text:
## Abstract

Motivation: Cluster analysis of gene expression profiles has been widely applied to clustering genes for gene function discovery. Many approaches have been proposed. The rationale is that the genes with the same biological function or involved in the same biological process are more likely to co-express, hence they are more likely to form a cluster with similar gene expression patterns. However, most existing methods, including model-based clustering, ignore known gene functions in clustering.

Results: To take advantage of accumulating gene functional annotations, we propose incorporating known gene functions as prior probabilities in model-based clustering. In contrast to a global mixture model applicable to all the genes in the standard model-based clustering, we use a stratified mixture model: one stratum corresponds to the genes of unknown function while each of the other ones corresponding to the genes sharing the same biological function or pathway; the genes from the same stratum are assumed to have the same prior probability of coming from a cluster while those from different strata are allowed to have different prior probabilities of coming from the same cluster. We derive a simple EM algorithm that can be used to fit the stratified model. A simulation study and an application to gene function prediction demonstrate the advantage of our proposal over the standard method.

Contact:weip@biostat.umn.edu

## 1 INTRODUCTION

This article concerns with clustering genes for gene function discovery using microarray gene expression data. It has been widely observed that genes with a similar function or involved in the same biological process are likely to co-express, hence clustering genes’ expression profiles provides a means for gene function discovery; see, e.g. Eisen et al. (1998), Brown et al. (2000), Wu et al. (2002), Xiao and Pan (2005) and references therein. However, most existing approaches all ignore known functions of some genes in the process of clustering; few exceptions in the context of non-model-based clustering include Hanisch et al. (2002), Cheng et al. (2004), Fang et al. (2006) and Huang and Pan (2006). For example, in model-based clustering, all the genes are treated equally a priori; in particular, all the genes are assumed to have an equal prior probability of being in a given cluster (e.g. Li and Hong, 2001; Ghosh and Chinnaiyan, 2002; Pan et al., 2002). As mentioned, if some genes are known to share the same function, it is more likely that they belong to the same cluster. Hence, it seems more plausible to model the genes sharing the same biological function to have an equal prior probability while allowing the genes with different functions to have varying prior probabilities. This provides a more efficient way to account for the association between gene function and co-expression. In this paper, we propose such an approach that uses gene functional annotations as priors for model-based clustering. Specifically, first, the genome is partitioned into several groups with one group containing the genes of unknown function and each of the other groups containing the genes sharing the same function. Gene functional annotations are readily available from many existing databases, such as the Gene Ontology (GO) (Ashburner et al., 2000) and MIPS (Mewes et al., 2004). Second, each group is treated as a stratum and a stratified mixture model is used: the genes from the same groups are assumed to have the same prior probability of coming from the same cluster while the prior probabilities for different groups are allowed to be unequal. Because of possible heterogeneity in each gene functional group, we do not assume that the genes from the same functional group come from the same cluster. In fact, for the genes in the group of unknown function, they may come from any cluster. With relatively high noise levels of genomic data, it is recognized that incorporating biological knowledge into statistical analysis is a reliable way to maximize statistical efficiency and enhance the interpretability of the analysis results.

This article is organized as follows. In Section 2, we first briefly review the standard method of model-based clustering with a global mixture model, then propose our stratified mixture model and associated stratified clustering. We derive a simple EM algorithm to fit our stratified model. In Section 3, we demonstrate the advantage of our proposal using simulated data, and then using real data for gene function prediction. We end the paper with a short discussion.

## 2 METHODS

### 2.1 Standard model-based clustering

In model-based clustering, it is assumed that each observation x, a p-dimensional vector, is drawn from a finite mixture distribution

(1)
$f(x;Θ)=∑i=1gπifi(x;θi),$
with the prior probability πi, component-specific distribution fi and its parameters θi. We use Θ = {(πi, θi) : i = 1, … , g} to denote all unknown parameters, with the restriction that 0 ≤ πi ≤ 1 for any i and that $∑i=1gπi=1$. Each component of the mixture distribution corresponds to a cluster. The number of clusters, g, has to be determined in practice; see Section 2.3.

The finite normal mixture model is most widely used: each component fi is a normal distribution; we use this model throughout this article. Because we are particularly interested in applications with ‘large p, small n’ (i.e. high-dimensional data with small sample sizes) encountered in genomic studies, we adopt a working independence model for the components of x as in the naive Bayesian, though other more sophisticated methods may be preferred (e.g. McLachlan et al., 2003). Specifically, we have

$fi(x;θi)=1(2π)p/2∣V∣1/2exp(−12(x−μi)′V−1(x−μi)),$
where $V=diag(σ12,σ22,…,σp2)$ and $∣V∣=∏q=1pσq2$.

Given a dataset xj for j = 1, … , n, the EM algorithm (Dempster et al., 1977) is used to estimate the parameters Θ iteratively in the standard model-based clustering (McLachlan and Peel, 2002; Fraley and Raftery, 2002). We use generic notation Θ(m) to represent the parameter estimates at iteration m, then the EM works by iterating the following:

(2)
$μ^i(m+1)=∑jτij(m+1)xj/∑jτij(m+1), σ^k2,(m+1)=∑i=1g∑j=1nτij(m)(xjk−μik(m))2/n,$

(3)
$π^i(m+1)=∑j=1nτij(m)/n,$
where
(4)
$τij(m)=πi(m)fi(xj;θi(m))∑i=1gπi(m)fi(xj;θi(m)).$
is the estimated posterior probability of xjs coming from component i.

The above iteration is repeated until convergence, resulting in maximum likelihood estimate (MLE) $Θ^$. Then we use (4) to calculate the posterior probability of any observation xjs belonging to any cluster i, and assign the observation to the cluster with the largest such probability. Because of possible existence of multiple local maxima, we need to start the algorithm multiple times with various starting values; in this paper, we use the result from K-means as starting values for the EM. We fit a series of models with various values of g, then use a model selection criterion to choose its appropriate value, as discussed in Section 2.3.

### 2.2 Stratified model-based clustering

In Section 2.1, all the genes are treated equally a priori. To take advantage of the known gene functions, we propose first partitioning the genome into several groups, say, G1, … , GK. GK contains the genes of unknown function while each of the other groups contains the genes sharing the same biological function. Second, rather than a global model (1), we propose using a stratified model: for any gene j in functional group Gk,

(5)
$f(k)(xj;Θ(k))=∑i=1gπ(k),ifi(xj;θi)$
for k = 1, … , K. Note that the K stratified models differ in using stratum-specific prior probabilities while sharing the same set of component distributions. Hence, we assume that genes from the same group Gk share the same prior probability π(k),i of coming from the same cluster i while allowing them to come from different clusters. It is easy to see that the above stratified model reduces to the standard model with K = 1.

In practice, Gk can be determined based on the GO (Ashburner et al., 2000), MIPS (Mewes et al., 2004) or other sources of biological knowledge; the choice of the database depends on the application at hand. For example, if the goal is to predict gene function while both data quality and terminology of GO are preferred, GO can be used. Because some genes may have multiple functions, we allow a gene to be in multiple groups. A simple method considered here is that, for example, if a gene has two functions corresponding to groups G1 and G2, we duplicate the gene expression profile for the gene and treat the two observations as two genes, one in G1 and one in G2, in the process of clustering; see the final section for more discussions on this issue.

Next we derive the EM algorithm for the above model-based clustering. Given data X = {xj : j = 1, … , n}, the log-likelihood is

$logL(Θ)=∑j=1nlog[∑i=1gπ(k),ifi(xj;θi)],$
where for simplicity we suppress the dependence of group index k = k(j) on gene j and we use this convention throughout. Maximization of the above log-likelihood is difficult, and it is common to use the EM algorithm (Dempster et al., 1977) by casting the problem in the framework of missing data. Define zij as the indicator of whether xj is from component i; i.e. zij = 1 if xj is indeed from component i and zij = 0 otherwise. If we could observe the missing data zijs, then the complete data log-likelihood is
$logLc(Θ)=∑i∑jzij[logπ(k),i+logfi(xj;θi)].$
It is easy to verify that the E-step of the EM yields
$Q(Θ;Θ(m))=EΘ(m)(logLc∣X)=∑i∑jτij(m)[logπ(k),i+logfi(xj;θi)],$
where for gene jGk,
(6)
$τij(m)=π(k),i(m)fi(xj;θi(m))∑i=1gπ(k),i(m)fi(xj;θi(m))$
is the estimated posterior probability of xjs coming from component i.

The M-step of the EM maximizes the above Q to update the parameter estimates. For a stratified model, updates for the mean and variance parameters are the same as in (2), but the update for the prior probability is different:

(7)
$π^(k),i(m+1)=∑j∈Gkτij(m)nk,$
where nk is the number of the genes in Gk.

As before, the above updates are iterated until convergence with resulting MLE $Θ^$. We fit a series of models with various values of g, then use a model selection criterion to choose an appropriate value.

We use (6) to calculate the posterior probability of any observation xjs belonging to each cluster, and assign the observation to the cluster with the largest such probability.

### 2.3 Model selection

In practice, we have to determine the number of components, g. This is realized by first fitting a series of models with various numbers of components, and then using a model selection criterion to choose the best one. In model-based clustering, it is common to use Bayesian information criterion (BIC) (Schwartz, 1978) defined as

$BIC=−2logL(Θ^)+log(n)d,$
where d is the total number of (effective) parameters (Fraley and Raftery, 1998). In the standard model, because we have three sets of parameters, πis, σqs and μiqs, and we have the constraint $∑i=1gπi=1$, we have d = p + gp + g − 1; in the stratified model with K strata, instead of a set of πis, we have K sets of π(k),is, thus d = p + gp + K(g − 1).

