text:
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Understanding tie strength in social networks using a local “bow tie” framework

## Abstract

Understanding factors associated with tie strength in social networks is essential in a wide variety of settings. With the internet and cellular phones providing additional avenues of communication, measuring and inferring tie strength has become much more complex. We introduce the social bow tie framework, which consists of a focal tie and all actors connected to either or both of the two focal nodes on either side of the focal tie. We also define several intuitive and interpretable metrics that quantify properties of the bow tie which enable us to investigate associations between the strength of the “central” tie and properties of the bow tie. We combine the bow tie framework with machine learning to investigate what aspects of the bow tie are most predictive of tie strength in two very different types of social networks, a collection of medium-sized social networks from 75 rural villages in India and a nationwide call network of European mobile phone users. Our results show that tie strength depends not only on the properties of shared friends, but also on non-shared friends, those observable to only one person in the tie, hence introducing a fundamental asymmetry to social interaction.

## Introduction

The strength of any kind of relationship between two individuals lies on a spectrum. People in general have a close relationship with only a few friends or family members, a somewhat weaker tie with a larger group of individuals with whom they interact less frequently, and an even weaker connection with a large number of casual acquaintances. This tradeoff between tie strength and the number of people a person is connected to through his or her ties was elegantly captured by Dunbar1. Measuring and predicting tie strength, and moreover, understanding the factors that drive tie strength, has been an expanding area of interest, with increasing utility and complexity in the digital age, i.e., the ever-increasing forms of communication via mobile phones and social media. Knowledge of the strength of a tie, as well as the social dynamics contributing to tie strength, has been shown to increase the accuracy of link prediction, enhance the modeling of the spread of disease and information, and lead to more targeted marketing2,3,4.

Several indicators of tie strength have been proposed, perhaps most notably by Mark Granovetter in his seminal work The Strength of Weak Ties5. Granovetter differentiated between strong and weak ties and proposed the weak ties hypothesis: the stronger the tie between any two people, the higher the fraction of friends they have in common5. Much of the current methodology centered on tie strength has stemmed from Granovetter’s weak ties hypothesis and his proposed four dimensions of tie strength: the amount of time spent interacting with someone, the level of intimacy, the level of emotional intensity, and the level of reciprocity. More recently, three additional dimensions of tie strength have been proposed: (1) emotional support6,7, (2) structural variables, i.e. network topology8,9,10, and (3) social distance, i.e. the difference in socioeconomic status, education level, political affiliation, race, and gender9,11. These categories have facilitated the definition and quantification of numerous possible predictors of tie strength; some generalizable to any network, and some specific to a limited number of social networks.

Another hypothesis of importance to this analysis is a corresponding perspective outlined by Elizabeth Bott12 that suggests that the tie strength between husband and wife varies inversely with the number of non-overlapping ties. That is, overlapping (common) friends support the tie strength between husband and wife, and non-overlapping friends, i.e. friends in each spouse’s separate social circle, detract from it. Several studies have tested Bott’s hypothesis with mixed findings. The studies that did not find evidence to support the hypothesis suffer from non-representative samples, a lack of statistical analysis, and confounding from age, social class and gender7,13,14,15.

Initially, highly generalizable similarity indices such as the number of common neighbors two nodes share, preferential attachment, and path distance were used to infer tie strength. These metrics were most commonly used for link prediction and were shown to provide some information regarding tie strength3,16. However, it was quickly discovered that the addition of nodal attributes and other metrics not solely based on network topology greatly enhanced the measurement and prediction of tie strength17,18. Gilbert and Karahalios defined indicators of tie strength specific to a network of Facebook users and built a predictive model that achieved 85% accuracy for binary tie strength (weak vs. strong) classification19. They found that the act of communicating once leads to a significant increase in tie strength, and that educational difference plays a role in determining tie strength. Pappalardo et al. introduced a measure of tie strength using multiple online social networks and found that the strength of a tie is related to the number of interactions between the two individuals16. In addition, several studies have shown that frequent communication, both online and offline, is positively related to tie strength6,20.

While previous studies have provided advances and valuable insights, they suffer from a binary definition of tie strength (weak vs strong), low diversity in the types of social networks studied (the vast majority being social media sites), and non-representative samples. In this work, we propose a decomposition of a social network into an ensemble of interconnected “social bow ties,” constellations consisting of nodes and ties that surround each network tie. We call any such subgraph a “social bow tie” because the topological structure that surrounds each tie resembles a bow tie. We also introduce several simple metrics that quantify properties of the bow tie. Further, we use random forests and linear regression to build models that predict categorical and continuous measures of tie strength from different properties of the bow tie, including nodal attributes (covariates) of the nodes included in the bow tie. We apply our framework to two social networks, a collection of 75 social networks from the villages of Karnataka, India, and a call network of European mobile phone subscribers. We find that the bow tie framework contributes to more accurate predictions of tie strength and provides insights on which metrics are the most informative of tie strength. Specifically, we find that the larger the proportion of shared friends, the stronger the tie, and the more clustered the individual friendship circles (consisting of non-overlapping friends), the weaker the tie. Consequently, these findings provide evidence to support both the weak ties hypothesis and a generalized version of the Bott hypothesis12.

## Methods

### Data Description

We analyzed two social network data sets. The first data set is social network data collected in 2006 from 75 villages located in 5 districts in rural southern Karnataka, India. The data were collected through household and individual surveys as part of a study by Banerjee et al.21. Of relevance for this study, the survey included social network data along 12 dimensions: friends or relatives who visit the respondent’s home, friends or relatives the respondent visits, any kin in the village, non-relatives with whom the respondent socializes, those from whom who the respondent receives medical advice, with whom who the respondent goes to temple to pray, from whom the respondent would borrow money, to whom the respondent would lend money, from whom the respondent would borrow material goods from, to whom the respondent would lend material goods, from whom the respondent gets advice, and to whom the respondent gives advice. It is worth noting that these forms of interaction are largely face-to-face, unlike the mediated material from the call detail records (CDRs) described below. Additionally, a proportion of villagers were given individual surveys that recorded age and sex, among other attributes.

For this data set, we define the strength of a tie as the number of distinct types of social relationships reported to exist between the two individuals. For example, if individual i borrows money from individual j and in addition gives advice to individual j, the weight of the (undirected) tie between i and j would be equal to 2. If i and j also attend temple together, their tie strength would be 3 and so on, with a minimum strength of 1 and a maximum strength of 12 for any tie. Note that a tie strength of 0 implies that the two individuals are not connected by any kind of social tie. We denote the strength of a tie between individuals i and j as w ij . Because we ignore the directionality of ties, our definition of tie strength is symmetric.

The second data set consists of call detail records (CDRs) from a mobile phone provider in an undisclosed European country where 68% of citizens own a smartphone and 85% own a cellular phone. The data examined here span a period of three months in 2013, and each record consists of the following daily aggregate communication summaries for pairs of individuals: the date, anonymized caller ID, anonymized callee ID, daily call duration (in minutes), daily number of calls, daily number of text messages (SMS), and daily number of multimedia messages (MMS). Age, sex, and billing zip codes were available for a large majority of individuals.

An undirected, weighted call network was created from the records by first summing the call durations between any two individuals over the three-month period. If two individuals spoke on the phone at least once during the period, we connected them with an edge of strength w ij , where the value of edge strength was set to the total amount of time spent on the phone with one another. Since tie strength is defined in terms of absolute time, it does not take into account the total amount of time each individual spends on the phone, which makes it somewhat difficult to quantify the relative strength of ties since the strength of a tie is not measured on the same scale either for individuals or pairs of individuals. We therefore normalized tie strength and represent it with two measurements: one that represents tie strength from the perspective of individual i, and one that represents tie strength from the perspective of individual j. Specifically, for each tie, the first measurement of tie strength is the total call duration (w ij ) divided by the total time individual i spends on the phone s i , the strength of node i. similarly, the second measurement of tie strength is the total call duration divided by the total time individual j spends on the phone s j , the strength of node j. Dividing total call duration by the strength of each focal node results in a consistent definition of tie strength. We denote these new tie strength measurements as y ij and y ji . We created another summary measure of tie strength by taking the average of y ij and y ji , and we denote this z ij  = (y ij  + y ji )/2.

### Bow Tie Framework

To introduce the “bow tie” structure, consider a weighted social network G, which may be directed or undirected, and consider a tie with weight w ij that connects two individuals i and j. We call these two individuals the focal nodes of the bow tie. We use the term focal tie to refer to the tie that links them. We start by partitioning i’s friends and j’s friends into three disjoint sets. Group i, denoted g i , contains the nodes that are connected to only i; group j, denoted g j , contains nodes that are connected to only j; and group ij, denoted g ij , contains nodes that are connected to both i and j. These three groups jointly make up the shared and non-shared friends of i and j. We call this structure the ij bow tie. Formally, the groups g i , g j and g ij are induced subgraphs, where the node sets that induce them are the neighbors of i, the neighbors of j, and the common neighbors of i and j, respectively. The bow tie ij, denoted by G ij , is the subgraph that is induced by the union of all neighbors of i and j. Note that G ij is more than the sum of g i , g j and g ij : in addition to containing the same set of nodes and ties as those subgraphs do, it also contains the inter-group ties among this set of nodes, i.e., the ties linking nodes across g i , g j and g ij . Important to our analysis below is the hierarchical structure of the bow tie: at the upper level of hierarchy we have the bow tie G ij ; at the intermediate level, we have the three groups, g i , g j and g ij ; and at the lowest level we have the nodes and ties from which each group is composed. A simple example of the bow tie structure surrounding nodes i and j is shown in Fig. 1. While we were inspired by the well-known WWW topology bow tie structure presented by Broder et al.22, the framework introduced here is quite different. Broder et al. view the internet at a global, macroscopic level, while the social bow tie is a local, microscopic structure.

