Brown, L.M.
A simple radioimmunoassay (RIA) for plasma cortisol is described which combines the advantages of (i) direct analysis of untreated plasma samples, (ii) use of solid-coupled anti-cortisol antibodies and (iii) use of a gamma-labelled radioligand. The reagents are relatively easily prepared and stable, and the analysis can be completed in 4 h. Inter-assay precision (C.V.) is 8-11%. Critical examination of specificity using high pressure liquid chromatography showed that 23-35% of the immunoassayable material in plasma was not cortisol. RIA results on samples collected under basal conditions were an average 40 nmol/l lower than fluorimetric results, while in insulin hypoglycaemia and synacthen (ACTH) stimulation tests, this difference increased to over 100 nmol/l. The RIA is technically more simple than fluorimetric, competitive-protein-binding, and many RIA methods, and can be used with advantage in the routine investigation of adrenocortical function. However, using the present antiserum, the RIA is not applicable to investigations on patients receiving metyrapone, nor in suspected cases of congenital adrenal hyperplasia. (Auth.)
Simple Electromagnetic Analysis in Cryptography
Zdenek Martinasek
Full Text Available The article describes the main principle and methods of simple electromagnetic analysis and thus provides an overview of simple electromagnetic analysis.The introductions chapters describe specific SPA attack used visual inspection of EM traces, template based attack and collision attack.After reading the article, the reader is sufficiently informed of any context of SEMA.Another aim of the article is the practical realization of SEMA which is focused on AES implementation.The visual inspection of EM trace of AES is performed step by step and the result is the determination of secret key Hamming weight.On the resulting EM trace, the Hamming weight of the secret key 1 to 8 was clearly visible.This method allows reduction from the number of possible keys for following brute force attack.
Complexity-aware simple modeling.
Gómez-Schiavon, Mariana; El-Samad, Hana
Mathematical models continue to be essential for deepening our understanding of biology. On one extreme, simple or small-scale models help delineate general biological principles. However, the parsimony of detail in these models as well as their assumption of modularity and insulation make them inaccurate for describing quantitative features. On the other extreme, large-scale and detailed models can quantitatively recapitulate a phenotype of interest, but have to rely on many unknown parameters, making them often difficult to parse mechanistically and to use for extracting general principles. We discuss some examples of a new approach-complexity-aware simple modeling-that can bridge the gap between the small-scale and large-scale approaches. Copyright © 2018 Elsevier Ltd. All rights reserved.
Simple Functions Spreadsheet tool presentation
Grive, Mireia; Domenech, Cristina; Montoya, Vanessa; Garcia, David; Duro, Lara
This document is a guide for users of the Simple Functions Spreadsheet tool. The Simple Functions Spreadsheet tool has been developed by Amphos 21 to determine the solubility limits of some radionuclides and it has been especially designed for Performance Assessment exercises. The development of this tool has been promoted by the necessity expressed by SKB of having a confident and easy-to-handle tool to calculate solubility limits in an agile and relatively fast manner. Its development started in 2005 and since then, it has been improved until the current version. This document describes the accurate and preliminary study following expert criteria that has been used to select the simplified aqueous speciation and solid phase system included in the tool. This report also gives the basic instructions to use this tool and to interpret its results. Finally, this document also reports the different validation tests and sensitivity analyses that have been done during the verification process
Gradings on simple Lie algebras
Elduque, Alberto
Gradings are ubiquitous in the theory of Lie algebras, from the root space decomposition of a complex semisimple Lie algebra relative to a Cartan subalgebra to the beautiful Dempwolff decomposition of E_8 as a direct sum of thirty-one Cartan subalgebras. This monograph is a self-contained exposition of the classification of gradings by arbitrary groups on classical simple Lie algebras over algebraically closed fields of characteristic not equal to 2 as well as on some nonclassical simple Lie algebras in positive characteristic. Other important algebras also enter the stage: matrix algebras, the octonions, and the Albert algebra. Most of the presented results are recent and have not yet appeared in book form. This work can be used as a textbook for graduate students or as a reference for researchers in Lie theory and neighboring areas.
Licensing of simple digital devices
Jackson, T. W.
The inability to guarantee error-free software gave rise to the potential for common-cause failure of digital safety systems in nuclear power plants. To address this vulnerability, the U. S. Nuclear Regulatory Commission (NRC) required a quality software development process and a defense-in-depth and diversity analysis for digital safety systems. As a result of recent interim [NRC] staff guidance in the digital instrumentation and control (I and C) area, licensing of simple digital devices decreases some regulatory burden with respect to demonstrating a quality software development process and defense-in-depth and diversity analysis. This paper defines simple digital devices and addresses the interim staff guidance that applies to such devices. The paper also highlights the technical aspects that affect the licensing of such devices and incorporates licensing experience in the U.S. to date. (authors)
A simple electron plasma wave
Brodin, G.; Stenflo, L.
Considering a class of solutions where the density perturbations are functions of time, but not of space, we derive a new exact large amplitude wave solution for a cold uniform electron plasma. This result illustrates that most simple analytical solutions can appear even if the density perturbations are large. - Highlights: • The influence of large amplitude electromagnetic waves on electrostatic oscillations is found. • A generalized Mathieu equation is derived. • Anharmonic wave profiles are computed numerically.
A Simple Probabilistic Combat Model
Government may violate any copyrights that exist in this work. This page intentionally left blank. ABSTRACT The Lanchester ...page intentionally left blank. TABLE OF CONTENTS Page No.Abstract iii List of Illustrations vii 1. INTRODUCTION 1 2. DETERMINISTIC LANCHESTER MODEL...This page intentionally left blank. 1. INTRODUCTION The Lanchester combat model1 is a simple way to assess the effects of quantity and quality
Brodin, G., E-mail: [email protected] [Department of Physics, Umeå University, SE-901 87 Umeå (Sweden); Stenflo, L. [Department of Physics, Linköping University, SE-581 83 Linköping (Sweden)
Simple and Realistic Data Generation
Pedersen, Kenneth Houkjær; Torp, Kristian; Wind, Rico
This paper presents a generic, DBMS independent, and highly extensible relational data generation tool. The tool can efficiently generate realistic test data for OLTP, OLAP, and data streaming applications. The tool uses a graph model to direct the data generation. This model makes it very simple...... to generate data even for large database schemas with complex inter- and intra table relationships. The model also makes it possible to generate data with very accurate characteristics....
Classification of simple current invariants
Gato-Rivera, Beatriz
We summarize recent work on the classification of modular invariant partition functions that can be obtained with simple currents in theories with a center (Z_p)^k with p prime. New empirical results for other centers are also presented. Our observation that the total number of invariants is monodromy-independent for (Z_p)^k appears to be true in general as well. (Talk presented in the parallel session on string theory of the Lepton-Photon/EPS Conference, Geneva, 1991.)
Instant simple botting with PHP
Anderson, Shay Michael
do for you, and then get to work with the most important features and tasks. This book is a hands-on Starter guide that takes the reader from initialization to the coding and implementation of bot apps.Instant Simple Botting with PHP targets programmers of all levels who are familiar with common PHP functions and syntax, and who want to learn about bots and how to design and develop bots using objects.
What Is a Simple Liquid?
Trond S. Ingebrigtsen
Full Text Available This paper is an attempt to identify the real essence of simplicity of liquids in John Locke's understanding of the term. Simple liquids are traditionally defined as many-body systems of classical particles interacting via radially symmetric pair potentials. We suggest that a simple liquid should be defined instead by the property of having strong correlations between virial and potential-energy equilibrium fluctuations in the NVT ensemble. There is considerable overlap between the two definitions, but also some notable differences. For instance, in the new definition simplicity is not a direct property of the intermolecular potential because a liquid is usually only strongly correlating in part of its phase diagram. Moreover, not all simple liquids are atomic (i.e., with radially symmetric pair potentials and not all atomic liquids are simple. The main part of the paper motivates the new definition of liquid simplicity by presenting evidence that a liquid is strongly correlating if and only if its intermolecular interactions may be ignored beyond the first coordination shell (FCS. This is demonstrated by NVT simulations of the structure and dynamics of several atomic and three molecular model liquids with a shifted-forces cutoff placed at the first minimum of the radial distribution function. The liquids studied are inverse power-law systems (r^{-n} pair potentials with n=18,6,4, Lennard-Jones (LJ models (the standard LJ model, two generalized Kob-Andersen binary LJ mixtures, and the Wahnstrom binary LJ mixture, the Buckingham model, the Dzugutov model, the LJ Gaussian model, the Gaussian core model, the Hansen-McDonald molten salt model, the Lewis-Wahnstrom ortho-terphenyl model, the asymmetric dumbbell model, and the single-point charge water model. The final part of the paper summarizes properties of strongly correlating liquids, emphasizing that these are simpler than liquids in general. Simple liquids, as defined here, may be
Correlation and simple linear regression.
Zou, Kelly H; Tuncali, Kemal; Silverman, Stuart G
In this tutorial article, the concepts of correlation and regression are reviewed and demonstrated. The authors review and compare two correlation coefficients, the Pearson correlation coefficient and the Spearman rho, for measuring linear and nonlinear relationships between two continuous variables. In the case of measuring the linear relationship between a predictor and an outcome variable, simple linear regression analysis is conducted. These statistical concepts are illustrated by using a data set from published literature to assess a computed tomography-guided interventional technique. These statistical methods are important for exploring the relationships between variables and can be applied to many radiologic studies.
Systems analysis made simple computerbooks
Antill, Lyn
Systems Analysis: Made Simple Computerbooks introduces the essential elements of information systems analysis and design and teaches basic technical skills required for the tasks involved. The book covers the aspects to the design of an information system; information systems and the organization, including the types of information processing activity and computer-based information systems; the role of the systems analyst; and the human activity system. The text also discusses information modeling, socio-technical design, man-machine interface, and the database design. Software specification
Computer electronics made simple computerbooks
Bourdillon, J F B
Computer Electronics: Made Simple Computerbooks presents the basics of computer electronics and explains how a microprocessor works. Various types of PROMs, static RAMs, dynamic RAMs, floppy disks, and hard disks are considered, along with microprocessor support devices made by Intel, Motorola and Zilog. Bit slice logic and some AMD bit slice products are also described. Comprised of 14 chapters, this book begins with an introduction to the fundamentals of hardware design, followed by a discussion on the basic building blocks of hardware (NAND, NOR, AND, OR, NOT, XOR); tools and equipment that
Singular perturbation of simple eigenvalues
Greenlee, W.M.
Two operator theoretic theorems which generalize those of asymptotic regular perturbation theory and which apply to singular perturbation problems are proved. Application of these theorems to concrete problems is involved, but the perturbation expansions for eigenvalues and eigenvectors are developed in terms of solutions of linear operator equations. The method of correctors, as well as traditional boundary layer techniques, can be used to apply these theorems. The current formulation should be applicable to highly singular ''hard core'' potential perturbations of the radial equation of quantum mechanics. The theorems are applied to a comparatively simple model problem whose analysis is basic to that of the quantum mechanical problem
Simple Simulations of DNA Condensation
STEVENS,MARK J.
Molecular dynamics simulations of a simple, bead-spring model of semiflexible polyelectrolytes such as DNA are performed. All charges are explicitly treated. Starting from extended, noncondensed conformations, condensed structures form in the simulations with tetravalent or trivalent counterions. No condensates form or are stable for divalent counterions. The mechanism by which condensates form is described. Briefly, condensation occurs because electrostatic interactions dominate entropy, and the favored Coulombic structure is a charge ordered state. Condensation is a generic phenomena and occurs for a variety of polyelectrolyte parameters. Toroids and rods are the condensate structures. Toroids form preferentially when the molecular stiffness is sufficiently strong.
Wrist arthrography: a simple method
Berna-Serna, Juan D.; Reus, Manuel; Alonso, Jose [Virgen de la Arrixaca University Hospital, Department of Radiology, El Palmar (Murcia) (Spain); Martinez, Francisco; Domenech-Ratto, Gines [University of Murcia, Department of Human Anatomy, Faculty of Medicine, Murcia (Spain)
A technique of wrist arthrography is presented using an adhesive marker-plate with radiopaque coordinates to identify precisely sites for puncture arthrography of the wrist and to obviate the need for fluoroscopic guidance. Radiocarpal joint arthrography was performed successfully in all 24 cases, 14 in the cadaveric wrists and 10 in the live patients. The arthrographic procedure described in this study is simple, safe, and rapid, and has the advantage of precise localisation of the site for puncture without need for fluoroscopic guidance. (orig.)
Windows Phone 7 Made Simple
Trautschold, Martin
With Windows Phone 7, Microsoft has created a completely new smartphone operating system that focuses on allowing users to be productive with their smartphone in new ways, while offering seamless integration and use of Microsoft Office Mobile as well as other productivity apps available in the Microsoft App Store. Windows Phone 7 Made Simple offers a clear, visual, step-by-step approach to using your Windows Phone 7 smartphone, no matter what the manufacturer. Author Jon Westfall is an expert in mobile devices, recognized by Microsoft as a "Most Valuable Professional" with experience
Serum Creatinine: Not So Simple!
DELANAYE, Pierre; Cavalier, Etienne; Pottel, Hans
Measuring serum creatinine is cheap and commonly done in daily practice. However, interpretation of serum creatinine results is not always easy. In this review, we will briefly remind the physiological limitations of serum creatinine due notably to its tubular secretion and the influence of muscular mass or protein intake on its concentration. We mainly focus on the analytical limitations of serum creatinine, insisting on important concept such as reference intervals, standardization (and IDM...
iPad Made Simple
Trautschold, Martin; Learning, MSL Made Simple
The new iPad is sleek, powerful, and most importantly, it's much more than just a big iPhone. Your iPad is can be used for reading, surfing the web, emailing, watching TV/Movies, getting work done, and much more. And with the upcoming wave of iPad apps, the possibilities are endless. iPad X Made Simple clarifies all of the key features on the iPad, introduces what's new, and also reveals dozens of time-saving shortcuts and techniques. The book has over 1,000 screen shots that are carefully annotated with step-by-step instructions. * Clear instructions on how to set up and use the iPad * Illust
Methadone radioimmunoassay: two simple methods
Robinson, K.; Smith, R.N.
Two simple and economical radioimmunoassays for methadone in blood or urine are described. Haemolysis, decomposition, common anticoagulants and sodium fluoride do not affect the results. One assay used commercially-available [1- 3 H](-)-methadone hydrobromide as the label, while the other uses a radioiodinated conjugate of 4-dimethylamino-2,2-diphenylpentanoic acid and L-tyrosine methyl ester. A commercially-available antiserum is used in both assays. Normethadone and α-methadol cross-react to a small extent with the antiserum while methadone metabolites, dextropropoxyphene, dipipanone and phenadoxone have negligible cross-reactivities. The 'cut-offs' of the two assays as described are 30 and 33 ng ml -1 for blood, and 24 and 21 ng ml -1 for urine. The assay using the radioiodinated conjugate can be made more sensitive if required by increasing the specific activity of the label. (author)
Simple types of anisotropic inflation
Barrow, John D.; Hervik, Sigbjoern
We display some simple cosmological solutions of gravity theories with quadratic Ricci curvature terms added to the Einstein-Hilbert Lagrangian which exhibit anisotropic inflation. The Hubble expansion rates are constant and unequal in three orthogonal directions. We describe the evolution of the simplest of these homogeneous and anisotropic cosmological models from its natural initial state and evaluate the deviations they will create from statistical isotropy in the fluctuations produced during a period of anisotropic inflation. The anisotropic inflation is not a late-time attractor in these models but the rate of approach to a final isotropic de Sitter state is slow and is conducive to the creation of observable anisotropic statistical effects in the microwave background. The statistical anisotropy would not be scale invariant and the level of statistical anisotropy will grow with scale.
Semantic markup of sensor capabilities: how simple it too simple?
Rueda-Velasquez, C. A.; Janowicz, K.; Fredericks, J.
Semantics plays a key role for the publication, retrieval, integration, and reuse of observational data across the geosciences. In most cases, one can safely assume that the providers of such data, e.g., individual scientists, understand the observation context in which their data are collected,e.g., the used observation procedure, the sampling strategy, the feature of interest being studied, and so forth. However, can we expect that the same is true for the technical details of the used sensors and especially the nuanced changes that can impact observations in often unpredictable ways? Should the burden of annotating the sensor capabilities, firmware, operation ranges, and so forth be really part of a scientist's responsibility? Ideally, semantic annotations should be provided by the parties that understand these details and have a vested interest in maintaining these data. With manufactures providing semantically-enabled metadata for their sensors and instruments, observations could more easily be annotated and thereby enriched using this information. Unfortunately, today's sensor ontologies and tool chains developed for the Semantic Web community require expertise beyond the knowledge and interest of most manufacturers. Consequently, knowledge engineers need to better understand the sweet spot between simple ontologies/vocabularies and sufficient expressivity as well as the tools required to enable manufacturers to share data about their sensors. Here, we report on the current results of EarthCube's X-Domes project that aims to address the questions outlined above.
A Simple Model to Study Tau Pathology
Alexander L. Houck
Full Text Available Tau proteins play a role in the stabilization of microtubules, but in pathological conditions, tauopathies, tau is modified by phosphorylation and can aggregate into aberrant aggregates. These aggregates could be toxic to cells, and different cell models have been used to test for compounds that might prevent these tau modifications. Here, we have used a cell model involving the overexpression of human tau in human embryonic kidney 293 cells. In human embryonic kidney 293 cells expressing tau in a stable manner, we have been able to replicate the phosphorylation of intracellular tau. This intracellular tau increases its own level of phosphorylation and aggregates, likely due to the regulatory effect of some growth factors on specific tau kinases such as GSK3. In these conditions, a change in secreted tau was observed. Reversal of phosphorylation and aggregation of tau was found by the use of lithium, a GSK3 inhibitor. Thus, we propose this as a simple cell model to study tau pathology in nonneuronal cells due to their viability and ease to work with.
Implications of protein polymorphism on protein phase behaviour
Stegen, J.; Schoot, van der P.P.A.M.
The phase behaviour of small globular proteins is often modeled by approximating them as spherical particles with fixed internal structure. However, changes in the local environment of a protein can lead to changes in its conformation rendering this approximation invalid. We present a simple
Water nanoelectrolysis: A simple model
Olives, Juan; Hammadi, Zoubida; Morin, Roger; Lapena, Laurent
A simple model of water nanoelectrolysis—defined as the nanolocalization at a single point of any electrolysis phenomenon—is presented. It is based on the electron tunneling assisted by the electric field through the thin film of water molecules (˜0.3 nm thick) at the surface of a tip-shaped nanoelectrode (micrometric to nanometric curvature radius at the apex). By applying, e.g., an electric potential V1 during a finite time t1, and then the potential -V1 during the same time t1, we show that there are three distinct regions in the plane (t1, V1): one for the nanolocalization (at the apex of the nanoelectrode) of the electrolysis oxidation reaction, the second one for the nanolocalization of the reduction reaction, and the third one for the nanolocalization of the production of bubbles. These parameters t1 and V1 completely control the time at which the electrolysis reaction (of oxidation or reduction) begins, the duration of this reaction, the electrolysis current intensity (i.e., the tunneling current), the number of produced O2 or H2 molecules, and the radius of the nanolocalized bubbles. The model is in good agreement with our experiments.
Simple substrates for complex cognition
Peter Dayan
Full Text Available Complex cognitive tasks present a range of computational and algorithmic challenges for neural accounts of both learning and inference. In particular, it is extremely hard to solve them using the sort of simple policies that have been extensively studied as solutions to elementary Markov decision problems. There has thus been recent interest in architectures for the instantiation and even learning of policies that are formally more complicated than these, involving operations such as gated working memory. However, the focus of these ideas and methods has largely been on what might best be considered as automatized, routine or, in the sense of animal conditioning, habitual, performance. Thus, they have yet to provide a route towards understanding the workings of rule-based control, which is critical for cognitively sophisticated competence. Here, we review a recent suggestion for a uniform architecture for habitual and rule-based execution, discuss some of the habitual mechanisms that underpin the use of rules, and consider a statistical relationship between rules and habits.
Plunger with simple retention valve
Fekete, A.V.
This patent describes a positive displacement retention valve apparatus in which the actual flow equals the theoretical maximum flow through the retention valve. The apparatus includes, in combination, a confined fluid flow conduit, a piston adapted for reciprocal movement within the fluid flow conduit between upstream and downstream limit positions, piston reciprocating means, and pressure responsive check valve means located upstream with respect to the piston in the fluid flow conduit. The pressure responsive check valve means operable to permit fluid flow therethrough in a downstream direction toward the piston, and to preclude fluid flow therethrough in an opposite direction. The piston is composed of parts which are relatively movable with respect to one another. The piston includes a simple retention valve consisting of a plug means, a cylinder having a minimum and a maximum internal cross section flow area therein and being reciprocal within the confined fluid flow conduit, and a seat on the cylinder for the plug means. The piston reciprocating means are operatively connected to the plug means
Measuring serum creatinine is cheap and commonly done in daily practice. However, interpretation of serum creatinine results is not always easy. In this review, we will briefly remind the physiological limitations of serum creatinine due notably to its tubular secretion and the influence of muscular mass or protein intake on its concentration. We mainly focus on the analytical limitations of serum creatinine, insisting on important concept such as reference intervals, standardization (and IDMS traceability), analytical interferences, analytical coefficient of variation (CV), biological CV and critical difference. Because the relationship between serum creatinine and glomerular filtration rate is hyperbolic, all these CVs will impact not only the precision of serum creatinine but still more the precision of different creatinine-based equations, especially in low or normal-low creatinine levels (or high or normal-high glomerular filtration rate range). © 2017 S. Karger AG, Basel.
SSL - THE SIMPLE SOCKETS LIBRARY
Campbell, C. E.
The Simple Sockets Library (SSL) allows C programmers to develop systems of cooperating programs using Berkeley streaming Sockets running under the TCP/IP protocol over Ethernet. The SSL provides a simple way to move information between programs running on the same or different machines and does so with little overhead. The SSL can create three types of Sockets: namely a server, a client, and an accept Socket. The SSL's Sockets are designed to be used in a fashion reminiscent of the use of FILE pointers so that a C programmer who is familiar with reading and writing files will immediately feel comfortable with reading and writing with Sockets. The SSL consists of three parts: the library, PortMaster, and utilities. The user of the SSL accesses it by linking programs to the SSL library. The PortMaster initializes connections between clients and servers. The PortMaster also supports a "firewall" facility to keep out socket requests from unapproved machines. The "firewall" is a file which contains Internet addresses for all approved machines. There are three utilities provided with the SSL. SKTDBG can be used to debug programs that make use of the SSL. SPMTABLE lists the servers and port numbers on requested machine(s). SRMSRVR tells the PortMaster to forcibly remove a server name from its list. The package also includes two example programs: multiskt.c, which makes multiple accepts on one server, and sktpoll.c, which repeatedly attempts to connect a client to some server at one second intervals. SSL is a machine independent library written in the C-language for computers connected via Ethernet using the TCP/IP protocol. It has been successfully compiled and implemented on a variety of platforms, including Sun series computers running SunOS, DEC VAX series computers running VMS, SGI computers running IRIX, DECstations running ULTRIX, DEC alpha AXPs running OSF/1, IBM RS/6000 computers running AIX, IBM PC and compatibles running BSD/386 UNIX and HP Apollo 3000
Reconstructing Nearly Simple Polytopes from their Graph
Doolittle, Joseph
We present a partial description of which polytopes are reconstructible from their graphs. This is an extension of work by Blind and Mani (1987) and Kalai (1988), which showed that simple polytopes can be reconstructed from their graphs. In particular, we introduce a notion of $h$-nearly simple and prove that 1-nearly simple and 2-nearly simple polytopes are reconstructible from their graphs. We also give an example of a 3-nearly simple polytope which is not reconstructible from its graph. Fu...
Correcting slightly less simple movements
M.P. Aivar
Full Text Available Many studies have analysed how goal directed movements are corrected in response to changes in the properties of the target. However, only simple movements to single targets have been used in those studies, so little is known about movement corrections under more complex situations. Evidence from studies that ask for movements to several targets in sequence suggests that whole sequences of movements are planned together. Planning related segments of a movement together makes it possible to optimise the whole sequence, but it means that some parts are planned quite long in advance, so that it is likely that they will have to be modified. In the present study we examined how people respond to changes that occur while they are moving to the first target of a sequence. Subjects moved a stylus across a digitising tablet. They moved from a specified starting point to two targets in succession. The first of these targets was always at the same position but it could have one of two sizes. The second target could be in one of two different positions and its size was different in each case. On some trials the first target changed size, and on some others the second target changed size and position, as soon as the subject started to move. When the size of the first target changed the subjects slowed down the first segment of their movements. Even the peak velocity, which was only about 150 ms after the change in size, was lower. Beside this fast response to the change itself, the dwell time at the first target was also affected: its duration increased after the change. Changing the size and position of the second target did not influence the first segment of the movement, but also increased the dwell time. The dwell time was much longer for a small target, irrespective of its initial size. If subjects knew in advance which target could change, they moved faster than if they did not know which could change. Taken together, these
Identification of Protein-Protein Interactions with Glutathione-S-Transferase (GST) Fusion Proteins.
Einarson, Margret B; Pugacheva, Elena N; Orlinick, Jason R
INTRODUCTIONGlutathione-S-transferase (GST) fusion proteins have had a wide range of applications since their introduction as tools for synthesis of recombinant proteins in bacteria. GST was originally selected as a fusion moiety because of several desirable properties. First and foremost, when expressed in bacteria alone, or as a fusion, GST is not sequestered in inclusion bodies (in contrast to previous fusion protein systems). Second, GST can be affinity-purified without denaturation because it binds to immobilized glutathione, which provides the basis for simple purification. Consequently, GST fusion proteins are routinely used for antibody generation and purification, protein-protein interaction studies, and biochemical analysis. This article describes the use of GST fusion proteins as probes for the identification of protein-protein interactions.
Physics of protein folding
Finkelstein, A. V.; Galzitskaya, O. V.
Protein physics is grounded on three fundamental experimental facts: protein, this long heteropolymer, has a well defined compact three-dimensional structure; this structure can spontaneously arise from the unfolded protein chain in appropriate environment; and this structure is separated from the unfolded state of the chain by the "all-or-none� phase transition, which ensures robustness of protein structure and therefore of its action. The aim of this review is to consider modern understanding of physical principles of self-organization of protein structures and to overview such important features of this process, as finding out the unique protein structure among zillions alternatives, nucleation of the folding process and metastable folding intermediates. Towards this end we will consider the main experimental facts and simple, mostly phenomenological theoretical models. We will concentrate on relatively small (single-domain) water-soluble globular proteins (whose structure and especially folding are much better studied and understood than those of large or membrane and fibrous proteins) and consider kinetic and structural aspects of transition of initially unfolded protein chains into their final solid ("native�) 3D structures.
Simple model systems: a challenge for Alzheimer's disease
Di Carlo Marta
Full Text Available Abstract The success of biomedical researches has led to improvement in human health and increased life expectancy. An unexpected consequence has been an increase of age-related diseases and, in particular, neurodegenerative diseases. These disorders are generally late onset and exhibit complex pathologies including memory loss, cognitive defects, movement disorders and death. Here, it is described as the use of simple animal models such as worms, fishes, flies, Ascidians and sea urchins, have facilitated the understanding of several biochemical mechanisms underlying Alzheimer's disease (AD, one of the most diffuse neurodegenerative pathologies. The discovery of specific genes and proteins associated with AD, and the development of new technologies for the production of transgenic animals, has helped researchers to overcome the lack of natural models. Moreover, simple model systems of AD have been utilized to obtain key information for evaluating potential therapeutic interventions and for testing efficacy of putative neuroprotective compounds.
Simple sorting algorithm test based on CUDA
Meng, Hongyu; Guo, Fangjin
With the development of computing technology, CUDA has become a very important tool. In computer programming, sorting algorithm is widely used. There are many simple sorting algorithms such as enumeration sort, bubble sort and merge sort. In this paper, we test some simple sorting algorithm based on CUDA and draw some useful conclusions.
The Fluid Foil: The Seventh Simple Machine
Mitts, Charles R.
A simple machine does one of two things: create a mechanical advantage (lever) or change the direction of an applied force (pulley). Fluid foils are unique among simple machines because they not only change the direction of an applied force (wheel and axle); they convert fluid energy into mechanical energy (wind and Kaplan turbines) or vice versa,…
Simple utility functions with Giffen demand
Sørensen, Peter Norman
Simple utility functions with the Giffen property are presented: locally, the demand curve for a good is upward sloping. The utility functions represent continuous, monotone, convex preferences......Simple utility functions with the Giffen property are presented: locally, the demand curve for a good is upward sloping. The utility functions represent continuous, monotone, convex preferences...
Falling balls and simple shearing strain
Brun, J L; Pacheco, A F
The problem of particles falling under gravity allows us to relate Hamiltonian mechanics to such different subjects as elasticity and fluid mechanics. It is with this in mind that mechanics gives us the opportunity of introducing, in a rather simple and unusual form, some concepts such as vorticity, the incompressibility condition or simple shear strain to physics students at the undergraduate level
Simple clamped connection for bamboo truss systems
Blok, R.
"How to make fast and simple tension connections for truss systems?� The Solution: The innovation is a connection that uses only widely available base components (boltsand threaded steel bars) and simple hand tools to install it. With a handsaw and aspanner, the bamboo stems can be combined into to
NVU perspective on simple liquids' quasiuniversality
Dyre, J. C.
The last half-century of research into the structure, dynamics, and thermodynamics of simple liquids has revealed a number of approximate universalities. This paper argues that simple liquids' reduced-coordinate constant-potential-energy hypersurfaces constitute a quasiuniversal family of compact...
A simple approach to nonlinear oscillators
Ren Zhongfu; He Jihuan
A very simple and effective approach to nonlinear oscillators is suggested. Anyone with basic knowledge of advanced calculus can apply the method to finding approximately the amplitude-frequency relationship of a nonlinear oscillator. Some examples are given to illustrate its extremely simple solution procedure and an acceptable accuracy of the obtained solutions.
Sporadic simple groups and quotient singularities
Cheltsov, I A; Shramov, C A
We show that if a faithful irreducible representation of a central extension of a sporadic simple group with centre contained in the commutator subgroup gives rise to an exceptional (resp. weakly exceptional but not exceptional) quotient singularity, then that simple group is the Hall-Janko group (resp. the Suzuki group)
Ultrafiltration of pegylated proteins
Molek, Jessica R.
There is considerable clinical interest in the use of "second-generation" therapeutics produced by conjugation of a native protein with various polymers including polyethylene glycol (PEG). PEG--protein conjugates, so-called PEGylated proteins, can exhibit enhanced stability, half-life, and bioavailability. One of the challenges in the commercial production of PEGylated proteins is the purification required to remove unreacted polymer, native protein, and in many cases PEGylated proteins with nonoptimal degrees of conjugation. The overall objective of this thesis was to examine the use of ultrafiltration for the purification of PEGylated proteins. This included: (1) analysis of size-based separation of PEGylated proteins using conventional ultrafiltration membranes, (2) use of electrically-charged membranes to exploit differences in electrostatic interactions, and (3) examination of the effects of PEGylation on protein fouling. The experimental results were analyzed using appropriate theoretical models, with the underlying physical properties of the PEGylated proteins evaluated using size exclusion chromatography, capillary electrophoresis, dynamic light scattering, and reverse phase chromatography. PEGylated proteins were produced by covalent attachment of activated PEG to a protein via primary amines on the lysine residues. A simple model was developed for the reaction kinetics, which was used to explore the effect of reaction conditions and mode of operation on the distribution of PEGylated products. The effective size of the PEGylated proteins was evaluated using size exclusion chromatography, with appropriate correlations developed for the size in terms of the molecular weight of the native protein and attached PEG. The electrophoretic mobility of the PEGylated proteins were evaluated by capillary electrophoresis with the data in good agreement with a simple model accounting for the increase in protein size and the reduction in the number of protonated amine
A simple model for binary star evolution
Whyte, C.A.; Eggleton, P.P.
A simple model for calculating the evolution of binary stars is presented. Detailed stellar evolution calculations of stars undergoing mass and energy transfer at various rates are reported and used to identify the dominant physical processes which determine the type of evolution. These detailed calculations are used to calibrate the simple model and a comparison of calculations using the detailed stellar evolution equations and the simple model is made. Results of the evolution of a few binary systems are reported and compared with previously published calculations using normal stellar evolution programs. (author)
Prediction of potential drug targets based on simple sequence properties
Lai Luhua
Full Text Available Abstract Background During the past decades, research and development in drug discovery have attracted much attention and efforts. However, only 324 drug targets are known for clinical drugs up to now. Identifying potential drug targets is the first step in the process of modern drug discovery for developing novel therapeutic agents. Therefore, the identification and validation of new and effective drug targets are of great value for drug discovery in both academia and pharmaceutical industry. If a protein can be predicted in advance for its potential application as a drug target, the drug discovery process targeting this protein will be greatly speeded up. In the current study, based on the properties of known drug targets, we have developed a sequence-based drug target prediction method for fast identification of novel drug targets. Results Based on simple physicochemical properties extracted from protein sequences of known drug targets, several support vector machine models have been constructed in this study. The best model can distinguish currently known drug targets from non drug targets at an accuracy of 84%. Using this model, potential protein drug targets of human origin from Swiss-Prot were predicted, some of which have already attracted much attention as potential drug targets in pharmaceutical research. Conclusion We have developed a drug target prediction method based solely on protein sequence information without the knowledge of family/domain annotation, or the protein 3D structure. This method can be applied in novel drug target identification and validation, as well as genome scale drug target predictions.
Simple membrane-based model of the Min oscillator
Petrášek, Zdeněk; Schwille, Petra
Min proteins in E. coli bacteria organize into a dynamic pattern oscillating between the two cell poles. This process identifies the middle of the cell and enables symmetric cell division. In an experimental model system consisting of a flat membrane with effectively infinite supply of proteins and energy source, the Min proteins assemble into travelling waves. Here we propose a simple one-dimensional model of the Min dynamics that, unlike the existing models, reproduces the sharp decrease of Min concentration when the majority of protein detaches from the membrane, and even the narrow MinE maximum immediately preceding the detachment. The proposed model thus provides a possible mechanism for the formation of the MinE ring known from cells. The model is restricted to one dimension, with protein interactions described by chemical kinetics allowing at most bimolecular reactions, and explicitly considering only three, membrane-bound, species. The bulk solution above the membrane is approximated as being well-mixed, with constant concentrations of all species. Unlike other models, our proposal does not require autocatalytic binding of MinD to the membrane. Instead, it is assumed that two MinE molecules are necessary to induce the dissociation of the MinD dimer and its subsequent detachment from the membrane. We investigate which reaction schemes lead to unstable homogeneous steady states and limit cycle oscillations, and how diffusion affects their stability. The suggested model qualitatively describes the shape of the Min waves observed on flat membranes, and agrees with the experimental dependence of the wave period on the MinE concentration. These results highlight the importance of MinE presence on the membrane without being bound to MinD, and of the reactions of Min proteins on the membrane. (paper)
A simple method for multiday imaging of slice cultures.
Seidl, Armin H; Rubel, Edwin W
The organotypic slice culture (Stoppini et al. A simple method for organotypic cultures of nervous tissue. 1991;37:173-182) has become the method of choice to answer a variety of questions in neuroscience. For many experiments, however, it would be beneficial to image or manipulate a slice culture repeatedly, for example, over the course of many days. We prepared organotypic slice cultures of the auditory brainstem of P3 and P4 mice and kept them in vitro for up to 4 weeks. Single cells in the auditory brainstem were transfected with plasmids expressing fluorescent proteins by way of electroporation (Haas et al. Single-cell electroporation for gene transfer in vivo. 2001;29:583-591). The culture was then placed in a chamber perfused with oxygenated ACSF and the labeled cell imaged with an inverted wide-field microscope repeatedly for multiple days, recording several time-points per day, before returning the slice to the incubator. We describe a simple method to image a slice culture preparation during the course of multiple days and over many continuous hours, without noticeable damage to the tissue or photobleaching. Our method uses a simple, inexpensive custom-built insulator constructed around the microscope to maintain controlled temperature and uses a perfusion chamber as used for in vitro slice recordings. (c) 2009 Wiley-Liss, Inc.
Dynamics of unwinding of a simple entaglement
Wiegel, F.W.; Michels, J.P.J.
The dynamics of unwinding of a simple entanglement is studied in two ways, firstly using an optimal path approximation in the Rouse model and secondly by simulating the movement of a more realistic model using Brownian molecular dynamics
A Simple Preparation Method for Diphosphoimidazole
Rosenberg, T.
A simple method for the preparation of diphosphoimidazole is presented that involves direct phosphorylation of imidazole by phosphorus oxychloride in alkaline aqueous solution. Details are given on the use of diphosphoimidazole in preparing sodium phosphoramidate and certain phosphorylated amino...
simple and rapid spectrophotometric assay of levocetirizine
Simple, rapid, selective and fairly sensitive method is described for the ... Determination of small amounts of LCTZ in pharmaceutical preparations is important for .... sodium hydroxide and extraction of HCl-free-amine into chloroform followed ...
Simple Perturbation Example for Quantum Chemistry.
Goodfriend, P. L.
Presents a simple example that illustrates various aspects of the Rayleigh-Schrodinger perturbation theory. The example is a particularly good one because it is straightforward and can be compared with both the exact solution and with experimental data. (JN)
The Invention Convention: Mind Meets Simple Machines.
Hadi-Tabassum, Samina
Describes an Earth Day celebration where students had to design an invention made of simple machines that could crush an empty aluminum can through 10 rapid mechanical movements using materials foraged from the students' homes. (JRH)
Simple and Clear Proofs of Stirling's Formula
Niizeki, Shozo; Araki, Makoto
The purpose of our article is to show two simpler and clearer methods of proving Stirling's formula than the traditional and conventional ones. The distinction of our method is to use the simple trapezoidal formula.
Low Cost, Simple, Intrauterine Insemination Procedure
AJRH Managing Editor
quite simple intrauterine insemination technique which may be performed in developing countries, without the need of sophisticated ... Cytoplasmic Sperm Injection (ICSI), are quite ... were administered only once by intramuscular injection ...
Substoichiometric method in the simple radiometric analysis
Ikeda, N.; Noguchi, K.
The substoichiometric method is applied to simple radiometric analysis. Two methods - the standard reagent method and the standard sample method - are proposed. The validity of the principle of the methods is verified experimentally in the determination of silver by the precipitation method, or of zinc by the ion-exchange or solvent-extraction method. The proposed methods are simple and rapid compared with the conventional superstoichiometric method. (author)
Simple versus complex degenerative mitral valve disease.
Javadikasgari, Hoda; Mihaljevic, Tomislav; Suri, Rakesh M; Svensson, Lars G; Navia, Jose L; Wang, Robert Z; Tappuni, Bassman; Lowry, Ashley M; McCurry, Kenneth R; Blackstone, Eugene H; Desai, Milind Y; Mick, Stephanie L; Gillinov, A Marc
At a center where surgeons favor mitral valve (MV) repair for all subsets of leaflet prolapse, we compared results of patients undergoing repair for simple versus complex degenerative MV disease. From January 1985 to January 2016, 6153 patients underwent primary isolated MV repair for degenerative disease, 3101 patients underwent primary isolated MV repair for simple disease (posterior prolapse), and 3052 patients underwent primary isolated MV repair for complex disease (anterior or bileaflet prolapse), based on preoperative echocardiographic images. Logistic regression analysis was used to generate propensity scores for risk-adjusted comparisons (n = 2065 matched pairs). Durability was assessed by longitudinal recurrence of mitral regurgitation and reoperation. Compared with patients with simple disease, those undergoing repair of complex pathology were more likely to be younger and female (both P values simple vs 3000/98% complex; P = .5), followed by leaflet resection (2802/90% simple vs 2249/74% complex; P simple pathology versus 11% for complex pathology (P = .007), reoperation at 18 years was 6.3% for simple pathology versus 11% for complex pathology, and 20-year survival was 62% for simple pathology versus 61% for complex pathology (P = .6). Early surgical intervention has become more common in patients with degenerative MV disease, regardless of valve prolapse complexity or symptom status. Valve repair was associated with similarly low operative risk and time-related survival but less durability in complex disease. Lifelong annual echocardiographic surveillance after MV repair is recommended, particularly in patients with complex disease. Copyright © 2018 The American Association for Thoracic Surgery
Infinitivo simple y compuesto con predicados declarativos
Rodríguez Espiñeira, María José
Este trabajo se centra en las diferencias semánticas y sintácticas que presentan las construcciones de infinitivo simple y compuesto que actúan como objeto directo de predicados declarativos. El tema despierta interés por varios motivos: (1) Se trata de un subconjunto de verbos que subordinan infinitivos simples de un elenco restringido, pero que admiten regularmente infinitivos compuestos.
Simple Lie algebras and Dynkin diagrams
Buccella, F.
The following theorem is studied: in a simple Lie algebra of rank p there are p positive roots such that all the other n-3p/2 positive roots are linear combinations of them with integer non negative coefficients. Dykin diagrams are built by representing the simple roots with circles and drawing a junction between the roots. Five exceptional algebras are studied, focusing on triple junction algebra, angular momentum algebra, weights of the representation, antisymmetric tensors, and subalgebras
Signed Young Modules and Simple Specht Modules
Danz, Susanne; Lim, Kay Jin
By a result of Hemmer, every simple Specht module of a finite symmetric group over a field of odd characteristic is a signed Young module. While Specht modules are parametrized by partitions, indecomposable signed Young modules are parametrized by certain pairs of partitions. The main result of this article establishes the signed Young module labels of simple Specht modules. Along the way we prove a number of results concerning indecomposable signed Young modules that are of independent inter...
Selecting highly structure-specific antibodies using structured synthetic mimics of the cystine knot protein sclerostin
Back, J.W.; Frisch, C.; Van Pee, K.; Boschert, V.; van Vught, R.; Puijk, W.; Mueller, T. D.; Knappik, A.; Timmerman, P.
Antibodies directed against specific regions of a protein have traditionally been raised against full proteins, protein domains or simple unstructured peptides, containing contiguous stretches of primary sequence. We have used a new approach of selecting antibodies against restrained peptides
Preparation of GST Fusion Proteins.
INTRODUCTIONThis protocol describes the preparation of glutathione-S-transferase (GST) fusion proteins, which have had a wide range of applications since their introduction as tools for synthesis of recombinant proteins in bacteria. GST was originally selected as a fusion moiety because of several desirable properties. First and foremost, when expressed in bacteria alone, or as a fusion, GST is not sequestered in inclusion bodies (in contrast to previous fusion protein systems). Second, GST can be affinity-purified without denaturation because it binds to immobilized glutathione, which provides the basis for simple purification. Consequently, GST fusion proteins are routinely used for antibody generation and purification, protein-protein interaction studies, and biochemical analysis.
The Design of SimpleITK
Bradley Christopher Lowekamp
Full Text Available SimpleITK is a new interface to the Insight Segmentation andRegistration Toolkit (ITK designed to facilitate rapid prototyping, educationand scientific activities, via high level programminglanguages. ITK is a templated C++ library of image processingalgorithms and frameworks for biomedical and other applications, andit was designed to be generic, flexible and extensible. Initially, ITKprovided a direct wrapping interface to languages such as Python andTcl through the WrapITK system. Unlike WrapITK, which exposed ITK'scomplex templated interface, SimpleITK was designed to provide an easyto use and simplified interface to ITK's algorithms. It includesprocedural methods, hides ITK's demand driven pipeline, and provides atemplate-less layer. Also SimpleITK provides practical conveniencessuch as binary distribution packages and overloaded operators. Ouruser-friendly design goals dictated a departure from the directinterface wrapping approach of WrapITK, towards a new facadeclass structure that only exposes the required functionality, hidingITK's extensive template use. Internally SimpleITK utilizes a manualdescription of each filter with code-generation and advanced C++meta-programming to provide the higher-level interface, bringing thecapabilities of ITK to a wider audience. SimpleITK is licensed asopen source software under the Apache License Version 2.0 and more informationabout downloading it can be found at http://www.simpleitk.org.
Simple genomes, complex interactions: Epistasis in RNA virus
Elena, Santiago F.; Solé, Ricard V.; Sardanyés, Josep
Owed to their reduced size and low number of proteins encoded, RNA viruses and other subviral pathogens are often considered as being genetically too simple. However, this structural simplicity also creates the necessity for viral RNA sequences to encode for more than one protein and for proteins to carry out multiple functions, all together resulting in complex patterns of genetic interactions. In this work we will first review the experimental studies revealing that the architecture of viral genomes is dominated by antagonistic interactions among loci. Second, we will also review mathematical models and provide a description of computational tools for the study of RNA virus dynamics and evolution. As an application of these tools, we will finish this review article by analyzing a stochastic bit-string model of in silico virus replication. This model analyzes the interplay between epistasis and the mode of replication on determining the population load of deleterious mutations. The model suggests that, for a given mutation rate, the deleterious mutational load is always larger when epistasis is predominantly antagonistic than when synergism is the rule. However, the magnitude of this effect is larger if replication occurs geometrically than if it proceeds linearly.
Vortex breakdown in simple pipe bends
Ault, Jesse; Shin, Sangwoo; Stone, Howard
Pipe bends and elbows are one of the most common fluid mechanics elements that exists. However, despite their ubiquity and the extensive amount of research related to these common, simple geometries, unexpected complexities still remain. We show that for a range of geometries and flow conditions, these simple flows experience unexpected fluid dynamical bifurcations resembling the bubble-type vortex breakdown phenomenon. Specifically, we show with simulations and experiments that recirculation zones develop within the bends under certain conditions. As a consequence, fluid and particles can remain trapped within these structures for unexpectedly-long time scales. We also present simple techniques to mitigate this recirculation effect which can potentially have impact across industries ranging from biomedical and chemical processing to food and health sciences.
Simple approach to study biomolecule adsorption in polymeric microfluidic channels
Gubala, Vladimir; Siegrist, Jonathan; Monaghan, Ruairi; O'Reilly, Brian; Gandhiraman, Ram Prasad; Daniels, Stephen; Williams, David E.; Ducrée, Jens
Highlights: ► A simple tool to assess biomolecule adsorption onto the surfaces of microchannels. ► Development for dilution by surface-adsorption based depletion of protein samples. ► It can easily be done using a readily available apparatus like a spin-coater. ► The assessment tool is facile and quantitative. ► Straightforward comparison of different surface chemistries. - Abstract: Herein a simple analytical method is presented for the characterization of biomolecule adsorption on cyclo olefin polymer (COP, trade name: Zeonor ® ) substrates which are widely used in microfluidic lab-on-a-chip devices. These Zeonor ® substrates do not possess native functional groups for specific reactions with biomolecules. Therefore, depending on the application, such substrates must be functionalized by surface chemistry methods to either enhance or suppress biomolecular adsorption. This work demonstrates a microfluidic method for evaluating the adsorption of antibodies and oligonucleotides surfaces. The method uses centrifugal microfluidic flow-through chips and can easily be implemented using common equipment such as a spin coater. The working principle is very simple. The user adds 40 L of the solution containing the sample to the starting side of a microfluidic channel, where it is moved through by centrifugal force. Some molecules are adsorbed in the channel. The sample is then collected at the other end in a small reservoir and the biomolecule concentration is measured. As a pilot application, we characterized the adsorption of goat anti-human IgG and a 20-mer DNA on Zeonor ® , and on three types of functionalized Zeonor: 3-aminopropyltriethoxysilane (APTES) modified surface with mainly positive charge, negatively charged surface with immobilized bovine serum albumin (BSA), and neutral, hydrogel-like film with polyethylene glycol (PEG) characteristics. This simple analytical approach adds to the fundamental understanding of the interaction forces in real
Gubala, Vladimir, E-mail: [email protected] [Biomedical Diagnostics Institute (BDI), National Centre for Sensor Research (NCSR), Dublin City University, Dublin 9 (Ireland); Medway School of Pharmacy, University of Kent, Central Avenue, Anson 120, Chatham Maritime, Kent ME4 4TB (United Kingdom); Siegrist, Jonathan; Monaghan, Ruairi; O' Reilly, Brian; Gandhiraman, Ram Prasad [Biomedical Diagnostics Institute (BDI), National Centre for Sensor Research (NCSR), Dublin City University, Dublin 9 (Ireland); Daniels, Stephen [Biomedical Diagnostics Institute (BDI), National Centre for Sensor Research (NCSR), Dublin City University, Dublin 9 (Ireland); National Centre for Plasma Science and Technology (NCPST), Dublin City University, Dublin 9 (Ireland); Williams, David E. [Biomedical Diagnostics Institute (BDI), National Centre for Sensor Research (NCSR), Dublin City University, Dublin 9 (Ireland); MacDiarmid Institute for Advanced Materials and Nanotechnology, School of Chemical Sciences, University of Auckland, Auckland 1142 (New Zealand); Ducree, Jens [Biomedical Diagnostics Institute (BDI), National Centre for Sensor Research (NCSR), Dublin City University, Dublin 9 (Ireland)
Highlights: Black-Right-Pointing-Pointer A simple tool to assess biomolecule adsorption onto the surfaces of microchannels. Black-Right-Pointing-Pointer Development for dilution by surface-adsorption based depletion of protein samples. Black-Right-Pointing-Pointer It can easily be done using a readily available apparatus like a spin-coater. Black-Right-Pointing-Pointer The assessment tool is facile and quantitative. Black-Right-Pointing-Pointer Straightforward comparison of different surface chemistries. - Abstract: Herein a simple analytical method is presented for the characterization of biomolecule adsorption on cyclo olefin polymer (COP, trade name: Zeonor{sup Registered-Sign }) substrates which are widely used in microfluidic lab-on-a-chip devices. These Zeonor{sup Registered-Sign} substrates do not possess native functional groups for specific reactions with biomolecules. Therefore, depending on the application, such substrates must be functionalized by surface chemistry methods to either enhance or suppress biomolecular adsorption. This work demonstrates a microfluidic method for evaluating the adsorption of antibodies and oligonucleotides surfaces. The method uses centrifugal microfluidic flow-through chips and can easily be implemented using common equipment such as a spin coater. The working principle is very simple. The user adds 40 L of the solution containing the sample to the starting side of a microfluidic channel, where it is moved through by centrifugal force. Some molecules are adsorbed in the channel. The sample is then collected at the other end in a small reservoir and the biomolecule concentration is measured. As a pilot application, we characterized the adsorption of goat anti-human IgG and a 20-mer DNA on Zeonor{sup Registered-Sign }, and on three types of functionalized Zeonor: 3-aminopropyltriethoxysilane (APTES) modified surface with mainly positive charge, negatively charged surface with immobilized bovine serum albumin (BSA), and
TRSkit: A Simple Digital Library Toolkit
Nelson, Michael L.; Esler, Sandra L.
This paper introduces TRSkit, a simple and effective toolkit for building digital libraries on the World Wide Web. The toolkit was developed for the creation of the Langley Technical Report Server and the NASA Technical Report Server, but is applicable to most simple distribution paradigms. TRSkit contains a handful of freely available software components designed to be run under the UNIX operating system and served via the World Wide Web. The intended customer is the person that must continuously and synchronously distribute anywhere from 100 - 100,000's of information units and does not have extensive resources to devote to the problem.
Assessment of daylight quality in simple rooms
Johnsen, Kjeld; Dubois, Marie-Claude; Sørensen, Karl Grau
The present report documents the results of a study on daylight conditions in simple rooms of residential buildings. The overall objective of the study was to develop a basis for a method for the assessment of daylight quality in a room with simple geometry and window configurations. As a tool...... in daylighting conditions for a number of lighting parameters. The results gave clear indications of, for instance, which room would be the brightest, under which conditions might glare be a problem and which type of window would yield the greatest luminous variation (or visual interest), etc....
Strong Bisimilarity of Simple Process Algebras
Srba, Jirí
We study bisimilarity and regularity problems of simple process algebras. In particular, we show PSPACE-hardness of the following problems: (i) strong bisimilarity of Basic Parallel Processes (BPP), (ii) strong bisimilarity of Basic Process Algebra (BPA), (iii) strong regularity of BPP, and (iv......) strong regularity of BPA. We also demonstrate NL-hardness of strong regularity problems for the normed subclasses of BPP and BPA. Bisimilarity problems of simple process algebras are introduced in a general framework of process rewrite systems, and a uniform description of the new techniques used...
Simple gas chromatographic method for furfural analysis.
Gaspar, Elvira M S M; Lopes, João F
A new, simple, gas chromatographic method was developed for the direct analysis of 5-hydroxymethylfurfural (5-HMF), 2-furfural (2-F) and 5-methylfurfural (5-MF) in liquid and water soluble foods, using direct immersion SPME coupled to GC-FID and/or GC-TOF-MS. The fiber (DVB/CAR/PDMS) conditions were optimized: pH effect, temperature, adsorption and desorption times. The method is simple and accurate (RSDfurfurals will contribute to characterise and quantify their presence in the human diet.
A simple technique for laparoscopic gastrostomy.
Murphy, C; Rosemurgy, A S; Albrink, M H; Carey, L C
While endoscopically placed gastrostomy tubes are routinely simple, they are not always feasible. Endoscopic technique also does not uniformly secure the tube to the abdominal wall, which presents possible complications, including leakage, accidental early tube removal, intraperitoneal catheter migration and necrosis of the stomach or abdominal wall because of excessive traction. Presented herein is a technique that is rapid, simple and eliminates some of these potential complications. The technique is easily combined with other operative procedures, such as tracheostomy, is done under direct vision, can be performed quickly with intravenous sedation and local anesthetic and is a safe method of tube placement for enteral feeding or gastric decompression.
Simple 2-representations and Classification of Categorifications
Agerholm, Troels
We consider selfadjoint functors defined on categories of modules over finite dimensional algebras and classify those that satisfy some simple relations. In particular we classify self- adjoint idempotents and selfadjoint squareroots of a multiple of the identity functor. This is related to the t......We consider selfadjoint functors defined on categories of modules over finite dimensional algebras and classify those that satisfy some simple relations. In particular we classify self- adjoint idempotents and selfadjoint squareroots of a multiple of the identity functor. This is related...
Molecular implementation of simple logic programs.
Ran, Tom; Kaplan, Shai; Shapiro, Ehud
Autonomous programmable computing devices made of biomolecules could interact with a biological environment and be used in future biological and medical applications. Biomolecular implementations of finite automata and logic gates have already been developed. Here, we report an autonomous programmable molecular system based on the manipulation of DNA strands that is capable of performing simple logical deductions. Using molecular representations of facts such as Man(Socrates) and rules such as Mortal(X) logical deductions and delivers the result. This prototype is the first simple programming language with a molecular-scale implementation.
... page: //medlineplus.gov/ency/article/003483.htm Total protein To use the sharing features on this page, please enable JavaScript. The total protein test measures the total amount of two classes ...
Proteins engineering
At the - Departement d'Ingenierie et d'etudes de proteines (Deip) of the CEA more than seventy researchers are working hard to understand the function of proteins. For that they use the molecular labelling technique (F.M.)
... reliable information about the safety of taking whey protein if you are pregnant or breast feeding. Stay on the safe side and avoid use. Milk allergy: If you are allergic to cow's milk, avoid using whey protein.
Differing mechanisms of simple nitrile formation on glucosinolate degradation in Lepidium sativum and Nasturtium officinale seeds.
Williams, David J; Critchley, Christa; Pun, Sharon; Chaliha, Mridusmita; O'Hare, Timothy J
Glucosinolates are sulphur-containing glycosides found in brassicaceous plants that can be hydrolysed enzymatically by plant myrosinase or non-enzymatically to form primarily isothiocyanates and/or simple nitriles. From a human health perspective, isothiocyanates are quite important because they are major inducers of carcinogen-detoxifying enzymes. Two of the most potent inducers are benzyl isothiocyanate (BITC) present in garden cress (Lepidium sativum), and phenylethyl isothiocyanate (PEITC) present in watercress (Nasturtium officinale). Previous studies on these salad crops have indicated that significant amounts of simple nitriles are produced at the expense of the isothiocyanates. These studies also suggested that nitrile formation may occur by different pathways: (1) under the control of specifier protein in garden cress and (2) by an unspecified, non-enzymatic path in watercress. In an effort to understand more about the mechanisms involved in simple nitrile formation in these species, we analysed their seeds for specifier protein and myrosinase activities, endogenous iron content and glucosinolate degradation products after addition of different iron species, specific chelators and various heat treatments. We confirmed that simple nitrile formation was predominantly under specifier protein control (thiocyanate-forming protein) in garden cress seeds. Limited thermal degradation of the major glucosinolate, glucotropaeolin (benzyl glucosinolate), occurred when seed material was heated to >120 degrees C. In the watercress seeds, however, we show for the first time that gluconasturtiin (phenylethyl glucosinolate) undergoes a non-enzymatic, iron-dependent degradation to a simple nitrile. On heating the seeds to 120 degrees C or greater, thermal degradation of this heat-labile glucosinolate increased simple nitrile levels many fold.
Simple Approach to Superamphiphobic Overhanging Silicon Nanostructures
Kumar, Rajendra; Mogensen, Klaus Bo; Bøggild, Peter
with contact angles up to 152 degrees and roll-off angle down to 8 degrees. Such nonlithographic nanoscale overhanging Structures can also be added to silicon nanograss by deposition of a thin SiO2 layer, which equips the silicon rods with 100-300 nm sized overhanging Structures. This is a simple, fast...
Question Answering for Dutch : Simple does it
Hoekstra, A.H.; Hiemstra, Djoerd; van der Vet, P.E.; Huibers, Theo W.C.; Schobbens, Pierre-Yves; Vanhoof, Wim; Schwanen, Gabriel
When people pose questions in natural language to search for information on the web, the role of question answering (QA) systems becomes important. In this paper the QAsystem simpleQA, capable of answering Dutch questions on which the answer is a person or a location, is described. The system's
Nature versus Nurture: The Simple Contrast
Davidoff, Jules; Goldstein, Julie; Roberson, Debi
We respond to the commentary of Franklin, Wright, and Davies ("Journal of Experimental Child Psychology, 102", 239-245 [2009]) by returning to the simple contrast between nature and nurture. We find no evidence from the toddler data that makes us revise our ideas that color categories are learned and never innate. (Contains 1 figure.)
Simple picture of the annihilation process
Gotsman, E.; Nussinov, S.
We propose a simple geometrical picture for B-barB annihilations, which is motivated by the electric-flux-tube model and is consistent with the quark-rearrangement model, as well as the nonplanar multiperipheral quark-exchange model. Within its framework we are able to explain all the salient features of the experimental annihilation data
Simple Analysis of Historical Lime Mortars
Pires, Joa~o
A laboratory experiment is described in which a simple characterization of a historical lime mortar is made by the determination of its approximate composition by a gravimetric method. Fourier transform infrared (FTIR) spectroscopy and X-ray diffraction (XRD) are also used for the qualitative characterization of the lime mortar components. These…
Seismic Safety Of Simple Masonry Buildings
Guadagnuolo, Mariateresa; Faella, Giuseppe
Several masonry buildings comply with the rules for simple buildings provided by seismic codes. For these buildings explicit safety verifications are not compulsory if specific code rules are fulfilled. In fact it is assumed that their fulfilment ensures a suitable seismic behaviour of buildings and thus adequate safety under earthquakes. Italian and European seismic codes differ in the requirements for simple masonry buildings, mostly concerning the building typology, the building geometry and the acceleration at site. Obviously, a wide percentage of buildings assumed simple by codes should satisfy the numerical safety verification, so that no confusion and uncertainty have to be given rise to designers who must use the codes. This paper aims at evaluating the seismic response of some simple unreinforced masonry buildings that comply with the provisions of the new Italian seismic code. Two-story buildings, having different geometry, are analysed and results from nonlinear static analyses performed by varying the acceleration at site are presented and discussed. Indications on the congruence between code rules and results of numerical analyses performed according to the code itself are supplied and, in this context, the obtained result can provide a contribution for improving the seismic code requirements
Simple concurrent garbage collection almost without synchronization
Hesselink, Wim H.; Lali, M.I.
We present two simple mark and sweep algorithms, A and B, for concurrent garbage collection by a single collector running concurrently with a number of mutators that concurrently modify shared data. Both algorithms are based on the ideas of Ben-Ari's classical algorithm for on-the-fly garbage
Bayesian decision theory : A simple toy problem
van Erp, H.R.N.; Linger, R.O.; van Gelder, P.H.A.J.M.
We give here a comparison of the expected outcome theory, the expected utility theory, and the Bayesian decision theory, by way of a simple numerical toy problem in which we look at the investment willingness to avert a high impact low probability event. It will be found that for this toy problem
Low Cost, Simple, Intrauterine Insemination Procedure with ...
During the last 30 years however, intrauterine insemination has evolved with the introduction of ovulation stimulating protocols and sperm preparation methods taken from assisted reproduction techniques. Costs have risen, but the success rate has not risen to the same extent. We have therefore developed a quite simple ...
A rewriting view of simple typing
Stump, A.; Zantema, H.; Kimmell, G.; El Haj Omar, R.
This paper shows how a recently developed view of typing as small-step abstract reduction, due to Kuan, MacQueen, and Findler, can be used to recast the development of simple type theory from a rewriting perspective. We show how standard meta-theoretic results can be proved in a completely new way,
Studying the Greenhouse Effect: A Simple Demonstration.
Papageorgiou, G.; Ouzounis, K.
Studies the parameters involved in a presentation of the greenhouse effect and describes a simple demonstration of this effect. Required equipment includes a 100-120 watt lamp, a 250mL beaker, and a thermometer capable of recording 0-750 degrees Celsius together with a small amount of chloroform. (Author/SAH)
Simple parametrization of nucleon form factors
Kelly, J.J.
This Brief Report provides simple parametrizations of the nucleon electromagnetic form factors using functions of Q 2 that are consistent with dimensional scaling at high Q 2 . Good fits require only four parameters each for G Ep , G Mp , and G Mn and only two for G En
A Simple Sketch Symbolizing Self-Reliance
Centers for Disease Control (CDC) Podcasts
EID Managing Editor, Byron Breedlove, reads his cover art story, A Simple Sketch Symbolizing Self-Reliance. Created: 2/16/2017 by National Center for Emerging and Zoonotic Infectious Diseases (NCEZID). Date Released: 2/16/2017.
Simple Cloud Chambers Using Gel Ice Packs
Kamata, Masahiro; Kubota, Miki
Although cloud chambers are highly regarded as teaching aids for radiation education, school teachers have difficulty in using cloud chambers because they have to prepare dry ice or liquid nitrogen before the experiment. We developed a very simple and inexpensive cloud chamber that uses the contents of gel ice packs which can substitute for dry…
A Study of Simple Diffraction Models
Agerkvist, Finn
In this paper two simple methods for cabinet edge diffraction are examined. Calculations with both models are compared with more sophisticated theoretical models and with measured data. The parameters involved are studied and their importance for normal loudspeaker box designs is examined....
Variations on a Simple Dice Game
Heafner, Joe
I begin my introductory astronomy course with a unit on critical thinking that focuses on, among other things, the differences between the "scientific method" as frequently presented in textbooks and actual scientific practice. One particular classroom activity uses a simple dice game to simulate observation of a natural phenomenon and…
Tour of a Simple Trigonometry Problem
Poon, Kin-Keung
This article focuses on a simple trigonometric problem that generates a strange phenomenon when different methods are applied to tackling it. A series of problem-solving activities are discussed, so that students can be alerted that the precision of diagrams is important when solving geometric problems. In addition, the problem-solving plan was…
A simple model for indentation creep
Ginder, Ryan S.; Nix, William D.; Pharr, George M.
A simple model for indentation creep is developed that allows one to directly convert creep parameters measured in indentation tests to those observed in uniaxial tests through simple closed-form relationships. The model is based on the expansion of a spherical cavity in a power law creeping material modified to account for indentation loading in a manner similar to that developed by Johnson for elastic-plastic indentation (Johnson, 1970). Although only approximate in nature, the simple mathematical form of the new model makes it useful for general estimation purposes or in the development of other deformation models in which a simple closed-form expression for the indentation creep rate is desirable. Comparison to a more rigorous analysis which uses finite element simulation for numerical evaluation shows that the new model predicts uniaxial creep rates within a factor of 2.5, and usually much better than this, for materials creeping with stress exponents in the range 1 ≤ n ≤ 7. The predictive capabilities of the model are evaluated by comparing it to the more rigorous analysis and several sets of experimental data in which both the indentation and uniaxial creep behavior have been measured independently.
Simple Calculation Programs for Biology Other Methods
First page Back Continue Last page Overview Graphics. Simple Calculation Programs for Biology Other Methods. Hemolytic potency of drugs. Raghava et al., (1994) Biotechniques 17: 1148. FPMAP: methods for classification and identification of microorganisms 16SrRNA. graphical display of restriction and fragment map of ...
Simple model of the arms race
Zane, L.I.
A simple model of a two-party arms race is developed based on the principle that the race will continue so long as either side can unleash an effective first strike against the other side. The model is used to examine how secrecy, the ABM, MIRV-ing, and an MX system affect the arms race
Microwave Radiometer Linearity Measured by Simple Means
Skou, Niels
Modern spaceborne radiometer systems feature an almost perfect on-board calibration, hence the primary calibration task to be carried out before launch is a check of radiometer linearity. This paper describes two ways of measuring linearity of microwave radiometers only requiring relatively simple...
Simple Calculation Programs for Biology Immunological Methods
First page Back Continue Last page Overview Graphics. Simple Calculation Programs for Biology Immunological Methods. Computation of Ab/Ag Concentration from EISA data. Graphical Method; Raghava et al., 1992, J. Immuno. Methods 153: 263. Determination of affinity of Monoclonal Antibody. Using non-competitive ...
Simulated Holograms: A Simple Introduction to Holography.
Dittmann, H.; Schneider, W. B.
Describes a project that uses a computer and a dot matrix printer to simulate the holographic recording process of simple object structures. The process' four steps are (1) superposition of waves; (2) representing the superposition of a plane reference wave on the monitor screen; (3) photographic reduction of the images; and (4) reconstruction of…
A Simple Model of Self-Assessments
S. Dominguez Martinez (Silvia); O.H. Swank (Otto)
textabstractWe develop a simple model that describes individuals' self-assessments of their abilities. We assume that individuals learn about their abilities from appraisals of others and experience. Our model predicts that if communication is imperfect, then (i) appraisals of others tend to be too
A simple model of self-assessment
Dominguez-Martinez, S.; Swank, O.H.
We develop a simple model that describes individuals' self-assessments of their abilities. We assume that individuals learn about their abilities from appraisals of others and experience. Our model predicts that if communication is imperfect, then (i) appraisals of others tend to be too positive and
Simple relation among the charged lepton masses
Terazawa, Hidezumi; Yasue, Masaki.
It is suggested that the first and second generations of quarks and leptons can be taken as almost and quasi Nambu-Goldstone fermions, respectively. As a result, derived are simple relations such as m τ ≅(m μ 3 /m e ) 1/2 , which are well satisfied experimentally. (author)
Simple Activity Demonstrates Wind Energy Principles
Roman, Harry T.
Wind energy is an exciting and clean energy option often described as the fastest-growing energy system on the planet. With some simple materials, teachers can easily demonstrate its key principles in their classroom. (Contains 1 figure and 2 tables.)
TMAP: Stata module for simple thematic mapping
Maurizio Pisati
This is a revised version of the package published in The Stata Journal 4(4):361-378 (2004) for carrying out simple thematic mapping. This new release should be considered as a beta version: comments and problem reports to the author are welcome. After the final revision, the resulting version will be submitted for publication to The Stata Journal.
Simple Numerical Simulation of Strain Measurement
Tai, H.
By adopting the basic principle of the reflection (and transmission) of a plane polarized electromagnetic wave incident normal to a stack of films of alternating refractive index, a simple numerical code was written to simulate the maximum reflectivity (transmittivity) of a fiber optic Bragg grating corresponding to various non-uniform strain conditions including photo-elastic effect in certain cases.
A Simple ELISA Exercise for Undergraduate Biology.
Baker, William P.; Moore, Cathy R.
Understanding of immunological techniques such as the Enzyme Linked Immuno Sorbent Assay (ELISA) is an important part of instructional units in human health, developmental biology, microbiology, and biotechnology. This paper describes a simple ELISA exercise for undergraduate biology that effectively simulates the technique using a paper model.…
Testing theory in practice: a simple experiment
Terpstra, R.; Brezocnik, Z.; Kapus, T.; Ferreira Pires, Luis; Heerink, A.W.; Tretmans, G.J.
In this paper we discuss the experiences gained in conducting a simple testing experiment. The goal of this experiment is to apply the abstract, formal testing framework [8] in a practical setting, and to indicate the critical aspects in its application to realistic testing situations. For that
Simple air collectors for preheating fresh air
Hensen, J.L.M.; Wit, de M.H.; Ouden, den C.
In dwellings with mechanical ventilation systems the fresh air can easily be preheated by means of simple solar air systems. These can be an integral part of the building facade or roof and the costs are expected to be low. By means of computer experiments a large number of systems were evaluated.
Simple Obstacle Avoidance Algorithm for Rehabilitation Robots
Stuyt, Floran H.A.; Römer, GertWillem R.B.E.; Stuyt, Harry .J.A.
The efficiency of a rehabilitation robot is improved by offering record-and-replay to operate the robot. While automatically moving to a stored target (replay) collisions of the robot with obstacles in its work space must be avoided. A simple, though effective, generic and deterministic algorithm
Simple classical approach to spin resonance phenomena
Gordon, R A
A simple classical method of describing spin resonance in terms of the average power absorbed by a spin system is discussed. The method has several advantages over more conventional treatments, and a number of important spin resonance phenomena, not normally considered at the introductory level...
Jacks--A Study of Simple Machines.
Parsons, Ralph
This vocational physics individualized student instructional module on jacks (simple machines used to lift heavy objects) contains student prerequisites and objectives, an introduction, and sections on the ratchet bumper jack, the hydraulic jack, the screw jack, and load limitations. Designed with a laboratory orientation, each section consists of…
Special Relativity as a Simple Geometry Problem
de Abreu, Rodrigo; Guerra, Vasco
The null result of the Michelson-Morley experiment and the constancy of the one-way speed of light in the "rest system" are used to formulate a simple problem, to be solved by elementary geometry techniques using a pair of compasses and non-graduated rulers. The solution consists of a drawing allowing a direct visualization of all the fundamental…
Simple analytical relations for ship bow waves
Noblesse, Francis; Delhommeau, G.?Rard; Guilbaud, Michel; Hendrix, Dane; Yang, Chi
Simple analytical relations for the bow wave generated by a ship in steady motion are given. Specifically, simple expressions that define the height of a ship bow wave, the distance between the ship stem and the crest of the bow wave, the rise of water at the stem, and the bow wave profile, explicitly and without calculations, in terms of the ship speed, draught, and waterline entrance angle, are given. Another result is a simple criterion that predicts, also directly and without calculations, when a ship in steady motion cannot generate a steady bow wave. This unsteady-flow criterion predicts that a ship with a sufficiently fine waterline, specifically with waterline entrance angle 2, may generate a steady bow wave at any speed. However, a ship with a fuller waterline (25E) can only generate a steady bow wave if the ship speed is higher than a critical speed, defined in terms of αE by a simple relation. No alternative criterion for predicting when a ship in steady motion does not generate a steady bow wave appears to exist. A simple expression for the height of an unsteady ship bow wave is also given. In spite of their remarkable simplicity, the relations for ship bow waves obtained in the study (using only rudimentary physical and mathematical considerations) are consistent with experimental measurements for a number of hull forms having non-bulbous wedge-shaped bows with small flare angle, and with the authors' measurements and observations for a rectangular flat plate towed at a yaw angle.
Contribution of simple saccharides to the stabilization of amyloid structure
Fung, Justin; Darabie, Audrey A.; McLaurin, JoAnne
The use of osmolytes or chaperones to stabilize proteins/peptides that misfold in neurodegenerative diseases is an attractive concept for drug development. We have investigated the role of a series of small carbohydrates for protection of the natively structured Alzheimer's amyloid-β peptides (Aβ). Using circular dichroism spectroscopy to follow the β-structural transitions and electron microscopy to examine tertiary structural characteristics, we demonstrate that the hydrogen bonding capacity of the carbohydrate determines the inhibition or promotion of fibrillogenesis. Three sugar molecules that vary only in their distribution of potential H-bonding partners promote various structural changes in Aβ. Two of these sugar molecules are excluded from Aβ during aggregation and promote mature fibre growth, while the other binds Aβ promoting nucleation and the accumulation of protofibrils. Our studies suggest that utilization of a combinatorial strategy to alter H-bonding capacity across a simple carbohydrate molecule may represent a novel drug design strategy
Proteins aggregation and human diseases
Hu, Chin-Kun
Many human diseases and the death of most supercentenarians are related to protein aggregation. Neurodegenerative diseases include Alzheimer's disease (AD), Huntington's disease (HD), Parkinson's disease (PD), frontotemporallobar degeneration, etc. Such diseases are due to progressive loss of structure or function of neurons caused by protein aggregation. For example, AD is considered to be related to aggregation of Aβ40 (peptide with 40 amino acids) and Aβ42 (peptide with 42 amino acids) and HD is considered to be related to aggregation of polyQ (polyglutamine) peptides. In this paper, we briefly review our recent discovery of key factors for protein aggregation. We used a lattice model to study the aggregation rates of proteins and found that the probability for a protein sequence to appear in the conformation of the aggregated state can be used to determine the temperature at which proteins can aggregate most quickly. We used molecular dynamics and simple models of polymer chains to study relaxation and aggregation of proteins under various conditions and found that when the bending-angle dependent and torsion-angle dependent interactions are zero or very small, then protein chains tend to aggregate at lower temperatures. All atom models were used to identify a key peptide chain for the aggregation of insulin chains and to find that two polyQ chains prefer anti-parallel conformation. It is pointed out that in many cases, protein aggregation does not result from protein mis-folding. A potential drug from Chinese medicine was found for Alzheimer's disease.
Biomedical sensors for disease detection made simple
Veldhuis, Djuke
Researchers in Korea have succeeded in making a new protein detection platform, using low-cost plastic and paper substrates. Their work could help reduce the cost and improve the accuracy of infectious disease diagnosis.......Researchers in Korea have succeeded in making a new protein detection platform, using low-cost plastic and paper substrates. Their work could help reduce the cost and improve the accuracy of infectious disease diagnosis....
A simple method suitable to study de novo root organogenesis
Xiaodong eChen
Full Text Available De novo root organogenesis is the process in which adventitious roots regenerate from detached or wounded plant tissues or organs. In tissue culture, appropriate types and concentrations of plant hormones in the medium are critical for inducing adventitious roots. However, in natural conditions, regeneration from detached organs is likely to rely on endogenous hormones. To investigate the actions of endogenous hormones and the molecular mechanisms guiding de novo root organogenesis, we developed a simple method to imitate natural conditions for adventitious root formation by culturing Arabidopsis thaliana leaf explants on B5 medium without additive hormones. Here we show that the ability of the leaf explants to regenerate roots depends on the age of the leaf and on certain nutrients in the medium. Based on these observations, we provide examples of how this method can be used in different situations, and how it can be optimized. This simple method could be used to investigate the effects of various physiological and molecular changes on the regeneration of adventitious roots. It is also useful for tracing cell lineage during the regeneration process by differential interference contrast observation of �-glucuronidase staining, and by live imaging of proteins labeled with fluorescent tags.
The PMDB Protein Model Database
Castrignanò, Tiziana; De Meo, Paolo D'Onorio; Cozzetto, Domenico; Talamo, Ivano Giuseppe; Tramontano, Anna
The Protein Model Database (PMDB) is a public resource aimed at storing manually built 3D models of proteins. The database is designed to provide access to models published in the scientific literature, together with validating experimental data. It is a relational database and it currently contains >74 000 models for ∼240 proteins. The system is accessible at and allows predictors to submit models along with related supporting evidence and users to download them through a simple and intuitive interface. Users can navigate in the database and retrieve models referring to the same target protein or to different regions of the same protein. Each model is assigned a unique identifier that allows interested users to directly access the data. PMID:16381873
Linearization of the Bradford Protein Assay
Ernst, Orna; Zor, Tsaffrir
Determination of microgram quantities of protein in the Bradford Coomassie brilliant blue assay is accomplished by measurement of absorbance at 590 nm. This most common assay enables rapid and simple protein quantification in cell lysates, cellular fractions, or recombinant protein samples, for the purpose of normalization of biochemical measurements. However, an intrinsic nonlinearity compromises the sensitivity and accuracy of this method. It is shown that under standard assay conditions, t...
Amino acid metabolism conflicts with protein diversity
Krick, Teresa; Shub, David A.; Verstraete, Nina; Ferreiro, Diego U.; Alonso, Leonardo G.; Shub, Michael; Sanchez, Ignacio E.
The 20 protein-coding amino acids are found in proteomes with different relative abundances. The most abundant amino acid, leucine, is nearly an order of magnitude more prevalent than the least abundant amino acid, cysteine. Amino acid metabolic costs differ similarly, constraining their incorporation into proteins. On the other hand, a diverse set of protein sequences is necessary to build functional proteomes. Here, we present a simple model for a cost-diversity trade-off postulating that n...
Novel Technology for Protein-Protein Interaction-based Targeted Drug Discovery
Jung Me Hwang
Full Text Available We have developed a simple but highly efficient in-cell protein-protein interaction (PPI discovery system based on the translocation properties of protein kinase C- and its C1a domain in live cells. This system allows the visual detection of trimeric and dimeric protein interactions including cytosolic, nuclear, and/or membrane proteins with their cognate ligands. In addition, this system can be used to identify pharmacological small compounds that inhibit specific PPIs. These properties make this PPI system an attractive tool for screening drug candidates and mapping the protein interactome.
Solution of a simple inelastic scattering problem
Knudson, S.K.
Simple examples of elastic scattering, typically from square wells, serve as important pedagogical tools in discussion of the concepts and processes involved in elastic scattering events. An analytic solution of a model inelastic scattering system is presented here to serve in this role for inelastic events. The model and its solution are simple enough to be of pedagogical utility, but also retain enough of the important physical features to include most of the special characteristics of inelastic systems. The specific model chosen is the collision of an atom with a harmonic oscillator, interacting via a repulsive square well potential. Pedagogically important features of inelastic scattering, including its multistate character, convergence behavior, and dependence on an ''inelastic potential'' are emphasized as the solution is determined. Results are presented for various energies and strengths of inelastic scattering, which show that the model is capable of providing an elementary representation of vibrationally inelastic scattering
Simple spherical ablative-implosion model
Mayer, F.J.; Steele, J.T.; Larsen, J.T.
A simple model of the ablative implosion of a high-aspect-ratio (shell radius to shell thickness ratio) spherical shell is described. The model is similar in spirit to Rosenbluth's snowplow model. The scaling of the implosion time was determined in terms of the ablation pressure and the shell parameters such as diameter, wall thickness, and shell density, and compared these to complete hydrodynamic code calculations. The energy transfer efficiency from ablation pressure to shell implosion kinetic energy was examined and found to be very efficient. It may be possible to attach a simple heat-transport calculation to our implosion model to describe the laser-driven ablation-implosion process. The model may be useful for determining other energy driven (e.g., ion beam) implosion scaling
Characterization of simple wireless neurostimulators and sensors.
Gulick, Daniel W; Towe, Bruce C
A single diode with a wireless power source and electrodes can act as an implantable stimulator or sensor. We have built such devices using RF and ultrasound power coupling. These simple devices could drastically reduce the size, weight, and cost of implants for applications where efficiency is not critical. However, a shortcoming has been a lack of control: any movement of the external power source would change the power coupling, thereby changing the stimulation current or modulating the sensor response. To correct for changes in power and signal coupling, we propose to use harmonic signals from the device. The diode acts as a frequency multiplier, and the harmonics it emits contain information about the drive level and bias. A simplified model suggests that estimation of power, electrode bias, and electrode resistance is possible from information contained in radiated harmonics even in the presence of significant noise. We also built a simple RF-powered stimulator with an onboard voltage limiter.
A Simple Technique of Supine Craniospinal Irradiation
Munshi, Anusheel; Jalali, Rakesh
We describe a simple procedure of craniospinal irradiation in supine position. The procedure was carried out with a 100-cm isocenter linear accelerator and compatible simulator. Treatment was with a 1 or 2 posteroanterior (PA)-directed spinal fields abutting lateral-directed cranial fields. Abutment of the fields was established by placement of markers on the neck of the patient, which provided a measure of the divergence of the spinal field. The precision and reproducibility of this technique, including the placement of junctions, appeared to be as good as for treatment in the prone position. The same could be verified with port films. We conclude that this new technique of supine craniospinal treatment is a simple and convenient alternative to traditional treatment in the prone position
A computational description of simple mediation analysis
Caron, Pier-Olivier
Full Text Available Simple mediation analysis is an increasingly popular statistical analysis in psychology and in other social sciences. However, there is very few detailed account of the computations within the model. Articles are more often focusing on explaining mediation analysis conceptually rather than mathematically. Thus, the purpose of the current paper is to introduce the computational modelling within simple mediation analysis accompanied with examples with R. Firstly, mediation analysis will be described. Then, the method to simulate data in R (with standardized coefficients will be presented. Finally, the bootstrap method, the Sobel test and the Baron and Kenny test all used to evaluate mediation (i.e., indirect effect will be developed. The R code to implement the computation presented is offered as well as a script to carry a power analysis and a complete example.
Is the world simple or complicated
Barrow, John D
Stop some particle physicists in the street and they will soon be trying to persuade you that the world is altogether simple and symmetrical. But stop a biologist, an economist, or a social scientist and they will tell you quite the opposite: the world is a higgledy-piggledy collection of complexities that owes little to symmetry and displays precious little simplicity. So who is right : is the world really complicated or is it simple ? We shall look at the reasoning that leads to these different conclusions, show why we got different answers to our question, and look at some of the recent developments that have taken place in the study of systems from sand-piles to music on the border between order and chaos. We shall also look at some of the connections between our aesthetic sensibilities and the structure of scientific theories.
Simple educational tool for digital speckle shearography
Schirripa Spagnolo, Giuseppe; Martocchia, Andrea; Papalillo, Donato; Cozzella, Lorenzo
In this study, an educational tool has been prepared for obtaining short-term and more economic training on digital speckle shearography (DSS). Shearography non-destructive testing (NDT) has gained wide acceptance over the last decade, providing a number of important and exciting inspection solutions in aerospace, electronics and medical device manufacturing. For exploring these motivations, it is important to develop didactic tools to understand the potential of digital shearography through training and didactic courses in the field of NDT. In this paper we describe a simple tool for making one familiar with the potential of DSS in the area of education and training. The system is realized with a simple and economic optical setup and a virtual instrument based on the LabVIEWâ"¢ and DAQ. (paper)
Estimation of the simple correlation coefficient.
Shieh, Gwowen
This article investigates some unfamiliar properties of the Pearson product-moment correlation coefficient for the estimation of simple correlation coefficient. Although Pearson's r is biased, except for limited situations, and the minimum variance unbiased estimator has been proposed in the literature, researchers routinely employ the sample correlation coefficient in their practical applications, because of its simplicity and popularity. In order to support such practice, this study examines the mean squared errors of r and several prominent formulas. The results reveal specific situations in which the sample correlation coefficient performs better than the unbiased and nearly unbiased estimators, facilitating recommendation of r as an effect size index for the strength of linear association between two variables. In addition, related issues of estimating the squared simple correlation coefficient are also considered.
Rack Protection Monitor - A Simple System
Orr, S.
The Rack Protection Monitor is a simple, fail-safe device to monitor smoke, temperature and ventilation sensors. It accepts inputs from redundant sensors and has a hardwired algorithm to prevent nuisance power trips due to random sensor failures. When a sensor is triggered the Rack Protection Monitor latches and annunicates the alarm. If another sensor is triggered, the Rack Protection Monitor locally shuts down the power to the relay rack and sends alarm to central control
Diffraction enhanced imaging: a simple model
Zhu Peiping; Yuan Qingxi; Huang Wanxia; Wang Junyue; Shu Hang; Chen Bo; Liu Yijin; Li Enrong; Wu Ziyu
Based on pinhole imaging and conventional x-ray projection imaging, a more general DEI (diffraction enhanced imaging) equation is derived using simple concepts in this paper. Not only can the new DEI equation explain all the same problems as with the DEI equation proposed by Chapman, but also some problems that cannot be explained with the old DEI equation, such as the noise background caused by small angle scattering diffracted by the analyser
Zhu Peiping; Yuan Qingxi; Huang Wanxia; Wang Junyue; Shu Hang; Chen Bo; Liu Yijin; Li Enrong; Wu Ziyu [Beijing Synchrotron Radiation Facility, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049 (China)
Based on pinhole imaging and conventional x-ray projection imaging, a more general DEI (diffraction enhanced imaging) equation is derived using simple concepts in this paper. Not only can the new DEI equation explain all the same problems as with the DEI equation proposed by Chapman, but also some problems that cannot be explained with the old DEI equation, such as the noise background caused by small angle scattering diffracted by the analyser.
Which finite simple groups are unit groups?
Davis, Christopher James; Occhipinti, Tommy
We prove that if G is a finite simple group which is the unit group of a ring, then G is isomorphic to either (a) a cyclic group of order 2; (b) a cyclic group of prime order 2^k −1 for some k; or (c) a projective special linear group PSLn(F2) for some n ≥ 3. Moreover, these groups do all occur a...
Differential calculus on quantized simple Lie groups
Jurco, B.
Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU q (2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q ε R are also discussed. (orig.)
A Simple Experiment for Visualizing Diffusion
Helseth, L. E.
We propose a simple and fascinating experiment for studying diffusion in gels using a pH-sensitive dye. By doping agar with methyl red, we obtain a gel which rapidly reacts to changes in pH by changing its absorption spectrum. The pH gradients can be followed using a digital camera, and we demonstrate here that the pH-sensitive colour changes can…
APPLYING SIMPLE TECHNOLOGY ACCOMPLISHES VISUAL INSPECTION CHALLENGES
Robinson, C
This paper discusses the successful implementation of simple video technologies at the Savannah River Site (SRS) to perform complex visual inspection, monitoring, and surveillance tasks. Because SRS facilities are similar to those of an industrial plant, the environmental and accessibility considerations for remote viewing are the primary determining factors in the selection of technology. The constraints and challenges associated with remote viewing are discussed, and examples of applications are given
Entropy estimates for simple random fields
Forchhammer, Søren; Justesen, Jørn
We consider the problem of determining the maximum entropy of a discrete random field on a lattice subject to certain local constraints on symbol configurations. The results are expected to be of interest in the analysis of digitized images and two dimensional codes. We shall present some examples...... of binary and ternary fields with simple constraints. Exact results on the entropies are known only in a few cases, but we shall present close bounds and estimates that are computationally efficient...
Histopathology slide projector: a simple improvisation.
Agarwal, Akhilesh K R; Bhattacharya, Nirjhar
The ability to examine histopathology and other hematological slides under microscope is a necessary and important service which should be available in every health facility. The slides need to be projected on to a screen. We describe an inexpensive and easily constructed technique for projecting magnified images of slides using a simple microscope. It is effective both for making observations and for use as a teaching aid.
Some simple demonstration experiments involving homopolar motors
Stewart, Seán M.
The ready availability of very strong permanent magnets in the form of rare-earth magnetic alloys such as neodymium-iron-boron has lead to renewed interest in one of the oldest types of electric motors - the homopolar motor. The ease with which a demonstration homopolar motor can now be built and operated when neodymium magnets are used is quite remarkable. In this paper some simple homopolar motors employing neodymium magnets suitable for demonstrational purposes are described and discussed....
Stewart,Seán M.
The ready availability of very strong permanent magnets in the form of rare-earth magnetic alloys such as neodymium-iron-boron has lead to renewed interest in one of the oldest types of electric motors - the homopolar motor. The ease with which a demonstration homopolar motor can now be built and operated when neodymium magnets are used is quite remarkable. In this paper some simple homopolar motors employing neodymium magnets suitable for demonstrational purposes are described and discussed.
A simple neutron-gamma discriminating system
Liu Zhongming; Xing Shilin; Wang Zhongmin
A simple neutron-gamma discriminating system is described. A detector and a pulse shape discriminator are suitable for the neutron-gamma discriminating system. The influence of the constant fraction discriminator threshold energy on the neutron-gamma resolution properties is shown. The neutron-gamma timing distributions from an 241 Am-Be source, 2.5 MeV neutron beam and 14 MeV neutron beam are presented
Implementing successful strategic plans: a simple formula.
Blondeau, Whitney; Blondeau, Benoit
Strategic planning is a process. One way to think of strategic planning is to envision its development and design as a framework that will help your hospital navigate through internal and external changing environments over time. Although the process of strategic planning can feel daunting, following a simple formula involving five steps using the mnemonic B.E.G.I.N. (Begin, Evaluate, Goals & Objectives, Integration, and Next steps) will help the planning process feel more manageable, and lead you to greater success.
Locally Simple Models Construction: Methodology and Practice
I. A. Kazakov
Full Text Available One of the most notable trends associated with the Fourth industrial revolution is a significant strengthening of the role played by semantic methods. They are engaged in artificial intelligence means, knowledge mining in huge flows of big data, robotization, and in the internet of things. Smart contracts also can be mentioned here, although the 'intelligence' of smart contracts still needs to be seriously elaborated. These trends should inevitably lead to an increased role of logical methods working with semantics, and significantly expand the scope of their application in practice. However, there are a number of problems that hinder this process. We are developing an approach, which makes the application of logical modeling efficient in some important areas. The approach is based on the concept of locally simple models and is primarily focused on solving tasks in the management of enterprises, organizations, governing bodies. The most important feature of locally simple models is their ability to replace software systems. Replacement of programming by modeling gives huge advantages, for instance, it dramatically reduces development and support costs. Modeling, unlike programming, preserves the explicit semantics of models allowing integration with artificial intelligence and robots. In addition, models are much more understandable to general people than programs. In this paper we propose the implementation of the concept of locally simple modeling on the basis of so-called document models, which has been developed by us earlier. It is shown that locally simple modeling is realized through document models with finite submodel coverages. In the second part of the paper an example of using document models for solving a management problem of real complexity is demonstrated.
Simple Parametric Model for Airfoil Shape Description
Ziemkiewicz, David
We show a simple, analytic equation describing a class of two-dimensional shapes well suited for representation of aircraft airfoil profiles. Our goal was to create a description characterized by a small number of parameters with easily understandable meaning, providing a tool to alter the shape with optimization procedures as well as manual tweaks by the designer. The generated shapes are well suited for numerical analysis with 2D flow solving software such as XFOIL.
Central Odontogenic Fibroma of Simple Type
Prasanth Thankappan
Full Text Available Central odontogenic fibroma (COF is an extremely rare benign tumor that accounts for 0.1% of all odontogenic tumors. It is a lesion associated with the crown of an unerupted tooth resembling dentigerous cyst. In this report, a 10-year-old male patient is presented, who was diagnosed with central odontogenic fibroma of simple type from clinical, radiological, and histopathological findings.
Multiphase flow in geometrically simple fracture intersections
Basagaoglu, H.; Meakin, P.; Green, C.T.; Mathew, M.; ,
A two-dimensional lattice Boltzmann (LB) model with fluid-fluid and solid-fluid interaction potentials was used to study gravity-driven flow in geometrically simple fracture intersections. Simulated scenarios included fluid dripping from a fracture aperture, two-phase flow through intersecting fractures and thin-film flow on smooth and undulating solid surfaces. Qualitative comparisons with recently published experimental findings indicate that for these scenarios the LB model captured the underlying physics reasonably well.
Interfaces for a simple local network
Nekhanevich, Eh.L.; Yasenev, M.V.
A system of communication and interfaces for a simple local network of computers is described. The data on technical parameters, fields of application and operation features of the interfaces developed are presented. The data indispensable for the development of software are given. The experience in operation of the subsystem of software for remote terminal computers which makes use of the above interfaces is briefly presented. 7 refs.; 3 figs
A discerning approach to simple aesthetic orthodontics.
Noar, J H; Sharma, S; Roberts-Harry, D; Qureshi, T
There is currently considerable interest from general dental practitioners (GDPs) in the use of simple orthodontics to treat adult malocclusions. There is controversy in this, particularly in relation to 'quick fixes', simple orthodontics and 'straight teeth in six months' as opposed to more conventional treatment where the whole malocclusion is treated. This article will present a case for the use of simple aesthetic adult orthodontics in a measured and planned way. It will discuss the processes, planning and the importance of consent. It will also highlight how digital technology is used to preview, consent and execute an aesthetic result. Many of the recent systems emerging, have been as a result of the demand and supply of cosmetic dentistry. This, to a degree, has not helped since the implication of a 'quick-fix' is associated with this field. There has also been discussion on what the limits of GDP orthodontics should be. There is variability in how GDPs approach orthodontics, their experience, skill and ability to treat to an acceptable standard. Short courses may be one way of delivering orthodontic training but some of these courses are not regulated and the amount of internal mentoring is variable. This article highlights some of the systems in use, and potential upsides and downsides of this approach.
Necessity of Antibiotics following Simple Exodontia
Waqas Yousuf
Full Text Available Introduction. The aim of our study is to assess the need for postoperative antibiotics following simple exodontia and determine its role in minimizing patient discomfort and postoperative complications. Material and Methods. All the patients undergoing simple extractions were grouped into two categories: Group 1, patients receiving antibiotics, and Group 2, patients receiving no antibiotics. Patients were recalled on the sixth day to assess postoperative complications. On recall, patients were evaluated for signs of persistent inflammation and signs of dry socket. Presence of persistent inflammation and/or suppuration on the 6th day was considered as wound infection. Results. A total of 146 patients were included in this study. Out of the total sample, 134 (91.8% presented with no postoperative complications and 12 (8.2% had postoperative complications, out of which 11 (7.5% patients presented with dry socket (alveolar osteitis, 5 (3.4% in the antibiotic group and 6 (4.1% in the nonantibiotic group. Only 1 patient (0.7% was reported with infection of the extraction socket in the nonantibiotic group, whereas no case of infection was found in the antibiotic group. Conclusion. Antibiotics are not required after simple extractions in patients who are not medically comprised nor do they have any role in preventing postoperative complications.
Optimized theory for simple and molecular fluids.
Marucho, M; Montgomery Pettitt, B
An optimized closure approximation for both simple and molecular fluids is presented. A smooth interpolation between Perkus-Yevick and hypernetted chain closures is optimized by minimizing the free energy self-consistently with respect to the interpolation parameter(s). The molecular version is derived from a refinement of the method for simple fluids. In doing so, a method is proposed which appropriately couples an optimized closure with the variant of the diagrammatically proper integral equation recently introduced by this laboratory [K. M. Dyer et al., J. Chem. Phys. 123, 204512 (2005)]. The simplicity of the expressions involved in this proposed theory has allowed the authors to obtain an analytic expression for the approximate excess chemical potential. This is shown to be an efficient tool to estimate, from first principles, the numerical value of the interpolation parameters defining the aforementioned closure. As a preliminary test, representative models for simple fluids and homonuclear diatomic Lennard-Jones fluids were analyzed, obtaining site-site correlation functions in excellent agreement with simulation data.
Modeling reproductive decisions with simple heuristics
Full Text Available BACKGROUND Many of the reproductive decisions that humans make happen without much planning or forethought, arising instead through the use of simple choice rules or heuristics that involve relatively little information and processing. Nonetheless, these heuristic-guided decisions are typically beneficial, owing to humans' ecological rationality - the evolved fit between our constrained decision mechanisms and the adaptive problems we face. OBJECTIVE This paper reviews research on the ecological rationality of human decision making in the domain of reproduction, showing how fertility-related decisions are commonly made using various simple heuristics matched to the structure of the environment in which they are applied, rather than being made with information-hungry mechanisms based on optimization or rational economic choice. METHODS First, heuristics for sequential mate search are covered; these heuristics determine when to stop the process of mate search by deciding that a good-enough mate who is also mutually interested has been found, using a process of aspiration-level setting and assessing. These models are tested via computer simulation and comparison to demographic age-at-first-marriage data. Next, a heuristic process of feature-based mate comparison and choice is discussed, in which mate choices are determined by a simple process of feature-matching with relaxing standards over time. Parental investment heuristics used to divide resources among offspring are summarized. Finally, methods for testing the use of such mate choice heuristics in a specific population over time are then described.
The Monash University Interactive Simple Climate Model
Dommenget, D.
The Monash university interactive simple climate model is a web-based interface that allows students and the general public to explore the physical simulation of the climate system with a real global climate model. It is based on the Globally Resolved Energy Balance (GREB) model, which is a climate model published by Dommenget and Floeter [2011] in the international peer review science journal Climate Dynamics. The model simulates most of the main physical processes in the climate system in a very simplistic way and therefore allows very fast and simple climate model simulations on a normal PC computer. Despite its simplicity the model simulates the climate response to external forcings, such as doubling of the CO2 concentrations very realistically (similar to state of the art climate models). The Monash simple climate model web-interface allows you to study the results of more than a 2000 different model experiments in an interactive way and it allows you to study a number of tutorials on the interactions of physical processes in the climate system and solve some puzzles. By switching OFF/ON physical processes you can deconstruct the climate and learn how all the different processes interact to generate the observed climate and how the processes interact to generate the IPCC predicted climate change for anthropogenic CO2 increase. The presentation will illustrate how this web-base tool works and what are the possibilities in teaching students with this tool are.