## 3 RESULTS

### 3.1 Simulated data

We did a simulation study to demonstrate the improved performance of our proposal. There were two univariate clusters with distributions N1, σ1) and N2, σ2) respectively. We had two gene functional groups with n1 and n2 genes respectively. For a gene in functional group k for k = 1 or 2, its prior probability of being in cluster 1 was π(k),1. Four simulation set-ups were used with various parameter values: three sets of π(k),1s were used; n1 = n2 = 50 in the first three set-ups while n1 = 25 and n2 = 75 in set-up 4; set-ups 2 and 4 were the same except for different n1 and n2.

The results were based on 1000 simulations for each set-up. We considered both the mean and variance of a parameter estimate over 1000 simulations; we also gave the average numbers of observations in a simulated dataset that were incorrectly assigned to clusters different from their true ones.

In the first simulation set-up, we had π(1),1 = π(2),1, hence the standard mixture model was correct, and the standard method was supposed to perform better. Although the stratified mixture model was still correct, it unnecessarily required to estimate two separate parameters π(1),1 and π(2),1. It was confirmed that the standard method worked better, however, more interestingly, its performance was quite close to that of our proposed method: the estimates of the mean and variance parameters from the stratified method were almost the same as that from the standard method, but as expected, the estimate of the prior probability from the former had slightly larger variability.

In set-ups 2–4, the stratified mixture model was correct with π(1),1 ≠ π(2),1. Hence, with the knowledge of the gene functional groups, it was better to use the stratified model. On the other hand, ignoring the gene functional groups, the global mixture model still held with a corresponding π1 = 0.5 or π1 = 0.35. Unsurprisingly, the standard method gave estimates with larger estimation errors e.g. in terms of a mean squared error, which is the sum of the squared bias and the variance; this was especially evident for the mean and variance parameters of the second cluster, both with much larger variances than those from the stratified model. In consequence, the standard method resulted in much larger numbers of misclassified genes. Comparing set-ups 2 and 3, it was noted that the performance difference between the two methods increased with $∣π1(1)−π1(2)∣$. With different numbers of genes in the two functional groups (set-up 4), the stratified method still maintained better performance (Table 1).

Table 1

Simulation results for the standard method and the new method

Set-up Method μ1 μ2 σ1 σ2 π1 $π1(1)$ $π1(2)$ #Err1 #Err2
True 0.80 0.80 0.80
Standard 0.91 2.86 1.01 1.39 0.764 — — 11.3 11.4
(0.48) (1.52) (0.30) (0.69) (0.213)   (8.6) (8.5)
New 0.91 2.80 1.01 1.39 — 0.748 0.749 12.1 12.1
(0.48) (1.54) (0.30) (0.69)  (0.243) (0.240) (9.5) (9.5)
True 0.50 0.80 0.20
Standard 1.01 2.66 0.99 1.74 0.588 — — 13.6 23.0
(0.30) (1.32) (0.29) (0.55) (0.224)   (8.4) (8.2)
New 1.00 2.23 0.98 1.92 — 0.822 0.246 9.0 14.3
(0.25) (0.78) (0.21) (0.37)  (0.152) (0.223) (2.8) (6.8)
True 0.50
Standard 0.95 2.68 1.01 1.72 0.575 — — 11.6 25.4
(0.52) (1.31) (0.36) (0.54) (0.227)   (14.0) (12.7)
New 0.98 2.17 0.97 1.99 — 0.986 0.101 0.3 2.6
(0.15) (0.47) (0.11) (0.27)  (0.036) (0.139) (0.9) (7.7)
True 0.35 0.80 0.20
Standard 0.99 2.43 1.00 1.81 0.445 — — 10.1 29.8
(0.53) (1.01) (0.43) (0.46) (0.228)   (5.8) (12.9)
New 1.00 2.18 0.98 1.92 — 0.829 0.253 5.0 21.4
(0.30) (0.66) (0.30) (0.34)  (0.191) (0.219) (2.6) (10.7)
Set-up Method μ1 μ2 σ1 σ2 π1 $π1(1)$ $π1(2)$ #Err1 #Err2
True 0.80 0.80 0.80
Standard 0.91 2.86 1.01 1.39 0.764 — — 11.3 11.4
(0.48) (1.52) (0.30) (0.69) (0.213)   (8.6) (8.5)
New 0.91 2.80 1.01 1.39 — 0.748 0.749 12.1 12.1
(0.48) (1.54) (0.30) (0.69)  (0.243) (0.240) (9.5) (9.5)
True 0.50 0.80 0.20
Standard 1.01 2.66 0.99 1.74 0.588 — — 13.6 23.0
(0.30) (1.32) (0.29) (0.55) (0.224)   (8.4) (8.2)
New 1.00 2.23 0.98 1.92 — 0.822 0.246 9.0 14.3
(0.25) (0.78) (0.21) (0.37)  (0.152) (0.223) (2.8) (6.8)
True 0.50
Standard 0.95 2.68 1.01 1.72 0.575 — — 11.6 25.4
(0.52) (1.31) (0.36) (0.54) (0.227)   (14.0) (12.7)
New 0.98 2.17 0.97 1.99 — 0.986 0.101 0.3 2.6
(0.15) (0.47) (0.11) (0.27)  (0.036) (0.139) (0.9) (7.7)
True 0.35 0.80 0.20
Standard 0.99 2.43 1.00 1.81 0.445 — — 10.1 29.8
(0.53) (1.01) (0.43) (0.46) (0.228)   (5.8) (12.9)
New 1.00 2.18 0.98 1.92 — 0.829 0.253 5.0 21.4
(0.30) (0.66) (0.30) (0.34)  (0.191) (0.219) (2.6) (10.7)

The means and standard deviations (in parentheses) of parameter estimates, and #Err1 and #Err2 for the numbers of misclassifications for clusters 1 and 2 respectively, are presented.

### 3.2 Gene function prediction using gene expression profiles

With the completion of the human genome and other sequencing projects, it has become compelling to learn functions of many newly discovered genes. An important approach is to cluster gene expression profiles drawn from microarray experiments under various conditions; see, e.g. Eisen et al. (1998), Brown et al. (2000), Wu et al. (2001), Zhou et al. (2002), Xiao and Pan (2005) and references therein. The rationale is that genes sharing the same biological function are likely to co-express.

We considered clustering gene expression data to predict gene functions for yeast Saccharomyces cerevisiae. We used a large dataset containing 300 microarray experiments with gene deletions and drug treatments (Hughes et al., 2000). The data were centered and scaled so that the mean and variance of the expression profile for each gene were 0 and 1, respectively. Gene functions were downloaded from the MIPS database (Mewes et al., 2004). For illustration, we only considered two gene functions, mitotic cell cycle and cell cycle control (with MIPS code 030301) and mitochondrion (with MIPS 4016), shortened as classes 1 and 2, respectively. There were three strata: the first two strata each contained 200 genes randomly selected from one of the two classes, while the third stratum was a mixture of 119 and 112 genes from the two classes respectively. Hence, the genes in the first two strata were treated as those with known functions whereas the third stratum consisted of the genes whose functions were to be predicted.

We did not use the class labels of the genes in the process of clustering. Based on the clustering results, we predicted the functions of the genes in the third stratum. There were two ways to accomplish prediction, called hard classification and soft classification. In hard classification, each gene was assigned to a cluster, and a cluster was classified to a class to which the majority of the genes from the first two strata that were assigned to the cluster belonged; if there was an equal number of the genes from each of the first two strata, we randomly assigned a class label to the cluster. Any gene from stratum 3 that was assigned to a cluster was predicted to be in the same class as that of the cluster. When there were an equal or almost equal number of the genes from the first two strata, there seemed to be a certain degree of randomness in the hard classification. Furthermore, it did not take advantage of the soft-clustering feature of model-based clustering: even two genes were both assigned to the same cluster, they might have quite different posterior probabilities of being in the cluster. Hence, as an alternative, we used soft classification: first, for each cluster i, we calculated the proportion of the genes in class c, $Pic$, for c = 1 and 2; second, for any gene j in stratum 3, the expectation of its being in class c is $∑i=1gPicτij$. Summing over all the genes from class c (or the other class), we obtained an expected number of the genes predicted to be in class c. To illustrate the effect of various prior probabilities, we also included results using the prior probabilities of the first two strata to calculate τij for each jG3; in a table (i.e. Tables 2 and 3), they were respectively denoted as ‘New: $π^(k)$’ for k = 1, 2, 3.

Table 2

BIC with various numbers (g) of clusters in the standard and stratified clustering for Hughes’ gene expression data with 300 microarray experiments

g Standard New
BIC BIC
162479 162479
150773 150786
144384 144388
139320 139230
136185 136103
134219 134137
133502 133426
132840 132785
132892 132826
10 133112 133092
g Standard New
BIC BIC
162479 162479
150773 150786
144384 144388
139320 139230
136185 136103
134219 134137
133502 133426
132840 132785
132892 132826
10 133112 133092

Italics give the minima

Table 3

Predictions for stratum 3 based on hard classification using standard clustering and new clustering with g = 8 for Hughes’ gene expression data with 300 microarray experiments

Standard  New: $π^(1)$  New: $π^(2)$  New: $π^(3)$  Truth  Pred = 1  Pred = 2  1  2 1 88 31 88 31 88 31 88 31 2 30 82 31 81 30 82 30 82 Accuracy 0.736 0.732 0.736 0.736
Standard  New: $π^(1)$  New: $π^(2)$  New: $π^(3)$  Truth  Pred = 1  Pred = 2  1  2 1 88 31 88 31 88 31 88 31 2 30 82 31 81 30 82 30 82 Accuracy 0.736 0.732 0.736 0.736

As a comparison, we also treated the problem as supervised learning: the 200 genes in the first two strata and their class labels were training data, while the genes in the third stratum were test data. We applied a classic linear discriminant analysis (LDA) and three new classifiers regarded as among the best: the nearest shrunken centroids (NSC) (Tibshirani et al., 2003), random forests (RF) (Breiman, 2001) and support vector machines (SVM) (Vapnik, 1998). Our purpose here was not to directly compare clustering analyses with these classifiers; rather, these classifiers provided some estimates of the upper bound of the predictive performance of clustering for this particular application. We used the default settings of R functions

lda()
,
pamr()
,
randomForest()
and
svm()
implementing the methods except that 5-fold cross-validation was used to select parameters for NSC and SVM.