The localized nature of the bow tie framework gives rise to several topological metrics that can be used to predict tie strength and find evidence for or against both the weak ties hypothesis and the Bott hypothesis. We include unweighted23 and weighted24 edge overlap, which we denote o ij and $${\tilde{o}}_{ij}$$, respectively. Unweighted overlap is defined as in (1), and weighted overlap as in (2).

$${o}_{ij}=\frac{{n}_{ij}}{{k}_{i}+{k}_{j}-2-{n}_{ij}}$$
(1)
$${\tilde{o}}_{ij}=\frac{\sum _{k=1}^{{n}_{ij}}({w}_{ik}+{w}_{jk})}{{s}_{i}+{s}_{j}-2{w}_{ij}}$$
(2)

Here, n ij is the number of common (shared) friends of nodes i and j, k i (k j ) denotes the degree, or number of connections, node i (j) has, w ij denotes the weight associated with the tie between nodes i and j, and s i (s j ) denotes the strength of node i (j). In accordance with the weak ties hypothesis, we expect both o ij and $${\tilde{o}}_{ij}$$ to be positively associated with tie strength, i.e., that tie strength w ij , increases as the number of shared friends increases. Metrics based on customized versions of the clustering coefficients of i and j are used, where the calculation of a clustering coefficient is limited to the non-shared friends of each node, i.e., for node i, the nodes and edges in g i are used to calculate the clustering coefficient of i, and similarly, g j is used for node j. We denote the sum and absolute difference of these quantities as $$c{c}_{ij}^{S}$$ and $$c{c}_{ij}^{D}$$ for the unweighted clustering coefficients, and $$\tilde{c}{c}_{ij}^{S}$$ and $$\tilde{c}{c}_{ij}^{D}$$ for the weighted clustering coefficients. Here, we use the definition of weighted clustering coefficient provided by Saramäki et al.25. Specifically, the weights of ties are considered and the metric reflects how large triangle weights are compared to a network maximum. Other predictors include the sum and absolute difference in the degrees of i and j ($${k}_{ij}^{S}$$ and $${k}_{ij}^{D}$$), the sum and absolute difference in the strengths of i and j ($${s}_{ij}^{S}$$ and $${s}_{ij}^{D}$$), the number of nodes and edges in g ij (n ij and e ij ), and the sum and absolute difference in the number of nodes and the number of edges in g i and g j ($${n}_{ij}^{S}$$, $${n}_{ij}^{D}$$, $${e}_{ij}^{S}$$ and $${e}_{ij}^{D}$$). With these definitions, we can represent a generalized version, i.e. one that applies to all ties in the network, of Bott’s hypothesis in two different ways; using $${s}_{ij}^{S}$$ and $$c{c}_{ij}^{S}$$. Bott suggests that the more close-knit the non-overlapping social circles of two connected individuals, the weaker the tie between them. Translating this to our setting, we expect tie strength to be negatively associated with $${s}_{ij}^{S}$$ and $$c{c}_{ij}^{S}$$. Specifically, as the clustering and strength of ties among individuals in g i and g j increases, tie strength (w ij ) decreases. Finally, predictors created from the attributes of i and j include the sum and absolute difference in the ages of i and j ($${a}_{ij}^{S}$$ and $${a}_{ij}^{D}$$), the paired sex category (male-male, female-female, female-male) denoted I MM , I FF and I FM respectively, and an indicator if i and j have the same billing zip code, denoted Z ij . See Table 1 for a detailed description of each variable.

To predict tie strength and study how it is associated with different metrics, we used regression as well as Random Forest (RF) regression and classification26. For the India social network, tie strength is discrete with w ij  {1, …, 12}. Thus, the weight of a tie can be viewed as a categorical outcome, allowing RF classification and Poisson regression to be used to predict tie strength, or as continuous with RF regression used for prediction. For the CDR call network, tie strength is most naturally treated as a continuous variable, and we used RF regression and linear regression to predict both measures of tie strength.

In addition to ordinary least squares (OLS) regression, least absolute shrinkage and selection operator (LASSO) and ridge regression were used to fit more parsimonious and interpretable models as well as increase prediction accuracy. Before using LASSO and ridge regression, all data was centered around the mean and 10-fold cross validation was performed to select the best tuning parameters; denoted λL for LASSO and λR for ridge regression. For RF classification, the number of trees used was 200, and the maximum number of features (covariates) considered when splitting a node was $$\sqrt{n}$$ where n is the total number of features. For RF regression, 200 trees were used and the maximum number of features considered when splitting a node was n.

### Data availability

The India social network data analyzed during the current study are available in the Harvard Dataverse repository, https://dataverse.harvard.edu/dataset.xhtml?persistentId=hdl:1902.1/21538. CDR data that support the findings of this study are available from Telenor, but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available.

## Results

### India Social Network

The India network contained 69,444 nodes, of which 16,984 (24.5%) had full attribute information available, and 294,778 edges after the removal of isolated ties. Of these, 37,714 (12.8%) edges were between two individuals with complete attribute information available. The amount of nodal attribute missingness in the India network was high, 75.5%, and we therefore determined that imputation might significantly impact the results, and decided not to impute nodal attributes for this data set. This is because imputation methods for network (correlated) data are not yet fully developed. Consequently, we only included node pairs that had no missing attributes as focal ties. However, all nodes and ties contained in the bow tie structure surrounding each focal tie were used in the calculations and analysis. This was possible since attribute information is only needed for the focal nodes, and not the nodes in the surrounding bow tie structure. Thus, the network topology was not disturbed. We discovered tie strength had a bimodal distribution with ≈46% of ties having a maximum strength of 12. This was due to the fact that the majority (96%) of ties between individuals living in the same household had a weight of 12. We decided to exclude ties between individuals from the same household and only included cross-household ties as focal ties. This resulted in a Poisson distribution of tie strength and a total of 21,945 ties. Similar to the reasoning above, including only cross-household focal ties does not disrupt the topology of the network, but rather the generalizability of the results. Excluding within-household ties as focal ties implies our results cannot be applied to within-household ties. However, in this data set, 96% of within-household ties have a tie strength of 12 and were therefore deterministic. Additionally, according to Banerjee et al.21, nodal attributes were collected from all individuals in a household from a random sample of households in each village, and are assumed to be representative of the population.

RF regression and classification were used to fit three models both before and after nodal attribute imputation, where ties with complete attribute information available were included in the analysis before imputation and all ties were included after imputation. Model 1 is the full model and includes all covariates described in Table 1 with the exception of Z ij since it is specific to the CDR data set; Model 2 includes all covariates except weighted overlap; and Model 3 includes all covariates except unweighted overlap. It has been shown that categorical predictors do not need to be split into multiple dichotomous covariates (referred to as dummy variables) when implementing RF if there are a small number of them and their cardinality is low26,27. Therefore, the variable Sex was not split into two separate dummy variables due to its low cardinality and it being the single categorical predictor. Accuracy was measured as the residual, the absolute difference between empirical tie strength (w ij ) and predicted tie strength ($${\tilde{o}}_{ij}$$). Figure 2 shows the accuracy of RF regression and classification for all models. Note that only two lines are visible, one for RF regression and one for RF classification since the accuracy of all models is indistinguishable. Within one unit of tie strength, an accuracy of 36.4% and 55.3% was achieved by RF regression and classification, respectively.

Feature importance for each of the three models for both RF regression and classification is shown in Fig. 2. The horizontal bars represent how informative the predictor is with a longer bar meaning more informative. The black vertical line represents the value of an equilibrium or null importance if every predictor were equally informative. For both classification and regression, weighted overlap ($${\tilde{o}}_{ij}$$) is the most informative variable in models 1 and 3, and the sum of the clustering coefficients ($$c{c}_{ij}^{S}$$) is the most informative in model 2, followed by the sum of the number of friends in the non-overlapping social circles ($${n}_{ij}^{S}$$). These results provide evidence that the proposed indicators of tie strength in the Weak Ties and Bott hypotheses (the overlap of friendship circles and the amount of clustering in the non-overlapping friendship circles) are predictive of tie strength.

Poisson regression was used to model the associations between tie strength and each of the predictors, and the coefficients of significant predictors with magnitude greater than 0.2 are reported in (3). The predictors with the largest magnitudes include $${\tilde{o}}_{ij}$$, $$c{c}_{ij}^{S}$$, and I FM . Weighted overlap is positively associated with tie strength, illustrating the greater the proportion of strength among overlapping friends of the focal nodes, the stronger the tie between the focal nodes, and showing evidence to support Granovetter’s hypothesis. The sum of the clustering coefficients of the focal nodes is positively associated with tie strength, meaning tie strength decreases as the amount of clustering in the non-overlapping friendship circles increases. This provides quantitative evidence of Bott’s hypothesis in a novel population. Finally, the predictor I FM is negatively associated with tie strength, indicating that on average, female-male ties are weaker than male-male ties, which were used a reference group.

$$log({\mathbb{E}}[{w}_{ij}])=1.62+2.41{\tilde{o}}_{ij}-1.38c{c}_{ij}^{S}-0.2{{\rm{I}}}_{{\rm{FM}}}$$
(3)

### CDR Call Network

The CDR call network contained 2,276,495 nodes and 12,345,848 edges. Age was available for 89.25% of the individuals and had a mean of 48.2 (sd = 18.2) years. Of the 89.03% of individuals whose sex was recorded, 52.51% were male. Billing zip code was available for 99.35% of individuals. Overall, only 7.5% of nodal attributes were missing for this data set, and we therefore decided to perform imputation. Individuals in the CDR call network could have any combination of age, sex and billing zip code information missing. We used RF classification to impute sex and RF regression to impute age. Because of the abundance of billing zip code possibilities, rather than imputing billing zip code directly, we created a paired billing zip code dichotomous variable equal to 1 if the two focal nodes had the same billing zip code and 0 if they did not. We then used RF classification to impute paired billing zip code. After imputation, we sampled 500,000 of the 12,345,848 edges to be used as focal ties, excluding isolated ties, to limit computational expense. This resulted in a total of 496,941 ties. We then calculated the bow tie metrics using all of the nodes and ties contained in the bow tie structure surrounding each focal tie. Because the bow tie is a local structure, and none of the metrics used rely on global network topology, the topology of the network was not changed for the computations and subsequent analyses. Additionally, because we took a random sample of all edges in the network, the focal ties and associated bow ties used in the analyses are representative of the network as a whole. Similar to the India data set, three models were fit with RF regression both before and after nodal attribute imputation for each measure of tie strength and are denoted Models 1–3. Figure 3 shows the accuracy for RF regression after imputation for all three models and each measure of tie strength. The difference in accuracy for all models is very minimal and only one curve is visible for each tie strength measure. Within 0.05 units (a 5% difference between empirical and predicted tie strength), an accuracy of 61% was achieved for normalized tie strength, and 56.7% for averaged tie strength. Within 0.1 units, an accuracy of 76.5% was achieved for normalized tie strength and 77.3% for averaged tie strength. Accuracy for all models and both tie strength measurements before and after imputation are shown in Supplementary Figs S1 and S2. Imputation has a smaller impact on accuracy for this data set in all cases.

Feature importance for each of the three models after imputation is shown in Fig. 3. The black vertical line represents the value of importance if every predictor were equally informative. The most informative predictors in each model are $${s}_{ij}^{S}$$, $${s}_{ij}^{D}$$, $${n}_{ij}^{S}$$ and $${k}_{ij}^{S}$$, with $${\tilde{o}}_{ij}$$ and $${a}_{ij}^{S}$$ slightly more informative than the null importance value in models 1 and 3. This suggests focal node strength, degree and number of non-overlapping friends are the aspects of the bow tie most predictive of tie strength in this network. Feature importance plots for all models and all tie strength measures before and after imputation are presented in Supplementary Figs S1 and S2.