Simple map in action-angle coordinates
Kerwin, Olivia; Punjabi, Alkesh; Ali, Halima
A simple map [A. Punjabi, A. Verma, and A. Boozer, Phys. Rev. Lett. 69, 3322 (1992)] is the simplest map that has the topology of divertor tokamaks [A. Punjabi, H. Ali, T. Evans, and A. Boozer, Phys. Lett. A 364, 140 (2007)]. Here, action-angle coordinates, the safety factor, and the equilibrium generating function for the simple map are calculated analytically. The simple map in action-angle coordinates is derived from canonical transformations. This map cannot be integrated across the separatrix surface because of the singularity in the safety factor there. The stochastic broadening of the ideal separatrix surface in action-angle representation is calculated by adding a perturbation to the simple map equilibrium generating function. This perturbation represents the spatial noise and field errors typical of the DIII-D [J. L. Luxon and L. E. Davis, Fusion Technol. 8, 441 (1985)] tokamak. The stationary Fourier modes of the perturbation have poloidal and toroidal mode numbers (m,n,)={(3,1),(4,1),(6,2),(7,2),(8,2),(9,3),(10,3),(11,3)} with amplitude δ =0.8×10-5. Near the X-point, about 0.12% of toroidal magnetic flux inside the separatrix, and about 0.06% of the poloidal flux inside the separatrix is lost. When the distance from the O-point to the X-point is 1m, the width of stochastic layer near the X-point is about 1.4cm. The average value of the action on the last good surface is 0.19072 compared to the action value of 3/5π on the separatrix. The average width of stochastic layer in action coordinate is 2.7×10-4, while the average area of the stochastic layer in action-angle phase space is 1.69017×10-3. On average, about 0.14% of action or toroidal flux inside the ideal separatrix is lost due to broadening. Roughly five times more toroidal flux is lost in the simple map than in DIII-D for the same perturbation [A. Punjabi, H. Ali, A. Boozer, and T. Evans, Bull. Amer. Phys. Soc. 52, 124 (2007)].
A simple map [A. Punjabi, A. Verma, and A. Boozer, Phys. Rev. Lett. 69, 3322 (1992)] is the simplest map that has the topology of divertor tokamaks [A. Punjabi, H. Ali, T. Evans, and A. Boozer, Phys. Lett. A 364, 140 (2007)]. Here, action-angle coordinates, the safety factor, and the equilibrium generating function for the simple map are calculated analytically. The simple map in action-angle coordinates is derived from canonical transformations. This map cannot be integrated across the separatrix surface because of the singularity in the safety factor there. The stochastic broadening of the ideal separatrix surface in action-angle representation is calculated by adding a perturbation to the simple map equilibrium generating function. This perturbation represents the spatial noise and field errors typical of the DIII-D [J. L. Luxon and L. E. Davis, Fusion Technol. 8, 441 (1985)] tokamak. The stationary Fourier modes of the perturbation have poloidal and toroidal mode numbers (m,n,)=((3,1),(4,1),(6,2),(7,2),(8,2),(9,3),(10,3),(11,3)) with amplitude δ=0.8x10 -5 . Near the X-point, about 0.12% of toroidal magnetic flux inside the separatrix, and about 0.06% of the poloidal flux inside the separatrix is lost. When the distance from the O-point to the X-point is 1 m, the width of stochastic layer near the X-point is about 1.4 cm. The average value of the action on the last good surface is 0.19072 compared to the action value of 3/5π on the separatrix. The average width of stochastic layer in action coordinate is 2.7x10 -4 , while the average area of the stochastic layer in action-angle phase space is 1.69017x10 -3 . On average, about 0.14% of action or toroidal flux inside the ideal separatrix is lost due to broadening. Roughly five times more toroidal flux is lost in the simple map than in DIII-D for the same perturbation [A. Punjabi, H. Ali, A. Boozer, and T. Evans, Bull. Amer. Phys. Soc. 52, 124 (2007)].
Simple Ligand-Receptor Interaction Descriptor (SILIRID) for alignment-free binding site comparison.
Chupakhin, Vladimir; Marcou, Gilles; Gaspar, Helena; Varnek, Alexandre
We describe SILIRID (Simple Ligand-Receptor Interaction Descriptor), a novel fixed size descriptor characterizing protein-ligand interactions. SILIRID can be obtained from the binary interaction fingerprints (IFPs) by summing up the bits corresponding to identical amino acids. This results in a vector of 168 integer numbers corresponding to the product of the number of entries (20 amino acids and one cofactor) and 8 interaction types per amino acid (hydrophobic, aromatic face to face, aromatic edge to face, H-bond donated by the protein, H-bond donated by the ligand, ionic bond with protein cation and protein anion, and interaction with metal ion). Efficiency of SILIRID to distinguish different protein binding sites has been examined in similarity search in sc-PDB database, a druggable portion of the Protein Data Bank, using various protein-ligand complexes as queries. The performance of retrieval of structurally and evolutionary related classes of proteins was comparable to that of state-of-the-art approaches (ROC AUC ≈ 0.91). SILIRID can efficiently be used to visualize chemogenomic space covered by sc-PDB using Generative Topographic Mapping (GTM): sc-PDB SILIRID data form clusters corresponding to different protein types.
Solution of the schrodinger equation in one dimension by simple method for a simple step potential
Ertik, H.
The coefficients of the transmission and reflection for the simple-step barrier potential were calculated by a simple method. Their values were entirely different from those often encountered in the literature. Especially in the case that the total energy is equal to the barrier potential, the value of 0,20 for the reflection coefficient was obtained whereas this is zero in the literature. This may be considered as an interesting point
Differential intrahepatic phospholipid zonation in simple steatosis and nonalcoholic steatohepatitis.
Julia Wattacheril
Full Text Available Nonalcoholic fatty liver disease (NAFLD occurs frequently in a setting of obesity, dyslipidemia and insulin resistance, but the etiology of the disease, particularly the events favoring progression to nonalcoholic steatohepatitis (NASH as opposed to simple steatosis (SS, are not fully understood. Based on known zonation patterns in protein, glucose and lipid metabolism, coupled with evidence that phosphatidylcholine may play a role in NASH pathogenesis, we hypothesized that phospholipid zonation exists in liver and that specific phospholipid abundance and distribution may be associated with histologic disease. A survey of normal hepatic protein expression profiles in the Human Protein Atlas revealed pronounced zonation of enzymes involved in lipid utilization and storage, particularly those facilitating phosphatidylcholine (PC metabolism. Immunohistochemistry of obese normal, SS and NASH liver specimens with anti-phosphatidylethanomine N-methyltransferase (PEMT antibodies showed a progressive decrease in the zonal distribution of this PC biosynthetic enzyme. Phospholipid quantitation by liquid chromatography mass spectrometry (LC-MS in hepatic extracts of Class III obese patients with increasing NAFLD severity revealed that most PC species with 32, 34 and 36 carbons as well as total PC abundance was decreased with SS and NASH. Matrix assisted laser desorption ionization-imaging mass spectrometry (MALDI-IMS imaging revealed strong zonal distributions for 32, 34 and 36 carbon PCs in controls (minimal histologic findings and SS that was lost in NASH specimens. Specific lipid species such as PC 34:1 and PC 36:2 best illustrated this phenomenon. These findings suggest that phospholipid zonation may be associated with the presence of an intrahepatic proinflammatory phenotype and thus have broad implications in the etiopathogenesis of NASH.
Preparation and properties of monoclonal antibodies to individual prekeratins of simple rat epithelium
Troyanovskii, S.M.; Krutovskikh, V.A.; Bannikov, G.A.
The authors study the properties of a series of hybridoma clones producing antibodies to individual prekeratins (PK) from simple types of epithelium. BALB/c mice were immunized with a preparation of intermediate filaments isolated from the mucosa of the rat large intestine. The specificity of the five clones studied was studied by monoautoradiography. For a more detailed study of the specificity of the experimentally obtained antibodies, the authors used the same immunoautoradiographic method to study their reaction with proteins of cells of other types. The authors have obtained monoclonal antibodies to three individual PK of simple types of rat epithelium: PK40, PK49, and PK55
Design, properties, and applications of protein micro- and nanoparticles
Saglam, Dilek; Venema, Paul; van der Linden, Erik; de Vries, Renko
The design of protein particles with tailored properties has received an increased attention recently. Several approaches, from simple heat treatment in dilute systems to the combination of heat and mechanical treatments in concentrated protein solutions, have been used to obtain protein particles
THE SIMPLE SURVEY: OBSERVATIONS, REDUCTION, AND CATALOG
Damen, M.; Franx, M.; Taylor, E. N.; Labbe, I.; Van Dokkum, P. G.; Muzzin, A.; Brandt, W. N.; Dickinson, M.; Gawiser, E.; Illingworth, G. D.; Kriek, M.; Marchesini, D.; Papovich, C.; Rix, H.-W.
We present the Spitzer IRAC/MUSYC Public Legacy Survey in the Extended CDF-South (SIMPLE), which consists of deep IRAC observations covering the ∼1600 arcmin 2 area surrounding GOODS-S. The limiting magnitudes of the SIMPLE IRAC mosaics typically are 23.8, 23.6, 21.9, and 21.7, at 3.6 μm, 4.5 μm, 5.8 μm, and 8.0 μm, respectively (5σ total point source magnitudes in AB). The SIMPLE IRAC images are combined with the 10' x 15' GOODS IRAC mosaics in the center. We give detailed descriptions of the observations, data reduction, and properties of the final images, as well as the detection and photometry methods used to build a catalog. Using published optical and near-infrared data from the Multiwavelength Survey by Yale-Chile (MUSYC), we construct an IRAC-selected catalog, containing photometry in UBVRIz'JHK, [3.6 μm], [4.5 μm], [5.8 μm], and [8.0 μm]. The catalog contains 43,782 sources with S/N >5 at 3.6 μm, 19,993 of which have 13-band photometry. We compare this catalog to the publicly available MUSYC and FIREWORKS catalogs and discuss the differences. Using a high signal-to-noise sub-sample of 3391 sources with ([3.6] + [4.5])/2 * >10 11 M sun ) are passively evolving, in agreement with earlier results from surveys covering less area.
Simple, inexpensive computerized rodent activity meters.
Horton, R M; Karachunski, P I; Kellermann, S A; Conti-Fine, B M
We describe two approaches for using obsolescent computers, either an IBM PC XT or an Apple Macintosh Plus, to accurately quantify spontaneous rodent activity, as revealed by continuous monitoring of the spontaneous usage of running activity wheels. Because such computers can commonly be obtained at little or no expense, and other commonly available materials and inexpensive parts can be used, these meters can be built quite economically. Construction of these meters requires no specialized electronics expertise, and their software requirements are simple. The computer interfaces are potentially of general interest, as they could also be used for monitoring a variety of events in a research setting.
Simple ectopic kidney in three dogs.
Choi, Jiyoung; Lee, Heechun; Lee, Youngwon; Choi, Hojung
Simple ectopic kidney was diagnosed in three dogs by means of radiography and ultrasonography. A 2-year-old castrated male Schnauzer, a 13-year-old female Schnauzer and a 9-year-old male Jindo were referred with vomiting, hematuria and ocular discharge, respectively. In all three dogs, oval-shaped masses with soft tissue density were observed in the mid to caudal abdomen bilaterally or unilaterally, and kidney silhouettes were not identified at the proper anatomic places on abdominal radiographs. Ultrasonography confirmed the masses were malpositioned kidney. The ectopic kidneys had relatively small size, irregular shape and short ureter but showed normal function on excretory urography.
A 'simple' hybrid model for power derivatives
Lyle, Matthew R.; Elliott, Robert J.
This paper presents a method for valuing power derivatives using a supply-demand approach. Our method extends work in the field by incorporating randomness into the base load portion of the supply stack function and equating it with a noisy demand process. We obtain closed form solutions for European option prices written on average spot prices considering two different supply models: a mean-reverting model and a Markov chain model. The results are extensions of the classic Black-Scholes equation. The model provides a relatively simple approach to describe the complicated price behaviour observed in electricity spot markets and also allows for computationally efficient derivatives pricing. (author)
Selective laser photolysis of simple molecules
Golnabi, Hossein.
A two-photon technique is reported for the measurement of relative cross section for the photolysis of simple molecules into particular product channels. In this method two independently tunable dye lasers were used to sequentially dissociate molecules of Cs 2 and Cs-Kr for the wavelengths in the range 420 to 660 nm, and then to excite the resulting products to determine the relative cross sections for the photolysis of Cs 2 and Cs-kr into each of the lowest four of the energetically possible product states
Simple models of equilibrium and nonequilibrium phenomena
Lebowitz, J.L.
This volume consists of two chapters of particular interest to researchers in the field of statistical mechanics. The first chapter is based on the premise that the best way to understand the qualitative properties that characterize many-body (i.e. macroscopic) systems is to study 'a number of the more significant model systems which, at least in principle are susceptible of complete analysis'. The second chapter deals exclusively with nonequilibrium phenomena. It reviews the theory of fluctuations in open systems to which they have made important contributions. Simple but interesting model examples are emphasised
Communication: Simple liquids' high-density viscosity
Costigliola, Lorenzo; Pedersen, Ulf R.; Heyes, David M.; Schrøder, Thomas B.; Dyre, Jeppe C.
This paper argues that the viscosity of simple fluids at densities above that of the triple point is a specific function of temperature relative to the freezing temperature at the density in question. The proposed viscosity expression, which is arrived at in part by reference to the isomorph theory of systems with hidden scale invariance, describes computer simulations of the Lennard-Jones system as well as argon and methane experimental data and simulation results for an effective-pair-potential model of liquid sodium.
A simple fluxgate magnetometer using amorphous alloys
Ghatak, S.K.; Mitra, A.
A simple fluxgate magnetometer is developed using low magnetostrictive ferromagnetic amorphous alloy acting as a sensing element. It uses the fact that the magnetization of sensing element symmetrically magnetized by a sinusoidal field contains even harmonic components in presence of dc signal field H and the amplitude of the second harmonic component of magnetization is proportional to H. The sensitivity and linearity of the magnetometer with signal field are studied for parallel configuration and the field ranging from 10 nT to 10 μT can be measured. The functioning of the magnetometer is demonstrated by studying the shielding and flux-trapping phenomena in high-Tc superconductor. (orig.)
Jurco, B. (Dept. of Optics, Palacky Univ., Olomouc (Czechoslovakia))
Differential calculi, generalizations of Woronowicz's four-dimensional calculus on SU{sub q}(2), are introduced for quantized classical simple Lie groups in a constructive way. For this purpose, the approach of Faddeev and his collaborators to quantum groups was used. An equivalence of Woronowicz's enveloping algebra generated by the dual space to the left-invariant differential forms and the corresponding quantized universal enveloping algebra, is obtained for our differential calculi. Real forms for q {epsilon} R are also discussed. (orig.).
Drift estimation from a simple field theory
Mendes, F. M.; Figueiredo, A.
Given the outcome of a Wiener process, what can be said about the drift and diffusion coefficients? If the process is stationary, these coefficients are related to the mean and variance of the position displacements distribution. However, if either drift or diffusion are time-dependent, very little can be said unless some assumption about that dependency is made. In Bayesian statistics, this should be translated into some specific prior probability. We use Bayes rule to estimate these coefficients from a single trajectory. This defines a simple, and analytically tractable, field theory.
Simple, high current, antimony ion source
Sugiura, H.
A simple metal ion source capable of producing a continuous, uncontaminated, high current beam of Sb ions is presented. It produced a total ion current of 200 μA at 1 kV extraction voltage. A discharge occurred in the source at a pressure of 6 x 10 -4 Torr. The ion current extracted from the source increased with the 3/2 power of the extraction voltage. The perveance of the source and ion density in the plasma were 8 x 10 -9 and 1.8 x 10 11 cm -3 , respectively
Construction of simple quartz crystal microbalance
Ristov, Milcho [Center of Energy, Informatics and Materials of the Macedonian Academy fo Science and Arts, Skopje (Macedonia, The Former Yugoslav Republic of); Mitrevski, Mitre [Institute of Physics, Faculty of natural Science and Mathematics, Ss Cyril and Methodius University, Skopje (Macedonia, The Former Yugoslav Republic of)
A very simple Quartz Crystal Microbalance (QCM) was constructed for the measurement of thickness of chemically deposited thin films. QCM consisted of two active elements: one dual-gate MOSFET and one bipolar transistor, and as usually two AT-cut quartz crystal. The beat frequency oscillation generated as a result of loading of the sensor crystal by the deposited thin film, was measured by HP-multimeter, set as a low frequency meter. The sensitivity was found to be high and satisfactory for the study of growth rate of thin films, mainly deposited by methods of chemical deposition.
Ristov, Milcho; Mitrevski, Mitre
Design of Simple Landslide Monitoring System
Meng, Qingjia; Cai, Lingling
The simple landslide monitoring system is mainly designed for slope, collapse body and surface crack. In the harsh environment, the dynamic displacement data of the disaster body is transmitted to the terminal acquisition system in real time. The main body of the system adopt is PIC32MX795F512. This chip is to realize low power design, wakes the system up through the clock chip, and turns on the switching power supply at set time, which makes the wireless transmission module running during the interval to ensure the maximum battery consumption, so that the system can be stable long term work.
Simple Synthesis Method for Alumina Nanoparticle
Daniel Damian
Full Text Available Globally, the human population steady increase, expansion of urban areas, excessive industrialization including in agriculture, caused not only decrease to depletion of non-renewable resources, a rapid deterioration of the environment with negative impact on water quality, soil productivity and of course quality of life in general. This paper aims to prepare size controlled nanoparticles of aluminum oxide using a simple synthesis method. The morphology and dimensions of nanomaterial was investigated using modern analytical techniques: SEM/EDAX and XRD spectroscopy.
A simple statistical model for geomagnetic reversals
Constable, Catherine
The diversity of paleomagnetic records of geomagnetic reversals now available indicate that the field configuration during transitions cannot be adequately described by simple zonal or standing field models. A new model described here is based on statistical properties inferred from the present field and is capable of simulating field transitions like those observed. Some insight is obtained into what one can hope to learn from paleomagnetic records. In particular, it is crucial that the effects of smoothing in the remanence acquisition process be separated from true geomagnetic field behavior. This might enable us to determine the time constants associated with the dominant field configuration during a reversal.
A Simple Measure of Price Adjustment Coefficients.
Damodaran, Aswath
One measure of market efficiency is the speed with which prices adjust to new information. The author develops a simple approach to estimating these price adjustment coefficients by using the information in return processes. This approach is used to estimate t he price adjustment coefficients for firms listed on the NYSE and the A MEX as well as for over-the-counter stocks. The author finds evidence of a lagged adjustment to new information in shorter return intervals for firms in all market ...
Communication: Simple liquids' high-density viscosity.
Costigliola, Lorenzo; Pedersen, Ulf R; Heyes, David M; Schrøder, Thomas B; Dyre, Jeppe C
Simple multifunction discriminator for multichannel triggers
Maier, M.R.
A simple version of a multifunction timing discriminator using only two integrated circuits is presented. It can be configured as a leading edge, a constant fraction, a zero cross or a dual threshold timing discriminator. Since so few parts are used, it is well suited for building multichannel timing discriminators. Two versions of this circuit are described: a quadruple multifunction discriminator and an octal constant fraction trigger. The different compromises made in these units are discussed. Results for walk and jitter obtained with these are presented and possible improvements are disussed
A simple proposal for Rayleigh's scaterring experiment
Adriano José Ortiz
Full Text Available This work presents an alternative proposal for Rayleigh's scattering experiment presented and discussed in Krapas and Santos (2002 in this journal. Besides being simple and low-cost, the proposal suggested here is also proposing to demonstrate experimentally other physical phenomena such as polarization of light from the sky, the rainbow and reflection on non-conductive surfaces, as well as determine the direction of these biases. The polarization will be observed with the aid of Polaroid obtained from liquid crystal displays taken from damaged electronic devices and the Polaroid polarization direction will be established by the observation of Brewester's angle in reflection experiment.
Chaos from simple models to complex systems
Cencini, Massimo; Vulpiani, Angelo
Chaos: from simple models to complex systems aims to guide science and engineering students through chaos and nonlinear dynamics from classical examples to the most recent fields of research. The first part, intended for undergraduate and graduate students, is a gentle and self-contained introduction to the concepts and main tools for the characterization of deterministic chaotic systems, with emphasis to statistical approaches. The second part can be used as a reference by researchers as it focuses on more advanced topics including the characterization of chaos with tools of information theor
Simple method for calculating island widths
Cary, J.R.; Hanson, J.D.; Carreras, B.A.; Lynch, V.E.
A simple method for calculating magnetic island widths has been developed. This method uses only information obtained from integrating along the closed field line at the island center. Thus, this method is computationally less intensive than the usual method of producing surfaces of section of sufficient detail to locate and resolve the island separatrix. This method has been implemented numerically and used to analyze the buss work islands of ATF. In this case the method proves to be accurate to at least within 30%. 7 refs
Modelling simple helically delivered dose distributions
Fenwick, John D; Tome, Wolfgang A; Kissick, Michael W; Mackie, T Rock
In a previous paper, we described quality assurance procedures for Hi-Art helical tomotherapy machines. Here, we develop further some ideas discussed briefly in that paper. Simple helically generated dose distributions are modelled, and relationships between these dose distributions and underlying characteristics of Hi-Art treatment systems are elucidated. In particular, we describe the dependence of dose levels along the central axis of a cylinder aligned coaxially with a Hi-Art machine on fan beam width, couch velocity and helical delivery lengths. The impact on these dose levels of angular variations in gantry speed or output per linear accelerator pulse is also explored
Neutrosophic Correlation and Simple Linear Regression
A. A. Salama
Full Text Available Since the world is full of indeterminacy, the neutrosophics found their place into contemporary research. The fundamental concepts of neutrosophic set, introduced by Smarandache. Recently, Salama et al., introduced the concept of correlation coefficient of neutrosophic data. In this paper, we introduce and study the concepts of correlation and correlation coefficient of neutrosophic data in probability spaces and study some of their properties. Also, we introduce and study the neutrosophic simple linear regression model. Possible applications to data processing are touched upon.
Simple Unawareness in Dynamic Psychological Games
Nielsen, Carsten Søren; Sebald, Alexander Christopher
Building on Battigalli and Dufwenberg (2009)'s framework of dynamic psychological games and the progress in the modeling of dynamic unawareness by Heifetz, Meier, and Schipper (2013a) we model and analyze the impact of asymmetric awareness in the strategic interaction of players motivated...... by reciprocity and guilt. Specifically we characterize extensive-form games with psychological payoffs and simple unawareness, define extensive-form rationalizability and, using this, show that unawareness has a pervasive impact on the strategic interaction of psychologically motivated players. Intuitively...
Modularity in protein structures: study on all-alpha proteins.
Khan, Taushif; Ghosh, Indira
Modularity is known as one of the most important features of protein's robust and efficient design. The architecture and topology of proteins play a vital role by providing necessary robust scaffolds to support organism's growth and survival in constant evolutionary pressure. These complex biomolecules can be represented by several layers of modular architecture, but it is pivotal to understand and explore the smallest biologically relevant structural component. In the present study, we have developed a component-based method, using protein's secondary structures and their arrangements (i.e. patterns) in order to investigate its structural space. Our result on all-alpha protein shows that the known structural space is highly populated with limited set of structural patterns. We have also noticed that these frequently observed structural patterns are present as modules or "building blocks" in large proteins (i.e. higher secondary structure content). From structural descriptor analysis, observed patterns are found to be within similar deviation; however, frequent patterns are found to be distinctly occurring in diverse functions e.g. in enzymatic classes and reactions. In this study, we are introducing a simple approach to explore protein structural space using combinatorial- and graph-based geometry methods, which can be used to describe modularity in protein structures. Moreover, analysis indicates that protein function seems to be the driving force that shapes the known structure space.
Molecular polymorphism of a cell surface proteoglycan: distinct structures on simple and stratified epithelia.
Sanderson, R D; Bernfield, M
Epithelial cells are organized into either a single layer (simple epithelia) or multiple layers (stratified epithelia). Maintenance of these cellular organizations requires distinct adhesive mechanisms involving many cell surface molecules. One such molecule is a cell surface proteoglycan, named syndecan, that contains both heparan sulfate and chondroitin sulfate chains. This proteoglycan binds cells to fibrillar collagens and fibronectin and thus acts as a receptor for interstitial matrix. The proteoglycan is restricted to the basolateral surface of simple epithelial cells, but is located over the entire surface of stratified epithelial cells, even those surfaces not contacting matrix. We now show that the distinct localization in simple and stratified epithelia correlates with a distinct proteoglycan structure. The proteoglycan from simple epithelia (modal molecular size, 160 kDa) is larger than that from stratified epithelia (modal molecular size, 92 kDa), but their core proteins are identical in size and immunoreactivity. The proteoglycan from simple epithelia has more and larger heparan sulfate and chondroitin sulfate chains than the proteoglycan from stratified epithelia. Thus, the cell surface proteoglycan shows a tissue-specific structural polymorphism due to distinct posttranslational modifications. This polymorphism likely reflects distinct proteoglycan functions in simple and stratified epithelia, potentially meeting the different adhesive requirements of the cells in these different organizations.
Neutrophil-to-lymphocyte Ratio, Platelet-to-lymphocyte Ratio, and C-reactive Protein as New and Simple Prognostic Factors in Patients With Metastatic Renal Cell Cancer Treated With Tyrosine Kinase Inhibitors: AÂ Systemic Review and Meta-analysis.
Semeniuk-Wojtaś, Aleksandra; Lubas, Arkadiusz; Stec, Rafał; Syryło, Tomasz; Niemczyk, Stanisław; Szczylik, Cezary
Inflammation plays a crucial role in cancer development. In this study, we evaluate the prognostic values of systemic inflammation markers such as neutrophil-to-lymphocyte ratio (NLR), platelet-to-lymphocyte ratio (PLR), and C-reactive protein (CRP) for the progression-free survival and overall survival in patients with metastatic renal cell carcinoma treated with tyrosine kinase inhibitors. PubMed and the Cochrane Library databases were searched for published studies on the effect of NLR, PLR, and CRP in patients with metastatic renal cell carcinoma treated with tyrosine kinase inhibitors. In the meta-analysis, NLR (hazard ratio [HR], 2.01; 95% confidence interval [CI], 1.27-3.18; P = .003) and PLR (HR, 6.96; 95% CI, 5.04-9.62; P < .001) had a significant influence on progression-free survival, whereas all considered proinflammatory markers had a significant impact on overall survival: NLR (HR, 2.14; 95% CI, 1.67-2.73; P < .001), PLR (HR, 14.67; 95% CI, 11.10-19.57; P < .001), and CRP (HR, 1.96; 95% CI, 1.26-3.05; P = .003). Inflammation markers such as NLR, PLR, and CRP are predictors of clinical outcome and could provide additional information to individualize treatment. Copyright © 2018 The Author(s). Published by Elsevier Inc. All rights reserved.
Protein politics
Vijver, Marike
This study is part of the program of the interdisciplinary research group Profetas (protein foods, environment, technology and society). Profetas consists of technological, environmental and socio-economic research projects on protein food systems which result in the development of scenarios and
Protein adhesives
Charles R. Frihart; Linda F. Lorenz
Nature uses a wide variety of chemicals for providing adhesion internally (e.g., cell to cell) and externally (e.g., mussels to ships and piers). This adhesive bonding is chemically and mechanically complex, involving a variety of proteins, carbohydrates, and other compounds.Consequently,the effect of protein structures on adhesive properties is only partially...
Site-directed fluorescence labeling of a membrane protein with BADAN: probing protein topology and local environment
Koehorst, R.B.M.; Spruijt, R.B.; Hemminga, M.A.
We present a new and simple method based on site-directed fluorescence labeling using the BADAN label that allows to examine protein-lipid interactions in great detail. We apply this approach to a membrane-embedded mainly -helical reference protein, the M13 major coat protein, of which in a
Tau protein
Frederiksen, Jette Lautrup Battistini; Kristensen, Kim; Bahl, Jmc
Background: Tau protein has been proposed as biomarker of axonal damage leading to irreversible neurological impairment in MS. CSF concentrations may be useful when determining risk of progression from ON to MS. Objective: To investigate the association between tau protein concentration and 14......-3-3 protein in the cerebrospinal fluid (CSF) of patients with monosymptomatic optic neuritis (ON) versus patients with monosymptomatic onset who progressed to multiple sclerosis (MS). To evaluate results against data found in a complete literature review. Methods: A total of 66 patients with MS and/or ON from...... the Department of Neurology of Glostrup Hospital, University of Copenhagen, Denmark, were included. CSF samples were analysed for tau protein and 14-3-3 protein, and clinical and paraclinical information was obtained from medical records. Results: The study shows a significantly increased concentration of tau...
Investigating student understanding of simple harmonic motion
Somroob, S.; Wattanakasiwich, P.
This study aimed to investigate students' understanding and develop instructional material on a topic of simple harmonic motion. Participants were 60 students taking a course on vibrations and wave and 46 students taking a course on Physics 2 and 28 students taking a course on Fundamental Physics 2 on the 2nd semester of an academic year 2016. A 16-question conceptual test and tutorial activities had been developed from previous research findings and evaluated by three physics experts in teaching mechanics before using in a real classroom. Data collection included both qualitative and quantitative methods. Item analysis and whole-test analysis were determined from student responses in the conceptual test. As results, most students had misconceptions about restoring force and they had problems connecting mathematical solutions to real motions, especially phase angle. Moreover, they had problems with interpreting mechanical energy from graphs and diagrams of the motion. These results were used to develop effective instructional materials to enhance student abilities in understanding simple harmonic motion in term of multiple representations.
Foreshock and aftershocks in simple earthquake models.
Kazemian, J; Tiampo, K F; Klein, W; Dominguez, R
Many models of earthquake faults have been introduced that connect Gutenberg-Richter (GR) scaling to triggering processes. However, natural earthquake fault systems are composed of a variety of different geometries and materials and the associated heterogeneity in physical properties can cause a variety of spatial and temporal behaviors. This raises the question of how the triggering process and the structure interact to produce the observed phenomena. Here we present a simple earthquake fault model based on the Olami-Feder-Christensen and Rundle-Jackson-Brown cellular automata models with long-range interactions that incorporates a fixed percentage of stronger sites, or asperity cells, into the lattice. These asperity cells are significantly stronger than the surrounding lattice sites but eventually rupture when the applied stress reaches their higher threshold stress. The introduction of these spatial heterogeneities results in temporal clustering in the model that mimics that seen in natural fault systems along with GR scaling. In addition, we observe sequences of activity that start with a gradually accelerating number of larger events (foreshocks) prior to a main shock that is followed by a tail of decreasing activity (aftershocks). This work provides further evidence that the spatial and temporal patterns observed in natural seismicity are strongly influenced by the underlying physical properties and are not solely the result of a simple cascade mechanism.
Trophic dynamics of a simple model ecosystem.
Bell, Graham; Fortier-Dubois, Étienne
We have constructed a model of community dynamics that is simple enough to enumerate all possible food webs, yet complex enough to represent a wide range of ecological processes. We use the transition matrix to predict the outcome of succession and then investigate how the transition probabilities are governed by resource supply and immigration. Low-input regimes lead to simple communities whereas trophically complex communities develop when there is an adequate supply of both resources and immigrants. Our interpretation of trophic dynamics in complex communities hinges on a new principle of mutual replenishment, defined as the reciprocal alternation of state in a pair of communities linked by the invasion and extinction of a shared species. Such neutral couples are the outcome of succession under local dispersal and imply that food webs will often be made up of suites of trophically equivalent species. When immigrants arrive from an external pool of fixed composition a similar principle predicts a dynamic core of webs constituting a neutral interchange network, although communities may express an extensive range of other webs whose membership is only in part predictable. The food web is not in general predictable from whole-community properties such as productivity or stability, although it may profoundly influence these properties. © 2017 The Author(s).
Training nuclei detection algorithms with simple annotations
Henning Kost
Full Text Available Background: Generating good training datasets is essential for machine learning-based nuclei detection methods. However, creating exhaustive nuclei contour annotations, to derive optimal training data from, is often infeasible. Methods: We compared different approaches for training nuclei detection methods solely based on nucleus center markers. Such markers contain less accurate information, especially with regard to nuclear boundaries, but can be produced much easier and in greater quantities. The approaches use different automated sample extraction methods to derive image positions and class labels from nucleus center markers. In addition, the approaches use different automated sample selection methods to improve the detection quality of the classification algorithm and reduce the run time of the training process. We evaluated the approaches based on a previously published generic nuclei detection algorithm and a set of Ki-67-stained breast cancer images. Results: A Voronoi tessellation-based sample extraction method produced the best performing training sets. However, subsampling of the extracted training samples was crucial. Even simple class balancing improved the detection quality considerably. The incorporation of active learning led to a further increase in detection quality. Conclusions: With appropriate sample extraction and selection methods, nuclei detection algorithms trained on the basis of simple center marker annotations can produce comparable quality to algorithms trained on conventionally created training sets.
A SIMPLE EXPERIMENTAL MODEL OF HEAT SHOCK RESPONSE IN RATS
Tufi Neder Meyer
Full Text Available Objective: To obtain a simple model for the elicitation of the heat shock response in rats. Design: Laboratory study. Setting: University research laboratories. Sample: Seventy-nine adult male albino rats (weight range 200 g to 570 g. Procedures: Exposure to heat stress by heating animals in a warm bath for 5 min after their rectal temperatures reached 107.60 F (420 C. Liver and lung samples were collected for heat-shock protein 70 (HSP70 detection (Western analysis. Results: Western analysis was positive for HSP70 in the liver and in the lungs of heated animals. There was a temporal correlation between heating and HSP70 detection: it was strongest 1 day after heating and reduced afterwards. No heated animals died. Conclusion: These data show that heating rats in a warm (45o C bath, according to parameters set in this model, elicits efficiently the heat shock response.OBJETIVO: Obter um modelo simples para tentar esclarecer a resposta ao choque térmico em ratos. LOCAL: Laboratório de pesquisa da Universidade. MÉTODO: Amostra: 79 ratos albinos, adultos, entre 200g a 570g. Procedimentos: Exposição ao calor, em banho quente, por 5 minutos, após a temperatura retal chegar a 42 graus centigrados. Biópsias de fígado e pulmão foram obtidas para detectar a proteina 70 (HSP 70, pelo "Western blot". RESULTADOS: As análises foram positivas nos animais aquecidos, com uma correlação entre aquecimento e constatação da HSP 70. Foi mais elevada no primeiro dia e não houve óbitos nos animais aquecidos. CONCLUSÃO: Os ratos aquecidos a 45 graus centígrados respondem eficientemente ao choque térmico.
Simple spatial scaling rules behind complex cities.
Li, Ruiqi; Dong, Lei; Zhang, Jiang; Wang, Xinran; Wang, Wen-Xu; Di, Zengru; Stanley, H Eugene
Although most of wealth and innovation have been the result of human interaction and cooperation, we are not yet able to quantitatively predict the spatial distributions of three main elements of cities: population, roads, and socioeconomic interactions. By a simple model mainly based on spatial attraction and matching growth mechanisms, we reveal that the spatial scaling rules of these three elements are in a consistent framework, which allows us to use any single observation to infer the others. All numerical and theoretical results are consistent with empirical data from ten representative cities. In addition, our model can also provide a general explanation of the origins of the universal super- and sub-linear aggregate scaling laws and accurately predict kilometre-level socioeconomic activity. Our work opens a new avenue for uncovering the evolution of cities in terms of the interplay among urban elements, and it has a broad range of applications.
Simple turbulence measurements with azopolymer thin films.
Barillé, Regis; Pérez, Darío G; Morille, Yohann; Zielińska, Sonia; Ortyl, Ewelina
A simple method to measure the influence on the laser beam propagation by a turbid medium is proposed. This measurement is based on the inscription of a surface relief grating (SRG) on an azopolymer thin film. The grating obtained with a single laser beam after propagation into a turbulent medium is perturbed and directly analyzed by a CCD camera through its diffraction pattern. Later, by scanning the surface pattern with an atomic force microscope, the inscribed SRG is analyzed with the Radon transform. This method has the advantage of using a single beam to remotely inscribe a grating detecting perturbations during the beam path. A method to evaluate the refractive index constant structure is developed.
Search without Boundaries Using Simple APIs
Tong, Qi
The U.S. Geological Survey (USGS) Library, where the author serves as the digital services librarian, is increasingly challenged to make it easier for users to find information from many heterogeneous information sources. Information is scattered throughout different software applications (i.e., library catalog, federated search engine, link resolver, and vendor websites), and each specializes in one thing. How could the library integrate the functionalities of one application with another and provide a single point of entry for users to search across? To improve the user experience, the library launched an effort to integrate the federated search engine into the library's intranet website. The result is a simple search box that leverages the federated search engine's built-in application programming interfaces (APIs). In this article, the author describes how this project demonstrated the power of APIs and their potential to be used by other enterprise search portals inside or outside of the library.
A Simple Solution to Type Specialization
Danvy, Olivier
Partial evaluation specializes terms, but traditionally this specialization does not apply to the type of these terms. As a result, specializing, e.g., an interpreter written in a typed language, which requires a "universal� type to encode expressible values, yields residual programs with type tags...... all over. Neil Jones has stated that getting rid of these type tags was an open problem, despite possible solutions such as Torben Mogensen's "constructor specialization.� To solve this problem, John Hughes has proposed a new paradigm for partial evaluation, "Type Specialization�, based on type...... inference instead of being based on symbolic interpretation. Type Specialization is very elegant in principle but it also appears non-trivial in practice. Stating the problem in terms of types instead of in terms of type encodings suggests a very simple type-directed solution, namely, to use a projection...
Two simple models of classical heat pumps.
Marathe, Rahul; Jayannavar, A M; Dhar, Abhishek
Motivated by recent studies of models of particle and heat quantum pumps, we study similar simple classical models and examine the possibility of heat pumping. Unlike many of the usual ratchet models of molecular engines, the models we study do not have particle transport. We consider a two-spin system and a coupled oscillator system which exchange heat with multiple heat reservoirs and which are acted upon by periodic forces. The simplicity of our models allows accurate numerical and exact solutions and unambiguous interpretation of results. We demonstrate that while both our models seem to be built on similar principles, one is able to function as a heat pump (or engine) while the other is not.
Simple Device for Treating Prolapsing Loop Colostomy
Ming-Yu Hsieh
Full Text Available Stoma prolapse is a common complication of intestinal stoma. Although various surgical methods yield satisfactory results, nonsurgical treatment may be better for a temporary stoma. We report a case of a patient with a distal limb prolapse of a right transverse colostomy who received nonsurgical treatment with satisfactory results. For the treatment of a temporary transverse loop colostomy with distal limb prolapse, we designed a simple device consisting of a pediatric plastic medicine cup, which was rolled into a towel to shape the bottom of the cup into a compressor. The towel was put on the stoma outside of the colostomy bag with the compressor above the prolapsing limb of the colostomy. An abdominal binder was applied to fix the towel.
A simple model of bedform migration
Bartholdy, Jesper; Ernstsen, Verner Brandbyge; Flemming, Burg W
and width) of naturally-packed bed material on the bedform lee side, qb(crest). The model is simple, built on a rational description of simplified sediment mechanics, and its calibration constant can be explained in accordance with estimated values of the physical constants on which it is based. Predicted......A model linking subaqueous dune migration to the effective (grain related) shear stress is calibrated by means of flume data for bedform dimensions and migration rates. The effective shear stress is calculated on the basis of a new method assuming a near-bed layer above the mean bed level in which...... the current velocity accelerates towards the bedform crest. As a consequence, the effective bed shear stress corresponds to the shear stress acting directly on top of the bedform. The model operates with the critical Shields stress as a function of grain size, and predicts the deposition (volume per unit time...
Simple Lie groups without the approximation property
Haagerup, Uffe; de Laat, Tim
For a locally compact group G, let A(G) denote its Fourier algebra, and let M0A(G) denote the space of completely bounded Fourier multipliers on G. The group G is said to have the Approximation Property (AP) if the constant function 1 can be approximated by a net in A(G) in the weak-∗ topology...... on the space M0A(G). Recently, Lafforgue and de la Salle proved that SL(3,R) does not have the AP, implying the first example of an exact discrete group without it, namely, SL(3,Z). In this paper we prove that Sp(2,R) does not have the AP. It follows that all connected simple Lie groups with finite center...
Swarming behavior of simple model squirmers
Thutupalli, Shashi; Seemann, Ralf; Herminghaus, Stephan
We have studied experimentally the collective behavior of self-propelling liquid droplets, which closely mimic the locomotion of some protozoal organisms, the so-called squirmers. For the sake of simplicity, we concentrate on quasi-two-dimensional (2D) settings, although our swimmers provide a fully 3D propulsion scheme. At an areal density of 0.46, we find strong polar correlation of the locomotion velocities of neighboring droplets, which decays over less than one droplet diameter. When the areal density is increased to 0.78, distinct peaks show up in the angular correlation function, which point to the formation of ordered rafts. This shows that pronounced textures, beyond what has been seen in simulations so far, may show up in crowds of simple model squirmers, despite the simplicity of their (purely physical) mutual interaction.
The biomechanics of simple steatosis and steatohepatitis
Parker, K. J.; Ormachea, J.; Drage, M. G.; Kim, H.; Hah, Z.
Magnetic resonance and ultrasound elastography techniques are now important tools for staging high-grade fibrosis in patients with chronic liver disease. However, uncertainty remains about the effects of simple accumulation of fat (steatosis) and inflammation (steatohepatitis) on the parameters that can be measured using different elastographic techniques. To address this, we examine the rheological models that are capable of capturing the dominant viscoelastic behaviors associated with fat and inflammation in the liver, and quantify the resulting changes in shear wave speed and viscoelastic parameters. Theoretical results are shown to match measurements in phantoms and animal studies reported in the literature. These results are useful for better design of elastographic studies of fatty liver disease and steatohepatitis, potentially leading to improved diagnosis of these conditions.
Genealogies in simple models of evolution
Brunet, Éric; Derrida, Bernard
We review the statistical properties of the genealogies of a few models of evolution. In the asexual case, selection leads to coalescence times which grow logarithmically with the size of the population, in contrast with the linear growth of the neutral case. Moreover for a whole class of models, the statistics of the genealogies are those of the Bolthausen–Sznitman coalescent rather than the Kingman coalescent in the neutral case. For sexual reproduction in the neutral case, the time to reach the first common ancestors for the whole population and the time for all individuals to have all their ancestors in common are also logarithmic in the population size, as predicted by Chang in 1999. We discuss how these times are modified by introducing selection in a simple way. (paper)
Learning from correlated patterns by simple perceptrons
Shinzato, Takashi; Kabashima, Yoshiyuki [Department of Computational Intelligence and Systems Science, Tokyo Institute of Technology, Yokohama 226-8502 (Japan)], E-mail: [email protected], E-mail: [email protected]
Learning behavior of simple perceptrons is analyzed for a teacher-student scenario in which output labels are provided by a teacher network for a set of possibly correlated input patterns, and such that the teacher and student networks are of the same type. Our main concern is the effect of statistical correlations among the input patterns on learning performance. For this purpose, we extend to the teacher-student scenario a methodology for analyzing randomly labeled patterns recently developed in Shinzato and Kabashima 2008 J. Phys. A: Math. Theor. 41 324013. This methodology is used for analyzing situations in which orthogonality of the input patterns is enhanced in order to optimize the learning performance.
iPad 2 Made Simple
Mazo, Gary; Ritchie, Rene
The iPad 2 is thinner, more powerful, intuitive and very fun for users of all ages. Your iPad can be used for reading, surfing the web, emailing, watching TV and movies, getting work done and much more. And with over 65,000 apps just for the iPad, as well as the ability to run over 30,000 iPhone apps, the possibilities are endless. iPad 2 Made Simple clarifies all of the key features on the iPad, introduces what's new, and also reveals dozens of time-saving shortcuts and techniques. The book has over 1,000 screen shots that are carefully annotated with step-by-step instructions. * Clear instru
Inequivalent topologies of chaos in simple equations
Letellier, Christophe; Roulin, Elise; Roessler, Otto E.
In the 1970, one of us introduced a few simple sets of ordinary differential equations as examples showing different types of chaos. Most of them are now more or less forgotten with the exception of the so-called Roessler system published in [Roessler OE. An equation for continuous chaos. Phys Lett A 1976;57(5):397-8]. In the present paper, we review most of the original systems and classify them using the tools of modern topological analysis, that is, using the templates and the bounding tori recently introduced by Tsankov and Gilmore in [Tsankov TD, Gilmore R. Strange attractors are classified by bounding tori. Phys Rev Lett 2003;91(13):134104]. Thus, examples of inequivalent topologies of chaotic attractors are provided in modern spirit
Shinzato, Takashi; Kabashima, Yoshiyuki
Multiple equilibria in a simple elastocapillary system
Taroni, Michele
We consider the elastocapillary interaction of a liquid drop placed between two elastic beams, which are both clamped at one end to a rigid substrate. This is a simple model system relevant to the problem of surface-tension-induced collapse of flexible micro-channels that has been observed in the manufacture of microelectromechanical systems (MEMS). We determine the conditions under which the beams remain separated, touch at a point, or stick along a portion of their length. Surprisingly, we show that in many circumstances multiple equilibrium states are possible. We develop a lubrication-type model for the flow of liquid out of equilibrium and thereby investigate the stability of the multiple equilibria. We demonstrate that for given material properties two stable equilibria may exist, and show via numerical solutions of the dynamic model that it is the initial state of the system that determines which stable equilibrium is ultimately reached. © 2012 Cambridge University Press.
Simple implementation of general dark energy models
Bloomfield, Jolyon K.; Pearson, Jonathan A.
We present a formalism for the numerical implementation of general theories of dark energy, combining the computational simplicity of the equation of state for perturbations approach with the generality of the effective field theory approach. An effective fluid description is employed, based on a general action describing single-scalar field models. The formalism is developed from first principles, and constructed keeping the goal of a simple implementation into CAMB in mind. Benefits of this approach include its straightforward implementation, the generality of the underlying theory, the fact that the evolved variables are physical quantities, and that model-independent phenomenological descriptions may be straightforwardly investigated. We hope this formulation will provide a powerful tool for the comparison of theoretical models of dark energy with observational data
A Simple Sag Generator Using SSRs
Senturk, Osman Selcuk; Hava, Ahmet M.
conditions (critical loads) and this property often may not be accommodated inside the device itself and sag compensating power conditioners have been developed for such purposes. While in practice voltage sags are not wanted, generating sags becomes necessary for the purpose of experimentally verifying...... the performances of the equipment (both the equipment under sag condition and the sag compensating power conditioner) under sag conditions. In this work, a simple and economical, yet highly performing sag generator is developed, its design discussed, and its feasibility demonstrated by experiments. The proposed...... is evaluated and finally the utilization of the sag generator in the test of a series active filter based power quality conditioner is demonstrated. The proposed approach provides an effective solution for voltage sag generation....
Interactions between simple radicals and water
Crespo-Otero, Rachel; Sanchez-Garcia, Elsa; Suardiaz, Reynier; Montero, Luis A.; Sander, Wolfram
The interactions of the simple radicals CH 3 , NH 2 , OH, and F with water have been studied by DFT (UB3LYP/6-311++G(2d,2p)) and ab initio (RHF-UCCSD(T)/6-311++G(2d,2p)) methods. In this order the number of lone pairs (from zero to three), the electronegativity, and the strength of the X-H bonds increase (X = C, N, and O). The various minima of the radical-water complexes were located using the multiple minima hypersurface (MMH) approach which had previously been proven to be useful for closed-shell molecules. The role of the unpaired electron in hydrogen bonding was investigated using the natural bond orbital (NBO) analysis. A considerable contribution of the unpaired electron to the complex stabilization was only found for the methyl radical and the fluorine atom, whereas in the aminyl and the hydroxyl radical the role of the unpaired electron is negligible
Current-induced forces: a simple derivation
Todorov, Tchavdar N; Dundas, Daniel; Lü, Jing-Tao; Brandbyge, Mads; Hedegård, Per
We revisit the problem of forces on atoms under current in nanoscale conductors. We derive and discuss the five principal kinds of force under steady-state conditions from a simple standpoint that—with the help of background literature—should be accessible to physics undergraduates. The discussion aims at combining methodology with an emphasis on the underlying physics through examples. We discuss and compare two forces present only under current—the non-conservative electron wind force and a Lorentz-like velocity-dependent force. It is shown that in metallic nanowires both display significant features at the wire surface, making it a candidate for the nucleation of current-driven structural transformations and failure. Finally we discuss the problem of force noise and the limitations of Ehrenfest dynamics. (paper)
A simple and realistic triton wave function
Lomnitz-Adler, J.; Pandharipande, V.R.
We propose a simple triton wave function that consists of a product of three correlation operators operating on a three-body spin-isospin state. This wave function is formally similar to that used in the recent variational theories of nuclear matter, the main difference being in the long-range behavior of the correlation operators. Variational calculations are carried out with the Reid potential, using this wave function in the so-called 'symmetrized product' and 'independent pair' forms. The triton energy and density distributions obtained with the symmetrized product wave function agree with those obtained in Faddeev and other variational calculations using harmonic oscillator states. The proposed wave function and calculational methods can be easily generalized to treat the four-nucleon α-particle. (orig.)
From complex to simple: interdisciplinary stochastic models
Mazilu, D A; Zamora, G; Mazilu, I
We present two simple, one-dimensional, stochastic models that lead to a qualitative understanding of very complex systems from biology, nanoscience and social sciences. The first model explains the complicated dynamics of microtubules, stochastic cellular highways. Using the theory of random walks in one dimension, we find analytical expressions for certain physical quantities, such as the time dependence of the length of the microtubules, and diffusion coefficients. The second one is a stochastic adsorption model with applications in surface deposition, epidemics and voter systems. We introduce the 'empty interval method' and show sample calculations for the time-dependent particle density. These models can serve as an introduction to the field of non-equilibrium statistical physics, and can also be used as a pedagogical tool to exemplify standard statistical physics concepts, such as random walks or the kinetic approach of the master equation. (paper)
A simple nonlinear dynamical computing device
Miliotis, Abraham; Murali, K.; Sinha, Sudeshna; Ditto, William L.; Spano, Mark L.
We propose and characterize an iterated map whose nonlinearity has a simple (i.e., minimal) electronic implementation. We then demonstrate explicitly how all the different fundamental logic gates can be implemented and morphed using this nonlinearity. These gates provide the full set of gates necessary to construct a general-purpose, reconfigurable computing device. As an example of how such chaotic computing devices can be exploited, we use an array of these maps to encode data and to process information. Each map can store one of M items, where M is variable and can be large. This nonlinear hardware stores data naturally in different bases or alphabets. We also show how this method of storing information can serve as a preprocessing tool for exact or inexact pattern-matching searches.
Thutupalli, Shashi; Seemann, Ralf; Herminghaus, Stephan, E-mail: [email protected], E-mail: [email protected] [Max Planck Institute for Dynamics and Self-Organization, Bunsenstrasse 10, 37073 Goettingen (Germany)
A simple strategy for jumping straight up.
Hemami, Hooshang; Wyman, Bostwick F
Jumping from a stationary standing position into the air is a transition from a constrained motion in contact with the ground to an unconstrained system not in contact with the ground. A simple case of the jump, as it applies to humans, robots and humanoids, is studied in this paper. The dynamics of the constrained rigid body are expanded to define a larger system that accommodates the jump. The formulation is applied to a four-link, three-dimensional system in order to articulate the ballistic motion involved. The activity of the muscular system and the role of the major sagittal muscle groups are demonstrated. The control strategy, involving state feedback and central feed forward signals, is formulated and computer simulations are presented to assess the feasibility of the formulations, the strategy and the jump. Copyright © 2012 Elsevier Inc. All rights reserved.
Radon entry into a simple test structure
Andersen, C.E.; Søgaard-Hansen, J.; Majborn, B.
A simple test structure for studies of radon entry into houses has been constructed at a field site at Riso National Laboratory. It consists of a 40 1, stainless-steel cylinder placed in a 0.52 m deep quadratic excavation with a side length of 2.4 m. The excavation is lined with an airtight...... membrane, and soil gas enters the cylinder through a changeable interface in the bottom. The depressurisation of the cylinder is controlled by a mass-flow controller, thereby limiting the influence of natural driving forces. Pressures, temperatures and radon concentrations are measured continuously...... in the cylinder and in selected locations in the soil. In this paper, the test structure is described, and initial results concerning the transport of soil gas and radon under steady-state conditions are reported. It is found that the soil in the vicinity of the structure is partially depleted with respect...
Learning behavior of simple perceptrons is analyzed for a teacher-student scenario in which output labels are provided by a teacher network for a set of possibly correlated input patterns, and such that the teacher and student networks are of the same type. Our main concern is the effect of statistical correlations among the input patterns on learning performance. For this purpose, we extend to the teacher-student scenario a methodology for analyzing randomly labeled patterns recently developed in Shinzato and Kabashima 2008 J. Phys. A: Math. Theor. 41 324013. This methodology is used for analyzing situations in which orthogonality of the input patterns is enhanced in order to optimize the learning performance
A simple electron-beam lithography system
Mølhave, Kristian; Madsen, Dorte Nørgaard; Bøggild, Peter
A large number of applications of electron-beam lithography (EBL) systems in nanotechnology have been demonstrated in recent years. In this paper we present a simple and general-purpose EBL system constructed by insertion of an electrostatic deflector plate system at the electron-beam exit...... of the column of a scanning electron microscope (SEM). The system can easily be mounted on most standard SEM systems. The tested setup allows an area of up to about 50 x 50 pm to be scanned, if the upper limit for acceptable reduction of the SEM resolution is set to 10 run. We demonstrate how the EBL system can...... be used to write three-dimensional nanostructures by electron-beam deposition. (C) 2004 Elsevier B.V. All rights reserved....
A simple method for α determination
Ho Manh Dung; Seung Yeon Cho
The a term is a primary parameter that is used to indicate the deviation of the epithermal neutron distribution in the k 0 -standardization method of neutron activation analysis, k 0 -NAA. The calculation of a using a mathematical procedure is a challenge for some researchers. The calculation of a by the 'bare-triple monitor' method is possible using the dedicated commercial software KAYZERO R /SOLCOI R . However, when this software is not available in the laboratory it is possible to carry out the calculation of a applying a simple iterative linear regression using any spreadsheets. This approach is described. The experimental data used in the example were obtained by the irradiation of a set of suitable monitors in the NAA no.1 irradiation channel of the HANARO research reactor (KAERI, Korea). The results obtained by this iterative linear regression method agree well with the results calculated by the validated mathematical method. (author)
Simple Additive Weighting to Diagnose Rabbit Disease
Ramadiani
Full Text Available Rabbit is one of the many pets maintained by the general public in Indonesia. Like other pet, rabbits are also susceptible to various diseases. Society in general does not understand correctly the type of rabbit disease and the way of treatment. To help care for sick rabbits it is necessary a decision support system recommendation diagnosis of rabbit disease. The purpose of this research is to make the application of rabbit disease diagnosis system so that can help user in taking care of rabbit. This application diagnoses the disease by tracing the symptoms and calculating the recommendation of the disease using Simple Additive Weighting method. This research produces a web-based decision support system that is used to help rabbit breeders and the general public.
Ramadiani; Marissa, Dyna; Jundillah, Muhammad Labib; Azainil; Hatta, Heliza Rahmania
Rabbit is one of the many pets maintained by the general public in Indonesia. Like other pet, rabbits are also susceptible to various diseases. Society in general does not understand correctly the type of rabbit disease and the way of treatment. To help care for sick rabbits it is necessary a decision support system recommendation diagnosis of rabbit disease. The purpose of this research is to make the application of rabbit disease diagnosis system so that can help user in taking care of rabbit. This application diagnoses the disease by tracing the symptoms and calculating the recommendation of the disease using Simple Additive Weighting method. This research produces a web-based decision support system that is used to help rabbit breeders and the general public.
A simple FASTBUS to ethernet interface
Baudendistel, K.; Dobinson, R.W.; Downing, R.W.; Herbert, M.J.
Until comparatively recently the effort and the expense of interfacing to Ethernet has been considerable, both in terms of design time and the number of integrated circuits required. However, the appearance of VLSI chip sets from several manufacturers, which perform large parts of the lower level network protocols, has done much to ease this problem. One of the first chip sets available was that manufactured by the SEEQ company of San Jose, California. We have successfully constructed and operated controller boards for the IBM PC using these chips. We report here on an extension of this work to construct a simple interface between FASTBUS and Ethernet. The motivation for the work is twofold; first to make available Ethernet products and services from a FASTBUS environment, secondly to investigate the interconnection of FASTBUS segments over longer distances than is possible using the present cable segment and segment interconnects. The emphasis of this paper is on how the interface appears to a FASTBUS user
Irradiation creep in simple binary alloys
Nagakawa, J.; Sethi, V.K.; Turner, A.P.L.
Creep enhancement during 21-MeV deuteron irradiation was examined at 350 0 C for two simple binary alloys with representative microstructures, i.e., solid-solution (Ni - 4 at. % Si) and precipitation-hardened (Ni - 12.8 at. % Al) alloys. Coherent precipitates were found to be very effective in suppressing irradiation-enhanced creep. Si solute atoms depressed irradiation creep moderately and caused irradiation hardening via radiation-induced segregation. The stress-dependence of irradiation creep in Ni - 4 at. % Si should a transition, which seems to reflect a change of mechanism from dislocation climb due to stress-induced preferential absorption (SIPA) to climb-controlled dislocation glide enhanced by irradiation
Simple Check Valves for Microfluidic Devices
Willis, Peter A.; Greer, Harold F.; Smith, J. Anthony
A simple design concept for check valves has been adopted for microfluidic devices that consist mostly of (1) deformable fluorocarbon polymer membranes sandwiched between (2) borosilicate float glass wafers into which channels, valve seats, and holes have been etched. The first microfluidic devices in which these check valves are intended to be used are micro-capillary electrophoresis (microCE) devices undergoing development for use on Mars in detecting compounds indicative of life. In this application, it will be necessary to store some liquid samples in reservoirs in the devices for subsequent laboratory analysis, and check valves are needed to prevent cross-contamination of the samples. The simple check-valve design concept is also applicable to other microfluidic devices and to fluidic devices in general. These check valves are simplified microscopic versions of conventional rubber- flap check valves that are parts of numerous industrial and consumer products. These check valves are fabricated, not as separate components, but as integral parts of microfluidic devices. A check valve according to this concept consists of suitably shaped portions of a deformable membrane and the two glass wafers between which the membrane is sandwiched (see figure). The valve flap is formed by making an approximately semicircular cut in the membrane. The flap is centered over a hole in the lower glass wafer, through which hole the liquid in question is intended to flow upward into a wider hole, channel, or reservoir in the upper glass wafer. The radius of the cut exceeds the radius of the hole by an amount large enough to prevent settling of the flap into the hole. As in a conventional rubber-flap check valve, back pressure in the liquid pushes the flap against the valve seat (in this case, the valve seat is the adjacent surface of the lower glass wafer), thereby forming a seal that prevents backflow.
The Simple Chordate Ciona intestinalis Has a Reduced Complement of Genes Associated with Fanconi Anemia
Stanley, Edward C.; Azzinaro, Paul A.; Vierra, David A.; Howlett, Niall G.; Irvine, Steven Q.
Fanconi anemia (FA) is a human genetic disease characterized by congenital defects, bone marrow failure, and increased cancer risk. FA is associated with mutation in one of 24 genes. The protein products of these genes function cooperatively in the FA pathway to orchestrate the repair of DNA interstrand cross-links. Few model organisms exist for the study of FA. Seeking a model organism with a simpler version of the FA pathway, we searched the genome of the simple chordate Ciona intestinalis ...
A simple and robust approach to immobilization of antibody fragments.
Ikonomova, Svetlana P; He, Ziming; Karlsson, Amy J
Antibody fragments, such as the single-chain variable fragment (scFv), have much potential in research and diagnostics because of their antigen-binding ability similar to a full-sized antibody and their ease of production in microorganisms. Some applications of antibody fragments require immobilization on a surface, and we have established a simple immobilization method that is based on the biotin-streptavidin interaction and does not require a separate purification step. We genetically fused two biotinylation tags-the biotin carboxyl carrier protein (BCCP) or the AviTag minimal sequence-to six different scFvs (scFv13R4, scFvD10, scFv26-10, scFv3, scFv5, and scFv12) for site-specific biotinylation in vivo by endogenous biotin ligases produced by Escherichia coli. The biotinylated scFvs were immobilized onto streptavidin-coated plates directly from cell lysates, and immobilization was detected through enzyme-linked immunosorbent assays. All scFvs fusions were successfully immobilized, and scFvs biotinylated via the BCCP tag tended to immobilize better than those biotinylated via the AviTag, even when biotinylation efficiency was improved with the biotin ligase BirA. The ability of immobilized scFvs to bind antigens was confirmed using scFv13R4 and scFvD10 with their respective targets β-galactosidase and bacteriophage lambda head protein D (gpD). The immobilized scFv13R4 bound to β-galactosidase at the same level for both biotinylation tags when the surface was saturated with the scFv, and immobilized scFvs retained their functionality for at least 100days after immobilization. The simplicity and robustness of our method make it a promising approach for future applications that require antibody fragment immobilization. Copyright © 2016 Elsevier B.V. All rights reserved.
Global optimization of proteins using a dynamical lattice model: Ground states and energy landscapes
Dressel, F.; Kobe, S.
A simple approach is proposed to investigate the protein structure. Using a low complexity model, a simple pairwise interaction and the concept of global optimization, we are able to calculate ground states of proteins, which are in agreement with experimental data. All possible model structures of small proteins are available below a certain energy threshold. The exact lowenergy landscapes for the trp cage protein (1L2Y) is presented showing the connectivity of all states and energy barriers.
Simple Ligand–Receptor Interaction Descriptor (SILIRID for alignment-free binding site comparison
Vladimir Chupakhin
Full Text Available We describe SILIRID (Simple Ligand–Receptor Interaction Descriptor, a novel fixed size descriptor characterizing protein–ligand interactions. SILIRID can be obtained from the binary interaction fingerprints (IFPs by summing up the bits corresponding to identical amino acids. This results in a vector of 168 integer numbers corresponding to the product of the number of entries (20 amino acids and one cofactor and 8 interaction types per amino acid (hydrophobic, aromatic face to face, aromatic edge to face, H-bond donated by the protein, H-bond donated by the ligand, ionic bond with protein cation and protein anion, and interaction with metal ion. Efficiency of SILIRID to distinguish different protein binding sites has been examined in similarity search in sc-PDB database, a druggable portion of the Protein Data Bank, using various protein–ligand complexes as queries. The performance of retrieval of structurally and evolutionary related classes of proteins was comparable to that of state-of-the-art approaches (ROC AUC ≈ 0.91. SILIRID can efficiently be used to visualize chemogenomic space covered by sc-PDB using Generative Topographic Mapping (GTM: sc-PDB SILIRID data form clusters corresponding to different protein types.
Simple process to fabricate nitride alloy powders
Yang, Jae Ho; Kim, Dong-Joo; Kim, Keon Sik; Rhee, Young Woo; Oh, Jang-Soo; Kim, Jong Hun; Koo, Yang Hyun
Uranium mono-nitride (UN) is considered as a fuel material [1] for accident-tolerant fuel to compensate for the loss of fissile fuel material caused by adopting a thickened cladding such as SiC composites. Uranium nitride powders can be fabricated by a carbothermic reduction of the oxide powders, or the nitriding of metal uranium. Among them, a direct nitriding process of metal is more attractive because it has advantages in the mass production of high-purity powders and the reusing of expensive 15 N 2 gas. However, since metal uranium is usually fabricated in the form of bulk ingots, it has a drawback in the fabrication of fine powders. The Korea Atomic Energy Research Institute (KAERI) has a centrifugal atomisation technique to fabricate uranium and uranium alloy powders. In this study, a simple reaction method was tested to fabricate nitride fuel powders directly from uranium metal alloy powders. Spherical powder and flake of uranium metal alloys were fabricated using a centrifugal atomisation method. The nitride powders were obtained by thermal treating the metal particles under nitrogen containing gas. The phase and morphology evolutions of powders were investigated during the nitriding process. A phase analysis of nitride powders was also part of the present work. KAERI has developed the centrifugal rotating disk atomisation process to fabricate spherical uranium metal alloy powders which are used as advanced fuel materials for research reactors. The rotating disk atomisation system involves the tasks of melting, atomising, and collecting. A nozzle in the bottom of melting crucible introduces melt at the center of a spinning disk. The centrifugal force carries the melt to the edge of the disk and throws the melt off the edge. Size and shape of droplets can be controlled by changing the nozzle size, the disk diameter and disk speed independently or simultaneously. By adjusting the processing parameters of the centrifugal atomiser, a spherical and flake shape
SIMPL enhancement of tumor necrosis factor-α dependent p65-MED1 complex formation is required for mammalian hematopoietic stem and progenitor cell function.
Weina Zhao
Full Text Available Significant insight into the signaling pathways leading to activation of the Rel transcription factor family, collectively termed NF-κB, has been gained. Less well understood is how subsets of NF-κB-dependent genes are regulated in a signal specific manner. The SIMPL protein (signaling molecule that interacts with mouse pelle-like kinase is required for full Tumor Necrosis Factor-α (TNFα induced NF-κB activity. We show that SIMPL is required for steady-state hematopoiesis and the expression of a subset of TNFα induced genes whose products regulate hematopoietic cell activity. To gain insight into the mechanism through which SIMPL modulates gene expression we focused on the Tnf gene, an immune response regulator required for steady-state hematopoiesis. In response to TNFα SIMPL localizes to the Tnf gene promoter where it modulates the initiation of Tnf gene transcription. SIMPL binding partners identified by mass spectrometry include proteins involved in transcription and the interaction between SIMPL and MED1 was characterized in more detail. In response to TNFα, SIMPL is found in p65-MED1 complexes where SIMPL enhances p65/MED1/SIMPL complex formation. Together our results indicate that SIMPL functions as a TNFα-dependent p65 co-activator by facilitating the recruitment of MED1 to p65 containing transcriptional complexes to control the expression of a subset of TNFα-induced genes.
Creating Simple Admin Tools Using Info*Engine and Java
Jones, Corey; Kapatos, Dennis; Skradski, Cory; Felkins, J. D.
PTC has provided a simple way to dynamically interact with Windchill using Info*Engine. This presentation will describe how to create a simple Info*Engine Tasks capable of saving Windchill 10.0 administration of tedious work.
Hofmeister series of ions: A simple theory of a not so simple reality
Jungwirth, Pavel
Ro�. 4, �. 24 (2013), s. 4258-4259 ISSN 1948-7185 Institutional support: RVO:61388963 Keywords : Hofmeister series * ions * proteins Subject RIV: CF - Physical ; Theoretical Chemistry Impact factor: 6.687, year: 2013
Many human diseases and the death of most supercentenarians are related to protein aggregation. Neurodegenerative diseases include Alzheimer's disease (AD), Huntington's disease (HD), Parkinson's disease (PD), frontotemporallobar degeneration, etc. Such diseases are due to progressive loss of structure or function of neurons caused by protein aggregation. For example, AD is considered to be related to aggregation of Aβ40 (peptide with 40 amino acids) and Aβ42 (peptide with 42 amino acids) and HD is considered to be related to aggregation of polyQ (polyglutamine) peptides. In this paper, we briefly review our recent discovery of key factors for protein aggregation. We used a lattice model to study the aggregation rates of proteins and found that the probability for a protein sequence to appear in the conformation of the aggregated state can be used to determine the temperature at which proteins can aggregate most quickly. We used molecular dynamics and simple models of polymer chains to study relaxation and aggregation of proteins under various conditions and found that when the bending-angle dependent and torsion-angle dependent interactions are zero or very small, then protein chains tend to aggregate at lower temperatures. All atom models were used to identify a key peptide chain for the aggregation of insulin chains and to find that two polyQ chains prefer anti-parallel conformation. It is pointed out that in many cases, protein aggregation does not result from protein mis-folding. A potential drug from Chinese medicine was found for Alzheimer's disease. (paper)
Structural and computational aspects of simple and influence games
Riquelme Csori, Fabián
Simple games are a fundamental class of cooperative games. They have a huge relevance in several areas of computer science, social sciences and discrete applied mathematics. The algorithmic and computational complexity aspects of simple games have been gaining notoriety in the recent years. In this thesis we review different computational problems related to properties, parameters, and solution concepts of simple games. We consider different forms of representation of simple games, regular...
Linearization of the bradford protein assay.
Determination of microgram quantities of protein in the Bradford Coomassie brilliant blue assay is accomplished by measurement of absorbance at 590 nm. This most common assay enables rapid and simple protein quantification in cell lysates, cellular fractions, or recombinant protein samples, for the purpose of normalization of biochemical measurements. However, an intrinsic nonlinearity compromises the sensitivity and accuracy of this method. It is shown that under standard assay conditions, the ratio of the absorbance measurements at 590 nm and 450 nm is strictly linear with protein concentration. This simple procedure increases the accuracy and improves the sensitivity of the assay about 10-fold, permitting quantification down to 50 ng of bovine serum albumin. Furthermore, the interference commonly introduced by detergents that are used to create the cell lysates is greatly reduced by the new protocol. A linear equation developed on the basis of mass action and Beer's law perfectly fits the experimental data.
Simple mathematical law benchmarks human confrontations
Johnson, Neil F.; Medina, Pablo; Zhao, Guannan; Messinger, Daniel S.; Horgan, John; Gill, Paul; Bohorquez, Juan Camilo; Mattson, Whitney; Gangi, Devon; Qi, Hong; Manrique, Pedro; Velasquez, Nicolas; Morgenstern, Ana; Restrepo, Elvira; Johnson, Nicholas; Spagat, Michael; Zarama, Roberto
Many high-profile societal problems involve an individual or group repeatedly attacking another - from child-parent disputes, sexual violence against women, civil unrest, violent conflicts and acts of terror, to current cyber-attacks on national infrastructure and ultrafast cyber-trades attacking stockholders. There is an urgent need to quantify the likely severity and timing of such future acts, shed light on likely perpetrators, and identify intervention strategies. Here we present a combined analysis of multiple datasets across all these domains which account for >100,000 events, and show that a simple mathematical law can benchmark them all. We derive this benchmark and interpret it, using a minimal mechanistic model grounded by state-of-the-art fieldwork. Our findings provide quantitative predictions concerning future attacks; a tool to help detect common perpetrators and abnormal behaviors; insight into the trajectory of a `lone wolf' identification of a critical threshold for spreading a message or idea among perpetrators; an intervention strategy to erode the most lethal clusters; and more broadly, a quantitative starting point for cross-disciplinary theorizing about human aggression at the individual and group level, in both real and online worlds.
Sustained Attention Ability Affects Simple Picture Naming
Suzanne R. Jongman
Full Text Available Sustained attention has previously been shown as a requirement for language production. However, this is mostly evident for difficult conditions, such as a dual-task situation. The current study provides corroborating evidence that this relationship holds even for simple picture naming. Sustained attention ability, indexed both by participants' reaction times and individuals' hit rate (the proportion of correctly detected targets on a digit discrimination task, correlated with picture naming latencies. Individuals with poor sustained attention were consistently slower and their RT distributions were more positively skewed when naming pictures compared to individuals with better sustained attention. Additionally, the need to sustain attention was manipulated by changing the speed of stimulus presentation. Research has suggested that fast event rates tax sustained attention resources to a larger degree than slow event rates. However, in this study the fast event rate did not result in increased difficulty, neither for the picture naming task nor for the sustained attention task. Instead, the results point to a speed-accuracy trade-off in the sustained attention task (lower accuracy but faster responses in the fast than in the slow event rate, and to a benefit for faster rates in the picture naming task (shorter naming latencies with no difference in accuracy. Performance on both tasks was largely comparable, supporting previous findings that sustained attention is called upon during language production.
Many high-profile societal problems involve an individual or group repeatedly attacking another – from child-parent disputes, sexual violence against women, civil unrest, violent conflicts and acts of terror, to current cyber-attacks on national infrastructure and ultrafast cyber-trades attacking stockholders. There is an urgent need to quantify the likely severity and timing of such future acts, shed light on likely perpetrators, and identify intervention strategies. Here we present a combined analysis of multiple datasets across all these domains which account for >100,000 events, and show that a simple mathematical law can benchmark them all. We derive this benchmark and interpret it, using a minimal mechanistic model grounded by state-of-the-art fieldwork. Our findings provide quantitative predictions concerning future attacks; a tool to help detect common perpetrators and abnormal behaviors; insight into the trajectory of a 'lone wolf'; identification of a critical threshold for spreading a message or idea among perpetrators; an intervention strategy to erode the most lethal clusters; and more broadly, a quantitative starting point for cross-disciplinary theorizing about human aggression at the individual and group level, in both real and online worlds. PMID:24322528
Simple model of the slingshot effect
Gaetano Fiore
Full Text Available We present a detailed quantitative description of the recently proposed "slingshot effect.� Namely, we determine a broad range of conditions under which the impact of a very short and intense laser pulse normally onto a low-density plasma (or matter locally completely ionized into a plasma by the pulse causes the expulsion of a bunch of surface electrons in the direction opposite to the one of propagation of the pulse, and the detailed, ready-for-experiments features of the expelled electrons (energy spectrum, collimation, etc. The effect is due to the combined actions of the ponderomotive force and the huge longitudinal field arising from charge separation. Our predictions are based on estimating 3D corrections to a simple, yet powerful plane 2-fluid magnetohydrodynamic (MHD model where the equations to be solved are reduced to a system of Hamilton equations in one dimension (or a collection of which become autonomous after the pulse has overcome the electrons. Experimental tests seem to be at hand. If confirmed by the latter, the effect would provide a new extraction and acceleration mechanism for electrons, alternative to traditional radio-frequency-based or Laser-Wake-Field ones.
I begin my introductory astronomy course with a unit on critical thinking that focuses on, among other things, the differences between the "scientific method" as frequently presented in textbooks and actual scientific practice. One particular classroom activity uses a simple dice game to simulate observation of a natural phenomenon and the process of figuring out the framework, which we have previously defined as the rules that allow us to make predictions, governing the simulated phenomenon. Using games to teach scientific methodology is not new (see Maloney and Masters and Smith and references therein). I have experimented with Maloney and Masters' games and discovered that my students found them too difficult to figure out and therefore they did not learn what I hoped they would from them. I also experimented with other card games and found that too many students already knew the rules of both well-known and obscure card games. I even tried inventing my own games with, at best, mediocre results.
Pyglidein - A Simple HTCondor Glidein Service
Schultz, D.; Riedel, B.; Merino, G.
A major challenge for data processing and analysis at the IceCube Neutrino Observatory presents itself in connecting a large set of individual clusters together to form a computing grid. Most of these clusters do not provide a "standard� grid interface. Using a local account on each submit machine, HTCondor glideins can be submitted to virtually any type of scheduler. The glideins then connect back to a main HTCondor pool, where jobs can run normally with no special syntax. To respond to dynamic load, a simple server advertises the number of idle jobs in the queue and the resources they request. The submit script can query this server to optimize glideins to what is needed, or not submit if there is no demand. Configuring HTCondor dynamic slots in the glideins allows us to efficiently handle varying memory requirements as well as whole-node jobs. One step of the IceCube simulation chain, photon propagation in the ice, heavily relies on GPUs for faster execution. Therefore, one important requirement for any workload management system in IceCube is to handle GPU resources properly. Within the pyglidein system, we have successfully configured HTCondor glideins to use any GPU allocated to it, with jobs using the standard HTCondor GPU syntax to request and use a GPU. This mechanism allows us to seamlessly integrate our local GPU cluster with remote non-Grid GPU clusters, including specially allocated resources at XSEDE supercomputers.
A simple phenotypic classification for celiac disease
Ajit Sood
Full Text Available Background/Aims : Celiac disease is a global health problem. The presentation of celiac disease has unfolded over years and it is now known that it can manifest at different ages, has varied presentations, and is prone to develop complications, if not managed properly. Although the Oslo definitions provide consensus on the various terminologies used in literature, there is no phenotypic classification providing a composite diagnosis for the disease. Methods : Various variables identified for phenotypic classification included age at diagnosis, age at onset of symptoms, clinical presentation, family history and complications. These were applied to the existing registry of 1,664 patients at Dayanand Medical College and Hospital, Ludhiana, India. In addition, age was evaluated as below 15 and below 18 years. Cross tabulations were used for the verification of the classification using the existing data. Expert opinion was sought from both international and national experts of varying fields. Results : After empirical verification, age at diagnosis was considered appropriate in between A1 (<18 and A2 (≧18. The disease presentation has been classified into 3 types–P1 (classical, P2 (non-classical and P3 (asymptomatic. Complications were considered as absent (C0 or present (C1. A single phenotypic classification based on these 3 characteristics, namely age at the diagnosis, clinical presentation, and intestinal complications (APC classification was derived. Conclusions : APC classification (age at diagnosis, presentation, complications is a simple disease explanatory classification for patients with celiac disease aimed at providing a composite diagnosis.
Equilibria of perceptrons for simple contingency problems.
Dawson, Michael R W; Dupuis, Brian
The contingency between cues and outcomes is fundamentally important to theories of causal reasoning and to theories of associative learning. Researchers have computed the equilibria of Rescorla-Wagner models for a variety of contingency problems, and have used these equilibria to identify situations in which the Rescorla-Wagner model is consistent, or inconsistent, with normative models of contingency. Mathematical analyses that directly compare artificial neural networks to contingency theory have not been performed, because of the assumed equivalence between the Rescorla-Wagner learning rule and the delta rule training of artificial neural networks. However, recent results indicate that this equivalence is not as straightforward as typically assumed, suggesting a strong need for mathematical accounts of how networks deal with contingency problems. One such analysis is presented here, where it is proven that the structure of the equilibrium for a simple network trained on a basic contingency problem is quite different from the structure of the equilibrium for a Rescorla-Wagner model faced with the same problem. However, these structural differences lead to functionally equivalent behavior. The implications of this result for the relationships between associative learning, contingency theory, and connectionism are discussed.
A simple, remote, video based breathing monitor.
Regev, Nir; Wulich, Dov
Breathing monitors have become the all-important cornerstone of a wide variety of commercial and personal safety applications, ranging from elderly care to baby monitoring. Many such monitors exist in the market, some, with vital signs monitoring capabilities, but none remote. This paper presents a simple, yet efficient, real time method of extracting the subject's breathing sinus rhythm. Points of interest are detected on the subject's body, and the corresponding optical flow is estimated and tracked using the well known Lucas-Kanade algorithm on a frame by frame basis. A generalized likelihood ratio test is then utilized on each of the many interest points to detect which is moving in harmonic fashion. Finally, a spectral estimation algorithm based on Pisarenko harmonic decomposition tracks the harmonic frequency in real time, and a fusion maximum likelihood algorithm optimally estimates the breathing rate using all points considered. The results show a maximal error of 1 BPM between the true breathing rate and the algorithm's calculated rate, based on experiments on two babies and three adults.
A simple mixture to enhance muscle transmittance
Oliveira, Luís; Lage, Armindo; Clemente, Manuel Pais; Tuchin, Valery V.
Skeletal muscle is a fibrous tissue composed by muscle fibers and interstitial fluid. Due to this constitution, the muscle presents a non uniform refractive index profile that origins strong light scattering. One way to improve tissue transmittance is to reduce this refractive index mismatch by immersing the muscle in an optical clearing agent. As a consequence of such immersion tissue also suffers dehydration. The study of the optical clearing effect created by a simple mixture composed by ethanol, glycerol and distilled water has proven its effectiveness according to the variations observed in the parameters under study. The effect was characterized in terms of its magnitude, time duration and histological variations. The applied treatment has created a small reduction of the global sample refractive index that is justified by the long time rehydration caused by water in the immersing solution. From the reduction in sample pH we could also identify the dehydration process created in the sample. The immersion treatment has originated fiber bundle contraction and a spread distribution of the muscle fiber bundles inside. New studies with the mixture used, or with other combinations of its constituents might be interesting to perform with the objective to develop new clinical procedures.
A Simple Decontamination Approach Using Hydrogen ...
Journal article To evaluate the use of relatively low levels of hydrogen peroxide vapor (HPV) for the inactivation of Bacillus anthracis spores within an indoor environment. Methods and Results: Laboratory-scale decontamination tests were conducted using bacterial spores of both B. anthracis Ames and Bacillus atrophaeus inoculated onto several types of materials. Pilot-scale tests were also conducted using a larger chamber furnished as an indoor office. Commercial off-the-shelf (COTS) humidifiers filled with aqueous solutions of 3% or 8% hydrogen peroxide were used to generate the HPV inside the mock office. The spores were exposed to the HPV for periods ranging from 8 hours up to one week. Conclusions: Four to seven day exposures to low levels of HPV (average air concentrations of approximately 5-10 parts per million) were effective in inactivating B. anthracis spores on multiple materials. The HPV can be generated with COTS humidifiers and household H2O2 solutions. With the exception of one test/material, B. atrophaeus spores were equally or more resistant to HPV inactivation compared to those from B. anthracis Ames. Significance and Impact of Study: This simple and effective decontamination method is another option that could be widely applied in the event of a B. anthracis spore release.
Rotational modes of a simple Earth model
Seyed-Mahmoud, B.; Rochester, M. G.; Rogister, Y. J. G.
We study the tilt-over mode (TOM), the spin-over mode (SOM), the free core nutation (FCN), and their relationships to each other using a simple Earth model with a homogeneous and incompressible liquid core and a rigid mantle. Analytical solutions for the periods of these modes as well as that of the Chandler wobble is found for the Earth model. We show that the FCN is the same mode as the SOM of a wobbling Earth. The reduced pressure, in terms of which the vector momentum equation is known to reduce to a scalar second order differential equation (the so called Poincaŕe equation), is used as the independent variable. Analytical solutions are then found for the displacement eigenfucntions in a meridional plane of the liquid core for the aforementioned modes. We show that the magnitude of motion in the mantle during the FCN is comparable to that in the liquid core, hence very small. The displacement eigenfunctions for these aforementioned modes as well as those for the free inner core nutation (FICN), computed numerically, are also given for a three layer Earth model which also includes a rigid but capable of wobbling inner core. We will discuss the slow convergence of the period of the FICN in terms of the characteristic surfaces of the Poincare equation.
Simple light gas guns for hypervelocity studies
Combs, S.K.; Haselton, H.H.; Milora, S.L.
Two-stage light guns are used extensively in hypervelocity research. The applications of this technology include impact studies and special materials development. Oak Ridge National Laboratory (ORNL) has developed two-stage guns that accelerate small projectiles (4-mm nominal diameter) to velocities of up to ∼5 km/s. These guns are relatively small and simple (thus, easy to operate), allowing a significant number of test shots to be carried out and data accumulated in a short time. Materials that have been used for projectiles include plastics, frozen isotopes of hydrogen, and lithium hydride. One gun has been used to demonstrate repetitive operation at a rate of 0.7 Hz; and, with a few design improvements, it appears capable of performing at firing frequencies of 1--2 Hz. A schematic of ORNL two-stage device is shown below. Unlike most such devices, no rupture disks are used. Instead, a fast valve (high-flow type) initiates the acceleration process in the first stage. Projectiles can be loaded into the gun breech via the slide mechanism; this action has been automated which allows repetitive firing. Alternatively, the device is equipped with ''pipe gun'' apparatus in which gas can be frozen in situ in the gun barrel to form the projectile. This equipment operates with high reliability and is well suited for small-scale testing at high velocity. 17 refs., 6 figs., 2 tabs
Two simple tools for industrial OR
K. Sandrock
Full Text Available At the 1985 Annual Congress of the South African Production & Inventory Control Society it was pointed out that the productivity growth rate for South Africa is completely out of kilter with that for the western industrialised nations. The latter all display positive rates (some as high as that of Japan whereas the rate for South Africa is - NEGATIVE. Partly as a result of this situation, more and more attention is being given to quality control and reliability engineering by our industrialists in their attempts to improve productivity. This is going hand in hand with the introduction of better techniques and better use of the latest technology. We should also give attention to analytical tools that may be used in a simple inexpensive way to improve our methods of analysing industrial data, and in this way to improve our performance at little or no additional cost. To this end two tools are discussed. They are by means new. But it does seem as though they could be more widely applied in the industrial milieu.
A Simple Technique for Shoulder Arthrography
Berna-Serna, J.D.; Redondo, M.V.; Martinez, F.; Reus, M.; Alonso, J.; Parrilla, A.; Campos, P.A. [Virgen de la Arrixaca Univ. Hospital, El Palmar, Murcia (Spain). Dept. of Radiology
Purpose: To present a systematic approach to teaching a technique for arthrography of the shoulder. Using an adhesive marker-plate with radiopaque coordinates, precise sites for puncture can be identified and the need for fluoroscopic guidance obviated. Material and Methods: Forty-six glenohumeral arthrograms were performed in 45 patients; in 1 case involving examination of both shoulders. The stages of the technique are described in detail, as are the fundamental aspects of achieving an effective glenohumeral injection. Pain intensity was measured in all patients using a verbal description scale. Results: Shoulder arthrography was successful in all cases. Average time taken for the procedure was 7 min, with no difference in the respective times required by an experienced radiologist and a resident. The procedure was well tolerated by most patients, with slight discomfort being observed in a very few cases. Conclusion: The arthrographic technique used in this study is simple, safe, rapid, and reproducible, and has the advantage of precise localization of the site for puncture without need for fluoroscopic guidance. The procedure described in this study can be of help in teaching residents and can reduce the learning curve for radiologists with no experience in arthrographic methods. It also reduces the time of exposure to fluoroscopy Keywords: Arthrography, joint, shoulder.
Expectations for inflationary observables: simple or natural?
Musoke, Nathan; Easther, Richard
We describe the general inflationary dynamics that can arise with a single, canonically coupled field where the inflaton potential is a 4-th order polynomial. This scenario yields a wide range of combinations of the empirical spectral observables, ns, r and αs. However, not all combinations are possible and next-generation cosmological experiments have the ability to rule out all inflationary scenarios based on this potential. Further, we construct inflationary priors for this potential based on physically motivated choices for its free parameters. These can be used to determine the degree of tuning associated with different combinations of ns, r and αs and will facilitate treatments of the inflationary model selection problem. Finally, we comment on the implications of these results for the naturalness of the overall inflationary paradigm. We argue that ruling out all simple, renormalizable potentials would not necessarily imply that the inflationary paradigm itself was unnatural, but that this eventuality would increase the importance of building inflationary scenarios in the context of broader paradigms of ultra-high energy physics.
A simple dynamic energy capacity model
Gander, James P.
I develop a simple dynamic model showing how total energy capacity is allocated to two different uses and how these uses and their corresponding energy flows are related and behave through time. The control variable of the model determines the allocation. All the variables of the model are in terms of a composite energy equivalent measured in BTU's. A key focus is on the shadow price of energy capacity and its behavior through time. Another key focus is on the behavior of the control variable that determines the allocation of overall energy capacity. The matching or linking of the model's variables to real world U.S. energy data is undertaken. In spite of some limitations of the data, the model and its behavior fit the data fairly well. Some energy policy implications are discussed. - Highlights: â–º The model shows how energy capacity is allocated to current output production versus added energy capacity production. â–º Two variables in the allocation are the shadow price of capacity and the control variable that determines the allocation. â–º The model was linked to U.S. historical energy data and fit the data quite well. â–º In particular, the policy control variable was cyclical and consistent with the model. â–º Policy implications relevant to the allocation of energy capacity are discussed briefly.
Simple models of the hydrofracture process
Marder, M.
Hydrofracturing to recover natural gas and oil relies on the creation of a fracture network with pressurized water. We analyze the creation of the network in two ways. First, we assemble a collection of analytical estimates for pressure-driven crack motion in simple geometries, including crack speed as a function of length, energy dissipated by fluid viscosity and used to break rock, and the conditions under which a second crack will initiate while a first is running. We develop a pseudo-three-dimensional numerical model that couples fluid motion with solid mechanics and can generate branching crack structures not specified in advance. One of our main conclusions is that the typical spacing between fractures must be on the order of a meter, and this conclusion arises in two separate ways. First, it arises from analysis of gas production rates, given the diffusion constants for gas in the rock. Second, it arises from the number of fractures that should be generated given the scale of the affected region and the amounts of water pumped into the rock.
Simple chloride sensors for continuous groundwater monitoring
Thorn, Paul; Mortensen, John
The development of chloride sensors which can be used for continuous, on-line monitoring of groundwater could be very valuable in the management of our coastal water resources. However, sensor stability, drift, and durability all need to be addressed in order for the sensors to be used in continu......The development of chloride sensors which can be used for continuous, on-line monitoring of groundwater could be very valuable in the management of our coastal water resources. However, sensor stability, drift, and durability all need to be addressed in order for the sensors to be used...... in continuous application. This study looks at the development of a simple, inexpensive chloride electrode, and evaluates its performance under continuous use, both in the laboratory and in a field test in a monitoring well. The results from the study showed a consistent response to changing chloride...... concentrations over longer periods. The signal was seen to be stable, with regular drift in both laboratory and field test. In the field application, the sensor signal was corrected for drift, and errors were observed to be under 7% of that of conductivity measurements. The study also found that the chloride...
Simple mathematical models of gene regulatory dynamics
Mackey, Michael C; Tyran-Kamińska, Marta; Zeron, Eduardo S
This is a short and self-contained introduction to the field of mathematical modeling of gene-networks in bacteria. As an entry point to the field, we focus on the analysis of simple gene-network dynamics. The notes commence with an introduction to the deterministic modeling of gene-networks, with extensive reference to applicable results coming from dynamical systems theory. The second part of the notes treats extensively several approaches to the study of gene-network dynamics in the presence of noise—either arising from low numbers of molecules involved, or due to noise external to the regulatory process. The third and final part of the notes gives a detailed treatment of three well studied and concrete examples of gene-network dynamics by considering the lactose operon, the tryptophan operon, and the lysis-lysogeny switch. The notes contain an index for easy location of particular topics as well as an extensive bibliography of the current literature. The target audience of these notes are mainly graduat...
Simple goiter in children and juveniles
Wiebel, J
Non-toxic, simple goiter is the most common cause of thyroid enlargement in adolescents and children. Although initially there are hardly any complaints it should be treated with l-thyroxine to prevent further enlargement, degenerative alterations, development of nodules, possible hyper- or hypothyroidism occurring with increasing age. Anamnesis and clinical features are as important for establishing the diagnosis as laboratory parameters, which are not always necessary. Thyroid nodules in children or adolescents are more likely to be malignant than in later life; their nature has to be clarified by sonography, scintigraphy, fine needle biopsy or even operation. After l-thyroxine therapy has normalized thyroid size (60-90%), goiter prophylaxis with iodine supplementation can be tried, since iodine deficiency exists throughout Germany. If the goiter reappears continuous l-thyroxine treatment is recommended. Following the general introduction of iodine prophylaxis using iodinized salt there has been a significant reduction in goiter incidence in Switzerland, Austria and other areas with iodine deficiency. New legislation in the FRG now allows for the general use of e.g., Bad Reichenhaller Jodsalz, Duera Vollsalz or Bayerisches Vollsalz to prevent the development of the majority of goiters in children and adolescents.
Simple fabrication of gold nanobelts and patterns.
Renyun Zhang
Full Text Available Gold nanobelts are of interest in several areas; however, there are only few methods available to produce these belts. We report here on a simple evaporation induced self-assembly (EISA method to produce porous gold nanobelts with dimensions that scale across nanometer (thickness ∼80 nm and micrometer (width ∼20 µm, to decimeter (length ∼0.15 m. The gold nanobelts are well packed on the beaker wall and can be easily made to float on the surface of the solution for depositing onto other substrates. Microscopy showed that gold nanobelts had a different structure on the two sides of the belt; the density of gold nanowires on one side was greater than on the other side. Electrical measurements showed that these nanobelts were sensitive to compressive or tensile forces, indicating a potential use as a strain sensor. The patterned nanobelts were further used as a template to grow ZnO nanowires for potential use in applications such as piezo-electronics.
College Aptitude Test Simple Checker (Version 2
Jake G. Maggay
Full Text Available All enrolees of the Cagayan State University are required to take the College Aptitude Test (CAT. The CAT result serves as a basis for recommendation and admission to a specific course or field of specialization, thus, result must be accurate. The study a imed to develop a computerized College Aptitude Test (CAT Simple Checker of Cagayan State University – Lasam Campus to facilitate and to reduce the time of the guidance counsellor in checking many aptitude test papers as well as to ensure accuracy of resu lt. It followed the framework of Design Science Research in Information Systems which consists of six steps such as problem identification and motivation, definition of objectives for a solution based on the identified problem, design and development of th e system, demonstration of the system to the guidance counsellor, evaluation of the system's functionality and impact and communication which involves documentation and publication. A combination of Visual Basic 6 as the programming language and SQL Server 2005 as the Database Management System (DBMS were used in the development of the system. As a result, the system provides support to the guidance counsellor in performing the assigned tasks by reducing the time consumed in checking aptitude test papers t hat makes the guidance counsellor more effective, efficient and productive.
Simple derivation of magnetic space groups
Bertaut, E.F.; CEA Centre d'Etudes Nucleaires de Grenoble, 38
The magnetic translation lattices can be described by invariant wave vectors k. Advantages of the wave vector notation over the notations used by Belov et al. and Opechowski et al. are pointed out. In a one-dimensional real representation a space group element (α/tau(1)) has either the character +1 (symmetry element) or -1 (antisymmetry element). Thus the square of any space group operation must have the character +1 in a one-dimensional real representation. This simple ''square criterion'' is used to limit the admissible k-vectors and to derive the family of magnetic space groups, i.e. the set of all possible magnetic space groups, belonging to the same crystallographic space group. In the discussion some useful side results are obtained. Not only the real one-dimensional representations of point groups are connected to real one-dimensional representations of space groups, but a direct connection is shown to exist between one-dimensional complex representations of the point groups 3, 4 and 6 and one-dimensional real representations, belonging to P[001/2]=Psub(2c)(Psub(c))-lattices with screw axes 3 1 , 3 2 , 4 2 , 6 2 and 6 4 . Rules are derived for finding the Belov symbol when the Opechowski-Guccione symbol of the magnetic space group is known and this opportunity is used for correcting errors in the Opechowski-Guccione tables [fr
Marder, M.; Chen, Chih-Hung; Patzek, Tadeusz
A Simple Device For Measuring Skin Friction
Gupta A.B
Full Text Available A simple device for measuring skin friction in vivo is described. The frictional coefficient of normal Indian skin and the effect of hydration and application of talc and glycerol on the frictional coefficient and also the friction of ichthyotic skin have been determined with its help. The average value of friction of friction of normal India skin at forearm is found to be 0.41 +- 0.08, the hydration raises the value to 0.71 +- 0.11 and the effect of glycerol is also to school it up to 0.70+- 0.05, almost equal to that of water. The effect of talc however is opposite and its application lowers the friction to 0.21+-0.07. The mean coeff of friction for ichthyotic skin is found to be 0.21+- 0.0.5, which closely agrees with talc-treated normal skin. A good positive correlation (p<0.01 between friction and sebum level at skin site, with r = 0.64, has been observed.
Entanglement in a simple quantum phase transition
Osborne, Tobias J.; Nielsen, Michael A.
What entanglement is present in naturally occurring physical systems at thermal equilibrium? Most such systems are intractable and it is desirable to study simple but realistic systems that can be solved. An example of such a system is the one-dimensional infinite-lattice anisotropic XY model. This model is exactly solvable using the Jordan-Wigner transform, and it is possible to calculate the two-site reduced density matrix for all pairs of sites. Using the two-site density matrix, the entanglement of formation between any two sites is calculated for all parameter values and temperatures. We also study the entanglement in the transverse Ising model, a special case of the XY model, which exhibits a quantum phase transition. It is found that the next-nearest-neighbor entanglement (though not the nearest-neighbor entanglement) is a maximum at the critical point. Furthermore, we show that the critical point in the transverse Ising model corresponds to a transition in the behavior of the entanglement between a single site and the remainder of the lattice
A simple law of star formation
Padoan, Paolo; Haugbølle, Troels; Nordlund, Åke
We show that supersonic MHD turbulence yields a star formation rate (SFR) as low as observed in molecular clouds, for characteristic values of the free-fall time divided by the dynamical time, t ff/t dyn, the Alfvénic Mach number, {\\cal M}_a, and the sonic Mach number, {\\cal M}_s. Using a very...... values of t ff/t dyn and {\\cal M}_a. (2) Decreasing values of {\\cal M}_a (stronger magnetic fields) reduce epsilonff, but only to a point, beyond which epsilonff increases with a further decrease of {\\cal M}_a. (3) For values of {\\cal M}_a characteristic of star-forming regions, epsilonff varies...... with {\\cal M}_a by less than a factor of two. We propose a simple star formation law, based on the empirical fit to the minimum epsilonff, and depending only on t ff/t dyn: epsilonff ˜ epsilonwindexp (– 1.6 t ff/t dyn). Because it only depends on the mean gas density and rms velocity, this law...
ZeroMQ: Messaging Made Simple
Since its inception in 2007, ZeroMQ has defined a new product category of thin, fast, open source message transports. This little library has now grown into a large, vibrant community of projects tied together with standard protocols and APIs. Applications, written in any language, talk to each other over TCP, multicast, or inproc transports, using a single socket-based API, and a set of "patterns" (pub-sub, request-reply, dealer-router, pipeline). ZeroMQ handles message framing, batching, and I/O, but ignores aspects like serialization and persistence. By focussing on the essentials, and acting as a toolkit rather than a pre-packaged solution, ZeroMQ turns the complex problem of distributed computing into a relatively simple recipe. About the speaker Pieter Hintjens is a writer, programmer, and public speaker who has spent decades building large software systems, organizations, and businesses. He designed the Advanced Message Queuing Protocol (AMQP) in 2006 for JPMorganChase and left ...
Crowding with conjunctions of simple features.
Põder, Endel; Wagemans, Johan
Several recent studies have related crowding with the feature integration stage in visual processing. In order to understand the mechanisms involved in this stage, it is important to use stimuli that have several features to integrate, and these features should be clearly defined and measurable. In this study, Gabor patches were used as target and distractor stimuli. The stimuli differed in three dimensions: spatial frequency, orientation, and color. A group of 3, 5, or 7 objects was presented briefly at 4 deg eccentricity of the visual field. The observers' task was to identify the object located in the center of the group. A strong effect of the number of distractors was observed, consistent with various spatial pooling models. The analysis of incorrect responses revealed that these were a mix of feature errors and mislocalizations of the target object. Feature errors were not purely random, but biased by the features of distractors. We propose a simple feature integration model that predicts most of the observed regularities.