#### 3.2.1 Using 300 microarray experiments

First, we considered the use of the full dataset with all 300 microarray experiments included. Table 4 gives the BIC values for the two methods with various numbers of clusters, based on which g = 8 that minimized BIC was selected for both methods. The predictive results based on hard classification for the genes in stratum 3 are given in Table 5, showing that the two methods gave almost the same result. As mentioned earlier, the result of ‘New: $π^(k)$’ was based on using the prior probability estimate $π^(k)$ to calculate the posterior probabilities and thus made predictions for the genes in stratum 3; in practice, we would just simply use ‘New: $π^(3)$’ for stratum 3. Furthermore, based on soft classification, the results from the two methods were still almost the same. Tables 6 and 7 give the estimated prior probabilities and predictive results of the five classifiers, respectively. For NSC, cross-validation selected the number of the microarray experiments in the constructed classifier at 85; as a comparison, we also included the result using all the 300 microarray experiments. All the supervised methods except LDA worked well; the inferior performance of LDA was likely because of a problematic estimation of a large (i.e. 300 × 300) covariance matrix with only a few hundred observations.

Table 4

Predictions for stratum 3 based on soft classification using standard clustering and new clustering with g = 8 for Hughes’ gene expression data with 300 microarray experiments

Truth Standard New: $π^(1)$ New: $π^(2)$ New: $π^(3)$
Pred = 1 Pred = 2
74.5 44.5 74.5 44.5 74.5 44.5 74.5 44.5
36.2 75.8 36.7 75.3 36.0 76.0 36.1 75.9
Accuracy 0.651 0.649 0.652 0.651
Truth Standard New: $π^(1)$ New: $π^(2)$ New: $π^(3)$
Pred = 1 Pred = 2
74.5 44.5 74.5 44.5 74.5 44.5 74.5 44.5
36.2 75.8 36.7 75.3 36.0 76.0 36.1 75.9
Accuracy 0.651 0.649 0.652 0.651
Table 5

Estimated prior probabilities with g = 8 for Hughes’ gene expression data with 300 microarray experiments

 $π^$ 0.135 0.154 0.114 0.105 0.084 0.215 0.117 0.077 $π^(1)$ 0.11 0.015 0.2 0.09 0.13 0.28 0.165 0.01 $π^(2)$ 0.149 0.245 0.045 0.13 0.04 0.176 0.05 0.165 $π^(3)$ 0.143 0.195 0.1 0.091 0.082 0.199 0.13 0.061
 $π^$ 0.135 0.154 0.114 0.105 0.084 0.215 0.117 0.077 $π^(1)$ 0.11 0.015 0.2 0.09 0.13 0.28 0.165 0.01 $π^(2)$ 0.149 0.245 0.045 0.13 0.04 0.176 0.05 0.165 $π^(3)$ 0.143 0.195 0.1 0.091 0.082 0.199 0.13 0.061
Table 6

Predictions for a separate test dataset (i.e. stratum 3) using the NSC, LDA, RF and SVM for Hughes’ gene expression data with 300 microarray experiments

Truth NSC (p = 85) NSC (p = 300) LDA RF SVM
Pred = 1 Pred = 2
100 19 97 22 70 49 100 19 99 20
21 91 23 89 30 82 21 91 19 93
Accuracy 0.827 0.805 0.658 0.827 0.831
Truth NSC (p = 85) NSC (p = 300) LDA RF SVM
Pred = 1 Pred = 2
100 19 97 22 70 49 100 19 99 20
21 91 23 89 30 82 21 91 19 93
Accuracy 0.827 0.805 0.658 0.827 0.831
Table 7

BIC with various numbers (g) of clusters in the standard and stratified clustering for Hughes’ gene expression data with only 10 microarray experiments

g Standard New
BIC BIC
5416 5416
5172 5115
4989 4913
4977 4912
4888 4830
4892 4860
4901 4871
g Standard New
BIC BIC
5416 5416
5172 5115
4989 4913
4977 4912
4888 4830
4892 4860
4901 4871

Italics give the minima

In summary, using the gene expression data with all the 300 microarray experiments, the standard clustering and our proposed method had similar performance. The close performance between the two methods was presumably because there was enough information in the data for predicting the two functional classes; with fewer microarray experiments or more functional classes, difference might appear. On the other hand, as in the null case (i.e. set-up 1) of simulation, it showed that, although our proposed method did not improve over the standard method, it did not result in any significant loss of efficiency either.

#### 3.2.2 Using 10 microarray experiments

We suspected that because of information redundancy in the gene expression data for predicting the two gene functional classes, the influence of the prior probability was negligible; e.g. the cluster centers were all far away from each other, dominating the resulting posterior probability calculations and thus final clustering results. Hence, to increase the difficulty or the noise level of the problem, next we only used the first 10 microarray experiments.

Based on BIC, we chose g = 5 clusters for both methods (Table 8). Using hard classification (Table 3), the overall accuracy rates from the two approaches were close, though the new method had a slight edge over the standard method. It seemed that using the estimated prior probabilities for strata 1 and 2 improved the predictive accuracy rates for the two classes respectively at the expense of a lower accuracy rate for the other class. This trend was more evident with soft classification (Table 9); more importantly, when using the estimated prior probability from stratum 3 as prescribed, our proposed method gave a higher overall accuracy rate than that of the standard method.

Table 8

Predictions for stratum 3 based on hard classification using standard clustering and new clustering with g = 5 for Hughes’ gene expression data with 10 microarray experiments

Truth Standard New: $π^(1)$ New: $π^(2)$ New: $π^(3)$
Pred = 1 Pred = 2
101 18 97 22 81 38 92 27
53 59 47 65 26 86 35 77
Accuracy 0.693 0.701 0.723 0.732
Truth Standard New: $π^(1)$ New: $π^(2)$ New: $π^(3)$
Pred = 1 Pred = 2
101 18 97 22 81 38 92 27
53 59 47 65 26 86 35 77
Accuracy 0.693 0.701 0.723 0.732
Table 9

Predictions for stratum 3 based on soft classification using standard clustering and new clustering with g = 5 for Hughes’ gene expression data with 10 microarray experiments

Truth Standard New: $π^(1)$ New: $π^(2)$ New: $π^(3)$
Pred = 1 Pred = 2
66.8 51.2 78.5 39.5 67.6 50.4 74.8 43.2
43.0 68.0 47.5 63.5 32.9 78.1 38.5 72.5
Accuracy 0.584 0.615 0.631 0.638
Truth Standard New: $π^(1)$ New: $π^(2)$ New: $π^(3)$
Pred = 1 Pred = 2
66.8 51.2 78.5 39.5 67.6 50.4 74.8 43.2
43.0 68.0 47.5 63.5 32.9 78.1 38.5 72.5
Accuracy 0.584 0.615 0.631 0.638

Table 10 gives prior probability estimates for the two methods. Most of these estimates were close to each other; in particular, $π^$ from the standard method was in general close to $π^(3)$ from the new method, perhaps because stratum 3 had similar proportions of the genes from the two classes as did the total dataset. However, some estimates were different; e.g. for cluster 2, the four prior estimates were 0.461, 0.535, 0.134 and 0.344 respectively. The overall closeness as well as some individual differences among the estimated prior probabilities offered an explanation on why the clustering results were close, but nevertheless different.

Table 10

Estimated prior probabilities with g = 5 for Hughes’ gene expression data with 10 microarray experiments

 $π^$ 0.219 0.461 0.206 0.111 0.003 $π^(1)$ 0.272 0.535 0.023 0.169 0 $π^(2)$ 0.14 0.134 0.329 0.397 0 $π^(3)$ 0.203 0.344 0.244 0.2 0.009
 $π^$ 0.219 0.461 0.206 0.111 0.003 $π^(1)$ 0.272 0.535 0.023 0.169 0 $π^(2)$ 0.14 0.134 0.329 0.397 0 $π^(3)$ 0.203 0.344 0.244 0.2 0.009

The predictive performances of the four classifiers are summarized in Table 11. Comparing Tables 11 and 3, we found that the two clustering methods, especially our proposed new one, worked well with results quite close to that of supervised learning methods, highlighting the promise of using clustering analysis for gene function prediction. Note that there are several other advantages of clustering analysis over supervised learning. First, clustering methods allow the possibility of discovering new classes while predicting for existing classes (Golub et al., 1999). Second, rather than using a hierarchical ontology to describe gene functions, gene associations or linkages have been argued to be an alternative (Fraser and Marcotte, 2004), for which clustering methods are more naturally applicable than supervised learning.