For each measure of tie strength, three different models, denoted Models A–C, were fit using linear regression methods following imputation. Model A denotes the full model that was fit using OLS regression. Model B was fit using LASSO and Model C using ridge regression. Because the distributions of normalized and averaged tie strength are highly skewed for this data set, we first log-transformed each measure of tie strength and then centered them around the mean. All predictors were standardized (centered around the mean with unit variance) before fitting models B and C. Implementing LASSO and ridge regression require the selection of tuning parameters that determine the extent of shrinkage administered when calculating coefficient estimates. As the tuning parameter approaches 0, the corresponding coefficient estimates match the OLS estimates. In this extreme, the amount of bias is minimal, if nonexistent, but the amount of variance is comparatively high. As the tuning parameter is increased, the values of the coefficients decrease and approach 0 once the tuning parameter is sufficiently large. In this extreme, bias is increased but variance in the estimates is decreased. The optimal choice for a tuning parameter balances the amount of bias and variance and can be selected via cross-validation. We performed 10-fold cross validation to select values of the tuning parameters λL and λR. The values of the LASSO coefficients as a function of λL and, as a more interpretable measure, the l1 penalty $$\parallel {\hat{\beta }}_{L}\parallel /\parallel \hat{\beta }{\parallel }_{1}$$ which represents the amount of shrinkage, are shown in Supplementary Figs S3 and S4. The values of the ridge regression coefficients as a function of λR and the l2 penalty $$\parallel {\hat{\beta }}_{R}\parallel /\parallel \hat{\beta }{\parallel }_{2}$$ are shown in Supplementary Figs S3 and S4. Significant predictors, their coefficients, adjusted R2 values and the values of the tuning parameters for models B and C are presented in Supplementary Table S1. Equations (46) show the fitted regression equations for normalized tie strength, y ij , for OLS, LASSO and ridge regression respectively. Similarly, (79) show the fitted regression equations for averaged tie strength, z ij , for OLS, LASSO and ridge regression respectively.

$${\mathbb{E}}{({y}_{ij})}_{OLS}=-\,0.35{k}_{ij}^{D}-0.25{s}_{ij}^{S}+0.29c{c}_{ij}^{D}+0.23{Z}_{ij}+0.27{o}_{ij}$$
(4)
$${\mathbb{E}}{({y}_{ij})}_{LASSO}=-\,0.33{k}_{ij}^{D}-0.25{s}_{ij}^{S}+0.23c{c}_{ij}^{D}+0.23{Z}_{ij}+0.21{o}_{ij}$$
(5)
$${\mathbb{E}}{({y}_{ij})}_{RIDGE}=-\,0.35{k}_{ij}^{D}-0.25{s}_{ij}^{S}+0.29c{c}_{ij}^{D}+0.23{Z}_{ij}+0.27{o}_{ij}$$
(6)
$${\mathbb{E}}{({z}_{ij})}_{OLS}=-\,0.35{k}_{ij}^{D}-0.25{s}_{ij}^{S}+0.29c{c}_{ij}^{D}+0.23{Z}_{ij}+0.27{o}_{ij}-0.2{s}_{ij}^{D}$$
(7)
$${\mathbb{E}}{({z}_{ij})}_{LASSO}=-\,0.21{k}_{ij}^{D}-0.39{s}_{ij}^{S}+0.24c{c}_{ij}^{D}+0.23{Z}_{ij}$$
(8)
$${\mathbb{E}}{({z}_{ij})}_{RIDGE}=-\,0.27{k}_{ij}^{D}-0.49{s}_{ij}^{S}+0.36c{c}_{ij}^{D}+0.24{Z}_{ij}+0.28{o}_{ij}+0.31{s}_{ij}^{D}$$
(9)

For normalized tie strength, λR was sufficiently large such that no shrinkage was implemented, and the estimated ridge regression coefficients are equivalent to the OLS estimates. The amount of LASSO shrinkage was approximately 12%, resulting in slightly different coefficient estimates. In all models, o ij , $${k}_{ij}^{D}$$, $${s}_{ij}^{S}$$, $$c{c}_{ij}^{D}$$ and Z ij were significantly associated with tie strength. Edge overlap is positively associated with tie strength in all models, showing that as the proportion of common friends two individuals share increases, so does the strength of the tie between the two individuals, supporting Granovetter’s hypothesis. Tie strength is negatively associated with $${s}_{ij}^{S}$$ which suggests that as the focal nodes expand their social circles and the time spent interacting with friends, the weaker the tie between them; more evidence to support Bott’s hypothesis. The positive association between Z ij and tie strength implies having the same billing zip code increases the strength of a tie and could suggest a geographical impact on tie strength.

Here, $$c{c}_{ij}^{D}$$ is positively associated with tie strength meaning the more dissimilar the non-overlapping clustering coefficients of the focal nodes, the stronger their tie. Lastly, the R2 values for these models are on the lower side (0.112 on average). This could be due to the network being constructed with phone-based communication rather than face-to-face interactions among highly clustered villagers. Furthermore, quantifying tie strength for CDR data is currently still rather ambiguous; the operationalization of using communication as a proxy for tie strength has not yet been validated20. An alternate measure of tie strength may increase the R2 values.

## Discussion

In this work, we introduce the social bow tie; a novel framework we use to perform a comprehensive analysis of the association between network structure and tie strength. Our framework decomposes a social network into a collection of nodes and ties immediately surrounding each network tie. This utilization of local structure produces easily interpretable metrics that quantify social perspectives of tie strength and allows for analyses that are computationally feasible for networks of any size. Through machine learning and regression methods including LASSO and ridge regression, we determine which properties of the bow tie structure are the most predictive of tie strength in two different types of social networks; a contact network of Indian villagers and a nationwide call network of European mobile phone users.

Overall, both data sets provide evidence to support the weak ties hypothesis and the Bott hypothesis. Following Granovetter, we find that the more friends two individuals share, the stronger their tie. Following Bott, the more tightly-knit their individual social circles, the weaker their tie. In addition, we find that the bow tie framework provides metrics that predict tie strength with high accuracy for both networks.

In future work, it would be interesting to apply the bow tie framework to a social network of married couples. In this case the dominant strong tie has properties that are not seen in more casual social ties, namely the individuals constitute a particularly strongly defined social institution that has both emotional (romantic attachment) as well as structural (e.g. common responsibility for children and common ownership of capital investments such as a home) elements that provide it resiliency. This would enable testing of the original version of Bott’s hypothesis, rather than a generalized form as we present here. It would also be interesting to test if the strength of in-person ties behaves similarly for the mobile phone call network.

## References

1. 1.

Dunbar, R. I. M. Neocortex size as a constraint on group size in primates. Journal of Human Evolution 22, 469–493 (1992).

2. 2.

Li, N., Feng, X., Ji, S. & Xu, K. Modeling Relationship Strength for Link Prediction, 62–74 (2013).

3. 3.

Linyuan, L. & Tao, Z. Link prediction in weighted networks: The role of weak ties. EPL Europhysics Letters 89, 18001 (2010).

4. 4.

Sá, H. R. d. & Prudêncio, R. B. C. Supervised link prediction in weighted networks. In paper presented at the 2011 International Joint Conference on Neural Networks, San Jose, CA (New York, NY: IEEE, 2011, October).

5. 5.

Granovetter, M. The strength of weak ties. American Journal of Sociology 78, 1360–1380 (1973).

6. 6.

Marsden, P. V. & Campbell, K. E. Measuring tie strength. Social Forces 63, 482–501 (1984).

7. 7.

Wellman, B. & Wortley, S. Different strokes from different folks: Community ties and social support. American Journal of Sociology 96, 558–588 (1990).

8. 8.

Ellison, N. B., Steinfield, C. & Lampe, C. The benefits of facebook ‘friends’: Social capital and college students’ use of online social network sites. Journal of Computer-Mediated Communication 12, 1143–1168 (2007).

9. 9.

Lin, N., Vaughn, J. & Ensel, W. Social resources and occupational status attainment. Social Forces 59, 1163–1181 (1981).

10. 10.

Xiang, R., Neville, J. & Rogati, M. Predicting tie strength in a new medium. In CSCW ‘12 Proceedings of the ACM 2012 conference on Computer Supported Cooperative Work, Seattle, WA (New York, NY, ACM, 2012, February).

11. 11.

He, J., Chu, W. W. & Liu, Z. Inferring privacy information from social networks. In Intelligence and Security Informatics, Lecture Notes in Computer Science, vol. 3975, 154–165 (Springer, Berlin, Heidelberg, 2006).

12. 12.

Bott, E. Family and Social Network: Roles, Norms and External Relationships in Ordinary Urban Families (Abingdon: Routledge, 1957).

13. 13.

Rogler, L. & Procidano, M. The effect of social networks on marital roles: A test of the bott hypothesis in an intergenerational context. Journal of Marriage and the Family 48, 693–701 (1986).

14. 14.

Udry, J. R. & Hall, M. Marital role segregation and social networks in middle-class middle-aged couples. Journal of Marriage and Family 27, 392–395 (1965).

15. 15.

Schonpflug, R. K. S., Ute & Schulz, J. Perceived decision-making influence in turkish migrant workers and german workers families:the impact of social support. Journal of Cross-Cultural Psychology 21, 261–282 (1990).

16. 16.

Pappalardo, L., Rossetti, G. & Pedreschi, D. How well do we know each other? detecting tie strength in multidimensional social networks. In Proceedings of the 2012 International Conference on Advances in Social Networks Analysis and Mining (ASONAM 2012), ASONAM 12, 1040–1045 (IEEE Computer Society, Washington, DC, USA, 2012).

17. 17.

Kahanda, I. & Neville, J. Using transactional information to predict link strength in online social networks. In Proceedings of the Third International ICWSM Conference, 74–81 (Association for the Advancement of Artificial Intelligence, 2009, June).

18. 18.

Luarn, P. & Chiu, Y.-P. Key variables to predict tie strength on social network sites. Internet Research 25, 218–238 (2015).

19. 19.

Gilbert, E. Predicting tie strength in a new medium. In Proceedings of the ACM 2012 Conference on Computer Supported Cooperative Work, CSCW 12, 1047–1056 (ACM, New York, NY, USA, 2012).

20. 20.

Wiese, J., Min, J.-K., Hong, J. & Zimmerman, J. Assessing Call and SMS Logs as an Indication of Tie Strength (2014). Unpublished manuscript.

21. 21.

Banerjee, A., Chandrasekhar, A., Duflo, E. & Jackson, M. The diffusion of microfinance. Science 341 (2013).

22. 22.

Broder, A. et al. Graph structure in the web. Comput. Netw. 33, 309–320 (2000).

23. 23.

Onnela, J.-P. et al. Structure and tie strengths in mobile communication networks. PNAS 104, 7332–7336 (2007).

24. 24.

Mattie, H. & Onnela, J.-P. Edge Overlap in Weighted and Directed Social Networks (2017). Unpublished manuscript.

25. 25.

Saramaki, J., M., Kivela, J.-P. O., Kaski, K. & Kertesz, J. Generalizations of the clustering coefficient to weighted complex networks. Physical Review E 75, 027105-1–027105-4 (2007).

26. 26.

Breiman, L. Random forests. Machine Learning 45, 5–32 (2001).

27. 27.