Eigenstructure of of singular systems. Perturbation analysis of simple eigenvalues
García Planas, María Isabel; Tarragona Romero, Sonia
The problem to study small perturbations of simple eigenvalues with a change of parameters is of general interest in applied mathematics. After to introduce a systematic way to know if an eigenvalue of a singular system is simple or not, the aim of this work is to study the behavior of a simple eigenvalue of singular linear system family
8 CFR 293.3 - Simple interest table.
... 8 Aliens and Nationality 1 2010-01-01 2010-01-01 false Simple interest table. 293.3 Section 293.3 Aliens and Nationality DEPARTMENT OF HOMELAND SECURITY IMMIGRATION REGULATIONS DEPOSIT OF AND INTEREST ON CASH RECEIVED TO SECURE IMMIGRATION BONDS § 293.3 Simple interest table. Following is a simple interest...
A simple Chelex protocol for DNA extraction from Anopheles spp.
Musapa, Mulenga; Kumwenda, Taida; Mkulama, Mtawa; Chishimba, Sandra; Norris, Douglas E; Thuma, Philip E; Mharakurwa, Sungano
Endemic countries are increasingly adopting molecular tools for efficient typing, identification and surveillance against malaria parasites and vector mosquitoes, as an integral part of their control programs. For sustainable establishment of these accurate approaches in operations research to strengthen malaria control and elimination efforts, simple and affordable methods, with parsimonious reagent and equipment requirements are essential. Here we present a simple Chelex-based technique for extracting malaria parasite and vector DNA from field collected mosquito specimens. We morphologically identified 72 Anopheles gambiae sl. from 156 mosquitoes captured by pyrethrum spray catches in sleeping rooms of households within a 2,000 km(2) vicinity of the Malaria Institute at Macha. After dissection to separate the head and thorax from the abdomen for all 72 Anopheles gambiae sl. mosquitoes, the two sections were individually placed in 1.5 ml microcentrifuge tubes and submerged in 20 μl of deionized water. Using a sterile pipette tip, each mosquito section was separately homogenized to a uniform suspension in the deionized water. Of the ensuing homogenate from each mosquito section, 10 μl was retained while the other 10 μl was transferred to a separate autoclaved 1.5 ml tube. The separate aliquots were subjected to DNA extraction by either the simplified Chelex or the standard salting out extraction protocol(9,10). The salting out protocol is so-called and widely used because it employs high salt concentrations in lieu of hazardous organic solvents (such as phenol and chloroform) for the protein precipitation step during DNA extraction(9). Extracts were used as templates for PCR amplification using primers targeting arthropod mitochondrial nicotinamide adenine dinucleotide dehydrogenase (NADH) subunit 4 gene (ND4) to check DNA quality, a PCR for identification of Anopheles gambiae sibling species(10) and a nested PCR for typing of Plasmodium falciparum infection
Keeping it simple: flowering plants tend to retain, and revert to, simple leaves.
Geeta, R; Dávalos, Liliana M; Levy, André; Bohs, Lynn; Lavin, Mathew; Mummenhoff, Klaus; Sinha, Neelima; Wojciechowski, Martin F
• A wide range of factors (developmental, physiological, ecological) with unpredictable interactions control variation in leaf form. Here, we examined the distribution of leaf morphologies (simple and complex forms) across angiosperms in a phylogenetic context to detect patterns in the directions of changes in leaf shape. • Seven datasets (diverse angiosperms and six nested clades, Sapindales, Apiales, Papaveraceae, Fabaceae, Lepidium, Solanum) were analysed using maximum likelihood and parsimony methods to estimate asymmetries in rates of change among character states. • Simple leaves are most frequent among angiosperm lineages today, were inferred to be ancestral in angiosperms and tended to be retained in evolution (stasis). Complex leaves slowly originated ('gains') and quickly reverted to simple leaves ('losses') multiple times, with a significantly greater rate of losses than gains. Lobed leaves may be a labile intermediate step between different forms. The nested clades showed mixed trends; Solanum, like the angiosperms in general, had higher rates of losses than gains, but the other clades had higher rates of gains than losses. • The angiosperm-wide pattern could be taken as a null model to test leaf evolution patterns in particular clades, in which patterns of variation suggest clade-specific processes that have yet to be investigated fully. © 2011 The Authors. New Phytologist © 2011 New Phytologist Trust.
An Arduino Investigation of Simple Harmonic Motion
Galeriu, Calin; Edwards, Scott; Esper, Geoffrey
We cannot hope for a new generation of scientists and engineers if we don't let our young students take ownership of their scientific and engineering explorations, if we don't let them enjoy the hands-on cycle of design and production, and if we don't let them implant their creativity into a technologically friendly environment. With this educational philosophy in mind, Massimo Banzi and his team have developed and popularized the open source Arduino microcontroller board. The Arduino board has helped countless people in their science, electronics, robotics, or engineering projects, allowing them to build things that we have not even dreamed of. Physics instructors have also realized the advantages of using Arduino boards for lab experiments. The schools are saving money because the homemade experimental equipment is much cheaper than the commercial alternatives. The students are thankful for an educational experience that is more interesting, more loaded with STEM content, and more fun. As further proof of this new trend in physics education, Vernier5 is now documenting the use of their probes with Arduino boards. This is why we have developed an Arduino-based physics investigation of the simple harmonic motion (SHM) of a mass on a spring. The experimental data are collected with the help of an ultrasonic distance sensor and an Arduino Uno board. The data are then graphed and analyzed using Origin 9. This rich cross-curricular STEM activity integrates electronics, computer programming, physics, and mathematics in a way that is both experimentally exciting and intellectually rewarding.
A simple tool for neuroimaging data sharing
Christian eHaselgrove
Full Text Available Data sharing is becoming increasingly common, but despite encouragement and facilitation by funding agencies, journals, and some research efforts, most neuroimaging data acquired today is still not shared due to political, financial, social, and technical barriers to sharing data that remain. In particular, technical solutions are few for researchers that are not a part of larger efforts with dedicated sharing infrastructures, and social barriers such as the time commitment required to share can keep data from becoming publicly available.We present a system for sharing neuroimaging data, designed to be simple to use and to provide benefit to the data provider. The system consists of a server at the International Neuroinformatics Coordinating Facility (INCF and user tools for uploading data to the server. The primary design principle for the user tools is ease of use: the user identifies a directory containing DICOM data, provides their INCF Portal authentication, and provides identifiers for the subject and imaging session. The user tool anonymizes the data and sends it to the server. The server then runs quality control routines on the data, and the data and the quality control reports are made public. The user retains control of the data and may change the sharing policy as they need. The result is that in a few minutes of the user's time, DICOM data can be anonymized and made publicly available, and an initial quality control assessment can be performed on the data. The system is currently functional, and user tools and access to the public image database are available at http://xnat.incf.org/.
Emergent biosynthetic capacity in simple microbial communities.
Hsuan-Chao Chiu
Full Text Available Microbes have an astonishing capacity to transform their environments. Yet, the metabolic capacity of a single species is limited and the vast majority of microorganisms form complex communities and join forces to exhibit capabilities far exceeding those achieved by any single species. Such enhanced metabolic capacities represent a promising route to many medical, environmental, and industrial applications and call for the development of a predictive, systems-level understanding of synergistic microbial capacity. Here we present a comprehensive computational framework, integrating high-quality metabolic models of multiple species, temporal dynamics, and flux variability analysis, to study the metabolic capacity and dynamics of simple two-species microbial ecosystems. We specifically focus on detecting emergent biosynthetic capacity--instances in which a community growing on some medium produces and secretes metabolites that are not secreted by any member species when growing in isolation on that same medium. Using this framework to model a large collection of two-species communities on multiple media, we demonstrate that emergent biosynthetic capacity is highly prevalent. We identify commonly observed emergent metabolites and metabolic reprogramming patterns, characterizing typical mechanisms of emergent capacity. We further find that emergent secretion tends to occur in two waves, the first as soon as the two organisms are introduced, and the second when the medium is depleted and nutrients become limited. Finally, aiming to identify global community determinants of emergent capacity, we find a marked association between the level of emergent biosynthetic capacity and the functional/phylogenetic distance between community members. Specifically, we demonstrate a "Goldilocks" principle, where high levels of emergent capacity are observed when the species comprising the community are functionally neither too close, nor too distant. Taken together
Transfer of Training in Simple Addition.
Chen, Yalin; Campbell, Jamie I D
In recent years several researchers have proposed that skilled adults may solve single-digit addition problems (e.g. 3 + 1 = 4, 4 + 3 = 7) using a fast counting procedure. Practicing a procedure, often leads to transfer of learning and faster performance of unpracticed items. Such transfer has been demonstrated using a counting-based alphabet arithmetic task (e.g., B + 4 = C D E F) that indicated robust generalization of practice (i.e., RT gains) when untrained transfer problems at test had been implicitly practiced (e.g., practice B + 3, test B + 2 or B + 1). Here we constructed analogous simple addition problems (practice 4 + 3, test 4 + 2 or 4 + 1). In each of three experiments (total n = 108) participants received six practice blocks followed by two test blocks of new problems to examine generalization effects. Practice of addition identity-rule problems (i.e., 0 + N = N) showed complete transfer of RT gains made during practice to unpracticed items at test. In contrast, the addition ties (2 + 2, 3 + 3, etc.) presented large RT costs for unpracticed problems at test, but sped up substantially in the second test block. This pattern is consistent with item-specific strengthening of associative memory. The critical items were small non-tie additions (sum ≤ 10) for which the test problems would be implicitly practiced if counting was employed during practice. In all three experiments (and collectively) there was no evidence of generalization for these items in the first test block, but there was robust speed up when the items were repeated in the second test block. Thus, there was no evidence of the generalization of practice that would be expected if counting procedures mediated our participants' performance on small non-tie addition problems.
A simple two-step purification procedure for the iC3b binding collectin conglutinin
Krogh-Meibom, Thomas; Ingvartsen, Klaus Lønne; Tornoe, Ida
Bovine conglutinin is a serum protein involved in innate immunity. It binds calcium dependently to iC3b, a product of the complement component C3 deposited on cell surfaces, immune complexes or artificial surfaces after complement activation. We here present a simple and efficient two-step proced...
A simple approach to the solvent reorganisation Gibbs free energy in electron transfer reactions of redox metalloproteins
Ulstrup, Jens
We discuss a simple model for the environmental reorganisation Gibbs free energy, E-r, in electron transfer between a metalloprotein and a small reaction partner. The protein is represented as a dielectric globule with low dielectric constant, the metal centres as conducting spheres, all embedded...
Production of structured soy-based meat analogues using simple shear and heat in a Couette Cell
Krintiras, G.A.; Gobel, T.W.; Goot, van der A.J.; Stefanidis, G.D.
A Couette Cell device was employed to provide proof of concept for the production of structured meat analogues by application of simple shear flow and heat to a 31 wt% Soy Protein Isolate (SPI)–Wheat Gluten (WG) dispersion. Three relevant process parameters (temperature, time and rotation rate) were
Simple and Reproducible Sample Preparation for Single-Shot Phosphoproteomics with High Sensitivity
Jersie-Christensen, Rosa R.; Sultan, Abida; Olsen, Jesper V
The traditional sample preparation workflow for mass spectrometry (MS)-based phosphoproteomics is time consuming and usually requires multiple steps, e.g., lysis, protein precipitation, reduction, alkylation, digestion, fractionation, and phosphopeptide enrichment. Each step can introduce chemical...... artifacts, in vitro protein and peptide modifications, and contaminations. Those often result in sample loss and affect the sensitivity, dynamic range and accuracy of the mass spectrometric analysis. Here we describe a simple and reproducible phosphoproteomics protocol, where lysis, denaturation, reduction......, and alkylation are performed in a single step, thus reducing sample loss and increasing reproducibility. Moreover, unlike standard cell lysis procedures the cell harvesting is performed at high temperatures (99 °C) and without detergents and subsequent need for protein precipitation. Phosphopeptides are enriched...
In vitro binding of germanium to proteins of rice shoots
Matsumoto, Hideaki; Takahashi, Eiichi
The possibility of in vitro binding between proteins of rice shoots and germanium (Ge) was investigated. The proteins in mixtures of aqueous extracts of rice shoots and radioactive germanium ( 68 GeO 2 ) were fractionated. The binding of radioactivity to the proteins was observed even after 5 successive fractionation steps from the original mixtures. At the final fractionation step using polyacrylamide gel electrophoresis, a constant proportionality between protein concentration and associated radioactivity was found in most samples although not all. These results indicate that the binding of 68 Ge to proteins is not due to the simple adsorption by proteins. (auth.)
Simple, miniaturized blood plasma extraction method.
Kim, Jin-Hee; Woenker, Timothy; Adamec, Jiri; Regnier, Fred E
A rapid plasma extraction technology that collects a 2.5 μL aliquot of plasma within three minutes from a finger-stick derived drop of blood was evaluated. The utility of the plasma extraction cards used was that a paper collection disc bearing plasma was produced that could be air-dried in fifteen minutes and placed in a mailing envelop for transport to an analytical laboratory. This circumvents the need for venipuncture and blood collection in specialized vials by a phlebotomist along with centrifugation and refrigerated storage. Plasma extraction was achieved by applying a blood drop to a membrane stack through which plasma was drawn by capillary action. During the course of plasma migration to a collection disc at the bottom of the membrane stack blood cells were removed by a combination of adsorption and filtration. After the collection disc filled with an aliquot of plasma the upper membranes were stripped from the collection card and the collection disc was air-dried. Intercard differences in the volume of plasma collected varied approximately 1% while volume variations of less than 2% were seen with hematocrit levels ranging from 20% to 71%. Dried samples bearing metabolites and proteins were then extracted from the disc and analyzed. 25-Hydroxy vitamin D was quantified by LC-MS/MS analysis following derivatization with a secosteroid signal enhancing tag that imparted a permanent positive charge to the vitamin and reduced the limit of quantification (LOQ) to 1 pg of collected vitamin on the disc; comparable to values observed with liquid-liquid extraction (LLE) of a venipuncture sample. A similar study using conventional proteomics methods and spectral counting for quantification was conducted with yeast enolase added to serum as an internal standard. The LOQ with extracted serum samples for enolase was 1 μM, linear from 1 to 40 μM, the highest concentration examined. In all respects protein quantification with extracted serum samples was comparable to
A simple and versatile system for the ATP-dependent assembly of chromatin.
Khuong, Mai T; Fei, Jia; Cruz-Becerra, Grisel; Kadonaga, James T
Chromatin is the natural form of DNA in the eukaryotic nucleus and is the substrate for diverse biological phenomena. The functional analysis of these processes ideally would be carried out with nucleosomal templates that are assembled with customized core histones, DNA sequences, and chromosomal proteins. Here we report a simple, reliable, and versatile method for the ATP-dependent assembly of evenly spaced nucleosome arrays. This minimal chromatin assembly system comprises the Drosophila nucleoplasmin-like protein (dNLP) histone chaperone, the imitation switch (ISWI) ATP-driven motor protein, core histones, template DNA, and ATP. The dNLP and ISWI components were synthesized in bacteria, and each protein could be purified in a single step by affinity chromatography. We show that the dNLP-ISWI system can be used with different DNA sequences, linear or circular DNA, bulk genomic DNA, recombinant or native Drosophila core histones, native human histones, the linker histone H1, the non-histone chromosomal protein HMGN2, and the core histone variants H3.3 and H2A.V. The dNLP-ISWI system should be accessible to a wide range of researchers and enable the assembly of customized chromatin with specifically desired DNA sequences, core histones, and other chromosomal proteins. © 2017 by The American Society for Biochemistry and Molecular Biology, Inc.
Protein nanoparticles for therapeutic protein delivery.
Herrera Estrada, L P; Champion, J A
Therapeutic proteins can face substantial challenges to their activity, requiring protein modification or use of a delivery vehicle. Nanoparticles can significantly enhance delivery of encapsulated cargo, but traditional small molecule carriers have some limitations in their use for protein delivery. Nanoparticles made from protein have been proposed as alternative carriers and have benefits specific to therapeutic protein delivery. This review describes protein nanoparticles made by self-assembly, including protein cages, protein polymers, and charged or amphipathic peptides, and by desolvation. It presents particle fabrication and delivery characterization for a variety of therapeutic and model proteins, as well as comparison of the features of different protein nanoparticles.
Analysis of Membrane Protein Topology in the Plant Secretory Pathway.
Guo, Jinya; Miao, Yansong; Cai, Yi
Topology of membrane proteins provides important information for the understanding of protein function and intermolecular associations. Integrate membrane proteins are generally transported from endoplasmic reticulum (ER) to Golgi and downstream compartments in the plant secretory pathway. Here, we describe a simple method to study membrane protein topology along the plant secretory pathway by transiently coexpressing a fluorescent protein (XFP)-tagged membrane protein and an ER export inhibitor protein, ARF1 (T31N), in tobacco BY-2 protoplast. By fractionation, microsome isolation, and trypsin digestion, membrane protein topology could be easily detected by either direct confocal microscopy imaging or western-blot analysis using specific XFP antibodies. A similar strategy in determining membrane protein topology could be widely adopted and applied to protein analysis in a broad range of eukaryotic systems, including yeast cells and mammalian cells.
Design and self-assembly of simple coat proteins for artificial viruses
Hernandez-Garcia, Armando; Kraft, Daniela J.; Janssen, Anne F J; Bomans, Paul H H; Sommerdijk, Nico A J M; Thies-Weesie, Dominique M E; Favretto, Marco E.; Brock, Roland; De Wolf, Frits A.; Werten, Marc W T; Van Der Schoot, Paul; Stuart, Martien Cohen; De Vries, Renko
Viruses are among the simplest biological systems and are highly effective vehicles for the delivery of genetic material into susceptible host cells1. Artificial viruses can be used as model systems for providing insights into natural viruses and can be considered a testing ground for developing
On characterization of anisotropic plant protein structures
Krintiras, G.A.; Göbel, J.; Bouwman, W.G.; Goot, van der A.J.; Stefanidis, G.D.
In this paper, a set of complementary techniques was used to characterize surface and bulk structures of an anisotropic Soy Protein Isolate (SPI)–vital wheat gluten blend after it was subjected to heat and simple shear flow in a Couette Cell. The structured biopolymer blend can form a basis for a
Protein-Protein Interaction Databases
Szklarczyk, Damian; Jensen, Lars Juhl
Years of meticulous curation of scientific literature and increasingly reliable computational predictions have resulted in creation of vast databases of protein interaction data. Over the years, these repositories have become a basic framework in which experiments are analyzed and new directions...
A Simple Metallothionein-Based Biosensor for Enhanced Detection of Arsenic and Mercury
Gordon W. Irvine
Full Text Available Metallothioneins (MTs are a family of cysteine-rich proteins whose biological roles include the regulation of essential metal ions and protection against the harmful effects of toxic metals. Due to its high affinity for many toxic, soft metals, recombinant human MT isoform 1a was incorporated into an electrochemical-based biosensor for the detection of As3+ and Hg2+. A simple design was chosen to maximize its potential in environmental monitoring and MT was physically adsorbed onto paper discs placed on screen-printed carbon electrodes (SPCEs. This system was tested with concentrations of arsenic and mercury typical of contaminated water sources ranging from 5 to 1000 ppb. The analytical performance of the MT-adsorbed paper discs on SPCEs demonstrated a greater than three-fold signal enhancement and a lower detection limit compared to blank SPCEs, 13 ppb for As3+ and 45 ppb for Hg2+. While not being as low as some of the recommended drinking water limits, the sensitivity of the simple MT-biosensor would be potentially useful in monitoring of areas of concern with a known contamination problem. This paper describes the ability of the metal binding protein metallothionein to enhance the effectiveness of a simple, low-cost electrochemical sensor.
Stabilization of Proteins and Noncovalent Protein Complexes during Electrospray Ionization by Amino Acid Additives.
Zhang, Hua; Lu, Haiyan; Chingin, Konstantin; Chen, Huanwen
Ionization of proteins and noncovalent protein complexes with minimal disturbance to their native structure presents a great challenge for biological mass spectrometry (MS). In living organisms, the native structure of intracellular proteins is commonly stabilized by solute amino acids (AAs) accumulated in cells at very high concentrations. Inspired by nature, we hypothesized that AAs could also pose a stabilizing effect on the native structure of proteins and noncovalent protein complexes during ionization. To test this hypothesis, here we explored MS response for various protein complexes upon the addition of free AAs at mM concentrations into the electrospray ionization (ESI) solution. Thermal activation of ESI droplets in the MS inlet capillary was employed as a model destabilizing factor during ionization. Our results indicate that certain AAs, in particular proline (Pro), pose considerable positive effect on the stability of noncovalent protein complexes in ESI-MS without affecting the signal intensity of protein ions and original protein-ligand equilibrium, even when added at the 20 mM concentration. The data suggest that the degree of protein stabilization is primarily determined by the osmolytic and ampholytic characteristics of AA solutes. The highest stability and visibility of noncovalent protein complexes in ESI-MS are achieved using AA additives with neutral isoelectric point, moderate proton affinity, and unfavorable interaction with the native protein state. Overall, our results indicate that the simple addition of free amino acids into the working solution can notably improve the stability and accuracy of protein analysis by native ESI-MS.
Protein detection using biobarcodes.
Müller, Uwe R
Over the past 50 years the development of assays for the detection of protein analytes has been driven by continuing demands for higher levels of sensitivity and multiplexing. The result has been a progression of sandwich-type immunoassays, starting with simple radioisotopic, colorimetric, or fluorescent labeling systems to include various enzymatic or nanostructure-based signal amplification schemes, with a concomitant sensitivity increase of over 1 million fold. Multiplexing of samples and tests has been enabled by microplate and microarray platforms, respectively, or lately by various molecular barcoding systems. Two different platforms have emerged as the current front-runners by combining a nucleic acid amplification step with the standard two-sided immunoassay. In both, the captured protein analyte is replaced by a multiplicity of oligonucleotides that serve as surrogate targets. One of these platforms employs DNA or RNA polymerases for the amplification step, while detection is by fluorescence. The other is based on gold nanoparticles for both amplification as well as detection. The latter technology, now termed Biobarcode, is completely enzyme-free and offers potentially much higher multiplexing power.
Biogas Digester with Simple Solar Heater
Kh S Karimov
Full Text Available ABSTRACT: In this research work, the design, fabrication and investigation of a biogas digester with simple solar heater are presented. For the solar heater, a built-in reverse absorber type heater was used. The maximum temperature (50°C inside the methane tank was taken as a main parameter for the design of the digester. Then, the energy balance equation for the case of a static mass of fluid being heated was used to model the process. The parameters of thermal insulation of the methane tank were also included in the calculations. The biogas digester consisted of a methane tank with built-in solar reverse absorber heater to harness the radiant solar energy for heating the slurry comprising of different organic wastes (dung, sewage, food wastes etc.. The methane tank was initially filled to 70% of its volume with organic wastes from the GIK institute's sewage. The remaining volume was filled with sewage and cow dung from other sources. During a three month period (October-December, 2009 and another two month period (February-March, 2010, the digester was investigated. The effects of solar radiation on the absorber, the slurry's temperature, and the ambient temperature were all measured during these investigations. It was found that using sewage only and sewage with cow dung in the slurry resulted in retention times of four and two weeks, respectively. The corresponding biogas produced was 0.4 m3 and 8.0 m3, respectively. Finally, this paper also elaborates on the upgradation of biogas through the removal of carbon dioxide, hydrogen sulphide and water vapour, and also the process of conversion of biogas energy into electric powerABSTRAK: Kajian ini membentangkan rekabentuk, fabrikasi dan penyelidikan tentang pencerna biogas dengan pemanas solar ringkas. Sebagai pemanas solar, ia dilengkapkan dengan penyerap pemanas beralik. Suhu maksimum(50oC di dalam tangki metana telah diambil sebagai parameter utama rekabentuk pencerna. Dengan menggunakan
Microstructural Fluids and Simple Fluids at Interfaces
Radlinska, Ewa Z.
studied using microcalorimetric specific heat measurements. The calorimetric technique is demonstrated to produce results consistent with the structural information derived from direct observations of the phase diagram and the DOC structural model. Two or possibly three separate cubic phases are distinguished in the cubic phase region. The results indicate that the values of specific heat per surfactant molecule reflect changes in both the surfactant film topology and geometry. The second part of this work is devoted to the study of adsorption of three simple liquids on molecularly smooth mica using an angle-averaging, refractive index-matching ellipsometric technique. A general outline is given in Chapter 8 followed by discussion of the ellipsometric method (Chapter 9). In Chapter 10 experimental results are presented and discussed. For water and cyclohexane the isotherms are structureless. For octamethylcyclotetrasiloxane a layering upon adsorption is observed, showing no qualitative change on passing through the triple point of the bulk liquid. The isotherms for all three liquids studied do not agree with the Lifshitz theory of van der Waals forces and are qualitatively interpreted using the surface phase diagrams of Pandit, Schick and Wortis. (author)
Tissue adhesives for simple traumatic lacerations.
Beam, Joel W
Farion K, Osmond MH, Hartling L, et al. Tissue adhesives for traumatic lacerations in children and adults. Cochrane Database Syst Rev. 2001(4);CD003326. What is the clinical evidence base for tissue adhesives in the management of simple traumatic lacerations? Studies were identified by searches of the following databases: Cochrane Wounds Group Specialized Trials Register (September 2003), Cochrane Central Register of Controlled Trials (CENTRAL) (CDROM 2003, issue 3), MEDLINE (1966 to September 2003, week 1), EMBASE (1988 to 2003, week 36), Web of Science Science Citation Index (1975 to September 13, 2003) and various clinical trials registers (September 2003). Investigators and product manufacturers were contacted to identify additional eligible studies. The search terms included wounds and injuries, laceration, face injury, nose injury, tissue adhesives, and acrylates. Each study fulfilled the following criteria: (1) The study was a randomized controlled trial that compared tissue adhesives with standard wound closure (SWC) (sutures, staples, adhesive strips) or tissue adhesive with tissue adhesive. (2) The wounds were acute, linear lacerations less than 12 hours old, resulting from blunt or sharp trauma. (3) The wound length, width, and depth allowed for approximation of the edges with minimal tension after deep sutures were placed, if required. Studies were included with no language or publication status restriction, with participants of any age recruited in an emergency department, outpatient clinic, walk-in clinic, or other primary care setting. Studies were excluded if the wounds were stellate lacerations, puncture wounds, mammalian bites, infected, heavily contaminated or devitalized, crossing joints or mucocutaneous junctions, in hair-bearing areas, or in patients with keloid formation or chronic illness. The characteristics of the study and participants, interventions, outcome measures, and findings were extracted by one author and verified by a second
Simple, economical heat-shock devices for zebrafish housing racks.
Duszynski, Robert J; Topczewski, Jacek; LeClair, Elizabeth E
One reason for the popularity of the zebrafish (Danio rerio) as a model vertebrate is the ability to manipulate gene expression in this organism. A common method is to induce gene expression transiently under control of a heat-shock promoter (e.g., hsp70l). By making simple mechanical adjustments to small aquarium heaters (25-50W), we were able to produce consistent and reliable heat-shock conditions within a conventional zebrafish housing system. Up to two heat-shock intervals per day (>37°C) could be maintained under conditions of continuous flow (5-25 mL/min). Temperature logging every 30 s indicated rapid warm up times, consistent heat-shock lengths, and accurate and precise peak water temperatures (mean±SD=38°C±0.2°C). The biological effects of these heat-shock treatments were confirmed by observing inducible expression of enhanced green fluorescent protein (EGFP) and inhibition of caudal fin regeneration in a transgenic fish line expressing a dominant negative fibroblast growth factor receptor (Tg(hsp70l:dnfgfr1-EGFP)(pd1)). These devices are inexpensive, easily modified, and can be calibrated to accommodate a variety of experimental designs. After setup on a programmable timer, the heaters require no intervention to produce consistent daily heat shocks, and all other standard care protocols can be followed in the fish facility. The simplicity and stability of these devices make them suitable for long-term heat shocks at any stage of the zebrafish lifecycle (>7 days postfertilization), and useful for both laboratory and classroom experiments on transgenic zebrafish.
Development of a simple estimation tool for LMFBR construction cost
Yoshida, Kazuo; Kinoshita, Izumi
A simple tool for estimating the construction costs of liquid-metal-cooled fast breeder reactors (LMFBRs), 'Simple Cost' was developed in this study. Simple Cost is based on a new estimation formula that can reduce the amount of design data required to estimate construction costs. Consequently, Simple cost can be used to estimate the construction costs of innovative LMFBR concepts for which detailed design has not been carried out. The results of test calculation show that Simple Cost provides cost estimations equivalent to those obtained with conventional methods within the range of plant power from 325 to 1500 MWe. Sensitivity analyses for typical design parameters were conducted using Simple Cost. The effects of four major parameters - reactor vessel diameter, core outlet temperature, sodium handling area and number of secondary loops - on the construction costs of LMFBRs were evaluated quantitatively. The results show that the reduction of sodium handling area is particularly effective in reducing construction costs. (author)
Feature generation and representations for protein-protein interaction classification.
Lan, Man; Tan, Chew Lim; Su, Jian
Automatic detecting protein-protein interaction (PPI) relevant articles is a crucial step for large-scale biological database curation. The previous work adopted POS tagging, shallow parsing and sentence splitting techniques, but they achieved worse performance than the simple bag-of-words representation. In this paper, we generated and investigated multiple types of feature representations in order to further improve the performance of PPI text classification task. Besides the traditional domain-independent bag-of-words approach and the term weighting methods, we also explored other domain-dependent features, i.e. protein-protein interaction trigger keywords, protein named entities and the advanced ways of incorporating Natural Language Processing (NLP) output. The integration of these multiple features has been evaluated on the BioCreAtIvE II corpus. The experimental results showed that both the advanced way of using NLP output and the integration of bag-of-words and NLP output improved the performance of text classification. Specifically, in comparison with the best performance achieved in the BioCreAtIvE II IAS, the feature-level and classifier-level integration of multiple features improved the performance of classification 2.71% and 3.95%, respectively.
SCRAED - Simple and Complex Random Assignment in Experimental Designs
Alferes, Valentim R.
SCRAED is a package of 37 self-contained SPSS syntax files that performs simple and complex random assignment in experimental designs. For between-subjects designs, SCRAED includes simple random assignment (no restrictions, forced equal sizes, forced unequal sizes, and unequal probabilities), block random assignment (simple and generalized blocks), and stratified random assignment (no restrictions, forced equal sizes, forced unequal sizes, and unequal probabilities). For within-subject...
A New Simple Approach for Entropy and Carnot Theorem
Veliev, E. V.
Entropy and Carnot theorem occupy central place in the typical Thermodynamics courses at the university level. In this work, we suggest a new simple approach for introducing the concept of entropy. Using simple procedure in TV plane, we proved that for reversible processes ∫dQ/T=0 and it is sufficient to define entropy. And also, using reversible processes in TS plane, we give an alternative simple proof for Carnot theorem
PRODt;CTION OF SINGLE CELL PROTEIN FROM BREWERY ...
customary food and feed sources of protein (agriculnrre and fishery) to ocher sources like single cell protein (SCP); whose production from hydrocarbons is one ... origin is unicellular or simple multicellular organism such as bacteria, yeasts, fungi, algae. protozoa, mid even bacterinphagcs generally cultivated on substrates ...
Directional Darwinian Selection in proteins.
McClellan, David A
Molecular evolution is a very active field of research, with several complementary approaches, including dN/dS, HON90, MM01, and others. Each has documented strengths and weaknesses, and no one approach provides a clear picture of how natural selection works at the molecular level. The purpose of this work is to present a simple new method that uses quantitative amino acid properties to identify and characterize directional selection in proteins. Inferred amino acid replacements are viewed through the prism of a single physicochemical property to determine the amount and direction of change caused by each replacement. This allows the calculation of the probability that the mean change in the single property associated with the amino acid replacements is equal to zero (H0: μ = 0; i.e., no net change) using a simple two-tailed t-test. Example data from calanoid and cyclopoid copepod cytochrome oxidase subunit I sequence pairs are presented to demonstrate how directional selection may be linked to major shifts in adaptive zones, and that convergent evolution at the whole organism level may be the result of convergent protein adaptations. Rather than replace previous methods, this new method further complements existing methods to provide a holistic glimpse of how natural selection shapes protein structure and function over evolutionary time.
Determination of Urine Albumin by New Simple High-Performance Liquid Chromatography Method.
Klapkova, Eva; Fortova, Magdalena; Prusa, Richard; Moravcova, Libuse; Kotaska, Karel
A simple high-performance liquid chromatography (HPLC) method was developed for the determination of albumin in patients' urine samples without coeluting proteins and was compared with the immunoturbidimetric determination of albumin. Urine albumin is important biomarker in diabetic patients, but part of it is immuno-nonreactive. Albumin was determined by high-performance liquid chromatography (HPLC), UV detection at 280 nm, Zorbax 300SB-C3 column. Immunoturbidimetric analysis was performed using commercial kit on automatic biochemistry analyzer COBAS INTEGRA ® 400, Roche Diagnostics GmbH, Manheim, Germany. The HLPC method was fully validated. No significant interference with other proteins (transferrin, α-1-acid glycoprotein, α-1-antichymotrypsin, antitrypsin, hemopexin) was found. The results from 301 urine samples were compared with immunochemical determination. We found a statistically significant difference between these methods (P = 0.0001, Mann-Whitney test). New simple HPLC method was developed for the determination of urine albumin without coeluting proteins. Our data indicate that the HPLC method is highly specific and more sensitive than immunoturbidimetry. © 2016 Wiley Periodicals, Inc.
Dependency on de novo protein synthesis and proteomic changes during metamorphosis of the marine bryozoan Bugula neritina
Wong, Yue Him; Arellano, Shawn M; Zhang, Huoming; Ravasi, Timothy; Qian, Pei-Yuan
synthesis of proteins and, instead, involves post-translational modifications of existing proteins, providing a simple mechanism to quickly initiate metamorphosis. To test our hypothesis, we challenged B. neritina larvae with transcription and translation
Black Holes Have Simple Feeding Habits
The biggest black holes may feed just like the smallest ones, according to data from NASA's Chandra X-ray Observatory and ground-based telescopes. This discovery supports the implication of Einstein's relativity theory that black holes of all sizes have similar properties, and will be useful for predicting the properties of a conjectured new class of black holes. The conclusion comes from a large observing campaign of the spiral galaxy M81, which is about 12 million light years from Earth. In the center of M81 is a black hole that is about 70 million times more massive than the Sun, and generates energy and radiation as it pulls gas in the central region of the galaxy inwards at high speed. In contrast, so-called stellar mass black holes, which have about 10 times more mass than the Sun, have a different source of food. These smaller black holes acquire new material by pulling gas from an orbiting companion star. Because the bigger and smaller black holes are found in different environments with different sources of material to feed from, a question has remained about whether they feed in the same way. Using these new observations and a detailed theoretical model, a research team compared the properties of M81's black hole with those of stellar mass black holes. The results show that either big or little, black holes indeed appear to eat similarly to each other, and produce a similar distribution of X-rays, optical and radio light. AnimationMulti-wavelength Images of M81 One of the implications of Einstein's theory of General Relativity is that black holes are simple objects and only their masses and spins determine their effect on space-time. The latest research indicates that this simplicity manifests itself in spite of complicated environmental effects. "This confirms that the feeding patterns for black holes of different sizes can be very similar," said Sera Markoff of the Astronomical Institute, University of Amsterdam in the Netherlands, who led the study
Simple Rules, Not So Simple: The Use of International Ovarian Tumor Analysis (IOTA) Terminology and Simple Rules in Inexperienced Hands in a Prospective Multicenter Cohort Study.
Meys, Evelyne; Rutten, Iris; Kruitwagen, Roy; Slangen, Brigitte; Lambrechts, Sandrina; Mertens, Helen; Nolting, Ernst; Boskamp, Dieuwke; Van Gorp, Toon
To analyze how well untrained examiners - without experience in the use of International Ovarian Tumor Analysis (IOTA) terminology or simple ultrasound-based rules (simple rules) - are able to apply IOTA terminology and simple rules and to assess the level of agreement between non-experts and an expert. This prospective multicenter cohort study enrolled women with ovarian masses. Ultrasound was performed by non-expert examiners and an expert. Ultrasound features were recorded using IOTA nomenclature, and used for classifying the mass by simple rules. Interobserver agreement was evaluated with Fleiss' kappa and percentage agreement between observers. 50 consecutive women were included. We observed 46 discrepancies in the description of ovarian masses when non-experts utilized IOTA terminology. Tumor type was misclassified often (n = 22), resulting in poor interobserver agreement between the non-experts and the expert (kappa = 0.39, 95 %-CI 0.244 - 0.529, percentage of agreement = 52.0 %). Misinterpretation of simple rules by non-experts was observed 57 times, resulting in an erroneous diagnosis in 15 patients (30 %). The agreement for classifying the mass as benign, malignant or inconclusive by simple rules was only moderate between the non-experts and the expert (kappa = 0.50, 95 %-CI 0.300 - 0.704, percentage of agreement = 70.0 %). The level of agreement for all 10 simple rules features varied greatly (kappa index range: -0.08 - 0.74, percentage of agreement 66 - 94 %). Although simple rules are useful to distinguish benign from malignant adnexal masses, they are not that simple for untrained examiners. Training with both IOTA terminology and simple rules is necessary before simple rules can be introduced into guidelines and daily clinical practice. © Georg Thieme Verlag KG Stuttgart · New York.
Aquaporin Protein-Protein Interactions
Jennifer Virginia Roche
Full Text Available Aquaporins are tetrameric membrane-bound channels that facilitate transport of water and other small solutes across cell membranes. In eukaryotes, they are frequently regulated by gating or trafficking, allowing for the cell to control membrane permeability in a specific manner. Protein–protein interactions play crucial roles in both regulatory processes and also mediate alternative functions such as cell adhesion. In this review, we summarize recent knowledge about aquaporin protein–protein interactions; dividing the interactions into three types: (1 interactions between aquaporin tetramers; (2 interactions between aquaporin monomers within a tetramer (hetero-tetramerization; and (3 transient interactions with regulatory proteins. We particularly focus on the structural aspects of the interactions, discussing the small differences within a conserved overall fold that allow for aquaporins to be differentially regulated in an organism-, tissue- and trigger-specific manner. A deep knowledge about these differences is needed to fully understand aquaporin function and regulation in many physiological processes, and may enable design of compounds targeting specific aquaporins for treatment of human disease.
Protein immobilization strategies for protein biochips
Rusmini, F.; Rusmini, Federica; Zhong, Zhiyuan; Feijen, Jan
In the past few years, protein biochips have emerged as promising proteomic and diagnostic tools for obtaining information about protein functions and interactions. Important technological innovations have been made. However, considerable development is still required, especially regarding protein
Hubungan antara konsumsi protein dengan produksi, protein dan laktosa susu kambing Peranakan Ettawa
Galuh Estu Prihatiningsih
Full Text Available The study aimed to determine a correlation between crude protein intake, milk production, milk protein and milk lactose. This study used purposive sampling method. The sample used in this study were 35 Etawa crossbred goats with months of lactation 4-5 and lactation periods 2-3. Parameters observed were crude protein intake, milk production, milk protein and milk lactose. Data were analyzed using correlation analysis and simple linear regression. The result showed that crude protein intake, total milk production concentrations of milk protein and lactose were 0.77 kg/day; 0.30 kg/day; 0.196% and 3.32% respectively. There was a medium positive linear correlation between the crude protein intake with total milk production, protein and lactose content of milk. The correlation coefficient (r were 0.258; 0.254 and 0,255 respectively. It could be concluded that the higher crude protein intake would increase the amount of milk production, protein and lactose contents. Keywords: crude protein intake, total milk production, milk protein, milk lactose
Metabolic behavior of cell surface biotinylated proteins
Hare, J.F.; Lee, E.
The turnover of proteins on the surface of cultured mammalian cells was measured by a new approach. Reactive free amino or sulfhydryl groups on surface-accessible proteins were derivatized with biotinyl reagents and the proteins solubilized from culture dishes with detergent. Solubilized, biotinylated proteins were then adsorbed onto streptavidin-agarose, released with sodium dodecyl sulfate and mercaptoethanol, and separated on polyacrylamide gels. Biotin-epsilon-aminocaproic acid N-hydroxysuccinimide ester (BNHS) or N-biotinoyl-N'-(maleimidohexanoyl)hydrazine (BM) were the derivatizing agents. Only 10-12 bands were adsorbed onto streptavidin-agarose from undervatized cells or from derivatized cells treated with free avidin at 4 degrees C. Two-dimensional isoelectric focusing-sodium dodecyl sulfate gel electrophoresis resolved greater than 100 BNHS-derivatized proteins and greater than 40 BM-derivatized proteins. There appeared to be little overlap between the two groups of derivatized proteins. Short-term pulse-chase studies showed an accumulation of label into both groups of biotinylated proteins up until 1-2 h of chase and a rapid decrease over the next 1-5 h. Delayed appearance of labeled protein at the cell surface was attributed to transit time from site of synthesis. The unexpected and unexplained rapid disappearance of pulse-labeled proteins from the cell surface was invariant for all two-dimensionally resolved proteins and was sensitive to temperature reduction to 18 degrees C. Long-term pulse-chase experiments beginning 4-8 h after the initiation of chase showed the disappearance of derivatized proteins to be a simple first-order process having a half-life of 115 h in the case of BNHS-derivatized proteins and 30 h in the case of BM-derivatized proteins
Protein Adsorption and Its Role in Bacterial Film Development
only the secondary antibody conjugated to alkaline phosphatase was used. Combined Amino Acids as Measured by HPLC We are interested in a simple, direct...specific assay for chitin that relies on the lectin, wheat germ agglutinin (WGA). Lectins are a general class of proteins that bind to carbohydrates. The...protein; 2) a new method for measuring combined amino acids (includes proteins) in seawater was shown to measure higher concentration than the old
Novel expressed sequence tag- simple sequence repeats (EST ...
Using different bioinformatic criteria, the SUCEST database was used to mine for simple sequence repeat (SSR) markers. Among 42,189 clusters, 1,425 expressed sequence tag- simple sequence repeats (EST-SSRs) were identified in silico. Trinucleotide repeats were the most abundant SSRs detected. Of 212 primer pairs ...
A simple method for human peripheral blood monocyte Isolation
Marcos C de Almeida
Full Text Available We describe a simple method using percoll gradient for isolation of highly enriched human monocytes. High numbers of fully functional cells are obtained from whole blood or buffy coat cells. The use of simple laboratory equipment and a relatively cheap reagent makes the described method a convenient approach to obtaining human monocytes.
Temperature dependence of transport coefficients of 'simple liquid ...
... has been investigated. The study carried out at two densities, r* = 0.60 and r* = 0.95. Result shows erratic variations of the shear viscosity in the two lattices structures. KeyWords: Temperature effect, face centred, simple cubic, transport properties, simple liquid. [Global Jnl Pure & Appl. Sci. Vol.9(3) 2003: 403-406]Â ...
A simple technique to increase profits in wood products marketing
George B. Harpole
Mathematical models can be used to solve quickly some simple day-to-day marketing problems. This note explains how a sawmill production manager, who has an essentially fixed-capacity mill, can solve several optimization problems by using pencil and paper, a forecast of market prices, and a simple algorithm. One such problem is to maximize profits in an operating period...
A SIMPLE AND EFFECTIVE CURSIVE WORD SEGMENTATION METHOD
nicchiotti, G.; Rimassa, S.; Scagliola, C.
A simple procedure for cursive word oversegmentation is presented, which is based on the analysis of the handwritten profiles and on the extraction of ``white holes\\'\\'. It follows the policy of using simple rules on complex data and sophisticated rules on simpler data. Experimental results show
Secondary School Students' Misconceptions about Simple Electric Circuits
Küçüközer, Hüseyin; Kocakülah, Sabri
The aim of this study is to reveal secondary school students' misconceptions about simple electric circuits and to define whether specific misconceptions peculiar to Turkish students exist within those identified. Data were obtained with a conceptual understanding test for simple electric circuits and semi-structured interviews. Conceptual…
A simple polarized-based diffused reflectance colour imaging system
A simple polarized-based diffuse reflectance imaging system has been developed. The system is designed for both in vivo and in vitro imaging of agricultural specimen in the visible region. The system uses a commercial web camera and a halogen lamp that makes it relatively simple and less expensive for diagnostic ...
Acute osteomyelitis complicating a simple fracture. A case report.
Drummond-Webb, J J; Schnaid, E
Osteomyelitis complicating a simple fracture is unusual, but the reason for its rarity is unknown. We report on a 9-year-old Black boy who developed acute osteomyelitis after sustaining an acute simple fracture of the femur. The causative role of trauma in acute osteomyelitis is discussed and 'the relative resistance of healthy bone' questioned.
Simple and Effective Algorithms: Computer-Adaptive Testing.
Linacre, John Michael
Computer-adaptive testing (CAT) allows improved security, greater scoring accuracy, shorter testing periods, quicker availability of results, and reduced guessing and other undesirable test behavior. Simple approaches can be applied by the classroom teacher, or other content specialist, who possesses simple computer equipment and elementary…
13 Comparative Effects of Cassava Starch and Simple Sugar in ...
Arc. Usman A. Jalam
Abstract. Comparative effects of simple laboratory quality sugar and cassava starch on grade C35 concrete were studied in the laboratory. The simple white sugar was used at concentrations of 0 to 1% by weight of cement in concrete cured at 3, 7, 14 and 28 days using ordinary Portland cement. Cassava starch of the same ...
CMOS Active-Pixel Image Sensor With Simple Floating Gates
Fossum, Eric R.; Nakamura, Junichi; Kemeny, Sabrina E.
Experimental complementary metal-oxide/semiconductor (CMOS) active-pixel image sensor integrated circuit features simple floating-gate structure, with metal-oxide/semiconductor field-effect transistor (MOSFET) as active circuit element in each pixel. Provides flexibility of readout modes, no kTC noise, and relatively simple structure suitable for high-density arrays. Features desirable for "smart sensor" applications.
A simple and rapid method to estimate radiocesium in man
Kindl, P.; Steger, F.
A simple and rapid method for monitoring internal contamination of radiocesium in man was developed. This method is based on measurements of the γ-rays emitted from the muscular parts between the thights by a simple NaJ(Tl)-system. The experimental procedure, the calibration, the estimation of the body activity and results are explained and discussed. (Authors)
Using "Tracker" to Prove the Simple Harmonic Motion Equation
Kinchin, John
Simple harmonic motion (SHM) is a common topic for many students to study. Using the free, though versatile, motion tracking software; "Tracker", we can extend the students experience and show that the general equation for SHM does lead to the correct period of a simple pendulum.
Can understanding the packing of side chains improve the design of protein-protein interactions?
Zhou, Alice; O'Hern, Corey; Regan, Lynne
With the long-term goal to improve the design of protein-protein interactions, we have begun extensive computational studies to understand how side-chains of key residues of binding partners geometrically fit together at protein-peptide interfaces, e.g. the tetratrico-peptide repeat protein and its cognate peptide). We describe simple atomic-scale models of hydrophobic dipeptides, which include hard-core repulsion, bond length and angle constraints, and Van der Waals attraction. By completely enumerating all minimal energy structures in these systems, we are able to reproduce important features of the probability distributions of side chain dihedral angles of hydrophic residues in the protein data bank. These results are the crucial first step in developing computational models that can predict the side chain conformations of residues at protein-peptide interfaces. CSO acknowledges support from NSF grant no. CMMT-1006527.
Emergence of modularity and disassortativity in protein-protein interaction networks.
Wan, Xi; Cai, Shuiming; Zhou, Jin; Liu, Zengrong
In this paper, we present a simple evolution model of protein-protein interaction networks by introducing a rule of small-preference duplication of a node, meaning that the probability of a node chosen to duplicate is inversely proportional to its degree, and subsequent divergence plus nonuniform heterodimerization based on some plausible mechanisms in biology. We show that our model cannot only reproduce scale-free connectivity and small-world pattern, but also exhibit hierarchical modularity and disassortativity. After comparing the features of our model with those of real protein-protein interaction networks, we believe that our model can provide relevant insights into the mechanism underlying the evolution of protein-protein interaction networks. © 2010 American Institute of Physics.
Relative quantification of protein-protein interactions using a dual luciferase reporter pull-down assay system.
Shuaizheng Jia
Full Text Available The identification and quantitative analysis of protein-protein interactions are essential to the functional characterization of proteins in the post-proteomics era. The methods currently available are generally time-consuming, technically complicated, insensitive and/or semi-quantitative. The lack of simple, sensitive approaches to precisely quantify protein-protein interactions still prevents our understanding of the functions of many proteins. Here, we develop a novel dual luciferase reporter pull-down assay by combining a biotinylated Firefly luciferase pull-down assay with a dual luciferase reporter assay. The biotinylated Firefly luciferase-tagged protein enables rapid and efficient isolation of a putative Renilla luciferase-tagged binding protein from a relatively small amount of sample. Both of these proteins can be quantitatively detected using the dual luciferase reporter assay system. Protein-protein interactions, including Fos-Jun located in the nucleus; MAVS-TRAF3 in cytoplasm; inducible IRF3 dimerization; viral protein-regulated interactions, such as MAVS-MAVS and MAVS-TRAF3; IRF3 dimerization; and protein interaction domain mapping, are studied using this novel assay system. Herein, we demonstrate that this dual luciferase reporter pull-down assay enables the quantification of the relative amounts of interacting proteins that bind to streptavidin-coupled beads for protein purification. This study provides a simple, rapid, sensitive, and efficient approach to identify and quantify relative protein-protein interactions. Importantly, the dual luciferase reporter pull-down method will facilitate the functional determination of proteins.
Targeted amino-terminal acetylation of recombinant proteins in E. coli.
Full Text Available One major limitation in the expression of eukaryotic proteins in bacteria is an inability to post-translationally modify the expressed protein. Amino-terminal acetylation is one such modification that can be essential for protein function. By co-expressing the fission yeast NatB complex with the target protein in E.coli, we report a simple and widely applicable method for the expression and purification of functional N-terminally acetylated eukaryotic proteins.
Depletion-induced instability in protein-DNA mixtures: Influence of protein charge and size
Vries, de R.J.
While there is abundant experimental and theoretical work on polymer-induced DNA condensation, it is still unclear whether globular proteins can condense linear DNA or not. We develop a simple analytical approximation for the depletion attraction between rodlike segments of semiflexible
Pseudo-simple heteroclinic cycles in R4
Chossat, Pascal; Lohse, Alexander; Podvigina, Olga
We study pseudo-simple heteroclinic cycles for a Γ-equivariant system in R4 with finite Γ ⊂ O(4) , and their nearby dynamics. In particular, in a first step towards a full classification - analogous to that which exists already for the class of simple cycles - we identify all finite subgroups of O(4) admitting pseudo-simple cycles. To this end we introduce a constructive method to build equivariant dynamical systems possessing a robust heteroclinic cycle. Extending a previous study we also investigate the existence of periodic orbits close to a pseudo-simple cycle, which depends on the symmetry groups of equilibria in the cycle. Moreover, we identify subgroups Γ ⊂ O(4) , Γ ⊄ SO(4) , admitting fragmentarily asymptotically stable pseudo-simple heteroclinic cycles. (It has been previously shown that for Γ ⊂ SO(4) pseudo-simple cycles generically are completely unstable.) Finally, we study a generalized heteroclinic cycle, which involves a pseudo-simple cycle as a subset.
Surface modification of protein enhances encapsulation in chitosan nanoparticles
Koyani, Rina D.; Andrade, Mariana; Quester, Katrin; Gaytán, Paul; Huerta-Saquero, Alejandro; Vazquez-Duhalt, Rafael
Chitosan nanoparticles have a huge potential as nanocarriers for environmental and biomedical purposes. Protein encapsulation in nano-sized chitosan provides protection against inactivation, proteolysis, and other alterations due to environmental conditions, as well as the possibility to be targeted to specific tissues by ligand functionalization. In this work, we demonstrate that the chemical modification of the protein surface enhances the protein loading in chitosan nanocarriers. Encapsulation of green fluorescent protein and the cytochrome P450 was studied. The increase of electrostatic interactions between the free amino groups of chitosan and the increased number of free carboxylic groups in the protein surface enhance the protein loading, protein retention, and, thus, the enzymatic activity of chitosan nanoparticles. The chemical modification of protein surface with malonic acid moieties reduced drastically the protein isoelectric point increasing the protein interaction with the polycationic biomaterial and chitosan. The chemical modification of protein does not alter the morphology of chitosan nanoparticles that showed an average diameter of 18 nm, spheroidal in shape, and smooth surfaced. The strategy of chemical modification of protein surface, shown here, is a simple and efficient technique to enhance the protein loading in chitosan nanoparticles. This technique could be used for other nanoparticles based on polycationic or polyanionic materials. The increase of protein loading improves, doubtless, the performance of protein-loaded chitosan nanoparticles for biotechnological and biomedical applications.
Combinatorial structures to modeling simple games and applications
Molinero, Xavier
We connect three different topics: combinatorial structures, game theory and chemistry. In particular, we establish the bases to represent some simple games, defined as influence games, and molecules, defined from atoms, by using combinatorial structures. First, we characterize simple games as influence games using influence graphs. It let us to modeling simple games as combinatorial structures (from the viewpoint of structures or graphs). Second, we formally define molecules as combinations of atoms. It let us to modeling molecules as combinatorial structures (from the viewpoint of combinations). It is open to generate such combinatorial structures using some specific techniques as genetic algorithms, (meta-)heuristics algorithms and parallel programming, among others.
Results of the Monte Carlo 'simple case' benchmark exercise
A new 'simple case' benchmark intercomparison exercise was launched, intended to study the importance of the fundamental nuclear data constants, physics treatments and geometry model approximations, employed by Monte Carlo codes in common use. The exercise was also directed at determining the level of agreement which can be expected between measured and calculated quantities, using current state or the art modelling codes and techniques. To this end, measurements and Monte Carlo calculations of the total (or gross) neutron count rates have been performed using a simple moderated 3 He cylindrical proportional counter array or 'slab monitor' counting geometry, deciding to select a very simple geometry for this exercise
Simple mucin-type carbohydrate antigens in major salivary glands
Therkildsen, M H; Mandel, U; Thorn, J
Simple mucin-type carbohydrate antigens Tn, sialosyl-Tn and T are often markers of neoplastic transformation and have very limited expression in normal tissues. We performed an immunohistological study of simple mucin-type carbohydrate antigens, including H and A variants, with well......-defined monoclonal antibodies (MAb) on frozen and paraffin-embedded normal salivary gland tissue from 22 parotid, 14 submandibular, six sublingual, and 13 labial glands to elucidate the simple mucin-type glycosylation pattern in relation to cyto- and histodifferentiation. The investigated carbohydrate structures...
Generalized Lotka—Volterra systems connected with simple Lie algebras
Charalambides, Stelios A; Damianou, Pantelis A; Evripidou, Charalambos A
We devise a new method for producing Hamiltonian systems by constructing the corresponding Lax pairs. This is achieved by considering a larger subset of the positive roots than the simple roots of the root system of a simple Lie algebra. We classify all subsets of the positive roots of the root system of type A n for which the corresponding Hamiltonian systems are transformed, via a simple change of variables, to Lotka-Volterra systems. For some special cases of subsets of the positive roots of the root system of type A n , we produce new integrable Hamiltonian systems. (paper)
Charalambides, Stelios A.; Damianou, Pantelis A.; Evripidou, Charalambos A.
We devise a new method for producing Hamiltonian systems by constructing the corresponding Lax pairs. This is achieved by considering a larger subset of the positive roots than the simple roots of the root system of a simple Lie algebra. We classify all subsets of the positive roots of the root system of type An for which the corresponding Hamiltonian systems are transformed, via a simple change of variables, to Lotka-Volterra systems. For some special cases of subsets of the positive roots of the root system of type An, we produce new integrable Hamiltonian systems.
Injectable nanocomposite cryogels for versatile protein drug delivery.
Koshy, Sandeep T; Zhang, David K Y; Grolman, Joshua M; Stafford, Alexander G; Mooney, David J
Sustained, localized protein delivery can enhance the safety and activity of protein drugs in diverse disease settings. While hydrogel systems are widely studied as vehicles for protein delivery, they often suffer from rapid release of encapsulated cargo, leading to a narrow duration of therapy, and protein cargo can be denatured by incompatibility with the hydrogel crosslinking chemistry. In this work, we describe injectable nanocomposite hydrogels that are capable of sustained, bioactive, release of a variety of encapsulated proteins. Injectable and porous cryogels were formed by bio-orthogonal crosslinking of alginate using tetrazine-norbornene coupling. To provide sustained release from these hydrogels, protein cargo was pre-adsorbed to charged Laponite nanoparticles that were incorporated within the walls of the cryogels. The presence of Laponite particles substantially hindered the release of a number of proteins that otherwise showed burst release from these hydrogels. By modifying the Laponite content within the hydrogels, the kinetics of protein release could be precisely tuned. This versatile strategy to control protein release simplifies the design of hydrogel drug delivery systems. Here we present an injectable nanocomposite hydrogel for simple and versatile controlled release of therapeutic proteins. Protein release from hydrogels often requires first entrapping the protein in particles and embedding these particles within the hydrogel to allow controlled protein release. This pre-encapsulation process can be cumbersome, can damage the protein's activity, and must be optimized for each protein of interest. The strategy presented in this work simply premixes the protein with charged nanoparticles that bind strongly with the protein. These protein-laden particles are then placed within a hydrogel and slowly release the protein into the surrounding environment. Using this method, tunable release from an injectable hydrogel can be achieved for a variety of | CommonCrawl |
# Introduction to SciPy and its optimization capabilities
SciPy is a powerful Python library that provides a wide range of mathematical and scientific computing tools. One of the key features of SciPy is its optimization module, which allows users to find the optimal values of functions, given certain constraints and criteria. In this textbook, we will explore the optimization capabilities of SciPy, with a focus on exponential and logarithmic functions.
SciPy's optimization module, `scipy.optimize`, provides a variety of algorithms and techniques for finding optimal values. These include gradient descent, Newton-Raphson, and trust-region methods, among others. It also includes specialized solvers for specific types of problems, such as linear programming and constrained optimization.
In this textbook, we will cover the following topics:
- Optimizing exponential functions using SciPy
- Optimizing logarithmic functions using SciPy
- Understanding the optimization process and methods
- Applying optimization to real-world problems
- Handling constraints and limitations in optimization
- Advanced optimization techniques and algorithms
- Integrating optimization into Python applications
- Best practices for optimizing functions with SciPy
- Conclusion: The importance of optimization in modern data analysis
- References and further reading
Let's start by exploring the optimization capabilities of SciPy in detail.
# Optimizing exponential functions using SciPy
Consider the following exponential function:
$$f(x) = 2e^x - 3$$
We want to find the value of $x$ that minimizes this function.
To optimize this function using SciPy, we can use the `scipy.optimize.minimize` function. This function requires two arguments: the function to optimize and the initial guess for the value of $x$.
Here's an example:
```python
import numpy as np
from scipy.optimize import minimize
def exponential_function(x):
return 2 * np.exp(x) - 3
result = minimize(exponential_function, 0)
print(result.x)
```
This code will output the value of $x$ that minimizes the exponential function.
## Exercise
Find the value of $x$ that minimizes the following exponential function:
$$f(x) = e^x - 2$$
To solve this exercise, you can use the same approach as before:
```python
def exponential_function(x):
return np.exp(x) - 2
result = minimize(exponential_function, 0)
print(result.x)
```
This code will output the value of $x$ that minimizes the exponential function.
# Optimizing logarithmic functions using SciPy
Consider the following logarithmic function:
$$f(x) = \ln(x) - 2$$
We want to find the value of $x$ that maximizes this function.
To optimize this function using SciPy, we can use the `scipy.optimize.minimize` function with the `method='SLSQP'` argument. This function requires two arguments: the function to optimize and the initial guess for the value of $x$.
Here's an example:
```python
import numpy as np
from scipy.optimize import minimize
def logarithmic_function(x):
return -np.log(x) + 2
result = minimize(logarithmic_function, 1, method='SLSQP')
print(result.x)
```
This code will output the value of $x$ that maximizes the logarithmic function.
## Exercise
Find the value of $x$ that maximizes the following logarithmic function:
$$f(x) = \ln(x) - 1$$
To solve this exercise, you can use the same approach as before:
```python
def logarithmic_function(x):
return -np.log(x) + 1
result = minimize(logarithmic_function, 1, method='SLSQP')
print(result.x)
```
This code will output the value of $x$ that maximizes the logarithmic function.
# Understanding the optimization process and methods
Gradient descent is a popular optimization method that uses the gradient of the function to find its optimal values. It works by iteratively updating the value of the variable based on the gradient.
Newton-Raphson is another optimization method that uses the second derivative of the function to find its optimal values. It works by iteratively updating the value of the variable based on the second derivative.
Trust-region methods are a class of optimization methods that use a trust region around the current value of the variable to find its optimal values. They are particularly useful for solving large-scale optimization problems.
SciPy provides functions for each of these methods, such as `scipy.optimize.minimize` for gradient descent and Newton-Raphson, and `scipy.optimize.minimize_scalar` for trust-region methods.
## Exercise
Compare the performance of gradient descent and Newton-Raphson on the following function:
$$f(x) = (x - 2)^2$$
To solve this exercise, you can use the following code:
```python
import numpy as np
from scipy.optimize import minimize
def quadratic_function(x):
return (x - 2) ** 2
result_gradient_descent = minimize(quadratic_function, 0)
result_newton_raphson = minimize(quadratic_function, 0, method='Newton-CG')
print(result_gradient_descent.x)
print(result_newton_raphson.x)
```
This code will output the optimal values found by gradient descent and Newton-Raphson.
# Applying optimization to real-world problems
Consider the following problem: You are a restaurant owner, and you want to maximize your profit by setting the price of your menu items. You have the following constraints and criteria:
- The price of each item is between $0 and $10.
- The total revenue is $100.
- The number of customers is given by the function $f(x) = 50 - 2x$, where $x$ is the price of the menu item.
To solve this problem, you can use the `scipy.optimize.minimize` function with the `method='SLSQP'` argument. This function requires two arguments: the function to optimize and the initial guess for the value of $x$.
Here's an example:
```python
import numpy as np
from scipy.optimize import minimize
def revenue_function(x):
return 50 - 2 * x
result = minimize(revenue_function, 5, method='SLSQP', bounds=[(0, 10)], constraints={'type': 'eq', 'fun': lambda x: 100 - x[0] * 50})
print(result.x)
```
This code will output the optimal price that maximizes the revenue, given the constraints and criteria.
## Exercise
Solve the following problem: You are a car manufacturer, and you want to minimize the cost of producing a car by optimizing the number of components and the time it takes to assemble each component. You have the following constraints and criteria:
- The number of components is between 1 and 10.
- The time to assemble each component is between 1 and 10.
- The total cost is $100.
- The cost of producing a car is given by the function $f(x) = 10 + 2x$, where $x$ is the number of components.
To solve this problem, you can use the `scipy.optimize.minimize` function with the `method='SLSQP'` argument. This function requires two arguments: the function to optimize and the initial guess for the value of $x$.
Here's an example:
```python
import numpy as np
from scipy.optimize import minimize
def cost_function(x):
return 10 + 2 * x
result = minimize(cost_function, 5, method='SLSQP', bounds=[(1, 10)], constraints={'type': 'eq', 'fun': lambda x: 100 - x[0] * (10 + 2 * x[1])})
print(result.x)
```
This code will output the optimal number of components and the time to assemble each component that minimizes the cost, given the constraints and criteria.
# Handling constraints and limitations in optimization
Consider the following optimization problem:
- Minimize the function $f(x) = x^2 - 4x + 3$.
- The constraint is $g(x) = x - 2 \le 0$.
To solve this problem, you can use the `scipy.optimize.minimize` function with the `method='SLSQP'` argument. This function requires two arguments: the function to optimize and the initial guess for the value of $x$.
Here's an example:
```python
import numpy as np
from scipy.optimize import minimize
def objective_function(x):
return x ** 2 - 4 * x + 3
def constraint_function(x):
return x - 2
result = minimize(objective_function, 0, method='SLSQP', constraints={'type': 'ineq', 'fun': constraint_function})
print(result.x)
```
This code will output the optimal value of $x$ that minimizes the function, given the constraint.
## Exercise
Solve the following optimization problem:
- Minimize the function $f(x) = x^2 - 4x + 3$.
- The constraints are $g_1(x) = x - 2 \le 0$ and $g_2(x) = 2x - 3 \le 0$.
To solve this problem, you can use the `scipy.optimize.minimize` function with the `method='SLSQP'` argument. This function requires two arguments: the function to optimize and the initial guess for the value of $x$.
Here's an example:
```python
import numpy as np
from scipy.optimize import minimize
def objective_function(x):
return x ** 2 - 4 * x + 3
def constraint_function_1(x):
return x - 2
def constraint_function_2(x):
return 2 * x - 3
result = minimize(objective_function, 0, method='SLSQP', constraints=[{'type': 'ineq', 'fun': constraint_function_1}, {'type': 'ineq', 'fun': constraint_function_2}])
print(result.x)
```
This code will output the optimal value of $x$ that minimizes the function, given the constraints.
# Advanced optimization techniques and algorithms
Consider the following optimization problem:
- Minimize the function $f(x) = x^2 - 4x + 3$.
- The constraint is $g(x) = x - 2 \le 0$.
To solve this problem, you can use the `scipy.optimize.minimize` function with the `method='CG'` argument. This function requires two arguments: the function to optimize and the initial guess for the value of $x$.
Here's an example:
```python
import numpy as np
from scipy.optimize import minimize
def objective_function(x):
return x ** 2 - 4 * x + 3
def constraint_function(x):
return x - 2
result = minimize(objective_function, 0, method='CG', constraints={'type': 'ineq', 'fun': constraint_function})
print(result.x)
```
This code will output the optimal value of $x$ that minimizes the function, given the constraint.
## Exercise
Solve the following optimization problem:
- Minimize the function $f(x) = x^2 - 4x + 3$.
- The constraints are $g_1(x) = x - 2 \le 0$ and $g_2(x) = 2x - 3 \le 0$.
To solve this problem, you can use the `scipy.optimize.minimize` function with the `method='CG'` argument. This function requires two arguments: the function to optimize and the initial guess for the value of $x$.
Here's an example:
```python
import numpy as np
from scipy.optimize import minimize
def objective_function(x):
return x ** 2 - 4 * x + 3
def constraint_function_1(x):
return x - 2
def constraint_function_2(x):
return 2 * x - 3
result = minimize(objective_function, 0, method='CG', constraints=[{'type': 'ineq', 'fun': constraint_function_1}, {'type': 'ineq', 'fun': constraint_function_2}])
print(result.x)
```
This code will output the optimal value of $x$ that minimizes the function, given the constraints.
# Integrating optimization into Python applications
Consider the following problem: You are a software developer, and you want to optimize the performance of a sorting algorithm by optimizing the input data. You have the following constraints and criteria:
- The input data is a list of integers.
- The total sum of the input data is $100$.
- The sorted sum of the input data is $100$.
- The sorted input data is given by the function $f(x) = \sum_{i=1}^n x_i$.
To solve this problem, you can use the `scipy.optimize.minimize` function with the `method='SLSQP'` argument. This function requires two arguments: the function to optimize and the initial guess for the value of $x$.
Here's an example:
```python
import numpy as np
from scipy.optimize import minimize
def sorted_sum_function(x):
return np.sum(x)
result = minimize(sorted_sum_function, np.array([0, 0, 100]), method='SLSQP', bounds=[(0, 100)], constraints={'type': 'eq', 'fun': lambda x: np.sum(x) * 2 - 100})
print(result.x)
```
This code will output the optimal input data that maximizes the sorted sum, given the constraints and criteria.
## Exercise
Solve the following problem: You are a machine learning researcher, and you want to optimize the performance of a neural network by optimizing its weights and biases. You have the following constraints and criteria:
- The weights and biases are between $-10$ and $10$.
- The total sum of the weights and biases is $0$.
- The loss function of the neural network is given by the function $f(x) = \sum_{i=1}^n x_i^2$.
To solve this problem, you can use the `scipy.optimize.minimize` function with the `method='SLSQP'` argument. This function requires two arguments: the function to optimize and the initial guess for the value of $x$.
Here's an example:
```python
import numpy as np
from scipy.optimize import minimize
def loss_function(x):
return np.sum(x ** 2)
result = minimize(loss_function, np.array([0, 0]), method='SLSQP', bounds=[(-10, 10), (-10, 10)], constraints={'type': 'eq', 'fun': lambda x: np.sum(x) * 2})
print(result.x)
```
This code will output the optimal weights and biases that minimize the loss function, given the constraints and criteria.
# Best practices for optimizing functions with SciPy
Consider the following optimization problem:
- Minimize the function $f(x) = x^2 - 4x + 3$.
- The constraints are $g_1(x) = x - 2 \le 0$ and $g_2(x) = 2x - 3 \le 0$.
To solve this problem, you can use the `scipy.optimize.minimize` function with the `method='SLSQP'` argument. This function requires two arguments: the function to optimize and the initial guess for the value of $x$.
Here's an example:
```python
import numpy as np
from scipy.optimize import minimize
def objective_function(x):
return x ** 2 - 4 * x + 3
def constraint_function_1(x):
return x - 2
def constraint_function_2(x):
return 2 * x - 3
result = minimize(objective_function, 0, method='SLSQP', constraints=[{'type': 'ineq', 'fun': constraint_function_1}, {'type': 'ineq', 'fun': constraint_function_2}])
print(result.x)
```
This code will output the optimal value of $x$ that minimizes the function, given the constraints.
## Exercise
Solve the following optimization problem:
- Minimize the function $f(x) = x^2 - 4x + 3$.
- The constraint is $g(x) = x - 2 \le 0$.
To solve this problem, you can use the `scipy.optimize.minimize` function with the `method='SLSQP'` argument. This function requires two arguments: the function to optimize and the initial guess for the value of $x$.
Here's an example:
```python
import numpy as np
from scipy.optimize import minimize
def objective_function(x):
return x ** 2 - 4 * x + 3
def constraint_function(x):
return x - 2
result = minimize(objective_function, 0, method='SLSQP', constraints={'type': 'ineq', 'fun': constraint_function})
print(result.x)
```
This code will output the optimal value of $x$ that minimizes the function, given the constraint.
# Conclusion: The importance of optimization in modern data analysis
In this textbook, we have explored the optimization capabilities of SciPy and how to use them to solve a wide range of problems. We have covered topics such as optimizing exponential and logarithmic functions, applying optimization to real-world problems, handling constraints and limitations, and using advanced optimization techniques and algorithms.
Optimization is a powerful tool that can be applied to a wide range of fields, including data analysis, machine learning, and engineering. It is essential for finding optimal values, making informed decisions, and improving the performance of algorithms and models.
## Exercise
Reflect on the importance of optimization in modern data analysis and how it can be applied to your own work.
In conclusion, optimization is a crucial aspect of modern data analysis and plays a vital role in improving the performance of algorithms, models, and decision-making processes. By understanding and applying optimization techniques effectively, we can unlock new insights and make a significant impact in various fields.
# References and further reading
To further explore the topics covered in this textbook, we recommend the following resources:
- SciPy documentation: https://docs.scipy.org/doc/scipy/reference/index.html
- SciPy optimization module: https://docs.scipy.org/doc/scipy/reference/optimize.html
- Optimization algorithms: https://en.wikipedia.org/wiki/List_of_algorithms#Optimization_algorithms
- Machine learning and optimization: https://www.coursera.org/learn/machine-learning
Additionally, we encourage you to explore other optimization libraries and tools available in Python, such as NumPy, Pandas, and TensorFlow. These tools can be used in conjunction with SciPy to enhance your optimization capabilities and apply optimization to a wider range of problems.
We hope that this textbook has provided you with a solid foundation in optimizing exponential and logarithmic functions with SciPy in Python. We wish you the best in your continued exploration of optimization and its applications in data analysis. | Textbooks |
\begin{document}
\title{On inequivalent factorizations of a cycle}
\author{G Berkolaiko$^1$, J M Harrison$^{2}$, M Novaes$^3$\\~\\
\small $^1$ Department of Mathematics, Texas A\&M University,\\
\small
College Station, TX 77843-3368, USA\\
\small
$^2$ Department of Mathematics, Baylor University,
Waco, TX 76798-7328, USA\\
\small
$^3$ Departamento de F\'isica, Universidade Federal de
S\~ao Carlos, 13565-905 S\~ao Carlos, SP, Brazil}
\maketitle
\begin{abstract}
We introduce a bijection between inequivalent minimal factorizations
of the $n$-cycle $(1\,2\ldots n)$ into a product of smaller cycles
of given length, on one side, and trees of a certain structure on
the other. We use this bijection to count the factorizations with a
given number of different commuting factors that can appear in the
first and in the last positions, a problem which has found
applications in physics. We also provide a necessary and sufficient
condition for a set of cycles to be arrangeable into a product
evaluating to $(1\,2\ldots n)$. \end{abstract}
\section{Introduction}
Counting factorizations of a permutation into a product of cycles of specified length is a problem with rich history, dating back at least to Hurwitz \cite{Hur1891}, and with many important applications, in particular in geometry (see e.g. \cite{EkeLanShaVai01}). Our interest in such problems is driven by applications encountered in physics, namely semiclassical trajectory-based analysis of quantum transport in chaotic systems \cite{MueHeuBraHaa07,BerHarNov08}. The main ingredient of this analysis is the existence of correlations between sets of long trajectories connecting an input channel of the quantum system to an output channel. The trajectories organize themselves into families, with the elements of a family differing among themselves only by their behavior in small regions (see Fig.~\ref{fig:corr}) in which some of them have crossings while others have anti-crossings. Enumerating possible configurations of crossing regions and their inter-connectivity is a question of combinatorial nature. In a certain special case it was found in \cite{BerHarNov08} to be connected to the {\em inequivalent} {\em minimal} factorizations of the $n$-cycle $(1\,2\ldots n)$ into a product of smaller cycles.
\begin{figure}
\caption{Two schematic examples of correlated sets of classical
trajectories through a quantum system. One set of trajectories is
represented by solid lines, while the other set is drawn in dashed
lines. The circles mark the crossing regions, where the
trajectories from different sets differ significantly: dashed lines
cross while solid lines narrowly avoid crossings.}
\label{fig:corr}
\end{figure}
The base for our results is a simple and highly pictorial bijection between said factorizations and plane trees. This bijection allows us to recover easily some already known results and to answer new questions about the structure of the set of factorizations. To be more specific we need to introduce some notation.
Let $\sigma_m \cdots \sigma_1$ be a factorization of the cycle $(1\,2\ldots n)$ into a product of smaller cycles. By convention, the first entry of a cycle is always its smallest element. We say the factorization is of type ${\boldsymbol{\alpha}}$ if among $\{\sigma_j\}$ there are exactly $\alpha_2$ 2-cycles (transpositions), $\alpha_3$ 3-cycles and so on. Let us define \begin{equation}
|{\boldsymbol{\alpha}}|=\sum_{j\ge 2} \alpha_j, \quad \langle {\boldsymbol{\alpha}}\rangle=\sum_{j\ge 2}
(j-1)\alpha_j. \end{equation} The quantity ${\boldsymbol{\alpha}}$ satisfies \begin{equation}
\label{rank}
\langle {\boldsymbol{\alpha}}\rangle\ge n-1. \end{equation} If the above relation becomes equality, the factorization is called {\em minimal}. We only consider minimal factorizations.
If two factorizations differ only in the order of commuting factors, they are said to be {\em equivalent}. An example of two equivalent factorizations is \begin{equation}
\label{eq:fact_example}
(1\,2\,3\,4) = (3\,4)(1\,2)(2\,4) = (1\,2)(3\,4)(2\,4). \end{equation} From now on we will refer to equivalence classes of factorizations simply as factorizations, unless the distinction is of particular importance. In Theorem 1 we establish a bijection between factorizations of type ${\boldsymbol{\alpha}}$ and plane trees with vertex degrees determined by ${\boldsymbol{\alpha}}$. In turn, the trees have been enumerated by Erd\'elyi and Etherington \cite{ErdEth41} (see also Tutte \cite{Tut64} and Stanley \cite{StanleyV2}, theorem 5.3.10). We are interested in counting the factorizations of type ${\boldsymbol{\alpha}}$. This number will be denoted\footnote{We use the same notation as \cite{Irv06}; there $H$
was counting \emph{all} factorizations} by $\widetilde{H}({\boldsymbol{\alpha}})$. In Theorem 2 we give an equation for its generating function and its relation to Catalan numbers.
Given an equivalence class of factorizations of the form $\sigma_m \cdots \sigma_1$, we refer to the number of different cycles that can appear in the position $\sigma_m$, the {\em number of heads\/} of the factorization. Similarly, the {\em number of tails\/} is the number of cycles that can appear in the position $\sigma_1$. For example, the factorization in (\ref{eq:fact_example}) has 2 heads (transpositions $(1\,2)$ and $(3\,4)$) and 1 tail. In Theorem 3 we derive a generating function for the number of inequivalent minimal factorizations with the specified number of heads and tails, denoted by $\widetilde{H}_{\boldsymbol{h},\boldsymbol{t}}({\boldsymbol{\alpha}})$. The vectors $\boldsymbol{h}=(h_2, h_3, \ldots)$ and $\boldsymbol{t}=(t_2, t_3, \ldots)$ characterize the number of heads and tails. Namely, $h_j$ is the number of $j$-cycle heads and $t_j$ is the number of $j$-cycle tails. The quantity $\widetilde{H}_{\boldsymbol{h},\boldsymbol{t}}({\boldsymbol{\alpha}})$ is of importance in applications to quantum chaotic transport \cite{BerHarNov08}. Looking again at Fig.~\ref{fig:corr}, the vector $\boldsymbol{h}$ (corresp.\ $\boldsymbol{t}$) counts the number of crossings that can happen close to the left (corresp.\ right) opening of the system. For example, $t_2=1$ on the diagram (a) while $t_2=0$ on the diagram (b), since the $3$-crossing prevents the $2$-crossing from getting close to the right opening.
Finally, we will give a complete characterization (necessary and sufficient conditions) for a set $\{\sigma_j\}$ of cycles to give rise to a minimal factorization of the $n$-cycle $(1\,2\ldots n)$. This characterization is given in Theorem~\ref{thm:struct_fact} (the corresponding result for the factorization into transpositions can be traced back to Eden and Sch{\"u}tzenberger \cite{EdeSch_mtamkik62}). Here we only mention one of its corollaries: a factorization equivalence class is completely determined by the factors. In other words, two factorizations composed of the same factors are equivalent.
Some of the results discussed in this paper are already known, although they have been derived using different methods. Namely, the number of inequivalent factorizations into a product of transpositions (i.e. ${\boldsymbol{\alpha}}=(n-1, \boldsymbol{0})$) has been obtained by Eidswick \cite{Eid89} and Longyear \cite{Lon89}. Springer \cite{Spr96} derived a formula for $\widetilde{H}({\boldsymbol{\alpha}})$ using a different bijection to trees of the same type. Irving \cite{Irv06} reproduced the result of Springer using more general machinery involving cacti. The novelty of our approach is in the type (and simplicity!) of the bijection used. Being very visual, our bijection allows us to obtain answers to new questions, namely to count factorizations with specified number of heads and tails, which proved to be invaluable in applications \cite{BerHarNov08,KuiWalPet_prl10,BerKui_jpa10}.
Of other related results we would like to mention Hurwitz \cite{Hur1891} who suggested a formula for the number of minimal transitive factorizations (counting equivalent factorizations as different) of a general permutation into a product of 2-cycles. A factorization is called {\em transitive} if the group generated by the factors $\sigma_1,\ldots, \sigma_m$ acts transitively on the set $1,\ldots,n$. However, Hurwitz gave only a sketch of a proof and his paper was largely unknown to the combinatorialists. For a special case of factorizations of the $n$-cycle, the formula was (re-)derived by D\'enes \cite{Den59}, with alternative proofs given by Lossers \cite{Los86}, Moszkowski \cite{Mos89}, Goulden and Pepper \cite{GouPep93}. For general permutations, Strehl \cite{Str96} reconstructed the original proof of Hurwitz, filling in the gaps, while Goulden and Jackson \cite{GouJac97} gave an independent proof. Generalizations of Hurwitz formula to factorizations into more general cycles were considered in Goulden and Jackson \cite{GouJac00} and Irving \cite{Irv06}. Finally, {\em inequivalent} minimal transitive factorizations of a permutation consisting of $m=2$ cycles have been counted in Goulden, Jackson and Latour \cite{GouJacLat01} (into transpositions) and in Irving \cite{Irv06} (into general cycles). For permutations with $m=3$ and $4$ cycles formulas have been found \cite{BerIrv_prep} using the technique presented in the current manuscript but generalizations to larger $m$ appear to be difficult.
\section{Visualizing a product of cycles} \label{sec:viz}
A particularly nice way to visualize a product of transpositions was suggested in \cite{Gar59} (see also \cite{Bog08}). A permutation from $S_n$ is represented as $n$ labeled horizontal lines with several vertical lines (``shuttles'') connecting some pairs of the horizontal lines, see Fig.~\ref{fig:diagram}. The right and left ends of a line $k$ are labeled with $t_k$ (for ``tail'') and $h_k$ (for ``head'') correspondingly. For every horizontal line, start at the right and trace the line to the left. Wherever an end of a shuttle is encountered, trace this shuttle vertically till its other end and then resume going to the left (towards $h$). Continue in this manner until you reach the left end of one of the horizontal lines. It is clear that the mapping ``right ends to left ends'' thus described is invertible and therefore one-to-one.
\begin{figure}
\caption{Visualizing a product of transpositions using a ``shuttle
diagram''. Each term $(k_1\ k_2)$ in the product corresponds to a
vertical edge (``shuttle'') connecting lines $k_1$ and $k_2$. The
lines are ordered in the same way as the terms in the product.
Part (a) depicts the product $(1\, 2)(2\, 4)(2\, 3)$ and finding
the image of $3$ under the resulting permutation (dashed line).
Part (b) is the representation of $(3\, 4)(1\,2\,4)$, where the
longer cycle is represented by a directed shuttle. Note that
these two shuttle diagrams correspond to the trajectory
reconnections depicted on Fig.~\ref{fig:corr}: the solid lines
connecting $h_j$ to $t_j$ correspond to the trajectories drawn in
solid lines and labelled $\gamma_j$ in Fig.~\ref{fig:corr}. The
dashed lines of Fig.~\ref{fig:corr} correspond to the paths from
$t_j$ to $h_{j+1}$ via the shuttles.}
\label{fig:diagram}
\end{figure}
In this construction, a shuttle connecting lines $k_1$ and $k_2$ represents the transposition $(k_1\, k_2)$. The transpositions are ordered in the same way as shuttles: right to left. If the two neighboring transpositions $(k_1\, k_2)$ and $(k_3\, k_4)$ commute (if and only if all four $k_j$ are distinct), the corresponding shuttles can be swapped around without affecting the dynamics. We can view the resulting diagram as a graph (with horizontal and vertical edges). If the diagram represents a factorization of an $n$-cycle, the graph is connected. By counting vertices and edges, one concludes that if a factorization is {\em minimal}, the resulting graph is a tree.
Suppose now that the diagram represents a minimal factorization of the cycle $(1\, 2 \ldots n)$. In addition to the right-to-left motion described above we define the left-to-right motion as going horizontally, ignoring the shuttles. Then, starting at $t_1$ and going left we arrive to $h_2$. Going right from there we arrive to $t_2$ and from there, to $h_3$. Continuing in this fashion, we obtain a closed walk with several important features. It visits the vertices $t_1, h_2, t_2, \ldots, h_n, t_n, h_1$ in this sequence. It traverses each edge of the graph exactly twice: once in each direction (this follows, for example, from the invertibility of the motion). Since the graph is a tree, we conclude that it goes from one vertex to the next one along the shortest possible route. This walk traversing the entire tree will play an important role in what follows.
\begin{figure}
\caption{Visualizing a product of transpositions as a directed
graph. Depicted are the steps in constructing the graph corresponding to
the product $(3\, 4)(2\, 4)(1\, 4)$. To read off the image of $k$ under
the resulting permutation we start at $t_k$ and follow the directions of
the edges, choosing the next edge in the counterclockwise order at each
vertex, until arriving to $h_{\pi(k)}$. The path traced starting with
$t_1$ is illustrated by the dashed line in (d).}
\label{fig:directed}
\end{figure}
Another way to visualize a product of transpositions as a directed plane graph is illustrated on Fig.~\ref{fig:directed}. We start with $n$ disjoint directed edges labeled $1$ to $n$. For a product $\pi = (k_{j-1}\, k_j)\cdots(k_3\, k_4)(k_1\, k_2)$, we start by joining the heads of the edges labeled $k_1$ and $k_2$ at a new vertex and add two more {\em outgoing\/} edges also labeled $k_1$ and $k_2$. We arrange them around the vertex so that, going counter-clockwise, the outgoing edge $k_1$ is followed by the incoming $k_1$, then by the outgoing $k_2$ and, finally, by the incoming $k_2$. At this and all later stages of the procedure for each $k=1,\ldots,n$ there is exactly one ``free'' head of an edge labeled $k$, and one free tail of possibly different edge also labeled $k$. We now repeat the procedure for the transposition $(k_3\, k_4)$, joining free heads of edges marked $k_3$ and $k_4$, adding new outgoing edges to new vertex and ordering the edges in the similar fashion: outgoing $k_3$, incoming $k_3$, outgoing $k_4$ and incoming $k_4$. On Fig.~\ref{fig:directed}(d) we identified the free heads and tails by labeling them with $h_j$ and $t_j$ correspondingly.
The resulting graph is closely related to the diagrams described earlier. Namely, the graph is obtained from the diagram by shrinking the shuttle edges and re-ordering the edges at the newly merged vertices, see Fig.~\ref{fig:shrinking}(a). Moreover, the ordering of edges has been designed so that, to determine the image of $k$ under the product permutation $\pi$, one would start at $t_k$ and travel along the direction of the edges, at each vertex taking the next edge in the counterclockwise order, finally arriving to $h_{\pi(k)}$. This is illustrated by the dashed line on Fig.~\ref{fig:directed}(d). Starting at $h_k$ and going {\em against\/} the direction of the edges, taking the next counterclockwise edge at each vertex, will get one to $t_k$.
\begin{figure}
\caption{(a) Transforming the diagram representation of a product of
transpositions into a directed graph representation. The ``shuttle'' edge
is shrunk and its end-vertices are merged. The edges on the left are
labeled in the order they are traversed by the walk $t_1, h_2, t_2,
\ldots, h_n, t_n, h_1, t_1$. (b) Visualization of the cycle $(1\,2\,3)$ as
a ``shuttle diagram'' and as a directed graph.}
\label{fig:shrinking}
\end{figure}
Thus, if $\pi$ is the cycle $(1\,2 \ldots n)$, the corresponding graph is a tree with $n-1$ vertices of total degree 4 (henceforth called {\em internal vertices\/}), $2n$ vertices of degree 1 (henceforth called {\em leaves\/}) and $3n-2$ edges. The walk $t_1, h_2, t_2, \ldots, h_n, t_n, h_1, t_1$, discussed in the context of diagrams, now circumnavigates the entire tree in the counter-clockwise direction. As before, it traverses each edge exactly once in each direction. The leaves of the tree are thus marked $h_1, t_1, h_2, t_2, \ldots, h_n, t_n$ going counter-clockwise, see Fig.~\ref{fig:directed}.
The generalization of this construction from a product of transpositions to a product of general cycles is straightforward. For a $m$-cycle $(k_1\, k_2 \ldots k_m)$, the corresponding shuttle is realized as $m$ directed edges indicating transitions from horizontal line $k_j$ to horizontal line $k_{j+1}$. In the directed graph visualization, the cycle corresponds to a vertex of total degree $2m$, with outgoing edge marked $k_1$ followed by the incoming edge $k_1$, then by outgoing edge $k_2$ and so on. We illustrate this in Fig.~\ref{fig:diagram}(b) using the cycle $(1\, 2\, 4)$ as an example.
\section{Main Results} \label{sec:main}
As described in section~\ref{sec:viz}, a factorization (up to equivalence) of the $n$-cycle into $n-1$ transpositions is naturally represented as a plane tree with $n-1$ internal vertices of total degree 4, $2n$ leaves of degree 1 and $3n-2$ edges. If we designate the leaf $h_1$ as the root of the tree, the labeling of all other leaves and the directions of edges can be reconstructed uniquely. This representation of a factorization as an undirected rooted plane tree turns out to be a bijection.
\begin{figure}
\caption{Reconstructing the factorization from an undirected rooted plane
tree. Note that at step (d) and (e) one could choose to remove the
vertex adjacent to leaves $t_4$ and $t_6$ instead.}
\label{fig:tree_to_fact}
\end{figure}
\begin{theorem}
\label{thm:bijection}
Inequivalent minimal factorizations of type ${\boldsymbol{\alpha}}$ of the $n$-cycle $(1\,2\ldots n)$ are in
one-to-one correspondence with undirected rooted plane trees having
$\alpha_j$ vertices of degree $2j$ and $2n$ leaves of degree 1. \end{theorem}
\begin{proof}
The mapping of factorization equivalence classes to trees, described
in section~\ref{sec:viz} is well defined. Indeed, the construction
steps corresponding to commuting factors also commute. We need to
show that this mapping is invertible and onto.
The mapping can be inverted by taking the following steps (see
Fig.~\ref{fig:tree_to_fact} for an example):
\begin{enumerate}
\item Label the leaves of the tree with $h_1, t_1, h_2, \ldots, t_n$ starting
with the root and going counterclockwise, Fig.~\ref{fig:tree_to_fact}(b)
(to avoid clutter we will omit the $h$-labels).
\item \label{itm:start_loop} For some value of $j$, choose a vertex with
degree $2j$ which has $j$ $t$-leaves adjacent to it. Such a vertex
exists by pigeonhole principle. The indices of the $t$-leaves give
the next (in the right to left order) factor in the expansion,
Fig.~\ref{fig:tree_to_fact}(c).
\item Remove the vertex. The edges connecting the vertex to leaves are
removed entirely. The edges connecting the vertex to other vertices, if
any, are cut in half. This creates one or more new leaves and their labels
are inherited from the leaves neighboring them in the counterclockwise
direction, Fig.~\ref{fig:tree_to_fact}(c).
\item Repeat from step \ref{itm:start_loop},
Fig.~\ref{fig:tree_to_fact}(d)-(f).
\end{enumerate}
Notice that the number of choices one has when first running step $2$ corresponds to the total number of tails.
To verify that the mapping is onto we have to check that the above inversion
applied to any tree produces a factorization of the $n$-cycle $(1\,2\ldots
n)$. To this end we observe that the deletion-relabeling process
coupled with the application of the cycles read at step \ref{itm:start_loop}
transports an object initially at $t_j$ to the leaf $h_{j+1}$ for all $j$
(assuming the convention $n+1\equiv 1$). \end{proof}
Before we proceed to counting trees, we would like to present a short corollary of the above theorem.
\begin{lemma}
\label{lem:increasing}
Let $\sigma=(s_1\ \ldots\ s_{|\sigma|})$ be a cycle in a
factorization of the $n$-cycle $(1\,2\ldots n)$. Then $\sigma$ is
increasing: $s_1 < s_2 < \ldots < s_{|\sigma|}$. \end{lemma}
\begin{proof}
When reading the factorization off the tree as described in the proof of
Theorem~\ref{thm:bijection}, the labels are assigned initially to the leaves
of the tree in the counterclockwise order. The operation of removing a
vertex and inheriting the labels preserves this ordering of the labels. If
a new connected component is created by the removal operation, its labels
are also ordered counterclockwise. Thus, when a vertex $\sigma$ is removed,
the labels of its leaves, $t_{s_1}, \ldots, t_{s_{|\sigma|}}$ satisfy $s_1 <
s_2 <\ldots < s_{|\sigma|}$ (provided the starting index $s_1$ is chosen
appropriately). Thus each factor read off the tree is an increasing cycle. \end{proof}
\begin{theorem}
\label{thm:gf_general}
The generating function of the number
$\widetilde{H}({\boldsymbol{\alpha}})$, defined by
\begin{equation}
\label{eq:gen_fun_simple}
\xi(\boldsymbol{x}) = \sum_{{\boldsymbol{\alpha}}} \widetilde{H}({\boldsymbol{\alpha}}) x_2^{\alpha_2}
x_3^{\alpha_3} \cdots,
\end{equation}
where the sum over ${\boldsymbol{\alpha}}$ is unrestricted, satisfies the
recurrence relation
\begin{equation}
\label{eq:gf_general}
\xi(\boldsymbol{x}) = 1 + x_2 \xi^3(\boldsymbol{x}) + x_3 \xi^5(\boldsymbol{x}) + \ldots
\end{equation}
It follows that
\begin{equation}
\sum_{{\boldsymbol{\alpha}}:\langle {\boldsymbol{\alpha}}\rangle=n}
(-1)^{|{\boldsymbol{\alpha}}|+n}\widetilde{H}({\boldsymbol{\alpha}})=\frac{1}{n+1}\binom{2n}{n}
= c_n,
\end{equation}
where $c_n$ is the $n$-th Catalan number. \end{theorem}
The above statement is a simple consequence of the bijection between factorizations and trees and the known results enumerating the trees, see Erd\'elyi and Etherington \cite{ErdEth41}, Tutte \cite{Tut64} or Stanley \cite[Theorem 5.3.10]{StanleyV2}. We will give a short proof in section~\ref{sec:counting} to introduce the methods used in the next result.
For factorizations with specified numbers of heads and tails we have \begin{theorem}
\label{thm:gf_heads_tails}
Let $g(\boldsymbol{x},\boldsymbol{v},\boldsymbol{u})$ be the generating function of the number
$\widetilde{H}_{\boldsymbol{h},\boldsymbol{t}}({\boldsymbol{\alpha}})$ of inequivalent minimal factorizations of
the $n$-cycle $(1\,2\ldots n)$ of type ${\boldsymbol{\alpha}}$ with specified number of heads and
tails, defined by
\begin{equation*}
g(\boldsymbol{x},\boldsymbol{v},\boldsymbol{u}) = \sum_{{\boldsymbol{\alpha}}}
\sum_{\boldsymbol{h}=(0,0,\ldots)}^{{\boldsymbol{\alpha}}} \sum_{\boldsymbol{t}=(0,0,\ldots)}^{{\boldsymbol{\alpha}}}
\widetilde{H}_{\boldsymbol{h},\boldsymbol{t}}({\boldsymbol{\alpha}})
x_2^{\alpha_2} u_2^{h_2} v_2^{t_2}
x_3^{\alpha_3} u_3^{h_3} v_3^{t_3} \cdots
\end{equation*}
Then $g(\boldsymbol{x},\boldsymbol{v},\boldsymbol{u})$ can be found as
\begin{align}
g &= f - \sum_{n\geq2} x_n (1-u_n) f^n \label{eq:rec_full_unsym}\\
\label{eq:rec_full_sym}
&= f\hat{f} - \sum_{n\geq2} x_n \left(f\hat{f}\right)^n,
\end{align}
where $f$ satisfies the recursion relation
\begin{equation}
\label{eq:reduced_rec}
f(\boldsymbol{x},\boldsymbol{v},\boldsymbol{u}) = 1
+ \sum_{n\geq2} x_n \left(f^n - 1 + v_n\right) \hat{f}^{n-1}
\end{equation}
and $\hat{f}$ is obtained from $f$ by exchanging the roles of $u$
and $v$,
\begin{equation*}
\hat{f}(\boldsymbol{x},\boldsymbol{v},\boldsymbol{u}) = f(\boldsymbol{x},\boldsymbol{u},\boldsymbol{v}).
\end{equation*} \end{theorem}
Moving on to the characterization of all possible sets of factors, we remind the reader that a cycle $(s_1\,s_2\ldots s_{|\sigma|})$ is called {\em increasing\/} if $s_1 < s_2 <\ldots < s_{|\sigma|}$. Here by $|\sigma|$ we denote the size of the cycle $\sigma$.
Another natural way to visualize factorizations of cycles is to draw the factors on a circle. We start by arranging the numbers $1,\ldots, n$ on the circle in the anti-clockwise direction. For a cycle
$(s_1\,s_2\ldots s_{|\sigma|})$ we will draw curves \emph{inside} the circle connecting $s_j$ to $s_{j+1}$ for $j=1,\ldots,|\sigma|$, with the last curve connecting $s_{|\sigma|}$ to $s_1$. We draw the curves without intersections (apart from at the vertices $s_j$), which is possible if and only if the cycle is increasing or decreasing. For uniformity, in case of a transposition $\sigma = (s_1, s_2)$, we draw two curves. An example with all cycles in a factorization drawn on a circle is shown on Fig.~\ref{fig:cactus}.
\begin{figure}
\caption{Cycles $(1\,5\,6\,8)$, $(2\,3\,5)$, $(4\,5)$ and $(6\,7)$
drawn on a circle. The cycles satisfy conditions of
Theorem~\ref{thm:struct_fact} and can be arranged into a
factorization $(1\,2\,3\,4\,5\,6\,7\,8) =
(4\,5)(2\,3\,5)(1\,5\,6\,8)(6\,7)$.}
\label{fig:cactus}
\end{figure}
\begin{theorem}
\label{thm:struct_fact}
The cycles $\sigma_1,\ \sigma_2,\ldots,\ \sigma_m$ can be arranged
into a factorization of the $n$-cycle $(1\,2\ldots n)$ if and only
if the following conditions hold
\begin{enumerate}
\item \label{itm:cover_all} any number $j=1,\ldots, n$ belongs to at
least one of the cycles,
\item \label{itm:increasing} all cycles are increasing,
\item \label{itm:no_intersect} the cycles can be drawn on a circle
without intersecting themselves and one another (apart from at the
vertices $j=1,\ldots,n$ on the circle),
\item \label{itm:simply_conn} the closed union of the resulting
curvilinear polygons is simply connected.
\end{enumerate}
All factorizations formed out of these cycles are equivalent. \end{theorem}
\begin{remark}
Either requirement number~\ref{itm:cover_all} or requirement number~\ref{itm:simply_conn}
can be substituted by the condition that
\begin{equation}
\label{eq:sizes}
1 + \sum_{j=1}^m (|\sigma_j|-1) = n.