Table 11

Predictions for a separate test dataset based on the NSC, LDA, RF and SVM for Hughes’ gene expression data with 10 microarray experiments

NSC (p = 9) NSC (p = 10) LDA RF SVM
Truth Pred = 1 Pred = 2
91 28 91 28 83 36 89 30 85 34
31 81 32 80 33 79 28 84 30 82
Accuracy 0.745 0.740 0.701 0.749 0.723
NSC (p = 9) NSC (p = 10) LDA RF SVM
Truth Pred = 1 Pred = 2
91 28 91 28 83 36 89 30 85 34
31 81 32 80 33 79 28 84 30 82
Accuracy 0.745 0.740 0.701 0.749 0.723

## 4 DISCUSSION

With relatively high noise levels in genomic data, the importance of incorporating biological knowledge into statistical analysis has been increasingly recognized. However, the current practice in this direction is mainly restricted to using biological knowledge as an evaluating criteria to validate the analysis results after the analysis is done. For example, many systems have been built to assess statistical enrichments of a list of user-supplied genes in any GO categories, as evidenced by two recent reviews by Handl et al. (2005) and Khatri and Draghici (2005); see references therein. There are a few exceptions: e.g. Mootha et al. (2003), Al-Shahrour et al. (2005) and Pan (2005) proposed different stratified/subgroup analysis approaches to detect differential gene expression, all based on the idea of using biological information to group genes to form strata; Lottaz and Spang (2005) considered incorporating GO categories into sample or tumor classifications, though their main motivation was to enhance interpretability of results, which is also an important factor applicable to genomic analyses. Here we extend the idea to clustering genes and demonstrate its improved performance using both simulated and real data.

In practice, one has to determine which gene functional groups to use. As in any serious statistical modeling, some careful thoughts and trade-offs are needed. There are two factors that need to be balanced. It is desirable to have any group Gk as homogeneous as possible, which, however, may contain only few genes. Furthermore, using smaller groups leads to a larger number of the groups, hence more parameters (i.e. π(k)s) are to be estimated based on smaller sample sizes. Here we suggest three approaches. The first is both simple and practical: to avoid using functional groups either too big or too small, we specify an acceptable range of the number of the genes a functional group can contain, plus possibly some other constraints, and thus determine eligible functional groups. A specific example is the use of ‘informative’ GO categories (Zhou et al., 2002); an informative GO category is the one satisfying the two conditions: (1) it contains at least γ genes and (2) it does not have any child category containing at least γ genes, where γ is a threshold in the range of 20–40. For example, with γ = 30, 73 informative MIPS categories were found (Xiao and Pan, 2005), which can be used as priors in our method. Second, among several candidate sets of functional groups, model-selection criteria, such as BIC, can be used to choose the one estimated to give the best predictive performance; this is an advantage of model-based clustering. Third, following the line of Huang et al. (2006), a weighted method can be applied to combine the results of using two sets of functional groups, or more generally, to take account of the hierarchical structure of a gene annotation system. More studies on genomic scales are needed.

In our example, to deal with genes with multiple functions, we simply replicated their expression profiles for each of their functional groups. This paralleled the usual treatment when the expression profile of a gene with multiple functions was used as a test observation. Nevertheless, a downside was that, owing to the introduced correlations among the expression profiles of the same genes, the resulting calculation of BIC was not exact. On the other hand, with only a small number of such genes as in our example, we suspect that the influence was limited. Furthermore, to avoid any potential problem, a simple alternative is to assign a gene randomly to only one of its known functional groups for training data; this will only influence the prior specification by ignoring multiple functions of some genes, but not the validity of final results.

In this article, we have only considered clustering genes, but the idea can be equally applied to model-based clustering of samples for tumor classifications (e.g. Yeung et al., 2001; Ghosh and Chinnaiyan, 2002; McLachlan et al., 2002) if the samples are known a priori to belong to different groups; these groups can be formed based on some diagnostic or prognostic factors. Our proposed method also bears some similarity to clustering partially classified data (McLachlan and Peel, 2000, Section 2.19); see Qu and Xu (2004) and Alexandridis et al. (2004) for the latter's two applications to gene expression data. However, there is a key difference: in partially classified data, all the genes known to have the same class label are assumed to be from the same cluster, whereas, we allow genes sharing the same class label to come from different clusters; we only assume that they share the same prior probability of coming from the same cluster. In other words, in partially classified data, a class label is assumed to be the same as a cluster membership, whereas we do not impose such a restriction. Considering the heterogeneity of various gene functional categories, our modeling assumption is not only weaker, but also more reasonable in the current context.

Many existing clustering methods are not model-based (e.g. Tamayo et al., 1999; Tseng and Wong, 2005). An advantage of model-based clustering is its explicit statistical modeling, which for instance facilitates incorporating subject-matter knowledge as priors into analysis, as we have shown here. In contrast, in non-model-based clustering, biological knowledge was directly incorporated into a distance metric (Hanisch et al., 2002; Cheng et al., 2004; Huang and Pan, 2006). The advantage of the former is its robustness: when the prior specification is incorrect, the final results may still be valid as long as there is enough information in the data (e.g. Carlin and Louis, 2000), while the latter may give completely wrong results. This also partially explains the results in our example: first, with the data containing 300 microarray experiments, the new method gave results similar to that of the standard method, presumably because there was enough information in the data and incorporating biological knowledge as prior would not make any difference; on the contrary, with the data consisting of only 10 microarray experiments, by incorporating biological knowledge, the new method was more efficient than the standard method and thus giving better results. We are currently further evaluating the difference between the two approaches of incorporating biological knowledge in clustering. In addition, our proposed method is an empirical Bayes approach; conceptually it seems possible to generalize our idea to fully Bayesian model-based clustering (Richardson and Green, 1997; Broet et al., 2002; Medvedovic and Sivaganesan, 2002; Fraley and Raftery, 2005, ), though computational challenges remain. The methodology may be also extended to other more elaborate modeling contexts, such as analysis of time-course microarray data (e.g. Ramoni et al., 2002; Luan and Li, 2003). These are interesting topics to be studied in the future.

The author thanks Guanghua Xiao and Peng Wei for helpful discussions and assistance with the gene expression data. The author is grateful to the reviewers for their constructive comments. This work was supported by NIH grant HL65462 and a UM AHC Development grant.

Conflict of Interest: none declared.

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## Author notes

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# Weak convergence and Subsequences

I am preparing for an exam and I saw this exercise that our teacher left for us to prove:

Let $\{x_n \}_{n=1}^{\infty}$ be a sequence in a Hilbert space $\mathcal{H}$ such that $x_n \rightharpoonup 0$ or $\langle x_n, u \rangle \rightarrow 0$ for all $u \in \mathcal{H}.$

1. Prove inductively that there exists a subsequence, $\{x_{n_k}\}_{k = 1}^{\infty}$ such that $\left| \langle x_{n_k}, x_{n_j} \rangle \right| \le \frac{1}{k}$ whenever $k > j.$
2. For $N \in \mathbb{N}$, define $$y_N = \frac{1}{N}\sum_{k=1}^N x_{n_k}.$$ Prove that $y_N \rightarrow 0.$

For 1, I was thinking that there exists a convergent subsequence. For 2, I'm rather lost.

For $(1)$:

Since $\langle x_n, x_1\rangle \xrightarrow{n\to\infty} 0$, there exists a strictly increasing sequence $(p_1(n))_{n=1}^\infty \in \mathbb{N}$ such that $\left|\langle x_{p_1(n)}, x_1\rangle\right| < \frac1{n}, \forall n \in \mathbb{N}$.

Define $x_{n_1} = x_{p_1(1)}$.

Since $\langle x_{p_1(n)}, x_{p_1(1)}\rangle \xrightarrow{n\to\infty} 0$, there exists a strictly increasing sequence $(p_2(n))_{n=1}^\infty \in \mathbb{N}$ such that $\left|\langle x_{p_2(p_1(n))}, x_{p_1(1)}\rangle\right| < \frac1{n}, \forall n \in \mathbb{N}$.

Define $x_{n_2} = x_{p_1(p_2(2))}$.

Since $\langle x_{p_2(p_1(n))}, x_{p_2(p_1(2))}\rangle \xrightarrow{n\to\infty} 0$, there exists a strictly increasing sequence $(p_3(n))_{n=1}^\infty \in \mathbb{N}$ such that $\left|\langle x_{p_3(p_2(p_1(n)))}, x_{p_2(p_1(2))}\rangle\right| < \frac1{n}, \forall n \in \mathbb{N}$.

Define $x_{n_3} = x_{p_3(p_2(p_1(3)))}$.

Continue inductively, and define $x_{n_k} = x_{p_1(p_2(\ldots p_k(k)))} = x_{(p_1 \circ \cdots \circ p_k)(k)}$ for all $k \in \mathbb{N}$.

For $k > j$ we have by construction of $p_{j}$

$$\left|\langle x_{n_k}, x_{n_j}\rangle\right| = \left|\langle x_{(p_1 \circ \cdots p_{j} \circ p_{j+1} \circ \cdots \circ p_k)(k)}, x_{(p_1 \circ \cdots \circ p_j)(j)}\rangle\right| < \frac1{(p_{j+1}\circ p_{j+2} \circ \cdots \circ p_k)(k)} < \frac1{k}$$

because all $p_i$ are strictly increasing.

For 2:
$\Vert y_N \Vert^2 = \frac{1}{N^2} \langle \sum_{k = 1}^N y_{n_k}, \sum_{k = 1}^N y_{n_k} \rangle = \frac{1}{N^2} \sum_{k = 1}^N \sum_{j = 1}^N \langle y_{n_k}, y_{n_j} \rangle \leq \frac{2}{N^2} \sum_{k = 1}^N \sum_{j = k + 1}^N \vert \langle y_{n_k}, y_{n_j} \rangle \vert$.
At the last estimate we use that $\vert \langle y_{n_k}, y_{n_j} \rangle \vert = \vert \langle y_{n_j}, y_{n_k} \rangle \vert$.

Now we can use the properties of the sequence and further estimate
$\frac{2}{N^2} \sum_{k = 1}^N \sum_{j = k + 1}^N \vert \langle y_{n_k}, y_{n_j} \rangle \vert \leq \frac{2}{N^2} \sum_{k = 1}^N \sum_{j = k + 1}^N \frac{1}{k} = \frac{2}{N^2} \sum_{k = 1}^N \frac{N - k}{k} = \frac{2}{N}(\sum_{k = 1}^N \frac{1}{k} - 1) \to 0$.