Hastie, T., Tibshirani, R. & Friedman, J. The Elements of Statistical Learning (Springer New York Inc., New York, NY, 2001).

## Acknowledgements

The authors thank Telenor for making available anonymized data for this study, and Banerjee et al. for making the India data set publicly available. H.M. was supported by the National Institute of Health, project number 5T32AI007358-28.

## Author information

Authors

### Contributions

J.-P.O. supervised all work; K.E.-M. processed raw mobile phone data. H.M., K.E.-M., R.L. and J.-P.O. performed research; H.M. and J.-P.O. analyzed data; H.M., K.E.-M., R.L. and J.-P.O. wrote the paper. All authors reviewed the manuscript.

### Corresponding author

Correspondence to Heather Mattie.

## Ethics declarations

### Competing Interests

The authors declare no competing interests.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Mattie, H., Engø-Monsen, K., Ling, R. et al. Understanding tie strength in social networks using a local “bow tie” framework. Sci Rep 8, 9349 (2018). https://doi.org/10.1038/s41598-018-27290-8

• Accepted:

• Published:

• ### Association between social asymmetry and depression in older adults: A phone Call Detail Records analysis

• Timothée Aubourg
• , Jacques Demongeot
• , Félix Renard
• , Hervé Provost
•  & Nicolas Vuillerme

Scientific Reports (2019)


text:
# Homework Help: Charles, Gay-Lussac and of course Boyle

1. Oct 4, 2008

### susan__t

I'm having a little bit of an issue trying to find an equation to use when the initial and final temperature, pressure and volume are all variable

I know $$\frac{V1}{T1}$$ =$$\frac{V2}{T2}$$

and $$\frac{P1}{T1}$$ =$$\frac{P2}{T2}$$

and P1V1=P2V2

But I want to combine them so that I can get the V2 when the Pressure and Temperature 1 and 2 both vary.

2. Oct 4, 2008

### susan__t

maybe P1V1/T1 = P2V2/T2 ? Sorry I used to know this but i can't seem to remember

3. Oct 4, 2008

### GCT

If I understood your question accurately ... yes , assuming that the moles are constant.

4. Oct 5, 2008

### Desh627

The combined gas law states that when moles are constant, P1V1T2=P2V2T1 (basically what has been said above, but getting rid of those fractions).

format: plain


text:
# XIII. Theorem of the Addition of Velocities. The Experiment of Fizeau

Relativity: The Special and General TheoryAlbert Einstein

Now in practice we can move clocks and measuring-rods only with velocities that are small compared with the velocity of light; hence we shall hardly be able to compare the results of the previous section directly with the reality. But, on the other hand, these results must strike you as being very singular, and for that reason I shall now draw another conclusion from the theory, one which can easily be derived from the foregoing considerations, and which has been most elegantly confirmed by experiment.

In Section VI we derived the theorem of the addition of velocities in one direction in the form which also results from the hypotheses of classical mechanics—This theorem can also be deduced readily horn the Galilei transformation (Section XI ). In place of the man walking inside the carriage, we introduce a point moving relatively to the co-ordinate system $K'$ in accordance with the equation

$x' = wt'.$

By means of the first and fourth equations of the Galilei transformation we can express $x'$ and $t'$ in terms of $x$ and $t$, and we then obtain

$x = (v + w)t.$

This equation expresses nothing else than the law of motion of the point with reference to the system $K$ (of the man with reference to the embankment). We denote this velocity by the symbol $W$, and we then obtain, as in Section VI,

$W=v+w \tag{A}$

But we can carry out this consideration just as well on the basis of the theory of relativity. In the equation

$x'=wt'$

we must then express $x'$and $t'$ in terms of $x$ and $t$, making use of the first and fourth equations of the Lorentz transformation. Instead of the equation (A) we then obtain the equation

$W = \frac{v+w}{I+\frac{vw}{c^2}} \tag{B}$

which corresponds to the theorem of addition for velocities in one direction according to the theory of relativity. The question now arises as to which of these two theorems is the better in accord with experience. On this point we axe enlightened by a most important experiment which the brilliant physicist Fizeau performed more than half a century ago, and which has been repeated since then by some of the best experimental physicists, so that there can be no doubt about its result. The experiment is concerned with the following question. Light travels in a motionless liquid with a particular velocity $w$. How quickly does it travel in the direction of the arrow in the tube T (see the accompanying diagram, Figure 3 when the liquid above mentioned is flowing through the tube with a velocity $v$?

Fig. 3

In accordance with the principle of relativity we shall certainly have to take for granted that the propagation of light always takes place with the same velocity w with respect to the liquid, whether the latter is in motion with reference to other bodies or not. The velocity of light relative to the liquid and the velocity of the latter relative to the tube are thus known, and we require the velocity of light relative to the tube.

It is clear that we have the problem of Section VI again before us. The tube plays the part of the railway embankment or of the co-ordinate system $K$, the liquid plays the part of the carriage or of the co-ordinate system $K'$, and finally, the light plays the part of the man walking along the carriage, or of the moving point in the present section. If we denote the velocity of the light relative to the tube by $W$, then this is given by the equation (A) or (B), according as the Galilei transformation or the Lorentz transformation corresponds to the facts. Experiment1 decides in favour of equation (B) derived from the theory of relativity, and the agreement is, indeed, very exact. According to recent and most excellent measurements by Zeeman, the influence of the velocity of flow $v$ on the propagation of light is represented by formula (B) to within one per cent.

Nevertheless we must now draw attention to the fact that a theory of this phenomenon was given by H. A. Lorentz long before the statement of the theory of relativity. This theory was of a purely electrodynamical nature, and was obtained by the use of particular hypotheses as to the electromagnetic structure of matter. This circumstance, however, does not in the least diminish the conclusiveness of the experiment as a crucial test in favour of the theory of relativity, for the electrodynamics of Maxwell-Lorentz, on which the original theory was based, in no way opposes the theory of relativity. Rather has the latter been developed trom electrodynamics as an astoundingly simple combination and generalisation of the hypotheses, formerly independent of each other, on which electrodynamics was built.

XIV. The Heuristic Value of the Theory of Relativity

1. Fizeau found $W=w+v\left(I-\frac{I}{n^2}\right)$, where $n=\frac{c}{w}$ is the index of refraction of the liquid. On the other hand, owing to the smallness of $\frac{vw}{c^2}$ as compared with $I$, we can replace (B) in the first place by $W=(w+v)\left(I-\frac{vw}{c^2}\right)$, or to the same order of approximation by $w+v\left(I-\frac{I}{n^2}\right)$, which agrees with Fizeau’s result.

### Subscribe to The Empty Robot

Get the latest posts delivered right to your inbox


text:
# Spherical segment

Calculate the volume of a spherical segment 18 cm high. The diameter of the lower base is 80 cm, the upper base 60 cm.

Correct result:

V =  73739.463 cm3

#### Solution:

$h=18 \ \text{cm} \ \\ D_{1}=80 \ \text{cm} \ \\ D_{2}=60 \ \text{cm} \ \\ \ \\ r_{1}=D_{1}/2=80/2=40 \ \text{cm} \ \\ r_{2}=D_{2}/2=60/2=30 \ \text{cm} \ \\ \ \\ V=\dfrac{ \pi \cdot \ h }{ 6 } \cdot \ (3 \cdot \ r_{1}^2+3 \cdot \ r_{2}^2+h^2)=\dfrac{ 3.1416 \cdot \ 18 }{ 6 } \cdot \ (3 \cdot \ 40^2+3 \cdot \ 30^2+18^2)=73739.463 \ \text{cm}^3$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Be the first to comment!

Tips to related online calculators
Tip: Our volume units converter will help you with the conversion of volume units.

#### You need to know the following knowledge to solve this word math problem:

We encourage you to watch this tutorial video on this math problem:

## Next similar math problems:

• Hemisphere cut
Calculate the volume of the spherical layer that remains from the hemisphere after the 3 cm section is cut. The height of the hemisphere is 10 cm.
• Spherical segment
Spherical segment with height h=5 has a volume V=117. Calculate the radius of the sphere of which is cut this segment.
• Gasholder
The gasholder has spherical shape with a diameter 20 m. How many m3 can hold in?
• Spherical tank
The tank of a water tower is a sphere of radius 35ft. If the tank is filled to one quarter of full, what is the height of the water?
• Hollow sphere
Calculate the weight of a hollow tungsten sphere (density 19.3 g/cm3), if the inner diameter is 14 cm and wall thickness is 3 mm.
• Cuboid height
What is the height of the cuboid if the edges of its base are 15 cm and 4 cm long and its volume is 420 cm cubic?
• Sphere growth
How many times grow volume of sphere if diameter rises 10×?
• Sphere fall
How many percent fall volume of sphere if diameter fall 10×?
• Gasoline tank cylindrical
What is the inner diameter of the tank, which is 8 m long and contains 40 cubic cubic meters of gasoline?
• The cylinder base
The cylinder with a base of 8 dm2 has a volume of 120 liters. From a cylinder fully filled with water, 40 liters of water was removed. At what height from the bottom /with precision to dm/ is the water level?
• Volume of ball
Find the volume of a volleyball that has a radius of 4 1/2 decimeters. Use 22/7 for π
• Sphere A2V
Surface of the sphere is 241 mm2. What is its volume?
• The volume 2
The volume of a cube is 27 cubic meters. Find the height of the cube.
• Cuboid enlargement
By how many percent increases the volume of cuboid if its every dimension increases by 30%?
• Volume increase
How many percent will increase in the pool 50 m, width 15m if the level rises from 1m to 150cm?
• Fit ball
What is the size of the surface of Gymball (FIT - ball) with a diameter of 65 cm?
• Holidays - on pool
Children's tickets to the swimming pool stands x € for an adult is € 2 more expensive. There was m children in the swimming pool and adults three times less. How many euros make treasurer for pool entry?


text:
# A fast routine of making a commutation matrix

R does not have this function by default, so I wrote a faster routine K() to create a commutation matrix.We always use the notation of commutation matrix on the paper. But when we put it into practice, we typically need the result of a dense matrix by pre- or post-multiplied by a commutation matrix. i.e., KX or YK., where K is the commutation matrix and X, Y are dense matrices. This could be a disaster when the dimensions are big in the usual way of multiplication. And I found a very easy and efficient way of doing this.

Please visit the git page for the updated code


text:
# Embed Images in Text Files

I’ve always wanted to take notes with images in Emacs: to actually embed images into the text file and manipulate and view them as easy as in other rich text editors. Org mode can display images but its image links are quite limited: 1) it is really a file path and you have to keep the file path in sync 2) changing the display size is not convenient, and you can’t control the size of individual images.

Since I’m moving from Notes.app to Deft to take notes, I have to have a comparable image support in Emacs. For that I wrote iimg.el. Now, if you think Org Mode’s link image are good enough and is the $$Right Way^{TM}$$ to do it, I hold no objection. My blogs are written in Org Mode and the images are linked. However, for my notes, I really want to pack all the stuff into one single file and not worry about keeping the paths intact.