\end{equation} \end{remark}
\begin{example}
The cycles $\{ (1\,4\,5),\ (1\,3),\ (2\,4) \}$ cannot be arranged
into a factorization since the curves connecting $(1\,3)$ and
$(2\,4)$ cannot be drawn inside a circle without intersecting. The
cycles $\{(1\,4\,5),\ (1\,2\,3),\ (3\,4) \}$ cannot be arranged into
a factorization since the resulting circle drawing has a
non-retractable loop $1, 4, 3, 1$. The cycles $\{(1\,4\,5),\
(1\,2),\ (2\,3) \}$ satisfy the conditions of
Theorem~\ref{thm:struct_fact} and yield the factorization
$(1\,4\,5)(1\,2)(2\,3)$. \end{example}
\begin{remark}
The result of drawing the cycles on the circle (after the circle has
been erased) is a cactus, precisely the object that was used by
Irving\cite{Irv06} to count factorizations. We will prove
Theorem~\ref{thm:struct_fact} in Section~\ref{sec:struct} by
establishing a bijection between our trees and cacti drawn on a
circle. An informal explanation for the bijection is rather simple:
while trees were obtained from the shuttle diagrams by shrinking
vertical edges, the cacti are obtained by shrinking horizontal edges
(provided the cycles are drawn as shown on
Fig.~\ref{fig:shrinking}(b)). \end{remark}
\section{Counting factorizations} \label{sec:counting}
While Theorem~\ref{thm:gf_general} is a simple consequence of Theorem~\ref{thm:bijection} and known counting results for trees (\cite{ErdEth41}, \cite{Tut64} or \cite[Theorem 5.3.10]{StanleyV2}), we provide a brief proof in order to introduce the methods used in the proof of Theorem~\ref{thm:gf_heads_tails}.
\begin{proof}[Proof of Theorem~\ref{thm:gf_general}]
We are going to enumerate the plane trees which have $\alpha_j$ vertices of
degree $2j$. The set of all such trees will be denoted by
$\mathcal{T}_{\boldsymbol{\alpha}}$, where ${\boldsymbol{\alpha}}=(\alpha_2,\alpha_3,\ldots)$.
\begin{figure}\label{fig:recur1}
\end{figure}
To derive a recurrence relation for $|\mathcal{T}_{\boldsymbol{\alpha}}|$ we break the
tree at the top vertex adjacent to the root. The top vertex has degree
$2(d+1)$ for some $d\geq1$ and, when splitting the tree, it becomes the root
of $2d+1$ subtrees $T_1,\dots,T_{2d+1}$, characterized by vectors
${\boldsymbol{\alpha}}_1,\dots,{\boldsymbol{\alpha}}_{2d+1}$ (some of them possibly empty). Clearly
${\boldsymbol{\alpha}}=\sum_{i=1}^{2d+1} {\boldsymbol{\alpha}}_i +\boldsymbol{e}_d$ where $\boldsymbol{e}_d$ has $1$ in its
$d$-th component\footnote{we remind the reader that the $k$-th component of
vector ${\boldsymbol{\alpha}}$ is $\alpha_{k+1}$} and zero elsewhere, representing the
top vertex that was removed. Figure \ref{fig:recur1}(a) shows a tree with
characteristic $(3,1,\boldsymbol{0})$. This tree splits at the top
vertex, degree six ($d=2$), into five subtrees. The number of all possible
trees with the top vertex of degree $2(d+1)$ is given by the number of
combinations of subtrees, $\prod_{j=1}^{2d+1}|\mathcal{T}_{{\boldsymbol{\alpha}}_j}|$,
where $\sum_{j=1}^{2d+1} {\boldsymbol{\alpha}}_j = {\boldsymbol{\alpha}}-\boldsymbol{e}_d$. Summing over the
possible degrees of the top vertex establishes the recursion relation,
\begin{equation}
\label{eq:recur_N2}
\widetilde{H}({\boldsymbol{\alpha}}) = |\mathcal{T}_{\boldsymbol{\alpha}}|
= \sum_{d\ge 1} \sum_{{\boldsymbol{\alpha}}_1\cdots{\boldsymbol{\alpha}}_{2d+1}}
\prod_{j=1}^{2d+1}\widetilde{H}({\boldsymbol{\alpha}})
\delta_{{\boldsymbol{\alpha}}_1+\ldots+{\boldsymbol{\alpha}}_{2d+1},\,{\boldsymbol{\alpha}}-\boldsymbol{e}_d} \ .
\end{equation}
Computing the generating function $\xi(\boldsymbol{x})$,
equation~(\ref{eq:gen_fun_simple}), we recover (\ref{eq:gf_general}).
To relate $\widetilde{H}({\boldsymbol{\alpha}})$ to Catalan numbers (something
important in applications, \cite{BerHarNov08}), we take
$x_j=-r^{j-1}$, $j\ge 2$. These particular values lead to
\begin{equation*}
x_2^{\alpha_2}x_3^{\alpha_3}\cdots=(-1)^{|{\boldsymbol{\alpha}}|}r^{\langle
{\boldsymbol{\alpha}}\rangle}.
\end{equation*}
On the other hand, recursion (\ref{eq:gf_general}) implies that
\begin{equation*}
\widetilde{\xi}=1-r\widetilde{\xi}^3-r^2\widetilde{\xi}^5-\cdots,\qquad
\mbox{ where }\quad
\widetilde{\xi}(r)=\xi(-r, -r^2, \ldots).
\end{equation*}
The right-hand side is almost a geometric series; we multiply the
equation by $1-r\widetilde{\xi}^2$ to arrive at
\begin{equation*}
r\widetilde{\xi}^2 + \widetilde{\xi} - 1 = 0.
\end{equation*}
This can be solved for $\widetilde{\xi}$ and results in the well known
generating function of $(-1)^nc_n$. \end{proof}
A recurrence relation for $\widetilde{H}_{\boldsymbol{h},\boldsymbol{t}}({\boldsymbol{\alpha}})$ can be established in a similar manner.
\begin{proof}[Proof of Theorem~\ref{thm:gf_heads_tails}]
We recap that we are counting the factorizations with a given number of
heads and tails. On a tree, a tail corresponds to a vertex of degree $2j$
which has $j$ free $t$-labeled edges attached to it. For example, on
Fig.~\ref{fig:recur1}, there are $t=2$ tails. Similarly a head is a degree
$2j$ vertex with $j$ free $h$-labeled edges attached (we omitted $h$ labels
from Fig.~\ref{fig:recur1} and other figures to avoid clutter). Note that
the top vertex can be both a tail and a head, although not simultaneously,
at least for trees with more than one vertex. The root counts as being
$h$-labeled and is always free. For example, the tree on
Fig.~\ref{fig:recur1}(a) has the top vertex as its only head, $h=1$. We
also introduce a variable $h'$ counting all heads excluding the top vertex.
We will refer to it as the {\em reduced head count\/} and for the tree on
Fig.~\ref{fig:recur1}(a) it is $h'=h-1=0$, while for the tree on
Fig.~\ref{fig:recur1}(b) $h'=h=1$. We will first derive a recursion
counting the trees with a given reduced head count and from there obtain the
number of trees with full head count.
The tail and (reduced) head count are further specialized to count the
number of heads and tails of a certain degree. Thus, in general, $\boldsymbol{h}$,
$\boldsymbol{h}'$ and $\boldsymbol{t}$ are infinite vectors with finitely many nonzero components.
Let $\phi$ be a partial generating function with respect to the tail and
reduced head count
\begin{equation*}
\phi({\boldsymbol{\alpha}},\boldsymbol{v},\boldsymbol{u}) =
\sum_{\boldsymbol{h}'=(0,0,\ldots)}^{{\boldsymbol{\alpha}}} \sum_{\boldsymbol{t}=(0,0,\ldots)}^{{\boldsymbol{\alpha}}}
\widetilde{H}_{\boldsymbol{h}',\boldsymbol{t}}({\boldsymbol{\alpha}})
u_2^{h_2'} v_2^{t_2}
u_3^{h_3'} v_3^{t_3} \cdots, \qquad \phi(\boldsymbol{0},\boldsymbol{v},\boldsymbol{u}) = 1.
\end{equation*}
To establish the recursion relation we again consider breaking the tree into
subtrees $T_1, \dots , T_{2d+1}$ at the top vertex of degree $2(d+1)$,
numbering the subtrees left to right. As before, the subtrees are
characterized by vectors ${\boldsymbol{\alpha}}_1,\dots,{\boldsymbol{\alpha}}_{2d+1}$. We introduce a
special notation for the sum of odd-indexed vectors and for the sum of
even-indexed ones,
\begin{equation}
\label{eq:alpha_odd_even}
{\boldsymbol{\alpha}}^o = \sum_{j=0}^d {\boldsymbol{\alpha}}_{2j+1}
\qquad {\boldsymbol{\alpha}}^e = \sum_{j=1}^d {\boldsymbol{\alpha}}_{2j}.
\end{equation}
The reduced head count $\boldsymbol{h}' = (h_2,h_3,\ldots)$ of the full tree
can be obtained by summing the appropriate counts for the subtrees,
namely
\begin{equation}
\label{eq:reduced_head_cnt}
\boldsymbol{h}'(T)=\boldsymbol{h}'(T_1) + \sum_{j=1}^{d} \big(\boldsymbol{t}(T_{2j})+ \boldsymbol{h}'(T_{2j+1})\big).
\end{equation}
Note that for the even-numbered subtrees, we need to add the number
of tails rather than heads. This corresponds to a change in the
labeling of the leaves on the subtrees with even index. On subtrees
with odd index the first (leftmost) leaf is always $t$-labeled,
while the first leaf of an even-numbered subtree is $h$-labeled,
see Fig.~\ref{fig:recur1}(b) for an example.
For the tail count of the complete tree, the procedure is analogous, with
the addition of the possible contribution of the top vertex. The top vertex
is a tail if all the odd subtrees are empty, i.e. ${\boldsymbol{\alpha}}_{2j+1} = \boldsymbol{0}$,
$j=0,\ldots,d$. Figure \ref{fig:recur1}(b) shows a tree where the top
vertex is a tail. Therefore,
\begin{equation*}
\boldsymbol{t}(T) = \boldsymbol{t}(T_1) + \sum_{j=1}^{d} \big(\boldsymbol{h}'(T_{2j}) + \boldsymbol{t}(T_{2j+1})\big)
+ \delta_{{\boldsymbol{\alpha}}^o, \boldsymbol{0}} \boldsymbol{e}_d.
\end{equation*}
Consequently $\phi({\boldsymbol{\alpha}},\boldsymbol{v},\boldsymbol{u})$ is expressed in terms of functions
$\phi({\boldsymbol{\alpha}}_j,\boldsymbol{v},\boldsymbol{u})$ generated by the subtrees,
\begin{multline}
\label{eq:phi}
\phi({\boldsymbol{\alpha}},\boldsymbol{v},\boldsymbol{u}) = \sum_{d\ge 1} \sum_{{\boldsymbol{\alpha}}_1\cdots{\boldsymbol{\alpha}}_{2d+1}}
\phi({\boldsymbol{\alpha}}_1,\boldsymbol{v},\boldsymbol{u})
\prod_{j=1}^{d}\phi({\boldsymbol{\alpha}}_{2j},\boldsymbol{u},\boldsymbol{v})\phi({\boldsymbol{\alpha}}_{2j+1},\boldsymbol{v},\boldsymbol{u})
\\ \times
\left(1-(1-v_{d+1})\delta_{{\boldsymbol{\alpha}}^o,\boldsymbol{0}}\right) \,
\delta_{{\boldsymbol{\alpha}}^o+{\boldsymbol{\alpha}}^e,\,{\boldsymbol{\alpha}}-\boldsymbol{e}_d}.
\end{multline}
It is important to observe that the functions $\phi$ with
even-indexed vectors ${\boldsymbol{\alpha}}_{2j}$ have their arguments $\boldsymbol{u}$ and
$\boldsymbol{v}$ switched around. Calculating the generating function
\begin{equation*}
f(\boldsymbol{x},\boldsymbol{u},\boldsymbol{v}) = \sum_{{\boldsymbol{\alpha}}} \phi({\boldsymbol{\alpha}},\boldsymbol{v},\boldsymbol{u}) x_2^{\alpha_2}
x_3^{\alpha_3} \cdots \, ,
\end{equation*}
we recover recurrence relation (\ref{eq:reduced_rec}).
The complete head count can be obtained from the appropriate counts
for the subtrees in a slight variation of
(\ref{eq:reduced_head_cnt}),
\begin{equation}
\label{eq:full_head_cnt}
\boldsymbol{h}(T) = \boldsymbol{h}'(T_1)
+ \sum_{j=1}^{d} \big(\boldsymbol{t}(T_{2j}) + \boldsymbol{h}'(T_{2j+1})\big)
+ \delta_{{\boldsymbol{\alpha}}^e, \boldsymbol{0}}\, \boldsymbol{e}_d,
\end{equation}
where ${\boldsymbol{\alpha}}^e$ was defined in equation~(\ref{eq:alpha_odd_even}).
The partial generating function with respect to the full head count
is then
\begin{multline*}
\psi({\boldsymbol{\alpha}},\boldsymbol{v},\boldsymbol{u}) =
\sum_{\boldsymbol{h}=(0,0,\ldots)}^{{\boldsymbol{\alpha}}} \sum_{\boldsymbol{t}=(0,0,\ldots)}^{{\boldsymbol{\alpha}}}
\widetilde{H}_{\boldsymbol{h},\boldsymbol{t}}({\boldsymbol{\alpha}})
u_2^{h_2} v_2^{t_2} u_3^{h_3} v_3^{t_3} \cdots \\
= \sum_{d\ge 1} \sum_{{\boldsymbol{\alpha}}_1\cdots{\boldsymbol{\alpha}}_{2d+1}}
\phi({\boldsymbol{\alpha}}_1,\boldsymbol{v},\boldsymbol{u})
\prod_{j=1}^{d}\phi({\boldsymbol{\alpha}}_{2j},\boldsymbol{u},\boldsymbol{v})\phi({\boldsymbol{\alpha}}_{2j+1},\boldsymbol{v},\boldsymbol{u})
\\ \times
\left[1-(1-v_{d+1})\delta_{{\boldsymbol{\alpha}}^o,\boldsymbol{0}}
- (1-u_{d+1})\delta_{{\boldsymbol{\alpha}}^e,\boldsymbol{0}}\right] \,
\delta_{{\boldsymbol{\alpha}}^o+{\boldsymbol{\alpha}}^e,\,{\boldsymbol{\alpha}}-\boldsymbol{e}_d}
\end{multline*}
Opening the square brackets, using the recursion (\ref{eq:phi}) for $\phi$, and the fact that ${\boldsymbol{\alpha}}^e=\boldsymbol{0}$ implies
$\phi({\boldsymbol{\alpha}}_{2j},\boldsymbol{u},\boldsymbol{v}) = 1$, we obtain
\begin{multline*}
\psi({\boldsymbol{\alpha}},\boldsymbol{v},\boldsymbol{u})
= \phi({\boldsymbol{\alpha}},\boldsymbol{v},\boldsymbol{u})
- \sum_{d\ge 1} \sum_{{\boldsymbol{\alpha}}_1\cdots{\boldsymbol{\alpha}}_{2d+1}}
\prod_{j=0}^{d}\phi({\boldsymbol{\alpha}}_{2j+1},\boldsymbol{v},\boldsymbol{u})
(1-u_{d+1})\, \delta_{{\boldsymbol{\alpha}}^o,\,{\boldsymbol{\alpha}}-\boldsymbol{e}_d}.
\end{multline*}
Calculating the full generating function $g(\boldsymbol{x},\boldsymbol{u},\boldsymbol{v})$ we obtain
\begin{equation*}
g(\boldsymbol{x},\boldsymbol{v},\boldsymbol{u}) = f - \sum_{d\geq1} x_{d+1} (1-u_{d+1}) f^{d+1},
\end{equation*}
which is the same as (\ref{eq:rec_full_unsym}) after the substitution
$n=d+1$. We now transform this relation to form (\ref{eq:rec_full_sym}),
which confirms that, in contrast to $f(\boldsymbol{x},\boldsymbol{v},\boldsymbol{u})$, the generating function
$g(\boldsymbol{x},\boldsymbol{v},\boldsymbol{u})$ is symmetric with respect to the exchange of $\boldsymbol{u}$ and
$\boldsymbol{v}$. We exchange $\boldsymbol{u}$ and $\boldsymbol{v}$ in (\ref{eq:reduced_rec}) to obtain a
recursion for $\hat{f}$,
\begin{equation*}
\hat{f} = 1
+ \sum_{n\geq2} x_n \left(\hat{f}^n - 1 + u_n\right) f^{n-1},
\end{equation*}
multiply it by $f$ and rearrange,
\begin{equation*}
f\hat{f} = f - \sum_{n\geq2} x_n(1-u_n) f^n
+ \sum_{n\geq2} x_n \hat{f}^n f^n,
\end{equation*}
from which (\ref{eq:rec_full_sym}) immediately follows. \end{proof}
\section{Trees and cacti} \label{sec:struct}
As seen in Section~\ref{sec:viz} and Theorem~\ref{thm:bijection}, a factorization can be visualized as a tree with vertices representing factors. An alternative visualization involves drawing cycles as (curvilinear) polygons inscribed in a circle, leading to an inscribed cactus. As we mentioned earlier, the two models can be viewed as two ways of removing redundant information from the shuttle diagrams of Section~\ref{sec:viz}. The trees are obtained by shrinking vertical edges (shuttles), while cacti are obtained by shrinking horizontal edges.
In this Section we will give a direct mapping between trees and cacti inscribed on a circle. Namely, we will show that cacti satisfying conditions \ref{itm:cover_all}-~\ref{itm:simply_conn} of Theorem~\ref{thm:struct_fact} are in one-to-one correspondence with the rooted plane trees of Theorem~\ref{thm:bijection}. Then the statement of Theorem~\ref{thm:struct_fact} follows directly from Theorem~\ref{thm:bijection} and Lemma~\ref{lem:increasing}. We will keep our exposition slightly informal since a formal proof of a similar result for transpositions is available, for example, in \cite{EdeSch_mtamkik62} (see also an exposition in Section 4.5 of \cite{Berge_principles}).
\subsection{From an inscribed cactus to a tree}
\begin{figure}
\caption{Turning a cactus into a tree: (a) original cactus inscribed
in a circle, (b) adding edges (bold) between touching corners of
polygons, (c) cutting circle between labels $j$ and $j+1$ and (d)
shrinking polygons to produce internal vertices of the tree.}
\label{fig:cactus_to_tree}
\end{figure}
To transform an inscribed cactus into a tree we start by separating on the circle the touching corners of polygons. This creates more edges (shown in bold lines on Fig.~\ref{fig:cactus_to_tree}). The new vertices on the circle will receive the same label as the original vertex. Thus, if $k$ polygons touched at vertex $j$ there are now $k$ vertices labeled $j$ in consecutive positions around the circle. The new edges will become the internal edges (i.e. not ending in a degree one vertex) of the resulting graph. We observe that after this step no two polygons have any points in common. Also, each corner of each polygon is connected to exactly two edges.
In the next step, every arc or the circle connecting vertices $j$ and $j+1$ (by $n+1$ we understand $1$) is cut in half. The new vertices receive labels $t_j$ and $h_{j+1}$, so that $t_j$ is connected to vertex $j$ and $h_{j+1}$ is connected to $j+1$, see an example on Fig.~\ref{fig:cactus_to_tree}, part (c).
Finally, all polygons are shrunk to form vertices, Fig.~\ref{fig:cactus_to_tree}(d). Since the original cactus was simply connected, the result is a tree with (by construction) the correct vertex degrees to satisfy Theorem~\ref{thm:bijection}. Thus it corresponds to a minimal factorization of the full $n$-cycle.
\subsection{From a tree to an inscribed cactus}
\begin{figure}
\caption{Expanding a tree vertex into a polygon. A vertex of degree
$2m$ becomes a polygon with $m$ corners. Each corner is connected
to a pair of edges. If the vertex has $t$-type, we group the
edges in pairs according to figure (b). Vertices of $h$-type
become polygons depicted on figure (c).}
\label{fig:tree_to_cactus}
\end{figure}
To make a cactus out of a tree we essentially reverse the procedure outlined in the previous section. The only difficult point is the ``inflation'' of internal vertices into polygons. Each vertex of degree $2m$ will become a polygon with $m$ corners and two edges attached to each corner. Since the circular ordering of the edges around the polygon is determined by the ordering around the tree vertex, there are only two possibilities to group edges into pairs (depicted in Fig.~\ref{fig:tree_to_cactus}).
To decide which vertex becomes which type of polygon, we determine the ``type'' of a vertex $v$. Imaging a counter-clockwise walk starting at the leaf $h_1$ and following the exterior of the tree. If the last leaf visited by the walk prior to coming to $v$ \emph{for the first time} was a $t$-leaf, the vertex $v$ is of type $h$. Otherwise, it is of type $t$. Note that this definition works for leafs as well as internal vertices, and produces for them the ``correct'' type. The type is also preserved in building a tree from the top down. Having determined the type of a vertex, the vertices of type $t$ become polygons of the type depicted on Fig.~\ref{fig:cactus_to_tree}(b) and vertices of type $h$ become polygons similar to Fig.~\ref{fig:cactus_to_tree}(c).
From this point, we merge pairs of vertices $t_j$ and $h_{j+1}$ to form the circle and then shrink the internal edges, obtaining an inscribed cactus.
\section{Conclusions and outlook}
The simple pictorial bijection introduced in Theorem~\ref{thm:bijection} has allowed us to perform an in-depth analysis of the set of inequivalent minimal factorizations of the $n$-cycle. The next logical step is to apply similar ideas to inequivalent minimal transitive factorizations of a general permutation. Our preliminary explorations showed that the ideas of the present manuscript provide a method for deriving a recursion for the generating function for any finite $m$, where $m$ is the number of cycles in the cycle representation of the target permutation ($m=1$ corresponds to an $n$-cycle). We have also found \cite{BerKui_prep10} that this question is directly applicable in computing non-linear moments of transmission probability through a chaotic quantum system. However, for the above application some information on the number of tails and heads in a factorization is again required.
\section*{Acknowledgment} The authors acknowledge the discussions they had with P.~Lima-Filho and F.~Sottile and thank them for making useful suggestions. The authors are extremely grateful to J.~Irving for sending us a copy of manuscript \cite{Spr96}.
\end{document} | arXiv |
\begin{document}
\title{\textbf{Robust Equilibria in General Competing Mechanism Games}} \author{Seungjin Han\thanks{ I thank Siyang Xiong, two anonymous referees and an associated editor for their comments and suggestions. Alex Sam provided excellent research assistance. I thankfully acknowledge financial support from the Social Sciences and Humanities Research Council of Canada. Address: Department of Economics, McMaster University, 1280 Main Street West, Hamilton, ON, CANADA. Email: [email protected].} \\
McMaster University} \date{November 6, 2022} \maketitle
\begin{abstract} This paper proposes a general competing mechanism game of incomplete information where a mechanism allows its designer to send a message to himself at the same time agents send messages. This paper introduces various notions of robust perfect Bayesian equilibrium (PBE). Given the additive separability of each agent's utility function with respect to principals' actions, this paper provides the full characterization of equilibrium allocations in terms of Bayesian incentive compatible (BIC) direct mechanisms for each notion of robust PBE, without reference to the set of arbitrary mechanisms allowed in the game. Szentes' Critique (Szentes (2010)) on the standard competing mechanism game of complete information is valid in a model with incomplete information, regarding the set of allocations supportable in robust PBEs where agents' strategies of communicating with non-deviating principals and their induced BIC direct mechanisms are responsive to market changes. \end{abstract}
\section{Introduction}
The Revelation Principle does not hold when multiple principals compete in the market (Epstein and Peters (1999), McAfee (1992)). When a principal (e.g., a seller) designs a mechanism, he is aware that agents (e.g., buyers) have not only private information on their payoff types but also market information on mechanisms offered by competing principals, their beliefs on the terms of trade determined in those mechanisms, etc. Therefore, it is generally not enough to ask agents to report their payoff types only.
The literature on competing mechanisms or contractible contracts has explored the big idea that principals can use mechanisms or contractible contracts that involve commitments to punish a competing principal's deviation. Punishment can take place when agents' messages reveal a competing principal's deviation (Epstein and Peters (1999), Yamashita (2010), Xiong (2013)), or it is prescribed in a contractible contract, directly conditional on a deviator's publicly observable contract (Szentes (2015)). This makes it possible for principals to sustain various collusive outcomes.
Epstein and Peters (1999) propose a class of universal mechanisms that allow agents to describe competing principals' mechanisms. However, the message spaces in these universal mechanisms that agents use in describing competing principals' mechanisms are quite complex due to the infinite regress problem: my mechanism depends on the competitor's mechanism, his mechanism depends on mine, and so on ad infinitum. A major contribution by Yamashita (2010) is that any equilibrium allocations can be understood as if a principal offers a recommendation mechanism that delegates a choice of his incentive compatible direct mechanism to agents. The incentive compatible direct mechanism chosen by a majority of agents is implemented. Recommendation mechanisms can be thought of as an extension of menus proposed for common agency (Martimort and Stole (2002), Page and Monterio (2003), and Peters (2001)).
On the other hand, Szentes (2015) studies a bilateral contracting game as in Han (2006), but augmented by contractible contracts (Peters and Szentes (2012)) in which a principal's ordinary contract is directly conditional on a deviator's ordinary contract described by the G\"{o}del language.
\subsection{Motivation}
The characterization of the equilibrium allocations in the current literature is however tied to the very set of arbitrary mechanisms allowed for each principal in the game. Therefore, the greatest lower bound of a principal's equilibrium utility is not expressed in terms of incentive compatible direct mechanisms.
Further, Szentes (2010) points out a restrictive feature in Yamashita (2010). Mechanisms adopted in Yamashita (2010) delegate a choice of an action to agents in the sense that agents' messages fully determine an action in a mechanism. This is the restriction imposed on mechanisms in the standard competing mechanism game in the literature. However, a principal may offer a \emph{set-valued} \emph{mechanism} in which a subset of actions is first determined by agents' messages and the principal subsequently chooses an action that he wants from the subset.
Szentes shows that if such mechanisms are allowed, the greatest lower bound of a principal's (pure-strategy) equilibrium utility in the competing mechanism game of \emph{complete information} can be expressed as his minmax value over actions regardless of the complexity of mechanisms allowed in the game. The reason is that a principal can deviate to a mechanism that leaves the whole set of actions for him regardless of agents' messages. The deviating principal can then choose an action that maximizes his utility given his belief on non-deviators' actions.\footnote{ In Szentes (2010), a principal cannot commit to an action schedule that conditions his action choice directly on action choices made by the other principals (This is the standard approach in the competing mechanism games. If it is possible, a contract is contractible (Peters and Szentes 2012)). Furthermore, a subset of actions determined by agents' messages is not observable by any other principals because messages are private. These imply that it does not matter whether or not a principal's action itself is observable or not given that each principal simultaneously chooses his action from a subset of actions determined by agents' messages to him. A key intuition behind Szentes' critique is that a principal can raise his payoff if he has a freedom to choose an action even after agents send messages.} Therefore, a principal's utility cannot be lower than his minmax value over actions in the case of the complete information. In the standard competing mechanism game, a principal is forced to offer a mechanism that takes away his freedom of choosing an action at a later stage of the game, inducing his maxmin value over actions as a principal's equilibrium utility. Szentes (2010) argues that this is hard to justify on economic grounds.
This paper proposes a general competing mechanism game of \emph{incomplete information} where a principal's general mechanism allows him to send a message to himself at the same time agents send messages and these messages altogether determine an action. A principal does not observe other principals' mechanisms when he sends a message to himself. A principal's mechanism that assigns a subset of actions conditional on agents' messages (Szentes (2010)) is strategically equivalent to a general mechanism. Therefore, the set of mechanisms in Szentes (2010) can be nested into the set of general mechanisms.
In a model with a single principal, the Revelation Principle holds for both set-valued mechanisms in Szentes (2010) and general mechanisms in this paper: For any equilibrium given any set valued mechanism or any general mechanism, there exists an incentive compatible direct mechanism where agents' messages uniquely determine the principal's action and the equilibrium allocation is preserved by a truth-telling equilibrium. Therefore, in a model with a single principal, the mechanism design with set-valued mechanisms in Szentes (2010) or general mechanisms in this paper is no different from the mechanism design with the single-valued mechanisms. Szentes shows that such an equivalence breaks down in a model of complete information with multiple principals.
\subsection{Main Results}
This paper first identifies the utility environment and various notions of robust PBE under which the set of equilibrium allocations is characterized by incentive compatible direct mechanisms. Further, it clearly shows the types of robust PBE allocations for which Szentes' critique is valid.
We are interested in an allocation that can be supportable in an equilibrium where any principal $j$ cannot gain regardless of a profile of agents' continuation equilibrium strategies of communicating with $j$ upon his deviation to any general mechanism, taking as given a profile of their contiuation equilibrium strategies of communicating with non-deviating principals. We call it a \emph{robust} \emph{perfect Bayesian equilibrium} (PBE).
One interesting variant of robust PBE is that each agent's strategy of communicating with a non-deviating principal does not depend on the other principals' mechanisms. That is, her communication with a non-deviating principal is invariant to market information. If a PBE is robust and invariant to market information, I call it a \emph{market-information invariant robust PBE}.
The notion of robust PBE is stronger than that of PBE because an allocation can be supported in a PBE as long as there exists a continuation equilibrium in which a principal cannot gain upon deviation. However, it is weaker than the notion of strongly robust PBE (Epstein and Peters (1999), Han (2007)) in which a principal cannot gain in all continuation equilibria agents may play upon his deviation. I formulate a notion of \emph{uniquely implementable robust PBE}. A robust PBE is uniquely implementable if any profile of agents' continuation equilibrium strategies of communicating with a non-deviating principal induces the same BIC direct mechanism on the path or off the path following a competing principal's deviation respectively. If a robust PBE is uniquely implementable, the notion of robust PBE \emph{ coincides} with that of strongly robust PBE. Once players reach a uniquely implementable robust PBE, not only is there no continuation equilibrium where a principal can gain off the equilibrium path following his deviation, but every possible continuation equilibrium on the equilibrium path induces the same equilibrium allocation.
If each agent's utility function is additively separable with respect to principals' actions, I can define an individually Bayesian incentive compatible (BIC) direct mechanism for each principal $j$. Then, a BIC profile of principals' direct mechanisms is a profile of principals' individually BIC direct mechanisms.
I first consider the competing BIC direct mechanism game where a principal is restricted to offer only a BIC direct mechanism. Not only does a (truth-telling) PBE in the competing BIC direct mechanism game exist but it is also robust. Furthermore, the set of market-information invariant robust PBE allocations in any general competing mechanism game is the same as the set of a (truth-telling) PBE allocation, denoted by $\Pi _{B}^{\ast }$, in the competing BIC direct mechanism game. Therefore, any (truth-telling) PBE in the competing BIC direct mechanism game is free from Szentes' critique.
Secondly, for the characterization of uniquely implementable robust PBE allocations, I extend the result from Bayesian implementation (Palfrey an Srivastava (2018)). That is, if a profile of direct mechanisms $\pi ^{\ast }=\left( \pi _{1}^{\ast },\ldots ,\pi _{J}^{\ast }\right) $ is uniquely implementable in a robust PBE, then (i) $\pi ^{\ast }$ is a (truth-telling) PBE allocation in the competing BIC direct mechanism game (i.e., $\pi ^{\ast }\in \Pi _{B}^{\ast }$) and (ii) each $\pi _{j}^{\ast }$ satisfies Bayesian monotonicity.\footnote{ Condition (i) in a single-principal case is just the incentive compatibility of the direct mechanism. See Definition 3.1 for Bayesian monotonicity in Palfrey and Srivastava (2018).} Let $\Pi _{UB}^{\ast }$ be the set of profiles BIC direct mechanisms that satisfies (i) and (ii). I can show $\Pi _{UB}^{\ast }$ is indeed the set of uniquely implementable robust PBE allocations in a general equilibrium when well-established additional conditions are satisfied.\footnote{ For the additional conditiona, see Palfrey an Srivastava (2018).} Furthermore, a uniquely implementable robust PBE is also free from Szentes' critique.
Generally, the set of robust PBE allocations is big. Specifically, it is the set of all profiles of BIC direct mechanisms that induces each principal $j$ 's utility no less than his \emph{minmax} utility value over BIC direct mechanisms (Let's denote it by $\Pi _{RB}^{\ast }$). Let us explain the key idea. In a general competing mechanism game, principal $j$ can offer a mechanism where the message space for himself is the set that includes all of his BIC direct mechanisms $\Pi _{j}^{B}$ and the message space for each agent is her type space. In this mechanism, principal $j$ chooses his BIC direct mechanism $\pi _{j}$ from $\Pi _{j}^{B}$ and agents send type messages to $j$ at the same time and subsequently $\pi _{j}$ determines $j$ 's action conditional on agents' messages. The notion of robust PBE makes it possible for $j$ to implement any BIC direct mechanism $\pi _{j}$ from $\Pi _{j}^{B}$. Given $j$'s belief on BIC direct mechanisms induced by non-deviators' mechanisms, principal $j$ can then choose his BIC direct mechanism $\pi _{j}$ from $\Pi _{j}^{B}$ that maximizes his utility. Therefore, principal $j$'s utility in a robust PBE of a general competing mechanism game cannot be lower than his minmax values over individually BIC direct mechanisms.
This paper then shows that any profile of BIC direct mechanisms that generates a utility for each principal no less than his minmax value can be supported in a robust PBE where each principal $j$ offers a deviator-reporting mechanism which asks agents to report only the identity of a deviating principal along with their payoff types. If a majority of agents report $j$, then deviator-reporting mechanisms assign individually BIC direct mechanisms that lead to $j$'s minmax value: Otherwise, they assign individually BIC direct mechanisms that are supposed to be implemented in an equilibrium.
Note that we have that \begin{equation*} \Pi _{UB}^{\ast }\subset \Pi _{B}^{\ast }\subset \Pi _{RB}^{\ast }. \end{equation*} All three sets are independent of the set of general mechanisms allowed in competing mechanism games, and they are all specified in terms of BIC direct mechanisms. One may view my paper as one that identifies the notions of PBEs that produce those sets of allocations as the sets of equilibrium allocations in a general competing mechanism game respectively.
For a robust PBE, the standard competing mechanism game is restrictive in the sense that it takes the freedom of choosing any BIC direct mechanism away from a principal at the same time agents send messages to him given his mechanism. In the standard competing mechanism game, a deviating principal's utility can be as low as his \emph{maxmin }value. The reason is that agents' communication with the deviator alone induces a BIC direct mechanism from the deviator's mechanism and non-deviators can choose their BIC direct mechanisms conditional on agents' reports on the deviator's BIC direct mechanism that they induce from the deviator's mechanism.
When random mechanisms are allowed as in our paper and there are only two principals, the restrictive feature of the standard competing mechanism game is not a problem because each principal's minmax value over individually BIC direct mechanisms is equal to his maxmin value. However, the equivalence between the minmax and maxmin values are not extended to the case with three or more principals. My results show that Szentes' critique on the standard competing mechanism game of complete information is also valid in a model of incomplete information, regarding the set of allocations supportable in robust PBEs where agents' strategies of communicating with non-deviating principals and their induced BIC direct mechanisms are responsive to market changes. However, allocations supportable in market-information invariant robust PBEs are free from Szentes' critique.
Section \ref{sec:literature} provides the related literature review. Section \ref{sec:model} sets out the model for general competing mechanism games. Section \ref{sec_equilibrium_notion} formalizes the three notions of robust PBE. In Section \ref{sec:competing_BIC}, I characterize the set of market-information invariant PBE allocations and the set of uniquely implementable robust PBE allocations. In Section \ref{sec: robust_PBE}, I characterize the set of robust PBE allocations. Section \ref{sec:discussion} provides discussions related to the results established in this paper.
\subsection{Related literature\label{sec:literature}}
While this paper imposes the separability on the agent's utility function, it allows for interdependent values of payoff types, an agent's utility depends on actions chosen by all principals, and there are no restrictions on the principal's utility function. In my companion paper (Han (2022)), I considered a particular environment where (i) an agent can choose only one principal for trading and her utility depends on only the action of the principal she chooses and her type (i.e. private value) and (ii) utility functions for all players are linear and there are also no externalities on the principal side. Han (2022) focuses on the specific (but tractable) competition model where a principal's general contract chooses his dominant-strategy incentive compatible direct mechanism as a function of messages sent by all agents before they choose their trading principals; each agent subsequently sends her true type to the principal she chooses. \footnote{ Extending the DIC-BIC equivalence in Gershkov, et al. (2013), Han (2022) shows that when all the other principals offer such general contracts, a principal has no incentive to deviate to any arbitrary general contract in the sense that there exists a continuation equilibrium that punishes a principal upon his deviation to an arbitrary general contract.} The focus in Han (2022) is to characterize the equilibrium allocations conditional on truthful type reporting to the trading principal in the specific competition model, show the validity of Szentes' critique based on such equilibria, and provide comparative statics on equilibrium profits and prices in applications.
In common agency (i.e., multiple principals and a single agent), Han (2012), Peters (2003, 2007), and Attar et al. (2008) identify the restrictions on the players' preference ordering with which there is no loss of generality for principals to focus on take-it-or-leave-it offers or incentive-compatible direct mechanisms. The \textquotedblleft no externalities\textquotedblright\ condition imposed in Peters (2003, 2007) focused on games where an agent's payoff type does not affect her preference ordering and it does not allow that a principal's utility depends on competing principals' actions. A separability on the agent's preference ordering is imposed in Attar et al. (2008) for common agency but I impose none of the following restrictions: (i) the no indifference condition in Attar et al. (2008) that an agent is never indifferent between any pair of actions from any principals\footnote{ We believe that for common agency as in Attar, et al. (2008), only the separability condition is needed for the Revelation Principle for robust equilibrium formulated in our paper without the no indifference condition.}, (ii) the \textquotedblleft no externalities\textquotedblright\ condition imposed on principals' preference ordering in Peters (2003, 2007), (iii) the monotonicity condition in Han (2012). In contrast to these papers, I formulate the notion of robust equilibrium such that it is sufficient to impose only the separability on agents' preference ordering for the characterization of the set of all robust equilibrium allocations in terms of BIC direct mechanisms in general competing mechanism games with multiple principals and multiple agents.
It is worthwhile to mention that mechanism design approaches considered in Szentes (2010) and this paper are different from contracting with limited commitment considered by Bester and Strausz (2000, 2010) and Dorval and Skreta (2018) in a model with a single principal. In Bester and Strausz (2000, 2010), a principal cannot specify the set of the actions that he can choose conditional on agents' messages. Because of that, the standard Revelation Principle does not hold even with a single principal (Bester and Stausz (2000)). What the principal can design is rather a communication device that generates a publicly observable output message conditional on an agent's private input message (Bester and Strausz (2010)). While the principal designs a communication device, he observes only the output message and this output message does not restrict the set of actions that the principal can choose. Given the posterior belief conditional on the output message, the principal chooses his action. Dorval and Skreta (2018) applies the limited commitment approach to a model with an agent's persistent type in multiple periods: The principal's action choice in the current period is independent of an agent's output messages observed from previous communication devices.
On the other hand, the class of mechanisms considered in Szentes (2010) and in this paper still allows a principal to commit to a mechanism that specifies a subset of a principal's actions as a function of agents' messages sent to him.\footnote{ These approaches assume the full commitment. A principal may want to offer a more general mechanism than a standard (single-valued) mechanism even if he can offer any standard mechanism. In fact, the set of standard mechanisms is a strict subset of the set of general mechanisms. On the other hand, the literature on contract design with limited commitment assumes that a principal can offer neither a general mechanism nor a standard mechanism.} As pointed out in footnote \ref{foot_RP}, when there is only one principal in a model, the standard Revelation Principle holds for the class of mechanisms considered in Szentes (2010) and my paper. Therefore, for a model with a single principal, the class of mechanisms considered in Szentes (2010) and this paper generates the same set of equilibrium allocations that are generated by the class of standard mechanisms in which agents' messages uniquely determines a principal's action. Szentes (2010) shows that's not the case in a model of complete information with multiple principals by showing that a principal can raise the greatest lower bound of his equilibrium utility by using the class of mechanisms in which agents' messages only determines a subset of actions from which a principal can choose his action. I show that this result can be extended to a model of incomplete information with multiple principals.
\section{General Competing Mechanisms\label{sec:model}}
Throughout the paper, I assume that a set is a compact metric space. When a measurable structure is necessary, the corresponding Borel $\sigma $ algebra is used. For a set $Z$, $\Delta \left( Z\right) $ denotes the set of Borel probability distributions on $X$. For any mapping $L$ from $S$ into $Q$, $ L(S)$ denotes the image set of $L$.
There are $J$ principals and $I$ agents. Let $\mathcal{J}=\{1,\ldots ,J\}$ be the collection of principals and $\mathcal{I}=\{1,\ldots ,I\}$ the the collection of agents with $J\geq 2$ and $I\geq 3.$ Agent $i$'s payoff type is denoted by $x_{i}\in X_{i}.$ Let $x=(x_{1},\ldots ,x_{I})\in X=X_{1}\times \cdots \times X_{I}$ denote a profile of types, which is drawn from a probability distribution $F\in \Delta (X).$ Each principal $j$ takes his action $a_{j}\in A_{j}.$ Let $a=(a_{1},\ldots ,a_{I})\in A=A_{1}\times \cdots \times A_{J}$ denote a profile of actions. For all $a\in A$ and all $ x\in X,$ principal $j$'s utility is $v_{j}(a,x)$ and agent $i$'s utility is $ u_{i}(a,x)$.
Each players's preference relation satisfies the von Neumann-Morgenstern expected utility property. I assume that each agent's utility function is \emph{additively separable} with respect to principals' actions. That is, for all $a\in A$ and all $x\in X,$ agent $i$'s utility takes the form of \begin{equation} u_{i}(a,x)=\sum_{k=1}^{J}u_{ik}(a_{k},x). \label{agent_payoff} \end{equation}
Notably, (\ref{agent_payoff}) allows a quasilinear representation of an agent's preference relation. For example, let $a_{j}=(y_{j},t_{j}),$ where $ y_{j}$ is a non-monetary (multidimensional) component of principal $j$'s action and $t_{j}=(t_{0j},t_{1j},\ldots ,t_{Ij})$ is a profile of monetary transfers with $t_{0j}$ for principal $j$ himself and $t_{ij}$ for agent $i$ for all $i\in \mathcal{I}$. Then, (\ref{agent_payoff}) allows us to express agent $i$'s utility as $u_{i}(a,x)=\sum_{k=1}^{J}h_{ik}(y_{k},x)+T_{i}$ with $T_{i}=\sum_{k=1}^{J}t_{ik}+m_{i}$ for agent $i$'s initial amount of money $ m_{i}\in
\mathbb{R}
_{+}$.
While (\ref{agent_payoff}) imposes the separability on the agent's utility function, it is quite general in terms of the nature of actions, how principals' actions affect a player's utility, and how a player's utility depends on all agents' types. The equivalence between dominant-strategy implementation and Bayesian implementation in Gershkov, et al. (2013) is based on the restrictions that (i) agents' utility functions are linear, (ii) each agent's type has only \emph{private value}, and (iii) a principal's action is $a=(n,m)$, where an alternative $n$ belongs to a discrete and finite set $\{1,\ldots ,N\}$ and $m$ is a vector of monetary transfers, one for each agent. Given that the agent's utility environment in my paper allows for \emph{interdependent values} and generality of a principal's action, working with dominant strategy incentive compatible direct mechanisms for the characterization of equilibrium allocations loses its generality even with linear utility functions and monetary transfers. \footnote{ One may consider ex-post incentive compatible direct mechanisms for the case of interdependent types. However they are generally not equivalent to Bayesian incentive compatible direct mechanisms as shown in Gershkov, et al (2013).}
For a profile of random actions $(\alpha _{1},\ldots ,\alpha _{J})\in \Delta (A_{1})\times \cdots \times \Delta (A_{J})$ and a profile of types $x\in X,$ expected utilitys for agent $i$ and principal $j$ are \begin{gather*} U_{i}(\alpha _{1},\ldots ,\alpha _{J},x):=\int_{A_{1}}\cdots \int_{A_{J}}u_{i}(a_{1},\ldots ,a_{J},x)d\alpha _{J}\times \cdots \times d\alpha _{1}, \\ V_{j}(\alpha _{1},\ldots ,\alpha _{J},x):=\int_{A_{1}}\cdots \int_{A_{J}}v_{j}(a_{1},\ldots ,a_{J},x)d\alpha _{J}\times \cdots \times d\alpha _{1}. \end{gather*} For each $i\in \mathcal{I}$, (\ref{agent_payoff}) allows us to express $ U_{i}(\alpha _{1},\ldots ,\alpha _{J},x)$ as \begin{equation} U_{i}(\alpha _{1},\ldots ,\alpha _{J},x)=\sum_{k=1}^{J}U_{ik}(\alpha _{k},x), \label{agent_payoff1} \end{equation} where $U_{ik}(\alpha _{k},x):=\int_{A_{k}}u_{ik}(a_{k},x)d\alpha _{k}$. Note that $U_{i}$ and $V_{j}$ are both linear in each $\Delta (A_{j})$ for all $ j\in \mathcal{J}$.
A major departure from the literature on competing mechanisms is to allow a principal to send a message to himself given his mechanism. Specifically, each principal $j$ can offer a \emph{general} \emph{mechanism} that determines his action contingent on agents' messages and his own message. Let $M_{ij}$ be the set of messages available for agent $i\in \mathcal{I}.$ Let $M_{0j}$ be the set of messages available for principal $j$ himself. Let $M_{j}=\times _{k=0}^{I}M_{kj}.$ A mechanism is then a measurable mapping \begin{equation*} \gamma _{j}:M_{j}\rightarrow \Delta (A_{j}). \end{equation*} Let $\Gamma _{j}$ be the set of mechanisms available for principal $j$. Let $ \Gamma =\Gamma _{1}\times \cdots \times \Gamma _{J}.$ Note that I allow for \emph{random} \emph{mechanisms}.
The class of mechanisms proposed in this paper is more general than a standard mechanism that specifies a principal's action contingent on agents' messages only. In terms of our formulation, a standard mechanism is the same as a mechanism $\gamma _{j}:M_{j}\rightarrow \Delta (A_{j})$ such that for all $m_{0j},m_{0j}^{\prime }\in M_{0j}$ and all $(m_{1j},\ldots ,m_{Ij})\in \times _{k\neq 0}M_{kj}$, \begin{equation} \gamma _{j}(m_{0j},m_{1j},\ldots ,m_{Ij})=\gamma _{j}(m_{0j}^{\prime },m_{1j},\ldots ,m_{Ij}). \label{restriction1} \end{equation} Let $\overline{\Gamma }_{j}$ be the set of mechanisms satisfying (\ref {restriction1}) for each $j\in \mathcal{J}$. Clearly $\overline{\Gamma }_{j}$ is a strict subset of $\Gamma _{j}$. Standard competing mechanism games (e.g., Epstein and Peters (1999), Han (2006), Yamashita (2010) among many) allow principal $j$ to offer only a mechanism in $\overline{\Gamma }_{j}$. For notational simplicity, I drop the message set $M_{0j}$ for any mechanism in $\overline{\Gamma }_{j}$ and denote a mechanism in $\overline{\Gamma } _{j} $ by a measurable mapping from $M_{1j}\times \cdots \times M_{Ij}$ into $\Delta (A_{j}).$
Mechanisms in $\Gamma $ or $\overline{\Gamma }$ are quite general in terms of the degree and nature of communication that they allow. For example, they are sufficiently general that there exists an embedding $\eta _{j}$ from the set of individually BIC direct mechanisms for principal $j$ defined in Section 3 into $\overline{\Gamma }_{j}$. $\overline{\Gamma }$ is also large enough to include mechanisms homeomorphic to deviator-reporting mechanisms and deviator's DM (direct mechanism) reporting mechanisms defined in Sections 4 and 5.
For a contract in Szentes (2010), there exists a strategically equivalent mechanism in $\Gamma _{j}$. Principal $j$'s contract proposed by Szentes (2010) is a measurable mapping $h_{j}:M_{1j}\times \cdots \times M_{Ij}\rightarrow \mathcal{P}(\Delta (A_{j}))$ in our environment, where $ \mathcal{P}(\Delta (A_{j}))$ is the set of all subsets of $\Delta (A_{j}).$ In this contract, a profile of agents' messages $(m_{1j},\ldots ,m_{Ij})$ first determines $h_{j}(m_{1j},\ldots ,m_{Ij}),$ a subset of $\Delta (A_{j}), $ from which principal $j$ subsequently chooses his action. For such a contract, I can construct a strategically equivalent mechanism $ \gamma _{j}:M_{j}\rightarrow \Delta (A_{j})$. Let $m_{0j}$ be a measurable mapping from $h_{j}(M_{1j}\times \cdots \times M_{Ij})$ (i.e., the image set of $h_{j}$) into $\Delta (A_{j})$ with the property that for all $ h_{j}(m_{1j},\ldots ,m_{Ij})\in h_{j}(M_{1j}\times \cdots \times M_{Ij})$, $ m_{0j}(h_{j}(m_{1j},\ldots ,m_{Ij}))\in h_{j}(m_{1j},\ldots ,m_{Ij}).$ Let $ M_{0j}$ be the set of all such measurable mappings. Let us define a mechanism $\gamma _{j}:M_{j}\rightarrow \Delta (A_{j})$ such that for a general contract $h_{j}$ and all $(m_{0j},m_{1j},\ldots ,m_{Ij})\in M_{j}$ \begin{equation*} \gamma _{j}(m_{0j},m_{1j},\ldots ,m_{Ij})=m_{0j}\left( h_{j}\left( m_{1j},\ldots ,m_{Ij}\right) \right) . \end{equation*} In a contract $h_{j},$ agents' messages determine a subset of actions and then principal $j$ chooses an action from the subset. In the corresponding mechanism $\gamma _{j}$, principal $j$ reports his contingency plan $m_{0j}$ to himself, at the same time agents send messages to him, as to what action to choose as a function of the subset of actions determined by agents' messages. Therefore, the set of contracts in Szentes (2010) can be nested into $\Gamma $.
A general competing mechanism game with $\Gamma $ starts when principals simultaneously offer their mechanisms from $\Gamma .$ Given a profile of mechanisms $\gamma =(\gamma _{1},\ldots ,\gamma _{J})$, each agent sends a message to every principal and every principal sends a message only to himself at the same time. When principals and agents send messages, each principal $j$'s mechanism is only observed by agents but not by the other principals. According to each principal $j$'s mechanism $\gamma _{j}$, a profile of messages $m_{j}=(m_{0j},m_{1j},\ldots ,m_{Ij})$ determines his random action as $\gamma _{j}(m_{j})\in \Delta (A_{j})$. Finally, utilities are realized.
One may consider an alternative game where a principal inputs his message after observing the other principals' mechanisms. In terms of characterizing equilibrium allocations, it does not matter whether or not the other principals' mechanisms are observable by a principal when he inputs his message in his mechanism given three or more agents. The reason is that the message spaces can be sufficiently general that (i) agents are able to describe the other principals' mechanisms with their messages and (ii) a principal can input a message in his mechanism that describes a contingency plan with respect to all possible messages that agents could send, as explained earlier.
\section{Notion of Equilibrium\label{sec_equilibrium_notion}}
I consider a pure-strategy equilibrium, as in Yamashita (2010), in the sense that principals use pure strategies for their mechanism choices. In this paper, mixed communication strategies are however allowed to such an extent that the existence of an equilibrium can be addressed, whereas Yamashita (2010) considers only pure communication strategies.
Let $c_{ij}:\Gamma \times X_{i}\rightarrow \Delta (M_{ij})$ denote agent $i$ 's mixed strategy of communicating with principal $j$. Let $c_{0j}:\Gamma _{j}\rightarrow \Delta (M_{0j})$ be principal $j$'s mixed strategy of communicating with himself. For all $\gamma \in \Gamma $ and all $x\in X$, let \begin{multline*} g_{j}(\gamma _{j},c_{0j}(\gamma _{j}),c_{1j}\left( \gamma ,x_{1}\right) ,\ldots ,c_{Ij}\left( \gamma ,x_{I}\right) ) \\ :=\int_{M_{0j}}\int_{M_{1j}}\cdots \int_{M_{Ij}}\gamma _{j}(m_{0j},m_{1j},\ldots ,m_{Ij})dc_{0j}(\gamma _{j})dc_{1j}(\gamma ,x_{1})\cdots dc_{Ij}(\gamma ,x_{I}). \end{multline*} Given $\gamma =(\gamma _{1},\ldots ,\gamma _{J})\in \Gamma ,$ $g_{j}(\gamma _{j},c_{0j}(\gamma _{j}),c_{1j}\left( \gamma ,x_{1}\right) ,\ldots ,c_{Ij}\left( \gamma ,x_{I}\right) )\in \Delta (A_{j})$ is the probability distribution over $A_{j}$ conditional on $x=(x_{1},\ldots ,x_{I})$ that is induced by (i) the strategies of communicating with principal $j$, $ c_{j}(\gamma )=(c_{0j}(\gamma _{j}),c_{1j}\left( \gamma ,\cdot \right) ,\ldots ,c_{Ij}\left( \gamma ,\cdot \right) )$ and (ii) principal $j$'s mechanism $\gamma _{j}$. Then, \begin{equation} g_{j}(\gamma _{j},c\left( \gamma \right) )=g_{j}(\gamma _{j},c_{0j}(\gamma _{j}),c_{1j}\left( \gamma ,\cdot \right) ,\ldots ,c_{Ij}\left( \gamma ,\cdot \right) ) \label{direct_mech} \end{equation} is a mapping from $X\rightarrow \Delta (A_{j})$, which is a \emph{direct mechanism} induced by the communication strategies $c_{j}(\gamma )$ and principal $j$'s mechanism $\gamma _{j}$.\footnote{ Suppose that there is only one principal in a model. Given a mechanism $ \gamma _{1}$ offered by principal 1 (i.e., single principal), a profile of continuation-equilibrium communication strategies $c_{1}\left( \gamma _{1}\right) $ induces a Bayesian incentive compatible direct mechanism $ g_{1} $ according to (\ref{direct_mech}) in which only agents send type messages and the principal does not send a message to himself. Therefore, the standard Revelation Principle holds. This implies that in a model of a single principal, the class of mechanisms considered in Szentes (2010) and my paper is no different from the class of standard mechanisms in terms of the set of equilibrium allocations that it generates.\label{foot_RP}} Let us denote principal $j$'s direct mechanism by $\pi _{j}:X\rightarrow \Delta (A_{j})$. Let $\Pi _{j}$ be the set of all possible (measurable) mappings $ \pi _{j}:X\rightarrow \Delta (A_{j}).$
For $\gamma \in \Gamma ,$ agent $i$'s (ex-ante) expected utility is expressed as \begin{equation*} \mathbb{E}_{x}\left[ U_{i}(g_{1}(\gamma _{1},c_{1}(\gamma ,x)),\ldots ,g_{J}(\gamma _{J},c_{J}(\gamma ,x)),x)\right] =\sum_{k=1}^{J}\mathbb{E}_{x} \left[ U_{ik}(g_{k}(\gamma _{k},c_{k}\left( \gamma ,x\right) ),x)\right] , \end{equation*} where $\mathbb{E}_{x}\left[ \cdot \right] \mathbb{\ }$the expectation operator based on the probability distribution $F$ over $X$ for all $k\in \mathcal{J}$. For $\gamma \in \Gamma $, principal $j$'s (ex-ante) expected utility is denoted by $\mathbb{E}_{x}\left[ V_{j}(g_{1}(\gamma _{1},c_{1}(\gamma ,x)),\ldots ,g_{J}(\gamma _{J},c_{J}(\gamma ,x)),x)\right] $.
\begin{definition} \label{def_cont_eq}For all $\gamma \in \Gamma $, $c\left( \gamma \right) =\left( c_{1}(\gamma ),\ldots ,c_{J}(\gamma )\right) $ is said to be a \emph{ continuation equilibrium} given $\gamma $ if
\begin{enumerate} \item[(i)] for all $i\in \mathcal{I}$ and all her communication strategies $ (c_{i1}^{\prime }(\gamma ,\cdot ),\ldots ,c_{iJ}^{\prime }(\gamma ,\cdot ))$ \begin{multline} \sum_{k=1}^{J}\mathbb{E}_{x}\left[ U_{ik}(g_{k}(\gamma _{k},c_{0k}(\gamma _{k}),c_{1k}\left( \gamma ,x_{1}\right) ,\ldots ,c_{Ik}\left( \gamma ,x_{I}\right) ),x)\right] \geq \\ \sum_{k=1}^{J}\mathbb{E}_{x}\left[ U_{ik}(g_{k}(\gamma _{k},c_{0k}(\gamma _{k}),c_{ik}^{\prime }\left( \gamma ,x_{i}\right) ,c_{-ik}\left( \gamma ,x_{-i}\right) ),x)\right] , \notag \end{multline} where $c_{-ik}\left( \gamma ,x_{-i}\right) =\left( c_{\ell k}\left( \gamma ,x_{-\ell }\right) \right) _{\ell \neq i}$ and
\item[(ii)] for all $j\in \mathcal{J}$ and all his communication strategies $ c_{0j}^{\prime }(\gamma )$, \begin{multline} \mathbb{E}_{x}\left[ V_{j}(g_{j}(\gamma _{j},c_{j}(\gamma ,x)),g_{-j}(\gamma _{-j},c_{-j}(\gamma ,x)),x)\right] \geq \\ \mathbb{E}_{x}\left[ V_{j}(g_{j}(\gamma _{j},c_{0j}^{\prime }(\gamma ),c_{-0,j}(\gamma ,x)),g_{-j}(\gamma _{-j},c_{-j}(\gamma ,x)),x)\right] , \label{principal_eq_payoff} \end{multline} where $c_{-0,j}(\gamma ,x)=(c_{ij}(\gamma ,x_{i}))_{i\in \mathcal{I}}$ and $ g_{-j}(\gamma _{-j},c_{-j}(\gamma ,x))=(g_{k}(\gamma _{k},c_{k}(\gamma ,x)))_{k\neq j}$. \end{enumerate} \end{definition}
Definition \ref{def_cont_eq} is based on ex-ante expected utilities. Condition (i) in Definition \ref{def_cont_eq} can be equivalently formulated with agents' interim expected utilities as follows: For all $i\in \mathcal{I} $, a.e. $x_{i}\in X_{i}$ and all $(\mu _{i1},\ldots ,\mu _{iJ})\in \Delta (M_{i1})\times \cdots \times \Delta (M_{iJ}),$ \begin{multline} \sum_{k=1}^{J}\mathbb{E}_{x_{-i}}\left[ U_{ik}(g_{k}(\gamma _{k},c_{0k}(\gamma _{k}),c_{1k}\left( \gamma ,x_{1}\right) ,\ldots
,c_{Ik}\left( \gamma ,x_{I}\right) ),x_{-i},x_{i})|x_{i}\right] \geq \\ \sum_{k=1}^{J}\mathbb{E}_{x_{-i}}\left[ U_{ik}(g_{k}(\gamma _{k},c_{0k}(\gamma _{k}),\mu _{ik},c_{-ik}\left( \gamma ,x_{-i}\right)
),x_{-i},x_{i})|x_{i}\right] , \label{agent_eq_payoff} \end{multline}
where $\mathbb{E}_{x_{-i}}\left[ \cdot |x_{i}\right] $ is the expectation operator based on the probability distribution over $X_{-i}$ conditional on $ x_{i}\in X_{i}$. Let $\mathcal{C}(\gamma )$ be the set of all continuation equilibria given $\gamma .$ For all $\gamma \in \Gamma $, let $\mathcal{C} _{j}(\gamma )$ be the projection of $\mathcal{C}(\gamma )$ onto the space of strategies of communicating with principal $j.$ For all $\gamma \in \Gamma $ , let $\mathcal{C}_{-j}(\gamma )$ be the projection of $\mathcal{C}(\gamma )$ onto the space of strategies of communicating with principals except for $j.$
\begin{definition} \label{def_eq}$(\gamma ^{\ast },c^{\ast })$ is a \emph{pure-strategy perfect Bayesian equilibrium} (henceforth simply PBE) of a general competing mechanism game with $\Gamma $ if (i) $c^{\ast }(\gamma )\in \mathcal{C} \left( \gamma \right) $ for all $\gamma \in \Gamma $ and (ii) for all $j\in \mathcal{J}$ and all $\gamma _{j}\in \Gamma _{j}$, \begin{multline} \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j}^{\ast },c_{j}^{\ast }(\gamma ^{\ast },x)\right) ,g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}^{\ast }(\gamma ^{\ast },x)\right) ,x\right) \right] \geq \\ \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j},c_{j}^{\ast }(\gamma _{j},\gamma _{-j}^{\ast },x)\right) ,g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}^{\ast }(\gamma _{j},\gamma _{-j}^{\ast },x)\right) ,x\right) \right] . \label{original_eq_cond} \end{multline} \end{definition}
Yamashita (2010) and Xiong (2013) employ the notions of sequential equilibrium and subgame perfect Nash equilibrium respectively with finite types, whereas Szentes (2015) employs the notion of perfect Bayesian equilibrium in the model of contractible contracts. All these notions, including one defined in Definition \ref{def_eq} above, are in fact identical except for the extent to which mixed strategies are allowed. A common requirement of these equilibrium notions is that principals have no incentive to deviate as long as there exists a continuation equilibrium upon deviation where the deviating principal cannot gain.
A strongly robust equilibrium (Epstein and Peters (1999) and Han (2007)) is an alternative notion of equilibrium. In our general competing mechanism game, strongly robust PBE can be defined as follows.
\begin{definition} \label{def_strong_eq}$(\gamma ^{\ast },c^{\ast })$ is a \emph{strongly robust PBE} of a general competing mechanism game with $\Gamma $ if (i) it is a PBE and (ii) for all $j\in \mathcal{J}$, all $\gamma _{j}\in \Gamma _{j} $, and all $[c_{j}(\gamma _{j},\gamma _{-j}^{\ast }),c_{-j}(\gamma _{j},\gamma _{-j}^{\ast })]\in \mathcal{C}_{j}(\gamma _{j},\gamma _{-j}^{\ast }),$ \begin{multline} \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j}^{\ast },c_{j}^{\ast }(\gamma ^{\ast },x)\right) ,g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}^{\ast }(\gamma ^{\ast },x)\right) ,x\right) \right] \geq \\ \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j},c_{j}(\gamma _{j},\gamma _{-j}^{\ast },x)\right) ,g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}(\gamma _{j},\gamma _{-j}^{\ast },x)\right) ,x\right) \right] . \label{strong_eq_cond} \end{multline} \end{definition}
Condition (ii) in Definition \ref{def_strong_eq} is equivalent to the following condition: For all $j$ and all $\gamma _{j}\in \Gamma _{j}$ \begin{multline*} \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j}^{\ast },c_{j}^{\ast }(\gamma ^{\ast },x)\right) ,g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}^{\ast }(\gamma ^{\ast },x)\right) ,x\right) \right] \geq \\ \sup_{\lbrack c_{j}(\gamma _{j},\gamma _{-j}^{\ast }),c_{-j}(\gamma _{j},\gamma _{-j}^{\ast })]\in \mathcal{C}(\gamma _{j},\gamma _{-j}^{\ast })} \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j},c_{j}(\gamma _{j},\gamma _{-j}^{\ast },x)\right) ,g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}(\gamma _{j},\gamma _{-j}^{\ast },x)\right) ,x\right) \right] \end{multline*} A principal's equilibrium utility in a strongly robust PBE is compared to his utility based on continuation equilibrium strategies of communicating with all principals that are \emph{best} for him upon his deviation to any mechanism. The literature has not identified the set of strongly robust PBE allocations nor the greatest lower bound of each principal's strongly robust PBE utility even in terms of mechanisms allowed in the game.
I propose a new notion of equilibrium. For all $\gamma \in \Gamma $, all $ c_{-j}\left( \gamma \right) \in \mathcal{C}_{-j}(\gamma )$, let us define the set of strategies of communicating with principal $j$ as follows \begin{equation*}
\mathcal{C}_{j}(\gamma |c_{-j}\left( \gamma \right) ):=\left\{ c_{j}(\gamma ):\left( c_{j}\left( \gamma \right) ,c_{-j}\left( \gamma \right) \right) \in \mathcal{C}(\gamma )\right\} . \end{equation*}
\begin{definition} \label{Def_robust_eq}$(\gamma ^{\ast },c^{\ast })$ is a \emph{robust PBE} of a general competing mechanism game with $\Gamma $ if
\begin{enumerate} \item[(i)] $c^{\ast }(\gamma )\in \mathcal{C}(\gamma )$ for all $\gamma \in \Gamma $ and
\item[(ii)] for all $j\in \mathcal{J}$, all $\gamma _{j}\in \Gamma _{j}$, and all $c_{j}(\gamma _{j},\gamma _{-j}^{\ast })\in \mathcal{C}_{j}(\gamma _{j},\gamma _{-j}^{\ast }|c_{-j}^{\ast }(\gamma _{j},\gamma _{-j}^{\ast }))$ , \begin{multline} \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j}^{\ast },c_{j}^{\ast }(\gamma ^{\ast },x)\right) ,g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}^{\ast }(\gamma ^{\ast },x)\right) ,x\right) \right] \geq \\ \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j},c_{j}(\gamma _{j},\gamma _{-j}^{\ast },x)\right) ,g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}^{\ast }(\gamma _{j},\gamma _{-j}^{\ast },x)\right) ,x\right) \right] . \label{robust_eq_cond} \end{multline} \end{enumerate} \end{definition}
A PBE $(\gamma ^{\ast },c^{\ast })$ is robust if any principal $j$ cannot gain regardless of a profile of agents' continuation equilibrium strategies $ c_{j}(\gamma _{j},\gamma _{-j}^{\ast })$ of communicating with $j$ upon his deviation to any general mechanism $\gamma _{j}$, \emph{taking as given} the profile of their contiuation equilibrium strategies $c_{-j}^{\ast }(\gamma _{j},\gamma _{-j}^{\ast })$ of communicating with non-deviating principals.
One interesting variation of robust PBE is one where an agent's communication with a non-deviating principal does not depend on any other principal's mechanisms. That is, \begin{equation} c_{j}^{\ast }(\gamma _{j}^{\ast },\gamma _{-j})=c_{j}^{\ast }(\gamma _{j}^{\ast },\gamma _{-j}^{\prime })\text{ }\forall j\in \mathcal{J}\text{, } \forall \gamma _{-j},\gamma _{-j}^{\prime }\in \Gamma _{-j} \label{invariant} \end{equation} given a robust PBE $(\gamma ^{\ast },c^{\ast })$.
\begin{definition} A robust PBE $(\gamma ^{\ast },c^{\ast })$ is a \emph{market-information invariant }if it satisfies (\ref{invariant}). \end{definition}
In a market-information invariant robust PBE, each agent's communication with a non-deviating principal only depends on her type and the principal's equilibrium mechanism.
I also formulate a uniquely implementable robust PBE as follows.
\begin{definition} A robust PBE $(\gamma ^{\ast },c^{\ast })$ is uniquely implementable if, for all $j\in \mathcal{J}$, all $k\neq j$ and all $\gamma _{j}\in \Gamma _{j}$ \begin{equation} g_{k}\left( \gamma _{k}^{\ast },c_{k}(\gamma _{j},\gamma _{-j}^{\ast })\right) =g_{k}\left( \gamma _{k}^{\ast },c_{k}^{\prime }(\gamma _{j},\gamma _{-j}^{\ast })\right) \in \Pi _{j}^{B}\text{, }\forall c_{k}(\gamma _{j},\gamma _{-j}^{\ast }),c_{k}^{\prime }(\gamma _{j},\gamma _{-j}^{\ast })\in \mathcal{C}_{k}\left( \gamma _{j},\gamma _{-j}^{\ast }\right) \label{unique_eq} \end{equation} \end{definition}
Fix a path in the game after principals' mechanism announcement: either equilibrium path $\gamma _{j}=\gamma _{j}^{\ast }$ or off the equilirium path following a principal $j$'s unilateral deviation to an arbitrary general mechanism $\gamma _{j}\neq \gamma _{j}^{\ast }$. Given such a path, any profile of continuation-equilibrium strategies of communicating with each $\ $non-deviating principal $k$ always induces the same BIC direct mechanism for $k$. This implies that if a robust PBE is uniquely implementable, it is also strongly robust. Furthermore, not only is there no continuation equilibrium where a principal can gain off the equilibrium path following his deviation, but there is a unique equilibrium allocation regardless of a continuation-equilibrium agents play on the equilibrium path.
\section{Competing BIC direct mechanisms\label{sec:competing_BIC}}
In this section, I provide the characterizations of market-information invariant robust PBE allocations and uniquely implementable robust PBE allocations respectively. Let us first define incentive compatible direct mechanisms. A profile of direct mechanisms $\pi =(\pi _{1},\ldots ,\pi _{J})$ is \emph{Bayesian incentive compatible} (BIC) if for all $i\in \mathcal{I}$, a.e. $x_{i}\in X_{i}$, and all $x_{i1},\ldots ,x_{iJ}\in X_{i}$ \begin{multline}
\mathbb{E}_{x_{-i}}\left[ U_{i}(\pi _{1}(x),\ldots ,\pi _{J}(x),x_{-i},x_{i})|x_{i}\right] \geq \\
\mathbb{E}_{x_{-i}}\left[ U_{i}(\pi _{1}(x_{i1},x_{-i}),\ldots ,\pi _{J}(x_{iJ},x_{-i}),x_{-i},x_{i})|x_{i}\right] . \label{BIC_whole} \end{multline} Let $\Pi ^{B}$ be the set of all BIC profiles of direct mechanisms.
Principal $j$'s direct mechanism is \emph{individually BIC} if for all $i\in \mathcal{I}$, a.e. $x_{i}\in X_{i}$, and $x_{ij}\in X_{i},$ \begin{equation} \mathbb{E}_{x_{-i}}\left[ U_{ij}(\pi _{j}\left( x_{i},x_{-i}\right)
,x_{-i},x_{i})|x_{i}\right] \geq \mathbb{E}_{x_{-i}}\left[ U_{ij}(\pi _{j}\left( x_{ij},x_{-i}\right) ,x_{-i},x_{i})|x_{i}\right] . \label{BIC_individual} \end{equation} Let $\Pi _{j}^{B}$ be the set of all individually BIC direct mechanisms for principal $j.$ Lemma \ref{lemma_BIC} below shows that the set of BIC profiles of direct mechanisms is the Cartesian product of individually BIC direct mechanisms. This is due to the additive separability of the agent's utility function with respect to principals' actions.
\begin{lemma} \label{lemma_BIC}$\Pi ^{B}=\Pi _{1}^{B}\times \cdots \times \Pi _{J}^{B}.$ \end{lemma}
\begin{proof} Consider any BIC profile of direct mechanisms $\pi =(\pi _{1},\ldots ,\pi _{J})\in \Pi ^{B}.$ Because $\pi $ is BIC as a whole, I have that for all $ j\in \mathcal{J}$, all $i\in \mathcal{I}$, a.e. $x_{i}\in X_{i}$, and all $ x_{ij}\in X_{i}$, \begin{multline*}
\sum_{k=1}^{J}\mathbb{E}_{x_{-i}}\left[ U_{ik}(\pi _{k}(x),x_{-i},x_{i})|x_{i}\right] \geq \\
\mathbb{E}_{x_{-i}}\left[ U_{ij}(\pi _{j}(x_{ij},x_{i}),x_{-i},x_{i})|x_{i}
\right] +\sum_{k\neq j}\mathbb{E}_{x_{-i}}\left[ U_{ik}(\pi _{k}(x),x_{-i},x_{i})|x_{i}\right] , \end{multline*} The inequality above implies that for all $i\in \mathcal{I}$, a.e. $x_{i}\in X_{i}$, and all $x_{ij}\in X_{i}$, \begin{equation*} \mathbb{E}_{x_{-i}}\left[ U_{ij}(\pi _{j}\left( x_{i},x_{-i}\right)
,x_{-i},x_{i})|x_{i}\right] \geq \mathbb{E}_{x_{-i}}\left[ U_{ij}(\pi _{j}\left( x_{ij},x_{-i}\right) ,x_{-i},x_{i})|x_{i}\right] . \end{equation*} This means that $\pi _{j}$ is individually BIC and hence $\pi _{j}\in \Pi _{j}^{B}$ for all $j\in \mathcal{J}$. Therefore, $\pi =(\pi _{1},\ldots ,\pi _{J})\in \Pi _{1}^{B}\times \cdots \times \Pi _{J}^{B}.$
It is straightforward to show that any profile of individually BIC direct mechanisms $\pi =(\pi _{1},\ldots ,\pi _{J})\in \Pi _{1}^{B}\times \cdots \times \Pi _{J}^{B}$ is also a BIC profile of direct mechanisms, i.e., $\pi =(\pi _{1},\ldots ,\pi _{J})\in \Pi ^{B}$. \end{proof}
I consider the competing BIC direct mechanism game where each principal is restricted to offer only a BIC direct mechanism in $\Pi _{j}^{B}$.
\begin{definition} $\pi =\left( \pi _{1},\ldots \pi _{J}\right) \in \Pi ^{B}$ is a profile of equilibrium BIC direct mechanisms in a (truth-telling) PBE of the competing mechanism game with $\Pi ^{B}$if for all $j\in \mathcal{J}$, \begin{equation} \mathbb{E}_{x}\left[ V_{j}(\pi _{j}(x),\pi _{-j}(x),x)\right] \geq \mathbb{E} _{x}\left[ V_{j}(\pi _{j}^{\prime }(x),\pi _{-j}(x),x)\right] \text{ } \forall \pi _{j}^{\prime }\in \Pi _{j}^{B} \label{BIC_eq} \end{equation} \end{definition}
Now I show that the strategy space for each principal $j$ in the the competing mechanism game with $\Pi ^{B}$ is non-empty, convex and compact.
\begin{proposition} \label{Prop_compact_BIC}$\Pi _{j}^{B}$ is non-empty, convex, and compact for all $j\in \mathcal{J}$. \end{proposition}
\begin{proof} $\Pi _{j}^{B}$ is non-empty because a constant direct mechanism is BIC. It is also a compact convex set. It is straightforward to show that $\Pi _{j}^{B}$ is a convex set because $\lambda \pi _{j}+(1-\lambda )\pi _{j}^{\prime }$ is also BIC for all $\pi _{j},\pi _{j}^{\prime }\in \Pi _{j}^{B}$ and all $\lambda \in \lbrack 0,1].$
$\Pi _{j}^{B}$ is compact if and only if it is totally bounded and complete (See Theorem 3.11.2.2 in Kolmogorov and Fomin (1975)). To show that $\Pi _{j}^{B}$ is totally bounded, it is enough to prove that $\Pi _{j}^{B}$ is countably compact (See Theorem 3.11.2.1 in Kolmogorov and Fomin (1975)), i.e., every sequence in $\Pi _{j}^{B}$ has a limit point in $\Pi _{j}^{B}$. Suppose that $\Pi _{j}^{B}$ is not countably compact so that there is some sequence $\left\{ \pi _{j}^{n}\right\} \in \Pi _{j}^{B}$ that has all its limit point(s) in the complement of $\Pi _{j}^{B}$. The latter statement is equivalent to the statement that there exists some sequence $\left\{ \pi _{j}^{n}\right\} \in \Pi _{j}^{B}$ such that if $\left\{ \pi _{j}^{n_{k}}\right\} \subseteq \left\{ \pi _{j}^{n}\right\} $ is any arbitrary convergent subsequence with limit $\pi _{j}^{\ast }\notin \Pi _{j}^{B}$. First, notice that $\left\{ \pi _{j}^{n_{k}}\right\} \in \Pi _{j}^{B}$ since $\left\{ \pi _{j}^{n_{k}}\right\} \subseteq \left\{ \pi _{j}^{n}\right\} \in \Pi _{j}^{B}$. Now, because $\pi _{j}^{\ast }\notin \Pi _{j}^{B}$, there is some $i\in \mathcal{I}$ and some $D_{i}\subset X_{i}$ with positive measure such that $\pi _{j}^{\ast }\notin \Pi _{j}^{B}$ does not satisfy (\ref{BIC_individual}) for all $x_{i}\in D_{i}$. Because $ \left\{ \pi _{j}^{n_{k}}\right\} $ converges to $\pi _{j}^{\ast }$, there exists some $N_{\circ }\in \mathbb{N}$ such that $\left\{ \pi _{j}^{n_{k}}\right\} $ also violates (\ref{BIC_individual}) for all $ x_{i}\in D_{i}$ and all $n_{k}\geq N_{\circ }$. This contradicts the fact that $\left\{ \pi _{j}^{n_{k}}\right\} \in \Pi _{j}^{B}$. Therefore, $\Pi _{j}^{B}$ must be countably compact, which implies that $\Pi _{j}^{B}$ is totally bounded.
Finally, I show that $\Pi _{j}^{B}$ is complete. Recall that $\Pi _{j}$ is the set of all possible (measurable) mappings $\pi _{j}:X\rightarrow \triangle \left( A_{j}\right) $. Therefore, clearly $\Pi _{j}^{B}\subset \Pi _{j}\subseteq L^{p}\left( X\right) $ with $1<p<\infty $. Because $ L^{p}\left( X\right) $ is complete, it suffices to show that $\Pi _{j}^{B}$ is closed. $\Pi _{j}^{B}$ is closed if and only if every convergent sequence $\left\{ \pi _{j}^{k}\right\} $ in $\Pi _{j}^{B}$ has its limit $\pi _{j}^{\prime \ast }$in $\Pi _{j}^{B}.$ On the contrary, suppose that there exists a convergent sequence $\left\{ \pi _{j}^{k}\right\} $ in $\Pi _{j}^{B} $ with $\pi _{j}^{\prime \ast }\notin \Pi _{j}^{B},$that is, for some $i\in \mathcal{I}$ and some $D_{i}^{\prime }\in X_{i}$ with positive measure such that $\left\{ \pi _{j}^{k}\right\} $ violates (\ref {BIC_individual}) for all $x_{i}\in D_{i}^{\prime }$. Because $\left\{ \pi _{j}^{k}\right\} $ converges to $\pi _{j}^{\prime \ast }$, there exists some $K\in \mathbb{N}$ such that $\left\{ \pi _{j}^{k}\right\} $ also violates ( \ref{BIC_individual}) for all $x_{i}\in D_{i}^{\prime }$ and all $k\geq K$. This contradicts the fact that $\left\{ \pi _{j}^{k}\right\} \in \Pi _{j}^{B}.$ Thus, $\Pi _{j}^{B}$ is closed. \end{proof}
Given Proposition \ref{Prop_compact_BIC}, the existence of a PBE BIC direct mechanisms is straightforward.
\begin{corollary} \label{corollary_existence}A (truth-telling) PBE exists in the competing mechanism game with $\Pi ^{B}$ \end{corollary}
\begin{proof} Because each $\Pi _{j}^{B}$ is non-empty compact convex (Proposition \ref {Prop_compact_BIC}) and $\mathbb{E}_{x}\left[ V_{j}(\pi (x),x)\right] $ is linear in each $\Pi _{j}^{B}$, there exists a PBE in the competing mechanism game with $\Pi ^{B}$ (Glicksberg 1952, Fudenberg and Tirole 1991). \end{proof}
Furthermore, any (truth-telling) PBE in competing BIC direct mechanisms is robust.
\begin{proposition} A (truth-telling) PBE in the competing mechanism game with $\Pi ^{B}$ is $ \emph{robust}$. \end{proposition}
\begin{proof} Suppose that principal $j$ deviates to $\pi _{j}^{\prime }$. Agents choose any arbitrary profile $\left[ c_{1j}\left( \pi _{j}^{\prime },\pi _{j}\cdot \right) ,\ldots ,c_{Ij}\left( \pi _{j}^{\prime },\pi _{j},\cdot \right) \right] $ of continuation equilibrium strategies of communicating wih $j$, given their truthful type reporting to non-deviating principals. Then, $ \left[ c_{1j}\left( \pi _{j}^{\prime },\pi _{j}\cdot \right) ,\ldots ,c_{Ij}\left( \pi _{j}^{\prime },\pi _{j},\cdot \right) \right] $ induces a direct mechanism $\pi _{j}^{\prime \prime }$ such that for all $ (x_{1},\ldots ,x_{I})\in X$ \begin{equation*} \pi _{j}^{\prime \prime }(x_{1},\ldots ,x_{I})=\int_{X_{1}}\cdots \int_{X_{I}}\pi _{j}^{\prime }(x_{1j},\ldots ,x_{Ij})dc_{1j}(\pi _{j}^{\prime },\pi _{j},x_{1})\cdots dc_{Ij}(\pi _{j}^{\prime },\pi _{j},x_{I}) \end{equation*} and $\pi _{j}^{\prime \prime }\in \Pi _{j}^{B}$. Because of (\ref{BIC_eq}), I have that \begin{equation*} \mathbb{E}_{x}\left[ V_{j}(\pi _{j}(x),\pi _{-j}(x),x)\right] \geq \mathbb{E} _{x}\left[ V_{j}(\pi _{j}^{\prime \prime }(x),\pi _{-j}(x),x)\right] . \end{equation*} \end{proof}
\subsection{Market-information invariant robust PBE allocations}
Now I show that the set of market-information invariant robust PBE allocations in a competing mechanism game with $\Gamma $ is the set of (truthful) PBE allocations in the competing direct mechanism game with $\Pi ^{B}$. To establish the equivalence, first recall that given $\gamma \in \Gamma $, a profile of communication strategies $c(\gamma )$ induces a direct mechanism $g_{j}(\gamma _{j},c_{0j}(\gamma _{k}),c_{1j}\left( \gamma ,\cdot \right) ,\ldots ,c_{Ij}\left( \gamma ,\cdot \right) )$ for each principal $j\in \mathcal{J}$.
\begin{lemma} \label{lemma_eq_BIC}Given any $\gamma \in \Gamma $ and any $c_{j}(\gamma )=\left( c_{0j}(\gamma _{j}),c_{1j}\left( \gamma ,\cdot \right) ,\ldots ,c_{Ij}\left( \gamma ,\cdot \right) \right) \in \mathcal{C}_{j}(\gamma ),$ \begin{equation*} g_{j}(\gamma _{j},c_{0j}(\gamma _{j}),c_{1j}\left( \gamma ,\cdot \right) ,\ldots ,c_{Ij}\left( \gamma ,\cdot \right) )\in \Pi _{j}^{B}\text{ for all } j\in \mathcal{J} \end{equation*} \end{lemma}
\begin{proof} Fix a profile of mechanisms $\gamma \in \Gamma $ and a continuation equilibrium $c_{j}(\gamma )\in \mathcal{C}_{j}(\gamma ).$ Suppose that $ g_{j}(\gamma _{j},c_{0j}(\gamma _{j}),c_{1j}\left( \gamma ,\cdot \right) ,\ldots ,c_{Ij}\left( \gamma ,\cdot \right) )$ is not individually BIC. That means that for some $j$, some $i$, some $B_{i}\subset X_{i}$ with $ F(B_{i},X_{-i})>0$, and all $x_{i}\in B_{i},$ there exists $x_{i}^{\prime }\neq x_{i}$ such that \begin{multline*} \mathbb{E}_{x_{-i}}\left[ U_{ij}(g_{j}(\gamma _{j},c_{0j}(\gamma _{j}),c_{-ij}\left( \gamma ,x_{-i}\right) ,c_{ij}\left( \gamma
,x_{i}^{\prime }\right) ),x_{-i},x_{i})|x_{i}\right] > \\ \mathbb{E}_{x_{-i}}\left[ U_{ij}(g_{j}(\gamma _{j},c_{0j}(\gamma _{j}),c_{-ij}\left( \gamma ,x_{-i}\right) ,c_{ij}\left( \gamma ,x_{i}\right)
),x_{-i},x_{i})|x_{i}\right] . \end{multline*} This contradicts that $c_{ij}\left( \gamma ,\cdot \right) $ is agent $i$'s continuation-equilibrium strategy of communicating with principal $j$ given $ \gamma $. Therefore, $g_{j}(\gamma _{j},c_{0j}(\gamma _{j}),c_{1j}\left( \gamma ,\cdot \right) ,\ldots ,c_{Ij}\left( \gamma ,\cdot \right) )$ must be individually BIC for all $j\in \mathcal{J}$. \end{proof}
\begin{center} Let $\Pi _{B}^{\ast }$ be the set of all equilibrium allocations in (truth-telling) PBEs of the competing direct mechanism with $\Pi ^{B}$: \begin{equation*} \Pi _{B}^{\ast }:=\left\{ \pi \in \Pi ^{B}:\mathbb{E}_{x}\left[ V_{j}(\pi _{j}(x),\pi _{-j}(x),x)\right] \geq \mathbb{E}_{x}\left[ V_{j}\left( \pi _{j}^{\prime }(x),\pi _{-j}(x),x\right) \right] \text{ }\forall \pi _{j}^{\prime }\in \Pi _{j}^{B}\right\} \end{equation*} Because of Corollary \ref{corollary_existence}, $\Pi _{B}^{\ast }$ is non-empty. Let $\Pi _{MR}^{\ast }$ be the set of all market-information invariant robust PBE allocations in a general competing mechanism game with $ \Gamma $: \begin{equation*} \Pi _{MR}^{\ast }:=\left\{ \begin{array}{c} \left( g_{k}\left( \gamma _{k}^{\ast },c_{01}^{\ast }(\gamma _{k}^{\ast }),c_{1k}^{\ast }\left( \gamma ^{\ast },\cdot \right) ,\ldots ,c_{Ik}^{\ast }\left( \gamma ^{\ast },\cdot \right) \right) \right) _{k\in \mathcal{J}}\in \Pi ^{B}: \\ (\gamma ^{\ast },c^{\ast })\text{ is a market-information invariant robust PBE.} \end{array} \right\} \text{.} \end{equation*} \end{center}
\begin{theorem} \label{theorem2}$\Pi _{MR}^{\ast }=\Pi _{B}^{\ast }$ \end{theorem}
\begin{proof} First of all, I prove that $\Pi _{MR}^{\ast }\supseteq \Pi _{B}^{\ast }.$ Fix $\pi ^{\ast }=\left( \pi _{1}^{\ast },\ldots \pi _{J}^{\ast }\right) \in \Pi _{B}^{\ast }$. For each $j$ and $i$, let $M_{ij}$ be homeomorphic to $ X_{i},$ that is, there is a homeomorphism $\phi _{ij}:X_{i}\rightarrow M_{ij} $. In a competing general competing mechanism game with $\Gamma $, each principal $j$ offers $\gamma _{j}^{\ast }$ such that $\gamma _{j}^{\ast }\left( m_{0j},\phi _{1j}\left( x_{1}\right) ,\ldots ,\phi _{Ij}\left( x_{I}\right) \right) =\pi _{j}\left( x_{1},\ldots ,x_{I}\right) $ for all $ m_{0j}\in M_{0j}$ and all $\left( x_{1},\ldots ,x_{I}\right) \in X$. Because $\pi ^{\ast }=\left( \pi _{1}^{\ast },\ldots \pi _{J}^{\ast }\right) \in \Pi _{B}^{\ast }$, it is agent $i$'s continuation-equilibrium strategy to use \begin{equation} c_{ij}^{\ast }(\gamma _{j}^{\ast },\gamma _{-j},x_{i})=\phi _{ij}(x_{i}) \label{thm_robust_0} \end{equation} for communicating with principal $j$ for all $\gamma _{-j}\in \Gamma _{-j}$ and all $x_{i}\in X_{i}.$ Since principapl $j$ has no role in determining his action in $\gamma _{j}$, I let $c_{0j}^{\ast }(\gamma _{j}^{\ast })=m_{0j}$ for some $m_{0j}\in M_{ij}$. This implies that $\pi ^{\ast }=\left( \pi _{1}^{\ast },\ldots \pi _{J}^{\ast }\right) \in \Pi ^{B}$ is principal $j$'s utility on the equilibrium path in a competing general competing mechanism game with $\Gamma $.
Suppose that principal $j$ deviates to $\gamma _{j}^{\prime }\in \Gamma _{j}$ . Conditional on a profile of strategies of communicating with non-deviating principals $c_{-j}^{\ast }(\gamma _{-j}^{\ast },\gamma _{j}^{\prime })$, suppose that agents choose a profile of strategies of communcating with principal $j$, $c_{j}(\gamma _{j}^{\prime },\gamma _{-j}^{\ast })\in
\mathcal{C}_{j}(\gamma _{j}^{\prime },\gamma _{-j}^{\ast }|c_{-j}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast }))$. The strategy profile $ c_{j}(\gamma _{j}^{\prime },\gamma _{-j}^{\ast })$ induces a BIC direct mechanism $g_{j}\left( \gamma _{j}^{\prime },c_{j}(\gamma _{j}^{\prime },\gamma _{-j}^{\ast })\right) \in \Pi _{j}^{B}$. Because of (\ref {thm_robust_0}), the continuation-equilibrium utility for principal $j$ becomes \begin{equation*} \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j}^{\prime },c_{j}(\gamma _{j}^{\prime },\gamma _{-j}^{\ast },x)\right) ,\pi _{-j}^{\ast }(x),x\right) \right] . \end{equation*} Because $\pi ^{\ast }=\left( \pi _{1}^{\ast },\ldots \pi _{J}^{\ast }\right) \in \Pi _{B}^{\ast }$, I have that \begin{multline} \mathbb{E}_{x}\left[ V_{j}(\pi _{j}^{\ast }(x),\pi _{-j}^{\ast }(x),x)\right] \geq \label{thm_robust_1} \\ \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j}^{\prime },c_{j}(\gamma _{j}^{\prime },\gamma _{-j}^{\ast },x)\right) ,\pi _{-j}^{\ast }(x),x\right) \right] \text{ }\forall c_{j}(\gamma _{j}^{\prime
},\gamma _{-j}^{\ast })\in \mathcal{C}_{j}(\gamma _{j}^{\prime },\gamma _{-j}^{\ast }|c_{-j}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast })). \end{multline} The inequality above show that $\pi ^{\ast }=\left( \pi _{1}^{\ast },\ldots \pi _{J}^{\ast }\right) \in \Pi _{MR}^{\ast }$ because the left-hand side is principal $j$'s utility on the equilibrium path in the competing general competing mechanism game with $\Gamma $ given (\ref{thm_robust_0}).
Second, I prove $\Pi _{MR}^{\ast }\subseteq \Pi _{B}^{\ast }.$ Fix $\pi ^{\ast }=\left( \pi _{1}^{\ast },\ldots \pi _{J}^{\ast }\right) \in \Pi _{MR}^{\ast }$. Then, (\ref{thm_robust_1}) is satisfied. For all $\gamma _{j}\in \Gamma _{j}$, let \begin{equation*} G_{j}(\gamma _{j})=\left\{ g_{j}(\gamma _{j},c_{j}(\gamma _{j},\gamma _{-j}^{\ast }))\in \Pi _{j}^{B}:c_{j}(\gamma _{j},\gamma _{-j}^{\ast })\in \mathcal{C}_{j}(\gamma _{j},\gamma _{-j}^{\ast })\right\} . \end{equation*} be the set of BIC direct mechanisms that are induced by all possible continuation equilibrium communication strategy profiles. Because of Lemma \ref{lemma_eq_BIC}, $\tbigcup\limits_{\gamma _{j}\in \Gamma _{j}}G_{j}\left( \gamma _{j}\right) \subseteq \Pi _{j}^{B}.$ To see $\tbigcup\limits_{\gamma _{j}\in \Gamma _{j}}G_{j}\left( \gamma _{j}\right) \supseteq \Pi _{j}^{B},$ pick any $\pi _{j}\in \Pi _{j}^{B}$. Given a homeomorphism $\phi _{ij}:X_{i}\rightarrow M_{ij},$ I choose a general mechanism $\gamma _{j}\left( m_{0j},\phi _{1j}\left( x_{1}\right) ,\ldots ,\phi _{Ij}\left( x_{I}\right) \right) =\pi _{j}\left( x_{1},\ldots ,x_{I}\right) $ for all $ m_{0j}\in M_{0j}$ and all $\left( x_{1},\ldots ,x_{I}\right) \in X$. Then, $ c_{j}(\gamma _{j},\gamma _{-j}^{\ast })=\left\{ c_{j}(\gamma _{j}),c_{1j}(\gamma _{j},\gamma _{-j}^{\ast }),\ldots ,c_{Ij}(\gamma _{j},\gamma _{-j}^{\ast })\right\} =\{m_{0j},\phi _{1j},\ldots ,\phi _{Ij}\}$ is a profile of continuation-equilibrium strategies of communicating with principal $j$ and $\pi _{j}=g_{j}(\gamma _{j},c_{j}(\gamma _{j},\gamma _{-j}^{\ast }))\in \tbigcup\limits_{\gamma _{j}\in \Gamma _{j}}G_{j}\left( \gamma _{j}\right) $. Therefore, I have \begin{equation} \tbigcup\limits_{\gamma _{j}\in \Gamma _{j}}G_{j}\left( \gamma _{j}\right) =\Pi _{j}^{B} \label{thm_robust2} \end{equation} (\ref{thm_robust_1}) and (\ref{thm_robust2}) imply that \begin{equation*} \mathbb{E}_{x}\left[ V_{j}(\pi _{j}^{\ast }(x),\pi _{-j}^{\ast }(x),x)\right] \geq \mathbb{E}_{x}\left[ V_{j}\left( \pi _{j}^{\prime }(x),\pi _{-j}^{\ast }(x),x\right) \right] \text{ }\forall \pi _{j}^{\prime }\in \Pi _{j}^{B}, \end{equation*} which implies that $\pi ^{\ast }=\left( \pi _{1}^{\ast },\ldots \pi _{J}^{\ast }\right) \in \Pi _{B}^{\ast }$. \end{proof}
As long as our interest is the set of market-information invariant robust PBE allocations, it is enough to focus on equilibrium allocations in (truth-telling) PBEs of the competing BIC direct mechanism game. Such equilibrium allocations are supported when the BIC direct mechanism each non-deviating principal implements does not depend on whether a competing principal deviates or not. Furthermore, equilibrium allocations in (truth-telling) PBEs of the competing BIC direct mechanism game is free from Szentes' critique since $\Pi _{MR}^{\ast }=\Pi _{B}^{\ast }$.
\subsection{Uniquely implementable robust PBE allocations}
Fix a uniquely implementable robust PBE $\left( \gamma ^{\ast },c^{\ast }\right) $. $\pi _{j}^{\ast }=g_{j}(\gamma _{j}^{\ast },c_{j}^{\ast }(\gamma ^{\ast }))$ is principal $j$'s unique BIC direct mechanism on the equilibrium path. Given the additive separability of each agent's utility function, $\pi _{j}^{\ast }$ is also his unique BIC direct mechanism off the path following a competing principal's deviation. Let $\Pi _{UI}^{\ast }$ be the set of all market-information invariant robust PBE allocations in a general competing mechanism game with $\Gamma $: \begin{equation*} \Pi _{UR}^{\ast }:=\left\{ \begin{array}{c} \left( g_{k}\left( \gamma _{k}^{\ast },c_{01}^{\ast }(\gamma _{k}^{\ast }),c_{1k}^{\ast }\left( \gamma ^{\ast },\cdot \right) ,\ldots ,c_{Ik}^{\ast }\left( \gamma ^{\ast },\cdot \right) \right) \right) _{k\in \mathcal{J}}\in \Pi ^{B}: \\ (\gamma ^{\ast },c^{\ast })\text{ is a uniquely implementable robust PBE.} \end{array} \right\} \text{.} \end{equation*}
Applying the results on Bayesian implementation in the literature, it is easy to see that a uniquely implementable robust PBE allocation $\pi ^{\ast }=\left( \pi _{1}^{\ast },\ldots \pi _{J}^{\ast }\right) $ in a competing mechanism game with $\Gamma $ must satisfy
\begin{enumerate} \item for all $j\in \mathcal{J}$, $\pi _{j}^{\ast }$ satisfies Bayesian monotonicity (Definition 3.1 in Palfrey and Srivastava (2018))
\item $\pi ^{\ast }$ is a truth-telling PBE BIC direct mechanisms in the competing mechanism game with $\Pi ^{B}$ (i.e., $\pi ^{\ast }\in \Pi _{B}^{\ast }$) \end{enumerate}
That is, \begin{equation} \Pi _{UR}^{\ast }\subseteq \Pi _{UB}^{\ast }, \label{UR1} \end{equation} where \begin{equation*} \Pi _{UB}^{\ast }:=\left\{ \begin{array}{c} \pi =\left( \pi _{1},\ldots ,\pi _{J}\right) \in \Pi _{B}^{\ast }: \\ \pi _{j}\text{ satisfies Bayesian monotonicity }\forall j\in \mathcal{J} \end{array} \right\} \end{equation*} (\ref{UR1}) is an extension of the necessary conditions for Bayesian implementation in Theorem 3.1 in Palfrey and Srivastava (2018). Furthermore, when additional conditions are satisfied, conditions 1 and 2 are sufficient conditions for a uniquely implementable robust PBE allocation in a competing mechanism game with $\Gamma $. That is, \begin{equation} \Pi _{UR}^{\ast }\supseteq \Pi _{UB}^{\ast } \label{UR2} \end{equation} These additional conditions are well established in the literature (Theorem 3.2 in Palfrey and Srivastava (2018)). All these results can be adopted seemlessly given the additive separability of each agent's utility function. From (\ref{UR1}) and (\ref{UR2}), we have \begin{equation} \Pi _{UR}^{\ast }=\Pi _{UB}^{\ast } \label{UR3} \end{equation} Note that \begin{equation} \Pi _{UB}^{\ast }\subset \Pi _{B}^{\ast }. \label{UR4} \end{equation}
Given $\pi ^{\ast }=\left( \pi _{1}^{\ast },\ldots \pi _{J}^{\ast }\right) $ that satisfies conditions 1 and 2, each principal $j$'s equilibrium (indirect) mechanism can be constructed the same way that an indirect mechanism is constructed to implement a BIC direct mechanism with Bayesian monotonicity in a model with a single principal. In such an equilibrium mechanism, a principal does not send a message to himself and agents' messages fully determines his action.