• Thanks, @eddie. This proof makes a lot of sense. I think you meant $x_{n_k}$ not $y+{n_k}$. – Harry Evans May 20 '18 at 23:45
• @HarryEvans that's exactely what i meant ;) – eddie May 21 '18 at 7:55


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Is there a machinery describing all the irreducible representations ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T20:57:58Z http://mathoverflow.net/feeds/question/18559 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/18559/is-there-a-machinery-describing-all-the-irreducible-representations Is there a machinery describing all the irreducible representations ? Peter Lee 2010-03-18T08:14:01Z 2010-03-18T18:47:12Z
Suppose we have a finite dimensional Lie algebra $g$， Is there a machinery to describe all the irreducible representation of $g$.

Consider toy example: $sl_{2}$ or $sl_{3}$, how do we describe all the irreducible representations of them.

Further, consider quantum case, Is there a machinery way(like algorithm)describing all the irreducible representations of $U_{q}(sl_{2})$

EDIT: What I am looking for is an "mechanical" and canonical machinery describing all the irreducible representations(of course, not only finite dimensional representations,not only unitary representations)

EDIT2: What I am looking for is some reference to describe them in explicitly(such as $sl_{3}$）
 http://mathoverflow.net/questions/18559/is-there-a-machinery-describing-all-the-irreducible-representations/18564#18564 Answer by Bruce Westbury for Is there a machinery describing all the irreducible representations ? Bruce Westbury 2010-03-18T09:38:45Z 2010-03-18T13:25:13Z
The short answer is no. There is a classification of primitive ideals in the enveloping algebra (and quantised enveloping algebra). This reduces the problem to primitive rings. However the representation theory of primitive rings which are not Artinian is complicated.

An example which I find easier is the Weyl algebra (or linear differential operators). This ring is primitive since the vector space of polynomials is an irreducible faithful representation. This ring is in fact simple (no proper ideal). However the representation theory encompasses the theory of linear differential equations with polynomial coefficients.

So speaking heuristically, the representation theory of semisimple Lie algebras is at least as complicated as the representation theory of the Weyl algebra and it is unreasonable to expect an answer in this case.

I don't know of a formal result that says this is an unreasonable request. For example: does this problem include the problem of classifying indecomposable representations of a wild algebra?

Edit I have just found this reference which solves the question for $sl(2)$.

MR0605353 (83c:17010) Block, Richard E. The irreducible representations of the Lie algebra $sl(2)$ and of the Weyl algebra. Adv. in Math. 39 (1981), no. 1, 69--110.
 http://mathoverflow.net/questions/18559/is-there-a-machinery-describing-all-the-irreducible-representations/18620#18620 Answer by Torsten Ekedahl for Is there a machinery describing all the irreducible representations ? Torsten Ekedahl 2010-03-18T16:20:02Z 2010-03-18T18:47:12Z
The problem of classifying irreducible $sl_2(\mathbb C)$-representations is essentially untractable as it contains a wild subproblem. Indeed, the action of the Casimir element $C$ on any irreducible representation is by a complex scalar (by a theorem of Quillen I believe). If we consider the case when $C$ acts by zero, by a result of Beilinson-Bernstein the category of $sl_2$-representations with $C=0$ is equivalent to the category of quasi-coherent $\mathcal D_{\mathbf P^1}$-modules. In this $1$-dimensional case every irreducible $\mathcal D_{\mathbf P^1}$-module is holonomic. If we restrict ourselves to irreducible regular holonomic modules we have two possibilities. One case is that they are supported at a single point and then the point is a complete invariant. In the other case they are classified by a finite collection of points of $\mathbf P^1$ and equivalence classes of irreducible representation of the fundamental group of the complement of the points which map the monodromy elements of the points non-trivially. In particular we can consider the case of three points in which case the fundamental group is free on two generators (they and the inverse of their product being the three monodromy elements). The irreducible representations where one of the monodromy elements act trivially correspond to removing the corresponding point and thinking of the representation as a reprentation of the fundamental group of that complement.

Hence, we can embed the category of finite-dimensional representations of the free group on two elements as a full subcategory closed under kernels and cokernels of the category of $sl_2(\mathbb C)$-modules. This makes the latter category wild in the technical sense. However, the irreducible representations of the free group on two letters are also more or less unclassifiable.

There is no contradiction between this and the result of Block. His result gives essentially a classification of irreducibles in terms of equivalence classes of irreducible polynomials in a twisted polynomial ring over $\mathbb C$. So the consequence is that such polynomials are essentially unclassifiable.

[Added] Intractable depends on your point of view. As an algebraic geometer I agree with Mumford making (lighthearted) fun of representation theorists that think that wild problems are intractable. After all we have a perfectly sensible moduli space (in the case of irreducible representations) or moduli stack (in the general case). One should not try to "understand" the points of an algebraic variety but instead try to understand the variety geometrically. Today, I think that this view point has been absorbed to a large degree by representation theory.


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# three circles inscribed in a big circle

• Jan 12th 2010, 05:24 AM
arenkun
three circles inscribed in a big circle
hello ! help me to solve this problem, please!!
its really hard for me ! thank you so much and God bless you richly !(Wink)

Three circles A,B and C are tangent externally to each other and each tangent internally to a larger circle having a radius of 10 cm. Radius of circle A is 5 cm.

a) compute the distance from the center of the larger circle to the point of tangency of the two circles B and C which are identical.

b) compute the radius of circles B and C.
• Jan 12th 2010, 08:10 AM
Hi arenkun,

For the radii, you can use a combination of Sin(angle) and the Cosine Rule
(Law of Cosines) to solve this.

In the diagram...

$\vartriangle{oc_1c_2}$ has sides 5, 5+r, 10-r.

Also $\sin\theta = \frac{r}{5+r}$

From the Law of Cosines...

$(10-r)^2=(5+r)^2+5^2-2(5)(5+r)\cos\theta$

$100-20r+r^2=25+10r+r^2+25-10(5+r)\cos\theta$

$50-30r=-(50+10r)\cos\theta$

$\cos\theta = \frac{50-30r}{-(50+10r)}=\frac{30r-50}{50+10r}$

Now use the trigonometric identity $Sin^{2}\theta + Cos^{2}\theta = 1.$

$\frac{(30r-50)^2}{(50+10r)^2}+\frac{r^2}{(5+r)^2} = 1$

$\frac{900r^2-3000r+2500}{2500+100r^2+1000r} + \frac{r^2}{25+10r+r^2}=1$

$\frac{9r^2-30r+25}{25+r^2+10r}+\frac{r^2}{25+10r+r^2}=1$

$10r^2-30r+25=r^2+10r+25$

$9r^2-40r=0$

$r(9r-40)=0$

As r is not zero, then $9r=40$

$r=\frac{40}{9}$
• Jan 12th 2010, 12:19 PM
Once you have "r", you can use it to calculate the answer
to the first question as shown on the attachment.
• Jan 12th 2010, 12:23 PM
However, you may have meant the large circle to be the green one!

Let us know if you did.
• Jan 12th 2010, 06:51 PM
arenkun


text:
Circle Geometry Problem

I was trying this problem and i couldnt come up with a solution, it doesnt seem to hard but I cant think of what to do.

There is a point on a wheel that is rotating at a constant speed. The bottom of the wheel is 1 meters above the ground. At a point Q the point on the wheel is 16 meters above the ground. After 4 seconds its reaches the top of the wheel. After another 8 seconds it reaches a point P. If the wheel has a diameter of 16 meters how do we find out how high the point is at point P.

Thanks!

-
If you can use coordinates, they might help greatly. Center the rotating circle on the vertical axis; trigonometry might be helpful. –  Guess who it is. Apr 15 '12 at 16:03

To follow the argument below, you will need to draw a diagram.

Let $O$ be the centre of the wheel, and $T$ the top of the wheel.

Draw a perpendicular from $Q$ to the vertical line $OT$, meeting that line at $X$. The height of the point $Q$ (and therefore of $X$) is $16$, and the height of $T$ is $17$. It follows that $XT=1$ and therefore $OX=7$.

Let $\theta=\angle QOX$. We have $\cos\theta=\frac{7}{8}$. In $4$ seconds the wheel travels through an angle $\theta$. So in $8$ seconds it travels through an angle $2\theta$.

We can find $\cos 2\theta$ approximately, using a calculator, or exactly, using $\cos 2\theta=2\cos^2\theta-1$. Let's go for exactly. We get $\cos 2\theta=\frac{34}{64}$.

Now we can find the height of point $P$. The picture is much like the one we worked with for $Q$. Draw a perpendicular from $P$ to the vertical line $OT$, meeting that line at $Y$. Then $OY=8\cos 2\theta$. So the height of $P$ above the centre of the wheel is $8\cos 2\theta$, and therefore the height above the ground is $9+8\cos 2\theta$.

-


text:
# integration

#### paulmdrdo

##### Active member
how would you integrate

1. $\displaystyle \int \left(\frac{x}{y}\right) d\left(\frac{x}{y}\right)$

2. $\displaystyle \int \frac{d(xy)}{(xy)^3}$

if the two is just in this form I could easily answer it

$\displaystyle \int d\left(\frac{x}{y}\right)=(\frac{x}{y})$

$\displaystyle \int d(xy)=(xy)$

but I don't know how to treat $\frac{x}{y}$ in 1 and the $(xy)^3$ in 2.

#### mente oscura

##### Well-known member
how would you integrate

1. $\displaystyle \int \left(\frac{x}{y}\right) d\left(\frac{x}{y}\right)$

2. $\displaystyle \int \frac{d(xy)}{(xy)^3}$

if the two is just in this form I could easily answer it

$\displaystyle \int d\left(\frac{x}{y}\right)=(\frac{x}{y})$

$\displaystyle \int d(xy)=(xy)$

but I don't know how to treat $\frac{x}{y}$ in 1 and the $(xy)^3$ in 2.