## Introducing iimg.el

iimg.el provides these features:

• embedding images as base64 text in text files
• easy control of the size of each individual image
• rendering the images

Now I can insert an image by iimg-resize and change its displayed size by typing s on the image, and toggle thumbnail display by typing t (inspired by Notes.app). I can resize a image’s width/height to be n characters, n pixels, or n percent of the window width/height. Drag-and-drop is also supported. Lastly, I can export the embedded images out if I want to.

Here is a demo (demo link) (sorry for the flickering, Emacs has bad image scrolling):

## Implementation details

At first I thought of simply inserting the base64 string, but then there will be this wall of nonsense between meaningful text if you don’t render the image. That doesn’t sound like a good idea: what if someone else needs to view the file and don’t have iimg.el, or I need to view the file on some other places where iimg.el or even Emacs isn’t available? So I split the image into two parts: link and data. Data are the base64 strings and are placed at the end of each file. Links are inline with other normal text, and are rendered as images. This design adds a bit of hair to the implementation but I think it’s worth it.

I store the meta data (size, thumbnail state, etc) as plists in the links. A link looks like this:

({iimg-link (:name "hooks" :size (width char 70) :thumbnail t)})

I can simply use read and get all the information about the image and render it accordingly. And all these settings are persistent because they are directly saved to the file.

Data looks similar:

({iimg-data (:name "hooks" :data "/9j/4AAQSkZJRgAAEaAAUA...")})

I stored the base64 string literally as a string, and let read do the hard work.

## 後日談：Better integration with Deft

Deft stores file contents to cache, and storing the gibberish multi-megabyte image data to cache probably isn’t a good idea—it takes up memory and slows down searching. This is what we do:

First, define a function that prunes image data in a buffer:

(defun iimg-clean-data ()
"Clear any iimg-data in current buffer."
(goto-char (point-min))
(while (re-search-forward iimg--data-regexp nil t)
(delete-region (match-beginning 0) (match-end 0)))))

Second, add a hook to Deft to run our function before saving the file to cache. I forked deft.el and modified deft-cache-newer-file to run a hook before saving buffer content.

(with-temp-buffer
(insert-file-contents file nil nil nil t)
(run-hook-with-args 'deft-cache-file-hook)
(setq contents (buffer-substring-no-properties
(point-min) (point-max))))

The definition of the hook:

(defvar deft-cache-file-hook nil
"Hook run before a file is saved to the cache.
Run in a temp buffer with the file’s content with no argument.")

Finally, put our function in that hook:

(add-hook 'deft-cache-newer-file #'iimg-clean-data)

## 後日談2：Smooth scrolling over images

Because Emacs cannot partially display a line, inline images jumps in and out of the screen, which is super annoying. One solution is to display the image in multiple lines, so each line displays a strip of the image. I added this feature to iimg.el and now you can toggle between single and multi-line display by typing m on the image. Here is a demo scrolling over multi-line images (video link):

My Emacs finally looks like a modern editor, yay!

Here is another local backup.

First Published on 2020-08-13 Thu 16:52


text:
# Why are both bias resistors considered parallel in a common emitter amplifier?

When calculating the input impedance of a voltage divider biased BJT common emitter amplifier, we calculate the parallel resistance of the bias resistors and the emitter resistance. But why do we consider both bias resistors? Looking from the amplifier input (the transistor base), the other resistor leads to the voltage source, while the other leads to ground. Shouldn't we only consider the one that leads to ground? Why would any of the input current go to the voltage source/battery?

• electronics.stackexchange.com/questions/298560/… – G36 Jan 4 at 18:52
• Because in"small signal analysis" or AC analysis, both supplies are considered 0V. So they ARE in parallel. In practice. the big supply decoupling capacitor guarantees that Vs has an AC component of 0V. – Brian Drummond Jan 4 at 21:19

When you determine an input impedance (which is a transfer function), you install a test generator (the stimulus) $$\I_T\$$ across the connecting ports and you determine the response voltage $$\V_T\$$ collected across the current source terminals. This configuration is called a DPI or driving point impedance as described by the fast analytical circuits techniques or FACTs. The circuit below shows the configuration for the bipolar amplifier you described:
When you do that, you imply that the source supplying your circuit is uncorrelated with the stimulus. It means that if you sweep the input impedance with $$\I_T\$$, you won't see any ripple over the $$\V_{cc}\$$ rail supplied by a perfect dc source (0 output impedance). The small-signal amplitude of the $$\V_{cc}\$$ rail is thus 0 V during the analysis and the source can be replaced by a short circuit for the dynamic analysis. You can also imagine an infinite decoupling capacitor from the $$\V_{cc}\$$ rail to ground perfectly decoupling the rail during the analysis. As the drawing shows, when the source is shorted, the biasing resistances naturally come in parallel.


text:
# Dreamgrove

## Guardian Tier 21 Analysis

#### Last updated 12 December 2018.

Let’s talk about the Guardian Tier 21 set bonuses.

First, what are they?

: When Gore is consumed, the cooldown of Barkskin is reduced by 1 second.

: When Barkskin fades, all healing done to you is increased by 10% for 20 seconds.

I want to talk about what each bonus does, how they may impact gameplay and talent choices, and then I’ll give my overall impressions of them.

The TL;DR of this document is that Tier 21 is terrible in the best case, and detrimental to survivability in the worst case. I write this with the goal of providing feedback on the current bonuses, with the hopes that they will be changed for the better.

## 2pc

The definition of “consuming Gore” is as follows: you have a proc, you cast Mangle, and Gore is consumed. This means it’s not enough to have Gore activate, you have to spend the proc to get the benefit. As it turns out this is not terribly important, since Gore has the same chance to proc regardless of target count (unlike , Gore’s chance is defined per cast, not per target hit), but it does slightly reduce the effectiveness since you have a small chance to overwrite existing procs when prioritizing Thrash over Mangle.

The most obvious benefit of this set bonus is that it gives you more flexibility with your usage, which may save you an external or trinket on certain encounters. If you synchronize Barkskin with , which is common if you’re talented into and focusing on DPS, this bonus gives you no benefit.

To discuss the value of the set bonus, let’s express it in terms of damage reduction gains. I’ll make the following assumptions:

1. The damage intake is regular and there are no mechanics you need to save Barkskin for. In other words, the maximum potential DR gain will come from using Barkskin on CD. Note that this is absolutely not a recommended playstyle; you will always want to use Barkskin to mitigate damage spikes. This assumption is purely for the sake of simplifying the math.
2. You are running Galactic Guardian, have 20% haste, and follow the recommended DPS rotation (Thrash > Mangle > GG-empowered Moonfire > Swipe) with no dead GCD time (time in which you are able to cast one of the above, but don’t).
3. You have 44 traits and 44 traits, and you are not using any tier set bonuses other than the T21 2pc.
4. 25% of your GCDs are Mangles (a generous average obtained from simulations run on the rotation), and you use 90% of the Gore procs you gain, accounting for overwritten procs.

First let’s calculate how many Gore procs we can consume per minute.

With 20% haste, the GCD is $$1.5 / (1 + 0.2) = 1.25\text{s}$$ , which gives us 48 GCDs per minute. If 25% of GCDs are mangles, we have 36 chances per minute to proc Gore. With a Gore chance of 20%, we have $$36 * 0.2 = 7.2$$ procs per minute, and consuming 90% of those gives us $$7.2 * 0.9 = 6.48$$ Gores consumed per minute.

Now we can compute the effective cooldown of Barkskin with the set bonus. We can express this relation in the following way:

$$procsPerBarkskin = \frac{reducedCooldown}{secondsPerProc} = \frac{cooldown}{reductionPerProc}$$

Most of the terms are self-explanatory, except for “reductionPerProc” which is the amount of time removed from the remaining cooldown after consuming one Gore proc, including the time elapsed since the last proc.

Consuming 6.48 Gores per minute is an average time between procs of 9.26 seconds. Since each proc reduces the cooldown by 1 second, we can say that the effective time for consuming 1 proc is 10.26 seconds.

Given that the original cooldown of Barkskin is 79.2 seconds, we have a reduced cooldown of $$(79.2 * 10.26) / 9.26 = 71.48$$ seconds. This is a reduction of 9.75%.

Now that we have a reduced cooldown, we can compute the average damage reduction gained.

With the original cooldown of 79.2 seconds, using Barkskin on cooldown will give you an uptime of 17.7%, for an average DR of 3.54%. With T21, the Barkskin cooldown becomes 71.48 seconds for an uptime of 19.6% and an average DR of 3.91%.

We can now say that the value of the T21 set bonus is a potential DR gain of $$1 - ((1 - 0.0354) / (1 - 0.0391)) = 0.4\%$$ . This value will increase marginally with haste, but not in any significant way.

Additionally, if you run , the cooldown on Barkskin can be reduced to 47.9s, for an average DR of 5.84%. Thus, the value of the set bonus with SotF is a DR gain of 0.6%.

There is also a very small DPS benefit to the set bonus if you use Barkskin on cooldown (Baseline is syncing Barkskin with RotS).

## 4pc

This bonus works similarly to the increased healing portion of our mastery, with the added bonus that it also works on and healing. It increases the healing done by our , but it does not double dip by directly increasing the heals generated by Mastery. That is to say, if you have 20% Mastery and you receive a heal of 100, the set bonus will increase the direct heal you receive to 110, and then Mastery will produce another heal for $$110 * 0.2 = 22$$ , but the Mastery heal will not be further increased by the set bonus.

Unlike the 2pc, the 4pc always has value regardless of your talents or rotation. You will gain more value out of the set bonus the more frequently you use Barkskin, due to the increased uptime on the buff.

We can equate this increased healing to damage reduction if we express them both in the form of healing required. In order to demonstrate this, imagine the following scenario:

You have 100 HP, and you take a melee swing that deals 50 damage pre-mitigation. You also have the 4pc set bonus, increasing your healing taken by 10%. Your health deficit after you are hit (the amount of healing before the set bonus buff required to return you to full HP) is $$50 / (1 + 0.1) = 45.45$$ .

Now how much DR would be needed to reach the same health deficit without the set bonus? Since you don’t have any increased healing (ignoring Mastery for the moment), the damage you take from the swing post-mitigation has to be equal to the healing deficit. Therefore, the increased healing taken from the 4pc is equivalent to $$1 - (45.45 / 50) = 9.10\%$$ DR.

Continuing from the math we did for the 2pc set bonus, using Barkskin on cooldown with a duration of 71.48 seconds the uptime on the set bonus will be $$20 / 71.48 = 28.0\%$$ for an average increased healing of 2.80%. This equates to an average DR of 2.72%.

If you use Barkskin every 90s, the uptime on the healing bonus will be 22.2%, qeuivalent to an average DR of 2.17%.

It’s worth noting that increased healing taken is not interchangeable with damage reduction. Increased healing taken cannot save you if you take damage equal to or greater than your HP in a single swing, whereas damage reduction can. In general, straight up damage reduction is always preferred.