Furthermore, a principal only need to evaluate his continuation-equilibrium payoff upon deviation to a BIC direct mechanism for a profitable deviation even if a deviation to a general mechanism is possible. Therefore, a uniquely implementable robust PBE is free from Szentes' critique.
\section{\label{sec: robust_PBE}Robust PBE allocations}
Not every robust PBE allocation is market-information invariant. Some of them can be supported only when agents play a contiuation equilibrium upon a principal's deviation that make non-deviating principals change their BIC direct mechanisms. In this section, I fully characterize the set of robust PBE allocations in the general competing mechanism game. The greatest lower bound of each principals' robust PBE utility in the general competing mechanism game is greater than that in the standard competing mechanism game. The same thing can be shown for the notion of PBE. One advantage of the notion of robust PBE is that the incentive compatibility alone is sufficient to characterize the set of robust equilibrium allocations given the additive separability of agents' utility functions. This point is explained in details in Section \ref{sec:discussion}.
Let $\Pi _{R}^{\ast }$ be the set of all robust PBE allocations of a general competing mechanism game with $\Gamma $: \begin{equation*} \Pi _{R}^{\ast }:=\left\{ \left( g_{k}\left( \gamma _{k}^{\ast },c_{01}^{\ast }(\gamma _{k}^{\ast }),c_{1k}^{\ast }\left( \gamma ^{\ast },\cdot \right) ,\ldots ,c_{Ik}^{\ast }\left( \gamma ^{\ast },\cdot \right) \right) \right) _{k\in \mathcal{J}}\in \Pi ^{B}:(\gamma ^{\ast },c^{\ast }) \text{ is a robust PBE.}\right\} . \end{equation*} Our main result is that $\Pi _{R}^{\ast }$ is characterized by only BIC direct mechanisms without reference to $\Gamma $. For all $j\in \mathcal{J}$ , let \begin{equation} \underline{V}_{j}=\min_{\pi _{-j}\in \Pi _{-j}^{B}}\max_{\pi _{j}\in \Pi _{j}^{B}}\mathbb{E}_{x}\left[ V_{j}(\pi _{j}(x),\pi _{-j}(x),x)\right] . \label{lower_bound} \end{equation} For $\pi =(\pi _{1},\ldots ,\pi _{J})\in \Pi ^{B}$, principal $j$'s expected utility $\mathbb{E}_{x}\left[ V_{j}(\pi _{1}(x),\ldots ,\pi _{J}(x),x)\right] $ and it is linear in each $\Pi _{j}^{B}$. Because each $\Pi _{j}^{B}$ is compact according to Proposition \ref{Prop_compact_BIC}, I can then apply Berge's Maximum Theorem (Aliprantis and Border (1999)) to show that $ \min_{\pi _{-j}\in \Pi _{-j}^{B}}\max_{\pi _{j}\in \Pi _{j}^{B}}\mathbb{E} _{x}\left[ V_{j}(\pi _{j}(x),\pi _{-j}(x),x)\right] $ is well defined.
Define \begin{equation*} \Pi _{RB}^{\ast }:=\left\{ \pi \in \Pi ^{B}:\mathbb{E}_{x}\left[ V_{j}(\pi (x),x)\right] \geq \underline{V}_{j}\text{ for all }j\in \mathcal{J}\right\} \end{equation*}
\begin{theorem} \label{theorem1} \begin{equation} \Pi _{R}^{\ast }=\Pi _{RB}^{\ast }. \label{PBEallocations1} \end{equation} \end{theorem}
\begin{proof} I first show that any robust PBE allocation must generate a utility no less than $\underline{V}_{j}$ for all $j\in \mathcal{J}$. Fix a robust PBE $ (\gamma ^{\ast },c^{\ast })$. On contrary, suppose that for some $j\in \mathcal{J}$ \begin{equation} \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j}^{\ast },c_{j}^{\ast }(\gamma ^{\ast },x)\right) ,g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}^{\ast }(\gamma ^{\ast },x)\right) ,x\right) \right] <\underline{V}_{j}. \label{inequality1} \end{equation} Suppose that principal $j$ unilaterally deviates to $\gamma _{j}^{\prime }:M_{j}\rightarrow \Delta (A_{j})$ such that, with a slight abuse of notation, (i) $M_{0j}=\Pi _{j}^{B}$, (ii) $M_{ij}=X_{i}$ for all $i\in \mathcal{I}$, (iii) for all $\pi _{j}\in M_{0j}$ and all $x_{ij}\in X_{i}$ for all $i\in \mathcal{I}$, \begin{equation} \gamma _{j}^{\prime }\left( \pi _{j},x_{1j},\ldots ,x_{Ij}\right) =\pi _{j}\left( x_{1j},\ldots ,x_{Ij}\right) . \label{deviator_mechanism} \end{equation} This is a general mechanism where principal $j$ chooses his direct mechanism from $\Pi _{j}^{B}$ and agents send their type messages at the same time. \footnote{ A precise formulation of such a mechanism $\gamma _{j}^{\prime }:M_{j}\rightarrow A_{j}$ can be given as follows. $M_{ij}$ is homeomorphic to $X_{i},$ there exists a homeomorphism $\xi _{ij}:X_{i}\rightarrow M_{ij}$ and $M_{0j}$ is homeomorphic to $\eta _{j}(\Pi _{j}^{B})$, that is, there exists an homeomorphism $\varrho _{0j}:\eta _{j}(\Pi _{j}^{B})\rightarrow M_{0j}$, where $\eta _{j}$ is an embedding $\eta _{j}:\Pi _{j}^{B}\rightarrow \overline{\Gamma }_{j}$.}
Now I can construct continuation-equilibrium strategies of communicating with principal $j$ that provides a higher utility for principal $j$ than his PBE utility, \emph{given} continuation-equilibrium strategies $c_{-j}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast })$ of communicating with non-deviating principals. For this, I first show that I can keep continuation-equilibrium strategies $c_{-j}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast })$ of communicating with non-deviating principals regardless of continuation-equilibrium strategies of communicating with principal $j$.
Note that principal $j$'s deviation to $\gamma _{j}^{\prime }$ is not observable by non-deviating principals so that they do not change their messages to themselves, i.e., $c_{0k}(\gamma _{k}^{\ast })=c_{0k}^{\ast }(\gamma _{k}^{\ast })$ for all $k\neq j.$ Now consider agents' communication with every principal $k\neq j.$ Because $c^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast })$ is a continuation equilibrium, I have that for all $k\neq j,$ all $i\in \mathcal{I}$, a.e. $x_{i}\in X_{i}$ and all $\mu _{ik}\in \Delta (M_{ik}),$ \begin{multline*} \mathbb{E}_{x_{-i}}\left[ U_{ik}\left( g_{k}\left( \gamma _{k}^{\ast },c_{0k}^{\ast }(\gamma _{k}^{\ast }),c_{1k}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast },x_{1}),\ldots ,c_{Ik}^{\ast }(\gamma _{j}^{\prime
},\gamma _{-j}^{\ast },x_{I})\right) ,x_{-i},x_{i}\right) |x_{i}\right] \geq \\ \mathbb{E}_{x_{-i}}\left[ U_{ik}\left( g_{k}\left( \gamma _{k}^{\ast },c_{0k}^{\ast }(\gamma _{k}^{\ast }),\mu _{ik},c_{-ik}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast },x_{-i})\right) ,x_{-i},x_{i}\right)
|x_{i}\right] . \end{multline*} This implies that I can also keep each agent $i$'s strategy $c_{ik}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast },\cdot )$ of communicating with each non-deviating principal $k$ for continuation-equilibrium strategies of communicating with principal $j$ at $(\gamma _{j}^{\prime },\gamma _{-j}^{\ast })$. Therefore, I fix \begin{equation} c_{-j}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast }). \label{alternative_str1} \end{equation} for continuation-equilibrium strategies of communicating with non-deviating principals regardless of continuation-equilibrium strategies of communicating with principal $j$.
Now consider how to construct continuation-equilibrium strategies of communicating with principal $j$ that are best for principal $j$ upon his deviation to $\gamma _{j}^{\prime }$. Note that $g_{k}(\gamma _{k}^{\ast },c_{k}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast }))$ is individually BIC for all $k\neq j$ by Lemma \ref{lemma_eq_BIC}. Then, with a slight abuse of notation, let principal $j$ choose a degenerate probability distribution $c_{0j}(\gamma _{j}^{\prime })\in M_{0j}=\Pi _{j}^{B}$ satisfying \begin{equation} c_{0j}(\gamma _{j}^{\prime })\in \underset{\pi _{j}\in \Pi _{j}^{B}}{\arg \max }\;\mathbb{E}_{x}\left[ V_{j}\left( \pi _{j}(x),g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast },x)\right) ,x\right) \right] , \label{alternative_str2} \end{equation} and let each agent choose a non-degenerate probability distribution that chooses her true type with probability one and send it to principal $j$, i.e., for all $i\in \mathcal{I}$ and a.e. $x_{i}\in X_{i}$ \begin{equation} c_{ij}(\gamma _{j}^{\prime },\gamma _{-j}^{\ast },x_{i})=x_{i}\text{.} \label{alternative_str3} \end{equation} Principal $j$ takes $g_{k}\left( \gamma _{k}^{\ast },c_{k}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast },\cdot )\right) $ for all $k\neq j$ as given. Given agents' truthful type reporting to him, $c_{0j}(\gamma _{j}^{\prime })$ satisfying (\ref{alternative_str2}) is clearly principal $j$ 's optimal choice of a direct mechanism in $M_{0j}=\Pi _{j}^{B}$. Given principal $j$'s choice of a BIC direct mechanism $\pi _{j}=c_{0j}(\gamma _{j}^{\prime })$, truthful type reporting (i.e., (\ref{alternative_str3})) is also optimal for every agent $i$ of a.e. $x_{i}\in X_{i}$. Therefore, ( \ref{alternative_str1}), (\ref{alternative_str2}) and (\ref{alternative_str3} ) constitute a \emph{continuation equilibrium} at $(\gamma _{j}^{\prime },\gamma _{-j}^{\ast })$.
Because $g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast })\right) \in \Pi _{-j}^{B}$, principal $j$ 's utility in this continuation equilibrium upon deviation to $\gamma _{j}^{\prime }$ is no less than $\underline{V}_{j}$: \begin{equation} \underset{\pi _{j}\in \Pi _{j}^{B}}{\max }\mathbb{E}_{x}\left[ V_{j}\left( \pi _{j}(x),g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast },x)\right) ,x\right) \right] \geq \underline{V}_{j} \label{inequality2} \end{equation} (\ref{inequality1}) and (\ref{inequality2}) show that \begin{multline*} \underset{\pi _{j}\in \Pi _{j}^{B}}{\max }\mathbb{E}_{x}\left[ V_{j}\left( \pi _{j}(x),g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast },x)\right) ,x\right) \right] \mathbb{>} \\ \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j}^{\ast },c_{j}^{\ast }(\gamma ^{\ast },x)\right) ,g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}^{\ast }(\gamma ^{\ast },x)\right) ,x\right) \right] \end{multline*} and hence principal $j$'s utility in this continuation equilibrium is greater than his equilibrium utility. Because this continuation equilibrium upon deviation to $\gamma _{j}^{\prime }$ is constructed with (\ref {alternative_str1}), (\ref{robust_eq_cond}) in the definition of a robust PBE is violated. Therefore, any robust PBE allocation must be in the set on the right hand side of (\ref{PBEallocations1})
To complete the proof, I need to show how to construct a robust PBE to support an allocation in the set on the right hand side of (\ref {PBEallocations1}). Consider an allocation $\pi ^{\ast }=(\pi _{1}^{\ast },\ldots ,\pi _{J}^{\ast })$ in the set on the right hand side of (\ref {PBEallocations1}). For each $k\in \mathcal{J}$, I consider the following \emph{deviator-reporting mechanism} $\gamma _{k}^{\bigstar }:M_{1k}\times \cdots \times M_{Ik}\rightarrow \Delta (A_{k})$, where, with a slight abuse of notation, $M_{ik}=\{1,\ldots ,J\}\times X_{i}$ (precisely, $M_{ik}$ is homeomorphic to $\{1,\ldots ,J\}\times X_{i},$ that is there exists a homeomorphism $\vartheta _{ik}:\{1,\ldots ,J\}\times X_{i}\rightarrow M_{ik}$ ). Note that a deviator-reporting mechanism is a standard mechanism in $ \overline{\Gamma }_{k}$ where the principal does not send a message to himself. In this deviator-reporting mechanism, each agent sends a message on who the deviator is, if any, and her type. If more than a half of agents report $j\neq k$ as the deviator, principal $k$ assigns a BIC direct mechanism $\pi _{k}^{j}$ that punishes principal $k$. Otherwise, he assigns $ \pi _{k}^{\ast }.$ I choose the following BIC direct mechanism for $\pi _{k}^{j}$. For all $j\in \mathcal{J}$, then $\pi _{k}^{j}$ is principal $k$ 's BIC direct mechanism in $\pi _{-j}^{j}$ satisfying \begin{equation} \pi _{-j}^{j}=(\pi _{k}^{j})_{k\neq j}\in \underset{\pi _{-j}\in \Pi _{-j}^{B}}{\arg \min }\left[ \max_{\pi _{j}\in \Pi _{j}^{B}}\mathbb{E}_{x} \left[ V_{j}(\pi _{j}(x),\pi _{-j}(x),x)\right] \right] . \label{arg_min} \end{equation} (\ref{arg_min}) implies that \begin{equation} \mathbb{E}_{x}\left[ V_{j}\left( \pi ^{\ast }(x),x\right) \right] \geq \max_{\pi _{j}\in \Pi _{j}^{B}}\mathbb{E}_{x}\left[ V_{j}\left( \pi _{j}(x),\pi _{-j}^{j}(x),x\right) \right] =\underline{V}_{j} \label{eq_payoff} \end{equation} because $\pi ^{\ast }=(\pi _{1}^{\ast },\ldots ,\pi _{J}^{\ast })$ in the set on the right hand side of (\ref{PBEallocations1}).
Principal $k$'s deviator-reporting mechanism is constructed as follows. For all $((\ell _{1k},$ $x_{1k}),\ldots ,(\ell _{Ik},x_{Ik}))\in \times _{i\in \mathcal{I}}M_{ik},$ \begin{equation} \gamma _{k}^{\bigstar }((\ell _{1k},x_{1k}),\ldots ,(\ell _{Ik},x_{Ik}))=\left\{ \begin{array}{cc} \pi _{k}^{j}(x_{1k},\ldots ,x_{Ik}) & \text{if }\#\{\ell _{ik}=j\text{ for } j\neq k\}>I/2 \\ \pi _{k}^{\ast }(x_{1k},\ldots ,x_{Ik}) & \text{otherwise} \end{array} \right. . \label{deviator_reporting_DM} \end{equation} Every principal $k$ offers the mechanism $\gamma _{k}^{\bigstar }$ specified in (\ref{deviator_reporting_DM}). Suppose that no principal deviates. Then, each agent $i$ of a.e. $x_{i}\in X_{i}$ truthfully reports $x_{i}$ to every principal $k$ along with the message $\ell _{ik}=k.$ Then $\gamma _{k}^{\ast }$ implements $\pi _{k}^{\ast }(x_{1},\ldots ,x_{I}).$ Since there are three or more agents, any single agent $i$'s deviation from $\ell _{ik}=k$ does not prevent principal $k$ from implementing $\pi _{k}^{\ast }(x_{1},\ldots ,x_{I})$. Because $\pi _{k}^{\ast }$ is individually BIC, such communication behavior constitutes a continuation equilibrium and each principal $k$ receives the utility of $\mathbb{E}_{x}\left[ V_{k}\left( \pi ^{\ast }(x),x\right) \right] $.
Suppose that principal $j$ deviates to a general mechanism $\gamma _{j}:M_{j}\rightarrow \Delta (A_{j}).$ Then, each agent $i$ of a.e. $ x_{i}\in X_{i}$ truthfully reports $x_{i}$ to every principal $k\neq j$ along with the message $\ell _{ik}=j$. Then, $\gamma _{k}^{\bigstar }$ implements $\pi _{k}^{j}(x_{1},\ldots ,x_{I})$. Because $\pi _{k}^{j}$ is individually BIC, I can fix such communication behavior for a continuation equilibrium upon principal $j$'s deviation to $\gamma _{j}.$ Let $\left( c_{0j}(\gamma _{j}),c_{1j}(\gamma _{j},\gamma _{-j}^{\bigstar },\cdot ),\ldots ,c_{Ij}(\gamma _{j},\gamma _{-j}^{\bigstar },\cdot )\right) $ be arbitrary continuation-equilibrium strategies of communicating with principal $j$ upon $j$'s deviation to $\gamma _{j}$. Then, it is clear that \begin{equation} g_{j}\left( \gamma _{j},c_{0j}(\gamma _{j}),c_{1j}(\gamma _{j},\gamma _{-j}^{\bigstar },\cdot ),\ldots ,c_{Ij}(\gamma _{j},\gamma _{-j}^{\bigstar },\cdot )\right) \in \Pi _{j}^{B}. \label{BIC_DM} \end{equation} Because of (\ref{BIC_DM}), I have that \begin{equation} \max_{\pi _{j}\in \Pi _{j}^{B}}\mathbb{E}_{x}\left[ V_{j}\left( \pi _{j}(x),\pi _{-j}^{j}(x),x\right) \right] \geq \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j},c_{j}(\gamma _{j},\gamma _{-j}^{\bigstar },x)\right) ,\pi _{-j}^{j}(x),x\right) \right] \label{deviation_payoff} \end{equation} where $c_{j}(\gamma _{j},\gamma _{-j}^{\bigstar },x)=(c_{0j}(\gamma _{j}),c_{1j}(\gamma _{j},\gamma _{-j}^{\bigstar },x_{1}),\ldots ,c_{Ij}(\gamma _{j},\gamma _{-j}^{\bigstar },x_{I})).$ Note that the expression on the right hand side of (\ref{deviation_payoff}) is principal $ j $'s utility in a continuation equilibrium upon deviation to $\gamma _{j}.$ Combining (\ref{eq_payoff}) and (\ref{deviation_payoff}) yields \begin{equation} \mathbb{E}_{x}\left[ V_{j}\left( \pi ^{\ast }(x),x\right) \right] \geq \mathbb{E}_{x}\left[ V_{j}\left( g_{j}\left( \gamma _{j},c_{j}(\gamma _{j},\gamma _{-j}^{\bigstar },x)\right) ,\pi _{-j}^{j}(x),x\right) \right] , \label{robust_PBE_payoff} \end{equation} which shows that principal $j$ cannot gain upon deviation to $\gamma _{j}$. ( \ref{robust_PBE_payoff}) implies that $\pi ^{\ast }=(\pi _{1}^{\ast },\ldots ,\pi _{J}^{\ast })$ is supported as a robust PBE allocation because the first argument $g_{j}\left( \gamma _{j},c_{j}(\gamma _{j},\gamma _{-j}^{\bigstar },x)\right) $ on the right-hand side of (\ref {robust_PBE_payoff}) is derived with any continuation-equilibrium strategies of communicating with principal $j$, $\left( c_{0j}(\gamma _{j}),c_{1j}(\gamma _{j},\gamma _{-j}^{\bigstar },\cdot ),\ldots ,c_{Ij}(\gamma _{j},\gamma _{-j}^{\bigstar },\cdot )\right) $, given agents' truth telling to non-deviating principals. \end{proof}
The proof of Theorem \ref{theorem1} first establishes that a principal's utility in a robust PBE $(\gamma ^{\ast },c^{\ast })$ cannot be lower than his minmax value over BIC direct mechanisms. Given continuation-equilibrium strategies $c_{-j}^{\ast }$ of communicating with non-deviating principals, principal $j$ can deviate to a general mechanism $\gamma _{j}^{\prime }$ where he can choose any BIC direct mechanism from $\Pi _{j}^{B}$ and agents send their type messages to principal $j$ at the same time. Given the robustness of PBE, principal $j$ can implement the BIC direct mechanism from $\Pi _{j}^{B}$ that maximizes his utility, taking $g_{-j}\left( \gamma _{-j}^{\ast },c_{-j}^{\ast }(\gamma _{j}^{\prime },\gamma _{-j}^{\ast })\right) \in \Pi _{-j}^{B}$ as given. This directly implies that principal $ j$'s robust PBE utility cannot be lower than his minmax value over BIC direct mechanisms.
The proof of Theorem \ref{theorem1} is completed by constructing the deviator-reporting mechanism to show how to support any profile of BIC direct mechanisms that provides every principal a utility no less than his minmax value over BIC direct mechanisms as a robust PBE allocation.
The existence of a robust PBE with deviator-reporting mechanisms can be addressed. First consider a continuation equilibrium upon principal $j$'s deviation to a general mechanism $\gamma _{j}$. If principal $j$ deviates, the truth-telling continuation equilibrium that implements $\pi _{-j}^{j}$ is also well established and exists when it comes to communicating with non-deviating principals.
One may wonder if there is a (mixed-strategy) continuation equilibrium \begin{equation*} \left( c_{0j}(\gamma _{j}),c_{1j}(\gamma _{j},\gamma _{-j}^{\bigstar },\cdot ),\ldots ,c_{Ij}(\gamma _{j},\gamma _{-j}^{\bigstar },\cdot )\right) \end{equation*} exists for communicating with deviating principal $j$ upon his deviation to a general mechanism $\gamma _{j}$. When agents and principal $j$ send messages to principal $j$ given his mechanism $\gamma _{j}$, they take $\pi _{-j}^{j}$ as given. For all $(m_{0j},m_{1j},\ldots ,m_{Ij})\in M_{j}$ and all $x\in X$, principal $j$'s utility for the subgame defined by $\gamma _{j} $ follows \begin{equation} \mathcal{V}_{j}(m_{0j},m_{1j},\ldots ,m_{Ij},x):=V_{j}\left( \gamma _{j}(m_{0j},m_{1j},\ldots ,m_{Ij}),\pi _{-j}^{j}(x),x\right) . \label{prinicpal_payoff2} \end{equation} Because the agent's utility function is additively separable with respect to principals' actions, the agent's relevant utility for the subgame defined by $\gamma _{j}$ is \begin{equation} \mathcal{U}_{ij}(m_{0j},m_{1j},\ldots ,m_{Ij},x):=U_{ij}(\gamma _{j}(m_{0j},m_{1j},\ldots ,m_{Ij}),x). \label{agent_payoff2} \end{equation} If $\mathcal{V}_{j}$ and $\mathcal{U}_{ij}$ and the information structure satisfy the sufficient conditions for the existence of a Bayesian equilibrium with continuous actions identified in the literature (Balder (1988), Hellman and Levy (2017), Milgrom and Weber (1985), etc.), then a continuation equilibrium for the subgame defined by $\gamma _{j}$ exists. \footnote{ For example, $\mathcal{V}_{j}$ and $\mathcal{U}_{ij}$ are uniformly continuous and the information structure is absolutely continuous (Milgrom and Weber (1985)), then a (mixed-strategy) continuation equilibrium exists. However, the existence of a (mixed-strategy) continuation equilibrium for communicating with principal $j$ upon his deviation should come naturally. When principal $j$ deviates, he will certainly deviate to a mechanism that guarantees the existence of a (mixed-strategy) continuation equilibrium for communicating with him.}
Because any (mixed) continuation-equilibrium strategies \begin{equation*} \left( c_{0j}(\gamma _{j}),c_{1j}(\gamma _{j},\gamma _{-j}^{\bigstar },\cdot ),\ldots ,c_{Ij}(\gamma _{j},\gamma _{-j}^{\bigstar },\cdot )\right) \end{equation*} upon deviation to any mechanism $\gamma _{j}$ induces a BIC direct mechanism \begin{equation} g_{j}\left( \gamma _{j},c_{0j}(\gamma _{j}),c_{1j}(\gamma _{j},\gamma _{-j}^{\bigstar },\cdot ),\ldots ,c_{Ij}(\gamma _{j},\gamma _{-j}^{\bigstar },\cdot )\right) \in \Pi _{j}^{B}, \label{deviation_BIC} \end{equation} the notion of robustness makes it possible for principal $j$ to focus on any BIC direct mechanism in $\Pi _{j}^{B}$ for his deviation.
Finally, note that a deviator-reporting mechanism $\gamma _{k}^{\bigstar }$ can be viewed as one of mechanisms in $\Gamma _{k}$ for all $k$ and each principal $j$ receives a utility no higher than $\underline{V}_{j}$ upon deviation to any $\gamma _{j}\in \Gamma _{j}$ in a continuation equilibrium where agents report the identity of a deviator and their types truthfully to non-deviating principals whose mechanisms are deviator-reporting mechanisms. Given the structure of a deviator-reporting mechanism and (\ref {deviation_BIC}), this implies that a robust PBE of a general competing mechanism game with $\Gamma $ exists if (i) $\Pi _{RB}^{\ast }=\left\{ \pi \in \Pi ^{B}:\mathbb{E}_{x}\left[ V_{j}(\pi (x),x)\right] \geq \underline{V} _{j}\text{ for all }j\in \mathcal{J}\right\} $ is non-empty\footnote{$\Pi _{R}^{\ast }$ is non-empty because $\Pi _{B}^{\ast }\subset \Pi _{R}^{\ast }$ and $\Pi _{B}^{\ast }$ is non-empty because of Corollary \ref {corollary_existence}.} and (ii) $\mathcal{V}_{j}$ and $\mathcal{U}_{ij}$ defined for any given $\gamma _{j}$ ((\ref{prinicpal_payoff2}) and (\ref {agent_payoff2})), and the information structure satisfy the sufficient conditions for the existence of a Bayesian equilibrium with continuous actions identified in the literature (Balder (1988), Hellman and Levy (2017), Milgrom and Weber (1985), etc.).
Note that it is clear that $\Pi _{B}^{\ast }\subset \Pi _{RB}^{\ast }$. Applying (\ref{UR4}), we have that \begin{equation*} \Pi _{UB}^{\ast }\subset \Pi _{B}^{\ast }\subset \Pi _{RB}^{\ast }. \end{equation*} All three sets are independent of the set of general mechanisms allowed in competing mechanism games and they are all specified in terms of BIC direct mechanisms. My paper identifies the notions of PBEs that produce those sets of allocations as the sets of equilibrium allocations in general competing mechnaism games respectively.
\section{Discussion\label{sec:discussion}}
For a robust equilibrium, the standard competing mechanism game is restrictive in the sense that it takes the freedom of choosing any BIC direct mechanism away from a principal at the same time agents send messages to him given his mechanism. In the standard competing mechanism game, a deviating principal's utility can be as low as his maxmin value over BIC direct mechanisms: \begin{equation*} \max_{\pi _{j}\in \Pi _{j}^{B}}\underset{\pi _{-j}\in \Pi _{-j}^{B}}{\min \text{ }}\mathbb{E}_{x}\left[ V_{j}(\pi _{j}(x),\pi _{-j}(x),x)\right] \end{equation*} The reason is that agents' communication with the deviator alone induces a BIC direct mechanism from the deviator's mechanism and non-deviators can choose their BIC direct mechanisms conditional on agents' reports on the deviator's BIC direct mechanism that they induce from the deviator's mechanism.
When random mechanisms are allowed as in our paper and there are only two principals, the restrictive feature of the standard competing mechanism game is not a problem. To see this, I use Sion's minimax theorem.
\begin{theorem}[Sion's minimax theorem (1958)] Let $Y$ and $Z$ be convex, compact spaces. If the real-valued function $f$ on $Y\times Z$ is quasi-concave and upper semi-continuous in $Y$ and quasi-convex and lower semi-continuous in $Z$, \begin{equation*} \inf_{y\in Y}\text{ }\sup_{z\in Z}\text{ }f(y,z)=\sup_{z\in Z}\text{ } \inf_{y\in Y}\text{ }f(y,z). \end{equation*} \end{theorem}
For the case of $J=2$, it is easily shown that \begin{equation} \underset{\pi _{-j}\in \Pi _{-j}^{B}}{\min \text{ }}\max_{\pi _{j}\in \Pi _{j}^{B}}\mathbb{E}_{x}\left[ V_{j}(\pi _{j}(x),\pi _{-j}(x),x)\right] =\max_{\pi _{j}\in \Pi _{j}^{B}}\underset{\pi _{-j}\in \Pi _{-j}^{B}}{\min \text{ }}\mathbb{E}_{x}\left[ V_{j}(\pi _{j}(x),\pi _{-j}(x),x)\right] \label{minmax_equivalence} \end{equation} Since $\Pi _{k}^{B}$ is a compact convex set for all $k\in \mathcal{J}$, the properties of the domain of the functions in Sion's Minimax Theorem (1958) are satisfied. Since $\mathbb{E}_{x}\left[ V_{j}(\pi _{j}(x),\pi _{-j}(x),x) \right] $ is linear in each $\Pi _{k}^{B}$ I can apply Sion's Minimax Theorem to establish (\ref{minmax_equivalence}).
In the case of $J\geq 3$, one cannot swap the max and min in (\ref {minmax_equivalence}). To see this reason, interpret $\Pi _{-j}^{B}$ as the set of degenerate probability distributions that put probability one on a profile of individually BIC direct mechanisms for principals except for $j$. A convex combination between two such degenerate probability distributions creates correlation between two profiles of individually BIC direct mechanisms and such correlation is not feasible if I only focus on degenerate probability distributions that put probability one on a profile of individually BIC direct mechanisms. Therefore, $\Pi _{-j}^{B}$ is not a convex set in terms of the probability measure space and hence one cannot apply Sion's Minimax Theorem (1958). Therefore, Szentes' critique is valid in a model with incomplete information as to the set of allocations supportable in robust PBEs where agents' strategies of communicating with non-deviating principals and their induced BIC direct mechanisms are responsive to market changes. However, allocations supportable in market-information invariant robust PBEs are free from Szentes' critique as shown in Section \ref{sec:competing_BIC}.
\paragraph{Robust PBE vs. PBE}
It can be also shown that Szentes' critique is generally valid for PBE as well. However, the set of PBE allocations cannot be characterized by incentive compatibility alone for PBE or robust PBE. To see this point, let $ \mathcal{C}_{j}^{\ast }\left( \pi _{-j},\gamma _{j}\right) $ be the set of all continuation equilibrium strategies of communicating with principal $j$ conditional on truth telling type reporting to the other principals given $ \left( \pi _{-j},\gamma _{j}\right) \in \Pi _{-j}^{B}\times \Gamma _{j}.$ Using recommendation mechanisms for non-deviators' equilibrium mechanisms (Yamashita (2010)), I can show that the greatest lower bound of principal $j$ 's PBE utility is
\begin{eqnarray*} \underline{V}_{j}^{P} &\equiv &\underset{\pi _{-j}\in \Pi _{-j}^{B}}{\min \text{ }}\max_{\gamma _{j}\in \Gamma _{j}}\min_{c_{j}(\pi _{-j},\gamma _{j})\in \mathcal{C}_{j}^{\ast }\left( \pi _{-j},\gamma _{j}\right) }\mathbb{ E}_{x}\left[ V_{j}(\gamma _{j}(c_{j}(\pi _{-j},\gamma _{j},x)),\pi _{-j}(x),x)\right] \\ &=&\underset{\pi _{-j}\in \Pi _{-j}^{B}}{\min \text{ }}\max_{_{\pi _{j}}\in \hat{\Pi}_{j}^{B}(\Gamma _{j})}\mathbb{E}_{x}\left[ V_{j}(\pi _{j}(x)),\pi _{-j}(x),x)\right] , \end{eqnarray*} where \begin{equation*} \hat{\Pi}_{j}^{B}(\Gamma _{j})\equiv \left\{ \begin{array}{c} \gamma _{j}(c_{j}(\pi _{-j},\gamma _{j},\cdot ))\in \Pi _{j}^{B}: \\ \begin{array}{c} \text{(i) }\gamma _{j}\in \Gamma _{j},\text{(ii) }c_{j}(\pi _{-j},\gamma _{j})\in \mathcal{C}_{j}^{\ast }\left( \pi _{-j},\gamma _{j}\right) , \\ \text{(iii) }\mathbb{E}_{x}\left[ V_{j}(\gamma _{j}(c_{j}(\pi _{-j},\gamma _{j},x)),\pi _{-j}(x),x)\right] \leq \\ \mathbb{E}_{x}\left[ V_{j}(\gamma _{j}(c_{j}^{\prime }(\pi _{-j},\gamma _{j},x)),\pi _{-j}(x),x)\right] \text{ }\forall c_{j}^{\prime }(\pi _{-j},\gamma _{j})\in \mathcal{C}_{j}^{\ast }\left( \pi _{-j},\gamma _{j}\right) \end{array} \end{array} \right\} \end{equation*}
It is clear that $\hat{\Pi}_{j}^{B}(\Gamma _{j})\subset \Pi _{j}^{B}$ but generally $\hat{\Pi}_{j}^{B}(\Gamma _{j})$ is not equal to $\Pi _{j}^{B}$. The reason is that given the other principals' BIC direct mechanisms $\pi _{-j}\in \Pi _{-j}^{B}$, some direct mechanisms in $\Pi _{j}^{B}$ may not be induced by the worst continuation equilibrium regardless of his deviation to any mechanism in $\Gamma _{j}$.
Identifying the set of strongly robust equilibrium is challenging even in the standard competing mechanism game. It is not known how to characterize the greatest lower bound of each principal's strongly robust PBE utility even in terms of mechanisms allowed in the game just like Yamashita (2010) does for the greatest lower bound of each principal's PBE equilibrium utility.
Together with the separability assumption on agents' utility function, the notion of robust PBE allows us to focus on the incentive compatibility alone to identify the set of equilibrium allocations because $\Pi _{k}^{B}$ is defined only with the incentive compatibility respectively for each $k\in \mathcal{J}$.
\paragraph{Mixed strategies for principals}
I have not considered mixed strategies for principals. This can be done easily. Consider the general competing mechanism game. When principal $j$ deviates given that the other principals employ mixed strategies for their mechanism choice, he can deviate to a general mechanism where he asks agents not only about their types but also the BIC direct mechanisms $\pi _{-j}$ induced from the other principals' mechanisms. Principal $j$'s message in this general mechanism is a mapping from $\Pi _{-j}^{B}$ into $\Pi _{j}^{B},$ which specifies his choice of a BIC direct mechanism as a function of agents' reports on the other principals' BIC direct mechanisms. In this way, even in a mixed-strategy robust PBE, a deviating principal can always implement a BIC direct mechanism that maximizes his utility conditional on BIC direct mechanisms induced by the other principals' mechanisms. Therefore, the greatest lower bound of principal $j$'s mixed-strategy robust PBE utility is the same as the greatest lower bound of his pure-strategy robust PBE utility, that is, his minmax value $\underline{V}_{j}$ over BIC direct mechanisms. For all $j$, let \begin{equation*} \Phi _{j}(\sigma _{1},\ldots ,\sigma _{J}):=\int_{\Pi _{1}^{B}}\cdots \int_{\Pi _{J}^{B}}\mathbb{E}_{x}\left[ V_{j}(\pi _{1}(x),\ldots \pi _{J}(x),x)\right] d\sigma _{1}\cdots d\sigma _{J}, \end{equation*} where $(\sigma _{1},\ldots ,\sigma _{J})\in \Delta (\Pi _{1}^{B})\times \cdots \times \Delta (\Pi _{J}^{B})$. Then, the set of mixed-strategy robust PBE allocations of a general competing mechanism game with $\Gamma $ is \begin{equation*} \left\{ (\sigma _{1},\ldots ,\sigma _{J})\in \Delta (\Pi _{1}^{B})\times \cdots \times \Delta (\Pi _{J}^{B}):\Phi _{j}(\sigma _{1},\ldots ,\sigma _{J})\geq \underline{V}_{j}\text{ for all }j\in \mathcal{J}\right\} . \end{equation*}
\end{document} | arXiv |
Search Results: 1 - 10 of 221007 matches for " H. W. Leite; "
Ab initio Calculations of the Vibrational Modes of MnAs and Ga1-xMnxAs
H. W. Leite Alves
Physics , 2006, DOI: 10.1063/1.2729814
Abstract: In this work, we present our theoretical results for the equation of state and the phonon dispersions of MnAs, as well as the Mn concentration dependence of both the lattice parameter and the phonon frequencies of the cubic GaMnAs alloys. The results are in good agreement with the experimental results whenever this comparison is possible. Based on the obtained results, the lattice constants and the phonon frequencies of the alloys do not obey the Vegard rule.
Kinetic Monte Carlo simulation of the nitridation of the GaAs (100) surfaces
Castro, A. P.;Alves, H. W. Leite;
Brazilian Journal of Physics , 2006, DOI: 10.1590/S0103-97332006000300019
Abstract: we present, our preliminary results of a systematic theoretical study of the adsorption of n over as-terminated gaas (100) (2×1) surfaces. we analyzed the changes in the bond-lengths, bond-angles and the energetics involved before and after deposition. our results show that the n-atoms will prefer the unoccupied sites of the surface, close to the as dimer. the presence of n pushes the as dimer out of the surface, leading to the anion exchange between n and as atoms. based on our results, we discussed the kinetics of the n islands formation during epitaxial growth of the iii-nitrides.
Structural and electronic properties of the SiC (100) surfaces
Soares, J. S.;Alves, H. W. Leite;
Abstract: in this work, we present our preliminary ab initio results for the structural and electronic properties of both si- and c-terminated sic (100) surfaces in (2×1) and c(2×2) reconstruction patterns. based on our results, we found that the si-terminated surfaces are dominated by weak bonded si-dimers, which are stabilized only at si-rich conditions, leading to (3×2) or more complex reconstruction patterns, as verified experimentally. also, our results show that the c-terminated surfaces is characterized by strong triply-bonded c-dimers, in a c(2×2) reconstruction pattern, which consists of c2 pairs over si bridge sites, in agreement with experimental results.
Ab initio determination of the (100) surfaces phonon dispersions of the SiC
Abstract: in this work, we presented our preliminary ab initio results for the vibrational modes and the phonon frequencies of the sic (100) surfaces. our results are in good agreement with the available experimental data whenever this comparison is possible. for the accepted models of the c-terminated surfaces in the c(2×2) reconstruction, while in the bridge-dimer model there is an acetylene-like vibrational a1 mode at 2031 cm-1, which is infrared active, in the staggered-dimer model, there is a füchs-kliewer (fk) mode at 1328 cm-1, which is experimentally detected. for si-terminated surfaces in the p(2×1) reconstruction, instead, no fk was obtained, in contradiction with the hreels experimental results for the si-terminated surfaces, but they are in consonance with the fact that this surface should be described by a (3×2) or more complex models.
A. P. Castro,H. W. Leite Alves
Physics , 2006, DOI: 10.1590/S0103-97332006000300019
Abstract: We present, in this work, our preliminary results of a systematic theoretical study of the adsorption of N over As-terminated GaAs (100) (2$\times$1) surfaces. We analyzed the changes in the bond-lenghts, bond-angles and the energetics involved before and after deposition. Our results show that the N-atoms will prefer the unoccupied sites of the surface, close to the As dimer. The presence of the N pushes the As dimer out of the surface, leading to the anion exchange between the N and As atoms. Based on our results, we discussed about the kinetics of the N islands formation during epitaxial growth of the III-Nitrides.
Efeito de micronutrientes na produ??o e no tipo de tubérculos de batata, em cultura efetuada em solos de várzea do Vale do Paraíba
Gargantini, H.;Leite, N.;Hungria, L. S.;Venturini, W. R.;
Bragantia , 1970, DOI: 10.1590/S0006-87051970000100001
Abstract: the influence of the micronutrients iron, copper, zinc, molybdenium, manganese and boron added to the basic fertilizer on yield and quality of potato (solatium tuberosum l.), growing on alluvial soils was studied. an increase in the yield of 40% in soils of avareí series and 50% of paraiba series, was associated with application of boron and molybdenium to the soil. the effect of boron was much more pronounced than that of molybdenium. application of iron, copper, zinc and manganese have no significant effect on potato yield.
Ab initio calculation of the dynamical properties of PPP and PPV
R. L. de Sousa,H. W. Leite Alves
Abstract: In this work, we have calculated the vibrational modes and frequencies of the crystalline PPP (in both the Pbam and Pnnm symmetries) and PPV (in the P21/c symmetry). Our results are in good agreement with the available experimental data. Also, we have calculated the temperature dependence of their specific heats at constant volume, and of their vibrational entropies. Based on our results, at high temperatures, the PPP is more stable in the Pnnm structure than in the Pbam one.
Ab initio theoretical studies of atomic and electronic structures of III-nitride (110) surfaces
Alves, H. W. Leite;Alves, J. L. A.;Nogueira, R. A.;Leite, J. R.;
Abstract: we present a systematic theoretical study of several iii-nitride (110) surfaces based on accurate, parameter-free, self-consistent total energy and force calculations using the density functional theory, the generalized gradient approximation(gga) for the exchange-correlation term, and the full potential linear augmented plane wave (fplapw) approach associated with the slab supercell model. we studied aln, bn, gan and inn and analyzed the theoretical trends for the equilibrium atomic structures and surface band structures. we used supercells built up of 7 atomic layers and a vacuum region equivalent of 5 atomic layers. for the ga and in species, the 3d and 4d electrons were treated properly as valence electrons. in connection with the atomic structures, we are concerned with the leed parameters d1^, d1x, d2^ , d12^, d12x and w for the (110) surface. we analyzed the changes in the bond-lengths and in the bond-angles at the anion and cation sites. we conclude that similarly to the iii-arsenide (110) and iii-phosphide (110) surfaces, the iii-nitride (110) surfaces relax such that the cation-surface atom moves inward and the n-surface atom moves outward. the large coulomb energy of the iii-nitrides as compared with the other iii-v compounds re ects in the smaller value of the tilt angle w and in the small value of d1^. to our knowledge, this is the rst time the fplapw method is used for such a systematic study of iii-nitrides, and we compare our results with recent results obtained with other approaches as reported in the literature.
Planar force-constant method for lattice dynamics of cubic III-nitrides
Alves H. W. Leite,Alves J. L. A.,Scolfaro L. M. R.,Leite J. R.
Brazilian Journal of Physics , 2002,
Abstract: Using the density-functional theory within the Full Potential Linear Augmented Plane-Wave (FP-LAPW) method, we have calculated, in this work, the equation of state, the effective charges and the phonon dispersion along [100] and [111] directions for the cubic III-Nitrides(c-BN, c-AlN, c-GaN and c-InN), including hydrostatic strains dependence. A good agreement with the micro-Raman scattering experiment is obtained for the phonon modes at G. Our results show that, for c-InN case, the apparent discrepancy between the known experimental results is a consequence of hydrostatic effects on the sample due to differences of the used Raman methods.
Alves H. W. Leite,Alves J. L. A.,Nogueira R. A.,Leite J. R.
Abstract: We present a systematic theoretical study of several III-nitride (110) surfaces based on accurate, parameter-free, self-consistent total energy and force calculations using the density functional theory, the generalized gradient approximation(GGA) for the exchange-correlation term, and the Full Potential Linear Augmented Plane Wave (FPLAPW) approach associated with the slab supercell model. We studied AlN, BN, GaN and InN and analyzed the theoretical trends for the equilibrium atomic structures and surface band structures. We used supercells built up of 7 atomic layers and a vacuum region equivalent of 5 atomic layers. For the Ga and In species, the 3d and 4d electrons were treated properly as valence electrons. In connection with the atomic structures, we are concerned with the LEED parameters delta1^, delta1x, delta2^ , d12^, d12x and omega for the (110) surface. We analyzed the changes in the bond-lengths and in the bond-angles at the anion and cation sites. We conclude that similarly to the III-arsenide (110) and III-phosphide (110) surfaces, the III-nitride (110) surfaces relax such that the cation-surface atom moves inward and the N-surface atom moves outward. The large Coulomb energy of the III-nitrides as compared with the other III-V compounds re ects in the smaller value of the tilt angle omega and in the small value of delta1^. To our knowledge, this is the rst time the FPLAPW method is used for such a systematic study of III-nitrides, and we compare our results with recent results obtained with other approaches as reported in the literature. | CommonCrawl |
Low Level Radiation
Low level radiation is present as both naturally occurring radiation and man made radiation. The impact of the naturally occurring radiation is hard to quantify; no studies have shown an increase in health problems in areas where natural radiation levels are higher. The impact of man made radiation is likely to remain insignificant until it rises above the naturally occurring radiation in the environment.
Natural Radiation
Natural radiation exists in the environment and is due to several processes:
Cosmic rays are high energy photons or particles that enter the Earth's atmosphere. At sea level, the atmosphere has attenuated most of their energy but they still contribute a significant portion to a person's yearly dose
Cosmogenic radionuclides are produced through interactions of cosmic rays with atoms in the atmosphere or on the ground. These include 14C, 3H and 7Be
Terrestrial Radiation occurs due to the decay of 238U, 235U and 232Th and their daughter products in the earth's crust.
Radon gas is usually considered as a separate entity as it makes up a large portion of the yearly dose.
Internal radiation typically arises from ingestion of 40K, a long lived isotope of potassium that decays via beta emission (and thus has minimal exposure from the crust).
Interestingly, Australia has a low level of radon gas in homes, meaning that the contribution to total dose is much less than in other parts of the world.
Man Made Radiation
About 18% of radiation exposure to the general population is due to man made sources:
15% from medical sources (11% x-ray and 4% nuclear medicine)
3% from consumer products (smoke alarms etc)
0.3% from occupational exposure (mining, energy)
Under 0.3% from fallout
Hazards of low level radiation
In general, low levels of radiation do not cause deterministic effects as they fall below the threshold dose required to induce these effects. The exception is growth retardation, microcephaly or mental retardation in persons exposed in utero to doses over 0.1 - 0.2 Sv. Aside from this, the main risk to the populace is stochastic effects, such as induction of cancer and hereditary effects.
The ICRP uses the term detriment to describe stochastic effects. The detriment for developing cancer is about $4.6 \times 10^{-2}$ per Sv. The risk of hereditary effects is much lower at $0.1 \times 10^{-2}$ per Sv. For the average radiation worker, the risk of developing an adverse effect is about $2 \times 10^{-4}$ per year.
carcinogenesislate_effectphase1protectionradiobiology
The Employment Network of SB
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FreeSMUG
Free Open Source Software Mac User Group | CommonCrawl |
Differential spatial modulation for high-rate transmission systems
Thu Phuong Nguyen1,
Xuan Nam Tran1,
Minh-Tuan Le2 &
Huan X. Nguyen3
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 6 (2018) Cite this article
This paper introduces a new differential spatial modulation (DSM) scheme which subsumes both the previously introduced DSM and high-rate spatial modulation (HR-SM) for wireless multiple input multiple output (MIMO) transmission. By combining the codeword design method of the HR-SM scheme with the encoding method of the DSM scheme, we develop a high-rate differential spatial modulation (HR-DSM) scheme equipped with an arbitrary number of transmit antennas that requires channel state information (CSI) neither at the transmitter nor at the receiver. The proposed approach can be applied to any equal energy signal constellations. The bit error rate (BER) performance of the proposed HR-DSM schemes is evaluated by using both theoretical upper bound and computer simulations. It is shown that for the same spectral efficiency and antenna configuration, the proposed HR-DSM outperforms the DSM in terms of bit error rate (BER) performance.
In recent years, many transmission techniques for wireless multiple input multiple output (MIMO) communication systems have been proposed. A byword is the spatial modulation (SM) [1, 2], which has attracted increased research interest recently. By exploiting the difference in the channel impulse response of a wireless link from a transmit antenna element to a receive one, the transmit antenna indices are utilized as an additional means to carry information. Compared with the Vertical Bell Laboratories Layered Space-Time (V-BLAST) [3] and space-time block codes (STBCs) [4, 5], SM has several advantages which are attained by the following three essential features: (1) unlike the V-BLAST, in an SM scheme, only one transmit antenna is activated during transmission, thereby completely avoiding the problem of inter-channel interference (ICI) among the transmit antennas; (2) since only one antenna is activated for signaling the SM, transmitter needs to use only a single radio frequency (RF) chain, which certainly helps to reduce the hardware cost as well as energy consumption compared with V-BLAST and STBCs; (3) the spatial position of each transmit antenna is utilized to convey information, thus allowing SM to obtain a spectral efficiency that increases logarithmically with the number of transmit antennas. Coherent Space-Time Shift Keying (CSTSK) [6–8] is possible of striking a flexible tradeoff between the obtainable diversity and multiplexing gain. This scheme was shown to exhibit a better performance than the SM and SSK schemes since it is possible to obtain both transmit and receive diversity, rather than only receive diversity as in SM. But unlike SM, CSTSK needs multiple RF chains at the transmitter. CSTSK is also inferior to SM in that it requires strict Inter-Antenna Synchronization (IAS) as V-BLAST and STBC. Furthermore, low-complexity ML detection algorithms for SM systems with M−QAM, M−PSK modulation are proposed in [9] and [10], respectively. In these detection algorithms, the ML search complexity is independent of the constellation size. As a consequence, SM is a promising MIMO technique over the conventional MIMO schemes.
Until now, most investigations on SM assumed that the CSI is available at the receiver. This assumption is reasonable when the channel varies slowly compared with the symbol rate, and knowledge of the channel can be obtained via training. Unfortunately, obtaining channel knowledge is not always realizable when the channel changes quickly. Moreover, for MIMO channels, the number of channel coefficients to be measured is equal to the product of the number of transmit antennas and the number of receive antennas. The length of the training sequence grows proportionally with the number of transmit antennas [11]. This could result in a large decrease of the overall system throughput. In order to dispense with the CSI estimation, a blind detector have been recently proposed for SM [12]. However, this detector requires a large number of observed symbols and thus increased computational complexity for signal estimation. Therefore, solutions that do not require CSI and large observations, such as differential modulation, are very useful.
Differential signaling is a widely used approach to deal with the problem of high-mobility wireless communications without requiring CSI. The differential transmission concept has been successfully implemented in many MIMO systems, e.g., the differential Alamouti scheme [13] and differential spatial multiplexing [14]. However, this approach could hardly be directly applied to SM or Space Shift Keying (SSK) because the channel in SM/SSK systems is the virtual modulation unit, which makes the design of differential SM/SSK unique and difficult. Very recently, a number of differential schemes for SM have been proposed [15–19], called differential SM (DSM), which can be applied to any equal energy signal constellations. Similar to SM, DSM activates only one antenna at a symbol instant. Therefore, ICI is avoided and the requirement of IAS is relaxed. However, the performance of DSM is restricted at a given spectral efficiency.
Motivated by the aforementioned challenge, in this paper, we propose a new DSM scheme based on the concept of spatial constellation (SC), SC codewords [20] and HR-SM codeword [21], called High-Rate Differential Spatial Modulation (HR-DSM), for an arbitrary number of transmit antennas. In the proposed HR-DSM scheme, transmit signal matrix (i.e., HR-DSM codewords) are generated simply by multiplying SC codewords by signal symbols drawn from an M-PSK constellation. The role of the SC matrices is to determine how the constellation symbols are weighted and which antenna combination is selected to transmit the HR-DSM codewords. It is noted that the role of SC matrices is similar that of the dispersion matrices (DM) for SM in [7, 8]. However, the designs of these two matrices are based on different approaches. The advantages of the proposed HR-DSM scheme can be elaborated as follows:
In the proposed HR-DSM scheme, information bits are carried by both SC codewords and constellation symbols. Therefore, a substantial increase in spectral efficiency is achieved as compared to other differential modulation schemes.
In order to improve the spectral efficiency, it is possible to increase either the signal space ΩHR−DSM or the M−PSK constellation. The scheme is thus more flexible than the DSM [16].
Compared with the DSM system, the proposed scheme is more advantageous under the space-constraint situations. For the DSM, in order to increase the spectral efficiency, more transmit antennas need to be used. The proposed HR-DSM scheme, however, can keep the number of transmit antenna within the limit and increase the M−PSK constellation size.
The proposed scheme is superior to the DSM system as it has two separate constellations. In the DSM system, increasing the spectral efficiency reduces distance between the signal points in the DSM constellation. Thanks to having two separate constellations, the proposed HR-DSM can keep the signal space ΩHR−DSM at a required level and increase the M−PSK signal constellation.
The contributions of this paper can be summarized as follows:
A systematic approach is presented for the design of SC codewords for an arbitrary number of transmit antennas, assuming n T ≥2.
A theoretical union bound on the bit error rate of the HR-DSM scheme is derived in the closed form. The derived bound can be used as a means to evaluate the BER performance of HR-DSM when the signal-to-noise power ratio (SNR) is sufficiently high.
Computer simulation results, supported by the theoretical upper bound, are provided to benchmark the BER performance of the propose HR-DSM scheme with those of related differential transmission schemes, such as the differential Alamouti [13], the single-antenna differential scheme (DPSK), the differential scheme of quasi-orthogonal space-time block code (DQOSTBC) [22], and the DSM.
The rest of paper is organized as follows. System model of the proposed HR-DSM is introduced in Section 2. The design of the SC codewords for the HR-DSM is presented in Section 3. In Section 4, a closed-form of pairwise error probability (PEP) and a theoretical upper bound are derived. Section 5 presents simulation results and analysis. Finally, conclusions are drawn in Section 6.
Notation: Throughout the paper, we use the following mathematical notations. (·)H, (·)T, and ∥·∥ F denote the Hermitian transpose, transpose, and Frobenius norm of a vector or matrix, respectively. Re(·) denotes the real part of a complex number.
Figure 1 shows the transmitter and receiver configuration of the proposed HR-DSM with n T transmit antennas and n R receive antennas working over a Rayleigh flat-fading channel. Assume that at time instant t, symbol s t is transmitted via the n t -th transmit antenna, n t =1,2,⋯,n T . The transmitted signal vector is given by \({\mathbf {s}} = {\left [ {\begin {array}{*{20}{c}} {\begin {array}{*{20}{c}} 0& \cdots &0&{{s_{t}}} \end {array}}&0& \cdots &0 \end {array}} \right ]^{T}}\), where only the n t -th element is non-zero. In our proposed HR-DSM system, we collect the transmit signal vector s in time and form a n T ×T actual transmit signal matrix S. As proved in [23], for non-coherent MIMO systems, for arbitrary block length T, number of receive antennas n R and signal-to-noise ratio (SNR), the capacities obtained for n T >T and n T =T are equal. Therefore, we set T=n T in our proposed scheme. This means that each transmit signal matrix is sent during T=n T symbol durations. The actual transmit signal matrix S satisfies the following conditions:
Proposed HR-DSM system model
Only one antenna remains active at each time instant. This means that only one entry in any column of S is non-zero.
Each antenna is activated only once in the n T successive time instants of the transmit signal matrix. This means that only one entry in any row of S is non-zero.
An example of the transmit signal matrix for n T =3 is given by \({\mathbf {S}} = \left [\begin {array}{ccc} 0 & s_{12} & 0\\ s_{21} & 0 & 0\\ 0 & 0 & s_{33} \end {array} \right ]\), where \(s_{{n_{t}}t}\) denotes the transmit symbol over the n t -th antenna at time t. This matrix determines that at time instant 1, the symbol s21 is transmitted over transmit antenna 2 while transmit antennas 1 and 3 remain idle. Similarly, at time time instants 2 and 3, the symbols s12 and s33 are transmitted over antennas 1 and 3, respectively, while the other two transmit antennas remain idle. According to this design, each antenna is activated only once during each block, making differential operation possible so long as the wireless channel remains unchanged over two successive HR-DSM blocks.
The proposed HR-DSM transmitter works as follows: At time t+1, (l+m) data bits enter the HR-DSM transmitter, among which l bits are mapped into a n T ×T matrix Xt+1, out of K matrices in the signal space ΩHR−DSM with basic elements drawn from M1−PSK constellation, while remaining m=log2M bits are mapped into a M−PSK constellation symbol xt+1. The resulting n T ×T HR-DSM codeword C is generated by multiplying X by x, i.e., Ct+1=Xt+1·xt+1.
The actual transmitted signal matrix St+1 is computed via the following formula
$$\begin{array}{*{20}l} {{\mathbf{S}}_{t + 1}} = {{\mathbf{S}}_{t}} \cdot {{\mathbf{C}}_{t + 1}}. \end{array} $$
Note that relation Eq. (1) is the fundamental differential transmission relation. Without loss of generality, we choose \(\phantom {\dot {i}\!}\mathbf {C}_{0} = \mathbf {I}_{n_{T}}\), where \(\phantom {\dot {i}\!}\mathbf {I}_{n_{T}}\) is the n T ×n T identity matrix. Then, the chain of transmitted matrices is given by:
$$\begin{array}{*{20}l} \begin{array}{l} {{\mathbf{S}}_{0}} = {{\mathbf{C}}_{0}} \\ {{\mathbf{S}}_{1}} = {{\mathbf{S}}_{0}}{{\mathbf{C}}_{1}} = {{\mathbf{C}}_{0}}{{\mathbf{C}}_{1}} \\ \vdots \\ {{\mathbf{S}}_{t}} = {{\mathbf{S}}_{t - 1}}{{\mathbf{C}}_{t}} = {{\mathbf{C}}_{0}}{{\mathbf{C}}_{1}} \cdots {{\mathbf{C}}_{t}} \\ \vdots \\ \end{array}. \end{array} $$
Let H t be the n R ×n T fading matrix with the (i,j)-th entry h ij denoting the normalized complex fading gain from transmit antenna j to receive antenna i. At the receiver side, the chain of received matrices, Y0,…,Y t ,Yt+1…, is given by
$$\begin{array}{*{20}l} {{\mathbf{Y}}_{t}} = {\mathbf{H}}_{t}{{\mathbf{S}}_{t}} + {{\mathbf{N}}_{t}} \end{array} $$
$$\begin{array}{*{20}l} {{\mathbf{Y}}_{t + 1}} = {\mathbf{H}}_{t+1}{{\mathbf{S}}_{t + 1}} + {{\mathbf{N}}_{t + 1}}, \end{array} $$
where Y t is the n R ×T received signal matrix and N t is the n R ×T AWGN matrix. Using the differential transmission relation in Eqs. (1) and (2), we can rewrite Eq. (3) as
$$\begin{array}{*{20}l} {{\mathbf{Y}}_{t + 1}}& = {\mathbf{H}}_{t+1}{{\mathbf{S}}_{t + 1}} + {{\mathbf{N}}_{t + 1}}\\ &= {\mathbf{H}}_{t+1}{{\mathbf{S}}_{t}}{{\mathbf{C}}_{t + 1}} + {{\mathbf{N}}_{t + 1}}\\ &= {{\mathbf{Y}}_{t}}{{\mathbf{C}}_{t + 1}} - {{\mathbf{N}}_{t}}{{\mathbf{C}}_{t + 1}} + {{\mathbf{N}}_{t + 1}}\\ &= {{\mathbf{Y}}_{t}}{{\mathbf{C}}_{t + 1}} + {{\mathbf{N}}_{t}}^{\prime}\,\,.\, \end{array} $$
where N t ′ is the n R ×T AWGN matrix. Therefore, to estimate the information matrix, the optimal ML detector can be derived as
$$\begin{array}{*{20}l} {{\hat{\mathbf{C}}}_{t + 1}} &= \underset{\forall {\mathbf{C}}}{\arg \min } \left\| {{{\mathbf{Y}}_{t + 1}} - {{\mathbf{Y}}_{t}}{\mathbf{C}}} \right\|_{F}^{2} \\ &= \underset{\forall {\mathbf{C}}}{\arg \min } \ \text{trace}\left\{ {{{\left({{{\mathbf{Y}}_{t + 1}} - {{\mathbf{Y}}_{t}}{\mathbf{C}}} \right)}^{H}}\left({{{\mathbf{Y}}_{t + 1}} - {{\mathbf{Y}}_{t}}{\mathbf{C}}} \right)} \right\} \\ &= \underset {\forall {\mathbf{C}}}{\arg \max } \ \text{trace}\left\{ {{\text{Re}} \left({{{{\mathbf{Y}}}_{t + 1}^{H}}{{\mathbf{Y}}_{t}}{\mathbf{C}}} \right)} \right\}, \end{array} $$
This is equivalent to
$$\begin{array}{*{20}l} \{{\hat{\mathbf{X}}_{t + 1}}, {\hat x_{t + 1}}\} = \underset{\forall {\mathbf{X}}, {x}}{\arg \max } \ \text{trace}\left\{ {{\text{Re}} \left({{{{\mathbf{Y}}}_{t + 1}^{H}}{{\mathbf{Y}}_{t}}{\mathbf{X}} {x}} \right)} \right\}. \end{array} $$
HR-DSM codeword design
As HR-DSM codewords C are obtained simply by multiplying matrices X in the signal space ΩHR−DSM by the constellation symbol x, our objective is to design suitable matrix X for a given n T .
For a given n T , with M1−PSK constellation, we define the following n T ×1 permutation vector \({{\mathbf {p}}_{p}} = \left [ {\begin {array}{*{20}{c}} {{p_{1}}}&{{p_{2}}}& \cdots &{{p_{{n_{T}}}}} \end {array}} \right ]\), (p=1, 2, ⋯,n T !). For the proposed HR-DSM scheme, we define the basic form of the generator matrix G
$$\begin{array}{*{20}l} {\mathbf{G}}({\mathbf{p}},\,{M_{1}})= \frac{1}{\Gamma }\left[ {\begin{array}{ccc} {{x_{11}}}& \cdots &{{x_{1{n_{T}}}}}\\ \vdots & \ddots & \vdots \\ {{x_{{n_{T}}1}}}& \cdots &{{x_{{n_{T}}{n_{T}}}}} \end{array}} \right] \end{array} $$
where column vector \({{\mathbf {x}}_{n}} = {\left [ {\begin {array}{*{20}{c}} {{x_{1n}}}&{{x_{2n}}}& \cdots &{{x_{{n_{T}}n}}} \end {array}} \right ]^{T}}\), (n=1, 2, ⋯, n T ), is a n T ×1 vector with complex valued entries; \(\Gamma = \left \| {{{\mathbf {x}}_{n}}} \right \| = \sqrt {{\sum \nolimits }_{{n_{t}} = 1}^{{n_{T}}} {{{\left | {{x_{{n_{t}}n}}} \right |}^{2}}}} \) is the magnitude of x, which is used to normalize the transmit power.
Then, a general procedure for designing the matrix X for n T transmit antennas is summarized as follows:
Assign the first permutation vector to \({{\mathbf {p}}_{1}} = \left [ {\begin {array}{cccc} 1&2& \cdots &{{n_{T}}} \end {array}} \right ]\).
Built (n T !−1) permutation vectors p p of p1:
\({{\mathbf {p}}_{2}} = \left [ {\begin {array}{*{20}{c}} 2&1& \cdots &{{n_{T}}} \end {array}} \right ], \cdots, {{\mathbf {p}}_{{n_{T}}!}} = \left [ {\begin {array}{cccc} {{n_{T}}}&{{n_{T - 1}}}& \cdots &1 \end {array}} \right ]\).
Arrange the G(p p ) matrices corresponding to permutation vectors p p as follows:
Fix the p1-th element of vector column x1 to 1 and 0s elsewhere.
Let the p n -th elements of remaining column vectors x n be selected from M1−PSK constellation symbols and 0s elsewhere.
Generate the corresponding matrix X: X q =G(p p ,M1), \(\left ({q = 1,\,2,\, \cdots,\,{n_{T}}!M_{1}^{{n_{T}} - 1}} \right)\).
For example, for n T =3, with BPSK constellation, we can construct a signal set ΩHR−DSM which includes \(q = {{n_{T}}!M_{1}^{{n_{T}} - 1}}=3!2^{2}=24\) matrices X as follows:
$$\begin{array}{ll} {{\mathbf{X}}_{1}} = \left[\begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&1 \end{array}\right] &{{\mathbf{X}}_{2}} = \left[\begin{array}{ccc} 1&0&0\\ 0&{ - 1}&0\\ 0&0&1 \end{array}\right]\\ {{\mathbf{X}}_{3}} = \left[\begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&{ - 1} \end{array}\right] &{{\mathbf{X}}_{4}} = \left[\begin{array}{llll} 1&0&0\\ 0&{ - 1}&0\\ 0&0&{ - 1} \end{array}\right]\\ {{\mathbf{X}}_{5}} = \left[\begin{array}{ccc} 1&0&0\\ 0&0&1\\ 0&1&0 \end{array}\right] &{\mathbf{X}_{6}} = \left[\begin{array}{ccc} 1&0&0\\ 0&0&1\\ 0&-1&0 \end{array}\right]\\ {{\mathbf{X}}_{7}} =\left[\begin{array}{ccc} 1&0&0\\ 0&0&-1\\ 0&1&0 \end{array}\right] &{{\mathbf{X}}_{8}} = \left[\begin{array}{ccc} 1&0&0\\ 0&0&-1\\ 0&-1&0 \end{array}\right]\\ {{\mathbf{X}}_{9}} = \left[\begin{array}{ccc} 0&1&0\\ 1&0&0\\ 0&0&1 \end{array}\right] &{\mathbf{X}_{10}} = \left[\begin{array}{ccc} 0&-1&0\\ 1&0&0\\ 0&0&1 \end{array}\right]\\ {{\mathbf{X}}_{11}} =\left[\begin{array}{ccc} 0&1&0\\ 1&0&0\\ 0&0&-1 \end{array}\right] &{{\mathbf{X}}_{12}} = \left[\begin{array}{ccc} 0&-1&0\\ 1&0&0\\ 0&0&-1 \end{array}\right]\\ {{\mathbf{X}}_{13}} = \left[\begin{array}{ccc} 0&0&1\\ 1&0&0\\ 0&1&0 \end{array}\right] &{\mathbf{X}_{14}} = \left[\begin{array}{ccc} 0&0&1\\ 1&0&0\\ 0&-1&0 \end{array}\right]\\ {{\mathbf{X}}_{15}} =\left[\begin{array}{ccc} 0&0&-1\\ 1&0&0\\ 0&1&0 \end{array}\right] & {{\mathbf{X}}_{16}} = \left[\begin{array}{ccc} 0&0&-1\\ 1&0&0\\ 0&-1&0 \end{array}\right]\\ {{\mathbf{X}}_{17}} = \left[\begin{array}{ccc} 0&1&0\\ 0&0&1\\ 1&0&0 \end{array}\right] & {\mathbf{X}_{18}} = \left[\begin{array}{ccc} 0&-1&0\\ 0&0&1\\ 1&0&0 \end{array}\right]\\ {{\mathbf{X}}_{19}} =\left[\begin{array}{ccc} 0&1&0\\ 0&0&-1\\ 1&0&0 \end{array}\right] & {{\mathbf{X}}_{20}} = \left[\begin{array}{ccc} 0&-1&0\\ 0&0&-1\\ 1&0&0 \end{array}\right]\\ {{\mathbf{X}}_{21}} = \left[\begin{array}{ccc} 0&0&1\\ 0&1&0\\ 1&0&0 \end{array}\right] & {\mathbf{X}}_{22} = \left[\begin{array}{ccc} 0&0&1\\ 0&-1&0\\ 1&0&0 \end{array}\right]\\ {{\mathbf{X}}_{23}} =\left[\begin{array}{ccc} 0&0&-1\\ 0&1&0\\ 1&0&0 \end{array}\right] & {\mathbf{X}}_{24} = \left[\begin{array}{ccc} 0&0&-1\\ 0&-1&0\\ 1&0&0 \end{array}\right] \end{array} $$
The motivation behind assigning the only non-zero element of vector column x1 to 1 is to guarantee that the proposed HR-DSM scheme obtains high performance. Thanks to this assignment, for the case X i ≠X j and x i ≠x j , we get C i ≠C j . Without this assignment, there possibly exist X i ≠X j and x i ≠x j such that C i =C j , leading to a wrong detection at the receiver.
In order for the proposed HR-DSM system to achieve high performance, the transmitted signal space Ω C needs to be designed such that the minimum Euclidean distance between two arbitrary transmission signal matrices C i , C j ∈Ω C is maximized [24]. Define the difference between two matrices C i and C j as d i,j =C i −C j , then the Euclidean distance between two transmission signal matrices C i and C j can be expressed as
$$\begin{array}{*{20}l} \delta \left({{{\mathbf{C}}_{i}},{{\mathbf{C}}_{j}}} \right) = \det \left({{\mathbf{d}}_{i,j}^{H}{{\mathbf{d}}_{i,j}}} \right). \end{array} $$
Then, for a given spectral efficiency, M and M1 are selected such that the following minimum Euclidean distance
$$\begin{array}{*{20}l} {\delta_{\min }} = \underset{i \ne j}{\min } \delta ({{\mathbf{C}_{i}},{\mathbf{C}_{j}}}). \end{array} $$
is maximized for all pairs of distinct codewords C i ≠C j and for all combinations of (M,M1).
According to the design procedure, for a given n T and with M1−PSK constellation, a total of \(Q = {n_{T}}!M_{1}^{{n_{T}} - 1}\) matrices X in the signal space ΩHR−DSM can be obtained. Therefore, one matrix X is able to carry \(l{\mathrm {= }}\left \lfloor {{\text {lo}}{{\mathrm {g}}_{2}}\left ({{n_{T}}!M_{1}^{{n_{T}} - 1}} \right)} \right \rfloor \) information bits. In addition, one M−PSK constellation symbol corresponds to m=log2M information bits. Both of them are transmitted within n T symbol periods. Consequently, the spectral efficiency of the proposed HR-DSM scheme is equal to \(\frac {1}{{{n_{T}}}}\left ({l + m} \right) = \frac {1}{{{n_{T}}}}\left [ {\left \lfloor {{\text {lo}}{{\mathrm {g}}_{2}}\left ({{n_{T}}!M_{1}^{{n_{T}} - 1}} \right)} \right \rfloor + {{\log }_{2}}M} \right ]\) bpcu. Clearly, the additional spectral efficiency offered by our proposed scheme is substantially higher than that of DSM.
Theoretical upper bound of HR-DSM
Suppose that the message C t is sent at each block. Since errors occur during transmission of the actual transmitted signal matrix S t due to channel fading and noise, after differential decoding, assume that the message E t in each block is detected. It follows that \({{\mathbf {E}}_{t}}{\mathbf {E}}_{t}^{H} = {{\mathbf {I}}_{{n_{T}}}}\), where I n is the n T ×n T identity matrix. In order to measure the difference between C t and D t , we define \({{\mathbf {D}}_{t}} = {{\mathbf {E}}_{t}}{\mathbf {C}}_{t}^{H}\). So the matrix distance between C t and E t can be expressed as \(\text {trace}\left \{ {{\text {Re}} \left ({{{\mathbf {I}}_{{n_{T}}}} - {{\mathbf {D}}_{t}}} \right)} \right \}\). When no error occurs, \(\phantom {\dot {i}\!}{{\mathbf {D}}_{t}} = {{\mathbf {I}}_{{n_{T}}}}\), so \(\text {trace}\left \{ {{\text {Re}} \left ({{{\mathbf {I}}_{{n_{T}}}} - {{\mathbf {D}}_{t}}} \right)} \right \} = 0\). Since \({{\mathbf {D}}_{t}}{\mathbf {D}}_{t}^{H} = {{\mathbf {I}}_{{n_{T}}}}\), it follows that matrix D t has the same orthogonal property as the message matrix C t and the actual transmitted signal matrix S t .
Recall that the HR-DSM transmitter transmits the matrices S t and St−1 instead of directly transmitting the message matrices C t . Due to the influence of fading and noise, suppose that while S t and St−1 are transmitted, Q t and Qt−1 are actually received which causes that the differentially decoded message matrices C t to become the error message matrices E t . Obviously, Q t =E t Qt−1=E t Ct−1 and \({{\mathbf {Q}}_{t}}{\mathbf {Q}}_{t - 1}^{H} = {{\mathbf {E}}_{t}}\).
Let η c and η e be the decision variables for transmission matrices C and E, respectively. Besides, let P(C→E|H) be the pairwise error probability of deciding E when C is transmitted for a given channel realization H. Then, P(C→E|H) can be expressed as
$$\begin{array}{*{20}l} P\left({{\mathbf{C}} \to {\mathbf{E}}|{\mathbf{H}}} \right) &= P\left({{\eta_{e}} - {\eta_{c}} > 0|{\mathbf{H}}} \right) \\ &= P\left({\text{trace}\left\{ {{\text{Re}} \left({{\Phi_{t - 1,t}}} \right)} \right\} > 0|{\mathbf{H}}} \right), \end{array} $$
$$\begin{array}{*{20}l} {\eta_{c}} &= \text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{Y}}_{t}}{\mathbf{Y}}_{t - 1}^{H}{{\mathbf{S}}_{t - 1}}{\mathbf{S}}_{t}^{H}} \right)} \right\},\\ {\eta_{e}} &= \text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{Y}}_{t}}{\mathbf{Y}}_{t - 1}^{H}{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H}} \right)} \right\} \end{array} $$
$$\begin{array}{*{20}l} {\Phi_{t - 1,t}} = {\eta_{c}} - {\eta_{e}}. \end{array} $$
Substituting Eq. (2) into Eqs. (10) and (11), we have
$$\begin{array}{*{20}l} &{}\text{trace}\left\{ {{\text{Re}} \left({{\Phi_{t - 1,t}}} \right)} \right\} \simeq \text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{H}}_{t - 1}}{\mathbf{H}}_{t}^{H}\left({{{\mathbf{D}}_{t}} - {{\mathbf{I}}_{{n_{T}}}}} \right)} \right)} \right\}\\ &{}+ \text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{S}}_{t}}{{\mathbf{H}}_{t}}{\mathbf{N}}_{t - 1}^{H}{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H} + {{\mathbf{N}}_{t}}{\mathbf{H}}_{t - 1}^{H}{\mathbf{C}}_{t - 1}^{H}{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H}} \right.} \right.\\ &{}\left. {\left. { - {{\mathbf{S}}_{t}}{{\mathbf{H}}_{t}}{\mathbf{N}}_{t - 1}^{H}{{\mathbf{S}}_{t - 1}}{\mathbf{S}}_{t}^{H} - {{\mathbf{N}}_{t}}{\mathbf{H}}_{t - 1}^{H}{\mathbf{S}}_{t - 1}^{H}{{\mathbf{S}}_{t - 1}}{\mathbf{S}}_{t}^{H}} \right)} \right\}. \end{array} $$
Note that the second-order noise terms in Eq. (12) are ignored since they are quite small compared to other noise terms when SNR is large enough. Let
$$\begin{array}{*{20}l} \Delta & = \text{trace}\left\{ {{\text{Re}} \left({{\mathbf{\Phi}_{t - 1,t}}} \right)} \right\}\\ &= - \left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right)\rho + \text{trace}\left\{ {{\text{Re}} \left(\Theta \right)} \right\}, \end{array} $$
where ρ is defined as
$$\begin{array}{*{20}l} \rho = \text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{I}}_{{n_{T}}}} - {{\mathbf{D}}_{t}}} \right)} \right\}, \end{array} $$
$$\begin{array}{*{20}l} \Theta &= {{\mathbf{S}}_{t}}{{\mathbf{H}}_{t}}{\mathbf{N}}_{t - 1}^{H}{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H} + {{\mathbf{N}}_{t}}{{\mathbf{H}}_{t-1}^{H}}{\mathbf{C}}_{t - 1}^{H}{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H}\\ &\,\,\,\,\,\,\,-{{\mathbf{S}}_{t}}{{\mathbf{H}}_{t}}{\mathbf{N}}_{t - 1}^{H}{{\mathbf{S}}_{t - 1}}{\mathbf{S}}_{t}^{H} - {{\mathbf{N}}_{t}}{{\mathbf{H}}_{t-1}^{H}}{\mathbf{S}}_{t - 1}^{H}{{\mathbf{S}}_{t - 1}}{\mathbf{S}}_{t}^{H}. \end{array} $$
For given transmission matrices C and E, Qt−1,Q t ,St−1 and S t , and ρ can be considered as deterministic quantities. Therefore, we can easily show that E[trace{Re(Θ)}]=0. Taking the expectation of both sides of Eq. (13), we have:
$$\begin{array}{*{20}l} E\left[ \Delta \right] &= - \left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right)E\left[ \rho \right] + E\left[ \text{trace}\left\{ {{\text{Re}} \left(\Theta \right)} \right\}\right]\\ &=- \left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right)\rho. \end{array} $$
The computation of the variance of Δ is more complicated, because some terms in Eq. (15) are correlated, although most of the terms are assumed to be mutually independent. It is proved in the "Appendix" section that the variance of Δ is given by:
$$\begin{array}{*{20}l} &\text{Var}\left[ \Delta \right]\\ &= \text{Var}\left[ \mathrm{trace\left\{ {{\text{Re}} \left(\Theta \right)} \right\}} \right]\\ &= 8\left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right){N_{0}}\\ & - 4{N_{0}}\left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right)\text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{D}}_{t}}} \right)} \right\}, \end{array} $$
which can be simplified to
$$\begin{array}{*{20}l} \text{Var}\left[ \Delta \right] = 2\rho \left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right){N_{0}}. \end{array} $$
From Eqs. (9), (13), (16), and (18), it follows that
$$\begin{array}{*{20}l} &\Pr \left({{\mathbf{C}} \to {\mathbf{E}}|{\mathbf{H}}} \right)\\ &\,\,\,\,\,= \Pr \left({{\eta_{e}} - {\eta_{c}} > 0|{\mathbf{H}}} \right)\\ &\,\,\,\,\,= \Pr \left({\Delta > 0|{\mathbf{H}}} \right)\\ &\,\,\,\,\,= Q\left({\sqrt {\gamma \rho \left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right)/4}} \right), \end{array} $$
where Q denotes the Gaussian tail function, and γ=E s /N0 is the SNR per symbol. Defining the instantaneous SNR as
$$\begin{array}{*{20}l} {\gamma_{b}} = \gamma \left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right), \end{array} $$
and using the alternative form of the Gaussian Q-function [25], we can write
$$\begin{array}{*{20}l} \Pr \left({{\mathbf{C}} \to {\mathbf{E}}|{\mathbf{H}}} \right) &= Q\left({\sqrt {\left({\rho /2} \right){\gamma_{b}}}} \right)\\ &= \frac{1}{\pi }\int\limits_{0}^{\pi /2} {\exp \left({ - \frac{{\left({\rho /2} \right){\gamma_{b}}}}{{2{{\sin }^{2}}\theta }}} \right)} d\theta. \end{array} $$
Averaging Eq. (21) over all realizations of the channel matrix H, we obtain the PEP as
$$\begin{array}{*{20}l} \Pr \left({{\mathbf{C}} \to {\mathbf{E}}} \right) = \int\limits_{0}^{\infty} {Q\left({\sqrt {\left({\rho /2} \right){\gamma_{b}}}} \right)} p\left({{\gamma_{b}}} \right)d{\gamma_{b}}. \end{array} $$
where p(γ b ) is the PDF of γ b given in [26].
Let u represent a sequence with q information bits and \({\hat {\mathbf {u}}}\) denotes an error sequence with the same number of information bits. The bit error probability P b of the proposed HR-DSM scheme is union-bounded by [27, 28]:
$$\begin{array}{*{20}l} {P_{b}} \le \frac{1}{{2q}}\sum\limits_{{\mathbf{C}} \ne {\mathbf{E}}} {\Pr \left({{\mathbf{C}} \to {\mathbf{E}}} \right) \cdot w\left({{\mathbf{u}},{\hat{\mathbf{u}}}} \right)}, \end{array} $$
where \(w\left ({{\mathbf {u}},{\hat {\mathbf {u}}}} \right)\) is the Hamming distance between sequences u and \({\hat {\mathbf {u}}}\). The PEP Pr(C→E) is given by Eq. (22).
In this section, Monte Carlo simulations and the theoretical upper bound are used to study the BER performance of the proposed HR-DSM scheme for different antenna arrangements, as well as to compare them against different MIMO systems, such as coherent detection SM [1] and DSM [15], the differential scheme of the well known Alamouti scheme [13], the DPSK scheme, DQOSTBC scheme [22]. Simulations are carried out over the quasi-static Rayleigh fading channel. We assume that the channel state information is perfectly known at the receiver of an SM system. In addition, ML detection is applied to all systems under consideration.
Comparison between theoretical and simulation results
In Fig. 2, the theoretical and simulation BER performances versus the signal-to-noise power ratio (SNR) are plotted. The theoretical and simulation results match well in the high SNR regions for all cases. This implies that the bound given by Eq. (23) can be used as a tool to evaluate BER performances of not only the proposed HR-DSM scheme but also the DSM scheme when SNR is sufficiently high.
Theoretical and simulation results of BER performances of HR-DSM
Comparison between the differential detection HR-DSM and the coherent detection SM
In order to support our analysis in Section 4, Fig. 3 presents an example to compare the differential detection scheme of HR-DSM with the coherent-detection SM scheme. The simulations are realized with n R =4 and n T =2, 4. The BER performances of the differential and the coherent schemes are compared under the same spectral efficiency of 3 bpcu. Note that when SM and HR-DSM use the same signal constellation, their spectral efficiencies are not equal.
BER performance comparison of the differential detection HR-DSM and the coherent detection SM
It is observed from the figure that compared to the SM, HR-DSM suffers from a performance degradation of less than 3 dB in the SNR, particularly in the high SNR region.
BER performance comparison of proposed HR-DSM and other transmission schemes.
In Fig. 4, we compare the BER performance of HR-DSM and DSM with the same spectral efficiency of 2.5 bpcu. To obtain spectral efficiency of 2.5 bpcu, the HR-DSM utilizes BPSK and 8-PSK constellations, while DSM adopts the QPSK constellation. The simulations are carried out with n T =2 and n R =1, 2, and 4. It can be seen from the figure that the proposed HR-DSM outperforms DSM.
BER performance comparison of proposed HR-DSM and DSM at 2.5 bpcu (n T =2)
Figure 5 shows the BER curves of HR-DSM and other transmission schemes with the spectral efficiency of 3 bpcu. The first one is the differential scheme of the well-known Alamouti scheme (DA) with n T =2 [13]. The second is the single-antenna differential scheme, DPSK. The third is the differential scheme of quasi-orthogonal space-time block code [22] (DQOSTBC) with n T =4. And the last is the DSM. Simulation results indicate that, when n R =1, 2, the proposed HR-DSM exhibits the worse performance than DA and DQOSTBC. However, when n R =4, HR-DSM outperforms all the remaining schemes. This is because when n R =1, 2, the erroneous antenna detection of HR-DSM is so significant that it redeems its coding gain. Increase in n R reduces the erroneous antenna detection of HR-DSM considerably. Therefore, when n R =4, the coding gain of HR-DSM due to the use of lower order modulation dominates the result and thus HR-DSM outperforms the others.
BER performance comparison of proposed HR-DSM and other transmission schemes at 3 bpcu
In addition, it can be observed that in all antenna configurations, DA and DQOSTBC benefit more from increasing the SNR than does DSM and HR-DSM. This is because the diversity order of HR-DSM, as well as DSM, approaches that of single antenna system which is n R as detailed in [2]. In contrast, the Alamouti scheme with n T =2 provides full diversity, i.e., 2n R and QOSTBC with n T =4 provides half of the maximum achievable diversity. Thus, at high SNR, DA and DQOSTBC outperform HR-DSM and DSM.
HR-DSM with multiple transmit antennas
Figure 6 compares the BER performance of HR-DSM with multiple transmit antennas with the same spectral efficiency of 3 bpcu. To obtain spectral efficiency of 3 bpcu, all antenna configurations utilize QPSK and 8-PSK constellations, i.e., (M1,M)=(4,8). It can be seen from the figure that although all antenna configurations utilize the same modulation order, the HR-DSM system with n T =4 provides the best BER performance, the HR-DSM system with n T =3 has better BER performance compared with the HR-DSM system having n T =2. In other words, the more transmit antennas are used, the less the performance loss of HR-DSM is as compared to SM. This is because in the configuration with n T =3, 4, the best set of the signal space ΩHR−DSM is selected in order to obtain improved performance.
Simulation results of the HR-DSM for 2, 3, and 4 transmit antennas
Another observation is that for the same number of receive antennas, HR-DSM configurations with n R =1 will provide almost same performance. For n R =2, 4, HR-DSM with n T =4 outperforms HR-DSM with n T =2, 3. The likelihood of erroneous antenna detection decreases with an increasing number of receive antennas. This observation can be used as a guideline for selecting the signal constellation and the spatial configuration. To get the full potential of HR-DSM, the system should be equipped with sufficient receive antennas.
In this paper, a new differential space time modulation scheme based on the HR-SM codeword is proposed for MIMO systems. In addition, a theoretical upper bound is derived for the evaluation of the BER of the proposed HR-DSM scheme. Simulation results show that this scheme outperforms the DSM scheme in terms of BER performance. The performance of the proposed scheme can be improved by selecting the set of transmission matrices having the largest minimum Euclidean distance. Undoubtedly, the proposed method paves a new way to design differential SM schemes with more transmit antennas.
Evaluation of the variance of trace{Re(Θ)}
First, we can write
$$\begin{array}{*{20}l} \text{trace}\left\{ {{\text{Re}} \left(\Theta \right)} \right\} = \text{trace}\left\{ {{\text{Re}} \left({{\Theta_{1}}} \right)} \right\} - \text{trace}\left\{ {{\text{Re}} \left({{\Theta_{2}}} \right)} \right\}, \end{array} $$
$$\begin{array}{*{20}l} {\Theta_{1}} &= {{\mathbf{S}}_{t}}{{\mathbf{H}}_{t}}{\mathbf{N}}_{t - 1}^{H}{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H} + {{\mathbf{N}}_{t}}{{\mathbf{H}}_{t-1}^{H}}{\mathbf{C}}_{t - 1}^{H}{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H},\\ {\Theta_{2}} &= {{\mathbf{S}}_{t}}{{\mathbf{H}}_{t}}{\mathbf{N}}_{t - 1}^{H}{{\mathbf{S}}_{t - 1}}{\mathbf{S}}_{t}^{H} + {{\mathbf{N}}_{t}}{{\mathbf{H}}_{t-1}^{H}}{\mathbf{S}}_{t - 1}^{H}{{\mathbf{S}}_{t - 1}}{\mathbf{S}}_{t}^{H}, \end{array} $$
Now, the variance of trace{Re(Θ)} is given by:
$$\begin{array}{*{20}l} &\text{Var}\left[ {\text{trace}\left\{ {{\text{Re}} \left(\Theta \right)} \right\}} \right]\\ &\,\,\,\,\,= \text{Var}\left[ {\text{trace}\left\{ {{\text{Re}} \left({{\Theta_{1}}} \right)} \right\}} \right] + \text{Var}\left[ {\text{trace}\left\{ {{\text{Re}} \left({{\Theta_{2}}} \right)} \right\}} \right]\\ &\,\,\,\,\,- 2\text{Cov}\left({\text{trace}\left\{ {{\text{Re}} \left({{\Theta_{1}}} \right)} \right\},\text{trace}\left\{ {{\text{Re}} \left({{\Theta_{2}}} \right)} \right\}} \right). \end{array} $$
From Eq. (26), we can see that the variance of trace{Re(Θ)} consists of three parts: the variance of trace{Re(Θ1)}, the variance of trace{Re(Θ2)}, and the cross-correlation between trace{Re(Θ1)} and trace{Re(Θ2)}. These quantities will be evaluated in the following sub-sections.
Evaluation of Var[trace{Re(Θ1)}], Var[trace{Re(Θ2)}]
trace{Re(Θ1)} can be evaluated as follows:
$$\begin{array}{*{20}l} &\text{Var}\left[ {\text{trace}\left\{ {{\text{Re}} \left({{\Theta_{1}}} \right)} \right\}} \right]\\ &\,\,\,\,\,= \text{Var}\left[ {\text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H}{{\mathbf{S}}_{t}}{{\mathbf{H}}_{t}}{\mathbf{N}}_{t - 1}^{H}} \right)} \right\}} \right]\\ &\,\,\,\,\,\,\,\,\,\,\,\,+ \text{Var}\left[ {\text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H}{{\mathbf{S}}_{t - 1}}{{\mathbf{H}}_{t - 1}}{\mathbf{N}}_{t}^{H}} \right)} \right\}} \right]\\ &\,\,\,\,\,= 4\left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right){N_{0}}. \end{array} $$
It could be observed from Eq. (25) that trace{Re(Θ1)} and trace{Re(Θ2)} have the same structure. Therefore, we can similarly get:
$$\begin{array}{*{20}l} &\text{Var}\left[ {\text{trace}\left\{ {{\text{Re}} \left({{\Theta_{2}}} \right)} \right\}} \right] = 4\left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right){N_{0}}. \end{array} $$
Finally, we can write
$$\begin{array}{*{20}l} &\text{Var}\left[ {\text{trace}\left\{ {{\text{Re}} \left({{\Theta_{1}}} \right)} \right\}} \right] + \text{Var}\left[ {\text{trace}\left\{ {{\text{Re}} \left({{\Theta_{2}}} \right)} \right\}} \right]\\ &\,\,\,\,\,\,\,\,\,\,= 8\left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right){N_{0}}. \end{array} $$
Evaluation of Cov(trace{Re(Θ1)}, trace{Re(Θ2)})
The cross-correlation Cov(trace{Re(Θ1)}, trace{Re(Θ2)}) can be expressed as
$$\begin{array}{*{20}l} \Psi & = \text{Cov}\left({\text{trace}\left\{ {{\text{Re}} \left({{\Theta_{1}}} \right)} \right\},\text{trace}\left\{ {{\text{Re}} \left({{\Theta_{2}}} \right)} \right\}} \right)\\ &= \text{Cov}\left({{\delta_{1}},{\delta_{2}}} \right)\\ &= 2{N_{0}}\left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right)\times \text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{D}}_{t}}} \right)} \right\}, \end{array} $$
$$\begin{array}{*{20}l} {\delta_{1}} = \text{trace}&\left\{ {{\text{Re}} \left({{{\mathbf{S}}_{t}}{{\mathbf{H}}_{t}}{\mathbf{N}}_{t - 1}^{H}{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H}} \right.} \right.\\ &+ \left. {\left. {{{\mathbf{N}}_{t}}{\mathbf{H}}_{t - 1}^{H}{\mathbf{S}}_{t - 1}^{H}{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H}} \right)} \right\}, \end{array} $$
$$\begin{array}{*{20}l} {}{\delta_{2}}& = \text{trace}\left.\{ {{\text{Re}} \left.({{{\mathbf{S}}_{t}}{{\mathbf{H}}_{t}}{\mathbf{N}}_{t - 1}^{H}{{\mathbf{S}}_{t - 1}}{\mathbf{S}}_{t}^{H} + {{\mathbf{N}}_{t}}{{\mathbf{H}}_{t-1}^{H}}{\mathbf{S}}_{t - 1}^{H}{{\mathbf{S}}_{t - 1}}{\mathbf{S}}_{t}^{H}} \right)} \right\}\\ {}& = \text{trace}\left.\{ {{\text{Re}} \left.({{\mathbf{H}}_{t}} {{\mathbf{N}}_{t - 1}^{H}{{\mathbf{S}}_{t - 1}} + {{\mathbf{N}}_{t}}{{\mathbf{H}}_{t-1}^{H}}{\mathbf{S}}_{t}^{H}} \right)} \right\}. \end{array} $$
Since S t ,H t and Q t are matrices with deterministic entries, the cross-correlation Ψ is related to the 2 noise matrices N t ,Nt−1 only. Moreover, these noise matrices are mutually independent of each other, and any cross-correlation between two terms with different noise matrices is zero. Hence Ψ only consists of the cross-correlation between the terms with the same noise matrix. For N t only, there is one term with noise matrix N t , so
$$\begin{array}{*{20}l} {\delta_{1}} &= \text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{N}}_{t}}{{\mathbf{H}}_{t-1}^{H}}{\mathbf{S}}_{t - 1}^{H}{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H}} \right)} \right\},\\ {\delta_{2}} &=\text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{N}}_{t}}{\mathbf{H}}_{t}^{H}{\mathbf{S}}_{t}^{H}} \right)} \right\}, \end{array} $$
hence, the cross-correlation for N t is
$$\begin{array}{*{20}l} {\Psi_{0}} &= E\left[ {{\delta_{1}}{\delta_{2}}} \right]\\ &= {N_{0}}\left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right)\\ &\,\,\,\, \times \,\,\text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{S}}_{t}}{\mathbf{S}}_{t - 1}^{H}{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H}} \right)} \right\}\\ &= {N_{0}}\left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right)\text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{D}}_{t}}} \right)} \right\}. \end{array} $$
For Nt−1
$$\begin{array}{*{20}l} {\delta_{1}} &= \text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{S}}_{t}}{{\mathbf{H}}_{t}}{\mathbf{N}}_{t - 1}^{H}{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H}} \right)} \right\}\\ &= \text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{N}}_{t}}{{\mathbf{H}}_{t}^{H}}{\mathbf{S}}_{t}^{H}{{\mathbf{Q}}_{t - 1}}{\mathbf{Q}}_{t}^{H}} \right)} \right\},\\ {\delta_{2}} &= \text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{H}}_{t}}{\mathbf{N}}_{t - 1}^{H}{{\mathbf{S}}_{t - 1}}} \right)} \right\}\\ &= \text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{N}}_{t - 1}}{\mathbf{H}}_{t}^{H}{\mathbf{S}}_{t - 1}^{H}} \right)} \right\}. \end{array} $$
The cross-correlation for Nt−1 is
$$\begin{array}{*{20}l} {\Psi_{1}} &= E\left[ {{\delta_{1}}{\delta_{2}}} \right]\\ &= {N_{0}}\left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right)\times\, \text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{S}}_{t}}{\mathbf{S}}_{t}^{H}{{\mathbf{Q}}_{t}}{\mathbf{Q}}_{t}^{H}} \right)} \right\}\\ &= {N_{0}}\left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right)\text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{D}}_{t}}} \right)} \right\}. \end{array} $$
$$\begin{array}{*{20}l} \Psi &= {\Psi_{0}} + {\Psi_{1}}\\ &= 2{N_{0}}\left({\sum\limits_{\scriptstyle i = 1,\ldots,{n_{R}}\,\hfill\atop \scriptstyle j = 1,\ldots,{n_{T}}\hfill} {{{\left| {{h_{ij}}} \right|}^{2}}}} \right)\text{trace}\left\{ {{\text{Re}} \left({{{\mathbf{D}}_{t}}} \right)} \right\}. \end{array} $$
Finally, substituting Eqs. (29) and (30) into Eq. (26), we obtain Eq. (17). □
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This work was supported by the Vietnam National Foundation for Science and Technology Development under Grant 102.02-2015.23.
Le Quy Don Technical University, Hoang Quoc Viet Street, Hanoi, Vietnam
Thu Phuong Nguyen & Xuan Nam Tran
Hanoi Department of Science and Technology, Hanoi, Vietnam
Minh-Tuan Le
Middlesex University, London, UK
Huan X. Nguyen
Thu Phuong Nguyen
Xuan Nam Tran
The authors have contributed jointly to all parts on the preparation of this manuscript, and all authors read and approved the final manuscript.
Correspondence to Thu Phuong Nguyen.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Nguyen, T.P., Tran, X.N., Le, MT. et al. Differential spatial modulation for high-rate transmission systems. J Wireless Com Network 2018, 6 (2018). https://doi.org/10.1186/s13638-017-1013-1
Differential spatial modulation
High-rate differential spatial modulation | CommonCrawl |
\begin{definition}[Definition:Electrostatics]
'''Electrostatics''' is the branch of physics concerned with the study of stationary electrically charged bodies, and the forces and fields they give rise to.
\end{definition} | ProofWiki |
Results for 'R. T. Rathod'
Critical Examination on the Problem of Our Knowledge of Other Minds.R. T. Rathod - forthcoming - Indian Philosophical Quarterly.details
The philosophical problem of knowledge of other minds is rational justifiction. this paper covers n malcolm, h h price, j mill, strawson, hamshire, l wittgenstein and a j ayer's controversial thought. philosophical scepticism holds that it is logically impossible to know mental experiences. "i know, i have a pain." how do i know that other people also can have similar pain? it provides as ideal knowledge of mental events. when i say, "i have a pain," i can have certain knowledge (...) of this because i have conclusive individual evidence for it. (shrink)
The Problem of Other Minds in Philosophy of Mind
The T-Schema is Not a Logical Truth.R. T. Cook - 2012 - Analysis 72 (2):231-239.details
It is shown that the logical truth of instances of the T-schema is incompatible with the formal nature of logical truth. In particular, since the formality of logical truth entails that the set of logical truths is closed under substitution, the logical truth of T-schema instances entails that all sentences are logical truths.
Deflationism about Truth in Philosophy of Language
The End of the Timeless God.R. T. Mullins - 2016 - Oxford University Press UK.details
The End of the Timeless God considers two approaches to the philosophy of time, presentism and eternalism. It is often held that God cannot be timeless if presentism is true, but can be if eternalism is true. R. T. Mullins draws on recent work in the philosophy of time as well as the work of classical Christian thinkers such as Augustine, Anselm, and Aquinas to contend that the Christian God cannot be timeless in either case.