Hello.

$$1) \ u=\dfrac{x}{y}$$

$$\displaystyle \int u \ d(u)=u= \dfrac{x}{y}$$

$$2) \ u=(xy)$$

$$\displaystyle \int \dfrac{d(u)}{u^3}=?$$

Can you follow?

Regards.

#### MarkFL

Staff member
Hello.

$$1) \ u=\dfrac{x}{y}$$

$$\displaystyle \int u \ d(u)=u= \dfrac{x}{y}$$
What we want here is:

$$\displaystyle \int u\,du=\frac{1}{2}u^2+C$$

#### mente oscura

##### Well-known member
What we want here is:

$$\displaystyle \int u\,du=\frac{1}{2}u^2+C$$
Right. I I'm wrong.

I wrote:

$$\int u\,du$$

And I thought in:

$$\int \,du$$

I'm sorry.

Regards.


text:
Try Gonometry

Algebra Level pending

Given: $$sin(x)\, +\, cos(x)\, =\, \frac { 1 }{ 2 }$$ And $${ sin }^{ 3 }(x)\, +\, { cos }^{ 3 }(x)\, =\, \frac { a }{ b }$$

Find: $$b\, -\, a$$ such that $$a$$ and $$b$$ are co-prime integers.

×

Problem Loading...

Note Loading...

Set Loading...


text:
If one just wants to test whether a given polynomial $f(x)$ is positive for all real $x$, this can be done without representing $f(x)$ as the sum of squares, using the following function:
def is_pol_positive(f):


text:
# Primes lying above a prime in a Galois extension

Assuming $$L$$ is Galois over $$K$$, and that $$O_L$$ and $$O_K$$ are their respective rings of integers. Let $$p$$ be a prime ideal of $$O_K$$, is there a classification of the prime ideals laying above $$p$$ in $$O_L$$?

I understand that if $$O_L$$ is generated by a unique element over $$O_K$$, i.e. $$O_L=O_K[\theta]$$, then there exists such a description, using the irreducible prime factors of $$\theta$$'s integral dependence equation, after "dividing by $$p$$". However, as there is no particular reason for this to happen in general, I am curious as to whether there exists a more general classification.

• The classification you are describing still holds even if $O_K[\theta] \subsetneq O_L$, provided that $p$ doesn't divide the conductor of $O_K[\theta]$ inside $O_L$ (this should be sufficient to consider primes that don't divide the discriminant of $f$, see here) – this misses only finitely many primes. See Neukirch's book on ANT, I.8.3, or also math.stanford.edu/~conrad/154Page/handouts/dedekindcrit.pdf. – Watson Dec 27 '18 at 20:51


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# Construct a linear map $M : V → V$ with the property that $K = \{v ∈ V\mid Mv = 0\}.$

"Suppose that V is a vector space and $L : V → V$ is a linear map.

(i) Let K ⊂ V be the set of all vectors $v ∈ V$ such that $L(v) = −v$. Show that K is a subspace of V .

(ii) Construct a linear map $M : V → V$ with the property that $K = \left\{v ∈ V | Mv = 0\right\}.$

For part (i), I just showed that K contained the zero vector, was closed under addition and closed under scalar multiplication.

However I'm really not sure what to do for part (ii). Would I have to use matrices?

• Note that $Lv=-v$ iff $(L+I)v=0$, where $I$ is the identity transformation. – symplectomorphic Jun 17 '16 at 4:00

For part (i) you're done with addition and scalar multiplication alone ($0$ then follows), and you can do it in one go with showing that $v_1,v_2 \in K$ and $\lambda$ a scalar implies $v_1 + \lambda \cdot v_2 \in K$.
For (ii) we always have the linear map $I: V \rightarrow V$ which has $I(v) = v$ for all $v \in V$, the identity. Then $L(v) = -v$ iff $L(v) + v = 0$ iff $L(v) + I(v) = 0$ iff $(L+I)(v) = 0$.
So the pointwise sum $M = L+I$ works. It's a standard exercise that the (pointwise) sum of linear maps is again a linear map. No need for matrices, this is true in the purely abstract.
Note that then (i) becomes a consequence of (ii), as $K$ is the kernel of $M$, and I suppose you have shown that every kernel of a linear map is a subspace (again true in general).


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# Math Help - System with floor functions

1. ## System with floor functions

For the following system I can figure out that $[x]=0, [y]=$ or $[x]=1,[y]=3$ but i can't go on. Sugestions please!

2. ## Re: System with floor functions

Hey mathmagix.

For this problem you should consider what possibilities can generate the following relations.

What is the value of the floor function in terms of its inputs to return 3 in the 2nd answer?

Based on this can you get an in-equality for xy? (Hint: if z = [n,n+1) for integer n then floor(z) = n. So the range for [n,n+1) is the range you need to consider for the input.


text:
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# Is k + 1/k < -1 ?

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Is k + 1/k < -1 ? [#permalink]

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Is k + 1/k < -1 ?

1) k^2 < 1
2) k^3 + k^2 + k < 0
[Reveal] Spoiler: OA

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Is k + 1/k < -1 ? [#permalink]

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10 Oct 2017, 06:43
MathRevolution wrote:
Is k + 1/k < -1 ?

1) k^2 < 1
2) k^3 + k^2 + k < 0

Hi..
$$k+\frac{1}{k}<-1......(\frac{1+k+1}{k})<0.....\frac{(k^2+k+1)}{k}<0$$...
So both numerator and denominator should be of opposite sign..

If $$k<0, k^2+k+1>0$$....
If $$k>0, k^2+k+1<0$$....not possible

1) $$k^2<1$$
Insufficient

2) $$k^3+k^2+k<0....... K(k^2+k+1)<0..$$
So $$k$$ and $$k^2+k+1$$ are of opposite sign.
This is same as the initial equation
Sufficient

B

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Re: Is k + 1/k < -1 ? [#permalink]

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12 Oct 2017, 01:44
Forget conventional ways of solving math questions. In DS, VA (Variable Approach) method is the easiest and quickest way to find the answer without actually solving the problem. Remember that equal number of variables and independent equations ensures a solution.

The first step of VA(Variable Approach) method is modifying the original condition and the question, and rechecking the number of variables and the number of equations.

k + 1/k < 1 is equivalent to k3 + k < -k2 or k3 + k2 + k < 0.
k(k2 + k + 1) < 0 implies k < 0, since k2 + k + 1 is always positive.
Thus, the simplified question is if k < 0.

Condition 1) k2< 1
It is equivalent to -1 < k < 1.
Thus,this is not sufficient.

Condition 2) k3 + k2 + k < 0
Thus, is equivalent to the question.
This is sufficient.

Therefore, the answer is B.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both con 1) and con 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Then, D is most likely to be the answer using con 1) and con 2) separately according to DS definition. Obviously, there may be cases where the answer is A, B, C or E.
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Re: Is k + 1/k < -1 ?   [#permalink] 12 Oct 2017, 01:44
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# Why, exactly, can't two field lines cross? [duplicate]

My physics textbook says two electric field lines never intersect. Their explanation runs somewhat like this:

If two field lines crossed, there would be two different directions to the electric field at the intersection point, which is impossible by definition.

A similar explanation is provided as to why two stream lines never cross.

However, mathematically, two intersecting curves can have the same direction of tangent at their intersection point. For example, consider the $x-$axis and the curve $y=x^3$ is the $xy$ plane. Or any two curves of the form $y=x^{2k+1}$ with $k\in\mathbb N$, plotted for $-1<x<1$.

Is the explanation then wrong, considering these counterexamples? If not, why not? If yes, then what's the correct explanation?

Edit: The answers to this Phys.SE question seem to focus on the very explanation I am having trouble understanding; I don't see how this is a duplicate of that one. I'd rather describe this as a follow-up question to that one.

## marked as duplicate by anna v, Yashas, Qmechanic♦Apr 3 '17 at 6:26

• Possible duplicate of Why can two (or more) electric field lines never cross? – anna v Apr 3 '17 at 5:14
• If you draw what you are describing, you are drawing a source at x=0 . Sources are charges in experimental observations and as modeled by classical electrodynamics. electric field lines theoreticlly cross on the point where a point charge is. – anna v Apr 3 '17 at 5:17
• This is really a mathematical question, related to the existence and uniqueness theorem for ODEs. The answer depends on how pathological you allow your situations to be. – knzhou Apr 3 '17 at 5:25
• In a real physical setup, this wouldn't be allowed. By allowing field lines to merge into one, you increase the local electric flux density to infinity. – knzhou Apr 3 '17 at 5:26
• @knzhou I'm interested to know how the existence and uniqueness theorem for ODEs solves my problem; mind elaborating? And FWIW, I don't mind pathological situations. :) – Ankoganit Apr 3 '17 at 6:09


text:
sábado, 9 de abril de 2011

Bureaucracy

bureaucracy |byoŏˈräkrəsē|
noun ( pl. -cies)
a system of government in which most of the important decisions are made by state officials rather than by elected representatives.
a state or organization governed or managed according to such a system.
the officials in such a system, considered as a group or hierarchy.
excessively complicated administrative procedure, seen as characteristic of such a system : the unnecessary bureaucracy in local government.
ORIGIN early 19th cent.: from French bureaucratie, from bureau (see bureau, -cracy ).


text:
• anonymous
When converting $\frac{31}{11111}$ to a decimal, the decimal turns out to be a repeating decimal. How many digits repeat in this repeating decimal? For example, if you get the repeating decimal $0.\overline{123},$ then your answer should be $3,$ and if you get $0.436\overline{7},$ your answer should be $1.$
Mathematics

Looking for something else?