## Legendaries

Currently the only legendary that has a direct interaction with T21 is . Without the 2pc set bonus, OPQ will generate 56.8 Rage per minute when using Barkskin on cooldown. With the set bonus, that becomes 62.9 Rage per minute, for a gain of 6.1 Rage per minute, or 0.1 Rage per second. Not terribly significant.

has an indirect interaction in that its heal is increased by the 4pc set bonus.

## Talents

The talents that gain value from the set bonuses are (increased Barkskin frequency -> increased Brambles DPS), and (reduced Barkskin cooldown -> increased uptime on 4pc).

With the 4pc, Survival of the Fittest may overtake for survivability on fights with significant magic damage, but GoE (and occasionally ) will still win out on physical damage fights. Offensively Survival of the Fittest has a minor advantage over the alternatives, however that was already true without T21. Defensive gains will heavily outweigh offensive ones on this row, without question.

## Impressions

Overall I think these set bonuses are highly underwhelming, especially as a final tier set for Legion. Not only do they actively encourage a harmful playstyle, they have next to no synergy with existing playstyles/legendaries/talents and don’t introduce any additional depth into what is already a simple class mechanically.

The main benefit of the 2pc is, as mentioned above, affording you more flexibility with your Barkskin usage. But Survival of the Fittest already does this job far better and also works on Survival Instincts to boot.

With respect to the 4pc, having the healing bonus granted after Barkskin ends (i.e. after the harmful part of most tank mechanics) makes the “optimal” way to play around the set bonus counterproductive. The most common use of Barkskin is as a defensive cooldown for big, infrequent damage spikes on bosses, and as such the duration of the buff is far less important than how much damage it actually mitigates.

In essence, the bonus extends the Barkskin “buff” well beyond its original duration, which incentivizes using it more like active mitigation than like a defensive cooldown (that is, casting it with high frequency and maintaining as high an uptime as possible). But this negates the entire purpose of the defensive cooldown portion in the first place!

If Barkskin is used on cooldown to maintain the set bonus and with little regard for the damage curve of an encounter, the 20% DR component of Barkskin itself will almost certainly be wasted on periods of low/consistent/predictable damage when it’s not needed. Not only this, it can potentially be dangerous to play this way if you discover later that you need Barkskin for an emergency and don’t have it. The vast majority of self-inflicted tank deaths are due to being unprepared for spike damage, so I can’t see any realistic situation in which a survivability-oriented player would forego a 20% DR on a large hit for a minor pseudo-mastery buff.

Numbers wise, the set bonuses seem well below the benchmarks set by T19 and T20.

The significantly improves your self-healing capability by increasing the number of FRs you have access to in an encounter, and gains value on harder content where you take more damage and spend more time at low HP. The T21 2pc grants you on average a 10% cooldown reduction on your Barkskin, which will either give you no value at all or a small increase in flexibility with your defensive timings that is worse than an existing talent.

The currently grants approximately 5% additional physical DR, in addition to creating opportunities to FR for 0 rage. (I say currently because the 4pc was actually far more valuable during Nighthold, due to being a higher % armor per stack and it affecting as well.) Meanwhile the T21 4pc, in the best possible circumstances (77 Perpetual Spring, 77 Ursoc’s Endurance, Survival of the Fittest) is equivalent to 4.4% DR, and in more practical terms is closer to 2% DR.

In conclusion, I’m very disappointed with this set, and I can’t recommend that anyone aim to use them in their current state. If these bonuses are what we end up with until the next expansion, I can’t say I’m looking forward to it. We’re still quite a few weeks out from Antorus release, so here’s hoping that we see some positive change between now and then.


text:
T:A:L:K:S

 close this window
 title: Variational Inequalities and Morozov\'s Discrepancy Principle name: Anzengruber first name: Stephan location/conference: cssip10 PRESENTATION-link: http://www.dfg-spp1324.de/nuhagtools/event/dateien/talks_cssip/anzengruber.pdf abstract: Variational Inequalities can be viewed as generalizations of classical source conditions and structural assumptions. In this talk we will show how, for non-linear ill-posed problems, Variational Inequalities can be combined with Morozov\'s Discrepancy principle to obtain convergence rates for Tikhonov Regularization with $\ell_p$ penalties, $p \geq 1$, which are well-known to promote sparsity.


text:
# zbMATH — the first resource for mathematics

Lattices in locally definable subgroups of $$\langle R^{n},+\rangle$$. (English) Zbl 1345.03072
Summary: Let $$\mathcal{M}$$ be an o-minimal expansion of a real closed field $$R$$. We define the notion of a lattice in a locally definable group and then prove that every connected, definably generated subgroup of $$\langle R^{n},+\rangle$$ contains a definable generic set and therefore admits a lattice.

##### MSC:
 03C64 Model theory of ordered structures; o-minimality 03C68 Other classical first-order model theory 22B99 Locally compact abelian groups (LCA groups)
##### Keywords:
o-minimality; locally definable groups; lattices; generic sets
Full Text:
##### References:
 [1] Baro, E., and M. J. Edmundo, “Corrigendum to ‘Locally definable groups in o-minimal structures,”’ by M. J. Edmundo, Journal of Algebra , vol. 320 (2008), pp. 3079-80. [2] Baro, E., and M. Otero, “Locally definable homotopy,” Annals of Pure and Applied Logic , vol. 161 (2010), pp. 488-503. · Zbl 1225.03043 [3] Edmundo, M. J., “Locally definable groups in o-minimal structures,” Journal of Algebra , vol. 301(2006), pp. 194-223. · Zbl 1104.03032 [4] Eleftheriou, P. E., and Y. Peterzil, “Definable quotients of locally definable groups,” Selecta Mathematica (N.S.) , vol. 18 (2012), pp. 885-903. · Zbl 1273.03130 [5] Eleftheriou, P. E., and Y. Peterzil, “Definable groups as homomorphic images of semilinear and field-definable groups,” Selecta Mathematica (N.S.) , vol. 18 (2012), pp. 905-40. · Zbl 1273.03129 [6] Hrushovski, E., Y. Peterzil, and A. Pillay, “Groups, measures, and the NIP,” Journal of the American Mathematical Society , vol. 21 (2008), pp. 563-96. · Zbl 1134.03024 [7] Peterzil, Y., and S. Starchenko, “Definable homomorphisms of abelian groups in o-minimal structures,” Annals of Pure and Applied Logic , vol. 101 (2000), pp. 1-27. · Zbl 0949.03033
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
[Tugindia] How to remove numbering in boxes.

E. Krishnan ekmath at md5.vsnl.net.in
Sun Apr 13 20:07:44 CEST 2003

On Sun, 13 Apr 2003 indrajit at cal2.vsnl.net.in wrote:

>              When we use \begin{thebibliography} \end{thebibliography}, the
> serial numbers of the references appear in boxes

............................................

> What should I do if I want the serial numbers not to appear in boxes ?

The "thebibibliography" environment is actually a general list, whose
labels are set by the "\@biblabel" command. So to change this you will
have redefine this command. Note that since it contains the @ character,
you will have to enclose it within "\makeatletter...\makeatother"
commands. For example

\makeatletter
\renewcommand{\@biblabel}[1]{#1.}
\makeatother

*before* the "\begin{thebibliography}" will give you serial numbers
followed by periods in your bibliography.

--
Krishnan


text:
Article

# Pattern formation in growing sandpiles

• ##### Samarth Chandra
EPL (Europhysics Letters) (Impact Factor: 2.26). 08/2008; DOI: 10.1209/0295-5075/85/48002
Source: arXiv

ABSTRACT Adding grains at a single site on a flat substrate in the Abelian sandpile models produce beautiful complex patterns. We study in detail the pattern produced by adding grains on a two-dimensional square lattice with directed edges (each site has two arrows directed inward and two outward), starting with a periodic background with half the sites occupied. The size of the pattern formed scales with the number of grains added $N$ as $\sqrt{N}$. We give exact characterization of the asymptotic pattern, in terms of the position and shape of different features of the pattern. Comment: 5 pages, 4 figures, submitted to Phys. Rev. Lett

0 Bookmarks
·
130 Views
• Source
##### Article: A sandpile model for proportionate growth
[Hide abstract]
ABSTRACT: An interesting feature of growth in animals is that different parts of the body grow at approximately the same rate. This property is called proportionate growth. In this paper, we review our recent work on patterns formed by adding $N$ grains at a single site in the abelian sandpile model. These simple models show very intricate patterns, show proportionate growth, and sometimes having a striking resemblance to natural forms. We give several examples of such patterns. We discuss the special cases where the asymptotic pattern can be determined exactly. The effect of noise in the background or in the rules on the patterns is also discussed.
Journal of Statistical Mechanics Theory and Experiment 10/2013; · 1.87 Impact Factor
• Source
##### Article: Abelian networks: foundations and examples
[Hide abstract]
ABSTRACT: In Dhar's model of abelian distributed processors, finite automata occupy the vertices of a graph and communicate via the edges. A local commutativity condition ensures that the final output does not depend on the order in which the automata process their inputs. We consider the halting problem for such networks and the critical group, an invariant that governs the behavior of the network on large inputs. Our main results are 1. A finite abelian network halts on all inputs if and only if all eigenvalues of its production matrix lie in the open unit disk; 2. The critical group of an irreducible abelian network acts freely and transitively on recurrent states of the network; 3. The critical group is a quotient of a free abelian group by a subgroup containing the image of the Laplacian, with equality in the case that the network is rectangular. We also estimate the running time of an abelian network up to a constant additive error.
09/2013;
• Source
##### Article: Proportionate growth in patterns formed in the rotor-router model
[Hide abstract]
ABSTRACT: We study the growing patterns in the rotor-router model formed by adding $N$ walkers at the center of a $L \times L$ two-dimensional square lattice, starting with a periodic background of arrows, and relaxing to a stable configuration. The pattern is made of large number of triangular regions, where in each region all arrows point in the same direction. The square circumscribing the region, where all the arrows have been rotated atleast one full circle, may be considered as made up of smaller squares of different sizes, all of which grow linearly with $N$, for $1 \ll N < 2 L$. We use the Brooks-Smith-Stone-Tutte theorem relating tilings of squares by smaller squares to resistor networks, to determine the exact relative sizes of the different elements of the asymptotic pattern. We also determine the scaling limit of the function describing the variation of number of visits to a site with its position in the pattern. We also present evidence that deviations of the sizes of different small squares from the linear growth for finite $N$ are bounded and quasiperiodic functions of $N$.
12/2013;


text:
# Prove that any two maps are homotopic.

I was trying to show the two following things: 1) any two maps from a topological space $$X$$ to $$I=[0,1]$$ are homotopic, 2) any two maps from $$I=[0,1]$$ to a path connected set $$Y$$ are homotopic.