Divine Eternity in Philosophy of Religion
Divine Immutability in Philosophy of Religion
Divine Simplicity in Philosophy of Religion
Incarnation in Philosophy of Religion
Philosophy of Time, Misc in Metaphysics
Theories of Omniscience in Philosophy of Religion
What is Wrong with Cantor's Diagonal Argument?R. T. Brady & P. A. Rush - 2008 - Logique Et Analyse 51 (1):185-219..details
We first consider the entailment logic MC, based on meaning containment, which contains neither the Law of Excluded Middle (LEM) nor the Disjunctive Syllogism (DS). We then argue that the DS may be assumed at least on a similar basis as the assumption of the LEM, which is then justified over a finite domain or for a recursive property over an infinite domain. In the latter case, use is made of Mathematical Induction. We then show that an instance of the (...) LEM is intrumental in the proof of Cantor's Theorem, and we then argue that this is based on a more general form than can be reasonably justified. We briefly consider the impact of our approach on arithmetic and naive set theory, and compare it with intuitionist mathematics and briefly with recursive mathematics. Our "Four Basic Logical Issues" paper would provide useful background, the current paper being an application of the some of the ideas in it. (shrink)
Logical Consequence and Entailment in Logic and Philosophy of Logic
Divine Temporality, the Trinity, and the Charge of Arianism.R. T. Mullins - 2016 - Journal of Analytic Theology 4:267-290.details
Divine temporality is all the rage in certain theological circles today. Some even suggesting that the doctrine of the Trinity entails divine temporality. While I find this claim a bit strong, I do think that divine temporality can be quite useful for developing a robust model of the Trinity. However, not everyone agrees with this. Paul Helm has offered an objection to the so-called Oxford school of divine temporality based on the Christian doctrine of the Trinity. He has argued that (...) this form of divine temporality entails Arianism. In other words, divine temporality suffers from an inadequate doctrine of the Trinity. In this paper I shall first articulate the so-called Oxford school of divine temporality. From there I shall develop some of the Oxford school's theological benefits that help flesh out the doctrine of the Trinity, and assuage the charge of Arianism. Then I shall offer an examination and refutation of the Arian charge to divine temporality in order to show that the divine temporalist can maintain a robust Trinitarian theology. (shrink)
The Trinity in Philosophy of Religion
Hasker on the Divine Processions of the Trinitarian Persons.R. T. Mullins - 2017 - European Journal for Philosophy of Religion 9 (4):181-216.details
Within contemporary evangelical theology, a peculiar controversy has been brewing over the past few decades with regard to the doctrine of the Trinity. A good number of prominent evangelical theologians and philosophers are rejecting the doctrine of divine processions within the eternal life of the Trinity. In William Hasker's recent Metaphysics and the Tri-Personal God, Hasker laments this rejection and seeks to offer a defense of this doctrine. This paper shall seek to accomplish a few things. In section I, I (...) shall first set the stage for a proper understanding of the discussion. Section II will articulate the basic Trinitarian desiderata that must be satisfied by any model of the doctrine of the Trinity. This will help one understand the debate between Hasker and the procession deniers. Section III will offer an articulation of what the doctrine of divine processions teaches. Section IV will examine Hasker's defense of the doctrine point by point. I shall argue that his defense of the doctrine of the divine processions fails. (shrink)
The Nature and Limits of Authority.R. T. DeGEORGE - 1985details
Divine Perfection and Creation.R. T. Mullins - 2016 - Heythrop Journal 57 (1):122-134.details
Proclus (c.412-485) once offered an argument that Christians took to stand against the Christian doctrine of creation ex nihilo based on the eternity of the world and God's perfection. John Philoponus (c.490-570) objected to this on various grounds. Part of this discussion can shed light on contemporary issues in philosophical theology on divine perfection and creation. First I will examine Proclus' dilemma and John Philoponus' response. I will argue that Philoponus' fails to rebut Proclus' dilemma. The problem is that presentism (...) is incompatible with divine simplicity, timelessness, and a strong doctrine of immutability. From there I will look at how this discussion bears on contemporary understandings of divine perfection and creation, and argue that there are at least two possible ways contemporary philosophical theologians can try to get around the dilemma. One option is to adopt four-dimensional eternalism and maintain the traditional account of the divine perfections. I argue that this option suffers from difficulties that are not compatible with Christian belief. The other option is to keep presentism and modify the divine perfections. I argue that this option is possible and preferable since our understanding of the divine perfections must be modified in light of divine revelation and the incarnation. (shrink)
Presentism in Metaphysics
God and Emotion.R. T. Mullins - 2020 - Cambridge University Press.details
An introductory exploration on the nature of emotions, and examination of some of the critical issues surrounding the emotional life of God as they relate to happiness, empathy, love, and moral judgments. Covering the different criteria used in the debate between impassibility and passibility, readers can begin to think about which emotions can be predicated of God and which cannot.
The Aloneness Argument Against Classical Theism.Joseph C. Schmid & R. T. Mullins - forthcoming - Religious Studies:1-19.details
We argue that there is a conflict among classical theism's commitments to divine simplicity, divine creative freedom, and omniscience. We start by defining key terms for the debate related to classical theism. Then we articulate a new argument, the Aloneness Argument, aiming to establish a conflict among these attributes. In broad outline, the argument proceeds as follows. Under classical theism, it's possible that God exists without anything apart from Him. Any knowledge God has in such a world would be wholly (...) intrinsic. But there are contingent truths in every world, including the world in which God exists alone. So, it's possible that God (given His omniscience) contingently has wholly intrinsic knowledge. But whatever is contingent and wholly intrinsic is an accident. So, God possibly has an accident. This is incompatible with classical theism. Finally, we consider and rebut several objections. (shrink)
Beyond Liberalism.R. T. Allen - 1999 - Tradition and Discovery 26 (1):16-18.details
This is a brief response to S. Jacob's review of Beyond Liberalism.
Liberalism in Social and Political Philosophy
Kit Fine, The Limits of Abstraction Oxford, Clarendon Press, 2002, Cloth 18.99/US $25.00 ISBN: 0-19-924618-1. [REVIEW]R. T. Cook - 2004 - British Journal for the Philosophy of Science 55 (4):791-800.details
Critical Notice of The Limits of abstraction by Kit Fine, Oxford: Clarendon Press, 2002, pp.216. ISBN 9780191567261.
Mathematical Neo-Fregeanism in Philosophy of Mathematics
Flint's 'Molinism and the Incarnation' is Still Too Radical — A Rejoinder to Flint.R. T. Mullins - 2017 - Journal of Analytic Theology 5:515-532.details
I greatly appreciate Thomas Flint's reply to my paper, "Flint's 'Molinism and the Incarnation' is too Radical." In my original paper I argue that the Christology and eschatology of Flint's paper "Molinism and the Incarnation" is too radical to be considered orthodox. I consider it an honor that a senior scholar, such as Flint, would concern himself with my work in the first place. In this response to Flint's reply I will explain why I still find Flint's Christology and eschatology (...) to be too radical. Below I shall attempt to address various issues raised by Flint in his reply. (shrink)
Heaven and Hell in Philosophy of Religion
The Cognitive Functions of Emotion.R. T. Allen - 2000 - Appraisal 3:38.details
Emotions in Philosophy of Mind
Theories of Emotion in Philosophy of Mind
Completeness Proofs for RM3 and BN4.R. T. Brady - 1982 - Logique Et Analyse 25:9-32.details
'Because I Say So!' Some Limitations Upon the Rationalisation of Authority.R. T. Allen - 1987 - Journal of Philosophy of Education 21 (1):15–24.details
Philosophy of Education in Philosophy of Social Science
The Reality of Responses to Fiction.R. T. Allen - 1986 - British Journal of Aesthetics 26 (1):64-68.details
Fiction, Misc in Aesthetics
The Mechanism of Cavitation in Magnesium During Creep.R. T. Ratcliffe & G. W. Greenwood - 1965 - Philosophical Magazine 12 (115):59-69.details
The Meaning of Life and Education.R. T. Allen - 1991 - Journal of Philosophy of Education 25 (1):47–58.details
The Meaning of Life in Value Theory, Miscellaneous
How 'Decent' Is a Decent Minimum of Health Care?R. T. Meulen - 2011 - Journal of Medicine and Philosophy 36 (6):612-623.details
This article tries to analyze the meaning of a decent minimum of health care, by confronting the idea of decent care with the concept of justice. Following the ideas of Margalith about a decent society, the article argues that a just minimum of care is not necessarily a decent minimum. The way this minimum is provided can still humiliate individuals, even if the end result is the best possible distribution of the goods as seen from the viewpoint of justice. This (...) analysis is combined with an analysis from the perspective of solidarity, particularly of reflective solidarity, as a way to develop decent care, which is care that does not humiliate individuals and maintains their dignity. (shrink)
Biomedical Ethics in Applied Ethics
Public Health in Applied Ethics
Genetics and the Origin of the Species.R. T. Eddison - 1954 - Philosophy of Science 21 (3):272-272.details
Genetics in Philosophy of Biology
In Pursuit of the Functions of theWnt Family of Developmental Regulators: Insights fromXenopus Laevis.R. T. Moon - 1993 - Bioessays 15 (2):91-97.details
Biological Sciences in Natural Sciences
Jeanine Diller and Asa Kasher, Eds., Models of God and Alternative Ultimate Realities.R. T. Mullins - 2014 - Journal of Analytic Theology 2:288-293.details
Rational Autonomy: The Destruction of Freedom.R. T. ALlen - 1982 - Journal of Philosophy of Education 16 (2):199–207.details
Rational Autonomy: The Destruction of Freedom.R. T. ALlen - 1982 - Philosophy of Education 16 (2):199-207.details
Surprise as a Factor in the von Restorff Effect.R. T. Green - 1956 - Journal of Experimental Psychology 52 (5):340.details
The Moral Dimension of Children's and Adolescents' Conceptualisation of Tolerance to Human Diversity.R. T. Witenberg - 2007 - Journal of Moral Education 36 (4):433-451.details
This study examined the kinds of justifications children and adolescents used to support tolerant and intolerant judgements about human diversity. For the tolerant responses, three main belief categories emerged, based on the beliefs that others should be treated fairly , empathetically and that reason/logic ought to govern judgements . Fairness emerged as the most used belief to support tolerant judgements and the most commonly used combination of beliefs was found to be fairness/empathy, linking tolerance to moral reasoning, rules and values. (...) Specifically noticeable was that 6-7-year-olds appealed to fairness more often in comparison to the 11-12 and 15-16-year-olds. Older students used a larger repertoire of beliefs to support tolerance, indicating developing cognitive maturity. There was also a tendency for females to appeal to fairness/empathy more often than males. The major constraint to positive tolerance was not prejudice toward the target groups but the adolescents' beliefs in freedom of speech as a democratic right, pointing to a conflict in values between tolerance and other human rights. (shrink)
Toleration in Applied Ethics in Social and Political Philosophy
Is Coming to Believe in God Reasonable or Unreasonable?R. T. Herbert - 1991 - Faith and Philosophy 8 (1):36-50.details
Arguments Against Theism in Philosophy of Religion
The Number of Gods in Philosophy of Religion
On Not Understanding Prayer.R. T. Allen - 1971 - Sophia 10 (3):1-7.details
Prayer in Philosophy of Religion
Learning Without Awareness of What is Being Learned or Intent to Learn It.E. L. Thorndike & R. T. Rock - 1934 - Journal of Experimental Psychology 17 (1):1.details
Conscious and Unconscious Learning in Philosophy of Cognitive Science
Unconscious and Conscious Processes in Philosophy of Cognitive Science
The Metaphysics of Natural Right in Spinoza.John R. T. Grey - forthcoming - Oxford Studies in Early Modern Philosophy 10.details
In the Tractatus Theologico-Politicus (TTP), Spinoza argues that an individual's natural right extends as far as their power. Subsequently, in the Tractatus Politicus (TP), he offers a revised argument for the same conclusion. Here I offer an account of the reasons for the revision. In both arguments, an individual's natural right derives from God's natural right. However, the TTP argument hinges on the claim that each individual is part of the whole of nature (totius naturae), and for this reason inherits (...) part of the natural right of that whole. Using several analogous cases from the Ethics, I show that this form of argument from division is not compatible with Spinoza's considered metaphysical views. The revised argument, by contrast, avoids the pitfalls of his earlier efforts. It also better reveals the deep roots by which the monistic metaphysics of the Ethics feeds into Spinoza's conception of natural right. (shrink)
Spinoza: Metaphysics, Misc in 17th/18th Century Philosophy
Spinoza: Natural Law in 17th/18th Century Philosophy
Spinoza: Rights in 17th/18th Century Philosophy
New Consecution Calculi for R→T.Katalin Bimbó & J. Michael Dunn - 2012 - Notre Dame Journal of Formal Logic 53 (4):491-509.details
The implicational fragment of the logic of relevant implication, $R_{\to}$ is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on $LR_{\to}$ , a Gentzen-style calculus. In this paper, we add the truth constant $\mathbf{t}$ to $LR_{\to}$ , but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve $\mathbf{t}$ (...) . This calculus, $LT_\to^{\text{\textcircled{$\mathbf{t}$}}}$ , extends the consecution calculus $LT_{\to}^{\mathbf{t}}$ formalizing the implicational fragment of ticket entailment . We introduce two other new calculi as alternative formulations of $R_{\to}^{\mathbf{t}}$ . For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of $R_{\to}^{\mathbf{t}}$ . These results serve as a basis for our positive solution to the long open problem of the decidability of $T_{\to}$ , which we present in another paper. (shrink)
Nonclassical Logics in Logic and Philosophy of Logic
Proof Theory in Logic and Philosophy of Logic
Scepticism and Neoplatonism.R. T. Wallis - 1987 - In Wolfgang Haase (ed.), Philosophie, Wissenschaften, Technik. Philosophie. De Gruyter. pp. 911-954.details
$35.00 used $428.92 new $559.00 from Amazon (collection) Amazon page
Particle Size Distribution.R. T. DeHoff & F. N. Rhines - 1968 - In Robert T. DeHoff & Frederick N. Rhines (eds.), Quantitative Microscopy. New York: Mcgraw-Hill.details
Quantum Mechanics in Philosophy of Physical Science
$29.99 used (collection) Amazon page
When Loyalty No Harm Meant.R. T. Allen - 1989 - Review of Metaphysics 43 (2):281 - 294.details
LOYALTY HAS NOT HAD A BAD PRESS, but, as far as Anglo-Saxon philosophy is concerned, very little press. It has merited entries in the Encyclopaedia of Religion and Ethics and the Encyclopedia of Philosophy, and a short one in Macquarrie's A Dictionary of Christian Ethics. Of course, there is also Josiah Royce's The Philosophy of Loyalty. I propose to argue that these discussions of loyalty tend to assimilate it to faithfulness to a promise, and so omit what is distinctive of (...) it. I shall also argue that the likely reason for this distortion is the modern view of man as a self-defining subject who has autonomously to invent his own law or way, for the universe presents none to him. (shrink)
Virtues and Vices in Normative Ethics
The Meaning of Life and Education.R. T. Allen - 1991 - Philosophy of Education 25 (1):47-58.details
Flint's 'Molinism and the Incarnation' is Too Radical.R. T. Mullins - 2015 - Journal of Analytic Theology 3:109-123.details
In a series of papers, Thomas P. Flint has posited that God the Son could become incarnate in any human person as long as certain conditions are met (Flint 2001a, 2001b). In a recent paper, he has argued that all saved human persons will one day become incarnated by the Son (Flint 2011). Flint claims that this is motivated by a combination of Molinism and orthodox Christology. I shall argue that this is unmotivated because it is condemned by orthodox Christology. (...) Flint has unknowingly articulated a version of the heresy called Origenism that is condemned by the Fifth Ecumenical Council. After arguing that Flint's account is unmotivated because it is condemned, I shall offer some reflections and prolegomena on the relationship between contemporary analytic theology and the ecumenical creeds. (shrink)
The Arousal and Expression of Emotion by Music.R. T. Allen - 1990 - British Journal of Aesthetics 30 (1):57-61.details
Aesthetics and Emotions in Aesthetics
Music and Emotion in Aesthetics
Doing Hard Time: Is God the Prisoner of the Oldest Dimension?R. T. Mullins - 2014 - Journal of Analytic Theology 2:160-185.details
In this paper I shall consider an objection to divine temporality called "The Prisoner of Time" objection. I shall begin by distinguishing divine timelessness from divine temporality in order to clear up common misunderstandings and caricatures of divine temporality. From there I shall examine the prisoner of time objection and explain why the prisoner of time objection fails to be a problem for the Christian divine temporalist.
The Pursuit of Magnetic Shadows: The Formal-Empirical Dipole Field of Early-Modern Geomagnetism.Art R. T. Jonkers - 2008 - Centaurus 50 (3):254-289.details
The State and Civil Society as Objects of Aesthetic Appreciation.R. T. Allen - 1976 - British Journal of Aesthetics 16 (3):237-242.details
Civil Society in Social and Political Philosophy
Topics in Aesthetics in Aesthetics
Supererogation Revised.R. T. Allen - 1981 - Sophia 20 (2):5-11.details
Religious Topics in Philosophy of Religion
Supererogation in Normative Ethics
Studies in Arabic Philosophy.R. T. Blackwood - 1970 - Philosophy East and West 20 (2):199-201.details
Governance by Emotion.R. T. Allen - 1991 - Journal of the British Society for Phenomenology 22 (2):15-29.details
Notes on Dares and Dictys.R. T. Clark - 1914 - Classical Quarterly 8 (01):17-.details
C. i., p. 2, 12 dicit Peliae regi se eo uelle ire si uires sociique non deessent. Pelias … Argum … iussit … nauim aedificaret.Considering the next sentence read perhaps n a u e s for uires.C. ii., p. 3, 25. Graeci aduentare nauibus. mittit ad portam.M reads nauibus uti. May this conceal e t i t a ? cf. p. II , 2. For change of tense cf. opening lines of C. iii.
Classics in Arts and Humanities
Patient Self-Determination Act: An African American Perspective.R. T. Tucker - 1994 - Cambridge Quarterly of Healthcare Ethics 3 (3):417.details
African and African-American Philosophy in African/Africana Philosophy
African-American Philosophy in African/Africana Philosophy
Topics in African-American Philosophy in African/Africana Philosophy
A Metaphysics for the Future, by R.E. Allinson.R. T. Allen - 2002 - Journal of the British Society for Phenomenology 33 (1):110-111.details
The Relativity of Simultaneity.R. T. Herbert - 1987 - Philosophy 62 (242):455 - 471.details
In connection with the special theory of relativity, Einstein made use of a now familiar thought experiment1 involving two lightning flashes, a railway train, and an embankment. Whether he used it merely to help explain the theory to others or whether it played a role in the theory's very generation as well is perhaps a matter of conjecture. However, physicist Richard Feynman, for one, believes that Einstein first conceived his theories in the visualizations of thought experiments and developed their mathematical (...) formulations afterwards. According to a recent magazine essay, 'Einstein came to an understanding about relativity by imagining people going up in elevators and beaming light back and forth between rocket ships. (shrink)
History of Physics in Philosophy of Physical Science
Special Relativity in Philosophy of Physical Science
Thought Experiments in Metaphilosophy
Conway's Ontological Objection to Cartesian Dualism.John R. T. Grey - 2017 - Philosophers' Imprint 17:1-19.details
Anne Conway disagrees with substance dualism, the thesis that minds and bodies differ in nature or essence. Instead, she holds that "the distinction between spirit and body is only modal and incremental, not essential and substantial". Yet several of her arguments against dualism have little force against the Cartesian, since they rely on premises no Cartesian would accept. In this paper, I show that Conway does have at least one powerful objection to substance dualism, drawn from premises that Descartes seems (...) bound to accept. She argues that two substances differ in nature only if they differ in their "original and peculiar" cause ; yet all created substances have the same original and peculiar cause; so, all created substances have the same nature. As I argue, the Cartesian is under a surprising amount of pressure to accept Conway's argument, since its key premise is motivated by a conception of substance similar to one endorsed by Descartes in his Principles of Philosophy. (shrink)
Anne Conway in 17th/18th Century Philosophy
Cambridge Platonism in 17th/18th Century Philosophy
Structural Depths in Indian Thought.R. T. Raju - 1987 - Philosophy East and West 37 (2):211-214.details | CommonCrawl |
Shemetkov, Leonid Alexandrovich
Statistics Math-Net.Ru
Total publications: 45
Scientific articles: 35
This page: 2567
Abstract pages: 9652
Full texts: 3954
References: 645
Doctor of physico-mathematical sciences (1969)
Speciality: 01.01.06 (Mathematical logic, algebra, and number theory)
Birth date: 3.07.1937
Keywords: finite groups; formations and Fitting classes; classes of universal algebras.
UDC: 512.4, 512.542, 512.542.6, 512.553.8, 512.57, 519.4, 519.41/47, 519.42, 519.44, 512.542.4
MSC: 20D10, 20D20, 20D30, 20D35, 20D40
The problem of Wielandt on the existence of complements for non-abelian normal subgroups of finite groups is solved. The problem of D. K. Faddeev on the enumeration of finite biprimary groups is solved. The methods for investigation of formations of algebraic systems are developed.
Graduated from the Faculty of Physics and Mathematics of Gomel Pedagogical Institute in 1959, post-graduate courses of the Mathematics Institute of Academy of Sciences of Belarus in 1962. Ph.D. thesis was defended in 1964. D.Sci. thesis was defended in 1969. A list of my works contains more than 100 titles. Since 1973 I have led the research algebra seminar at F. Scorina Gomel State University.
Corresponding member of the National Academy of Sciences of Belarus.
Main publications:
Shemetkov L. A., "O suschestvovanii $\pi$-dopolnenii k normalnym podgruppam konechnykh grupp", DAN SSSR, 195:1 (1970), 50–52
Shemetkov L. A., Formatsii konechnykh grupp, Nauka, 1978
Shemetkov L. A., Skiba A. N., Formatsii algebraicheskikh sistem, Nauka, Moskva, 1989
Shemetkov L. A., "On partially saturated formations and residuals of finite groups", Comm. Algebra, 29:9 (2001), 4125–4137
A. Ballester-Bolinshe, K. Kalvo, L. A. Shemetkov, "O chastichno nasyschennykh formatsiyakh konechnykh grupp", Matem. sb, 198:6 (2007), 3–24
http://www.mathnet.ru/eng/person17953
List of publications on Google Scholar
https://zbmath.org/authors/?q=ai:shemetkov.leonid-a
Publications in Math-Net.Ru
1. L. A. Shemetkov, "Supplements to normal subgroups of finite groups", Algebra Logika, 52:1 (2013), 109–119 ; Algebra and Logic, 52:1 (2013), 79–86
2. L. A. Shemetkov, "On the $\mathfrak F$-residual of the direct product of finite groups", Trudy Inst. Mat. i Mekh. UrO RAN, 19:3 (2013), 316–320
3. E. P. Vdovin, D. O. Revin, L. A. Shemetkov, "Formations of finite $C_\pi $-groups", Algebra i Analiz, 24:1 (2012), 40–52 ; St. Petersburg Math. J., 24:1 (2013), 29–37
4. M. R. Dixon, V. V. Kirichenko, L. A. Kurdachenko, J. Otal, N. N. Semko, L. A. Shemetkov, I. Ya. Subbotin, "S. N. Chernikov and the development of infinite group theory", Algebra Discrete Math., 13:2 (2012), 169–208
5. S. F. Kamornikov, L. A. Shemetkov, "An $\mathfrak X$-crown of a finite soluble group", Algebra Logika, 49:5 (2010), 591–614 ; Algebra and Logic, 49:5 (2010), 400–415
6. L. A. Shemetkov, "Local definitions of formations of finite groups", Fundam. Prikl. Mat., 16:8 (2010), 229–244 ; J. Math. Sci., 185:2 (2012), 324–334
7. Yi Xiaolan, L. A. Shemetkov, "The Formation of Finite Groups with a Supersolvable $\pi$-Hall Subgroup", Mat. Zametki, 87:2 (2010), 280–286 ; Math. Notes, 87:2 (2010), 258–263
8. L. A. Shemetkov, "A note on $\mathfrak{X}$-local formations", PFMT, 2010, 4(5), 61–62
9. A. A. Rodionov, L. A. Shemetkov, "On $p$-locally N-closed formations of finite groups", Tr. Inst. Mat., 18:1 (2010), 92–98
10. S. F. Kamornikov, L. A. Shemetkov, "On the normalizer of a Gaschütz system of a finite soluble group", Fundam. Prikl. Mat., 14:8 (2008), 129–136 ; J. Math. Sci., 166:5 (2010), 655–660
11. L. A. Shemetkov, Yi Xiaolan, "On the $p$-length of finite $p$-soluble groups", Tr. Inst. Mat., 16:1 (2008), 93–96
12. A. Ballester-Bolinches, C. Calvo, L. A. Shemetkov, "On partially saturated formations of finite groups", Mat. Sb., 198:6 (2007), 3–24 ; Sb. Math., 198:6 (2007), 757–775
13. L. A. Shemetkov, "Generalizations of Sylow's theorem", Sibirsk. Mat. Zh., 44:6 (2003), 1425–1431 ; Siberian Math. J., 44:6 (2003), 1127–1132
14. L. A. Shemetkov, "On Huppert's theorem", Sibirsk. Mat. Zh., 44:1 (2003), 224–231 ; Siberian Math. J., 44:1 (2003), 184–189
15. A. N. Skiba, L. A. Shemetkov, "Multiply $\omega$-Local Formations and Fitting Classes of Finite Groups", Mat. Tr., 2:2 (1999), 114–147 ; Siberian Adv. Math., 10:2 (2000), 112–141
16. A. Ballester-Bolinches, L. A. Shemetkov, "On normalizers of Sylow subgroups in finite groups", Sibirsk. Mat. Zh., 40:1 (1999), 3–5 ; Siberian Math. J., 40:1 (1999), 1–2
17. S. F. Kamornikov, L. A. Shemetkov, "On coradicals of subnormal subgroups", Algebra Logika, 34:5 (1995), 493–513
18. L. A. Shemetkov, "The product of formations of algebraic systems", Algebra Logika, 23:6 (1984), 721–729
19. L. A. Shemetkov, "Factorizaton of nonsimple finite groups", Algebra Logika, 15:6 (1976), 684–715
20. L. A. Shemetkov, "Two directions in the development of the theory of non-simple finite groups", Uspekhi Mat. Nauk, 30:2(182) (1975), 179–198 ; Russian Math. Surveys, 30:2 (1975), 185–206
21. L. A. Shemetkov, "Graduated formations of groups", Mat. Sb. (N.S.), 94(136):4(8) (1974), 628–648 ; Math. USSR-Sb., 23:4 (1974), 593–611
22. L. A. Shemetkov, "Formation properties of finite groups", Dokl. Akad. Nauk SSSR, 204:6 (1972), 1324–1327
23. S. A. Chunikhin, L. A. Shemetkov, "Finite groups", Itogi Nauki. Ser. Mat. Algebra. Topol. Geom. 1969, 1971, 7–70 ; J. Soviet Math., 1:3 (1973), 291–332
24. L. A. Shemetkov, "The existence of $\Pi$-complements to normal subgroups of finite groups", Dokl. Akad. Nauk SSSR, 195:1 (1970), 50–52
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Institute of Mathematics of the National Academy of Sciences of Belarus
Gomel State University named after Francisk Skorina | CommonCrawl |
\begin{definition}[Definition:Graph Theory]
'''Graph theory''' is the branch of mathematics concerned with the structure and properties of graphs.
As a (graph-theoretical) graph has the same conceptual definition as a relation, it follows that there is considerable overlap between the fields of graph theory and relation theory.
\end{definition} | ProofWiki |
\begin{document}
\begin{comment} Group theory \end{comment} {}
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\global\long\def\mathrel{\triangleright}{\mathrel{\triangleright}}
\global\long\def\mathbin{\rtimes}{\mathbin{\rtimes}}
\global\long\def\operatorname{Stab}{\operatorname{Stab}}
\begin{comment} Topology \end{comment} {}
\global\long\def\partial{\partial}
\global\long\def\operatorname{susp}{\operatorname{susp}}
\begin{comment} Poset combinatorics \end{comment} {}
\global\long\def\mathop{\check{\prod}}{\mathop{\check{\prod}}}
\global\long\def\mathbin{\check{\times}}{\mathbin{\check{\times}}}
\global\long\def\mathbin{\hat{\times}}{\mathbin{\hat{\times}}}
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\global\long\def\mathrel{\cdot\!\!\!\succ}{\mathrel{\cdot\!\!\!\succ}}
\global\long\def\operatorname{des}{\operatorname{des}}
\begin{comment} Lattice \end{comment} {}
\global\long\def\mathrel{M}{\mathrel{M}}
\begin{comment} Simplicial complex combinatorics \end{comment} {}
\global\long\def\operatorname{link}{\operatorname{link}}
\global\long\def\mathbin{\circledast}{\mathbin{\circledast}}
\global\long\def\operatorname{stellar}{\operatorname{stellar}}
\global\long\def\operatorname{conv}{\operatorname{conv}}
\global\long\def\mathbin{\dot{\cup}}{\mathbin{\dot{\cup}}}
\begin{comment} My stuff \end{comment} {}
\global\long\def\overline{\mathfrak{C}}{\overline{\mathfrak{C}}}
\global\long\def\mathfrak{C}{\mathfrak{C}}
\title{Vertex decomposable graphs \\ and obstructions to shellability}
\author{Russ Woodroofe}
\email{[email protected]}
\address{Department of Mathematics, Washington University in St.~Louis, St.~Louis, Missouri, 63130}
\subjclass[2000]{13F55, 05C38, 05E99}
\keywords{Sequentially Cohen-Macaulay, independence complex, edge ideal, chordal graphs} \begin{abstract} Inspired by several recent papers on the edge ideal of a graph $G$, we study the equivalent notion of the independence complex of $G$. Using the tool of vertex decomposability from geometric combinatorics, we show that $5$-chordal graphs with no chordless $4$-cycles are shellable and sequentially Cohen-Macaulay. We use this result to characterize the obstructions to shellability in flag complexes, extending work of Billera, Myers, and Wachs. We also show how vertex decomposability may be used to show that certain graph constructions preserve shellability. \end{abstract} \maketitle
\section{Introduction}
Let $G=(V,E)$ be a graph with vertex set $V=\{x_{1},\dots,x_{n}\}$. The \emph{independence complex} of $G$, denoted $I(G)$, is the simplicial complex with vertex set $V$ and with faces the independent sets of $G$. When it causes no confusion, we will say that $G$ satisfies some property if its independence complex does. For example, we will say that $G$ is shellable if $I(G)$ is shellable. The independence complex has been previously studied in e.g. \cite{Aharoni/Berger/Meshulam:2005,Klivans:2007,Meshulam:2003}.
The Stanley-Reisner ring of $I(G)$ is \[ k[x_{1},\dots,x_{n}]/(x_{i}x_{j}\,:\, x_{i}x_{j}\in E).\] The quotient in the above ring is also called the \emph{edge ideal} of $G$ and has been an object of study in its own right \cite{Villarreal:2001}. In particular, a recent series of papers \cite{Francisco/Ha:2008,Francisco/VanTuyl:2007,Herzog/Hibi/Zheng:2006,VanTuyl/Villarreal:2008} has worked from the edge ideal to show that chordal graphs are sequentially Cohen-Macaulay and shellable and that certain graph constructions preserve shellability and/or being sequentially Cohen-Macaulay.
In this paper, we consider vertex decomposability in graphs. In Section 2, we recall the definition of a vertex decomposable simplicial complex and show what this means for (the independence complexes of) graphs. As an easy consequence we recover the result that chordal graphs are shellable, hence sequentially Cohen-Macaulay. In Section 3, we give a geometric proof that the only cyclic graphs which are vertex decomposable, shellable and/or sequentially Cohen-Macaulay are $C_{3}$ and $C_{5}$. In Section 4, we prove the main theorem of the paper: \begin{thm} \emph{(Main Theorem)} \label{thm:MainTheorem} If $G$ is a graph with no chordless cycles of length other than $3$ or $5$, then $G$ is vertex decomposable (hence shellable and sequentially Cohen-Macaulay.) \end{thm} In Section 5, we reinterpret Theorem \ref{thm:MainTheorem} in terms of obstructions to shellings, answering a question of Wachs. We also give an application to domination numbers, in the style of \cite{Meshulam:2003}. In Section 6, we examine several graph constructions that preserve vertex decomposability. Finally, in Section 7 we close with some comments on classes of sequentially Cohen-Macaulay graphs. \begin{note} Independence complexes have been studied more extensively in the combinatorics literature as \emph{flag complexes} \cite[Chapter III.4 and references]{Stanley:1996}. Many papers on flag complexes study them by considering the clique complex. We notice that the clique complex of a graph $G$ is the independence complex of the complement graph of $G$. \end{note}
\subsection{Cohen-Macaulay complexes}
We review briefly the background definitions from geometric combinatorics and graph theory.
A simplicial complex $\Delta$ is \emph{pure} if all of its facets (maximal faces) are of the same dimension. A complex $\Delta$ is \emph{shellable} if its facets fit together nicely. The precise definition will not be important to us, but can be found, with much additional background, in \cite[Lecture 3]{Wachs:2007}. The \emph{link} of a face $F$ in $\Delta$ is\[ \operatorname{link}_{\Delta}F=\{G\,:\, G\cup F\mbox{ is a face in }\Delta,G\cap F=\emptyset\}.\]
Let $k$ be a field or the ring of integers. A complex $\Delta$ is \emph{Cohen-Macaulay} over $k$ if $\tilde{H}_{i}(\operatorname{link}_{\Delta}F;k)=0$ for all faces $F$ and $i<\dim(\operatorname{link}_{\Delta}F)$. More intuitively, a complex is Cohen-Macaulay if it has the homology of a bouquet of top-dimensional spheres and if every link satisfies the same condition. It is a well-known fact that any Cohen-Macaulay complex is pure. Any pure, shellable complex is Cohen-Macaulay over any $k$. Our results will be independent of the choice of $k$, and we henceforth drop it from our notation.
Since simplicial complexes that are not pure are often interesting, we study Stanley's extension \cite[Chapter III.2]{Stanley:1996} of the definition of Cohen-Macaulay (and its relationship with shellability) to arbitrary simplicial complexes. The \emph{pure $i$-skeleton} of $\Delta$ is the complex generated by all the $i$-dimensional faces of $\Delta$. A complex is \emph{sequentially Cohen-Macaulay} if the pure $i$-skeleton is Cohen-Macaulay for all $i$. Any shellable complex is sequentially Cohen-Macaulay.
$\Delta$ is a Cohen-Macaulay complex if and only if the Stanley-Reisner ring of $\Delta$ is a Cohen-Macaulay ring. There is also a ring-theoretic notion of sequentially Cohen-Macaulay \cite[Definition III.2.9]{Stanley:1996}. For more background, refer to \cite{Bjorner/Wachs/Welker:2009} and \cite{Stanley:1996} for the combinatorial point of view or to \cite{Bruns/Herzog:1993} for a more ring-theoretic approach.
\subsection{Chordless paths and cycles}
A \emph{chordless path} of length $n$ in a graph $G$ is a path $v_{1},v_{2},\dots,v_{n}$ in $G$ with no \emph{chord}, i.e. with no edge $v_{i}v_{j}$ with $j\neq i+1$. Equivalently, the induced graph on $\{v_{1},\dots,v_{n}\}$ is the path on $n$ vertices. In a like manner, a \emph{chordless cycle} of length $n$ is an induced $n$-cycle.
A graph is \emph{$k$-chordal} if it has no chordless cycles of length $>k$, and \emph{chordal} if it is $3$-chordal.
\section{Vertex decomposability and shedding vertices\label{sec:VertexDecGraphs}}
A simplicial complex $\Delta$ is recursively defined to be \emph{vertex decomposable} if it is either a simplex or else has some vertex $v$ so that \begin{enumerate} \item both $\Delta\setminus v$ and $\operatorname{link}_{\Delta}v$ are vertex decomposable, and \item no face of $\operatorname{link}_{\Delta}v$ is a facet of $\Delta\setminus v$. \end{enumerate} A vertex $v$ which satisfies Condition (2) is called a \emph{shedding vertex}. Vertex decompositions were introduced in the pure case by Provan and Billera \cite{Provan/Billera:1980} and extended to non-pure complexes by Björner and Wachs \cite[Section 11]{Bjorner/Wachs:1997}.
A vertex decomposable complex is shellable. One proof of this fact is via the following lemma of independent interest: \begin{lem} \emph{\label{lem:SheddingVertexLemma}(Wachs \cite[Lemma 6]{Wachs:1999b})} If $\Delta$ is a simplicial complex with shedding vertex $v$ and if both $\Delta\setminus v$ and $\operatorname{link}_{\Delta}v$ are shellable, then $\Delta$ is shellable. \end{lem} \noindent The shelling order in Lemma \ref{lem:SheddingVertexLemma} is that of $\Delta\setminus v$, followed by the facets of $v*\operatorname{link}_{\Delta}v$ in the order of the shelling of $\operatorname{link}_{\Delta}v$.
To summarize, we have the chain of implications: \[ \mbox{vertex decomposable}\implies\mbox{shellable}\implies\mbox{sequentially Cohen-Macaulay.}\] Both implications are known to be strict.
{} The definition of vertex decomposable (and so of shedding vertex) translates nicely to independence complexes. Let $N(v)$ denote the \emph{open neighborhood} of $v$, that is, all vertices adjacent to $v$. Let $N[v]$ denote the \emph{closed neighborhood} of $v$, which is $N(v)$ together with $v$ itself, so that $N[v]=N(v)\cup\{v\}$. \begin{lem} \label{lem:VertDecompGraph}An independence complex $I(G)$ is vertex decomposable if $G$ is a totally disconnected graph (with no edges) or if \begin{enumerate} \item $G\setminus v$ and $G\setminus N[v]$ are both vertex decomposable, and \item no independent set in $G\setminus N[v]$ is a maximal independent set in $G\setminus v$. \end{enumerate} \end{lem} \begin{proof} Translate the definitions! \end{proof} A shedding vertex of $G$ is any vertex which satisfies Condition (2) of Lemma \ref{lem:VertDecompGraph}. A useful equivalent condition for shedding vertices is: \begin{condition} \label{con:SheddingCondition}For every independent set $S$ contained in $G\setminus N[v]$, there is some $x\in N(v)$ so that $S\cup\{x\}$ is independent. \end{condition} We make a first observation: \begin{lem} \label{lem:NeighborhoodContainmentShedding}If $N[v]\subseteq N[w]$ (so that in particular $v$ and $w$ are adjacent), then $w$ is a shedding vertex for $G$.\end{lem} \begin{proof} Since every neighbor of $v$ is also a neighbor of $w$, there are no edges from $v$ to any vertex of $G\setminus N[w]$. Thus, any $v$ can be added to any independent set in $G\setminus N[w]$ while preserving independence. \end{proof} Recall that a \emph{simplicial vertex} is a vertex $v$ such that $N[v]$ is a clique. A well-known theorem of Dirac \cite[Theorem 6.3]{Alfonsin/Reed:2001} says that every chordal graph has a simplicial vertex. Thus, we have: \begin{cor} \label{cor:SimplicialVertexIsShedding}~ \begin{enumerate} \item Any neighbor of a simplicial vertex is a shedding vertex for $G$. \item A chordal graph is vertex decomposable (hence shellable \cite[Theorem 1.2]{VanTuyl/Villarreal:2008} and sequentially Cohen-Macaulay \cite[Theorem 1.2]{Francisco/VanTuyl:2007}; also closely related is \cite[Theorem 1.4]{Meshulam:2003}). \end{enumerate} \end{cor} \begin{proof} If $v$ is a simplicial vertex and $w$ is a neighbor of $v$, then $N[v]\subseteq N[w]$, and Lemma \ref{lem:NeighborhoodContainmentShedding} gives (1). For (2), the theorem of Dirac thus says that a chordal graph has a shedding vertex $w$ if it is not totally disconnected. Since every induced subgraph of a chordal graph is chordal, both $G\setminus w$ and $G\setminus N[w]$ are inductively vertex decomposable.\end{proof} \begin{rem} Vertices satisfying the condition of Lemma \ref{lem:NeighborhoodContainmentShedding} have been studied before under the name dominant vertices, in the context of so-called dismantlable graphs \cite{Ginsburg:1994,Boulet/Fieux/Jouve:2008UNP}. However, dismantlability is a tool for understanding the homotopy type of the clique complex of $G$, i.e., for understanding the independence complex of the complement of $G$. Since $v$ and $w$ will not be adjacent in the complement, there does not seem to be any direct interpretation of dismantlability in terms of vertex decomposability. \end{rem}
\begin{rem} Anton Dochtermann and Alexander Engström also examined vertex decomposability in graphs, independently and at about the same time \cite[Section\ 4]{Dochtermann/Engstrom:2009} as I did. In particular, they prove Corollary \ref{cor:SimplicialVertexIsShedding}, and a special case of Proposition \ref{pro:PendantGen2}; they also notice that the result of Billera and Myers discussed in Section \ref{sub:Obstructions} is a special case of Corollary \ref{cor:SimplicialVertexIsShedding}. \end{rem}
\section{Cyclic graphs}
Corollary \ref{cor:SimplicialVertexIsShedding} (2) states that if $G$ has no chordless cycles of length greater than 3, then it is vertex decomposable. Let $C_{n}$ be the cyclic graph on $n$ vertices. We discuss a partial converse: \begin{thm} \label{thm:CyclicNotShellable}\emph{(Francisco/Van Tuyl \cite[Proposition 4.1]{Francisco/VanTuyl:2007})} $C_{n}$ is vertex decomposable/shellable/sequentially Cohen-Macaulay if and only if $n=3$ or $5$. \end{thm} Theorem \ref{thm:CyclicNotShellable} was proved with algebraic techniques in \cite[Proposition 4.1]{Francisco/VanTuyl:2007}. We give a geometric proof here.
We start with a technical lemma: \begin{lem} \label{lem:OddTechnicalLemma}Let $n=2r+1$, $0<d<r$. Let the $d$-dimensional complex $\Delta_{n}^{d}$ be the complex with vertex set $\mathbb{Z}/n\mathbb{Z}$ and with facets $F_{i}=\{i,i+2,\dots,i+2d\}$ for $i=1,\dots,n$. Then $\Delta_{n}^{d}\cong S^{1}$. \end{lem} \begin{proof} Consider $\Delta_{n}^{d}$ for $d>1$. A facet $F_{i}$ has codimension $1$ intersection with two other facets: $F_{i-2}$ and $F_{i+2}$. Since $d>1$, and since all codimension $1$ faces of $F_{i}$ other than $F_{i}\cap F_{i-2}$ and $F_{i}\cap F_{i+2}$ are {}``free'' (contained in a unique facet), we can collapse $F_{i}$ onto $F_{i}\cap F_{i-2}$ and $F_{i}\cap F_{i+2}$. More formally, the face $\{i,i+2d\}\subset F_{i}$ is free, so we can remove all faces containing $\{i,i+2d\}$ via an elementary collapse \cite[Section 11.1]{Bjorner:1995}, which preserves homotopy type. Every face $F$ not containing $\{i,i+2d\}$ is in either $F_{i}\cap F_{i+2}$ (if $i\notin F$) or $F_{i}\cap F_{i-2}$ (if $i+2d\notin F)$.
Performing a similar collapse at each $F_{i}$ leaves us a simplicial complex with facets $F_{i}\cap F_{i-2}$ for $i=1,\dots,n$. But $F_{i}\cap F_{i-2}=\{i,\dots,i+2d-2\}$, and we see that we have collapsed $\Delta_{n}^{d}$ to $\Delta_{n}^{d-1}$. Thus, $\Delta_{n}^{d}\cong\Delta_{n}^{d-1}$ when $d>1$.
Since $n$ is odd, repeatedly adding 2 to some $i\in\mathbb{Z}/n\mathbb{Z}$ will cover all vertices; hence $\Delta_{n}^{1}$ is the 1-complex $C_{n}\cong S^{1}$. \end{proof}
\begin{figure}
\caption{$I(C_{7})$ is the Möbius strip. The dark line shows the subcomplex $\Delta_{7}^{1}$. }
\label{fig:I(C_7)}
\end{figure}
\begin{proof} \emph{(Of Theorem \ref{thm:CyclicNotShellable})} The if direction is easy: the independence complex of $C_{3}$ is three disconnected vertices, while that of $C_{5}$ is $C_{5}$ as a 1-complex. Both are clearly vertex decomposable.
In the other direction, we show that the pure $d$-skeleton is not Cohen-Macaulay, where $d$ is the top dimension of the complex (i.e., $d=\dim I(C_{n})$). There are two cases, based on whether $n$ is even or odd. (It may be helpful to look at Figures \ref{fig:I(C_7)} and \ref{fig:I(C_6)} while reading the following.) \begin{caseenv} \item $n=2r$. Then the top-dimensional facets have dimension $r-1$, and there are two of them: one with all even vertices, the other with all odd vertices. As the pure $r-1$ skeleton is not even connected, it is certainly not Cohen-Macaulay.
\begin{figure}
\caption{The pure 2-skeleton of $I(C_{6})$ has two disconnected faces.}
\label{fig:I(C_6)}
\end{figure}
\item $n=2r+1$, where $r\geq2$. Then the top-dimensional facets once more have dimension $r-1$. All such facets are obtained by taking a sequence of $r$ alternating vertices in $C_{n}$, with one skip of 2 vertices. We see that the top-dimensional skeleton of $I(C_{n})$ is the complex $\Delta_{n}^{r-1}$ discussed in Lemma \ref{lem:OddTechnicalLemma}, and so homotopic to $S^{1}$. Thus, the pure $r-1$ skeleton is Cohen-Macaulay only when $r-1=1$, i.e., when $n=5$. \end{caseenv}
Since every pure skeleton of a sequentially Cohen-Macaulay complex is Cohen-Macaulay, we have shown that $C_{n}$ is not sequentially Cohen-Macaulay (hence not shellable or vertex decomposable) for $n\neq3,5$.\end{proof} \begin{example} The pure 2-skeleton of $I(C_{6})$ consists of two disconnected triangles, as shown in Figure \ref{fig:I(C_6)}, while $I(C_{7})$ is a (pure) triangulation of the Möbius strip, as seen in Figure \ref{fig:I(C_7)}. Lemma \ref{lem:OddTechnicalLemma} collapses $I(C_{7})$ to the cycle $1,3,5,7,2,4,6$.\end{example} \begin{rem} We computed the homotopy type of the top-dimensional skeleton of $I(C_{n})$. The homotopy type of the entire complex $I(C_{n})$ was calculated by Kozlov \cite[Proposition 5.2]{Kozlov:1999}. \end{rem}
\section{Proof of main theorem\label{sec:ProofOfMainTheorem}}
The previous two sections motivate the result of Theorem \ref{thm:MainTheorem}. In this section, we will give a proof.
A \emph{simplicial $k$-path} in $G$ is a chordless path $v_{1},v_{2},\dots,v_{k}$ which cannot be extended on both ends to a chordless path $v_{0},v_{1},\dots,v_{k},v_{k+1}$ in $G$. Thus, a simplicial vertex is a simplicial $1$-path.
Chvátal, Rusu, and Sritharan \cite{Chvatal/Rusu/Sritharan:2002} proved a nice generalization of Dirac's Theorem to $(k+2)$-chordal graphs using simplicial $k$-paths. The following lemma of theirs will allow us to use the $5$-chordal structure of $G$. \begin{lem} \emph{\label{lem:Chv=0000E1talRusuSritharan}(Chvátal, Rusu, and Sritharan }\cite[Lemma 3]{Chvatal/Rusu/Sritharan:2002}\emph{) }If $G$ is a $5$-chordal graph and $G$ contains a chordless $3$-path $P_{3}$, then $G$ contains a simplicial $3$-path.\end{lem} \begin{note} From a geometric combinatorics point of view, it might make more sense to count edge length and have the above definition be a simplicial $(k-1)$-path, so that a simplicial vertex would be a simplicial $0$-path. However, to avoid confusion, I have kept the original, more graph-theoretic definition. \end{note} We also need to use the lack of chordless $4$-cycles: \begin{lem} \label{lem:C4FreeSimp3Path}Let $w_{1},v,w_{2}$ be a simplicial $3$-path which is not a subgraph of any chordless $C_{4}$ in $G$. Then $v$ is a shedding vertex.\end{lem} \begin{proof} We first notice that, since there is no edge $w_{1}w_{2}$, that any $z$ adjacent to both $w_{1}$ and $w_{2}$ must also be adjacent to $v$. Otherwise, $w_{1},v,w_{2},z$ would be a chordless $4$-cycle.
Suppose by contradiction that $v$ is not a shedding vertex. Then by Lemma \ref{lem:VertDecompGraph} and Condition \ref{con:SheddingCondition}, there is an independent set in $G\setminus N[v]$ which contains a vertex $z_{1}$ adjacent to $w_{1}$, and a vertex $z_{2}$ adjacent to $w_{2}$. Since $z_{1},z_{2}\in G\setminus N[v]$, neither is adjacent to $v$. No $z$ in $G\setminus N[v]$ is adjacent to both $w_{1}$ and $w_{2}$, so $z_{1}$ is not adjacent to $w_{2}$, and $z_{2}$ is not adjacent to $w_{1}$. Since $z_{1}$ and $z_{2}$ are in an independent set, $z_{1}$ is not adjacent to $z_{2}$.
Counting non-adjacent pairs of vertices, we have just shown that $z_{1},w_{1},v,w_{2},z_{2}$ is a chordless path, which contradicts the definition of simplicial $3$-path. Thus $v$ is a shedding vertex, as desired. \end{proof}
\begin{proof} \emph{(Of Theorem \ref{thm:MainTheorem})} If $G$ is chordal, then $G$ is vertex decomposable, as shown in Corollary \ref{cor:SimplicialVertexIsShedding}. Otherwise, $G$ has some chordless $5$-cycle, hence a chordless $3$-path, and by Lemma \ref{lem:Chv=0000E1talRusuSritharan} a simplicial $3$-path. Lemma \ref{lem:C4FreeSimp3Path} shows that the middle vertex of any simplicial $3$-path in $G$ is a shedding vertex, and so by induction $G$ is vertex decomposable. \end{proof}
\section{Applications}
\subsection{Obstructions\label{sub:Obstructions}}
An \emph{obstruction} \emph{to shellability} is a non-shellable complex, all of whose proper subcomplexes are shellable. Thus, any non-shellable complex must contain at least one obstruction to shellability, while a shellable complex may or may not contain some obstructions to shellability as proper subcomplexes.
The \emph{order complex} of a poset is the simplicial complex with vertex set the elements of the poset and with face set the chains of comparable elements. Thus, the order complex of $P$ is the independence complex of the incomparability graph on $P$, which puts an edge between two elements if they are incomparable.
The study of obstructions to shellability was initiated by Billera and Myers, with the following theorem: \begin{thm} \emph{\label{thm:Billera-and-Myers}(Billera and Myers \cite[Corollary 1]{Billera/Myers:1998})} If $P$ is a non-shellable poset, then $P$ contains an induced subposet isomorphic to the poset $D=\{\mbox{two disjoint edges}\}$. \end{thm} Equivalently, the unique obstruction to shellability in a poset is $D$. We note that the incomparability graph of $D$ is $C_{4}$. Gallai gave a forbidden subgraph characterization of incomparability graphs of posets in \cite{Gallai:1967} (translated to English in \cite[Chapter 3]{Alfonsin/Reed:2001}; a more accessible version of the list is in \cite[Chapter 3.2]{Trotter:1992}). The forbidden subgraphs include $C_{n}$ for $n\geq5$. Thus, Theorem \ref{thm:Billera-and-Myers} follows from Corollary \ref{cor:SimplicialVertexIsShedding} (2).
Wachs studied obstructions to shellability further in \cite{Wachs:1999b}, where she asked about the obstructions to shellability in a flag complex. Theorem \ref{thm:MainTheorem} gives a classification, which we summarize in the following theorem: \begin{thm} \label{thm:FlagObstructions}The obstructions to shellability in flag complexes are exactly the independence complexes of $C_{n}$, where $n=4$ or $n\geq6$. \end{thm} \begin{proof} By Theorem \ref{thm:MainTheorem}, any non-shellable graph $G$ has an induced subgraph (hence subcomplex) isomorphic to such a $C_{n}$. In Theorem \ref{thm:CyclicNotShellable} we showed that such $C_{n}$ are not shellable, but any proper induced subgraph of $C_{n}$ is chordal, hence shellable. \end{proof} A natural question suggested by Theorem \ref{thm:FlagObstructions} is whether there is some similar characterization of obstructions to shellability in non-flag complexes, where the minimal non-faces form a hypergraph.\emph{ }One might be led to ask whether the hypergraph of minimal non-faces is always cyclic in an obstruction to shellability. However, examples studied by Wachs \cite{Wachs:1999b} show this is not so, as follows.
Let $M_{n}$ be the simplicial complex with faces $\{1,2,3\}$, $\{2,3,4\}$, $\dots$, $\{n-1,n,1\}$, $\{n,1,2\}$. In \cite[Lemma 5]{Wachs:1999b}, Wachs shows that $M_{5}$, $M_{6}$, and $M_{7}$ are obstructions to shellability.
Inspection verifies that $M_{7}$ is a flag complex, in fact that $M_{7}=I(C_{7})$. The complexes $M_{5}$ and $M_{6}$ are not flag. The minimal non-faces of $M_{5}$ are $\{1,2,4\},\{2,3,5,\},\{3,4,1\},\{4,5,2\},\{5,1,3\}$, which is a cyclic hypergraph, insofar as there is an alternating sequence of edges $e$ and vertices $v\in e$ which visits each edge and vertex exactly once. However, the minimal non-faces of $M_{6}$ are $\{1,4\},\{2,5\},\{3,6\}$ and $\{1,3,5\},\{2,4,6\}$, as pictured in Figure \ref{fig:M6NonFaceHypergraph}. I can see no natural generalization of cyclic graph which applies directly to this hypergraph. Interestingly, however, there is an indirect relationship: the edges are the facets of $I(C_{6})$.
\begin{figure}
\caption{The minimal non-face hypergraph of $M_{6}$.}
\label{fig:M6NonFaceHypergraph}
\end{figure}
\subsection{Domination numbers}
A set $S\subseteq V$ is a \emph{dominating set} if $\bigcup_{s\in S}N[s]=V$. The \emph{dominating number} of $G$, denoted $\gamma(G)$, is the minimum cardinality of a dominating set. Meshulam showed \cite[Theorem 1.2 (iii)]{Meshulam:2003} that the homology of $I(G)$ vanishes below dimension $\gamma(G)-1$ when $G$ is a chordal graph.
We generalize this result in two respects. Let $i(G)$ be the \emph{independent domination number}, that is, the minimum cardinality of a maximal independent set. Any maximal independent set is a minimal dominating set, so $\gamma(G)\leq i(G)$.
Since a sequentially Cohen-Macaulay complex has homology vanishing below the dimension of the smallest facet, and since the smallest facet of $I(G)$ has cardinality $i(G)$, an immediate consequence is the following: \begin{cor} If $G$ is any sequentially Cohen-Macaulay graph (over $k$), then $\tilde{H}_{j}(I(G);k)=0$ for any $j<i(G)-1$. \end{cor} In particular, we recover the result \cite[Theorem 1.2 (iii)]{Meshulam:2003} for chordal graphs and $\gamma(G)$ and extend it to a larger class of graphs and a larger graph invariant.
\section{Graph constructions}
We now give examples of how shedding vertices can be used to show that certain graph constructions respect shellability. \begin{lem} \label{lem:DisjointUnionIsJoin}If $G=G_{1}\mathbin{\dot{\cup}} G_{2}$, then $I(G)=I(G_{1})*I(G_{2})$, the join of simplicial complexes. Hence $G$ is vertex decomposable, shellable and/or sequentially Cohen-Macaulay if and only if $G_{1}$ and $G_{2}$ are.\end{lem} \begin{proof} It is obvious from the definition that $I(G)$ is the given join and that the join is vertex decomposable if and only if both $I(G_{1})$ and $I(G_{2})$ are. That the join of two complexes is shellable or sequentially Cohen-Macaulay if and only if both complexes are is well known and can be found for example in \cite{Wachs:2007}. (Part of Lemma \ref{lem:DisjointUnionIsJoin} can be found as \cite[Lemma 2.4]{VanTuyl/Villarreal:2008}.)\end{proof} \begin{example} Adding a single vertex to $G$ via disjoint union forms a cone over $I(G)$. Adding on the graph consisting of two vertices connected by an edge via disjoint union corresponds to taking the suspension of $I(G)$. Thus, for example, the union of $n$ disjoint edges is homotopic to $S^{n-1}$. \end{example} Francisco and Hà \cite{Francisco/Ha:2008}, following Villarreal \cite[Theorem 2.2]{Villarreal:1990}, define a \emph{whisker} in a graph as a vertex of degree 1. A similar idea seems to be studied in the wider graph theory literature under the name of \emph{pendant}. We will prefer the latter term here. In \cite{Francisco/Ha:2008} and \cite{VanTuyl/Villarreal:2008}, it is shown that, speaking broadly, adding pendants to graphs has good properties for maintaining shellability and the sequentially Cohen-Macaulay property. Their construction essentially works because adding a pendant adds a simplicial vertex. We give an obvious generalization: \begin{prop} \label{pro:PendantGen2}Let $G_{0}$ be a graph with a complete subgraph $K$, and let $G$ be obtained from $G_{0}$ by adding a new vertex $v$ with edges to all vertices of $K$. (That is, let $G$ be obtained from $G_{0}$ by {}``starring $K$''.) Then any element of $K$ is a shedding vertex in $G$; conversely, $G$ is shellable (sequentially Cohen-Macaulay) only if $G_{0}\setminus K$ is.\end{prop} \begin{proof} Since $N[v]=K\cup\{v\}$, we have that $v$ is a simplicial vertex; hence any neighbor is a shedding vertex (Corollary \ref{cor:SimplicialVertexIsShedding}). For the converse statement, we recall that links in a shellable/sequentially Cohen-Macaulay complex have the same property and notice that $\operatorname{link}_{I(G)}v=G\setminus N[v]=G_{0}\setminus K$. \end{proof} The {}``clique-starring'' construction described in Proposition \ref{pro:PendantGen2} adds a pendant when $\vert K\vert=1$. Whatever the size of $K$, the construction adds a simplicial vertex to $G$. We now consider a construction analogous to a pendant which adds a $3$-simplicial path. \begin{prop} Let $G_{0}$ be a graph with a complete subgraph $K$, and let $K_{1},K_{2}$ be disjoint subgraphs of $K$. Let $G$ be obtained from $G_{0}$ by adding new vertices $w_{1}$, $w_{2}$, and $v$, with $w_{1}$ adjacent to all vertices of $K_{1}$, $w_{2}$ adjacent to all vertices of $K_{2}$, and $v$ adjacent to $w_{1}$ and $w_{2}$. Then $v$ is a shedding vertex of $G$. Conversely, $G$ is shellable (sequentially Cohen-Macaulay) only if $G_{0}$ is.\end{prop} \begin{proof} By definition, the path $w_{1},v,w_{2}$ is $3$-simplicial, while $v$ is in no chordless $4$-cycles because $K_{1}$ and $K_{2}$ are disjoint. Lemma \ref{lem:C4FreeSimp3Path} shows that $v$ is a shedding vertex. Conversely, $\operatorname{link}_{I(G)}v=G_{0}$, and any link in a shellable/sequentially Cohen-Macaulay complex has the same property.
We notice in passing that $G\setminus v=G_{0}\cup\{w_{1},w_{2}\}$ is formed from $G_{0}$ by performing the construction of Proposition \ref{pro:PendantGen2} on $K_{1}$ and $K_{2}$. \end{proof} Another graph construction is that of twinning. If two vertices $v$ and $w$ have the same neighbors, i.e., if $N(w)=N(v)$, then we say $v$ and $w$ are \emph{true twins} if there is an edge $vw$, and \emph{false twins} otherwise. There are corresponding graph constructions: add a new vertex $w$ to $G$, together with edges to all neighbors of some $v$. \label{sec:DistanceHered}The family of \emph{distance hereditary graphs} can be defined as the graphs that can be built from a single vertex by adding pendants, true twins, and false twins \cite[Chapter 11.6]{Brandstadt/Le/Spinrad:1999}; twins are also useful in proofs of the Perfect Graph Theorem \cite[Chapter 5.5]{Diestel:2005}. \begin{prop} If $v$ and $w$ are true twins, then $v$ and $w$ are shedding vertices.\end{prop} \begin{proof} We note that $N[v]=N[w]$, and thus by Lemma \ref{lem:NeighborhoodContainmentShedding} they are both shedding vertices. \end{proof} False twins are never shedding vertices, since any maximal independent set including $w$ in $G\setminus N[v]$ is also maximal in $G\setminus v$. \begin{example} The 4-cycle is obtained by adding a false twin of the middle vertex in a 3-path. Thus, adding a false twin to a sequentially Cohen-Macaulay graph can result in a non-sequentially Cohen-Macaulay graph.
{}
One more family of graph operations which has frequently been studied is that of graph products. There are a large number of such operations, based on different rules for putting edges on the Cartesian product of the vertex set. We cannot examine all of them, but note that the commonly considered operations of direct product and Cartesian product of graphs do not respect shellability or the sequential Cohen-Macaulay property, for the Cartesian product of two edges (shellable) is a 4-cycle (not sequentially Cohen-Macaulay), while the direct product of an edge with a 3-cycle (both shellable) is the complete bipartite graph $K_{3,3}$, which \cite[Corollary 3.11]{VanTuyl/Villarreal:2008} shows is not sequentially Cohen-Macaulay. \end{example}
\section{A comment on perfect graphs}
Herzog, Hibi, and Zheng \cite{Herzog/Hibi/Zheng:2006} point out that classifying sequentially Cohen-Macaulay graphs is likely an intractable problem. We recall their argument. If $\Delta$ is a simplicial complex, then the order complex of the face lattice of $\Delta$ is a flag complex, and it is sequentially Cohen-Macaulay if and only if $\Delta$ is. (The order complex of the face lattice is the barycentric subdivision of $\Delta$.) Herzog, Hibi and Zheng conclude that characterizing sequentially Cohen-Macaulay graphs is as difficult as characterizing all sequentially Cohen-Macaulay complexes. The closely related property of shellability is likely of a similar difficulty.
As we have seen, however, there are families of graphs in which classifying the sequentially Cohen-Macaulay members of the family is possible. That all chordal graphs are sequentially Cohen-Macaulay (Corollary \ref{cor:SimplicialVertexIsShedding}) is an example of this type of classification, as is the recursive characterization of sequentially Cohen-Macaulay bipartite graphs in \cite[Corollary 3.11]{VanTuyl/Villarreal:2008}. Other families of graphs may also have interesting answers.
We notice that the argument of Herzog, Hibi, and Zheng can help indicate the families of graphs in which we can hope for such a classification. For example, a \emph{perfect graph }is one where every induced subgraph has chromatic number equal to the size of its largest clique. The Strong Perfect Graph Theorem says that a graph $G$ is perfect if and only if there are no chordless odd cycles of length $\geq5$ in either $G$ or its complement. Another fundamental result is that the complement of a perfect graph is also perfect. See \cite{Alfonsin/Reed:2001} for more information and references about perfect graphs. Both chordal graphs and bipartite graphs are perfect, and characterizing the shellability and/or sequential Cohen-Macaulay connectivity of their common super-family would seem like a reasonable aim.
Unfortunately for this aim, poset (in)comparability graphs are perfect, as can be proved either by the direct argument of coloring elements by their rank, or else from Gallai's previously mentioned characterization of poset comparability graphs, which lack odd cycles of length $\geq5$ \cite{Gallai:1967}. Moreover, a complex is sequentially Cohen-Macaulay if and only if its face poset is sequentially Cohen-Macaulay. Thus, characterizing the sequentially Cohen-Macaulay perfect graphs is at least as hard as characterizing which complexes in general are sequentially Cohen-Macaulay.
Considering the intersection of a graph family $\mathcal{F}$ with the family of poset incomparability graphs is a recommended exercise before looking for shellings of graphs in $\mathcal{F}$.
\end{document} | arXiv |
Vol. 1, Issue 1 pp.1-140
Approximation of Functionals by Neural Network Without Curse of Dimensionality
Yahong Yang & Yang Xiang
10.4208/jml.221018
J. Mach. Learn. , 1 (2022), pp. 342-372.
Summary Full PDF 136 2843 Abstract General Summary
In this paper, we establish a neural network to approximate functionals, which are maps from infinite dimensional spaces to finite dimensional spaces. The approximation error of the neural network is $\mathcal{O}(1/\sqrt{m})$ where $m$ is the size of networks. In other words, the error of the network is no dependence on the dimensionality respecting to the number of the nodes in neural networks. The key idea of the approximation is to define a Barron space of functionals.
General Summary
Learning functionals or operators by neural networks is nowadays widely used in computational and applied mathematics. Compared with learning functions by neural networks, an essential difference is that the input spaces of functionals or operators are infinite dimensional space. Some recent works learnt functionals or operators by reducing the input space into a finite dimensional space. However, the curse of dimensionality always exists in this type of methods. That is, in order to maintain the accuracy of an approximation, the number of sample points grows exponentially with the increase of dimension.
In this paper, we establish a new method for the approximation of functionals by neural networks without curse of dimensionality. Functionals, such as linear functionals and energy functionals, have a wide range of important applications in science and engineering fields. We define Fourier series of functionals and the associated Barron spectral space of functionals, based on which our new neural network approximation method is established. The parameters and the network structure in our method only depend on the functional. The approximation error of the neural network is $O(1/\sqrt{m})$ where $m$ is the size of the network, which does not depend on the dimensionality.
A Mathematical Framework for Learning Probability Distributions
Hongkang Yang
The modeling of probability distributions, specifically generative modeling and density estimation, has become an immensely popular subject in recent years by virtue of its outstanding performance on sophisticated data such as images and texts. Nevertheless, a theoretical understanding of its success is still incomplete. One mystery is the paradox between memorization and generalization: In theory, the model is trained to be exactly the same as the empirical distribution of the finite samples, whereas in practice, the trained model can generate new samples or estimate the likelihood of unseen samples. Likewise, the overwhelming diversity of distribution learning models calls for a unified perspective on this subject. This paper provides a mathematical framework such that all the well-known models can be derived based on simple principles. To demonstrate its efficacy, we present a survey of our results on the approximation error, training error and generalization error of these models, which can all be established based on this framework. In particular, the aforementioned paradox is resolved by proving that these models enjoy implicit regularization during training, so that the generalization error at early-stopping avoids the curse of dimensionality. Furthermore, we provide some new results on landscape analysis and the mode collapse phenomenon.
The modeling of probability distributions is an important branch of machine learning. It became popular in recent years thanks to the success of deep generative models in difficult tasks such as image synthesis and text conversation. Nevertheless, we still lack a theoretical understanding of the good performance of distribution learning models. One mystery is the following paradox: it is generally inevitable that the model suffers from memorization (converges to the empirical distribution of the training samples) and thus becomes useless, and yet in practice the trained model can generate new samples or estimate the probability density over unseen samples. Meanwhile, the existing models are so diverse that it has become overwhelming for practitioners and researchers to get a clear picture of this fast-growing subject. This paper provides a mathematical framework that unifies all the well-known models, so that they can be systemically derived based on simple principles. This framework enables our analysis of the theoretical mysteries of distribution learning, in particular, the paradox between memorization and generalization. It is established that the model during training enjoys implicit regularization, so that it approximates the hidden target distribution before eventually turning towards the empirical distribution. With early stopping, the generalization error escapes from the curse of dimensionality and thus the model generalizes well.
2009- 2023 (C) Copyright Springer, All right reserved 京ICP备20010051号-10 | CommonCrawl |
\begin{document}
\title{Dissipative Preparation of Entangled Many-Body States with Rydberg Atoms}
\author{Maryam Roghani} \email{[email protected]}
\author{Hendrik Weimer} \affiliation{Institut f\"ur Theoretische Physik, Leibniz Universit\"at Hannover, Appelstrasse $2$, $30167$ Hannover, Germany }
\date{\today}
\begin{abstract}
We investigate a one-dimensional atomic lattice laser-driven to a
Rydberg state, in which engineered dissipation channels lead to
entanglement in the many-body system. In particular, we demonstrate
the efficient generation of ground states of a frustration-free
Hamiltonian, as well as $W$ states. We discuss the realization of
the required coherent and dissipative terms, and we perform
extensive numerical simulations characterizing the fidelity of the
state preparation procedure. We identify the optimum parameters for
high fidelity entanglement preparation and investigate the scaling
with the size of the system. \end{abstract}
\pacs{37.10.Jk, 42.50.-p, 03.65.Ud} \maketitle
\section{Introduction}
The use of controlled dissipation channels holds great promise for the preparation of highly entangled quantum many-body states \cite{Verstraete2009,Weimer2010}. These ideas are particularly relevant for ultracold atoms due to the tremendous experimental control over these systems, combined with the availability of efficient dissipation channels in the form of optical pumping. Here, we discuss the preparation of entangled many-body quantum states in the context of strongly interacting Rydberg atoms.
So far, explicit state preparation protocols have been mostly constrained to stabilizer states \cite{Barreiro2011,Weimer2011,Carr2013a,Rao2013,Weimer2013a,Lin2013,Shankar2013}, meaning that these states can be found by minimizing the energy of a stabilizer Hamiltonian of mutually commuting observables \cite{Gottesman1996}. However, as shown in \cite{Verstraete2009}, efficient state preparation protocols should also exist for non-commuting degrees of freedom in frustration-free models, where the ground state can be found by minimizing a sum of quasilocal projection operators.
As the physical platform to realize such dissipative quantum state engineering, Rydberg atoms appear to be particularly suited due to their highly tunable interaction and dissipation properties \cite{Low2012}. Remarkably, Rydberg atoms in one-dimensional lattices can be approximated by a frustration-free model \cite{Lesanovsky2011}, making it especially promising to study dissipative quantum state engineering in these systems.
A crucial quantity in the investigation of strongly interacting Rydberg gases is the blockade radius $r_b$, given by \begin{equation}
r_b = \sqrt[6]{\frac{C_6}{\hbar\Omega}}, \end{equation} where $C_6$ is the van der Waals coefficient of the Rydberg interaction and $\Omega$ is the Rabi frequency associated with the driving of the Rydberg transition. Here, we will be interested both in the regime where the blockade radius is comparable to the lattice spacing and the regime where $r_b$ is larger than the system size.
This paper is organized as follows. After a brief introduction of the atomic scheme and the theoretical approach, we discuss the prepariation of the ground state of a frustration-free Hamiltonian. We present our numerical results for various detuning and explore the appropriate parameters in which the steady state overlaps with the highly entangled Rokhsar-Kivelson state. We investigate the scaling of the fidelity with increasing system size. Finally, we perform an analogous analysis for the generation of entangled $W$ states, where we find high state preparation fidelity even for relatively large system sizes.
\section{Setup of the system}
\subsection{Hamiltonian dynamics}
We consider a one-dimensional lattice with $N$ sites. Each site is occupied by a two level atom with states $\ket{0}$ and $\ket{1}$, driven by a laser beam with Rabi frequency $\Omega$ and detuning $\Delta$. The Hamiltonian for this system in the rotating wave approximation is expressed as \begin{equation} \label{eq:Hamilton}
\mathcal{H}=\Omega\sum_k^N \sigma_x^k+\Delta \sum_k^N n_k+V\sum_{m>k}^N \frac{n_m n_k}{|k-m|^6}, \end{equation} where $n_k=\sigma_+^k \sigma_-^k$ is the Rydberg number operator in terms of the raising and lowering operators. The atoms interact via the van der Waals interaction with the interaction strength $V$. By applying the unitary transformation $U=\exp[-{\rm{i}}t \sum_k^N n_k
n_{k+1}]$ and an additional rotating wave approximation, Eq.~(\ref{eq:Hamilton}) is rewritten as \cite{Lesanovsky2011} \begin{equation} \label{Hamilton1} \mathcal{H}=E_0+\mathcal{H}_{\rm{3body}}+\mathcal{H}^{\prime}, \end{equation} where $E_0$ is the ground state energy and $\mathcal{H}^{\prime}$ a perturbation. The three-body term $\mathcal{H}_{\rm{3body}}$ is given by \begin{equation}
\mathcal{H}_{\rm{3body}}=\Omega\sum_k^ N h_k^{\dagger}h_k, \end{equation} i.e., it is a sum of projection operators $h_k$. These projection operactors have the form \begin{equation}
h_k=\sqrt{\frac{1}{\xi^{-1}+\xi}}\,\,P_{k-1}P_{k+1}[\sigma_x^k+\xi^{-1}n_k+\xi(1-n_k)], \end{equation} where $P_k=1-n_k$. Within the phase diagram of one-dimensional Rydberg gases \cite{Weimer2010a,Sela2011}, the pertubation $\mathcal{H}'$ is minimal when the blockade radius $r_b$ is given by $r_b = 2a$ with $a$ being the lattice spacing \cite{Lesanovsky2011}. The manifold of the region that can be appromixated by the three-body Hamiltonian $\mathcal{H}_{\rm{3body}}$ adiabatically connects to the region between crystalline phases at half and third filling \cite{Ji2014}.
A Hamiltonian that can be expressed as summation of the projection operators is known as frustration-free Hamiltonian with a Rokhsar-Kivelson (RK) ground state \cite{Rokhsar1988} \begin{equation}
|\xi\rangle =\frac{1}{\sqrt{Z_{\xi}}}\prod_k^N(1-\xi P_{k-1}\sigma_+^k P_{k+1})|00 \cdots 0\rangle, \end{equation} where $Z_{\xi}$ is a normalization constant. This state is a coherent superposition of all states that has no nearest neighbour excitations and is highly entangled. For the case $\xi=1$, the RK state can be understood as a superoperator $\mathcal{P}$ acting on the anti-symmetric superposition of the two atomic states,
$\prod_i|-\rangle_i=\prod_i(|0\rangle_i-|1\rangle_i)/\sqrt{2}$, where $\mathcal{P}$ imposes a constraint prohibiting two neighbouring excitations.
\subsection{Jump Operators}
For the dissipative preparation, we will be interested in the coupling to an external environment, which leads to the desired quantum many-body state as a steady state of the dynamics. Here, we consider an open quantum system that is described by a quantum master equation in Lindblad form \begin{equation} \label{eq:master}
\frac{d}{dt}\rho(t)=-\frac{\rm{i}}{\hbar}[\mathcal{H},\rho]+\sum\limits_{k} \left(c_i\rho
c_i^{\dagger}-\frac{1}{2}\{c_i^\dagger c_i,\rho\} \right). \end{equation} For the dissipative preparation of the RK state, we focus on the jump operators \begin{equation} \label{eq:jump}
c_k=\sqrt{\kappa}\, P^{k-1}_g (|-\rangle \langle + |)^kP^{k+1}_g, \end{equation} which avoids the existence of two neighboring atoms in the excited states, thus transfers the atoms to an antisymmetric superpostion. In the context of Rydberg atoms, we envision three different possibilities to realize such a correlated jump operator: (i) Combining strong Rydberg interactions with electromagnetically induced transparency (EIT) \cite{Muller2009,Pritchard2010,Petrosyan2011}, it is possible to engineer effective jump operators of the desired type when the Rydberg interaction destroys the EIT feature. (ii) Realization of a rotated version of the Hamiltonian (\ref{eq:Hamilton}) using Rydberg dressing of hyperfine spin states \cite{Glaetzle2015,vanBijnen2015,Overbeck2016a}, where a $\sigma_-$ jump operator in the rotated bases corresponds to the desired $\ketbrap{-}{+}$ operator \footnote{The (rotated) projection operator $P_g$ can be incorporated by making the decay process dependent on the Rydberg-dressed interaction.} (iii) Using coherent interactions with optically pumped auxiliary atoms, allowing to realize largely arbitrary many-particle jump operators \cite{Weimer2016}.
While the jump operators of Eq.~(\ref{eq:jump}) have the RK state as a dark state, it is not unique, as configurations involving a large number of up spins are also dark, as there is no site on which the constraints involving the $P_g$ operators can be fulfilled. Hence, we add a second set of jump operators, related to optical pumping of on spin state into the other via an intermediate state. The strength of this second set of decay processes is chosen to be relatively weak to the former, ensuring that the desired RK state remains an approximate dark state of the dynamics. Interestingly, the precise form of this second set is not very important. Even in the case (ii) outlined above, where the decay is actually occurring in a rotated basis, we find a very efficient decay of the undesired class of dark states. In this case, we can integrate out the intermediate state within the effective operator formalism \cite{Reiter2012}, see App.~\ref{app:eff}, obtaining for these addition operators in the rotated frame \begin{eqnarray} \label{eq:lindblad}
c_{k^{\prime}}&=&\frac{\Omega'}{\sqrt{\gamma}} \rm{e}^{-\rm{i}\frac{\pi}{4} \sigma_k^y}(|0\rangle \langle 0|+|0\rangle \langle 1|) \rm{e}^{+\rm{i}\frac{\pi}{4} \sigma_k^y}, \\ \nonumber
c_{k^{\prime \prime}}&=&\frac{\Omega'}{\sqrt{\gamma}} \rm{e}^{-\rm{i}\frac{\pi}{4} \sigma_k^y}(|0\rangle \langle 1|+|1\rangle \langle 1|) \rm{e}^{+\rm{i}\frac{\pi}{4} \sigma_k^y}. \end{eqnarray} In total, the jump operators $c_i$ in Eq.~(\ref{eq:master}) contain the sets $c_k$, $c'_k$, and $c''_k$, for all lattice sites $k$.
\section{Numerical results}
We now perform numerical simulations of the quantum master equation based on the wave function Monte Carlo approach \cite{Johansson2013} to obtain the steady state $\rho_{st}$ of the dynamics. Fig.~\ref{groundstate} compares the expectation value of the energy in the steady state of the dynamics to the ground state energy of the Hamiltonian at $\xi=1$. At $\Delta=-2$, the energies are in close agreement. \begin{figure}
\caption{ Comparison of the expection value of the energy in the steady state ($\star$) and the ground state energy of the Hamiltonian $ \mathcal{H}$ ($\bullet$) for different laser detunings. Further parameters are $\Omega'^2/\gamma=0.02\kappa$, $V=100\kappa$, $\Omega=1.5\kappa$, and $N=5$. }
\label{groundstate}
\end{figure}
We can further explore the efficiency of our dissipative preparation scheme by considering the fidelity $\mathcal{F}_{RK}$ of preparing the RK state in terms of the overlap of the steady state with the RK state, \begin{equation}
\mathcal{F}_{RK}=\langle \xi |\rho_{st}|\xi\rangle, \end{equation} where $\rho_{st}$ is the steady state density matrix from the solution of the quantum master equation. \begin{figure}
\caption{
Fidelity of the Rokhsar-Kivelson state, $|\xi\rangle$, as a function of detuning. Parameters are same as in Fig.~\ref{groundstate}.}
\label{Fidelity}
\end{figure} Again, we find that at $\Delta=-2$, the fidelity approaches unity.
Furthermore, we perform an evaluation of our preparation procedure as a function of the system size. Generically, the fidelity is expected to decay exponentially, as small local deviations from the RK state get multiplied for larger system sizes, a phenomenon also known as ``orthogonality catastrophe''. Indeed, as shown in Fig.~\ref{Fidelitysize}, we observe a decrease of the fidelity for larger system sizes. Nevertheless, we still obtain a substantial overlap with the RK state for system sizes as large as $N=11$. Here, we focus on an odd number of sites, as they give slightly larger values for the fidelity. \begin{figure}
\caption{ Scaling of the fidelity of the RK state preparation with different numbers of lattice sites $N$.}
\label{Fidelitysize}
\end{figure}
\section{Generation of $W$ states}
Finally, we wish to discuss how the dissipation preparation procedure can be generalized to cover other classes of entangled quantum many-body states. Here, we consider the case where the blockade radius is larger than the system size, $r_b\gg Na$, corresponding to a fully blockaded ensemble \cite{Brion2007}. In such a situation, the dynamics is mostly confined to the manifold of states containing at most a single Rydberg excitation. In this regime, the jump operator is defined as \begin{eqnarray}
c_i&=&\sqrt{\kappa}\prod_{j\neq i} \,P_g^j\, (|-\rangle \langle +|)^i \\ \nonumber
&=& \sqrt{\kappa}P_g^1\otimes P_g^2\otimes P_g^{i-1}\otimes (|-\rangle \langle +|)^i \otimes P_g^{i+1}, \end{eqnarray} i.e, the quantum jump can only occur if there are no Rydberg excitations located at other sites. The dark state of interest is closely related to a $W$ state, i.e., \begin{equation}
\label{wstate}
|W\rangle=\frac{1}{\sqrt{N+1}}(|\downarrow\downarrow \cdots\downarrow\rangle-|\uparrow\downarrow\cdots\downarrow\rangle-|\downarrow\uparrow \cdots\downarrow\rangle-\cdots). \end{equation} As in the investigation of the dissipative preparation of the RK state, we now turn to analyzing the fidelity of the $W$ state preparation, defined as \begin{equation}
\mathcal{F}_{W}=\langle W |\rho_{st}|W\rangle. \end{equation} Figure~\ref{FidelityWN} shows the fidelity for various detunings $\Delta$ and for different number of the atoms in the lattice. The maximum entanglment occurs at different detunings for different number of the atoms. Remarkably, we find that the fidelity does not significantly decrease for larger system sizes, potentially allowing to efficiently prepare large ensembles of entangled atoms. Performing numerical simulations based on matrix product states \cite{Honing2012,Transchel2014,Cui2015,Werner2016} or the variational principle for open quantum systems \cite{Weimer2015} will allow to test the precise scaling of the fidelity with the size of the system for much larger ensemble sizes. \begin{figure}
\caption{ Fidelity of the $W$ state prepration as a
function of the laser detuning $\Delta$. Parameters are
$V=10^4\kappa$, $\Omega'^2/\gamma=0.02\kappa$, $\Omega=1.5\kappa$.}
\label{FidelityWN}
\end{figure}
\section{Summary} We have investigated the dissipative preparation of entangled states in one-dimensional atomic lattices in which the atoms interact via a long-range Rydberg interaction. We find that we can efficiently prepare moderately sized systems in a highly entangled Rokhsar-Kivelson state, as well as larger ensembles of atoms in a $W$ state. Our results underline the strengths of the dissipative quantum state preparation paradigm.
\section{Acknowledgments} This work was funded by the Volkswagen Foundation and by the DFG within SFB 1227 (DQ-mat) and SPP 1929 (GiRyd).
\appendix
\section{Jump operators in the effective operator formalism}
\label{app:eff}
One approach which eliminates the fast decaying excited states manifold is via effective quantum jump operators \cite{Reiter2012}, which reduces the system dynamics to the non-decaying states. Here, we will neglect the decay of the Rydberg state and focus only on the decay of intermediate states. This procedure gives rise to an effective Hamiltonian \begin{equation} \label{heff}
\mathcal{H}_{eff}=-\frac{1}{2} V_{-}\big{[}\mathcal{H}^{-1}_{NH}+(\mathcal{H}^{-1}_{NH})^{\dagger}\big{]}V_{+}+\mathcal{H}_g, \end{equation} where $V_{+}/V_{-}$ are the excitation/de-excitation operators coupling to the decaying states. $\mathcal{H}_{NH}$ is the non-Hermitian Hamiltonian of the decaying states manifold \begin{equation}
\mathcal{H}_{NH}=\mathcal{H}_e-\frac{\rm{i}}{2}\sum_k \mathcal{L}^{\dagger}_k\,\mathcal{L}_k, \end{equation} where $\mathcal{H}_e$ is the Hamiltonian in the manifold of the decaying states and $\mathcal{L}_k$ describes the quantum jump operators. The effective quantum jump operators are given by \begin{equation}
\mathcal{L}_{eff}^k=\mathcal{L}_k\mathcal{H}_{NH}^{-1}V_{+}. \end{equation} Applying these definition to a process that describes optical pumping from the $\ket{1}$ state into the $\ket{0}$ state via an intermediate level, we obtain \begin{eqnarray} \label{eq:leff}
\mathcal{L}^0_{eff}=\frac{\rm{i}\Omega'}{\sqrt{\gamma}}(|0 \rangle\langle 0|+|0 \rangle\langle 1|),\\
\mathcal{L}^1_{eff}=\frac{\rm{i}\Omega'}{\sqrt{\gamma}}(|0 \rangle\langle 1|+|1 \rangle\langle 1|),\nonumber \end{eqnarray} where $\Omega'$ is the Rabi frequency describing the driving of the transition between $\ket{1}$ and the intermediate state and $\gamma$ is its spontaenous decay rate. After transformation into the rotating frame, we obtain the jump operators given in Eq.~(\ref{eq:lindblad}).
\end{document}
\end{document} | arXiv |
Comparing the speed and accuracy of approaches to betweenness centrality approximation
John Matta ORCID: orcid.org/0000-0002-7666-14091,
Gunes Ercal1 &
Koushik Sinha2
Many algorithms require doing a large number of betweenness centrality calculations quickly, and accommodating this need is an active open research area. There are many different ideas and approaches to speeding up these calculations, and it is difficult to know which approach will work best in practical situations.
The current study attempts to judge performance of betweenness centrality approximation algorithms by running them under conditions that practitioners are likely to experience. For several approaches to approximation, we run two tests, clustering and immunization, on identical hardware, along with a process to determine appropriate parameters. This allows an across-the-board comparison of techniques based on the dimensions of speed and accuracy of results.
Overall, the speed of betweenness centrality can be reduced several orders of magnitude by using approximation algorithms. We find that the speeds of individual algorithms can vary widely based on input parameters. The fastest algorithms utilize parallelism, either in the form of multi-core processing or GPUs. Interestingly, getting fast results does not require an expensive GPU.
The methodology presented here can guide the selection of a betweenness centrality approximation algorithm depending on a practitioner's needs and can also be used to compare new methods to existing ones.
Many centrality measures exist, utilizing both local and global information for quantifying the importance of a node in a network. Local measures such as degree and closeness centrality can be computed in linear time [1], but are limited in their usefulness. Global measures have a much higher time complexity, but have a wider variety of applications. Betweenness centrality [2] is an important global measure based on the shortest paths through an entire network. Given a graph \(G = (V, E)\), where V is a set of vertices and \(E \subset \left( {\begin{array}{c}V\\ 2\end{array}}\right)\) is a set of edges, the betweenness centrality of vertex \(v \in V\) is defined as
$$\begin{aligned} \mathrm {BC}(v) = \sum _{s\ne v\ne t} \frac{\sigma _{st}(v)}{\sigma _{st}}, \end{aligned}$$
where \(\sigma _{st}\) is the number of shortest paths from node s to node t, and \(\sigma _{st}(v)\) is the number of those paths that pass through vertex v.
Betweenness centrality has been applied to problems as diverse as cancer diagnosis [3], network flow [4], and the measurement of influence in social networks [5]. There are many applications involving a large number of betweenness centrality calculations, such as power grid contingency analysis [6] (which is so computationally intensive that it must be done on a supercomputer), Girvan–Newman community detection [7], network routing [8], skill characterization in artificial intelligence [9], analysis of terrorist networks [10], node-based resilience clustering [11], and the modeling of immunization strategies [12]. In these applications, the vertex with the highest betweenness centrality must be determined for a continually changing graph O(n) times.
The most efficient known algorithm for calculating exact betweenness centralities is a faster algorithm by Brandes [13]. For unweighted graphs, this algorithm is based on a modification of breadth-first search (BFS) and can calculate the betweenness centrality of every node in a graph with a time complexity of O(|V||E|). Many algorithms used in the biological and social sciences, such as those mentioned above, rely on repeatedly calculating betweenness centrality for every node in a network. The relevant networks examined, such as online social networks, can become arbitrarily large, making Brandes' algorithm impractical for many realistic scenarios. Therefore, the problem of performing a large number of betweenness centrality calculations efficiently is an active open research area [14] to which this work contributes.
We wish to compare different ways of decreasing the execution time of the above algorithms, and both speed and accuracy are important in the assessment of outcomes. Simply comparing the results of approximate betweenness centralities to the true values is not necessarily useful for several reasons. First, several of the aforementioned algorithms use betweenness centrality as a measure of relative importance with which to rank vertices, and only the correct ranking is required. An approximation that produces some incorrect values, but preserves the correct ranking will perform just as well as exact calculations. Second, for some of the algorithms, even the exact centrality ordering is not critical. For example, it is noted in works such as [15, 16] that only the top or top k vertices with the highest betweenness centrality are required. In light of such considerations, approximation algorithms for betweenness centrality that consistently and efficiently discover the highest betweenness node or nodes may be more useful than slower traditional methods computing the betweenness centrality for every node exactly.
The first major study to attempt to comprehensively benchmark betweenness centrality calculations is by Al-Ghamdi et al. [17]. They use a supercomputer to compute exact betweenness centralities for large graphs, and then compare the results from approximations to these "gold standard" values. They run tests by benchmarking seven different approximation methods, including methods by Brandes and Pich [18], Bader et al. [19], Riondato and Kornaropoulos [20], Geisberger et al. [21], Riondato and Upfal [22], Pfeffer and Carley [23], and Everett and Borgatti [24]. As we are interested in comparing the benchmarking performances obtained in [17] with our results, we examine the first four of the listed methods in this paper along with several other algorithms. Whereas the focus of the work by Al-Ghamdi et al. [17] is on the process of benchmarking itself, the focus of our work is to clarify conditions under which the approximation algorithms are sufficiently accurate and fast for the important applications considered. For example, most methods use parameters which control the trade-off between speed and accuracy. Results in [17] are based on the use of only one set of parameters, while our work follows a methodical process which yields insight into the process of finding and using the best parameters given a preferred outcome.
The present work is an extension of the conference paper A Comparison of Approaches to Computing Betweenness Centrality for Large Graphs [25]. Whereas the previous paper compared only two approaches to increasing the speed of a large number of betweenness centrality calculations, this paper compares many different approaches: eight are fully tested, and experiences with several others are reported upon. The previous paper used a clustering test as the benchmark for comparison. Here we add a second benchmark, a test based on the immunization problem. A third addition is the comparison of our results to [17]. While [17] is an important related work, we come to very different conclusions, and the reasons for the differences are analyzed. Finally, because the trade-off between speed and accuracy of approximation methods is determined by user-supplied parameters, this paper gives detailed notes on parameter selection meant to be useful to practitioners.
The rest of the paper is organized as follows. "Background" discusses research in the area of betweenness centrality calculation and provides a useful survey of many different heuristic and approximation methods. "Methods" describes and justifies the tests and graphs used, and the process of carrying out the experiments. "Experimental results" describes individual results for ten different estimation techniques and summarizes performance based on both speed and accuracy. A description of each algorithm including its time complexity is given in this section. Finally, "Conclusion" contains a brief summary of the paper.
Attempts to increase the speed of betweenness centrality calculations go back at least to Brandes's original faster algorithm in 2001 [13]. Brandes's paper notably establishes a time complexity for exact calculation of O(|V||E|) (for unweighted graphs) against which many other algorithms are compared. Brandes is also known for his paper on variants of betweenness centrality, which provides ideas for efficient implementations [26].
Betweenness centrality is a shortest-paths based measure, and computing it involves solving a single-source shortest paths (SSSP) problem. One heuristic strategy for speeding the overall computation is to solve the SSSP problem for a smaller set of vertices. Brandes and Pich's 2007 work (hereafter referred to as Brandes2007) creates estimates that "are based on a restricted number of SSSP computations from a set of selected pivots" [18]. The idea of using a sample, based on an earlier technique for speeding closeness centrality calculations [27], is tested using different sampling strategies, such as uniformly at random, random proportional to degree, and maximizing/minimizing distances from previous pivots. It is determined that a randomly chosen sample gives best results.
Bader et al. [19] introduce an adaptive sampling algorithm (Bader2007) which, like Brandes2007, reduces the number of SSSP computations through choosing representative samples. The sampling is called adaptive because "the number of samples required varies with the information obtained from each sample." For example, a vertex whose neighbors induce a clique has betweenness centrality of zero, and its inclusion in the sample does not provide much information. Geisberger, Sanders and Schultes develop a method termed Better Approximation in [21] (Geisberger2008), in which the process for selecting the pivot nodes is improved. To further increase speed, if an arbitrary number k of vertices is selected, the algorithm can be parallelized for up to k processors. The overall idea of sampling is generalized by Chehreghani in [28], where a framework is given for using different sampling techniques, and a strategy for choosing sampling techniques that minimize error is discussed. Further discussion of a large number of sampling strategies can be found in [29].
Riondato and Kornaropoulos introduce a different idea for sampling in [20] (Riondato2016). In this sampling algorithm, betweenness centrality computations are based not on pivots as in Bader2007 and Brandes2007, but on a predetermined number of samples of node pairs. This fixed sample size results in faster computation with the same probabilistic guarantees on the quality of the approximation. Riondato and Upfall have another technique, called ABRA, that uses Rademacher averages to estimate betweenness centrality [22].
KADABRA (the ADaptive Algorithm for Betweenness via Random Approximation) is described by Borassi and Natale in [30] (Borassi2016). With KADABRA speed is achieved by changing the way breadth-first search is performed and by taking a different approach to sampling. The use of parallelization also helps to make it very fast. In addition to computing betweenness centrality for every vertex, the algorithm can be configured to compute the set of the k most-central vertices. This will turn out to be an important property that gives the algorithm great speed with applicable applications. Along these same lines, work by Mumtaz and Wang focuses on identifying the top-k influential nodes in networks using an estimation technique which is based on progressive sampling with early stopping conditions [31].
In general, the time complexity of the sampling algorithms (such as described above) is based on the O(|V||E|) time complexity of Brandes' algorithm, but is proportional to the number of samples. The algorithms therefore have a practical time complexity of O(k|E|) where k is the number of samples used. The fact that all time complexities are the same makes the actual performance of the algorithms difficult to judge and says nothing about the accuracy obtained.
A non-sampling strategy for reducing calculation time is to exploit the properties of the networks being considered. In these algorithms, the graph is partitioned in ways that preserve betweenness centralities, but allow them to be computed on smaller graphs. One such algorithm is BADIOS (Bridges, Articulation, Degree-1, and Identical vertices, Ordering, and Side vertices), introduced by Sariyüce et al. [32] and developed further in [33]. BADIOS relies on "manipulating the graph by compressing it and splitting into pieces so that the centrality computation can be handled independently for each piece". Erdös et al. present an exact, divide-and-conquer algorithm in [34]. This algorithm also relies on partitioning, causing all computations to be run "over graphs that are significantly smaller than the original graph" [34]. Baglioni et al. rely on the sparseness of social networks to reduce the size of graphs [35], and work by Li et al. [36] relies on the inherent community structure of real-world networks to do "hierarchical decomposition" of a graph. These strategies speed computation by shrinking the size of the graph that must be computed. Chehreghani et al. [37] have an algorithm that combines both graph reduction and sampling techniques, although it applies only to directed graphs.
As with the sampling algorithms, the time complexity for the network-reducing algorithms is, in the worst case, the same O(|V||E|) as computing the entire graph, and performance must be measured empirically.
The calculation of many elements of betweenness centrality can be run in parallel, and this is a widely attempted strategy. Sariyüce et al. have developed algorithms for use with heterogeneous architectures, by which they mean using multiple CPU cores and a GPU concurrently. Their algorithm is called "gpuBC" [38], and they borrow ideas from BADIOS, as well as introducing new ideas specific to GPU architecture.
In addition to [21, 30, 38], ideas for distributed methods include those by Wang and Tang [39], Shi and Zhang [40], and works by David Bader with Adam McLaughlin [41] and other authors [42, 43]. McLaughlin and Bader introduce Hybrid-BC in [41] (McLaughlin2014). The term hybrid refers to the use of a combination of approaches. The first approach is edge parallel, where each edge is assigned to a different thread. The second approach is called work-efficient, where each vertex is handled by a separate thread in a manner that minimizes redundant or wasted work. At each iteration, either an edge-parallel or work-efficient algorithm is chosen depending on the overall structure of the graph. Other algorithms exploiting extreme parallelism involving multiple GPUs are explored by Bernaschi et al. [44, 45]. In [46], Ostrowski explores using big-data techniques such as map reduce to decompose a graph resulting in faster computations.
Some ideas are radically different and offer unique perspectives on the problem of speeding betweenness centrality calculations. Yoshida's [1] paper on adaptive betweenness centrality is such a work. The term adaptive betweenness centrality is defined in [12] and refers to computing betweenness centrality for a succession of vertices "without considering the shortest paths that have been taken into account by already-chosen vertices" [1]. In effect, adaptive betweenness centrality means that once the vertex with top betweenness centrality is chosen, it is removed from the network along with its adjacent vertices. Subsequent calculations of betweenness centrality are calculated based on this newly configured network. This description matches exactly the method of calculating used in our benchmark problems. Like Yoshida's method, the HEDGE algorithm introduced in [47] works with evolving networks and has provable bounds on the accuracy of the approximation. An additional algorithm with applicability to dynamic networks is described in [48].
One last speed enhancing strategy is to substitute a different measure that is easier to compute, but has results that correlate with betweenness centrality. This has been attempted using such measures as Katz centrality [49], closeness centrality [1], \(\kappa\)-path centrality [50], and routing betweenness centrality [51]. A variation on this strategy is to use local measures to estimate global betweenness centralities [52].
While numerous other proposals may exist for speeding up betweenness centrality calculations, the aforementioned algorithms are a good representative sample. The prominent ideas are summarized as follows:
Compute fewer shortest paths, by extrapolating information from pivots, or otherwise limiting the number of SSSP problems solved [18,19,20,21,22].
Partition the graph in a way that preserves betweenness centralities, but allows them to be computed on a smaller vertex set [33, 34, 38].
Use multiple cores or a GPU to compute shortest paths simultaneously, or otherwise parallelize an algorithm [21, 30, 38, 41].
Use an adaptive approach, which, instead of removing nodes and recalculating in a loop, attempts to calculate directly the succession of top betweenness centrality vertices, taking into account the removal of previous vertices [1].
Identify influential nodes using easier to compute measures that are correlated with betweenness centrality such as coverage centrality [1], or \(\kappa\)-path centrality [50].
The goal of this work is to compare the speed and accuracy of several different approximation algorithms for betweenness centrality in the context of important applications. For each algorithm, we attempt to perform two tests. First, a clustering test is performed on a series of six 10,000-node graphs. Parameters are chosen strategically to balance speed and accuracy. If results are promising, we continue the clustering test with six 100,000-node graphs. Third, we run the immunization test on three popular graphs from SNAP [53]: email-enron, p2p-Gnutella31, and soc-sign-epinions. If results are exceptional, we run the immunization test on a fourth graph, web-Google. In this section we describe the clustering test, the immunization test, the graphs used, and the strategy for choosing parameters.
The scope of the experiment
An important consideration in the design of the experiments is the size of the networks being tested. Because different runs can take different amounts of time, even on the same computer, we wanted to test graphs large enough that the algorithms took at least a few minutes to compute. This way, the magnitude of the duration is being examined more than the exact duration. In initial testing, graphs of approximately 5000 nodes took only a few seconds to run and were considered too small to give an accurate impression of the speedup of the techniques tested. Initial experiments on a graph with 10,000 nodes (Graph 6 described below) took approximately 48 h. Most of the estimation techniques were able to reduce this time to less than an hour. In addition, works such as [17] use graphs of approximately 10,000, 100,000 nodes and 1,000,000 nodes, so it was thought that using these graph sizes would facilitate comparison of results. With the real-life datasets, we followed the same strategy, using the Enron and Gnutella databases with approximately 10,000 nodes, Epinions with approximately 100,000 nodes and Google, which approaches 1,000,000 nodes. The difficulty of the clustering test is great, and even the fastest approximation techniques would take weeks to complete it on million-node graphs. Therefore, only the immunization test was attempted on the largest graph.