Not the answer you are looking for? Search for more explanations.


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## Publicações > Artigos em Revistas Internacionais

### Stochastic ordering in the qualitative assessment of the performance of simultaneous schemes for bivariate processes

Ramos, Patrícia F.; Morais, M. C. ; Pacheco, António; Schmid, Wolfgang

Sequential Analysis-Design Methods and Applications, 32(2) (2013), 214-229
http://dx.doi.org/10.1080/07474946.2013.774619

Misleading signals (MS) are likely to happen while using a simultaneous scheme to control the mean vector (\mu) and the covariance matrix (\Sigma) of a bivariate process. They correspond to valid signals that lead to a misinterpretation of a shift in \mu (resp. \Sigma) as a shift in \Sigma (resp. \mu). Following previous work, focused on the quantitative assessment of the probabilities of misleading signals (PMS) in simultaneous schemes for bivariate processes, we now make use of stochastic ordering to qualitatively assess the impact of changes in \mu and \Sigma in those probabilities.


text:
Question
Two coils are placed close together in a physics lab to demonstrate Faraday’s law of induction. A current of 5.00 A in one is switched off in 1.00 ms, inducing a 9.00 V emf in the other. What is their mutual inductance?

$1.80 \textrm{ mH}$

Solution Video

# OpenStax College Physics Solution, Chapter 23, Problem 55 (Problems & Exercises) (1:01)

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Video Transcript
This is College Physics Answers with Shaun Dychko. We are going to find the mutual inductance between these two coils by taking the absolute value of induced EMF because we don’t really care about directions here, so we will just take the EMF induced in the second coil equal the mutual inductance times the rate of change of current in the first coil and so we will multiply both sides by delta T over I and then we will solve for M and so the mutual inductance is the induced EMF times the times in which the current changes divided by the amount which current changes. So, its nine volts induced and then that happens in one milli-second and change in current is five amps. Because it gets switched off and so its goes from five amps it goes to zero is what implied by saying “switched off” and this makes 1.80 milli-henry is the mutual inductance.


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MainPage.dox 3.09 KB
 PIACIBELLO Cyrille committed Nov 29, 2013 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 // This page contains the special doxygen pages and mainpage. /*! * @mainpage ScalFmm * * \section overV Overview * * ScalFMM is a software library to simulate N-body interactions using * the Fast Multipole Method. * * The library offers two methods to compute interactions between * bodies when the potential decays like 1/r. The first method is the * classical FMM based on spherical harmonic expansions and the second is * the Black-Box method which is an independent kernel formulation * (introduced by E. Darve @ Stanford). With this method, we can now * easily add new non oscillatory kernels in our library. For the * classical method, two approaches are used to decrease the complexity * of the operators. We consider either matrix formulation that allows us * to use BLAS routines or rotation matrices to speed up the M2L * operator. * * ScalFMM intends to offer all the functionalities needed to perform * large parallel simulations while enabling an easy customization of * the simulation components: kernels, particles and cells. It works * in parallel in a shared/distributed memory model using OpenMP and * MPI. The software architecture has been designed with two major * objectives: being easy to maintain and easy to understand. There is * two main parts: *
*
• the management of the octree and the * parallelization of the method;
• *
• The kernels. This new * architecture allow us to easily add new FMM algorithm or kernels * and new paradigm of parallelization.
• *
* * * To download build and install the application, please see \ref * install. *  41  * The \ref quick can help you to understand the architecture of the  PIACIBELLO Cyrille committed Nov 29, 2013 42 43  * library. *  44 45  * The \ref mainexamples page can help you to understand how the test are made. *  PIACIBELLO Cyrille committed Nov 29, 2013 46 47 48 49 50 51  * This software is distributed under a specific License. For more * informations, see \ref License. * * If you want to cite the project and/or the team, please contact * us. See \ref contacts page. *  COULAUD Olivier committed Sep 12, 2014 52 53 54 55 56 57 58 59  * * \subsection Publications *
*
• Task-based FMM for heterogeneous architectures, Emmanuel Agullo; Berenger Bramas; Olivier Coulaud,;Eric Darve; Matthias Messner; Toru Takahashi, Inria Research Report, 2014, pp. 29. RR-8513 *
• Task-Based FMM for Multicore Architectures, Emmanuel Agullo; Berenger Bramas; Olivier Coulaud , Eric Darve; Matthias Messner; Toru Takahashi SIAM Journal on Scientific Computing, SIAM, 2014, 36 (1), pp. 66-93 DOI : 10.1137/130915662
• *
• Optimized M2L Kernels for the Chebyshev Interpolation based Fast Multipole Method, Matthias Messner; Berenger Bramas; Olivier Coulaud ; Eric Darve Research Report, 2012
•  COULAUD Olivier committed Sep 12, 2014 60  *
• Pipelining the Fast Multipole Method over a Runtime System, Emmanuel Agullo; Berenger Bramas; Olivier Coulaud ; Eric Darve ; Matthias Messner; Toru Takahashi, 2012, pp. 24. RR-7981  COULAUD Olivier committed Sep 12, 2014 61 62 63  * *
*  PIACIBELLO Cyrille committed Nov 29, 2013 64 65 66 67  */


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ID 683463
Date 10/04/2021
Public

## 4.10. Simulating Platform Designer Systems

You can simulate your Platform Designer system in a supported third-party simulator to verify and debug system operation. When you Generate HDL for your system or IP component, Platform Designer can also optionally generate the corresponding simulation models, along with scripts to set up your supported simulator.

To generate the simulation model and simulator setup scripts for your Platform Designer system or component, follow these steps:

1. In Platform Designer, click Generate > Generate HDL. The Generation dialog box appears.
2. Under Simulation, specify Verilog or VHDL for the Create simulation model option.
3. Turn on or off the ModelSim, VCS-MX, VCS, Riviera-Pro, or Xcelium option to generate simulator setup scripts for the simulation tool. If you turn on no simulator options, the scripts generate for all simulators.
4. Click the Generate button. Platform Designer generates the simulation models and setup scripts for your system or IP component in the following directory:
<top-level system name>/<system name>/<sim>/<simulator>
Figure 54. Simulation Options in Generation Dialog Box
Figure 55. Generated Simulation Files Location

By default, Platform Designer generates the simulation scripts for the currently loaded system and all subsystems. Alternatively, you can open a subsystem to generate a simulation script only for that subsystem.

You can use scripts to compile the required device libraries and system design files in the correct order and elaborate or load the top-level system for simulation.

Table 20.  Simulation Script VariablesThe simulation scripts provide variables that allow flexibility in your simulation environment.
Variable Description
TOP_LEVEL_NAME If the testbench Platform Designer system is not the top‑level instance in your simulation environment because you instantiate the Platform Designer testbench within your own top-level simulation file, set the TOP_LEVEL_NAME variable to the top-level hierarchy name.
QSYS_SIMDIR If the simulation files generated by Platform Designer are not in the simulation working directory, use the QSYS_SIMDIR variable to specify the directory location of the Platform Designer simulation files.
QUARTUS_INSTALL_DIR Points to the Quartus installation directory that contains the device family library.

### Top-Level Simulation HDL File for a Testbench System

The example below shows the pattern_generator_tb generated for a Platform Designer system called pattern_generator. The top.sv file defines the top-level module that instantiates the pattern_generator_tb simulation model, as well as a custom SystemVerilog test program with BFM transactions, called test_program.

module top();
pattern_generator_tb tb();
test_program pgm();
endmodule


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# Problem inlining methods in C++

This topic is 4945 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Interesting question (at least to me :) ). I like to keep the classes in my C++ project somewhat like Java (with just one class per file), and a header file to that class: wheel.hpp
#ifndef _WHEEL_HPP_
#define _WHEEL_HPP_

class C_Wheel {
private:
bool isEnabled;
public:
bool getIsEnabled();
};

#endif

wheel.cpp
#include "wheel.hpp"

// ctors and dtor
bool C_Wheel::getIsEnabled() {
return isEnabled;
}

This raises two questions: 1- Is this a good pratice? What is the pros and cons of such approach? 2- As I don't want to declare nothing in the headers (methods are inlined by default), how do I inline methods declared in the cpp file? Is it possible? I've read that YES (http://www.cs.utexas.edu/users/downing/cs378/Elements/Methods.html), using "inline boll C_MyClass::getIsEnabled()" in the cpp, but Visual C++ 6 drops a "error LNK2001: unresolved external symbol 'public: void __thiscall C_Wheel::roll(float)'". Thanks!!

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Thats how most commercial games i've seen done it. Look at the Half Life 2 SDK.( Half Life 1 works as well)

They sometimes implement the classes in the header but rarely.

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VC6 is not an ISO compliant C++ compiler. You will have to declare all your inline methods in the header. You can 'cheat' a little by seperating them into different files, though (common practise):
// header:#ifndef _WHEEL_HPP_#define _WHEEL_HPP_class C_Wheel {private:	bool isEnabled;public:	bool getIsEnabled() const;	};#include "wheel.inl"#endif

// wheel.inl:// ctors and dtorinline bool C_Wheel::getIsEnabled() const {    return isEnabled;}

Sidenote: 'getIsEnabled()' is not a good method name. 'IsEnabled()' is sufficient.

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Side note: Rampant inlining of everything is not necessarily a good thing.

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Quote:
 Original post by SumDudeThats how most commercial games i've seen done it. Look at the Half Life 2 SDK.( Half Life 1 works as well) They sometimes implement the classes in the header but rarely.

I'm in the right way :)

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Quote:
 Original post by MiserableSide note: Rampant inlining of everything is not necessarily a good thing.

Yet it is not a Bad Thing™ esp. for such accessor methods.
'inline' is just a hint to the compiler, anyway...