For the second I thought I could do it like this:

Let $$f:I\rightarrow Y$$ en $$g:I\rightarrow Y$$ continuous maps. Now define a function $$F:I\times I\rightarrow Y$$ such that $$F(x,t)=(1-t)f((1-t) x) + t g(tx)$$. Then we see that $$F(x,0)=f((1-0) x) + 0\cdot g(0\cdot x)=f(x)$$ and $$F(x,1)=(1-1)f((1-1) x)+1\cdot g(1\cdot x)=f(x)$$. Furthermore, $$F$$ is continuous since it's a composition of continuous functions. Now we can conclude that $$f$$ and $$g$$ are homotopic.

However, I don't use the fact that $$Y$$ is path connected, so I think this can't be right. I already found this, however, I don't understand what the $$e_{f(0)}$$ and $$e_{g(0)}$$ are. Furthermore, I found this, but in the answers they don't create the $$F$$-function.

Could someone help me find the right $$F$$-function to show this? And also for the first part?

• The reason the function $F$ you have written down does not work is because you use addition and multiplication. For a space $Y$ and two points $a,b\in Y$, what does $a+b$ mean? Unless $Y$ has a group structure, it means nothing. In particular, the expression you have written down doesn't quite make sense. – Brian Shin May 12 at 14:58
• @BrianShin Thanks, I didn't know that! – user665297 May 13 at 6:33

For the first question it is enough to show that the map $$f$$ is homotopic to the constant map $$g(x)=0$$, define $$H_t(x)=tf(x)$$.
For the second, you can also show that any map from $$I\rightarrow Y$$ is homotopic to $$g(t)=y_0, y_0\in Y$$. Firstly $$f$$ is homotopic to $$g_0(x)=f(0)$$ by using $$H_t(x)=f(tx)$$, consider a path $$c:[0,1]\rightarrow Y$$ such that $$c(0)=f(0), c(1)=y_0$$, write $$H'_t(x)=c(t)$$, $$H'$$ defines an homotopy between $$g$$ and $$g_0$$.
• Thanks! You said 'it is not to show' but you meant 'it is to show' right? In that case I understand that part. For the second, you have $f\simeq g_0$ and $g_0\simeq g$, can we then conclude $f\simeq g$ by the fact that homotopy is an equivalence relation? – user665297 May 12 at 13:51


text:
# (a) To calculate: How to rewrite any polynomial without powers of x. (b) To calculate: The relation betweeb the new form to synthetix division and the remainder theorem.

(a) To calculate: How to rewrite any polynomial without powers of x. (b) To calculate: The relation betweeb the new form to synthetix division and the remainder theorem.
You can still ask an expert for help

## Want to know more about Polynomial arithmetic?

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

avortarF
(a)A polynomial is written without powers of x by factoring out x from thenon-constant terms.Given:A fourth-degree polynomial in x such as $3{x}^{4}+5{x}^{3}+4{x}^{2}+3x+1$contains all of the powers of x from the first through the fourth.However, any polynomial can be written without powers of x. Evaluatinga polynomial without powers of x (Horner's method) is somewhat easierthan evaluating a polynomial with powers.Calculation:A polynomial is written in the form,$P\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+{a}_{n-2}{x}^{n-2}+.....+{a}_{0},$ Which is arranged from thehighest to the lowest degree of x.Now, x is factored out from the non-constant terms of the polynomial,and proceeding in this manner, a polynomial can be written as withoutthe power of x.Therefore, a polynomial is written without powers of x by factoring out x from the non-constant terms.(b)The new form uses addition and multiplication to evaluate a polynomial.Given:A fourth-degree polynomial in x such as $3{x}^{4}+5{x}^{3}+4{x}^{2}+3x+1$contains all of the powers of x from the first through the fourth.However, any polynomial can be written without powers of x. Evaluatinga polynomial without powers of x (Horner's method) is somewhat easierthan evaluating a polynomial with powers.Calculation:A polynomial is written in the form,$P\left(x\right)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+{a}_{n-2}{x}^{n-2}+.....+{a}_{0},$ Which is arranged from thehighest to the lowest degree of x.Now, the new form uses precisely addition and multiplication in thesame manner as that of the synthetic division to evaluate a polynomial.Therefore, the new form uses addition and multiplication to evaluate apolynomial.


text:
Intercept is in the error term (dropping?)

The model is $$y_{it} = \delta_0d2_t + \delta_1 crm_{it} + (\alpha_i+u_{it})$$

Here the intercept is placed in the error term. Therefore if $\alpha_i$ is correlated with an independent variable would cause bias problems.

• What are the consequences of dropping $\alpha_i$?
• Why is it important to keep it?

• Why do we delete it under fixed effects or first differencing $\alpha_i$ would be eliminated.

When buidling a sum you can easily change the order (Cummutative property of addition) and the brackets (Associative property of addition).

$y_{it} = \delta_0d2_t + \delta_1 crm_{it} + (\alpha_i+u_{it})$

is the same as

$y_{it} = \delta_0d2_t + \delta_1 crm_{it} + \alpha_i+u_{it}$

and the same as

$y_{it} = \alpha_i + \delta_0d2_t + \delta_1 crm_{it} +u_{it}$.

Nonetheless I would like to outline why theequation is written in the way it is written:

These are random effects which depend on the individual and not on the time component. The first part of the equation equals the expected value und certain conditions.

$y_{it} = \delta_0d2_t + \delta_1 crm_{it}$

However the part in the bracket represents the noise. These are random effects which depend on the individual and not on the time component. The variance which does not depend on the regression $\delta_0$ and $\delta_1$ is captured by the term in the brackets. Thus neither time nor crm have an impact on the dependent variable.

$\alpha_i+u_{it}$

In other words if $\delta_0$ and $\delta_1$ are both zero than the following equation is always true.

$y_{it} = \alpha_i+u_{it}$

Consequences of dropping $\alpha_i$:

You delete any effect which is typical for the individual. For example is your sample consists of Sweden, Italy, Spain, Portugal and Malta and you delete the individual specific component. Than you do not have the "Sweden effect" (or the effect for the other countries any more. So you will not anymore capture the effect that Sweden is a northern european, protestant country with long winters while the other countries are different.

If we apply a FD estimation we are not interest in the effect of Sweden, but in the effect of the difference between the years. Therefore we do not have a close look at the Sweden effect at a certain point of the procedure.


text:
HTTP-related tools to for PipeStack

## Summary

HTTP-related tools to for PipeStack

James Gardner

## Changes

### 2010-11-14

• Changed the behaviour of the input service to get more access to environ

• Raise an AttributeError if you set an attribute of the http_response service which doesn’t exist

### 0.2.0

2010-10-13

• Updated so that we use bag.environ and bag.http_response instead of bag.http.environ and bag.http.response

• Changed bag.http.input to bag.input and bag.http.query to bag.query so that nested pipe names are no longer needed

### 0.1.1

2009-12-24

• BuildKit support

### 0.1.0

#### 2009-11-12

• Completed basic documentation.

• Re-implemented WSGIService.

#### 2009-11-10

• Updated so that HTTPService doesn’t depend on WSGI.

#### 2009-10-04

• Updated to use the new service format.

#### 2009-10-02

• Fixed a bug with __setattr__() missing the value argument.

#### 2009-09-30

• First version.

HTTPKit - HTTP-related tools for PipeStack

This program is free software: you can redistribute it and/or modify it under the terms of the GNU Affero General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Affero General Public License for more details.

You should have received a copy of the GNU Affero General Public License along with this program. If not, see <http://www.gnu.org/licenses/>.

The httpkit.helper.cookie module is based on code by Ian Bicking and contributers from WebOb and can be used under this license as an alternative: http://pythonpaste.org/webob/license.html

## Project details

Uploaded source


text:
# Time Shifting property of the Fourier Transform

If $F(\omega)=\mathcal{F}[f(t)]$, then
$$\mathcal{F}\left[f\left(t-t_{0}\right)\right]=e^{-j \omega t_{0}} F(\omega)$$
that is, a delay in the time domain corresponds to a phase shift in the frequency domain. To derive the time shifting property, we note that
$$\mathcal{F}\left[f\left(t-t_{0}\right)\right]=\int_{-\infty}^{\infty} f\left(t-t_{0}\right) e^{-j \omega t} d t$$
If we let $x=t-t_{0}$ so that $d x=d t$ and $t=x+t_{0}$, then
\begin{aligned}\mathcal{F}\left[f\left(t-t_{0}\right)\right] &=\int_{-\infty}^{\infty} f(x) e^{-j \omega\left(x+t_{0}\right)} d x \\&=e^{-j \omega t_{0}} \int_{-\infty}^{\infty} f(x) e^{-j \omega x} d x=e^{-j \omega t_{0}} F(\omega)\end{aligned}
Similarly, $\mathcal{F}\left[f\left(t+t_{0}\right)\right]=e^{j \omega t_{0}} F(\omega)$. For example, $$\mathcal{F}\left[e^{-a t} u(t)\right]=\frac{1}{a+j \omega}$$ The transform of $f(t)=e^{-(t-2)} u(t-2)$ is
$$F(\omega)=\mathcal{F}\left[e^{-(t-2)} u(t-2)\right]=\frac{e^{-j 2 \omega}}{1+j \omega}$$


text:
Using 'expand' in snakemake target rule with constraints
1
0
Entering edit mode
7 days ago
cfos4698 ▴ 270

Dear all,

I'm working on a snakemake workflow where I need to use glob_wildcards (or something similar) to work on all samples in a directory.

SAMPLES = glob_wildcards(os.path.join(READSDIR,"{sample}_L001_R1_001.fastq.gz")).sample
wildcard_constraints:
sample="(?!Undet).*"

rule all:
input:
expand(os.path.join(RESULT_DIR, "fastp/{sample}/trimmed/{sample}_trimmed_R1.fq.gz"), sample = SAMPLES),
expand(os.path.join(RESULT_DIR, "fastp/{sample}/trimmed/{sample}_trimmed_R2.fq.gz"), sample = SAMPLES)

The input/output files for normal/work rules (sorry, don't know the proper name for them) populate correctly based on the {sample} wildcard. All rules finish as expected. However, I expect there to be some samples in the directory that I don't want. I can get around this by adding a global wildcards constraint at the beginning like so:

SAMPLES = glob_wildcards(os.path.join(READSDIR,"{sample}_L001_R1_001.fastq.gz")).sample
wildcard_constraints:
sample="(?!Undet).*"

However, the issue then is a 'MissingInputException':

Missing input files for rule all:
results2/fastp/Undetermined_S0/trimmed/Undetermined_S0_trimmed_R1.fq.gz
results2/fastp/Undetermined_S0/trimmed/Undetermined_S0_trimmed_R2.fq.gz

How can I change the expand function for the target rule input so that it behaves in the same way as the wildcards_constraint (i.e., ignoring files beginning with 'Undet')?