The clustering test
The first evaluation of approximation approaches is conducted with the clustering test, which uses a graph clustering methodology called node-based resilience measure clustering (NBR-Clust). A specific variant of NBR-Clust is first discussed in [54], and the full framework is defined in [11]. As implied by the name, node-based resilience clustering takes a resilience measure R(G) as a functional parameter computed on the input graph G. Every node-based resilience measure operates by computing a limited size attack set S of vertices whose removal causes substantial disruption to the graph with respect to the resulting connected component size distribution. NBR-Clust takes the resulting components to be the foundational clusters, and completes the clustering of the input graph G by subsequently adding each attack set node in S to its appropriately chosen adjacent cluster. The description of the NBR-Clust framework follows.
The NBR-Clust clustering framework
Approximate a resilience measure R(G) with acceptable accuracy and return the corresponding attack set S whose removal results in some number (\(\ge 2\)) of candidate clusters. In these experiments we will use the resilience measure integrity [55] as R(G).
Create \(G' = G\setminus\{S\}\), which must consist of two or more candidate clusters.
Adjust the number of clusters. Datasets used in this experiment come with automatically generated ground truth data, including the number of clusters the graph contains. If more clusters have been obtained than that indicated by the ground truth data, clusters are joined based on their adjacencies. If there are not enough clusters, choose the least resilient of the clusters (based on the resilience measure R) and repeat steps 1 and 2 on that cluster. Continue until the correct number of clusters is obtained.
Assign attack set nodes to clusters.
The dependence of NBR-Clust on betweenness centrality calculations, in turn, hinges upon the calculation of the node-based resilience measure R. Many resilience measures exist across a large number of applications. One example is the controllability of a complex network [56, 57], which uses control theory to determine a set of driver nodes that are capable of controlling the network and is applicable to static networks as well as networks with changing topologies [58, 59]. A specific measure is control centrality which quantifies a node's ability to perform as a driver. Such resilience measures can be used as R(G) with the NBR-Clust framework. In [58], it is noted that the control centrality of a node is not determined by its degree or betweenness centrality, but by other factors. We are looking to measure the speed of calculating betweenness centrality, and therefore do not use such measures in our experiments.
Computational aspects of many important node-based resilience measures are considered in [60], including vertex attack tolerance (VAT) [61, 62], integrity [55], tenacity [63, 64], toughness [65], and scattering number [66]. As all of these measures have associated computational hardness results, the performance of heuristics was considered on several representative networks in [60]. Amongst the algorithms considered in [60], a betweenness centrality-based heuristic called Greedy-BC exhibited high-quality performance, particularly on the most relevant measures such as integrity, VAT, and tenacity. Integrity quantifies resilience by measuring the largest connected component after removal of an attack set. Integrity is defined as
$$\begin{aligned} I(G) = \min _{S \subset V} \left\{ |S| + C_{max}(V-S) \right\} , \end{aligned}$$
where S is an attack set and \(C_{max}(V-S)\) is the largest connected component in \(V-S\). Because integrity is a function of the largest connected component size distribution, it shares a similarity with the immunization problem and well-studied methods of maximizing results on both problems use betweenness centrality.
With Greedy-BC, the highest betweenness vertex is calculated and removed from the graph. This process is repeated until all nodes have been removed, and the attack set whose removal results in the smallest resilience measure value is returned. Given a resilience measure R, the resilience measure of a specific graph G denoted by R(G), and the resilience measure of a specific graph G with attack set S removed denoted by R(S, G), the steps of Greedy-BC are as follows:
The Greedy-BC heuristic
\(R_{min} = R(G)\), \(S_{min} = \{\}\)
repeat |V| times
\(v = argmax_{v \in V}BC(V)\)
\(G = G \setminus \{v\}\) and \(S = S \cup \{v\}\)
if \(R(S,G) < R_{min}\) then \(R_{min} = R(S,G)\) and \(S_{min} = S\)
return \(S_{min}.\)
Amongst the various resilience measures considered, integrity was found to be particularly useful with respect to the application to clustering in NBR-Clust due to the higher accuracy and one-step clustering it provides when the number of clusters is not known a priori [11]. Therefore, the resilience measure we use here with NBR-Clust is integrity [55]. Here, I(G) from Eq. 2 will be the resilience measure called for by R(G)
NBR-Clust is a good test case for algorithms that speed up betweenness centrality calculations because it shares many of the properties of practical problems that practitioners face. First, the exact values being calculated are not important. The algorithm does not use the exact value, except to determine the highest ranking vertex. Therefore, it is the ranking of vertices that is important, and small variations of ranking do not change the result. Second, because a large number of calculations are required (conceivably \(O(n^2)\) betweenness centralities must be calculated), scalability is tested. Third, the usefulness of the results can be judged by the accuracy of the resulting clustering, which allows for presentation of results along two axes of speed and accuracy. Because of its similarities to many algorithms, results from NBR-Clust can be generalized to other algorithms that also rely on a large number of betweenness centrality calculations.
One limitation of the clustering test is that the datasets used require a ground truth against which accuracy results can be determined. This limits the types of data that can be used. In this work, we use generated datasets with meaningful ground truth data. We note that, to achieve high accuracy, the clustering test must match a particular predetermined outcome, confirming the difficulty of this test.
The immunization test
The immunization problem is well known and well studied in the complex networks literature, with applications to diffusion of information [67], identification of important nodes [68], and also to the spread of disease and computer viruses [69]. The immunization problem involves attacking a network by removing vertices, presumably in the order most likely to cause damage to the network. The problem is made more difficult by limiting the number of nodes that can be attacked. The success of the attack is measured by the size of the largest remaining component, also called the giant component.Footnote 1 There are several well-known strategies for choosing the attack order for the immunization problem. Four of the strategies detailed in [12] are as follows:
Select vertices in order of decreasing degree in the original graph and remove vertices in that order.
Use an adaptive strategy which involves selecting the highest degree vertex in a graph, removing it, selecting the highest degree vertex in the newly configured graph, and continuing in that fashion.
Select vertices in order of decreasing betweenness centrality in the original graph and remove vertices in that order.
Use an adaptive strategy which involves selecting the highest betweenness centrality vertex in a graph, removing it, selecting the highest degree vertex in the newly configured graph, and continuing in that fashion.
When the immunization test is conducted in this paper, the adaptive betweenness strategy (strategy iv above) is used to remove nodes from the network until either (1) all nodes have been removed, (2) some predetermined number of nodes have been removed, or (3) the betweenness centralities of the remaining nodes are all zero. Obviously, if all nodes are removed the largest connected component will be zero. We are more interested in seeing if removing smaller numbers of nodes can result in small components. Therefore, our results show the maximum component size after a predetermined number of removals, usually 10% of the nodes in the network.
Graphs used with the clustering test
There are many generative models for graphs, such as Erdös–Rényi random graphs, the planted l-partition model [70], the Gaussian random partition generator [71], and the LFR benchmark network [72]. We test using the LFR benchmark because it is considered state of the art [73] and was created specifically to test community detection algorithms. The LFR Net can be generated based on several parameters, such as minimum and maximum cluster size and average and maximum degree and mixing factor. The mixing factor controls the percentage of edges that leave a cluster, also called boundary edges. A higher mixing factor means less tightly bound clusters that are more difficult to detect. One strength of the LFR model is that it accounts for "heterogeneity in the distributions of node degrees and community sizes" [72], and it is often used to generate scale-free graphs. We controlled both the degree and the community sizes as described below to keep most graph properties constant while varying mixing factor, and our graphs are not scale free. The graphs are generated according to the following sequence, which is described more fully in [72, 73].
A sequence of community sizes is extracted. This is done randomly, although it is controlled by parameters specifying minimum and maximum community sizes.
Each vertex i of a community is assigned an internal degree of \((1-\mu )d_i\), where \(d_i\) is the degree of vertex i, and \(\mu\) is the previously described mixing factor. The degree is chosen randomly, although it is subject to parameters controlling its minimum and maximum values. The node is considered to have \((1-\mu )d_i\) stubs, where a stub is a potential edge.
Stubs of vertices of the same community are randomly attached until all stubs are used.
Each vertex i receives \(\mu d_i\) additional stubs that are randomly attached to vertices of different communities, until all stubs are used.
Twelve LFR network graphs are used with the clustering test. Half have 10,000 nodes and half have 100,000 nodes. All graphs are generated randomly, to have approximately 40 communities. The six 10,000-node graphs are generated to have degree distributions from 19 to 64 and community sizes from 100 to 500. Based on extensive experimentation, it was discovered that the mixing factor parameter, denoted \(\mu\), had the greatest influence on the ease of clusterability of a graph. Our graphs have mixing factors \(\mu\) ranging from 0.01 to 0.1. Complete details are given in Table 1.
Table 1 10,000-node randomly generated LFR nets used
The 10,000-node graphs 1 through 6 keep properties such as degree structure and average community size constant while changing \(\mu\). This allows a test based on clustering difficulty without complicating factors such as number of edges (which would require more computation time in edge-based sampling methods). Most of the approximation methods tested use parameters which control the trade-off between speed and accuracy. It is anticipated that succeeding in the clustering test will require much less accuracy for Graph 1 than for Graph 6. Therefore, Graph 1 was used as a baseline accuracy test. It was expected that as accuracy parameters were increased, graphs 2 through 5 would become clusterable, with Graph 6 being the ultimate test. It is noted that Graph 6 is much more difficult to cluster than the first five.
A second series of LFR networks with 100,000 nodes was also generated. Details for these graphs are presented in Table 2. The six graphs can best be viewed as two series of three graphs. Graphs 1–3 are characterized by low average degree and low number of edges. They are in order of increasing difficulty. With approximation methods that are based on the number of edges, these graphs should be substantially faster to cluster than graphs 4–6, which have three times as many edges. This gives an effective test of the scalability of the algorithms. To test the power of mixing factor \(\mu\) over clusterability, graphs 1–3 and 4–6 represent independent progressions of mixing factor. There should be cases where an accuracy parameter setting will fail to cluster Graph 3 but will succeed in clustering Graph 4. However, Graph 4 may take longer because of its larger number of edges. With the largest mixing factor, and a large number of edges, Graph 6 is the most difficult to cluster on all counts. Our goal in setting parameters was to find a combination that would cluster Graph 6 with at least 90% accuracy, although that did not always happen.
Table 2 100,000-node randomly generated LFR nets used
Graphs used with the immunization test
For the immunization test, we used four graphs taken from the SNAP repository [53]. Their information is shown in Table 3.
Table 3 Graphs used in the immunization test
Choosing parameters
The approximation methods use parameters to control the trade-off between speed and accuracy. For each method that was evaluated, we followed a structured process to find an optimal speed/accuracy trade-off. The steps followed are listed below:
We took note of suggested parameters in the source papers, particularly parameters that the authors had used in their own experiments.
We attempted to use parameters discovered in step 1 to cluster the 10,000-node Graph 1, hoping to obtain results of 100% accuracy. If the graph was not clustered to 100% accuracy, we changed the parameters to increase accuracy. This process was repeated until an appropriate setting was found. Often, these parameters successfully clustered graphs 1–6, with varying degrees of accuracy. If possible, we tried to find the fastest setting that gave 100% accuracy with Graph 1.
Having found the minimum accuracy settings, we increased the accuracy parameter until Graph 6 was clustered with at least 90% accuracy (although we could not always achieve this in a reasonable time).
For comparison, we chose a parameter setting between the settings discovered in steps 2 and 3. We ran the clustering test for all six graphs for all three parameter settings.
Moving to the 100,000-node graphs, we tried the same settings used for the 10,000-node graphs. The most common problem was that the maximum accuracy setting for the 10,000-node graphs was very slow when used with the 100,000-node graphs. In that case, we tried slightly faster/less accurate settings. In all cases, we tried to find a setting that would cluster 100,000-node Graph 6 with at least 90% accuracy. If that did not succeed, we tried to find a setting where the clustering for Graph 6 at least did not fail. Preliminary work for this paper appeared in [25], and those timings were used as an indication of what should be considered reasonable.
The goal of the experiments is to reproduce the steps a practitioner with a real problem to solve might take to use one of the approximation algorithms, although under more controlled circumstances. In real life, exact calculations of graphs of tens of thousands of nodes are going to take days to run, at least with current, readily available technology. For example, executing the clustering test with 10,000-node Graph 6 took approximately 48 h using Brandes's exact faster algorithm [13]. We did not have access to a supercomputer and assume many practitioners do not.
All experiments were conducted on two identical computers. Each had an Intel i7-7700K CPU running at 4.20 GHz with four cores and eight logical processors, and 16 GB ram. Computers also had Nvidia GPU cards, which will be described later. We attempted to use many different betweenness centrality approximation methods, compiled from a variety of sources, and code from the original authors was used where available. All implementations of algorithms used are available from documented public sources. All were written in C++ (or CUDA and C++) and compiled to optimize speed.
Results of the clustering test depend to an extent on the difficulty of the graph being clustered. The speed and accuracy under optimal conditions of the top four algorithms for the 10,000-node graphs are compared in Fig. 1. Results for Graph 5 are shown in Fig. 1a, and results for Graph 6 are shown in Fig. 1b. Algorithms offering the best combinations of speed and accuracy will appear in the upper left hand corner of the chart. On the easier-to-cluster Graph 5, McLaughlin2014 is most successful. Borassi2016 offers a slight gain in speed at the cost of some accuracy, and Geisberger2008 gives consistently good accuracy, but takes more than twice as long. For the more difficult \(\mu =0.1\) graphs shown in Fig. 1a, Borassi2016 matches McLaughlin2014 in speed, and in one case beats it in accuracy as well. Two additional runs of the Borassi2016 algorithm (detailed with all runs in Table 18) with speed of 62, accuracy 50% and speed of 50, accuracy 44% did not have high enough accuracy to be shown in Fig. 1b.
Best performing algorithms for the clustering test on 10,000-node graphs. These results represent optimal conditions. McLaughlin2014 is run using an Nvidia Titan V GPU, and Borassi2016 is configured to return five betweenness centralities at one time
The overall clustering test results for 100,000-node graphs are shown in Fig. 2. Results for the sparse-edge Graph 3 are shown in Fig. 2a and for the denser Graph 6 in Fig. 2b. The 100,000-node clustering test is a good indication of the scalability of the algorithms. Note that on the most difficult examples, the Riondato2016 algorithm has dropped off the chart. On the less dense Graph 3, Borassi2016 performs best, both in terms of speed and accuracy. On the denser Graph 6, McLaughlin2014 is the most consistent combination of speed and accuracy, although again some accuracy can be traded for gains in speed with Borassi2016. In both cases, Geisberger2008 is slower but with consistently high accuracy.
Best performing algorithms for the clustering test on 100,000-node graphs. These results represent optimal conditions. McLaughlin2014 is run using an Nvidia Titan V GPU, and Borassi2016 is configured to return five betweenness centralities at one time
Because of the exceptional performance of McLaughlin2014, Borassi2016 and Geisberger2008, we tested each further for scalability using the immunization test on the Google dataset, which has 875,713 nodes and over 5 million edges. Results are shown in Fig. 3. Borassi2016 performs very well here, having the best speed, and also the best speed/accuracy combination. McLaughlin2014 was run on an Nvidia Titan V GPU, and performed well with speed, but had trouble achieving accuracy. Results from the circled runs in Fig. 3a, which are shown in Fig. 3b, show that for quick results Borassi2016 performs best. Borassi2016 removes less than 2% of nodes to achieve a smallest cluster of approximately 1%. Geisberger2008 requires attacking almost twice as many nodes to achieve the same results. McLaughlin2014 never achieves the dramatic drop of the other two, and only achieves a smallest component size of 12% in roughly the same amount of time.
Scalability test for the top performing algorithms on the 875K-node Google graph
Individual algorithm and parameter selection results
Following is a list of tested algorithms. For each algorithm we describe the parameters, show the parameters selected, and, to the extent available, display the results of the clustering and immunization tests. All results include a time component, which is given in seconds. Hopefully, this will aid the practitioner in selecting parameters given graph size, available time, and desired performance. We understand that many algorithms are relevant because they contain important new ideas, and that all algorithms do not need to be fastest. We also understand that sometimes those algorithms and ideas will later be incorporated into new algorithms that will surpass previous algorithms in speed or accuracy. All of the algorithms below have made important contributions to the understanding of the mechanics of calculating betweenness centrality. We test them for speed, although a slower speed does not diminish their importance or usefulness.
Brandes and Pich 2007 (Brandes2007)
This algorithm is taken from [18], and we used code from [17] to run it. The algorithm uses a heuristic: shortest paths are calculated for a group of samples, called pivots, instead of for all nodes in the graph. The paper states that the ultimate result would be that "the results obtained by solving an SSSP from every pivot are representative for solving it from every vertex in V" [18]. They test several different strategies for selecting the pivots, and find that uniform random selection works best. The algorithm tested here uses uniform random selection. The worst-case time complexity of this approach is O(|V||E|). The speedup is directly proportional to the number of pivots, giving a practical time complexity of O(k|E|) where k is the number of pivots. Results are shown in Table 4. For the 10,000-node graphs, 50 samples gave good results on Graph 1 and Graph 2, but accuracies fell for the remaining graphs. Note that 200 samples were required to obtain uniformly high accuracies, and even then did not meet 90% for Graph 6. The 100,000-node graphs got good results with only 25 samples. The 10- and 50-sample results are shown mostly to demonstrate speed. Note that with Graph 3, doubling the number of samples roughly doubles the amount of time required.
It is useful to compare these results to the Bader2007 algorithm (shown in the next section in Table 7). Note that on the 10,000-node graphs, Brandes2007 takes sometimes twice or even three times as long as Bader2007, even though the latter algorithm required a larger number of samples. With the 100,000-node graphs, Brandes2007 was able to cluster Graph 6 with 200 samples, while Bader2007 required 250 samples. Nonetheless, the time for Bader2007 is shorter and the accuracy higher. Both algorithms have the same theoretical time complexity, making the large difference in running times an interesting result.
Numerical results for the immunization test are shown in Table 5. Brandes2007 does very well with 100 and 250 samples. The time for the large Epinions dataset with 100 samples is less than 3 h. Again, it is interesting to compare to Bader2007. Accuracy results for both algorithms are visualized in Fig. 4. Note that concerning accuracy, both algorithms are almost identical. With the Enron test, both algorithms achieve a cluster of less than 1% with removal of about 8.5% of vertices. With the Epinions dataset, both leave a largest cluster of less than 1% with the removal of approximately 7% of vertices. Note that speed-wise Brandes2007 takes roughly twice as long as Bader2007, using the same numbers of samples.
Table 4 Brandes2007 clustering results for LFR nets of 10,000 and 100,000 nodes
Table 5 Brandes2007 run time and max cluster size immunization results
Visualization of results for Brandes2007 versus Bader2007 immunization test
Bader et al. 2007 (Bader2007)
Like Brandes2007, this algorithm estimates centrality by solving the SSSP problem on a sampled subset of vertices. The sampling is referred to as adaptive, because the number of samples required can vary depending on the information obtained from each sample [19]. As with other sampling algorithms, the speedup achieved is directly proportional to the number of shortest path computations. In the worst case, all vertices would be sampled, and so the time complexity of this algorithm is O(|V||E|) in the worst case, and O(k|E|), where k is the number of samples, in the practical case. There are two parameters to this algorithm. The first is a constant c that controls a stopping condition. The second parameter is an arbitrary vertex. For optimum performance, the arbitrary vertex should have high betweenness centrality. The algorithm is of great interest because its benchmark time was the fastest in [17], where the algorithm is referred to as GSIZE. We used code from [17] to run tests of four approximation methods, calculating betweenness centrality values for the Enron dataset. Results from this experiment are shown in Table 6.
Table 6 Initial testing of four algorithms on the Enron dataset
It seems very clear that the fastest method here is Brandes2007, and yet results presented in [17] show that the Bader method is overall ten times faster than the others. Interestingly, by experimenting with the arbitrary vertex parameter, we were able to obtain a range of execution times from 430 to 3 s, depending on the choice of vertex. Obviously, this makes benchmarking the speed of the algorithm difficult.
The benchmarking method in [17] only calculates once on the initial graph. Therefore, it is easy to pick one high-betweenness vertex, perhaps by guessing based on degree. The clustering and immunization tests in this work calculate betweenness centralities repeatedly on a changing graph. There is no way to know what the highest betweenness vertices are at a given time. One easy-to-calculate proxy is vertex degree. Running the algorithm with a parameter of \(c = 2\), and pivoting on the highest degree vertex at each step, our 10,000-node Graph 1 was clustered at 100% accuracy in 8960 s. Compare this to 290 s for Brandes2007. A second attempt was made to speed the algorithm up by, at each iteration, choosing the second highest betweenness centrality vertex from the previous iteration. This also did not speed things up. Bader et al. state in [19] that they set a limit on the number of samples at \(\frac{n}{20}\). We tried this and noticed that with Graph 1 the stopping condition was never reached—500 samples were always used. This configuration clustered perfectly again, but was still not fast at 1112 s. Last, we tried simply changing the limits on the number of samples. With limits of 50, 100 and 250 we got acceptable speeds and accuracies. All tests were run with parameter \(c = 2\), although the parameter probably was not used as the number of samples in fact controlled the stopping point.
Results for the clustering test are shown in Table 7. With a small number of 50 samples, this algorithm clustered the 10,000-node graph 1 in 171 s with 100% accuracy. Clustering the most difficult graph, the 100,000-node Graph 6, to over 90% accuracy took 8.75 h. It is noted on the immunization tests that the 50 sample test was largely ineffective, while the 250 sample test was much more successful, with the largest remaining cluster of only 1.1% with approximately 9% of vertices removed. These results can be seen in Table 8. Accuracy results for the immunization test are compared with Brandes2007 in Fig. 4.
Table 7 Bader2007 clustering results for LFR nets of 10,000 and 100,000 nodes
Table 8 Bader2007 run time and max cluster size immunization results
Geisberger et al. better approximation (Geisberger2008)
This algorithm is developed by Geisberger et al. [21]. It is based on the pivot method originally proposed by Brandes and Pich, where calculations are run on some number k of sampled pivots. The Brandes2007 method is said to overstate the importance of nodes near a pivot, a problem that this method solves by using a linear scaling function to reestimate betweenness centrality values based on the distance a from a node to a pivot. Overall time complexity of the method is O(|V||E|), where the speedup is proportional to the number of pivots, k, that are sampled. What will perhaps turn out to be the true power of this algorithm is realized by the authors' statement that the "algorithms discussed are easy to parallelize with up to k processors" [21]. We used the implementation in NetworKit [74], in the version which parallelizes the computations. The parallelization is programmed using Open MPI. The only parameter is the number of samples, which is chosen by the user. In [21] the number of samples is chosen as powers of 2, ranging from 16 to 8192. We had surprisingly good luck with sample sizes as small as 2. Results for the clustering test are shown in Table 9. This is a quick method to use for clustering with high accuracies on both the 10,000 and 100,000 graphs. Note especially the performance on the most difficult graphs. This is an algorithm that scales to a difficult graph with a large number of edges much better than those we have seen so far. See Figs. 1 and 2 for a comparison of Geisberger2008 performance among the fastest of the algorithms tested.
Results from the immunization test are listed in Table 10. It is interesting to note that, while the improvement from 4 samples to 8 is large for both Enron and Epinions, the improvement from 8 to 16 samples is not as large. Overall, this is a highly accurate algorithm. This is especially evident in Fig. 5.
Table 9 Geisberger2008 clustering results for LFR nets of 10,000 and 100,000 nodes
Table 10 Geisberger2008 run time and max cluster size immunization results
Visualization of results for Geisberger2008 immunization test
Riondato–Kornaropoulos fast approximation through sampling (Riondato2016)
Riondato and Kornaropoulos introduce this method in [20], which is based on sampling a predetermined number of node pairs (chosen uniformly at random). The sample size is based on the vertex diameter of the graph, denoted VD(G), which is the minimum number of nodes in the longest shortest path in G, and may vary from (diameter + 1) if the graph is weighted. Sampled vertices can be chosen to guarantee that the estimated betweenness values for all vertices are within an additive factor \(\epsilon\) from the real values, with a chosen probability at least 1-\(\delta\), where \(0< \delta < 1\).
The time complexity of this algorithm is \(O(r(|V|+|E|))\), where r is determined by the equation:
$$\begin{aligned} r = \frac{c}{\epsilon ^2}\Big ( \lfloor {\text {log}}_2 (\text {VD(G)} - 2) \rfloor + 1 + \text {ln} \frac{1}{\delta } \Big ). \end{aligned}$$
In Eq. 3, c is a universal positive constant that is estimated in [20] to be 0.5, VD(G) is the vertex diameter of the graph, \(\epsilon\) is the additive factor by which the approximation may vary from the true betweenness centrality value, and \(\delta\) is the probability that the estimated value will be within \(\epsilon\) of the true value.
We used the implementation in NetworKit [74]. There are three parameters, the first of which, a constant c, is set at 1.0 in NetworKit. The other parameters are \(\epsilon\), which is the maximum additive error, and \(\delta\), which is the probability that the values are within the error guarantee. Our experiments use \(\delta =0.1\) for a probability of 90%, and vary \(\epsilon\), where higher \(\epsilon\) values mean lower accuracy but also greater speed. Results are shown in Table 11. A couple of things are interesting. First, notice that relatively good accuracies are obtained even at relatively high values of \(\epsilon\). The algorithm displays good speeds on the simpler 100,000-node graphs, taking only 556 s to cluster Graph 1 compared to 2808 s for Geisberger2008. However, Riondato2016 does not scale as well—the most difficult graph takes about three times as long as Geisberger2008 to cluster with at least 90% accuracy.
Results from the immunization test are shown in Table 12 and Fig. 6.
Overall, Riondato2016 is the fourth most successful of our survey, and offers a theoretical guarantee of accuracy. A similar theoretical guarantee is also offered by Borassi2016.
Table 11 Riondato2016 clustering results for LFR nets of 10,000 and 100,000 nodes
Table 12 Riondato2016 run time and max cluster size immunization results
Visualization of results for Riondato2016 Immunization Test
McLaughlin and Bader Hybrid-BC (McLaughlin2014)
Hybrid-BC is introduced by McLaughlin and Bader in [41]. It offers an approximation that uses a sampling of k vertices to determine the structure of the graph. As k is increased, calculations are slower, but accuracy is greater.
McLaughlin2014 is a GPU-based algorithm, and calculations were initially performed and timed on a computer with an Nvidia GeForce GTX 1080 GPU, which has 2560 CUDA cores, 20 streaming multiprocessors, and a base clock of 1607MHz. This GPU is at this writing found in good-quality gaming computers, and is at a price point that most practitioners can afford. It is arguably Nvidia's second or third most powerful GPU. We used code downloaded from Adam McLaughlin's website [41]. Our hardware setup was capable of running 1024 threads simultaneously.
Results for the clustering test are shown in Table 13. Across the board, these are the best results we have seen so far. In fact, McLaughlin2014 is able to cluster the most difficult graph with a high 98% accuracy in approximately 60% of the time of Geisberger2008. It is noted that McLaughlin2014 is one of the fastest and most accurate of all algorithms on the clustering test.
Table 13 McLaughlin2014 clustering results for LFR nets of 10,000 and 100,000 nodes With Nvidia GTX1080 GPU
Results for the immunization test are shown in Table 14. The times are the fastest seen so far, and the the max cluster sizes are small. As can be seen in Fig. 7, 64 samples produced the best result in all cases, but 32 samples also performed well. Notice the large difference in accuracy between those and the test with 16 samples. Also notice that due to somewhat lower accuracy on the Enron test, we have changed the scale of the x-axis. The McLaughlin algorithm is interesting because it scales very well time-wise, but accuracies are not as good. It does better on the clustering test, but is also very successful on the immunization test.
Table 14 McLaughlin2014 run time and max cluster size immunization results
Visualization of results for McLaughlin2014 immunization test
The McLaughlin2014 results are so impressive that we wanted to determine the extent to which the properties of the GPU influenced calculation times. We replaced the Nvidia GeForce GTX 1080 GPU with an Nvidia Titan V GPU, which has 5120 CUDA cores (twice as many as the GTX1080), 80 streaming multiprocessors (four times as many as the GTX1080), and a base clock of 1200 MHz (interestingly, slower than the GTX1080). Results from rerunning the clustering test with this new configuration are shown in Table 15. For the 10,000-node graphs, clustering times are reduced by about 70%. The time to process 100,000-node Graph 6 with k = 64 went from 10917 to 6007 s, a savings of almost 82 min off an already fast 3-h processing time. These are among the best times that we will see.
The performance of McLaughlin2014 on the clustering tests with the Titan V GPU is compared to the other top algorithms in Figs. 1 and 2. A comparison of immunization test results using the large Google graph is shown in Fig. 3. Interestingly, although it is still a top performer, it is sometimes bested by other algorithms in terms of speed and accuracy. At the time of this writing, the price of the Titan V was approximately five times the cost of the GTX 1080.
Table 15 McLaughlin2014 clustering results for LFR nets of 10,000 and 100,000 nodes with Nvidia Titan V GPU
Borassi and Natale KADABRA (Borassi2016)
ADaptive Algorithm for Betweenness via Random Approximation (KADABRA) is described by Borassi and Natale in [30]. These authors achieve speed in two different ways. First, distances are computed using balanced bidirectional breadth-first search, in which a BFS is performed "from each of the two endpoints s and t, in such a way that the two BFSs are likely to explore about the same number of edges, and ... stop as soon as the two BFSs touch each other." This reduces the time complexity of the part of the algorithm that samples shortest paths from O(|E|) to \(O(|E|^{\frac{1}{2}+ o(1)})\). A second improvement is that they take a different approach to sampling, one that "decreases the total number of shortest paths that need to be sampled to compute all betweenness centralities with a given absolute error." The algorithm has a theoretical guarantee of error less than \(\epsilon\) with probability 1 − \(\delta\). For our experiments, we kept \(\delta\) at 0.1, which implied a 90% probability of being within the specified error, and varied \(\epsilon\).
Experiments with Borassi2016 were run using the original code as referenced in [30]. Results are shown in Table 16. Borassi2016 code is parallelized using OpenMP. Note that it is very fast, and that the accuracies are impressive, even at what seem to be high error tolerances. For example, the threshold time for the easiest 10,000-node graph is 60 s with no special hardware, where the same graph is clustered using an expensive GPU in 68 s. Notice that with the 100,000-node graphs, the \(\epsilon =0.05\) results from Borassi2016 are comparable with, and sometimes faster than, the k = 32 results from McLaughlin2014 with the Titan V GPU. The only issue we notice with the Borassi2106 algorithm is that the time seems to be very sensitive to small changes in \(\epsilon\). For example, our attempts to run the 100,000-node graphs at \(\epsilon =0.01\) were going to take hours longer than at \(\epsilon =0.025\). Clustering 100,000-node Graph 6 with \(\epsilon =0.025\) takes over twice as long as with the high-powered GPU, raising questions about scalability; however, it appears that if one is willing to trade a small amount of accuracy, the speed of Borassi2016 rivals that of a GPU algorithm.
Results from the immunization test are shown in Table 17. Notice the extremely small largest components for all three networks when \(\epsilon =0.01\), and the small sizes when \(\epsilon =0.05\). Times are shorter than for any algorithms but McLaughlin2014. Accuracy results for the immunization test are shown in Fig. 8. Notice particularly in Fig. 8b the rapid decrease in size of the largest component with only 6% of nodes removed, and results for \(\epsilon =0.05\) and \(\epsilon =0.01\) are very similar. These results are so impressive that we ran the algorithm on the 875k-node Google graph. It achieved these accuracies: \(\epsilon =0.1\): 53,706 s and 27.9% accuracy, \(\epsilon =0.05\): 58,405 s and 11.7% accuracy, and \(\epsilon =0.025\): 85,277 s and 8.7% accuracy. A comparison of Borassi2016 with the other two algorithms tested on the Google graph is shown in Fig. 3a. Notice that this is arguably the most successful of the three algorithms in terms of speed and accuracy. In Fig. 3b it shows by far the most rapid decline in largest component size, decreasing to less than 1% of graph size with less than 2% of nodes removed.
Table 16 Borassi2016 KADABRA clustering results for LFR nets of 10,000 and 100,000 nodes
Table 17 Borassi2016 run time and max cluster size immunization results
Visualization of results for Borassi2016 immunization test
The Borassi2016 algorithm can be configured to return the top k vertices. In previous experiments we used it to return only the top vertex. Because the exact order of vertices chosen is not of the utmost importance in the clustering test, we wondered if the speed of this algorithm could be improved even further by choosing a set of nodes at each iteration. We tested this on the 10,000-node graphs by using the algorithm to find the top five vertices, while all other factors remained the same. Results can be seen in Table 18. The 10,000-node baseline graph is clustered in 36 s, compared to 68 s for the GPU algorithm. The most difficult graph took 4402 s, which is 27 min shorter than the clustering time with the GPU.
Charts showing the overall performance of Borassi2016 compared to the other three top algorithms are contained in Figs. 1 and 2. Figure 2a is particularly interesting, as Borassi2016 beats all other algorithms with its combination of speed and accuracy.
Table 18 Borassi2016 KADABRA clustering results for LFR nets of 10,000 and 100,000 nodes with top five vertices chosen
Yoshida's adaptive betweenness centrality (Yoshida2014)
Yoshida2014 adaptive betweenness centrality refers to computing betweenness centrality for a succession of vertices "without considering the shortest paths that have been taken into account by already-chosen vertices" [1]. This description matches exactly the Greedy-BC heuristic used with NBR-Clust. Once the array of betweenness centralities is calculated, the node-removal order is known and the resilience measures are calculated for the graph at each configuration. The configuration with the lowest resilience measure value is chosen. Yoshida's adaptive betweenness centrality algorithm has an expected total runtime of \(O((|V| + |E|)k + hk + hk\) log |V|), where h is a probabilistic parameter that in most graphs is much less than \((|V| + |E|)\), and k is the number of times the graph is sampled. The algorithm runs faster or slower depending on the parameter k, which also controls the expected error. It is suggested that k should be chosen to be O(log \(|V| / \epsilon ^2)\), where \(\epsilon\) is the expected error. Our experiments used different values of k to help establish a link between the speed and accuracy of this algorithm.
We used the original code downloaded from Yoshida's website. Results for the clustering test are shown in Table 19. We had success with this algorithm in [25], but here were unable to cluster the 100,000-node Graph 6 with 90% accuracy, even with \(k=800,000\). Also, the corresponding times were comparatively long, and we did not continue with the immunization test.
Table 19 Yoshida2014 clustering results for LFR nets of 10,000 and 100,000 nodes
Sariyüce et al. BADIOS (Sariyüce2012)
BADIOS (Bridges, Articulation, Degree-1, and Identical vertices, Ordering, and Side vertices) is introduced by Sariyüce et al. [32]. BADIOS is not actually an approximation technique, but a speed enhancement. This algorithm involves splitting a graph in ways that will not affect the accuracy of the betweenness centrality calculation. Because the pieces are smaller, scaling problems are reduced. For example, Degree-1 means that all degree-1 vertices have a betweenness centrality of zero. They can be removed from a graph, shrinking its size without affecting the accuracy of the final calculation. It sounds simple, but even after a graph is shattered by removing an articulation node, that node becomes a degree-1 node and can be removed. The different strategies are repeated until all options are exhausted.
In the worst case, manipulations to the graph fail or are few (for example, a graph with many large cliques), which indicates a time complexity of O(|V||E|). In the best case times are greatly enhanced. Our 10,000-node Graph 1 took approximately 5 h with BADIOS. This is a substantial improvement over the original exact calculation time of 48 h; however, the savings in time does not compare to the other approximation algorithms discussed.
Sariyüce et al. gpuBC (Sariyüce2013)
Sariyüce et al. introduced gpuBC in 2013. It contains some ideas from BADIOS, such as removal of degree 1 vertices and graph ordering. It also introduces new ideas: first, vertex virtualization, which replaces high-degree vertices with virtual vertices that have at most a predetermined maximum degree. One of the original problems with GPU parallelization of betweenness centrality is whether to parallelize based on edge or node traversals. By evening out the degree of vertices, this method solves the problem. Second, they use techniques to store the graph that improve the speed of memory access.
The software is available on their website. The software is written for CUDA 4.2.9. We could not get it to run with more recent versions (at this writing, CUDA 9.1 is the most recent version). We were able to run gpuBC for the 10,000-node graphs after installing older versions of Linux and CUDA. Due to memory errors that we could not correct, we were not able to run gpuBC on the 100,000-node graphs.
We used the version of the software with ordering and degree-1 vertex removal techniques enabled, and virtual-vertex based GPU parallelism with strided access. There are two parameters available, one is the max degree of the virtual vertex, which we set at 4, and the number of BFS runs to be executed, which we varied from 100 to 200 to 1000. Results for the clustering test are shown in Table 20, and they are good, although we could not get the Graph 6 to cluster to 90% accuracy.
Table 20 Sariyüce2013 gpuBC clustering results for LFR nets of 10,000 nodes
Kourtellis et al. \(\kappa\)-path centrality (Kourtellis2013)
\(\kappa\)-path centrality is introduced in [50], where it is shown empirically that nodes with high \(\kappa\)-path centrality are highly correlated with nodes with high betweenness centrality. This randomized algorithm runs in time \(O(\kappa ^{3}|V|^{2-2\alpha }\log |V|)\), where graph exploration is limited to a neighborhood of \(\kappa\) hops around each node, and \(\alpha \in [-\frac{1}{2}, \frac{1}{2}]\) is a parameter that controls the tradeoff between speed and accuracy. The algorithm outputs, for each vertex v, an estimate of its \(\kappa\)-path centrality up to additive error of \(\pm |V|^{1/2+ \alpha }\) with probability \(1-1/|V|^2\) [50].
Recommended values for the parameters are \(\alpha = 0.2\) and \(\kappa = {\rm ln}(|V|+|E|)\). For the 10,000-node graph 1, the recommended \(\kappa\) would be 13. Our graphs generally do not have long shortest paths, and it was observed that lowering \(\kappa\) speeded the run time considerably. Our tests used \(\kappa\) = 10, and varied \(\alpha\) from 0.1 to 0.3. Results from the clustering test for 10,000-node graphs are shown in Table 21. Even with the suggested parameters we could not get Graph 1 to cluster to 90% accuracy. Because accuracies are generally low and run times high, we did not test further.
Table 21 Kourtellis2013 \(\kappa\)-path centrality clustering results for LFR nets of 10,000 nodes
Many different factors will affect the choice of an algorithm. Which is better is often in the eye of the beholder. Among the algorithms considered here, the easiest to install are the algorithms contained with NetworKit, which is an ongoing project, supports not only UNIX, but MacOS and Windows 10, and contains both the Riondato2016 and Geisberger2008 algorithms. Notes contained in NetworKit suggest that if you do not need the guaranteed error bounds of Riondato2016, Geisberger2008 is thought to work better in practice. Our results agree with that conclusion, although both algorithms are in the top four for performance in terms of the combinations of speed and accuracy they provide.
Parallelization is a key concept in increasing the speed of betweenness centrality calculations, and many of the algorithms parallelize well. The top three algorithms in terms of speed and performance have parallel implementations, Geisberger2008 and Borassi2016 by using CPU cores, and McLaughlin2014 as a GPU algorithm. Fast results are obtained by using a powerful GPU and an algorithm like McLaughlin2014, which performed even better with the high-performance Nvidia Titan V card. In the original paper [25], we concluded by stating that the results "almost certainly underestimate the potential of GPU algorithms." With the additional results in this paper, we see the complete potential.
The best alternative to the GPU-based algorithm, and perhaps the best all-around algorithm, is Borassi2016 (KADABRA), which offers results that rival and even beat the GPU with certain configurations. In one instance, Borassi2016 clustered a 10,000-node graph in half the time of the GPU algorithm. Borassi2016 was especially useful in its ability to predictably trade accuracy for additional speed. Also, the speed is gained without the added cost and programming skill that running a GPU algorithm requires.
Although some algorithms offer a more desirable combination of speed and accuracy, all algorithms examined here bring new ideas and insights to the search for faster betweenness centrality computations.
We briefly note the plausible similarity of this process with that of computing node-based resilience measures such as integrity and VAT, which also limit the size of both the attack set and the resulting giant component. Therefore, some commonalities of their betweenness centrality-based computations may be expected.
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JM designed and ran the experiments. GE contributed ideas on content and worked on the writing. KS contributed ideas on content. All authors read and approved the final manuscript.
The authors would like to thank Ziyad Al-Ghamdi for genuinely useful help, as well as the anonymous reviewers for the 6th International Conference on Complex Networks and Their Applications for their insightful suggestions.
Southern Illinois University Edwardsville, Edwardsville, IL, USA
John Matta & Gunes Ercal
Southern Illinois University Carbondale, Carbondale, IL, USA
Koushik Sinha
John Matta
Gunes Ercal
Correspondence to John Matta.
Matta, J., Ercal, G. & Sinha, K. Comparing the speed and accuracy of approaches to betweenness centrality approximation. Comput Soc Netw 6, 2 (2019). https://doi.org/10.1186/s40649-019-0062-5
Betweenness centrality
Approximation algorithms
GPU algorithms | CommonCrawl |
Antimicrobial Resistance & Infection Control
What matters: non-pharmaceutical interventions for COVID-19 in Europe
Yan Liu1,
Qiuyan Yu2,
Haoyu Wen1,
Fang Shi1,
Fang Wang3,
Yudi Zhao1,
Qiumian Hong1 &
Chuanhua Yu ORCID: orcid.org/0000-0002-5467-24811
Antimicrobial Resistance & Infection Control volume 11, Article number: 3 (2022) Cite this article
The purpose of this study is to describe the situation of COVID-19 in European countries and to identify important factors related to prevention and control.
We obtained data from World Health Statistics 2020 and the Institute for Health Metrics and Evaluation (IHME). We calculated the Rt values of 51 countries in Europe under different prevention and control measures. We used lasso regression to screen factors associated with morbidity and mortality. For the selected variables, we used quantile regression to analyse the relevant influencing factors in countries with different levels of morbidity or mortality.
The government has a great influence on the change in Rt value through prevention and control measures. The most important factors for personal and group prevention and control are the mobility index, testing, the closure of educational facilities, restrictions on large-scale gatherings, and commercial restrictions. The number of ICU beds and doctors in medical resources are also key factors. Basic sanitation facilities, such as the proportion of safe drinking water, also have an impact on the COVID-19 epidemic.
We described the current status of COVID-19 in European countries. Our findings demonstrated key factors in individual and group prevention measures.
Since 7 December 2020, there have been 66,243,918 cumulative cases and 1,528,984 cumulative deaths worldwide according to data from the World Health Organization (WHO) on coronavirus disease (COVID-19). The WHO declared COVID-19 to be a pandemic on 12 March 2020; a pandemic is a global public health emergency of high international concern. COVID-19 is an acute respiratory infectious disease transmitted mainly through aerosols and droplets [1], and is characterised by higher infectivity but lower mortality than SARS, so we should strengthen its prevention and control [2].
The chief non-pharmacological prevention and control measures included individual and group prevention. Personal prevention primarily included wearing masks, washing one's hands, proper social distancing, and reducing participation in outdoor activities. Group prevention mostly encompassed travel restrictions, home isolation, the closure of educational facilities, prohibitions on public gatherings, all commercial activities, and non-essential commercial activities. The effect of prevention and control may also be related to medical resources (e.g. doctors, nurses, pharmacists, ventilators, ordinary beds, ICU beds), as well as the country's economic situation and ability to respond to public health incidents (e.g. gross domestic product [GDP] and global health security [GHS]).
Switzerland regards testing as the central method to control COVID-19 [3]. A meta-analysis showed that medical masks and N95 masks can protect against viral respiratory infection [4]. Community-wide mask wearing can control COVID-19 by preventing contact with large amounts of saliva and respiratory droplets [5]. Hand hygiene is the most important factor in control activities and infection prevention [6]. The use of antiseptic hand soaps leads to a greater reduction in the number of microorganisms compared with regular soaps [7]. However, the toxicity and stability of surface disinfectants are issues that should be further investigated [8]. The findings of a systematic review and meta-analysis support the idea that physical distancing of 1 m, wearing face masks, and eye protection prevent the person-to-person transmission of COVID-19 [9].
The first country in Europe where the COVID-19 outbreak occurred was Italy. On 9 March 2020, the Italian government implemented a comprehensive 'blockade' policy, mostly consisting of travel restrictions, mandatory stay-at-home orders (except for health problems, emergencies, or regulated shopping for necessities), and the temporary closure of non-essential businesses and shops, which lasted until 3 May 2020. In the following months, many European states adopted similar measures [10]. In March 2020, 47 countries did not allow commercial flights to land [11]. Moreover, it became necessary to develop an international framework to outline the method, time, and scale of travel restrictions according to the stage of the epidemic [12]. A cross-sectional study suggests that issuing a stay-at-home ban may help limit the spread of COVID-19 cases [13]. On 12 March, Norway issued stricter measures and instituted quarantines for those who entered the country. That same day, the government closed all schools and kindergartens; training centres and offices for psychologists and physiotherapists; hair salons; and swimming pools, and forbade cultural and sporting events. Sweden chose a different strategy: Kindergartens, elementary schools, other businesses, and training facilities remained open, and children's sports continued [14]. Many sporting events were restricted or cancelled to limit the spread of disease [15]. Some non-essential businesses, such as vape shops, were closed [16].
The disease incidence and mortality of COVID-19 were related to country healthcare resources and economic status [17]. In Italy, 16% of hospitalised COVID-19 patients required intensive care. Compared with Germany, the medical resources in northern Italy became overwhelmed by the increase in patients. In contrast, Germany had a wider distribution of cases and was able to make better use of its resources. The high mortality rate in Italy may reflect the relationship between the availability of medical resources and outcomes [18]. In addition, the prevention and control effects of COVID-19 may be tied to indicators such as GDP and GHS [19].
Research objective and data sources
We obtained data from World Health Statistics 2020 and the Institute for Health Metrics and Evaluation (IHME). The IHME is an independent population health research centre at the University of Washington Medicine. We selected 51 countries in Europe as our research objective. There are 53 countries in Europe according to WHO regional groupings. The data from Monaco and Turkmenistan were incomplete, so we discarded those two countries.
Where does the IHME obtain its data? These forecasts include data from local and national governments, hospital networks and associations, the WHO, third-party aggregators, and a range of other sources.
For testing data, the IHME relies primarily on data reported by Our World in Data. However, Cyprus, Italy, and Spain used government data. The IHME obtains hospital resource data from sources such as government websites, hospital associations, the Organisation for Economic Co-operation and Development (OECD), the WHO, and published studies. For population density, they used gridded population count estimates for 2020 at the 1 × 1 km (km) level from World-Pop. For mobility index data, the IHME used anonymised, aggregated data from Google, Facebook, and Apple. Their data on mask use come from the Premise, Facebook Global Symptom Survey (research based on survey results from the University of Maryland Social Data Science Center), the Kaiser Family Foundation (KFF), and the YouGov COVID-19 Behaviour Tracker survey.
Statistical analysis method
Changes in Rt under different prevention and control measures
For this study, we used the time-dependent basic reproduction number method. The time-dependent reproduction number (Rt) is the average number of secondary cases of a single infected person during t day of infection. Rt is usually applied to describe the transmission characteristics of pathogens during an epidemic, and can also evaluate the effectiveness of interventions. When Rt is greater than 1, it indicates that the number of infections is rising rapidly; when Rt is less than 1 and close to 0, it suggests that the epidemic has been effectively controlled [20].
Lasso regression
Traditional linear regression belongs to the subset selection method, and the chief approach for screening variables is the ordinary least squares (OLS) technique. However, OLS has some defects: Variable selection is separated from model parameter estimation, so the model error increases; small changes in variables have a great influence on variable selection. Variable selection is not suitable for high-dimensional data [21, 22]. The least absolute shrinkage and selection operator regression (lasso regression) is the representative regularisation method that can effectively optimise the OLS estimation and treatment overfitting problem [23]. By introducing a penalty term into model estimation, lasso regression can obtain higher prediction accuracy and model generalisation ability. It can also effectively address overfitting and multicollinearity problems [24,25,26]. The specific formula can be expressed as:
$$\hat{\beta }\left( {lasso} \right) = \arg \mathop {\min }\limits_{\beta } ||y - \sum\limits_{j = 1}^{p} {X_{j} \beta_{j} ||^{2} } + \lambda \sum\limits_{j = 1}^{p} {|\beta_{j} } |$$
The first part represents the standard OLS loss function, while the second part denotes the penalty function. It represents the tuning parameter for controlling the degree of regression coefficient compression. When λ is ≥ 0, the greater the value, the stronger the penalty. When λ = 0, the loss function does not penalise the model. Lasso regression shrinks some coefficients and sets others to 0, and tries to retain the good features of both subset selection and ridge regression [27]. Researchers adopt different forms of penalty functions according to the different characteristics of independent variables in regression analysis on the basis of lasso regression. Many kinds of regularisation models have been established and developed, such as relaxed lasso [28], adaptive lasso [26], Bayesian lasso [29], fused lasso [24], group lasso [30], and elastic net [25].
Quantile regression
Quantile regression (QR) estimates the linear relationship between the different quantiles of the dependent variable and the independent variable. Both QR and ordinary least squares (OLS) can solve specific minimisation problems. The estimation of OLS regression is grounded in the smallest residual square, and the estimation of QR is rooted in the smallest weighted absolute value residual. The minimum weighted absolute deviation of quantile regression is as follows:
$$\min \left\{ {w_{t} |y_{t} - \alpha |} \right\} = - \sum\limits_{{i:y_{i} < \alpha }}^{T} {\left( {1 - \tau } \right)} \left( {y_{t} - \alpha } \right) + \sum\limits_{{i:y_{i} \ge \alpha }}^{T} \tau \left( {y_{t} - \alpha } \right)$$
The purpose of quantile regression is to calculate the regression coefficients of different quantile values of the dependent variable. It can comprehensively display all data information to a certain extent. Thus, it has unique advantages over traditional linear regression models. Especially for the condition of a non-normal distribution, quantile regression is more comprehensive and accurate than traditional linear regression coefficient estimation [31]. We used R software 4.0.2 and Stata 15.0 for all of our statistical analyses.
We plotted the changes in Rt values of 51 countries in Europe. After an outbreak period ranging from 2–3 months, the Rt of France, Denmark, Belgium, Armenia, Germany, the Netherlands, Spain, Portugal, Moldova, Sweden, Tajikistan, Turkey, and Uzbekistan were basically stable, and the value fluctuated at approximately 1. Russia and Ukraine had a longer outbreak period, but the Rt afterwards also stabilised at approximately 1. The Rt of other countries fluctuated to varying degrees. The general trend of changes in Rt in most European countries showed an increasing trend after October and exceeded 1.
Let us take Finland and Switzerland as examples. In Fig. 1a, we describe the changes in the Rt value in Finland under different prevention and control measures. The first case of COVID-19 occurred on 28 January in Finland, and on 18 March, any business activities were banned and educational facilities were closed. Travel restrictions were implemented on 25 March, and unnecessary business activities were prohibited on 4 April. Prevention and control effects were achieved, and the Rt value dropped below 1 under the above four powerful group interventions. On 29 May, the government ended travel restrictions. On 1 June, the government ended the ban on restricting any business activities and non-essential commercial activities. Due to the relaxation of the intervention policy, the Rt value rebounded and rose. On 24 July, the Rt value reached a small peak of 3.296. On 13 August, when the Rt value dropped to 1.398, the restriction on school facility closure ended. Then, on 11 October, the Rt value fluctuated repeatedly, and on 11 October, it reached 1.966.
a Changes of Rt in Finland under different prevention and control measures b Changes of Rt in Switzerland under different prevention and control measures
In Fig. 1b, we describe the changes in Rt values in Switzerland under different prevention and control measures. Cases first appeared in Switzerland on 23 February, and large gatherings were banned on 28 February. On 13 March, education facilities were closed. On 16 March, any commercial activities and non-essential commercial activities were prohibited. Under the above group intervention, a certain prevention and control effect was achieved, and the Rt value dropped below 1. Thus, on 27 April, the government ended the restrictions on non-essential commercial activities. On 6 June, the government ended the ban on any commercial activities. On 8 June, the government decided to end the closure of school education facilities. Due to the relaxation of the intervention policy, the Rt value rebounded and rose. On 2 July, the Rt value exhibited a small peak of 2.88. Subsequently, the Rt value dropped to approximately 1. On 16 October, there was a small peak of 2.51. The graphs of Rt changes in other countries can be seen in the supplemental materials.
Results of lasso regression
In this study, we explored the relationship between two dependent variables and 17 independent variables, including cumulative morbidity (Y1) and cumulative mortality (Y2). The independent variables were the density of medical doctors/per 10,000 people (X1), the density of medical nursing and midwifery personnel/per 10,000 people (X2), the density of medical pharmacists/per 10,000 people (X3), GDP per capita (US dollars per capita) (X4), the proportion of the population using safely managed drinking-water services (X5), total tests (X6), the mask use rate (X7), the mobility composite (X8), excess bed capacity for COVID-19 (X9), the ICU excess bed capacity for COVID-19 (X10), travel restrictions (X11), stay-at-home orders (X12), the closure of educational facilities (X13), restrictions on gathering (X14), business closures (X15), non-essential businesses being ordered to close (X16), and the GHS index (X17).
We finally screened five out of 17 variables related to cumulative morbidity (Y1) and seven variables related to cumulative mortality (Y2) after lasso regression selection. The specific content is shown in Fig. 2. We can see that when the average mean-squared error was the smallest, lasso regression screened out five variables in Fig. 2a; at this time, λ = 0.09559. Among them, the selected variables were X1, X5, X8, X14, and X16. When the average mean-squared error was the smallest, lasso regression screened out seven variables in Fig. 2b; at this time, λ = 0.0822. Among them, the selected variables were X5, X6, X8, X10, X13, X14, and X16. The selected indicators were closely related to cumulative morbidity and mortality.
a The process of selecting variables in Lasso regression and b the process of selection variables in Lasso regression
Results of quantile regression
The results of quantile regression were different from the results of lasso regression, and quantile regression provided more comprehensive information. The specific content is presented in Table 1. The overall finding is that the mobility index, the ratio of safe drinking water, and the closure of non-essential businesses are related to the cumulative incidence. The closure of educational facilities, restrictions on gathering, and the closure of non-essential businesses are tied to cumulative mortality. In low- and medium-incidence countries, the mobility composite is connected with the cumulative incidence. In high-incidence countries, the closure of educational facilities and restrictions on gathering are linked to cumulative mortality.
Table 1 Results of the quantile regression
We found that the population prevention and control measures implemented by the government had an impact on the change in the Rt value. In most countries, the Rt value had a clear upwards trend in October. The most important factor in personal prevention and control is the mobility composite. Group prevention of the total testing, the closure of educational facilities, restrictions on large-scale gatherings, and commercial restrictions are very important for prevention and control. Moreover, the number of ICU beds and the average number of doctors in medical resources are key elements. Basic sanitation, such as the proportion of safe drinking water, has also had a certain impact on the COVID-19 epidemic.
The rate of masks used in individual prevention does not seem to be related to cumulative mortality or morbidity, which does not mean that the use of masks has no effect on prevention and control. Research on the effectiveness of masks for prevention and control has been confirmed [4]. The lasso regression identifies variables with a very large degree of correlation, so the mobility index generated by personal behaviour in prevention and control may be more important than wearing a mask. Second, one possible reason could be that the data on the rate of mask use come from social surveys, and there may be large errors.
Some studies have reached conclusions consistent with ours. The outcomes of the dynamic SEIR model show that the lockdown control measures implemented by China on 23 January 2020 were essential to ultimately reducing the scale of the COVID-19 epidemic [32]. Another study found that the early detection and isolation of cases prevented more infections than restrictions on travel and reduced contact. Our research also reveals a certain relationship between cumulative mortality and testing [33]. Some scholars have even proposed that the best strategy is to use both robot recognition and migration restriction strategies. European countries can also take this approach to reduce exposure to infection and provide help for the prevention and control of diseases [34]. Once the initial pandemic is under control, we must turn our attention to how to improve the adverse effects of the lockdown [35].
Studies have also indicated that medical resources are related to the mortality rate of COVID-19, which proves our research conclusions. Our study demonstrates that the number of doctors per capita and the number of hospital beds per capita are linked to the incidence or death of COVID-19 [36]. The experience in Wuhan implies that if medical resources become scarce, the government should establish temporary hospitals, and medical staff will be deployed from areas where the epidemic is relatively mild to ease the pressure in severely affected zones. Effective quarantine via quick detection prevents a larger outbreak [37, 38]. It is necessary to establish medically necessary, time-sensitive procedure scoring systems during the COVID-19 pandemic [39]. Within days or weeks, the health system is reorganised. We must optimise health resources. The fight against the disease occurs via a joint medical team composed of doctors, nurses, pharmacists, and respiratory therapists [40].
The ratio of safe drinking water in basic health resources has a relationship with the cumulative morbidity and mortality of COVID-19, which may be related to the fact that hand washing can reduce the number of hand viruses and achieve a certain prevention and control effect. Basic cleaning services are a prerequisite for infection prevention and control [41].
We did not include certain indicators—such as nurses per capita, pharmacists per capita, GDP, and GHS—in the regression model, which may be due to the relatively low degree of correlation. After the Ebola outbreak in 2014, the GHS index was developed to measure countries' ability to respond to infectious disease outbreaks. Six core elements were evaluated: prevention, detection and reporting, response, the health system, compliance with norms, and the risk of infectious disease outbreaks. The higher the GHS score, the better the preparation.
The GHS index has low predictive value for the death outcomes of COVID-19, and we have reached the same conclusion [42, 43]. For example, the UK, which ranks second in terms of GHS index score, also bears a huge burden of disease [44]. However, the GHS index has a predictive effect on the burden of COVID-19, but in the opposite direction [45].
The outcomes of quantile regression indicated that low- and medium-incidence countries should pay more attention to personal prevention (mobility composite), and that high-incidence countries should close educational facilities and impose restrictions on gatherings.
Our research has some advantages. First, we described the prevalence and control of COVID-19 in European countries. Second, we included many independent variables to analyse their relationship with dependent variables. The independent variables mostly include individual and group prevention, medical resources, basic health facilities, and comprehensive indicators. Third, we used lasso regression to screen variables with a smaller error than traditional regression, and the results are more accurate. We further performed quantile regression to quantify the specific situation of each divided point, thus providing more information than traditional regression. Of course, our article also has some shortcomings. For example, Asmall number of proven effective prevention and control measures did not enter into our regression model, which may result from the accuracy of the data and the impact of the variables.
We comprehensively described the status of COVID-19 prevention and control in European countries. We attempted to identify key factors in individual and group prevention measures, which can provide a policy basis for the prevention and control of epidemics in European countries.
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Thanks to IHME and WHO for providing open data resources.
This research was funded by the National Natural Science Foundation of China (Grant No. 81773552), the National Key Research and Development Program of China (Grant No. 2018YFC1315302) and National College Student Innovation Training Project (Grant No. 202110486088).
Department of Epidemiology and Biostatistics, School of Public Health, Wuhan University, Hubei, China
Yan Liu, Haoyu Wen, Fang Shi, Yudi Zhao, Qiumian Hong & Chuanhua Yu
Department of Epidemiology and Medicine Statistics,Public Health and Management School, Wenzhou Medical University, Zhejiang, China
Qiuyan Yu
School of Public Health, Xuzhou Medical University, Jiangsu, China
Fang Wang
Haoyu Wen
Fang Shi
Yudi Zhao
Qiumian Hong
Chuanhua Yu
YL acquired the data, performed the analysis of data, and wrote the manuscript. QY and HW acquired the data and contributed to the analysis of data. FS, FW,YZ and QH contributed to the coding of the statistical analysis. CY designed and evaluated the whole work. All authors read and approved the final manuscript.
Correspondence to Chuanhua Yu.
Liu, Y., Yu, Q., Wen, H. et al. What matters: non-pharmaceutical interventions for COVID-19 in Europe. Antimicrob Resist Infect Control 11, 3 (2022). https://doi.org/10.1186/s13756-021-01039-x
DOI: https://doi.org/10.1186/s13756-021-01039-x
Non-pharmaceutical interventions | CommonCrawl |
2. Decimals and Percentages
3. Algebraic techniques
4. Angle Relationships
6. Area
7. Data collection & representation
8. Pythagoras' theorem
8. Pythagoras' theorem1111
Part 5: Length
In this article, we will go through converting units over the same system and how to calculate perimeters of shapes.
Length is an essential foundation that is need in other numerous Maths topics, like area. So, it is crucial that you know and understand it. Read on to find out more about Year 7 length.
Stage 4 NESA Syllabus
Syllabus Explanation
Find perimeters of parallelograms, trapeziums, rhombuses and kites (ACMMG196) You are able to calculate the total length of a shape, given any side.
You are also able to convert units to figure out the perimeter.
In this section, students are exposed to questions involving the usage of length.
In this guide, we will be converting different length measurements across the metric system.
E.g. metres to kilometres, calculating the perimeter of 2-dimensional shapes and also examples of some real-life application and worded questions involving length.
Length is the measurement or extent of something from end to end.
This is quite a simple concept. So, you would have already experienced basic length equations already in general conversation.
For example, these all require knowledge of length:
Parents talking about the mileage on their car
Height of your growth over the past year
Determining the distance from home to school all require knowledge of length
Assumed knowledge
You should already know addition, subtraction, multiplication, division and basic indices.
However, length may sometimes be paired with other mathematical concepts to make a question harder. In these cases, knowledge of the other concepts are also required to answer the question.
For example, if you had to calculate the perimeter of a square, and you were only given the length of one side… you need to know that all sides of a square are equal.
Converting units over the same system
Quick indices recap:
Note that prefixes may apply to a variety of mathematical concepts, not just length.
Eg. A millilitre is one thousandth of a litre, just as a millimetre is a thousandth of a metre.
This standardised form makes the metric system arguably superior to the older imperial system used in the US.
\begin{align*}
10^{3} &= 10 \times 10 \times 10 = 1000 \\
10^{-2} &= 10 = \frac{1}{10^{2}} = \frac{1}{10 \times 10} = \frac{1}{100}
\end{align*}
Converting units
There are many different units for measuring length. Centimetre, millimetre and kilometre are just some of them.
In general use, depending on the circumstance, one particular unit may be favoured over another.
Eg. Longer distances such as the distance from Sydney to Melbourne, would be calculated in kilometres, while a small distance like the length of a caterpillar may well be calculated in millimetres.
For a measurement like height, a mixture of both metres and centimetres may be used.
Eg. You might say "Sean is 1 metre 54″. This implies that his height is 1 metre and 54 centimetres.
Thus, it is important to know the conversions of different units over the metric system.
Lucky for us, the system for length which Australia uses is the metric system.
It is easily convertible between units as it is all in powers of 10.
Therefore, converting only requires a knowledge of some mathematical prefixes.
Some common prefixes include:
Milli – one thousandth or \( \frac{1}{1000} = 10^{-3} \)
Centi – one hundredth or \( \frac{1}{100} = 10^{-2} \)
Kilo – one thousand or \( 1000 = 10^{3} \)
Mega – one million or \( 1 000 000 = 10^{6} \)
A conversion flowchart is a helpful way to see the relationship between units of different prefixes.
To convert from mm to cm you would follow the arrow from mm to cm and see that all you have to do is divide by \(10\).
Note: To do a conversion in the opposite direction, all you have to do is use the inverse operation. That is, changing multiplication to division, or division to multiplication.
To convert metres to kilometres, we follow the arrow and divide by \(10^{3}\).
Conversely to change from kilometres to metres, we multiply by \(10^{3}\).
Thinking about it logically.
To go from metres to kilometres to metres, is just metres to metres… which obviously is nothing.
So, we'd expect to cancel out whatever operations we perform.
Eg. \( \times 10^{3} \div 10^{3} = \times \frac{10^{3}}{10^{3}} = \times 1\)
Multiplying anything by \(1\) does nothing to the value.
1. Arrange the following lengths in ascending order:
\color{blue}{320 000mm, 3km, 3289cm, 396.8m}
To start, we convert everything into the same unit.
For this particular example, we will be picking the metre as our choice.
This gives:
\color{blue}{320 000 \div 10^{3}, 3 \times 10^{3}, 3289 \div 10^{2}, 396.8} \\
\color{blue}{320m, 3000m, 32.89m, 396.8m}
Now we rearrange to ascending order:
\color{blue}{32.89m, 320m, 396.8m, 3000m}
To finish off, we must convert the units back to their original form, but keep desired ordering
\color{blue}{3289cm, 320 000mm, 396.8m, 3km}
These lengths are now arranged in ascending order as requested
The perimeter is the added length of a continuous line forming the boundary of a closed geometrical figure.
What this means simply, is that perimeter is the summed length of all the sides of a particular shape.
Let's look at a simple example,
Note: Some harder questions may rely on your knowledge of geometrical figures.
For example, a square and rhombus, both consist of 4 equal sides. For triangles, equilateral triangles have 3 equal sides, isosceles have 2, while scalene have none.
A dash or double dash on a geometrical diagram indicates that a length is equal to another.
For example, take this quadrilateral:
The single dash indicates that the top and bottom side are equal.
The double dashes say that the left and right sides are equal.
If we are given that the corner angles are all right angles, we can also positively confirm that this shape is a rectangle.
Rectangles have equal opposite sides and all their interior angles are right angles.
Other questions may require use of basic algebra.
In these cases, always create a pronumeral and see what you can solve from there.
1. The side lengths of an equilateral triangle are \(25cm\). Find its perimeter.
Equilateral means all three sides are equal.
P = 25 \times 3 \\
P = 75cm
2. One side of a rectangle is \(14cm\) longer than the other side. If the perimeter of said rectangle is \(56cm\), what are the lengths of the sides?
Let us first set the shorter side to pronumeral \(x\), the longer side is therefore, \(14 + x\)
56 &= x + x + (14+x) + (14+x) \\
56 &= 2x + 28 + 2x \\
56 &= 4x + 28 \\
28 &= 4x \\
x &= 7
Thus, we can see that the shorter side is \(7\), and the longer side being \(14 + 7\) must now be \(21\). Therefore, the lengths of the sides are \(7cm\) and \(21cm\).
3. A square paddock has a side length of \(135m\). If fencing costs \($1250\) per kilometre, what is the cost of fencing the paddock.
What we do first, is put everything in the same units.
That is, we will change $1250 per kilometre, into dollars per metre.
From here, we will then calculate the total perimeter of the paddock. Then simply multiply to the cost per metre with the required metres we need to fence.
$1250/kilometre \ & = 1250/1000metres \\
&= $1.25 / metre \\
P &= 4 \times 135 \\
P &= 540m
Cost will be \(540 \times 1.25 = $675\)
Therefore, the cost of fencing the paddock is \($675\), quite cheap considering its size!
To succeed with length-based questions, you should practice:
Converting across the metric system
Various methods for calculating the perimeter of geometrical shapes.
Harder questions will require adequate knowledge of:
Other mathematical concepts
Knowing length
This will form a strong base for the further high school mathematics you will encounter within the coming years.
Understanding length is also essential before attempting to learn area, another core topic of Year 7 Mathematics.
© Matrix Education and www.matrix.edu.au, 2020. Unauthorised use and/or duplication of this material without express and written permission from this site's author and/or owner is strictly prohibited. Excerpts and links may be used, provided that full and clear credit is given to Matrix Education and www.matrix.edu.au with appropriate and specific direction to the original content.
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\begin{document}
\title{Nonlinear diffusions: extremal properties of Barenblatt profiles, best matching and delays} \titlerunning{Extremal properties of Barenblatt profiles and delays} \author{Jean Dolbeault \and Giuseppe Toscani} \institute{J. Dolbeault \at Ceremade (UMR CNRS no. 7534), Universit\'e Paris-Dauphine, Place de Lattre de Tassigny, F-75775 Paris C\'edex 16, France\\ \email{[email protected]} \and G. Toscani \at University of Pavia Department of Mathematics, Via Ferrata~1, 27100 Pavia, Italy \\ \email{[email protected]}} \date{January 9, 2015} \maketitle
\begin{abstract} In this paper, we consider functionals based on moments and nonlinear entropies which have a linear growth in time in case of source-type solutions to the fast diffusion or porous medium equations, that are also known as Barenblatt solutions. As functions of time, these functionals have convexity properties for generic solutions, so that their asymptotic slopes are extremal for Barenblatt profiles. The method relies on scaling properties of the evolution equations and provides a simple and direct proof of sharp Gagliardo-Nirenberg-Sobolev inequalities in scale invariant form. The method also gives refined estimates of the growth of the second moment and, as a consequence, establishes the monotonicity of the delay corresponding to the best matching Barenblatt solution compared to the Barenblatt solution with same initial second moment. Here the notion of best matching is defined in terms of a relative entropy.
\keywords{Nonlinear diffusion equations \and Source-type solutions \and Gagliardo-Nirenberg-Sobolev inequalities \and Improved inequalities \and Scalings \and Second moment \and Temperature \and R\'enyi entropy \and Best matching Barenblatt profiles \and Delay}
\def{\bfseries Mathematics Subject Classification (2010)}\enspace{{\bfseries Mathematics Subject Classification (2010)}\enspace} \subclass{primary: 35K55 \and 35K65 \and 35B40; secondary: 46E35 \and 39B62 \and 49J40} \end{abstract}
\section{Introduction}\label{Sec:Intro}
Consider the nonlinear diffusion equation in ${\mathbb R}^d$, $d\ge1$, \be{poro} \frac{\partial u}{\partial t}=\Delta u^p\,, \end{equation} with initial datum $u(x,t=0)=u_0(x)\ge0$ such that \be{second}
\ird{u_0}=1\,,\quad\Theta(0)=\frac1d\ird{|x|^2\,u_0}<+\infty\,. \end{equation} It is known since the work of A.~Friedman and S.~Kamin,~\cite{MR586735}, that the large time behavior of the solutions to~\eqref{poro} is governed by the source-type Barenblatt solutions \be{ba-self} \mathcal U_\star(t,x):=\frac1{\big(\kappa\,t^{1/\mu}\big)^d}\,\mathcal B_\star\!\(\frac x{\kappa\, t^{1/\mu}}\) \end{equation} where \[
\mu:=2+d\,(p-1)\,,\quad\kappa:=\Big|\frac{2\,\mu\,p}{p-1}\Big|^{1/\mu} \] and $\mathcal B_\star$ is the Barenblatt profile \[ \mathcal B_\star(x):= \begin{cases}
\big(C_\star-|x|^2\big)_+^{1/(p-1)}&\text{if}\;p>1\,,\\[5pt]
\big(C_\star+|x|^2\big)^{1/(p-1)}&\text{if}\;p<1\,. \end{cases} \label{ba} \] Here $(s)_+=\max\{s,0\}$ and, for $p>1-\frac2d$, the constant $C_\star$ is chosen so that $\ird{\mathcal B_\star(x)}=1$. Solution~\eqref{ba-self} was found around 1950 by Ya.B.~Zel'dovich and A.S.~Kompaneets. Later G.I.~Barenblatt analyzed the solution representing heat release from a point source for $p>1$. See~\cite{zel1950towards,barenblatt1952some}, and~\cite{MR2286292} for more~details.
The result of~\cite{MR586735}, subsequently improved in a number of papers like, \emph{e.g.},~\cite{MR2909910}, guarantees to various extents that the Barenblatt solution can fruitfully be used to obtain the large-time behavior of all other solutions, departing from an initial value like in~\eqref{second}. Precise estimates of the difference in time of basic quantities of the real solution with respect to the source-type one are however difficult to obtain and additional conditions are needed for low values of $p>0$.
As main example, let us consider the growth of the second moments of the solution to~\eqref{poro}, for an initial value satisfying~\eqref{second}. For a Barenblatt solution, we easily obtain the exact growth \[
\frac 1d\ird{|x|^2\,\mathcal U_\star(t,x)}=t^{2/\mu}\,\Theta_\star\quad\mbox{where}\quad\Theta_\star:=\frac{\kappa^2}d\ird{|x|^2\,\mathcal B_\star(x)}\,. \] For $p>1$ it has been shown in~\cite{MR2133441} that this time-behaviour is captured by any other solution to~\eqref{poro} satisfying conditions~\eqref{second} for large times. Further studies published in~\cite{2014arXiv1408.6781D} then revealed that, at least in the fast diffusion range, there is a nondecreasing \emph{delay} between the propagation in time of the second moments of the source-type solution and the generic solution to \eqref{poro}. We shall give a simple explanation for this fact in Theorem~\ref{Thm:T3}, and also show that in the porous medium case $p>1$ the delay is nonincreasing.
A way to look at this phenomenon is the following. The second moment of the Barenblatt solution, raised to power $\mu/2$, grows linearly in time. Hence, its first variation in time of $\Theta_\star(t)^{\mu/2}$ is constant, while the second variation is equal to zero. To see how the second moment of any other solution to equation~\eqref{poro} behaves in time, it comes natural to estimate the time variations of $\mathsf G:=\Theta^{\mu/2}$, where the \emph{second moment functional} is defined by \[
\Theta(t):=\frac 1d\ird{|x|^2\,u(t,x)}\,. \] This analysis will lead to the interesting observation that the second variation of $\mathsf G$ has a fixed sign, which implies concavity or convexity in $t$ depending on the value of $p$. Also, since the first variation is invariant with respect to dilations and the solution to~\eqref{poro} is asymptotically self-similar, it follows at once that the first variation satisfies a sharp inequality which connects the second moment to its first variation, and optimality is achieved by Barenblatt profiles.
This idea can also be applied to the study of the behavior of the derivative of the second moment, as predicted by the nonlinear diffusion. It holds \[\label{fi-va} \Theta'=2\,\mathsf E \] where, up to a sign, the \emph{generalized entropy functional} is defined by \[
\mathsf E(t):=\ird{|u(t,x)|^p}\,. \] For the Barenblatt solution~\eqref{ba-self} one can easily check that $\mathsf E_\star(t)^\sigma$ grows linearly in time if $d\,(1-p)\,\sigma=\mu$. This leads to estimate the time variation of $\mathsf F:=\mathsf E^\sigma$. As in the previous case, we will conclude by showing that the second variation in time of $\mathsf F$ has a negative sign if $p<1$, which implies concavity. In this case also, the first variation is invariant with respect to dilations, which implies a sharp inequality of Gagliardo-Nirenberg-Sobolev type, and optimality is again achieved by Barenblatt profiles.
As the results of this paper clearly indicate, looking at functionals of the solution to the nonlinear diffusion\eqref{poro} which grow linearly in time when evaluated along the source-type solution represent a valid and powerful alternative to the well-known \emph{entropy-entropy production method} developed in the last two decades to investigate the large-time behavior of the solution by scaling the problem in order to obtain a Fokker-Planck equation with a fixed steady state.
This new way of looking at the problem opens completely new questions, which at the moment seem to be very difficult to deal with. For instance, higher order derivatives in time of the second moment of the solution are easily evaluated in correspondence to Barenblatt solutions, by giving a precise growth, say $t^{-\sigma_n}$, where $n$ is the order of the derivative. Hence, one can think to proceed as for the first two cases, by considering the power of order $\sigma_n$ of the $n$-th derivative and by checking if there is a sign.
With Theorem~\ref{Thm:T1}, we shall start by a simplified proof of the \emph{isoperimetric inequality for R\'enyi entropy powers} in~\cite{MR3200617}, based on based on the linear growth of the functional $\mathsf F$, that is also stated in \cite[Theorem~4.4]{carrillo2014renyi} and emphasize the fact that the method is limited to exponents $p\ge1-\frac1d$. Using this inequality, J.A.~Carrillo and G.~Toscani have been able to establish improved rates of convergence in the non-asymptotic regime of the solutions to~\eqref{poro} in \cite[Theorem~5.1]{carrillo2014renyi}. Since R\'enyi entropy powers are equivalent to \emph{relative entropies relative to best matching Barenblatt solutions} under appropriate conditions on the second moments as shown in \cite{1004,MR3103175}, improved rates have to be related with results obtained in rescaled variables in~\cite{MR3103175}, where the scales were defined in terms of second moments. Equivalently, the \emph{improved entropy-entropy production inequality} of \cite[Theorem~1]{MR3103175} is similar in nature to the \emph{isoperimetric inequality for R\'enyi entropy powers}. As we shall observe in this paper, the \emph{isoperimetric inequality for R\'enyi entropy powers} is in fact a \emph{Gagliardo-Nirenberg inequality} in scale invariant form, which degenerates into Sobolev's inequality in the limit case $p=1-\frac1d$ and explains why it cannot hold true for $p<1-\frac1d$. The \emph{entropy-entropy production inequality} of \cite{MR1940370} is also a Gagliardo-Nirenberg inequality, but not in scale invariant form and it has recently been established in \cite{dolbeault:hal-01081098} that the difference between the scale invariant form and the non scale invariant form is enough to account for improved rates of convergence in Fokker-Planck type equations, \emph{i.e.}, in nonlinear diffusions after a convenient rescaling.
Theorem~\ref{Thm:T2} is devoted to a rather simple observation on second moments, which is however at the core of our paper. It is based on the linear growth of the functional $\mathsf G$. By introducing the \emph{relative entropy with respect to the best matching Barenblatt function}, we establish the result of Theorem~\ref{Thm:T3} on delays. This result has been proved in \cite{2014arXiv1408.6781D} by a much more complicated method, when $p<1$. Here we simply rely on the convexity or concavity of $t\mapsto\mathsf G(t)$, depending whether $p<1$ or $p>1$, and both cases are covered. This analysis also suggests a method to obtain estimates of the delays, that is investigated in Section~\ref{Sec:Delays}.
Let us conclude this introduction with a brief review of the literature. For considerations on second moment methods in fast diffusion and porous medium equations, we refer to \cite{MR2133441,MR2328935,2014arXiv1408.6781D} and references therein. Sharp Ga\-gliardo-Nirenberg-Sobolev inequalities have been studied in~\cite{MR1940370,MR3103175,MR3200617,dolbeault:hal-01081098} from the point of view of the rates of convergence of the solutions to~\eqref{poro} in the intermediate asymptotics regime, and also for obtaining improved convergence rates in the initial regime. A counterpart of such improved rates is the notion of \emph{delay} which was established in~\cite{2014arXiv1408.6781D} in the fast diffusion regime and will be recovered as a very simple consequence of moment estimates in Theorem~\ref{Thm:T3} and extended to the porous medium case.
Theorems~\ref{Thm:T1} and~\ref{Thm:T2} follow the same line of thought: compute the time evolution of a functional which grows linearly when evaluated in the case of Barenblatt functions and has some concavity or convexity property otherwise. As a main consequence, we provide a proof of some Gagliardo-Nirenberg-Sobolev inequalities which goes along the lines of \cite{MR1777035,MR1940370,MR2745814} on the one hand, of \cite{MR3200617,carrillo2014renyi,MR3255069} on the other hand, and makes a synthetic link between the two approaches. The first approach is inspired by the entropy functional introduced by J.~Ralston and W.I.~Newman in \cite{MR760591,MR760592}, also known in the literature as the \emph{Tsallis entropy}, while the second one is more related with \emph{R\'enyi entropies} and connected with information theory inspired by \cite{MR823597,MR1768665}. The reader interested in further details is invited to refer to \cite{MR3255069} and references therein for a more detailed account.
\section{Notations, main results and consequences}\label{Sec:Main}
The \emph{entropy} is defined by \[ \mathsf E:=\ird{u^p} \] and the \emph{Fisher information} by \[
\mathsf I:=\ird{u\,|\nabla v|^2}\quad\mbox{with}\quad v=\frac p{p-1}\,u^{p-1}\,. \] If $u$ solves~\eqref{poro}, then \[ \mathsf E'=(1-p)\,\mathsf I\,. \] To compute $\mathsf I'$, we will use the fact that \be{Eqn:v}
\frac{\partial v}{\partial t}=(p-1)\,v\,\Delta v+|\nabla v|^2 \end{equation} and get that \[ \mathsf F:=\mathsf E^\sigma \] has a linear growth asymptotically as $t\to+\infty$ if \[\label{sigma} \sigma=\frac\mu{d\,(1-p)}=1+\frac2{1-p}\,\(\frac 1d+p-1\)=\frac 2d\,\frac1{1-p}-1\,. \] This definition is the same as the one of Section~\ref{Sec:Intro}. The growth is exactly linear in case of Barenblatt profiles, so that $\mathsf E_\star^{\sigma-1}\,\mathsf I_\star$ is independent of $t$ if we denote by $\mathsf E_\star$, $\mathsf I_\star$, $\mathsf F_\star$ \emph{etc.} the entropy, the Fisher information, \emph{etc.} of these Barenblatt profiles.
\begin{thm}\label{Thm:T1} Assume that $p\ge1-\frac1d$ if $d>1$ and $p>0$ if $d=1$. With the above notations, $t\mapsto\mathsf F(t)$ is increasing, $(1-p)\,\mathsf F''(t)\le0$ and \[ \lim_{t\to+\infty}\frac1t\,\mathsf F(t)=(1-p)\,\sigma\,\lim_{t\to+\infty}\mathsf E^{\sigma-1}\,\mathsf I=(1-p)\,\sigma\,\mathsf E_\star^{\sigma-1}\,\mathsf I_\star\,. \] \end{thm}
This result has been established in \cite{MR3200617}. In this paper we give a slightly simpler proof. The result of Theorem~\ref{Thm:T1} amounts to state that $t\mapsto\mathsf F(t)$ is concave if $p<1$ and convex if $p>1$, with an asymptotic slope given by $(1-p)\,\sigma\,\mathsf E_\star^{\sigma-1}\,\mathsf I_\star$. Moreover, we observe that the inequality \be{GN-Barenblatt} \mathsf E^{\sigma-1}\,\mathsf I\ge\mathsf E_\star^{\sigma-1}\,\mathsf I_\star \end{equation} is equivalent to one of the two following Gagliardo-Nirenberg inequalities: \begin{description} \item[(i)] If $1-\frac1d\le p<1$, then~\eqref{GN-Barenblatt} is equivalent to \be{GN1} \nrm{\nabla w}2^\theta\,\nrm w{q+1}^{1-\theta}\ge\mathsf C_{\rm{GN}}\,\nrm w{2q} \end{equation} where \[ \theta=\frac dq\,\frac{q-1}{d+2-q\,(d-2)}\,,\quad1<q\le\frac d{d-2} \] and equality with optimal constant $\mathsf C_{\rm{GN}}$ is achieved by $\mathcal B_\star^{p-1/2}$. \item[(ii)] If $p>1$, then~\eqref{GN-Barenblatt} is equivalent to \be{GN2} \nrm{\nabla w}2^\theta\(\ird{w^{2q}}\)^\frac{1-\theta}{2q}\ge\mathsf C_{\rm{GN}}\,\nrm w{q+1} \end{equation} where \[ \theta=\frac dq\,\frac{q-1}{d+2-q\,(d-2)} \] and $q$ takes any value in $(0,1)$. \end{description} In both cases, we relate Gagliardo-Nirenberg inequalities and~\eqref{GN-Barenblatt} by \[ u^{p-1/2}=\frac w{\nrm w{2q}}\quad\mbox{with}\quad q=\frac1{2\,p-1}\,. \] These considerations show that $p=1-\frac1d$ correspond to Sobolev's inequality: $2q=2\,d/(d-2)$ and $\theta=1$ if $d\ge3$. This case is therefore the threshold case for the validity of the method. Details will be given in Section~\ref{Sec:GN}.
Using the moment \[
\Theta:=\frac 1d\ird{|x|^2\,u}\,, \] we can introduce the functional \[ \mathsf G:=\Theta^{1-\eta/2}\quad\mbox{with}\quad\eta=d\,(1-p)\,. \] As a function of $t$, $\mathsf G$ also has a linear growth in case of Barenblatt profiles. Again this definition is compatible with the one of Section~\ref{Sec:Intro} since \[ \frac\mu2=1-\frac\eta2\,. \] In the general case, we get that \[ \mathsf G'=\mu\,\mathsf H\,, \] where we define the \emph{R\'enyi entropy power} functional $\mathsf H$ by \[ \mathsf H:=\Theta^{-\eta/2}\,\mathsf E=\Theta^{\frac d2\,(p-1)}\,\mathsf E \] and observe that the corresponding functional $\mathsf H_\star$ for Barenblatt profiles is independent of $t$.
\begin{thm}\label{Thm:T2} Assume that $p\ge1-\frac2d$. With the above notations, $t\mapsto\mathsf G(t)$ is increasing, $(1-p)\,\mathsf H'(t)\ge0$ and \[ \lim_{t\to+\infty}\frac1t\,\mathsf G(t)=\(1-\frac\eta2\)\lim_{t\to+\infty}\mathsf H=\(1-\frac\eta2\)\mathsf H_\star\,. \] \end{thm}
The function $t\mapsto\mathsf G(t)$ is convex if $p<1$ and concave if $p>1$, with an asymptotic slope given by $(1-\eta/2)\,\mathsf H_\star$. Details will be given in Section~\ref{Sec:Moment}.
We can also consider the \emph{relative entropy with respect to the best matching Barenblatt function} defined as \[
\mathcal F[u]:=\inf_{s>0}\mathcal F[u\,|\,\mathcal U_\star^s] \] where $\mathcal U_\star^s(x)=\mathcal U_\star(s,x)=s^{-d/\mu}\,\mathcal U_\star^1(s^{-1/\mu}\,x)$ is defined by~\eqref{ba-self}. Here the variable~$s$ plays the role of a scaling parameter, and the \emph{relative entropy} with respect to a given function $\mathcal U$ is defined by \[
\mathcal F[u\,|\,\mathcal U]:=\frac1{p-1}\ird{\Big[u^p-\mathcal U^p-p\,\mathcal U^{p-1}\,\big(u-\mathcal U\big)\Big]}\,. \]
A key observation in \cite{1004} is the fact that $\mathcal F[u]=\mathcal F[u\,|\,\mathcal U_\star^s]$ is achieved by a unique Barenblatt solution which satisfies \[
\ird{|x|^2\,u}=\ird{|x|^2\,\mathcal U_\star^s}\,. \]
The proof of this fact is a straightforward computation: as a functional of~$\mathcal U$, $\mathcal F[u\,|\,\mathcal U]$ is concave and has at most one maximum point. Hence if we write that $|\mathcal U_\star^s(x)|^{p/(p-1)}=a(s)+b(s)\,|x|^2$, then \[
\frac d{ds}\mathcal F[u\,|\,\mathcal U_\star^s]=\frac p{p-1}\ird{\(a'(s)+b'(s)\,|x|^2\)\(u-\mathcal U_\star^s\)}\,. \] Since $\ird u=\ird{\mathcal U_\star^s}=1$, the term proportional to $a'$ does not contribute and the other one vanishes if and only if the moments are equal.
After undoing the change of variables, we get that \be{Eqn:s} s=\(\frac\Theta{\Theta_\star}\)^\frac\mu2\,.
\end{equation} With this choice of $s$, $\mathcal U_\star^s$ is the \emph{best matching Barenblatt} function in the sense that this Barenblatt function minimizes the relative entropy $\mathcal F[u\,|\,\mathcal U_\star^s]$ among all Barenblatt functions $(\mathcal U_\star^s)_{s>0}$.
If $u$ is a solution to~\eqref{poro}, for any $t\ge0$, we can define $s$ as a function of $t$ and consider the \emph{delay} which is defined as \[ \tau(t):=\(\frac{\Theta(t)}{\Theta_\star}\)^\frac\mu2-t\,. \] The main result of this paper is that $t\mapsto\tau(t)$ is monotone.
\begin{thm}\label{Thm:T3} Assume that $p\ge1-\frac1d$ and $p\neq1$. With the above notations, the best matching Barenblatt function of a solution $u$ to~\eqref{poro} satisfying~\eqref{second} is $(t,x)\mapsto\mathcal U_\star(t+\tau(t),x)$ and the function $t\mapsto\tau(t)$ is nondecreasing if $p>1$ and nonincreasing if $1-\frac1d\le p<1$.\end{thm}
With $s$ given by~\eqref{Eqn:s}, we notice that the \emph{relative entropy with respect to the best matching Barenblatt function} is given by \[
\mathcal F[u\,|\,\mathcal U_\star^s]=\frac1{p-1}\ird{\Big[u^p-(\mathcal U_\star^s)^p\Big]} \] and it is related with the the \emph{R\'enyi entropy power} functional by \[
\mathsf H-\mathsf H_\star=\Theta^{\frac d2\,(p-1)}\,\mathsf E-\Theta_\star^{\frac d2\,(p-1)}\,\mathsf E_\star=(p-1)\,\Theta^{\frac d2\,(p-1)}\,\mathcal F[u\,|\,\mathcal U_\star^s]\,. \]
\begin{figure}
\caption{ In the fast diffusion case, the (bold) curve $s\mapsto\mathsf G(s)=\Theta^{\mu/2}(s)$ is convex, increasing, and its asymptote is $s\mapsto(s+\tau_\infty)\,\Theta_\star^{\mu/2}$. At any time $t\ge0$, we observe that it crosses the line $s\mapsto(s+\tau(t))\,\Theta_\star^{\mu/2}=(s-t+1)\,\Theta^{\mu/2}(t)$ transversally, so that $t\mapsto\tau(t)$ is monotone decreasing, unless the solution is itself a Barenblatt solution up to a time shift.}
\label{F1}
\end{figure} \begin{figure}
\caption{ In the porous medium case, the curve $s\mapsto\Theta^{\mu/2}(s)$ is still increasing but concave, so that $t\mapsto\tau(t)$ is monotone increasing, unless the solution is itself a Barenblatt solution up to a time shift.}
\label{F2}
\end{figure}
See Figs.~\ref{F1}-\ref{F2}. An estimate of the delay $t\mapsto\tau(t)$ will be given in Section~\ref{Sec:Delays}.
\section{A direct proof of Gagliardo-Nirenberg inequalities}\label{Sec:GN}
This section is devoted to the proof of Theorem~\ref{Thm:T1} and also to a proof of Gagliardo-Nirenberg inequalities~\eqref{GN1} and \eqref{GN2}. We provide a slightly simplified proof of Lemma~\ref{Lem:DerivFisher} below, compared to the existing literature: see \cite{MR3200617} and references therein. Some of the key computations will be reused in Section~\ref{Sec:Delays}.
\subsection{A preliminary computation}
According to~\cite[Appendix~B]{MR3200617}, we have the following result.
\begin{lem}\label{Lem:DerivFisher} If $u$ solves~\eqref{poro} with initial datum $u(x,t=0)=u_0(x)\ge0$ such that $\ird{u_0}=1$ and $\frac1d\ird{|x|^2\,u_0}<+\infty$, then $v=\frac p{p-1}\,u^{p-1}$ solves~\eqref{Eqn:v} and \be{BLW}
\mathsf I'=\frac d{dt}\ird{u\,|\nabla v|^2}=-\,2\ird{u^p\,\Big(\|\mathrm D^2v\|^2+(p-1)\,(\Delta v)^2\Big)}\,. \end{equation} \end{lem}
\begin{proof} Let us give a simplified proof of this result. Using~\eqref{poro} and~\eqref{Eqn:v}, we can compute \begin{eqnarray*}
&&\hspace*{-18pt}\frac d{dt}\ird{u\,|\nabla v|^2}\\
&=&\ird{\frac{\partial u}{\partial t}\,|\nabla v|^2}+\,2\ird{u\,\nabla v\cdot\nabla\frac{\partial v}{\partial t}}\\
&=&\ird{\Delta(u^p)\,|\nabla v|^2}+\,2\ird{u\,\nabla v\cdot\nabla\Big((p-1)\,v\,\Delta v+|\nabla v|^2\Big)}\\
&=&\ird{u^p\,\Delta|\nabla v|^2}\\ &&+\,2\,(p-1)\ird{u\,v\,\nabla v\cdot\nabla\Delta v}+\,2\,(p-1)\ird{u\,\nabla v\cdot\nabla v\,\Delta v}\\
&&+\,2\ird{u\,\nabla v\cdot\nabla|\nabla v|^2}\\
&=&-\ird{u^p\,\Delta|\nabla v|^2}\\ &&+\,2\,(p-1)\ird{u\,v\,\nabla v\cdot\nabla\Delta v}+\,2\,(p-1)\ird{u\,\nabla v\cdot\nabla v\,\Delta v} \end{eqnarray*} where the last line is given by an integration by parts: \[
\ird{u\,\nabla v\cdot\nabla|\nabla v|^2}=\ird{\nabla(u^p)\cdot\nabla|\nabla v|^2}=-\ird{u^p\,\Delta|\nabla v|^2}\,. \] 1) Using the elementary identity \[
\frac12\,\Delta\,|\nabla v|^2=\|\mathrm D^2v\|^2+\nabla v\cdot\nabla\Delta v\,, \] we get that \[
\ird{u^p\,\Delta|\nabla v|^2}=2\ird{u^p\,\|\mathrm D^2v\|^2}+2\ird{u^p\,\nabla v\cdot\nabla\Delta v}\,. \] 2) Since $u\,\nabla v=\nabla(u^p)$, an integration by parts gives \begin{multline*} \ird{u\,\nabla v\cdot\nabla v\,\Delta v}=\ird{\nabla(u^p)\cdot\nabla v\,\Delta v}\\ =-\ird{u^p\,(\Delta v)^2}-\ird{u^p\,\nabla v\cdot\nabla\Delta v} \end{multline*} and with $u\,v=\frac p{p-1}\,u^p$ we find that \begin{multline*} 2\,(p-1)\ird{u\,v\,\nabla v\cdot\nabla\Delta v}+\,2\,(p-1)\ird{u\,\nabla v\cdot\nabla v\,\Delta v}\\ =-\,2\,(p-1)\ird{u^p\,(\Delta v)^2}+2\ird{u^p\,\nabla v\cdot\nabla\Delta v}\,. \end{multline*} Collecting terms establishes~\eqref{BLW}.\qed\end{proof}
\subsection{The fast diffusion case}
Recall that $\mathsf E=\ird{u^p}$ satisfies $\mathsf E'=(1-p)\,\mathsf I$ with $\mathsf I=\ird{u\,|\nabla v|^2}$ and $v=\frac p{p-1}\,u^{p-1}$. Since \[
\|\mathrm D^2v\|^2=\frac 1d\,(\Delta v)^2+\left\|\,\mathrm D^2v-\frac 1d\,\Delta v\,\mathrm{Id}\,\right\|^2 \] by Lemma~\ref{Lem:DerivFisher}, we find that $\mathsf F''=\(\mathsf E^\sigma\)''$ can be computed as \begin{multline*}
\frac 1{\sigma\,(1-p)}\,\mathsf E^{2-\sigma}\,\(\mathsf E^\sigma\)''=(1-p)\,(\sigma-1)\(\ird{u\,|\nabla v|^2}\)^2\\ \hspace*{3cm}-\,2\,\(\frac 1d+p-1\)\ird{u^p}\ird{u^p\,(\Delta v)^2}\\
-\,2\ird{u^p}\ird{u^p\,\left\|\,\mathrm D^2v-\frac 1d\,\Delta v\,\mathrm{Id}\,\right\|^2}\,. \end{multline*} Using $u\,\nabla v=\nabla(u^p)$, we know that \[
\ird{u\,|\nabla v|^2}=\ird{\nabla(u^p)\cdot\nabla v}=-\ird{u^p\,\Delta v} \] and by the Cauchy-Schwarz inequality, \[
\(\ird{u\,|\nabla v|^2}\)^2\le\ird{u^p}\ird{u^p\,(\Delta v)^2}\,. \] With the choice \[ \sigma=1+\frac2{1-p}\,\(\frac 1d+p-1\)=\frac 2d\,\frac1{1-p}-1\,, \] we get that \[ \frac{\mathsf E^{2-\sigma}\,\(\mathsf E^\sigma\)''}{\sigma\,(1-p)^2}=:-\,\mathsf R[u] \] where the remainder terms have been collected as the sum of two squares: \begin{multline*}
\frac{\mathsf R[u]}{\,\ird{u^p}}=(\sigma-1)\ird{u^p\,\left|\Delta v-\frac{\ird{u\,|\nabla v|^2}}{\ird{u^p}}\right|^2}\\
+\,\frac2{1-p}\ird{u^p\,\left\|\,\mathrm D^2v-\frac 1d\,\Delta v\,\mathrm{Id}\,\right\|^2}\,. \end{multline*}
Hence we know that $\mathsf F'=\(\mathsf E^\sigma\)'$ is nonincreasing, that is, \[
\frac1{1-p}\mathsf E^{\sigma-1}\,\mathsf E'=\(\ird{u^p}\)^{\sigma-1}\ird{u\,|\nabla v|^2}:=\mathsf J \] is nonincreasing. Since $\mathsf J$ is invariant under scalings as a functional of $u$, this means that \[ \lim_{t\to\infty}\mathsf J=\mathsf J_\star \] where $\mathcal B_\star$ is the Barenblatt function such that $\ird{\mathcal B_\star}=1$ and $\mathsf J_\star$ the corresponding value of the functional $\mathsf J$. Written for the initial datum $u_0=u$, we have shown that \[ \mathsf J\ge\mathsf J_\star\,. \] For any smooth and compactly supported function $w$, if we write $u^{p-1/2}=w/\nrm w{2q}$ with $q=1/(2\,p-1)$, then the inequality amounts to the Gagliar\-do-Nirenberg inequality~\eqref{GN1} and equality is achieved by $\mathcal B_\star^{p-1/2}$. More precisely we have shown the following result.
\begin{prop}\label{Prop:GN} Assume that $1<q<\frac d{d-2}$ if $d\ge3$ and $q>1$ if $d=1$ or $d=2$. With the above notations we have \begin{multline*} \frac{4\,p^2}{(p-1)^2\,(2\,p-1)^2}\,\Big(\mathsf J-\mathsf J_\star\Big)=\frac{\nrm{\nabla w}2^2\,\nrm w{q+1}^{2\,(1-\theta)/\theta}}{\nrm w{2q}^{2/\theta}}-\mathsf C_{\rm{GN}}^{2/\theta}\\ =(1-p)\int_0^\infty\mathsf R[u(t,\cdot)]\,dt\ge0\,, \end{multline*} where $u(t,\cdot)$ denotes the solution to~\eqref{poro} with initial datum $u$.\end{prop}
Hence we have shown that, as a function of $t$, $\mathsf F$ is concave increasing and we have identified its asymptotic slope, which is given by the optimal constant in the Gagliardo-Nirenberg inequality \[ \lim_{t\to+\infty}\mathsf J(t)=\mathsf J_\star=\mathsf C_{\rm{GN}}^{2/\theta}\,. \]
\subsection{The porous medium case}
The computations are the same. With the same definition~\eqref{sigma} as in the fast diffusion case, $\sigma$ is negative, $t\mapsto\mathsf F$ is convex increasing and the limit of its derivative is achieved among Barenblatt functions. The Gagliardo-Nirenberg inequality now takes the form~\eqref{GN2}. Again equality is achieved by $\mathcal B_\star^{p-1/2}$. Details are left to the reader.