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Quote:
 Original post by darookieVC6 is not an ISO compliant C++ compiler. You will have to declare all your inline methods in the header. You can 'cheat' a little by seperating them into different files, though (common practise):

Very interesting! It's a so used tech, yet not supported by MSVC++...
Ah, your cheat is very creative although the code itself is not so beatiful this way [wink]

Quote:
 Sidenote: 'getIsEnabled()' is not a good method name. 'IsEnabled()' is sufficient.

Agreed! Certainly a bad example :)

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The Simulink® model `watertank` includes the nonlinear Water-Tank System plant and a PI controller in a single-loop feedback system.

The Water-Tank System is shown in the following figure.

Water enters the tank from the top at a rate proportional to the voltage, V, applied to the pump. The water leaves through an opening in the tank base at a rate that is proportional to the square root of the water height, H, in the tank. The presence of the square root in the water flow rate results in a nonlinear plant.

The following table describes the variables, parameters, differential equations, states, inputs, and outputs of the Water-Tank System.

 Variables H is the height of water in the tank.Vol is the volume of water in the tank.V is the voltage applied to the pump. Parameters A is the cross-sectional area of the tank.b is a constant related to the flow rate into the tank.a is a constant related to the flow rate out of the tank. Differential equation `$\frac{d}{dt}Vol=A\frac{dH}{dt}=bV-a\sqrt{H}$` States H Inputs V Outputs H


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MFEM  v4.1.0 Finite element discretization library
mfem::L2ProjectionGridTransfer Class Reference

Transfer data between a coarse mesh and an embedded refined mesh using L2 projection. More...

#include <fespace.hpp>

Inheritance diagram for mfem::L2ProjectionGridTransfer:
[legend]
Collaboration diagram for mfem::L2ProjectionGridTransfer:
[legend]

## Classes

class  L2Projection

class  L2Prolongation

## Public Member Functions

L2ProjectionGridTransfer (FiniteElementSpace &coarse_fes, FiniteElementSpace &fine_fes)

virtual const OperatorForwardOperator ()
Return an Operator that transfers GridFunctions from the domain FE space to GridFunctions in the range FE space. More...

virtual const OperatorBackwardOperator ()
Return an Operator that transfers GridFunctions from the range FE space back to GridFunctions in the domain FE space. More...

Public Member Functions inherited from mfem::GridTransfer
GridTransfer (FiniteElementSpace &dom_fes_, FiniteElementSpace &ran_fes_)

virtual ~GridTransfer ()
Virtual destructor. More...

void SetOperatorType (Operator::Type type)
Set the desired Operator::Type for the construction of all operators defined by the underlying transfer algorithm. More...

virtual const OperatorTrueForwardOperator ()
Return an Operator that transfers true-dof Vectors from the domain FE space to true-dof Vectors in the range FE space. More...

virtual const OperatorTrueBackwardOperator ()
Return an Operator that transfers true-dof Vectors from the range FE space back to true-dof Vectors in the domain FE space. More...

## Protected Attributes

L2ProjectionF
Forward, coarse-to-fine, operator. More...

L2ProlongationB
Backward, fine-to-coarse, operator. More...

Protected Attributes inherited from mfem::GridTransfer
FiniteElementSpacedom_fes
Domain FE space. More...

FiniteElementSpaceran_fes
Range FE space. More...

Operator::Type oper_type
Desired Operator::Type for the construction of all operators defined by the underlying transfer algorithm. It can be ignored by derived classes. More...

OperatorHandle fw_t_oper
Forward true-dof operator. More...

OperatorHandle bw_t_oper
Backward true-dof operator. More...

bool parallel

Protected Member Functions inherited from mfem::GridTransfer
bool Parallel () const

const OperatorMakeTrueOperator (FiniteElementSpace &fes_in, FiniteElementSpace &fes_out, const Operator &oper, OperatorHandle &t_oper)

## Detailed Description

Transfer data between a coarse mesh and an embedded refined mesh using L2 projection.

The forward, coarse-to-fine, transfer uses L2 projection. The backward, fine-to-coarse, transfer is defined locally (on a coarse element) as B = (F^t M_f F)^{-1} F^t M_f, where F is the forward transfer matrix, and M_f is the mass matrix on the union of all fine elements comprising the coarse element. Note that the backward transfer operator, B, is a left inverse of the forward transfer operator, F, i.e. B F = I. Both F and B are defined in physical space and, generally, vary between different mesh elements.

This class currently only fully supports L2 finite element spaces and fine meshes that are a uniform refinement of the coarse mesh. Generally, the coarse and fine FE spaces can have different orders, however, in order for the backward operator to be well-defined, the number of the fine dofs (in a coarse element) should not be smaller than the number of coarse dofs.

If used on H1 finite element spaces, the transfer will be performed locally, and the value of shared (interface) degrees of freedom will be determined by the value of the last transfer to be performed (according to the element numbering in the finite element space). As a consequence, the mass conservation properties for this operator from the L2 case do not carry over to H1 spaces.

Definition at line 859 of file fespace.hpp.

## Constructor & Destructor Documentation

 mfem::L2ProjectionGridTransfer::L2ProjectionGridTransfer ( FiniteElementSpace & coarse_fes, FiniteElementSpace & fine_fes )
inline

Definition at line 908 of file fespace.hpp.

## Member Function Documentation

 const Operator & mfem::L2ProjectionGridTransfer::BackwardOperator ( )
virtual

Return an Operator that transfers GridFunctions from the range FE space back to GridFunctions in the domain FE space.

Implements mfem::GridTransfer.

Definition at line 2745 of file fespace.cpp.

 const Operator & mfem::L2ProjectionGridTransfer::ForwardOperator ( )
virtual

Return an Operator that transfers GridFunctions from the domain FE space to GridFunctions in the range FE space.

Implements mfem::GridTransfer.

Definition at line 2739 of file fespace.cpp.

## Member Data Documentation

 L2Prolongation* mfem::L2ProjectionGridTransfer::B
protected

Backward, fine-to-coarse, operator.

Definition at line 905 of file fespace.hpp.

 L2Projection* mfem::L2ProjectionGridTransfer::F
protected

Forward, coarse-to-fine, operator.

Definition at line 904 of file fespace.hpp.

The documentation for this class was generated from the following files:


text:
US Postal Service Money Order: This is an eleven digit number using digits 1,2,...9 where the sum of the first ten digits is congruent to the eleventh digit modulo 9. That is $a_1a_2\cdots a_{11}$ where $a_1+ \cdots +a_{10} \equiv a_{11}$ mod $9$.
That is $a_1\cdots a_na_{n+1}$ where $a_1a_2\cdots a_n \equiv a_{n+1}$ mod 7.


text:
Talks
Fall 2014

# Inequalities and Reductions for Nonzero Kronecker Coefficients

Tuesday, September 16th, 2014 1:20 pm2:10 pm

The Kronecker coefficients $g_{\alpha\beta\gamma}$ and the Littlewood-Richardson coefficients $c_{\alpha\beta}^\gamma$ are nonnegative integers depending on three partitions $\alpha$, $\beta$, and $\gamma$. By definition, $g_{\alpha\beta\gamma}$ (resp. $c_{\alpha\beta}^\gamma$) are the multiplicities of the tensor product decomposition of two irreducible representations of symmetric groups (resp. linear groups). By a classical Littlewood-Murnaghan's result the Kronecker coefficients extend the Littlewood-Richardson ones.
The nonvanishing of the Littlewood-Richardson coefficient $c_{\alpha\beta}^\gamma$ implies that $(\alpha, \beta, \gamma)$ satisfies some linear inequalities called Horn inequalities. We extend some Horn inequalities to the triples of partitions corresponding to a nonzero Kronecker coefficient.


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Bcftools equivalent of vcf-merge R?
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Entering edit mode
5.0 years ago
CuriousGuy ▴ 70

Hi everyone,

I need to merge two VCF files which contain different variants, something I can do with either vcf-merge (VCFtools) or BCFtools merge. By default, both assume that information is missing in positions listed in one file, but not in the other, which is represented as a dot (.) in the merger of both. However, vcf-merge has an option (-R) to use REF allele (0/0) instead of the default missing genotype. Unfortunately, I have not been able to find the same in bcftools, which is a shame because bcftools is much faster than VCFtools. And I am handling some pretty big VCF files...

That being said, is there any fast and easy way to do the same using bcftools? I put emphasis on the speed because this is precisely the reason why I am interested in bcftools in the first place.

Thanks for your help, it's much appreciated.

vcf bcftools variation genotype • 2.8k views
3
Entering edit mode
5.0 years ago
CuriousGuy ▴ 70

I posted the same message on the official development repository and, shortly afterwards, Petr Danecek kindly added an experimental feature to BCFtools that achieves that same thing. I leave the link here for those who might be in the same situation as me:

OK, I added the option -0, --missing-to-ref, it is available from http://pd3.github.io/bcftools/

https://github.com/samtools/bcftools/issues/402

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Entering edit mode

nice move, great to know

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Entering edit mode
5.0 years ago

sure a similar option to the vcf-merge -R embedded in bcftools merge would be much faster, but you may go for a pipe to bcftools plugin missing2ref, which will change missing genotypes for homozygous reference alleles as stated here.

you will need to have the plugins perfectly installed, otherwise you will have to go for a pipe to something similar to sed 's/\t\.\/\./\t0\/0/g'

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Entering edit mode

Thanks for the suggestion. However, I need to change only those missing genotypes that are added during the merge because that position is not present in one of the two VCF files. Both the pipe and the plugin replace all missing genotypes, even when they were genuinely missing in the original files and I want to keep them that way. It may come in handy in other situations, though, so thanks.

I have posted the solution I received from the developer, for those interested.