Thanks!

snakemake glob_wildcards expand • 212 views
2
Entering edit mode
7 days ago
yztxwd ▴ 490
SAMPLES = glob_wildcards(os.path.join(READSDIR,"{sample, (?!Undet).*}_L001_R1_001.fastq.gz")).sample

0
Entering edit mode

Thanks, works a charm. I'd gone down the rabbit hole of:

expand(os.path.join(RESULT_DIR, "fastp/{sample}/trimmed/{sample}_trimmed_R1.fq.gz"), sample = [x for x in SAMPLES if 'Undet' not in x])

And it seemed to work, but I prefer the neatness of yours.


text:
# IV international conference on particle physics and astrophysics

22-26 October 2018
Hotel Intourist Kolomenskoye 4*
Europe/Moscow timezone

## The Structure of UV Divergences in Maximally Supersymmetric Gauge Theories

24 Oct 2018, 16:15
15m
Alekseevskiy hall (Hotel Intourist Kolomenskoye 4*)

### Alekseevskiy hall

#### Hotel Intourist Kolomenskoye 4*

Kashyrskoye shosse, 39B, Moscow, Russia, 115409
Plenary/section talk Particle physics: hep theory

### Speaker

Mr. Denis Tolkachou

### Description

We consider the UV divergences up to sub-sub leading order for the four-point on-shell scattering amplitudes in D=8 supersymmetric Yang-Mills theory in the planar limit. We trace how the leading, subleading, etc divergences appear in all orders of perturbation theory. The structure of these divergences is typical for any local quantum field theory independently on renormalizability. We show how the generalized RG equations allow one to evaluate the leading, subleading, etc. contributions in all orders of PT starting from one-, two-, etc. loop diagrams respectively. We focus then on subtraction scheme dependence of the results and show that in full analogy with renormalizable theories the scheme dependence can be absorbed into the redefinition of the couplings. The only difference is that the role of the couplings play dimensionless combinations like $g^2s^2$ or $g^2t^2$, where $s$ and $t$ are the Mandelstam variables.

### Presentation Materials

There are no materials yet.
###### Your browser is out of date!

Update your browser to view this website correctly. Update my browser now

×


text:
# Soft midrule in table

3 small questions:

1) I'd like to put a soft \midrule in this table between Care type and the headers, but I'd like it to not quite reach the edge of the table: (at your discretion to mess with the layout of table )

\documentclass{article}
\usepackage{tabularx}
\usepackage{booktabs}
%\usepckage{Something extra}

\begin{document}
\begin{table}[!ht]
\centering\makebox[\textwidth]
{\resizebox{1.1\textwidth}{!}{%
\begin{tabularx}{1.2\textwidth}{*6X}
\toprule
\multicolumn{6}{c}{Care Type} \\
\midrule
\vspace{3pt}
\textbf{Direct care} & \textbf{Housekeeping} & \textbf{Mealtimes} & \textbf{Medication rounds  } & \textbf{Miscellaneous} & \textbf{Personal care} \\
\midrule
Blood pressure measurement & Equipment cleaning& Dispensing meals& Distributing medication&Call requests&Toiletting\\
& &&&&\\
Weights & Cleaning patient surfaces& &Injections&Bed making &Changing\\
& &&&&\\
SATs~\footnote{Blood sugar saturation}  & &&&&\\
\bottomrule
\end{tabularx}}}
\smallskip
\caption{Activity type and examples of each}
\label{tab:activities}\end{table}
\end{document}

\midrule dashed and colored in booktabs -- two kinds of midrules in one booktab

2) Also is there an out-of-the box to get table footnotes in, as I don't know where mine have gone? Footnotes tabularx-environment LaTeX

3) How to avoid the text in each cell being stretched to fill it and hence making odd hyphenations (eg. cleaning pa-tient surfaces) and or (eg. Hygiene prod-ucts)?

• I would suggest making this three separate questions or showing how they are all related (or stemming from one base question, perhaps?). – Sean Allred Mar 28 '13 at 14:37
• See tex.stackexchange.com/questions/2890. As for the stretching, I haven't tested this one bit - but try putting it in a box, maybe? (Also, what is Something extra? If you can't say and you're sure it isn't important, just remove it -- otherwise stick it back in there...) – Sean Allred Mar 28 '13 at 14:39
• just a comment on the format: the multi-line entries would look much nicer with \raggedright (or \RaggedRight using ragged2e if you want to retain hyphenation). – barbara beeton Mar 28 '13 at 14:44
• @barbarabeeton this might do the trick. Where do I put that \RaggedRight command in the table context? on ever line? Just somewhere inside the environment? – HCAI Mar 28 '13 at 14:59
• @HCAI -- david carlisle has included \raggedright in his answer, along with a method of getting the rule you asked for, so i think you should go with that. (the \raggedright is placed in every "paragraph" cell in the table, but since all of the cells are defined the same way, this is done with code that is automatically replicated 6 times, once for each column. it's nice clean, compact code; recommended.) – barbara beeton Mar 28 '13 at 16:02

I think you are looking for \cmidrule but I also removed tabularx and the scale box commands.

\documentclass{article}
\usepackage{array}
\usepackage{booktabs}
\usepackage{graphicx}
%\usepckage{Something extra}

\begin{document}
\begin{table}[htp]
\centering
\begin{tabular}{{}*6{>{\hspace{0pt}\raggedright\arraybackslash}p{\dimexpr(\textwidth-10\tabcolsep)/6\relax}}@{}}
\toprule
\multicolumn{6}{c}{Care Type} \\
\cmidrule(l{2em}r{2em}){1-6}
\textbf{Direct care} & \textbf{House\-keeping} & \textbf{Meal\-times} & \textbf{Medi\-cation rounds  } & \textbf{Misc\-ellaneous} & \textbf{Personal care} \\
\midrule
Blood pressure measurement & Equipment cleaning& Dispensing meals& Distributing medication&Call requests&Toiletting\\
& &&&&\\
Weights & Cleaning patient surfaces& &Injections&Bed making &Changing\\
& &&&&\\
SATs~\footnote{Blood sugar saturation}  & &&&&\\
\bottomrule
\end{tabular}
\smallskip
\caption{Activity type and examples of each}
\label{tab:activities}
\end{table}

\noindent X\dotfill X
\end{document}

• This is absolutely superb, many many thanks indeed! Incidentally the code in my question actually from this very forum where I asked how to make the table fit centrally on the page. What is the 10\tabcolsep)/6 bit? this looks insanely useful at preventing the whole kit and caboodle from flooding off the page! I wish there was a gallery where people could showcase this sort of professional looking code. Is there? – HCAI Mar 29 '13 at 18:42
• @HCAI tabcolsep is the padding latex puts either side of each column. The outer padding is suppressed with @{} so for 6 columns you just have the 5 inner separations of 2\tabcolsep so the width available for columns is \textwidth-10\tabcolsep so the width of each column is 1/6 of that. – David Carlisle Mar 29 '13 at 19:13


text:
## A trailer for p-adic analysis, second half: Mahler coefficients

In the previous post we defined $latex {p}&fg=000000$-adic numbers. This post will state (mostly without proof) some more surprising results about continuous functions $latex {f \colon \mathbb Z_p \rightarrow \mathbb Q_p}&fg=000000$. Then we give the famous proof of the Skolem-Mahler-Lech theorem using $latex {p}&fg=000000$-adic analysis. 1. Digression on $latex {\mathbb C_p}&fg=000000$ Before I go on,… Continue reading A trailer for p-adic analysis, second half: Mahler coefficients

## A trailer for p-adic analysis, first half: USA TST 2003

I think this post is more than two years late in coming, but anywhow... This post introduces the $latex {p}&fg=000000$-adic integers $latex {\mathbb Z_p}&fg=000000$, and the $latex {p}&fg=000000$-adic numbers $latex {\mathbb Q_p}&fg=000000$. The one-sentence description is that these are integers/rationals carrying full mod $latex {p^e}&fg=000000$ information'' (and only that information). The first four sections will… Continue reading A trailer for p-adic analysis, first half: USA TST 2003


text:
The Fibonacci numbers are a unique sequence of numbers where each new term is found by adding the two previous terms.

The Fibonacci Numbers are as follows:

$1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...$

## Fibonacci Sequences

A Fibonacci sequence refers to any sequence where each new term is found by adding the previous two terms, given any two starting values.

### Example 1

Which of the following are Fibonacci sequences?

a) $1, 4, 5, 9, 14, ...$

Calculate whether each new term is found by adding the previous two terms.

$t_3=t_1+t_2$
$t_3=1+4$
$t_3=5$

Check that \$t_3\$ is indeed equal to 5 by checking with the question.

This is correct.

$t_4=t_2+t_3$
$t_4=4+5$
$t_4=9$

This is correct.

$t_5=t_3+t_4$
$t_5=5+9$
$t_5=14$

This is correct.

This is a Fibonacci sequence as each new term is found by adding the two previous terms.

b) $3, 7, 10, 13, 23, ...$

Calculate whether each new term is found by adding the previous two terms.

$t_3=t_1+t_2$
$t_3=3+7$
$t_3=10$

Check that \$t_3\$ is indeed equal to 10 by checking with the question.

This is correct.

$t_4=t_2+t_3$
$t_4=7+10$
$t_4=17$

$t_4$ is supposed to be 13.

This is wrong.

This is not a Fibonacci sequence as each new term is not found by adding the previous term.

## Second Order Difference Equations for a Fibonacci Sequence

Second order difference equations for  Fibonacci sequences follows the following equation:

$f_{n+2}=f_n+f_{n+1}$   given $f_1$ and $f_2$

### Example 2

Find the first five terms of the following Fibonacci sequence given by the second order difference equation:

$f_{n+2}=f_n+f_{n+1}$   $f_1=2$ $f_2=1$

The question defines the first two terms so use these in the second order difference equation to calculate the remaining terms.

$f_1=2$ and $f_2=1$
$f_{n+2}=f_n+f_{n+1}$

$f_{3}=f_1+f_2$
$f_3=2+1$
$f_3=3$

$f_4=f_2+f_3$
$f_4=1+3$
$f_4=4$

$f_5=f_3+f_4$
$f_5=3+4$
$f_5=7$

The first five terms of the sequence are 2, 1, 3, 4 and 7.

### Example 3

Find the value of $t_2$ given $t_1=5, t_4=7$ and $t_5=13$.

We are given two consecutive values so all we need to do is to work backwards to find $t_3$.

$t_3=t_5-t_4$
$t_3=13-7$
$t_3=6$

Now we know $t_3$ we can work backwards to find $t_2$.

$t_2=t_4-t_3$
$t_2=7-6$
$t_2=1$

The value of $t_2$ is 1.


text:
This R-package implements the CSD algorithm presented by Voigt et al. 2017 in an efficient manner.

• Requrirements: Requirements: R (version 4.1.0 or higher) with packages WGCNA, optparse, glue, magrittr and Rcpp (and of course a C++ compiler) installed. Additionally, having an optimized Blas library such as openBlas is highly recommended for performance reasons (see this link for more info).

Installation

In order to install the release version from Bioconductor, do type in the R terminal:

if(!requireNamespace("BiocManager")){
install.packages("BiocManager")
}
BiocManager::install("phyloseq")

If you want to install the development version instead:

# install.packages("devtools")
devtools::install_github("AlmaasLab/csdR", ref = "main")