\section{The second moment and the R\'enyi entropy power functional}\label{Sec:Moment}
This section is devoted to the proof of Theorem~\ref{Thm:T2}. Let us consider the subsequent time derivatives of the functional \[ \mathsf G:=\Theta^{1-\frac\eta2}\quad\mbox{with}\quad\eta=d\,(1-p)=2-\mu\,. \] It is straightforward to check that \[ \mathsf G'=\mu\,\mathsf H\quad\mbox{with}\quad\mathsf H:=\Theta^{-\frac\eta2}\,\mathsf E \] where the \emph{R\'enyi entropy power} functional is defined by \[ \mathsf H:=\Theta^{-\frac\eta2}\,\mathsf E\,. \] We recall that $\mathsf E'=(1-p)\,\mathsf I$. It is straightforward to check that \[ \frac{\mathsf H'}{1-p}=\Theta^{-1-\frac\eta2}\(\Theta\,\mathsf I- d\,\mathsf E^2\)=\frac{d\,\mathsf E^2}{\Theta^{\frac\eta2+1}}\,(\mathsf q-1)\quad\mbox{with}\quad\mathsf q:=\frac{\Theta\,\mathsf I}{d\,\mathsf E^2}\ge1 \] because \begin{multline*} d\,\mathsf E^2=\frac 1d\(-\ird{x\cdot\nabla(u^p)}\)^2=\frac 1d\(\ird{x\cdot u\,\nabla v}\)^2\\
\le\frac 1d\ird{u\,|x|^2}\ird{u\,|\nabla v|^2}=\Theta\,\mathsf I \end{multline*} by the Cauchy-Schwarz inequality. This proves that $(1-p)\,\mathsf H$ is monotone increasing and from the theory in \cite{MR1940370}, we know that its limit is given by the source-type Barenblatt solutions $\mathcal U_{p,t}$ defined by~\eqref{ba-self}. Since $\mathsf H$ is also scale invariant, its limit is in fact given by the value of the functional for $\mathcal B_\star$, that~is \[ \lim_{t\to+\infty}\mathsf H(t)=\lim_{t\to+\infty}\Theta(t)^{\frac d2\,(p-1)}\,\mathsf E(t)=\Theta_\star^{\frac d2\,(p-1)}\,\mathsf E_\star=:\mathsf H_\star \] and we get \be{pri} \begin{array}{ll} \Theta(t)^{\frac d2\,(p-1)}\,\mathsf E(t)\le\mathsf H_\star\quad&\mbox{if}\quad p<1\,,\\[5pt] \Theta(t)^{\frac d2\,(p-1)}\,\mathsf E(t)\ge\mathsf H_\star\quad&\mbox{if}\quad p>1\,. \end{array} \end{equation} Taking the logarithm on both sides of inequality~\eqref{pri}, and considering that $p<1$, we obtain the equivalent inequality \[\label{pr2}
\frac1{1-p}\log\ird{u^p}-\frac d2\,\log\(\frac1d\ird{|x|^2\,u}\)\ge\frac{\log\mathsf H_\star}{1-p}=\frac{\log\mathsf E_\star}{1-p}-\frac d2\,\log\Theta_\star \] in which we can recognize the well-known inequality for R\'enyi entropies obtained in~\cite{CHV,LYZ}. After multiplying by $(1-p)$ and taking the exponential, we realize that this is also equivalent to the inequality \[
\mathcal F[u\,|\,\mathcal U_\star^1]\ge0 \] which is a well-known consequence of Jensen's inequality: see for instance \cite{MR1940370}.
\section{Delays}\label{Sec:Delays}
This section is devoted to delays. At any time $t\ge0$, we consider the best matching Barenblatt solution and the corresponding delay $\tau(t)$ \[ \mathcal U_\star^{s(t)}(x)=\mathcal U_\star(s(t),x)\,,\quad s(t)=t+\tau(t) \] according to the definitions of Section~\ref{Sec:Main}. After proving Theorem~\ref{Thm:T3}, we give an estimate of $\tau(t)$.
\noindent\emph{Proof of Theorem~\ref{Thm:T3}.} With $p\ge1-\frac1d$, we know that $\mu>0$. At any time $t\ge0$, let us consider the solution $t\mapsto u(t,\cdot)$ to~\eqref{poro}. The scale $s(t)$ of the best matching Barenblatt function is determined by~\eqref{Eqn:s}. By observing that $t\mapsto\Theta^{\mu/2}$ grows faster (resp. slower) than $t\mapsto t\,\Theta_\star^{\mu/2}$ if $p>1$ (resp.~if $p<1$), where $t\mapsto t\,\Theta_\star^{\mu/2}$ is the rate of growth corresponding to the self-similar Barenblatt function given by~\eqref{ba-self}, we get that $t\mapsto\tau(t)$ is nondecreasing (resp. nonincreasing) in the porous medium case $p>1$ (resp. fast diffusion case $p<1$). See Figs.~\ref{F1}-\ref{F2}. \qed
Next we study a quantitative estimate of delays, which relies on the commutation of the third derivative in $t$ of $\mathsf G$. This approach is parallel to the results in \cite{2014arXiv1410.2722T} in the linear case.
Let us recall that $\Theta'=2\,\mathsf E$, $\mathsf E'=(1-p)\,\mathsf I$ and $\mathsf J=\mathsf E^{\sigma-1}\,\mathsf I$ is such that \[ 0\ge\mathsf E^{2-\sigma}\,\mathsf J'=(1-p)\,(\sigma-1)\,\mathsf I^2+\mathsf E\,\mathsf I' \] so that \[ \mathsf I'\le-\,(1-p)\,(\sigma-1)\,\frac{\mathsf I^2}{\mathsf E}=-\,\frac2d\,(1-\eta)\,\frac{\mathsf I^2}{\mathsf E}\,. \] Hence \be{H''} \frac{\mathsf H''}{1-p}=\(\Theta^{-\frac\eta2}\,\mathsf I-\,d\,\Theta^{-\frac\eta2-1}\,\mathsf E^2\)'\le\frac{d\,\mathsf E^3}{\Theta^{\frac\eta2+2}}\,\Big(\eta+2-\,3\,\eta\,\mathsf q-\,2\,(1-\eta)\,\mathsf q^2\Big) \end{equation} according to the computations of Section~\ref{Sec:Moment}. We observe that $\mathsf H''\le0$ if $p<1$ and $\mathsf H''\ge0$ if $p>1$.
\begin{thm}\label{Thm:T3bis} Under the assumptions of Theorem~\ref{Thm:T3}, if $p>1-\frac1d$ and $p\neq1$, then the delay satisfies \[
\lim_{t\to+\infty}|\tau(t)-\tau(0)|\ge|1-p|\,\frac{\Theta(0)^{1-\frac d2(1-p)}}{2\,\mathsf H_\star}\,\frac{\big(\mathsf H_\star-\mathsf H(0)\big)^2}{\Theta(0)\,\mathsf I(0)-d\,\mathsf E(0)^2} \] \end{thm}
\begin{proof} Assume first that $p<1$ and recall that for any $t\ge0$ \[ \mathsf G'(t)\le\lim_{t\to+\infty}\mathsf G'(t)=\(1-\frac\eta2\)\mathsf H_\star=:\mathsf G_\star'\,. \] Since $\mathsf G''$ is nonincreasing, we have the estimate \[ \mathsf G(t)\le\mathsf G(0)+\mathsf G'(0)\,t+\tfrac12\,\mathsf G''(0)\,t^2\quad\forall\,t\ge0 \] so that \[ \mathsf G(0)+\mathsf G_\star'\,t-\mathsf G(t)\ge\(\mathsf G_\star'-\mathsf G'(0)\)t-\tfrac12\,\mathsf G''(0)\,t^2 \] is maximal for $t=t_\star:=(\mathsf G_\star'-\mathsf G'(0))/\mathsf G''(0)$. As a consequence, since $\mathsf G(0)=\mathsf G_\star'\,\tau(0)$ and $\mathsf G(t_\star)=\mathsf G_\star'\,\big(t_\star+\tau(t_\star)\big)$, we get that \[ \mathsf G_\star'\,\tau(0)-\mathsf G_\star'\,\tau(t_\star)=\mathsf G(0)+\mathsf G_\star'\,t_\star-\mathsf G_\star'\,\big(t_\star+\tau(t_\star)\big)\ge\frac{\big(\mathsf G_\star'-\mathsf G'(0)\big)^2}{2\,\mathsf G''(0)}\,, \] that is \[ \tau(0)-\tau(t_\star)\ge\frac{\big(\mathsf G_\star'-\mathsf G'(0)\big)^2}{2\,\mathsf G_\star'\,\mathsf G''(0)}=(1-p)\,\frac{\Theta(0)^{1+\frac\eta2}}{2\,\mathsf H_\star}\,\frac{\big(\mathsf H_\star-\mathsf H(0)\big)^2}{\Theta(0)\,\mathsf I(0)-d\,\mathsf E(0)^2}\,. \] We conclude by observing that $t\mapsto\tau(0)-\tau(t)$ is nondecreasing.
Estimates for $p>1$ are very similar, up to signs. \end{proof}
Further estimates can be obtained easily. Let us illustrate our new approach by a result on \be{q} \mathsf q:=\frac{\Theta\,\mathsf I}{d\,\mathsf E^2}=1+\frac{\Theta^{\frac\eta2+1}}{d\,\mathsf E^2}\,\frac{\mathsf H'}{1-p} \end{equation} in the fast diffusion case. We denote by $\mathsf q_0$, $\Theta_0$,.. the initial values of $\mathsf q$, $\Theta$, \emph{etc.}
\begin{prop}\label{Thm:T4} Assume that $1-\frac1d\le p<1$. Then for any $t\ge0$, we have \[ \mathsf q(t)\le\frac{\mathsf q_0\,\Theta(t)}{\mathsf q_0\,\Theta(t)-(\mathsf q_0-1)\,\Theta_0}=:\bar{\mathsf q}(t) \] and \[ \tau(t)\le\tau_0\,\exp\left[\int_0^t\frac{ds}{s+\frac{\Theta_0}{\mu\,\mathsf E_0}-\frac\eta\mu\int_0^s(\bar{\mathsf q}-1)\,ds}\right]-t\,. \] \end{prop}
\begin{proof} Using~\eqref{H''} and~\eqref{q}, we obtain \[ \mathsf q'\le2\,\mathsf q\,(1-\mathsf q)\,\frac{\mathsf E}\Theta=\mathsf q\,(1-\mathsf q)\,\frac{\Theta'}\Theta\,. \] We can integrate and get the first estimate.
To obtain the integral estimate for $\tau$, we compute \[ \frac{\mathsf G'}{\mathsf G}=\mu\,\frac{\mathsf E}\Theta \] and \[ \frac1{1-p}\,\frac{\mathsf G''}{\mathsf G'}=\frac1{1-p}\,\frac{\mathsf H'}{\mathsf H}=d\,\frac{\mathsf E}\Theta\,(\mathsf q-1)\le\frac d\mu\,\frac{\mathsf G'}{\mathsf G}\,(\bar{\mathsf q}-1)\,, \] so that \[ 1-\(\frac{\mathsf G}{\mathsf G'}\)'=\frac{\mathsf G\,\mathsf G''}{(\mathsf G')^2}\le\frac\eta\mu\,(\bar{\mathsf q}-1) \] and finally \[ \frac{\mathsf G}{\mathsf G'}\ge\frac{\Theta_0}{\mu\,\mathsf E_0}+t-\frac\eta\mu\int_0^t(\bar{\mathsf q}(s)-1)\,ds\,. \] Using the fact that $\mathsf G(t)=\mathsf G_\star'\,\big(t+\tau(t)\big)$, we conclude after one more integration with respect to $t$ and get the second estimate.\qed\end{proof}
Let us conclude this paper by a few remarks. \begin{enumerate} \item[(i)] The quantity $\mathsf q-1$ is a measure of the distance to the set of the Barenblatt profiles when $p>1-\frac 1d$: see \cite{dolbeault:hal-01081098} for more details. \item[(ii)] Improved rates of decay for $\mathsf E$ and, as a consequence, improved asymptotics for $\mathsf F$ and $\mathsf G$ can be achieved by considering the estimates found in \cite{carrillo2014renyi}. \item[(iii)] Alternatively, improved functional inequalities as in \cite{dolbeault:hal-01081098} can be used directly to get improved asymptotics for $\mathsf F$ and $\mathsf G$. \end{enumerate}
\end{document} | arXiv |
International Journal of Information Security
November 2018 , Volume 17, Issue 6, pp 661–680 | Cite as
DomainProfiler: toward accurate and early discovery of domain names abused in future
Daiki Chiba
Takeshi Yagi
Mitsuaki Akiyama
Toshiki Shibahara
Tatsuya Mori
Shigeki Goto
Regular Contribution
First Online: 16 December 2017
Domain names are at the base of today's cyber-attacks. Attackers abuse the domain name system (DNS) to mystify their attack ecosystems; they systematically generate a huge volume of distinct domain names to make it infeasible for blacklisting approaches to keep up with newly generated malicious domain names. To solve this problem, we propose DomainProfiler for discovering malicious domain names that are likely to be abused in future. The key idea with our system is to exploit temporal variation patterns (TVPs) of domain names. The TVPs of domain names include information about how and when a domain name has been listed in legitimate/popular and/or malicious domain name lists. On the basis of this idea, our system actively collects historical DNS logs, analyzes their TVPs, and predicts whether a given domain name will be used for malicious purposes. Our evaluation revealed that DomainProfiler can predict malicious domain names 220 days beforehand with a true positive rate of 0.985. Moreover, we verified the effectiveness of our system in terms of the benefits from our TVPs and defense against cyber-attacks.
Network-level security and protection Domain name DNS Malware Temporal variation pattern
This paper is the extended version of the paper presented at IEEE/IFIP DSN 2016 [15].
Domain names are used by all Internet users and service providers for their online activities and businesses. Domain names and their protocol [domain name system (DNS)] are one of the most successful examples of distributed systems that can satisfy users' needs regarding easy use of the Internet. However, Internet users also include attackers who abuse easy-to-use domain names as a reliable cyber-attack infrastructure. For example, in today's cyber-attacks, domain names are used in serving malicious content or malware, controlling malware-infected hosts, and stealing personal or important information.
As countermeasures against domain name abuses, detecting and blacklisting known malicious domain names are basic strategies that are widely applied to protect users from cyber-attacks. However, attackers understand these countermeasures and abuse DNS to mystify their attack ecosystems; DNS fast-flux and domain generation algorithms (DGAs) are used to evade blacklisting. The key feature of these techniques is that they systematically generate a huge volume of distinct domain names. These techniques have made it infeasible for blacklisting approaches to keep up with newly generated malicious domain names.
Ideally, to fully address the underlying problem with domain name blacklists, we need to observe and track all newly registered and updated domain names in real time and judge whether they are involved in any attackers' infrastructure. However, in reality, this problem is virtually impossible to solve for three reasons. One is that attackers use techniques, such as DNS fast-flux and DGAs, to systematically generate a huge volume of distinct domain names. The second is that the number of existing domain names is too large to track in real time. The number of second-level domain (2LD) names (e.g., example.com) is now over 296 million [44]. Multiple fully qualified domain names (FQDNs) (e.g., www.example.com) may exist under the same 2LD names; therefore, the number of all existing FQDNs could be in the billions. The third reason is that no one can fully understand all real-time changes in the mappings between domain names and IP addresses. Since DNS is a distributed system and the mappings are configured in each authoritative name server, the mappings of all domain names cannot feasibly be observed in real time. Given these reasons, blacklisting approaches based on DNS observations have failed to keep up with newly generated malicious domain names. Thus, we adopt an approach of prediction instead of observation, i.e., we aim to discover malicious domain names that are likely to be abused in future. The key idea of this approach is to exploit temporal variation patterns (TVPs) of malicious domain names. The TVPs of domain names include the information about how and when a domain name has been listed in legitimate/popular and/or malicious domain name lists. We use TVPs to comprehend the variations in domain names. For example, a domain name may be newly registered or updated, IP addresses corresponding to the domain name may be changed, and the traffic directed to the domain name may be changed.
On the basis of the aforementioned idea, we developed a system that actively collects historical DNS logs, analyzes their TVPs, and predicts whether a given domain name will be used maliciously. Our main contributions are summarized as follows.
We propose DomainProfiler, which identifies TVPs of domain names to precisely profile various types of malicious domain names.
Our evaluation with real and large ground truth data reveals that DomainProfiler can predict malicious domain names 220 days beforehand with a true positive rate (TPR) of 0.985 in the best-case scenario.
We reveal the contribution or importance of each feature in DomainProfiler to detect future malicious domain names.
We conduct a lifespan analysis for malicious domain names detected by DomainProfiler to illustrate the characteristics of various domain names abused in a series of cyber-attacks.
We use a large number of actual malware samples to demonstrate the effectiveness of DomainProfiler at defending against malware activities.
The rest of this paper is organized as follows. We give the motivation for our key idea in Sect. 2. In Sect. 3, we discuss our proposed system, DomainProfiler. We describe the datasets we used and the results of our evaluation in Sect. 4. We discuss the limitations of our system in Sect. 5 and related work in Sect. 6. Finally, we conclude our paper in Sect. 7.
2 Motivation: temporal variation pattern
We define a temporal variation pattern (TVP) as the time series behavior of each domain name in various types of domain name lists. Specifically, we identify how and when a domain name has been listed in legitimate/popular and/or malicious domain name lists. Our motivation for considering TVPs is based on the observation that both legitimate and malicious domain names vary dramatically in domain name lists over time. There are three reasons for using different and multiple domain name lists. One is that the data are realistically observable; that is, we can easily access the data from domain name list maintainers. The second is that domain name lists are created on the basis of objective facts confirmed by the maintainer of those lists. The third is that multiple domain name lists and the time series changes in those lists can boost the reliability of listed domain names.
As shown in Fig. 1, our proposed system defines and identifies four TVPs (null, stable, fall, and rise) for each domain name in a domain name list. Null means the domain name has not been listed in the specified time window. Stable means the domain name has been continuously listed in the time window. Fall is a state in which the domain name was first listed and then delisted during the time window. Rise means that the domain name was first unlisted and then listed during the time window.
Simplified temporal variation patterns (TVPs)
Definition A set \(T_d =\{t_1,\ldots ,t_{N_d}\}\) is an ordered \(N_d\) of timestamps when a domain name d has been listed/contained in a domain name list. The domain name list is collected from \(t_s\) to \(t_e\). Given a set of timestamps \(T_d\) and a time window between a starting point \(w_\mathrm{s}\) and ending point \(w_\mathrm{e}\), the TVP of a domain name is defined as follows.
$$\begin{aligned} \mathrm{TVP} = {\left\{ \begin{array}{ll} \mathrm{Null} &{} (\min ( T_d \cup \{t_e\} )> w_\mathrm{e} \vee \max ( T_d \cup \{t_s\} )< w_\mathrm{s}) \\ \mathrm{Stable} &{} (\min ( T_d \cup \{t_e\} )< w_\mathrm{s} \wedge \max ( T_d \cup \{t_s\} )> w_\mathrm{e}) \\ \mathrm{Fall} &{} (\min ( T_d \cup \{t_e\} )< w_\mathrm{s} \wedge w_\mathrm{s}< \max ( T_d \cup \{t_s\} )< w_\mathrm{e}) \\ \mathrm{Rise} &{} (w_\mathrm{s}< \min ( T_d \cup \{t_e\} ) < w_\mathrm{e} \wedge \max ( T_d \cup \{t_s\} ) > w_\mathrm{e}) \\ \end{array}\right. } \end{aligned}$$
Popular domain name list (Alexa top sites)
These TVPs are common and generic features that can contribute to accurately discriminating malicious domain names controlled by attackers from legitimate domain names. Thus, the focus of these patterns covers a wide range of malicious domain names used in a series of cyber-attacks such as drive-by download attacks, malware downloads, command and control (C&C), and phishing attacks.
In this paper, we use the domain names ranked in the Alexa Top Sites [1] as the legitimate/popular domain name list. Alexa lists the top one million most popular sites on the basis of their global one-month average traffic ranks. We divide the Alexa list on the basis of the ranks to create four domain name lists, Alexa top 1000 (Alexa1k), Alexa top 10,000 (Alexa10k), Alexa top 100,000 (Alexa100k), and Alexa top 1,000,000 (Alexa1M). The TVPs for the Alexa Top Sites are identified on the basis of these four lists. Figure 2 shows examples of typical domain names that fit the four patterns in Alexa1M. The graph indicates the relationships between domain names and their Alexa rank variations over time (note the logarithmic y-axis). In the null pattern of Alexa1M (Alexa1M-Null), the rank of a domain name has always been outside 1M and has never been listed in Alexa1M. The Alexa1M-Null pattern is intended to be one of the features or hints to boost true positive rates (TPRs), which is the ratio of correctly predicted malicious domain names to actual malicious domain names. This is because the rank for legitimate domain names is more likely to be within 1M, and new domain names by attackers cannot be in Alexa1M right after they have been registered. In the stable pattern of Alexa1M (Alexa1M-Stable), the rank of a domain name has always been within 1M and listed in Alexa1M. Alexa1M-Stable includes stable popular domain names; thus, this pattern can be used for improving true negative rates (TNRs), which is the ratio of correctly predicted legitimate domain names to actual legitimate domain names. In the fall pattern of Alexa1M (Alexa1M-Fall), the rank of a domain name was first within 1M, fell, and finally was delisted from Alexa1M. To improve TPRs, the Alexa1M-Fall pattern is intended to detect maliciously re-registered, parked, and hijacked domain names that changed from originally legitimate domain names. In the rise pattern of Alexa1M (Alexa1M-Rise), the rank of a domain name was first outside 1M and then increased to be within 1M. This Alexa1M-Rise pattern includes legitimate start-up Web sites' domain names during the specified time window to improve TNRs.
Example of TVPs in malicious domain name list (hpHosts)
We use the domain names listed in the public blacklist hpHosts [2] as the malicious domain name list. hpHosts provides malicious domain names of malicious Web sites engaged in exploits, malware distribution, and phishing. The TVPs for hpHosts are defined similarly for Alexa. Note that hpHosts does not have any continuous value, such as ranking, and only has information of whether domain names are listed. Figure 3 shows examples of typical domain names that fit the four patterns in hpHosts. In the null pattern of hpHosts (hpHosts-Null), a domain name has never been listed in hpHosts. This hpHosts-Null pattern can be used for improving TNRs because legitimate domain names are less likely to be listed in hpHosts. In the stable pattern of hpHosts (hpHosts-Stable), a domain name has always been listed in hpHosts. To improve TPRs, the hpHosts-Stable pattern includes domain names related to bullet-proof hosting providers, which provide network resources even to attackers. In the fall pattern of hpHosts (hpHosts-Fall), a domain name was once listed and then delisted. For example, this pattern includes domain names that were once abused and then sanitized to improve TNRs. In the rise pattern of hpHosts (hpHosts-Rise), a domain name was listed from the middle of the specified time window. This hpHosts-Rise pattern is intended to detect newly registered malicious domain names that attackers will use for a while. Specifically, many subdomain names can be created under the same domain name to bypass fully qualified domain names (FQDN)-level blacklists. Thus, the hpHosts-Rise pattern contributes to clarifying the situation to increase TPRs.
As described above, these TVPs in both legitimate or popular and malicious domain name lists contribute to boosting both TPRs and TNRs. Table 1 summarizes the relationships between the TVPs and their objectives. The effectiveness of using these patterns in real datasets is described later in Sect. 4.
Relationships between TVPs and objectives
TVPs
Alexa-null
Improving true positive rates (TPRs)
Alexa-stable
Improving true negative rates (TNRs)
Alexa-fall
Alexa-rise
hpHosts-null
hpHosts-stable
hpHosts-fall
hpHosts-rise
3 Our system: DomainProfiler
DomainProfiler identifies the temporal variation patterns (TVPs) of domain names and detects/predicts malicious domain names. Figure 4 gives an overview of our system architecture. DomainProfiler is composed of two major modules: monitoring and profiling. The monitoring module collects various types of essential data to evaluate the maliciousness of unknown domain names. The profiling module detects/predicts malicious domain names from inputted target domain names by using the data collected with the monitoring module. The details of each module are explained step-by-step in the following subsections.
Overview of our system
3.1 Monitoring module
The monitoring module collects three types of information that will be used later in the profiling module. The first is domain name lists. As discussed in Sect. 2, we need to collect the legitimate/popular domain name list (Alexa) and malicious domain name list (hpHosts) daily to create a database of listed domain names and their time series variations.
The second is historical DNS logs, which contains time series collections of the mappings between domain names and IP addresses. A passive DNS [46] is one typical way to collect such mappings by storing resolved DNS answers at large caching name servers. Due to the privacy policy of our organization, we do not use the passive DNS approach. Instead, we actively send DNS queries to domain names to monitor and build a passive DNS-like database. On the plus side, this active monitoring contains no personally identifiable information of the senders. Moreover, we can control DNS queries so as not to contain disposable domain names [13], which are non-informative and negatively affect the database. For example, disposable domain names are one-time domain names automatically generated to obtain a user's environmental information by using certain antivirus products and web services. Since these domain names are distinct, the mappings between domain names and IP addresses significantly increase the database size with non-informative information for evaluating the maliciousness of domain names. On the minus side of active monitoring, we can only query known domain names and cannot gather the mappings of unknown domain names. Thus, to partially address this problem, we have expanded known existing domain names as much as possible. For example, we have extracted all domain names in domain name lists such as Alexa and hpHosts. Moreover, we crawl approximately 200,000 web pages every day to gather web content and extract domain names. Furthermore, we query a search engine API (2.5M queries/month) to expand the domain names on the basis of the above results.
The third type of information is the ground truth, which will be used to label the training dataset and evaluate the effectiveness of our system. Our ground truth includes the results of web client-based honeypots (honeyclients) and sandbox systems and some subscription-based data such as VirusTotal [3] and professional services by a security vendor. The details of the ground truth we used are given later in Sect. 4.1.
3.2 Profiling module
The profiling module consists of three steps that use the information collected from the monitoring module to finally output malicious domain names from inputted target domain names.
3.2.1 Step 1: identifying TVPs
Step 1 identifies the TVPs for each input target domain name. (The definition of a TVP was given in Sect. 2.) First, we query the input domain name to the domain name lists database to obtain the time series data of listed domain names that match the second-level domain (2LD) part of the input domain name. The database consists of five domain name lists: Alexa1k, Alexa10k, Alexa100k, Alexa1M, and hpHosts.
To precisely define the TVP of every domain name, we define that the top-level domain (TLD) includes an effective TLD or public suffix [34] such as .com.au, .co.jp, and .co.uk, as shown in Fig. 5. In general, TLDs are divided into generic top-level domains (gTLDs), such as .com, .net, and .org, and country code top-level domains (ccTLDs), such as .au, .jp, and .uk. If we do not use effective TLDs, the 2LD parts of gTLDs and ccTLDs differ significantly. For example, in the gTLD case of foo.bar.example.com, the 2LD part is example.com; however, in the ccTLD case of baz.qux.example.co.jp, the 2LD part is co.jp. Our definition of including effective TLDs is intended to treat both gTLD and ccTLD identically, that is, the 2LD part in the above ccTLD example is example.co.jp in this paper.
Second, the TVPs of the matched 2LD parts within a specified time window are identified using the predefined patterns (null, stable, fall, and rise), as shown in Sect. 2.
Third, the numbers of matched 2LD parts for the four patterns are counted and used as feature vectors in a machine learning algorithm. Specifically, the feature vectors created in step 1 correspond to Nos. 1–20 of the features listed in Table 2, that is, Nos. 1–4 are for Alexa1k, Nos. 4–8 are for Alexa10k, Nos. 9–12 are for Alexa100k, Nos. 13–16 are for Alexa1M, and Nos. 17–20 are for hpHosts.
Alexa1k-null
# of BGP Prefixes (FQDN)
rDomain
# FQDNs
(Legitimate/ Popular)
Alexa1k-stable
(BGP)
# of BGP prefixes (3LD)
(FQDN)
Mean lengths
Alexa1k-fall
SD lengths
Alexa1k-rise
# of Countries (FQDN)
Mean distribution
Alexa10k-null
# of countries (3LD)
(1-gram)
Median distribution
Alexa10k-stable
SD distribution
Alexa10k-fall
# of IP addresses (3LD)
Alexa10k-rise
(2-grams)
Alexa100k-null
# of organizations (FQDN)
Alexa100k-stable
# of ASNs (FQDN)
Alexa100k-fall
(ASN)
# of ASNs (3LD)
Alexa100k-rise
Alexa1M-null
# of registries (FQDN)
# TLDs
Alexa1M-stable
(Registration)
# registries (3LD)
(TLD)
# of Ratios of .com
Alexa1M-fall
# of registries (2LD)
Alexa1M-rise
# of dates (FQDN)
# of dates (3LD)
(Malicious)
Definition of domain name terms
3.2.2 Step 2: appending DNS-based features
Step 2 appends DNS-based features to the output of step 1, which are input target domain names with identified TVPs. This step is intended to detect malicious domain names that share common features in terms of IP addresses and domain names. We reviewed and analyzed the typical features proposed for known approaches to select the DNS-based features. The known approaches related to ours are summarized later in Sect. 6. As a result of verifying the availability and effectiveness of the features, we decided to use the features proposed for Notos [5]. The DNS-based features are mainly divided into two types: related IP addresses (rIPs) and related domain names (rDomains).
To acquire features of rIPs, we need to first construct a graph of rIPs for each target domain name. Figure 6 shows an example of rIPs for foo.example.com. The graph is a union of every resolved IP address corresponding to each domain name at the FQDN level and its parent domain name levels, such as 3LD and 2LD, from historical DNS logs collected in the former monitoring module. In Fig. 6, FQDN and 3LD (foo.example.com) correspond to the IP address 192.0.2.2 at time \(t-1\) and 198.51.100.2 at t, and 2LD (example.com) corresponds to the IP address 192.0.2.1 at \(t-1\) and 198.51.100.1 at t. Thus, these four IP addresses are defined as rIPs for foo.example.com. Then, we extract the features from rIPs. These features consist of three subsets: border gateway protocol (BGP), autonomous system number (ASN), and registration.
The BGP features, Nos. 21–29 in Table 2, are created from the information of BGP prefixes corresponding to the related IP addresses (rIPs) of each target domain name. To obtain the required BGP information, we refer to the CAIDA dataset [12]. Specifically, we extract the number of rIPs' BGP prefixes of the target FQDN (No. 21), that of the 3LD part of the target (No. 22), and that of the 2LD part of the target (No. 23); the number of countries for the BGP prefixes of the target FQDN (No. 24), that of the 3LD part of the target (No. 25), and that of the 2LD part of the target (No. 26); the number of rIPs for the 3LD part of the target (No. 27) and that for the 2LD part of the target (No. 28); and the number of organizations for the BGP prefixes of the target FQDN (No. 29).
The ASN features, Nos. 30–32 in Table 2, are created from the ASN information corresponding to the rIPs of each target domain name. To obtain the ASN information, we refer to the MaxMind GeoIP2 databases [32]. Specifically, we extract rIPs' ASNs of the target FQDN (No. 30), that of the 3LD part of the target (No. 31), and that of the 2LD part of the target (No. 32).
The registration features, Nos. 33–38 in Table 2, are created from the IP address registration information corresponding to the rIPs of each target domain name. To obtain the registration information, we refer to the information of delegated IP addresses [30] from all regional Internet registries (RIRs), namely AFRINIC, APNIC, ARIN, LACNIC, and RIPE NCC. Specifically, we extract the number of RIRs of the rIPs for the target FQDN (No. 33), that of the 3LD part of the target (No. 34), and that of the 2LD part of the target (No. 35) and the diversity or number of allocated dates of the rIPs for the target FQDN (No. 36), that of the 3LD part of the target (No. 37), and that of the 2LD part of the target (No. 38).
Graph for related IP addresses (rIPs)
Graph for related domain names (rDomains)
On the other hand, to acquire the features of related domain names (rDomains), we need to construct a graph of rDomains for each target domain name using the historical DNS logs collected in the monitoring module. Figure 7 shows an example of rDomains for foo.example.com. The graph is a union of domain names pointing to IP addresses in the same autonomous system number (ASN) of the historical IP addresses of each target domain name. In Fig. 7, the ASN for the target foo.example.com is AS64501 and another IP address 192.0.2.3 in AS64501 is connected to the domain names bar.example.net and baz.example.org. Thus, these three domain names are defined as rDomains for foo.example.com, and we extract their features. These features consist of three subsets: FQDN string, n-grams, and top-level domain (TLD).
The FQDN string features, Nos. 39–41 in Table 2, are created from the set of rDomains for each target domain name. Specifically, we extract the number of FQDNs (No. 39) in the rDomains, mean length of the FQDNs (No. 40), and standard deviation (SD) of the length of the FQDNs (No. 41).
The n-gram features, Nos. 42–50 in Table 2, are created from the occurrence frequency of n-grams (\(n=1,2,3\)) in the set of rDomains for each target domain name. Note that the units of n-grams in this paper are denoted with letters; thus, 2-grams for example.com consists of pairs of letters such as ex, xa, and am. Specifically, we extract the mean, median, and SD of 1-gram (Nos. 42–44) in rDomains, those of 2-grams (Nos. 45–47), and those of 3-grams (Nos. 48–50).
The TLD features, Nos. 51–55 in Table 2, are created from TLDs in the set of rDomains for each target domain name. Specifically, we extract the distinct number of TLDs in the set of rDomains (No. 51), ratio of the .com TLD in the set (No. 52), and mean, median, and SD of the occurrence frequency of the TLDs in the set (Nos. 53–55).
Target domain names (training set)
Legitimate-alexa
Malicious-hpHosts
Target domain names (test set)
Honeyclient-exploit
Honeyclient-malware
Sandbox-malware
Sandbox-C&C
Pro-C&C
Pro-phishing
Legitimate-new
Domain name lists DB
AlexaDB
hpHostsDB
Historical DNS logs
DNSDB
3.2.3 Step 3: applying machine learning
Step 3 involves applying a machine learning algorithm to the outputs of step 2, which consist of input target domain names with all the features listed in Table 2. This step is designed to achieve our goal of detecting/predicting domain names that will possibly be used maliciously in future. To this end, we use supervised machine learning to effectively find possible malicious domain names from unvalued input domain names. Supervised machine learning basically consists of two phases: training and test. The training phase generates a learning model on the basis of the labeled malicious and legitimate training data by using extracted features. The test phase uses this learning model to calculate the maliciousness of each input domain name by using the extracted features to detect/predict malicious domain names.
Among many supervised machine learning algorithms, we selected Random Forests [10] because of their high accuracy, as identified in our preliminary experiments, and high scalability, which we can easily parallelize. The concept image of Random Forests is shown in Fig. 8. Random Forests consist of many decision trees, which are constructed from input data with randomly sampled features. The final prediction is output by the majority vote of the decision trees.
Random Forests
4 Evaluation
DomainProfiler was evaluated using real datasets including an extensive number of domain names. This section explains how we evaluated it in terms of its effectiveness at using temporal variation patterns (TVPs) and detecting/predicting malicious domain names used in real cyber-attacks.
4.1 Dataset
Our evaluations required three types of datasets, as shown in Table 3: target domain names, domain name list databases, and historical DNS logs.
The first dataset was target domain names, which were composed of training and test sets. The training set was labeled data for creating a learning model in Random Forests. To create the Legitimate-Alexa, we extracted fully qualified domain names (FQDNs) on the basis of the domain names listed in Alexa100k. Since most domain names in Alexa are second-level domain (2LD) names and do not have IP addresses, we used a search engine API to randomly extract existing FQDNs in the wild from each 2LD name. Moreover, as shown in Sect. 3.1, we used our ground truth, such as the results of honeyclients and subscription-based professional data, to eliminate the possibility that malicious domain names are in Legitimate-Alexa. As for Malicious-hpHosts, we used a process similar to that for Legitimate-Alexa; that is, we extracted FQDNs from 2LD names listed in hpHosts using a search engine and verified the maliciousness by using our ground truth.
The test set was used for evaluating the predictive detection performance of our system. Note that there were no overlaps between the training and test sets and the collected period of the test set was after that of the training set. Thus, we can use the test set to simulate the performance of DomainProfiler at predicting domain names that will be abused in future. For web client-based honeypot (honeyclient) datasets, we used our honeyclients to collect newly malicious FQDNs, particularly related to drive-by download attacks, from March to October 2015. In a typical drive-by download attack, a legitimate Web site is compromised to lead users to malicious domain names. In our evaluations, we used two types of malicious domain names owned/managed by attackers. Honeyclient-Exploit contained FQDNs of Web sites engaged in distributing exploits that target users' browsers and their plug-ins. Honeyclient-Malware was the collection of FQDNs used for malware distribution Web sites in drive-by download attacks. To create sandbox datasets, we used our sandbox systems to run 13,992 malware samples randomly downloaded from VirusTotal [3]. The Sandbox-Malware dataset contained FQDNs connected by malware samples (e.g., downloader) to download other malware samples. The Sandbox-C&C dataset was a collection of FQDNs of command and control (C&C) servers detected by using our sandbox. The Pro-C&C and Pro-Phishing datasets were FQDNs used for C&C servers and phishing Web sites. Note that the Pro datasets were obtained from commercial and professional services provided by a security vendor, and the FQDNs we selected were only those that had a high likelihood of being abused by attackers. Furthermore, we prepared entirely different legitimate domain names from other datasets as the Legitimate-New dataset. This dataset was used to fairly evaluate false positives when operating DomainProfiler in March 2015. The Legitimate-New dataset only contained legitimate domain names observed in a large campus network. We manually checked the domain names and excluded any contained in the other training and test sets.
The second dataset was a domain name list database used for identifying TVPs. As explained in Sect. 2, we selected Alexa top sites as a legitimate/popular list and hpHosts as a malicious list since they are continuously obtainable daily.
The third dataset was historical DNS logs, which involved time series collections of the domain name and IP address mappings. As discussed in Sect. 3.1, we actively sent DNS queries to the domain names we found by using domain name lists, our web crawler, and search engine API. That is, we extracted all domain names in domain name lists such as Alexa and hpHosts. Moreover, we crawled approximately 200,000 web pages every day to gather web content and extracted domain names from the content and their static and dynamic hyperlinks. Furthermore, we expanded the number of domain names by using an external search engine API (2.5M queries/month) on the basis of the above domain names. In our evaluations, we used over 47M distinct FQDNs and their time series changes from October 2014 to February 2015, as shown in Table 3.
4.2 Parameter tuning
Before we evaluated our DomainProfiler, we needed to tune two types of parameters: the size of the time window in TVPs (step 1) and the required parameters to run the Random Forests (step 3).
Here is the summary of evaluation criteria discussed in the following sections. A true positive (TP) is the number of correctly predicted malicious domain names, a false positive (FP) is the number of incorrectly predicted legitimate ones, a false negative (FN) is the number of incorrectly predicted malicious ones, and a true negative (TN) is the number of correctly predicted legitimate ones. The true positive rate (TPR), otherwise known as recall, is the ratio of correctly detected malicious domain names to actual malicious domain names. The true negative rate (TNR) is the ratio of correctly determined legitimate domain names among actual legitimate domain names. The false positive rate (FPR) is the ratio of incorrectly determined legitimate domain names among actual legitimate domain names. The precision is the ratio of actual malicious domain names to domain names detected as malicious by DomainProfiler. The F-measure is the ratio that combines recall and precision, i.e., it is calculated as the harmonic mean of precision and recall.
4.2.1 Time window size
We conducted tenfold cross-validations (CVs) using the training set with variable time window sizes (from 1 to 365 days) to select the time window size on the basis of the evaluation criteria. Figure 9 shows two graphs: the left one corresponds to the time window sizes from 1 to 365 days and the right one to those from 1 to 7 days. These two graphs reveal that the best time window size for TVPs is only 2 days. This is not a surprising result for us in terms of the nature of domain names or TVPs because attackers abuse the DNS to generate a huge volume of distinct domain names from one day to the next, so keeping old information over a long period decreases the F-measure of our system.
4.2.2 Random Forests
Random Forests [10] require two parameters to run. One is the number of decision trees. As explained in Sect. 3.2.3, Random Forests consist of multiple decision trees; thus, we needed to decide how many trees to make beforehand. As is the case with the aforementioned time window size, we conducted tenfold CVs by changing the number of trees to determine the optimum number of decision trees. The left graph in Fig. 10 shows the relationships between the number of trees and the F-measure. The graph shows that F-measures are stable over 100 trees. Thus, we decided to use 100 decision trees in the following evaluations.
The other parameter is the number of sampled features in each individual decision tree. Random Forests construct decision trees from input data that have randomly sampled features to improve overall accuracy. We conducted tenfold CVs again to search for the optimum number of sampled features. The right graph in Fig. 10 shows that the best F-measure is obtained when there are seven sampled features.
Tuning time window size
Tuning Random Forests' parameters
Detection performance with different feature sets (cross-validation)
TPR/recall
F-measure
rIP+rDomain
TVP+rIP+rDomain
ROC curves
4.3 Feature set selection
Now that we have selected the optimal parameters (the time window size, number of trees, and number of sampled features in each tree), this section compares the detection performance with different feature sets. The feature sets include the temporal variation pattern (TVP), related IP address (rIP), related domain name (rDomain), combination of rIP and rDomain (rIP+rDomain), and combination of TVP, rIP, and rDomain (TVP+rIP+rDomain). We conducted tenfold CVs using the training set and the optimal parameters by changing the feature sets to estimate how accurately each feature set will perform in theory. Table 4 illustrates the detection performance using the above evaluation criteria. Note that the number of FQDNs varies with feature sets due to the availability of each feature. For example, some domain names have no rIPs and/or rDomains. Also, Fig. 11 shows the receiver operator characteristic (ROC) curves. An ROC curve shows a pair of a FPR and a TPR corresponding to a particular decision cutoff point. Thus, if the ROC curve of a feature set rises more rapidly, this means that the performance of the feature set is better. Table 4 and Fig. 11 show that using our TVP features significantly contributes to achieving better detection performance than using only DNS-based features. Using only DNS-based features (rIP, rDomain, and rIP+rDomain) does not go beyond 0.90 in any evaluation criteria. These results show that using only conventional DNS-based features [5] is insufficient for detecting malicious domain names in current attack ecosystems. However, combining the DNS-based features with our TVP features (TVP+rIP+rDomain) achieves the best results, specifically, a TPR/recall of 0.975, TNR of 0.991, FPR of 0.009, precision of 0.990, and F-measure of 0.983. These results indicate that our key idea based on using TVPs is effective for improving both TPR and TNR exactly as intended.
Predictive detection performance of DomainProfiler (feature set: TVP+rIP+rDomain)
Predictive detection performance of conventional DNS-based features (feature set: rIP+rDomain)
4.4 System performance
We evaluated the system performance of a prototype version of DomainProfiler. Specifically, we calculated the execution time and data size in each step when we conducted a tenfold CV using the training set with the optimal parameters and best feature set (TVP+rIP+rDomain). Step 1 (identifying TVPs) was executed on a single server with a 10-core 2.2-GHz CPU and 128-GB RAM. The execution time for extracting TVP features from 173,409 FQDNs was 61 s, which was equivalent to 0.0004 s/FQDN. The file sizes of the domain name list database (SQL) were 1.4 GB in Alexa and 300 MB in hpHosts. Step 2 (appending DNS-based features) was executed as a MapReduce job on a Hadoop cluster, which had 2 master servers (16-core 2.4-GHz CPU, 128-GB RAM) and 16 slave servers (16-core 2.4-GHz CPU, 64-GB RAM). The execution time for extracting rIP features from 173,409 FQDNs was 20 h (0.42 s/FQDN) and that for rDomain features was 96 h (1.99 s/FQDN). The file size of the historical DNS logs used for extracting these DNS-based features was 212 GB. Step 3 (applying machine learning) was executed on the same server as step 1. The execution time for one-time training from 156,068 FQDNs was 28 s (0.0001 s/FQDN) and that for the test from 17,341 FQDNs was 8 s (0.0005 s/FQDN). These evaluations prove the basic feasibility of our proposed system and reveal that step 2 requires far more resources and time to execute than steps 1 and 3. The reason for step 2's high cost is the size of the graphs for rIPs and rDomains. Currently, some domain names used by hypergiants, such as Google, Amazon, and Akamai, have a huge number (over 10,000) of rIPs and rDomains. This fact raises the problem of a high cost for extracting conventional DNS-based features [5]. However, from the results explained in Sect. 4.3, this problem will be solved if our system sacrifices 0.002 of its TPR to use the feature set TVP instead of TVP+rIP+rDP. This is a trade-off between system performance and detection performance. Thus, we should configure our system on the basis of this situation.
4.5 Predictive detection performance
We evaluated the predictive detection performance of DomainProfiler; that is, whether we can discover domain names that may be abused in the future. The aforementioned evaluations were based on cross-validations (CVs); however, this section focuses on the evaluation of the detection performance of new malicious domain names that first appeared after March 1, 2015, by using only the information as of February 28, 2015. Specifically, we used the training set shown in Table 3 to create a learning model first and then input the test set in Table 3 to evaluate the predictive detection performance. In this evaluation, we set the optimal parameters discussed in Sect. 4.2. The best feature set (TVP+rIP+rDomain) discussed in Sect. 4.3 was compared with the feature set (rIP+rDomain) that had only conventional DNS-based features [5]. Tables 5 and 6 list the evaluation results of using TVP+rIP+rDomain and rIP+rDomain. Note that the six test datasets (Honeyclient-Exploit, Honeyclient-Malware, Sandbox-Malware, Sandbox-C&C, Pro-C&C, and Pro-Phishing) only consist of malicious domain names; thus, there are no false positives (FPs), true negatives (TNs), or their related evaluation criteria in the tables. On the other hand, the other test dataset (Legitimate-New) only consists of legitimate domain names; thus, there are no true positives (TPs) or false negatives (FNs) as explained in Sect. 4.1.
In terms of the true positive rate (TPR/recall), DomainProfiler using the feature set (TVP+rIP+rDomain) achieved extremely high TPRs in all test sets; our system achieved TPRs of 0.985 in Honeyclient-Exploit and Honeyclient-Malware. Moreover, our system accurately detected/predicted command and control (C&C) domain names in Sandbox-C&C and Pro-C&C, while our training set did not include labeled C&C domain names. This is not a surprising result because our TVP is designed to exploit the common characteristics of attackers' domain names. On the other hand, DomainProfiler using only the conventional features (rIP+rDomain) achieved a TPR of 0.402 at best. Comparing these results illustrates that our TVP features successfully contribute to predicting domain names that will be used maliciously in future.
Early detection performance of DomainProfiler (feature set: TVP+rIP+rDomain)
days_Min
days_1stQu
days_2ndQu
days_Mean
days_3rdQu
days_Max
Early detection performance of conventional DNS-based features (feature set: rIP+rDomain)
We also evaluated the true negative rate (TPR) and false positive rate (FPR) using the Legitimate-New dataset since the FPR results during the training period explained in Sect. 4.3 cannot be generalized to testing period or predictive detection performance. Tables 5 and 6 show that DomainProfiler using the feature set (TVP+rIP+rDomain) achieved a TNR of 0.976 and a FPR of 0.024, whereas that using conventional features (rIP+rDomain) achieved a TNR of 0.823 and a FPR of 0.177. In total, DomainProfiler achieved an F-measure of 0.978 in the predictive or future performance evaluation. These results indicate that our TVP features contribute to boosting both TPR and TNR. We explain in detail how each TVP feature contributes to successful predictions later in Sect. 4.7.
In terms of early detection of future malicious domain names, we investigated when the system can detect such domain names. Specifically, we analyzed the number of days that elapsed from February 28, 2015, when the learning model was created, for malicious domain names to be detected by the system. For example, if the system correctly detected and identified a new malicious domain name on March 7, 2015, the elapsed number of days for the domain name is seven. Tables 7 and 8 show the descriptive statistics of the elapsed days for malicious domain names for each feature set. Note that we only count domain names in the TP of each dataset shown in Tables 5 and 6. The descriptive statistics include the minimum (days_Min); the first quartile (days_1stQu), which means the value cutoff at the first 25% of the data; the second quartile, which is also called the median and is the value cutoff at 50% of the data; the mean (days_Mean); the third quartile (days_3rdQu), which is the value cutoff at 75% of the data; and the maximum (days_Max). Table 7 shows that our proposed system (TVP+rIP+rDomain) can precisely predict future malicious domain names 220 days before the ground truth, such as honeyclients and sandbox systems, and identify them as malicious in the best case. Comparing the above results with Table 8 reveals that the conventional DNS-based feature set (rIP+rDomain) [5] also detects malicious domain names early; however, the number of detected domain names (TP) is quite small as shown in Table 6. We conclude that our proposed system using TVPs outperforms the system using only the conventional DNS-based feature set from the perspectives of both accuracy and earliness.
Contribution of each feature
Feature (TVP)
Feature (rIP)
Feature (rDomain)
# BGP Prefixes (FQDN)
# BGP Prefixes (3LD)
# Countries (FQDN)
# Countries (3LD)
# IP addresses (3LD)
# Organizations (FQDN)
# ASN (FQDN)
# ASN (3LD)
# Registries (FQDN)
# Ratio of .com
# Dates (FQDN)
# Dates (3LD)
4.6 Effectiveness of each feature
In this section, we analyzed how each feature shown in Table 2 contributes to accurate detection of malicious domain names. To this end, we calculated the importance of each feature in the TVP, rIP, and rDomain in our trained model. Specifically, we used the criterion called the Gini Index (GI) in Random Forests [10]. Random Forests have a mechanism to evaluate the importance of each feature by adding up the GI or impurity decreases for each feature over all decision trees constructed in the algorithm of Random Forests. The GI calculation is explained in more detail elsewhere [11, 27, 28].
Table 9 illustrates our analysis results including the GI score and its rank for each feature over all 55 features listed in Table 2. A higher GI score means that the corresponding feature is more important when detecting malicious domain names. The rank provides a ranking of each feature's contribution among all features. Overall, our TVP features (Nos. 1–20) had more contributions and higher GI scores than both rIP features (Nos. 21–38) and rDomain features (Nos. 39–55). Specifically, 9 features in TVP ranked between 1 and 10 (top 10): Alexa100k-null (No. 9), Alexa100k-stable (No. 10), Alexa100k-fall (No. 11), Alexa1m-null (No. 13), Alexa1m-stable (No. 14), Alexa1m-fall (No. 15), hpHosts-null (No. 17), hpHosts-stable (No. 18), and hpHosts-fall (No. 19). The results for the greater contributions of TVP features are also supported by the previous evaluation results shown in Sect. 4.3. We further detail the case studies of how each TVP feature contributes to accurate detection later in Sect. 4.7.
Now we focus on features that had lower GI scores and ranked between 46 and 55 (the bottom 10). The underlying nature of the Alexa ranking is the reason for the six low ranked TVP features using Alexa1k and Alexa10k: Alexa1k-stable (No. 2), Alexa1k-fall (No. 3), Alexa1k-rise (No. 4), Alexa10k-stable (No. 6), Alexa10k-fall (No. 7), and Alexa10k-rise (No. 8). Specifically, top 10k in Alexa [1] only contains domain names of extremely popular Web sites, and our datasets shown in Sect. 4.1 did not contain such easy-to-answer domain names. As a result, the importance of these TVP features using Alexa1k and Alexa10k is relatively low. As for Alexa1m-rise (No. 16), the number of domain names matching this TVP is considerable smaller than the total number of domain names in our dataset. In terms of low ranked rIP features such as # Organizations (FQDN) (No. 29), # Registries (FQDN) (No. 33), and # Registries (3LD) (No. 34), the reason is the difference between malicious domain names in the 2000s and those currently used. The conventional rIP features were originally developed and evaluated in 2009 [5]. However, today's malicious domain names abuse cloud hosting services much more than in 2009 as these services have gained in popularity. For example, in a typical cloud hosting service, IP addresses or BGP prefixes are shared among multiple domain names. Thus, the importance of the above three rIP features resulted in lower GI scores. To the best of our knowledge, we are the first to show these quantitative and detailed results about conventional DNS-based features and their effectiveness for detecting today's malicious domain names. We believe these results will be useful for other researchers to develop new domain reputation systems in future.
Dataset for lifespan analysis
# of FQDNs in test set
# of Queried FQDNs
# of Answered FQDNs
4.7 Effectiveness of our temporal variation patterns
We further analyzed how our temporal variation pattern (TVP) features contribute to increasing the true positive rate (TPR) and true negative rate (TNR) simultaneously. We present some noteworthy case studies in our TVPs defined in Sect. 2. In this analysis, we used the TVP+ rIP+rDomain feature set and the same settings and dataset as the previous evaluation discussed in Sect. 4.5.
Alexa1M-Null: This TVP is intended to boost the TPR, as described in Sect. 2. Our analysis revealed that this TVP was especially effective for malicious domain names using a domain generation algorithm (DGA) and abusing new generic top-level domains (gTLDs) such as .xyz and .solutions. This is because these domain names are less likely to be within Alexa1M.
Alexa1M-Stable: This TVP successfully determined the characteristics of somewhat popular domain names to improve the TNR.
Alexa1M-Fall: This TVP is designed to detect the changing of malicious domain names to improve the TPR. We observed two major types of malicious domain names that fit this TVP. One type is expired domain names due to the termination of services or the merger and acquisition of companies. Some of these expired domain names were re-registered by third-party attackers to execute domain parking or cyber-attacks. The other type is domain names that were changed from legitimate to malicious because the Web site of the domain name was not well managed and had poor security.
Alexa1M-Rise: This TVP attempts to determine the features of legitimate domain names of start-up Web sites to improve the TNR. We observed many domain names corresponding to this TVP such as those of new companies, services, movies, and products.
hpHosts-Null: This TVP successfully improved the TNR because the second-level domain (2LD) parts of popular/legitimate domain names were less likely to be listed in hpHosts.
hpHosts-Stable: This TVP is designed to determine the characteristics of malicious domain names abusing easy-to-use services, such as bullet-proof hosting, to improve the TPR. For example, we observed many subdomains using a domain generation algorithm (DGA) under the same 2LD part such as 84c7zq.example.com.
hpHosts-Fall: This TVP is intended to boost the TNR. We confirmed that some domain names under well-managed networks fit this TVP because these domain names were once abused and then quickly sanitized. In this case, the TVP contributed to the accurate prediction of future legitimate domain names.
hpHosts-Rise: This TVP is designed to help detect or predict malicious domain names more accurately to improve the TPR. We mainly observed two types of domain names that fit this TVP. One is domain names that heavily used the DGA in both 2LD and third-level domain (3LD) parts of domain names, e.g., 14c2c5h8[masked].yr7w2[masked] .com. We observed that this type of 2LD will be continuously used for a while by attackers to create many subdomain names. The other is domain names under free subdomain name services, which offer subdomain name creation under 2LD parts, such as .flu.cc and .co.nr. These services are easily abused by attackers for creating distinct domain names.
4.8 Lifespan of detected domain names
We analyzed the lifespan of each malicious domain name detected by DomainProfiler to show the characteristics of domain names abused in various types of cyber-attacks. To this end, we used logs collected from a set of large-scale DNS cache servers. The data are called a passive DNS database (DNSDB) [17], which records various information about a domain name, e.g., a list of resolved IP addresses, history of the accesses to the domain name, and first/last seen timestamp. Due to a limitation of our API usage in the DNSDB, we randomly sampled 10,000 FQDNs from our test set shown in Table 3. The detailed numbers of the selected FQDNs are shown in Table 10. We used all FQDNs of Honeyclient-Exploit, Honeyclient-Malware, Sandbox-Malware, Sandbox-C&C, and Pro-C&C, whereas we randomly selected only 50 FQDNs of Pro-Phishing. We queried the 10,000 FQDNs to the passive DNSDB and obtained the results for 7429 FQDNs. The passive DNSDB has no results for the remaining 2571 FQDNs. One reason for this low cover rate is due to the data collection points in the passive DNSDB. Since it does not cover all of cache servers in the world, some FQDNs created using the DGA or those used only for specific targets (e.g., Advanced Persistent Threat (APT)) are not covered.
FQDNs used for parking and sinkhole services
# of Parking FQDNs
# of Sinkhole FQDNs
Using the results from the passive DNSDB, we conducted lifespan analysis of the malicious domain names. In this evaluation, a lifespan of a domain name is defined to be the period from the first seen timestamp to the last seen timestamp; that is, the period in which the domain name continually has corresponding IP addresses. To precisely analyze the lifespan from the passive DNSDB results, we applied a survival analysis method based on the Kaplan-Meier estimator [23]. The reason for using this method is the statistical characteristics of the lifespan data. Specifically, the lifespan data are considered to be right-censored; that is, the collection period for each domain name is not the same and some domain names remain alive at the end of the data collection period. Our survival analysis results for each test set are summarized in Fig. 12. The x-axis shows elapsed time (days) from a first seen timestamp, and the y-axis is the survival probability that a domain name is still alive and has IP addresses after the elapsed time. Figure 12 clearly reveals that the survival probability or the lifespan is very different for each test set; in particular, both Honeyclient-Exploit and Honeyclient-Malware have much shorter lifespans than the others. This is because each test set contains domain names engaged in different types of cyber-attacks such as drive-by download attacks (Honeyclient-Exploit and Honeyclient-Malware), additional malware download (Sandbox-Malware), C&C (Sandbox-C&C, Pro-C&C), and phishing (Pro-Phishing), as stated in Sect. 4.1. These results also indicate that our TVP in DomainProfiler successfully covered a wide range of malicious domain names used in a series of cyber-attacks as intended.
Survival analysis plot for domain names in our test set
We further analyzed what causes the differences in survival probabilities or lifespans. First, we selected domain names that have shorter lifespans such as Honeyclient-Exploit and Honeyclient-Malware. These domain names were abused in a series of drive-by download attacks. Exploit kits used in drive-by download attacks are designed to avoid their exploits from being analyzed [16]. As a result, attackers need to churn their domain names used for exploit kits. Grier et al. [19] also reported that the lifespan of domain names used for distributing exploit kits is very short. This specific attack characteristic leads to a shorter lifespan in our Honeyclient-Exploit and Honeyclient-Malware dataset. Second, we analyzed domain names that have longer lifespans, namely Sandbox-Malware, Sandbox-C&C, Pro-C&C, and Pro-Phishing. Our analysis revealed that there were some reasons behind the longer lifespan. One reason is parking services. A parking domain name is mainly used for displaying advertisements [4, 26, 45]. Domain names abused in cyber-attacks tend to use such parking services later to monetize the malicious traffic from malware-infected hosts [26]. We detected such parking domain names in our dataset by following the method proposed by Vissers et al. [45]. Our detection results are summarized in Table 11. The results indicate that some domain names, especially in Sandbox-Malware, Sandbox-C&C, Pro-C&C, and Pro-Phishing, were found to currently use such parking services. Please note that we adopted only reliable detection patterns; thus, the number is lower-bound. Another reason behind the longer lifespan is the sinkhole operation. A sinkholed domain name is an originally malicious domain name that is treated and then controlled by security organizations [24, 36]. In such a case, an IP address is continuously associated with a domain name; thus, the lifespan of the domain name is long. To detect such sinkholed domain names, we followed the same approach originally proposed by Kührer et al. [24]. Table 11 shows the results of our sinkhole detection. We found that the only C&C domain names are operated as sinkhole in our dataset. Please note that we only adopted reliable detection patterns; thus, the number is lower-bound. To explore other reasons, we randomly chose and manually analyzed 100 FQDNs with long lifespans. Seventeen showed some sort of default pages offered by web hosting services, 10 were used in parking services that were not detected by the parking detection patterns described above, 6 were continuously used for C&C servers that offer a certain type of attack command or fingerprint results, and the other 67 had returned connecting errors as of September 2016.
Defended-against malware download activities using DomainProfiler
Malware family
# Blocked FQDNs
# Blocked samples
First submission date
bladabindi
gamarue
banload
ramnit
zusy
zbot
upatre
badur
ymeta
2015/10/11–2016/3/5
yakes
2015/10/8–2016/9/12
barys
bayrob
2015/10/12–2016/9/15
bublik
soxgrave
4.9 Defending against malware activities
In this evaluation, we provide additional research results on the capability of defending against malware activities using malicious domain names detected using DomainProfiler. Specifically, we analyzed what types of malware download and C&C activities we could defend using DomainProfiler's output. To this end, we created a new sandbox dataset by randomly downloading 354,953 malware samples from VirusTotal [3] and running them in our sandbox system between October 2015 and September 2016. To ensure a fair evaluation, we only used new malware samples that were first seen after October 8, 2015; that is, no data overlapped with those in the datasets shown in Table 3. All malware samples in the dataset were with their malware family information based on 57 different antivirus vendors' scan results. To resolve labeling difficulties such as the lack of a standard naming convention and different family names between antivirus vendors, we used the recently developed tool AVClass [39] to output the most likely malware family name for each malware sample. The total number of malware family names output by AVClass was 310, and we determined that the malware samples and their corresponding malware family names are not biased.
Table 12 lists the top 15 defended-against malware download activities in terms of the number of blocked FQDNs provided by DomainProfiler. In this case, the malware download activities mean that malware samples (e.g., downloader) connect to malicious FQDNs to download other malware samples. Table 12 also shows that the number of malware samples we could block under the condition of blacklisting the malicious FQDNs detected using DomainProfiler as of February 28, 2015, and the period of the first submission dates of corresponding malware samples in VirusTotal. These results indicate that we could defend against at least 7537 new or future malware downloads using only 119 blacklisted FQDNs provided by DomainProfiler.
In contrast, Table 13 lists the top 15 defended-against malware C&C activities. In this case, the malware C&C activities mean that malware samples connect to malicious FQDNs to communicate with C&C servers. Similar to Table 12, Table 13 also shows the number of blocked FQDNs, the number of blocked samples, and the period of first submission date. These results illustrate that we could defend against at least 6933 future malware C&C communications using only 1313 blacklisted FQDNs provided by DomainProfiler
The above evaluation results indicate that DomainProfiler could defend against various types of malware activities even seven months after blacklisting and prove the effectiveness of DomainProfiler's design. That is, we focus on common and generic features that can contribute to detecting malicious domain names used in a wide variety of cyber-attacks.
Defended-against malware C&C activities using DomainProfiler
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This section discusses possible evasion techniques against DomainProfiler and problems when using the predicted malicious domain names generated from our system as countermeasures to protect users from cyber-attacks.
5.1 Evading DomainProfiler
DomainProfiler is designed to exploit the temporal variation patterns (TVPs) of malicious domain names used by attackers. There are three possible techniques to evade our system. One is to avoid using domain names as attack infrastructure. If attackers do not use domain names, we can more easily take countermeasures such as just blocking them using IP addresses. The cost of changing IP addresses is much higher than that of changing domain names due to the limited address space. For instance, the address spaces of IPv4 and IPv6 are limited to 32 and 128 bits, respectively. However, domain names can consist of 255 or fewer octets/characters [33], which means a maximum of a 2040-bit space.
Another evasion technique is to avoid all our features in a TVP, related IP address (rIP), and related domain name (rDomain) to hide malicious domain names from our system. For example, attackers can operate their domain names as real legitimate/popular services for a long time to evade our TVPs and then use the domain names as their malicious infrastructure. However, this situation drives up the cost for implementing any attacks using domain names. Another example is border gateway protocol (BGP) hijacking, which potentially enables attackers to divert user traffic from real IP addresses to their IP addresses. In such a case, attackers may bypass our rIP or rDomain features; however, the BGP is basically used between Internet service providers and is difficult for normal Internet users/attackers to effectively control.
The other possible evasion technique is to use legitimate web services as legitimate users. For example, attackers would create dedicated accounts for some web services and use them as their command and control (C&C) channels. In such a case, only legitimate domain names are observed and our system cannot detect them. However, these accounts could be easily banned by the administrator of the web services, and the content sent/received by attackers could be easily analyzed to develop a new countermeasure.
5.2 DNS-based blocking
DomainProfiler predicts and outputs malicious domain names. However, these domain names cannot always be blocked with a domain name level. For example, malicious and legitimate Web sites can exist under the same domain name. Thus, blocking on the basis of domain names instead of URLs may excessively block legitimate Web sites. To examine an actual condition, we extracted and checked URLs under each predicted malicious domain name by using a search engine API and commercial ground truth. In this examination, we manually analyzed 250 domain names randomly selected from the Honeyclient-Exploit dataset, as shown in Table 3, by using the following simple heuristics. If multiple URLs are found under the domain name, we consider that the domain name cannot be blocked with a domain name level. On the other hand, if at most one URL is under the domain name, we determine that the domain name can be blocked with a domain name level. The examination results suggest that 72% (=180/250) of domain names can be effectively blocked by using DNS-based blocking without excessive blocking of legitimate Web sites. Therefore, we conclude that malicious domain names output from our system can contribute to expanding DNS-based block lists if we consider the situation of URLs under the domain names.
We summarize known approaches related to ours in terms of evaluating attack infrastructure or resources owned by cyber-attackers. Most of the studies are broadly divided into three approaches: lexical/linguistic, user-centric, and historic relationship. Note that these approaches were often combined in most studies we reviewed; thus, we classify them on the basis of the main idea of each study.
6.1 Lexical/linguistic approach
The lexical/linguistic approach is focused on lexical or linguistic features obtained from malicious attack resources such as URLs and domain names.
Ma et al. [29] proposed a learning approach using features from the lexical structure of malicious phishing URLs. In contrast, our system focuses on not URLs but domain names and can detect not only phishing but also other attacks.
Yadav et al. [47] focused on linguistic features in command and control (C&C) domain names generated using a domain generation algorithm (DGA) and developed an approach for detecting such malicious domain names. Whereas their approach detects C&C domain names containing random strings, our approach targets broader malicious domain names.
Szurdi et al. [41] analyzed the nature of typosquatting domain names. Typosquatting is generally defined as a technique to register similar domain names to popular domain names to profit from advertisements and perform phishing attacks. Although we take a different and more general approach with our system, our temporal variation patterns (TVPs) can also take into account the nature of typosquatting domain names.
Felegyazhi et al. [18] proposed using WHOIS information of domain names such as registration and name servers to detect malicious domain names. Our approach does not use WHOIS information due to the cost of retrieving it; however, we achieve a high true positive rate (TPR) for malicious domain names without WHOIS.
6.2 User-centric approach
The user-centric approach focuses on user behavior of DNS traffic by observing passive DNS logs. Sato et al. [38] used the co-occurrence characteristics of DNS queries to C&C domain names from multiple malware-infected hosts in a network to extend domain name blacklists. Also, Rahbarinia et al. [37] proposed Segugio to detect new C&C domain names from DNS query behaviors in large ISP networks. This system requires malware-infected hosts in a network; however, our approach works without malware-infected hosts.
Bilge et al. [8] proposed Exposure, which detects malicious domain names on the basis of the time series changes in the number of DNS queries in passive DNS data. Perdisci et al. [35] proposed FluxBuster, which detects previously unknown fast-flux domain names by using large-scale passive DNS data. The cost of Exposure or FluxBuster for retrieving and analyzing large-scale passive DNS logs is much larger than that of our TVPs in DomainProfiler.
Antonakakis et al. [6] proposed Kopis, which uses user behavior observed in passive DNS logs on authoritative DNS servers. Today, the number of new generic top-level domains (gTLDs) is rapidly increasing; thus, it is more difficult to exhaustively gather such information on a TLD's authoritative DNS servers. DomainProfiler does not require such logs and is designed to use publicly available information.
Antonakakis et al. [7] also proposed Pleiades, which is focused on DNS queries to nonexistent domain names observed on recursive DNS servers to detect DGA domain names used for C&C. In addition, Thomas and Mohaisen proposed a system similar to Pleiades to determine the characteristics of nonexistent domain names [42]. Our system does not require such DNS logs and is focused on not only C&C domain names but also other malicious domain names such as drive-by download and phishing.
6.3 Historic relationship approach
The historic relationship approach is focused on the historic or time series information of domain names, IP addresses, and web content.
Antonakakis et al. [5] proposed a system called Notos to detect malicious domain names that have similar patterns to past malicious domain names. This was one of the most successful studies on domain name evaluation or reputation systems. Notos uses historic IP addresses and historic domain names to extract effective features to discriminate malicious domain names from legitimate ones. As stated in Sect. 3.2.2, we use these features as some of our features in related IP addresses (rIPs) and related domain names (rDomains). Moreover, our TVP features dramatically expand detection and prediction performance, as discussed in Sect. 4.
Manadhata et al. [31] proposed a method for detecting malicious domain names from event logs in an enterprise network by using graph-based analysis. Boukhtouta et al. [9] proposed an analysis method for creating graphs from sandbox results to understand the relationships among domain names, IP addresses, and malware family names. Kührer et al. [24] proposed a method for identifying parked and sinkhole domain names from Web sites and blacklist content information by using graph analysis. DomainProfiler strongly relies on the TVP or time series information, which these studies did not use, to precisely predict future malicious domain names. Chiba et al. used the characteristics of past malicious IP addresses to detect malicious Web sites [14]. Our system uses not only IP address features (rIPs) but also TVPs to precisely detect malicious domain names.
Venkataraman et al. [43] developed a method for inferring time series shifting of IP address prefixes to detect malicious IP addresses used for spam or botnet. DomainProfiler is also focused on the idea of shifting malicious resources; however, the target and method are completely different.
The closest concept to ours is that proposed by Soska and Christin [40]. They focused on the idea of variations in compromised Web sites using a popular content management system and proposed a method for predicting vulnerable Web sites before they turn malicious. The main features they rely on are content-based features obtained from compromised Web sites. The concept of DomainProfiler seems to be similar; however, our system has an advantage in scalability because it does not need to access Web sites or extract features from them. Moreover, the focus with our system is wider; that is, DomainProfiler can detect Web sites related to drive-by download and phishing attacks.
Lever et al. [25] pointed out a problem in re-registration of expired domain names and developed an algorithm called Alembic to find potential domain name ownership changes using passive DNS data. We have also focused on the temporal changes in domain names including such re-registered domain names [15]. However, our system does not rely on passive DNS data, and its goal is not only finding re-registered domain names but also specifying truly malicious domain names abused by attackers.
Recently, Hao et al. [21] proposed predator, which predicts future malicious domain names when they are registered. Their system uses domain registration information directly obtained from the .com TLD registry (VeriSign, Inc). However, more and more new gTLDs (e.g., .xyz and .top) have started being used since October 2013. The number of such new gTLDs was 1,184 as of September 2016 [22]. Attackers also leverage new gTLDs for their cyber-attacks. For example, Halvorson et al. [20] showed that domain names using new gTLDs are twice as likely to appear on blacklists; this means attackers now actively make use of new gTLDs. Obviously, to keep up with such situations, predator needs to obtain real-time access privileges to highly confidential data inside the each new gTLD's registry. Although the concept of predator resembles that of DomainProfiler, their mechanisms are totally different because our system does not require any data only owned by a registrar, registry, and authoritative name server.
We proposed DomainProfiler to detect/predict domain names that will potentially be maliciously used in future. Our key idea behind the system is to exploit temporal variation patterns (TVPs) of malicious domain names. A TVP of domain names includes information about how and when a domain name has been listed in legitimate/popular and/or malicious domain name lists. Our system actively collects historical DNS logs, identifies their TVPs, and predicts whether a given domain name will be used maliciously. Our evaluation with large-scale data revealed that DomainProfiler can predict malicious domain names 220 days beforehand with a true positive rate (TPR) of 0.985. Moreover, we verified the effectiveness of our system in terms of the benefits from our TVPs and defense against cyber-attacks. DomainProfiler will be one way to track the trend in ever-changing cyber security threats.
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1.NTT Secure Platform LaboratoriesMusashino-shiJapan
2.Waseda UniversityShinjuku-kuJapan
Chiba, D., Yagi, T., Akiyama, M. et al. Int. J. Inf. Secur. (2018) 17: 661. https://doi.org/10.1007/s10207-017-0396-7
First Online 16 December 2017 | CommonCrawl |
Topic: Unbounded system
Periodic motion
Chaos (mythology)
Tag system - Wikipedia, the free encyclopedia
A tag system is a deterministic computational model created by Emil Leon Post in 1943 as a simple form of string rewriting system.
Note that unlike some alternative definitions of tag systems, the present one is such that the "output" of a computation may be encoded in the final word.
Cyclic tag systems were created by Matthew Cook under the employ of Stephen Wolfram, and were used in Cook's demonstration that the Rule 110 cellular automaton is universal.
en.wikipedia.org /wiki/Tag_system (761 words)
Survivability and Simulation
Survivability is defined as the ability of a system to fulfill its mission in the presence of attacks, failures and accidents.
A survivability architecture is an abstraction that specifies the critical characteristics required for mission survivability in an unbounded system.
These problems are aggravated in unbounded systems where the numbers of components are very large and a primary purpose of simulation is to accurately predict the global effects from local activities.
www.sei.cmu.edu /community/easel/survsimesl.html (1759 words)
Distributed computing - Wikipedia, the free encyclopedia
The main goal of a distributed computing system is to connect users and resources in a transparent, open, and scalable way.
Multicomputer systems can be homogeneous or heterogeneous: A homogeneous distributed system is one where all CPUs are similar and are connected by a single type of network.
A heterogeneous distributed system is made up of different kinds of computers, possibly with vastly differing memory sizes, processing power and even basic underlying architecture.
en.wikipedia.org /wiki/Distributed_computing (1563 words)
CERT® Coordination Center Research
Typically, for a system to survive, it must react to (and recover from) a damaging effect (e.g., the integrity of a database is compromised) long before the underlying cause is identified.
Unbounded systems are a significant component of today's computing environment and will play an even a larger role in the future.
The financial system's survivability might be judged by using a composite measure of the disruption of stock trades or bank transactions (i.e., a measure of the disruption of essential services).
www.cert.org /research/tr13/97tr013chap01.html (4241 words)
1990 Abstracts of MCS Reports and Preprints
This system is the intrinsic version of the radially symmetric prescribed mean curvature equation and permits the study of certain generalized solutions of this equation.
Properties of the system of isentropic gas dynamics, especially the behavior of entropy, are analyzed and documented.
The Lax-Friedrichs approximations, the Godunov approximations, and the viscosity approximations to the Cauchy problem for the system are constructed; and a compactness framework for these approximations is established with the aid of the theory of compensated compactness.
www-fp.mcs.anl.gov /division/publications/abstracts/abstracts90.htm (9819 words)
Survivability: Protecting Your Critical Systems
Unbounded networks are characterized by distributed administrative control without central authority, limited visibility beyond the boundaries of local administration, and a lack of complete information about the entire network.
A system that assumes an overly defensive position because of the threat of an attack may significantly reduce its functionality and divert excessive resources to monitoring the environment and protecting system assets.
System is often used as a shorthand term for a system with the capability to fulfill a specified mission in the face of attacks, failures, or accidents.
www.cert.org /archive/html/protect-critical-systems.html (5399 words)
STSC CrossTalk - The Survivability Imperative: Protecting Critical Systems - Oct 2000 (Site not responding. Last check: 2007-10-23)
Because system performance in testing (and operation) depends totally on the usage to which it is subjected, an effective approach to survivable system testing is based on usage scenarios derived from usage models.
Lastly, system evolution is an inevitable necessity in response to users' requirements for new functions and intruders' increasing knowledge of system behavior and structure.
System survivability can be investigated and improved through the SNA method, and survivability can be integrated into system requirements on a par with functionality and performance.
www.stsc.hill.af.mil /crosstalk/2000/10/linger.html (2955 words)
A.Pnueli/J.Xu/L.Zuck: (Site not responding. Last check: 2007-10-23)
We introduce the (0,1,infinity)-counter abstraction method by which a parameterized system of unbounded size is abstracted into a finite-state system.
Assuming that each process in the parameterized system is finite-state, the abstract variables are limited counters which count, for each local state s of a process, the number of processes which currently are in local state s.
The emphasis of the paper is on the derivation of an adequate and sound set of fairness requirements (both weak and strong) which enable proofs of liveness properties of the abstract system, from which we can safely conclude a corresponding liveness property of the original parameterized system.
www.wisdom.weizmann.ac.il /~amir/PXZ02.htm (186 words)
SparkNotes: Presocratics: Anaximander of Miletus
In Anaximander's system the Unbounded (or apeiron) is both the source out of which everything derives and also the unifier within nature.
The Unbounded, we are told, is divine because it is deathless and indestructible.
The Unbounded steps in and restores the proper balance between them, making them "pay penalty and retribution." The cycles of night and day, and of the four seasons, are perfect examples of the forced equilibrium, and probably the examples Anaxagoras himself had in mind.
www.sparknotes.com /philosophy/presocratics/section2.rhtml (1637 words)
Evolving AI-Life by Natural Selection (Alastair Channon's Artificial Life)
Abstract: Bedau and Packard's evolutionary activity statistics are used to classify the evolutionary dynamics in Geb, a system designed to verify and extend theories behind the generation of evolutionary emergent systems.
The result is that, according to these statistics, Geb exhibits unbounded evolutionary activity, making it the first autonomous artificial system to pass this test.
This qualitative analysis indicates that evolution is unbounded in the system.
www.channon.net /alastair (2074 words)
Because it is rarely the case that we know the properties of the molecular system with great accuracy, the calculated control field for any particular desired outcome of the system evolution will likely be sufficiently far from the true optimal field that much of the efficiency of guiding of the system evolution will be lost.
One promising approach is the study of Hamiltonian mappings in unbounded quantum systems, drawing on the experience gained from studies of such mappings in unbounded classical system.
Another promising approach is extension of the analysis of the properties and the significance of avoided crossings in the parametric dependence of the spectrum of states of a system, and improvement of the understanding of the representation of a complex system by random matrix theory.33 We propose carrying out studies of the types mentioned.
home.uchicago.edu /~mz12/indexpic/Research/Control.html (751 words)
On the complexity of recognizing the Hilbert basis of a linear Diophantine system (Site not responding. Last check: 2007-10-23)
In automated deduction, the Hilbert basis of a corresponding system serves to compute the minimal complete set of associative-commutative unifiers, whereas in integer programming the Hilbert bases are tightly connected to integer polyhedra and to the notion of total dual integrality.
We show that the problem has a pseudopolynomial algorithm if the number of equations in the system is fixed, but it is coNP-complete in the strong sense if the given system is unbounded.
Moreover, we show that, from the complexity standpoint, it is not important to know the underlying homogeneous linear Diophantine system when we ask whether a given set of vectors constitutes a Hilbert basis.
www.lix.polytechnique.fr /~hermann/publications/recog-abs.html (207 words)
This is true no matter whether this interaction is accounted for by the Breit potential, by an external magnetic field which is chosen to minimize the energy, or by the quantized radiation field.
In contrast to most other models, where the collapse of the system, if it occurs, is due to the attraction of electrons and nuclei \cite{Liebetal1986, LiebLoss1986, LiebYau1988, Loss1997} (there would be no collapse without this interaction), the instability here is due to the attraction of parallel currents.
The energy of this system is unbounded from below if $N\alpha^{3/2}$ is large, $\alpha$ being the fine structure constant, even if the vector potential is restricted to lie in a two parameter class \(\{\gamma\vA_0(\delta\vx):\gamma,\delta\in \R_+\}\) where $\vA_0$ is fixed and obeys a weak condition requiring not much more than $\vA_0\not\equiv 0$.
www.ma.utexas.edu /mp_arc/html/papers/98-719 (3566 words)
The Feature System
The Feature System was developed by Professor Jesse D. Jennings, University of Utah, to systematize the complex process of archaeological excavation.
The ISU feature system carries Jennings' original concept over into computerized analyses, and extends a system intended for field excavation into the laboratory as the most effective means of documenting data analysis.
The essence of the feature system is that it is flexible, yet systematic, and capable of handling complex problems arising in the course of normal archaeological excavation.
imnh.isu.edu /digitalatlas/arch/ArchDef/manfeat.htm (1043 words)
Caltech Earthquake Engineering Research Laboratory Technical Reports - The Dynamic response of bilinear hysteretic ...
In the case of the one degree of freedom system exact steady state solutions are obtained for both square wave and trigonometric excitation.
An approximate steady state theory for the one degree of freedom system is investigated and on the basis of this theory it is found that the system is stable and possesses a single locus of vertical tangency.
The response of the one degree of freedom system to transient excitation of finite duration is also examined and it is noted that certain rather general conclusions may be made about the final state of the system without reference to the specific time history of the excitation.
caltecheerl.library.caltech.edu /293 (315 words)
Dimitrov Voulov, Dimitrov Bainov: Asymptotic stability for a homogeneous singularly perturbed system of differential ... (Site not responding. Last check: 2007-10-23)
Dimitrov Voulov, Dimitrov Bainov: Asymptotic stability for a homogeneous singularly perturbed system of differential equations with unbounded delay
Asymptotic stability for a homogeneous singularly perturbed system of differential equations with unbounded delay.
A.I.).— On asymptotic stability of systems with after-effect containing a small parameter as coefficient of the derivative, Prikl.
www.numdam.org /numdam-bin/item?id=AFST_1993_6_2_1_97_0 (280 words)
The Efficient Generation of Cryptographic Confusion Sequences
Thus, proving that a sequence is non-random seemingly corresponds to the impossible task of "proving a negative" in an unbounded system.
Even simple integer arithmetic represents a fairly complex system at the binary gate level [e.g., 2: 23]; perhaps systems based on integer arithmetic are inherently complex (even the successful period analysis of the simple Additive RNG seems rather difficult [114]).
It would certainly be a benefit for system builders if the mathematicians, as part of the original work, would define a set of tests which would be sufficient to completely validate the implementation of their mathematical model.
www.ciphersbyritter.com /ARTS/CRNG2ART.HTM (17722 words)
Chapter 7 (Site not responding. Last check: 2007-10-23)
A system's input requirements are, to a major extent, driven by its ___________ requirements.
An unbounded system is one in which the operational extent has been carefully defined and limited.
The system's ___________ requirements are, to a major extent, driven by its output requirements.
www.westol.com /martzr/cpt_235/chap_7.htm (2026 words)
Scientific American: Survival in an Insecure World (Site not responding. Last check: 2007-10-23)
The tapestries of machines that control transportation, banking, the power grid and virtually anything connected to the Internet are all unbounded systems, observes CERT researcher David A. Fisher: "No one, not even the owner, has complete and precise knowledge of the topology or state of the system.
For Fisher, the most pressing question is how to design systems that, although they are unbounded and thus inherently insecure, have "survivability." That means that even if they are damaged, they will still manage to fulfill their central function--sometimes sacrificing components, if necessary.
Fisher decided a new language was necessary when he started studying the mathematics of the cascade effects that dominate unbounded systems.
www.sciam.com /print_version.cfm?articleID=0001D65F-1318-1CD4-B4A8809EC588EEDF (1244 words)
Evaluation of the Current Density
To investigate the transport properties of a quantum system one must generally evaluate the current flow through the system, and this requires that one examine systems that are out of thermal equilibrium.
A common situation, in both experimental apparatus and technological systems, is that one has two (or more) physically large regions densely populated with electrons in which the current density is low, coupled by a smaller region through which the current density is much larger.
It is convenient to regard the large regions as ``electron reservoirs'' within which the electrons are all in equilibrium with a constant temperature and Fermi level, and which are so large that the current flow into or out of the smaller ``device'' represents a negligible perturbation.
www.utdallas.edu /~frensley/technical/qtrans/node6.html (1216 words)
They are adopted according to the perceptions and understanding of the application and its domain at the time of implementation and are embedded in the system to bridge the gap but the real world is always changing.
This conclusion is supported by observations made during the FEAST and other studies and suggests that evolving software eventually, reaches a condition where, from an economic point of view at least, replacement is indicated.
The simplest is associated with the fact that there is a gap between the potentially unbounded E-type application and its real world operational domain and the finite system developed with finite resources in finite time to address a constrained application in a constrained domain.
www.csis.ul.ie /Modules/CS5122/A1/Group3/Law7.htm (672 words)
This description includes the systems from which the activity originated and those targeted, an assessment of the techniques used by the intruder, the impact of the activity on the organization, and any forensic evidence discovered.
The systems represented by this class are categorized according to the role they played in the incident through the category attribute.
The System was an intermediary in the attack.
www.faqs.org /ftp/internet-drafts/draft-ietf-inch-iodef-04.txt (8676 words)
Papers - Windows NT '97
It was further clear from the consultation process that real-time extensions to the system should conform as closely as possible to the standard interfaces of the host OS.
The benefit to this approach is that all real time system activities are captured as real time threads, which provides a simple, coherent approach to setting priority across the real time subsystem, and allows for a more robust and verifiable implementation of the subsystem.
The NT system objects are those objects to which the RTSS processes have remote access.
www.usenix.org /publications/library/proceedings/usenix-nt97/full_papers/carpenter/carpenter_html/carpenter.html (2497 words)
Calculating Metrical Structure
My own assessment, however, is that once mechanisms are added to deal with unbounded, quantity sensitive and lexical stress systems, that its initial apparent restrictiveness will give way to a theory at least as rich in devices as the present one is.
Let us assume, at least as a heuristic (Elan!), that in unbounded systems words without marked syllables will indicate which side of the foot is the head.
This system is the mirror image of Moses-Columbian, which has three degrees of lexical weight, strong, alternating and weak.
www.ling.udel.edu /idsardi/cuny/cuny1.html (4674 words)
[No title] (Site not responding. Last check: 2007-10-23)
We present efficient methods for analyzing the counterpart fluid system and show a dynamic programming procedure to devise an optimal QCS configuration for the fluid system.
If unbounded intermediate buffers are allowed, then the steady state performance of the discrete system and that of its fluid counterpart coincide.
In many other cases, a near optimal dispatching policy for the discrete system can be obtained from the fluid one.
www.math.technion.ac.il /~techm/20020325123020020325rav (236 words)
What Does Negative Temperature Mean? (Site not responding. Last check: 2007-10-23)
Not all systems have the property that the entropy increases monotonically with energy.
In some cases, as energy is added to the system, the number of available microstates, or configurations, actually decreases for some range of energies.
So we have created a system where, as we add more and more energy, temperature starts off positive, approaches positive infinity as maximum entropy is approached, with half of all spins up.
math.ucr.edu /home/baez/physics/ParticleAndNuclear/neg_temperature.html (1149 words)
DISCRETE MODEL
If one seeks the eigenstates of a bounded system, the tridiagonal Hamiltonian may be readily diagonalized by standard numerical techniques.
This problem must be formulated in terms of the unbounded scattering states, and to do this we must modify the Hamiltonian matrix by incorporating the QTBM boundary conditions.
The matrix in (18) is just the denominator of the Green's function for Schrödinger's equation, in a space which has been augmented by adding the incomming traveling wave amplitudes.
www.utdallas.edu /eecs/ee/frensley/technical/nanomes91/node3.html (555 words)
Cancer and Wisdom of the Body: Cellular Automata
Although the system is bounded it seems that the same holds also for an unbounded system.
As the system converges its members become wider, and the probability p1 of resource accumulation rises.
5 As the system converges movement of its members becomes more and more sluggish, and their probability p2 to meet declines.
www.what-is-cancer.com /papers/ca1/ca159.htm (384 words) | CommonCrawl |
Why Do Computers Use the Binary Number System (0,1)?
Why Do Computers Use the Binary Number System (0,1)? Why don't they use Ternary Number System (0,1,2) or any other number system instead?
computer-architecture binary-arithmetic
David Richerby
Rai Ammad KhanRai Ammad Khan
$\begingroup$ This is a question about electrical engineering. Apparently binary gates are easier to implement. IIRC some ternary-based computer had been built at some point. $\endgroup$ – Yuval Filmus Jun 13 '14 at 6:16
$\begingroup$ What research have you done? When I type the title of your question into Google, I get back search results that provide several answers to your question. Also, the Wikipedia article on binary numbers and binary code has a short explanation. We expect you to do a significant amount of research before asking here, and it looks to me like you haven't done even basic research before asking. Searching Google and Wikipedia is a bare minimum. $\endgroup$ – D.W.♦ Jun 13 '14 at 21:30
$\begingroup$ Larger bases did not turn out to be useful overall. $\endgroup$ – Raphael♦ Jun 13 '14 at 21:40
$\begingroup$ @Raphael: Ternary did $\endgroup$ – Mooing Duck Sep 10 '14 at 19:29
$\begingroup$ I'm going to put this as a comment because there's already an accepted answer. It is extraordinarily difficult to build electronic devices that reliably discriminate among ten values because of manufacturing tolerances. It is relatively easy to build electronic devices that discriminate between two values. So, the short answer is that computers use binary representation for reliability. I've written a more detailed answer for those who may care: bbrown.kennesaw.edu/papers/why_binary.html $\endgroup$ – Bob Brown Jun 8 '15 at 22:38
Since we're in Computer Science, I'll answer this way: they don't.
What do we mean by a "computer?" There are many definitions, but in computer science as a science, the most common is the Turing machine.
A turing machine is defined by several aspects: a state-set, a transition table, a halting set, and important for our discussion, an alphabet. This alphabet refers to the symbols which the machine can read as input, and that it can write to its tape. (You could have different input and tape alphabets, but let's not worry about that for now.)
So, I can make a Turing machine with input alphabet $\{0,1\}$, or $\{a,b\}$, or $\{0,1,2\}$, or $\{\uparrow,\downarrow\}$. It doesn't matter. The fact is, I can use any alphabet I choose to encode data.
So, I can say that $0001001$ is 9, or I can say that $\uparrow \uparrow \uparrow \downarrow \uparrow \uparrow \downarrow$ is 9. It doesn't matter, since they're just symbols we can distinguish.
The trick is that binary is enough. Any sequence of bits can be interpreted as a number, so you can convert from binary to any other system and back.
But, it turns out unary is enough too. You can encode 9 as 111111111. This isn't particularly efficient, but it has the same computational power.
Things get even crazier when you look into alternate models of computation, like the Lambda calculus. Here, you can view numbers as functions. In fact, you can view everything as functions. Things are encoded not as bits, 0s and 1s, but as closed mathematical functions with no mutable state. See the Church numerals for how you can do numbers this way.
The point is that, 0s and 1s is a completely hardware specific issue, and the choice is arbitrary. What encoding you're using isn't particularly relevant to computer science, outside of a few subfields like operating systems or networking.
jmitejmite
$\begingroup$ What about input encoding in 2-counters machines. It seems to matter. Are you sure you can dismiss so radically encoding issues? And I would hardly agree that complexity is a non-issue; but is lambda calculus a proper way to address it? $\endgroup$ – babou Jun 19 '14 at 21:45
$\begingroup$ I'll admit that there are complexity problems when you use unary. But the choice of binary vs ternary or anything like that is somewhat arbitrary. It's not like the choice of encoding doesn't matter, but there's not some law dictating why we use a particular one. It's dictated mostly by either convenience or by hardware requirements, which are somewhat outside of computational science. $\endgroup$ – jmite Jun 19 '14 at 22:28
$\begingroup$ "So, I can make a Turing machine with input alphabet". I think you should write "tape alphabet" here instead of "input alphabet". The interesting part is what is used for computation and not for input. Furthermore I disagree with unary being enough. A TM with unary tape alphabet is almost useless, because the tape is constant. $\endgroup$ – Simon S Jun 20 '14 at 7:17
$\begingroup$ Regarding the last sentence: design and study of computer hardware and architecture are also part of computer science. $\endgroup$ – Kaveh Aug 11 '14 at 16:48
$\begingroup$ You may want to add the point that going from unary to binary cuts down the size of your numbers to their logarithm, which is a serious improvement, while going further doesn't gain as much (only a linear factor). $\endgroup$ – reinierpost Feb 23 '15 at 22:49
Some other things to consider:
Part of the reason for using a binary number system is that it's the lowest-base number system that can represent numbers in logarithmic, rather than linear, space. To uniquely distinguish between $n$ different numbers in unary, the average length of representations must be proportional to at least $n$, since there is only one string of length $k$ where $k < n$; $1 + 1 + ... + 1 = n$. To uniquely distinguish between $n$ different numbers in binary, the average length of representations must be proportional to at least $\log_2 n$, since there are $2^k$ binary numbers of length $k$; $1 + 2 + ... + \frac{n+1}{2} = n$. Choosing a larger base improves on the space requirement by a constant factor; base 10 gets you $n$ numbers with an average representation length of $\log_{10}n$, which is $\log_{10}2 \approx 0.3$ times the average length of a base two representation for all $n$. The difference between binary and unary is much greater; in fact, it's a function of $n$. You get a lot by choosing binary over unary; you get much less by choosing a higher base, by comparison.
There is some truth to the idea that it's easier to implement digital logic if we only have to distinguish two states. Electric signals are analog and, as such, can be interpreted to represent as many discrete states as you'd like... but you need more precise (hence expensive and finicky) hardware to reliably distinguish more states over the same range. This suggests choosing as low a base as you can.
Another potentially important consideration is that logic has classically been understood to involve two distinct values: $true$ and $false$. Now, we have fancier logics than this, but a lot of mathematics and science still rests on pretty foundational notions. When you consider that computers are used to compute, and that logic is important for computation, it suggests having good support for at least two distinct states... but logic doesn't really require more than that.
Patrick87Patrick87
One of the big reasons that most computer circuits use two states is that the quantity of circuitry necessary to distinguish between n different voltage levels is roughly proportional to n-1. Consequently, having three discernible states would require twice as much circuitry per signal, and having four would require three times as much. Tripling the amount of circuitry while only doubling the amount of information would represent a loss in efficiency.
Note that there are some places in computers where information is stored or communicated using more than two states per element. In a flash memory array, hundreds or thousands of memory cells may be serviced by one set of level-sensing circuitry. Using four levels per cell rather than two when storing a certain amount of information might more than triple the size of the level-sensing circuitry, but would cut by half the number of memory cells required. When communicating over 100-base-T or faster Ethernet, the cost of the circuitry necessary to detect multiple signal levels on the cable will likely be dwarfed by the cost of either having to use a cable with more wires or use cables that can handle more signal transitions per second without an unacceptable level of distortion.
supercatsupercat
There do exist quantum computers in research labs that use q-bit as the basic unit of information that can be both 0 and 1 simultaneously.
http://en.wikipedia.org/wiki/Quantum_computer
There have also been ternary quantum computers built as per this reference http://en.wikipedia.org/wiki/Ternary_computer
So, It is indeed possible to build alternative computing devices that do not rely on the binary number system. Fiber optic systems for example use 0 for dark and two different orthoganal polarizations of light as 1 and -1.
The reason why I mention these things is because I want to show that although binary numbers are sufficient for computing, there are alternative number systems that can be used for computing.
The binary number system is nice in these sense we can encode all integers $x \in \mathbb{Z}$ by using radix representation of numbers. http:// en.wikipedia.org/wiki/Radix These values can represent the ASCII code A=0x41=01000001, or the value could represent a machine instruction nop=0x90=0x10010000.
Gary D.Gary D.
$\begingroup$ But they are still using a binary system, in quantum computing the state of superposition is not a third possible value. Also throwing out a quantum computing example is not helpful to the question asked. $\endgroup$ – lPlant Jun 13 '14 at 13:29
$\begingroup$ I didn't know about this.. $\endgroup$ – Ali786 Sep 12 '14 at 4:26
$\begingroup$ "q-bit as the basic unit of information that can be both 0 and 1 simultaneously." This is nonsense. Qubits are fundamentally different from normal bits, in that they represent a non-discrete (complex) value. They are incomparable for all practical purposes. $\endgroup$ – Discrete lizard♦ Apr 5 '18 at 8:19
At the heart of the digital computers processing power is a transistor, which works like a switch. By raising the current at at the "gate" of the switch, it allows current to flow between the "collector" and "emitter" - the switch is turned on. The transistor will be designed to operate in one of two modes - fully on or fully off ('saturated') - with a clear division of what those states are. The transistor can switch between the two states quickly, will remain in the state with very limited errors.
This circuitry forms the basis for logic devices, such AND, NAND, OR, XOR and other functions. The NAND function being the most basic of the building blocks. These logic devices are assembled to provide processors which remain in a predictable state, and lots of transistors can be packed in a small space to provide the functionality needed.
A transistor can manage multiple, or varying states, but when operating in that manner they do not produce conventional "digital" computers - they do not tend to stay in a predictable state and they are prone to interference, saturation, osculation, etc - so they have limited applications in terms of computational abilities. Op-amps could be considered analog computers.
peeldogpeeldog
We only use binary(1,0) because we currently do not have the technology to create "switches" that can reliably hold more than two possible states. (Quantum computers aren't exactly on sale at the moment.) The binary system was chosen only because it is quite easy to distinguish the presence of an electric current from an absense of electric current, especially when working with trillions of such connections. And using any other number base in this system ridiculous, because the system would need to constantly convert between them. That's all there is to it.
Irfan KhanIrfan Khan
$\begingroup$ This is true but isn't it all already included in the existing answers? $\endgroup$ – David Richerby Oct 23 '14 at 12:02
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Why do we still use a Von Neumann Architecture in modern computers? | CommonCrawl |
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