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main-content

## Über dieses Buch

This book constitutes revised selected papers from the 9th International Conference on Computational Advances in Bio and Medical Sciences, ICCABS 2019, held in Miami, Florida, USA in November 2019.

The 15 papers presented in this volume were carefully reviewed and selected from 30 submissions. They deal with topics such as computational biology; biomedical image analysis; biological networks; cancer genomics; gene enrichment analysis; functional genomics; interaction networks; protein structure prediction; dynamic programming; and microbiome analysis.

## Inhaltsverzeichnis

### Detecting De Novo Plasmodesmata Targeting Signals and Identifying PD Targeting Proteins

Abstract
Subcellular localization plays important roles in protein’s functioning. In this paper, we developed a hidden Markov model to detect de novo signals in protein sequences that target at a particular cellular location: plasmodesmata. We also developed a support vector machine to classify plasmodesmata located proteins (PDLPs) in Arabidopsis, and devised a decision-tree approach to combine the SVM and HMM for better classification performance. The methods achieved high performance with ROC score 0.99 in cross-validation test on a set of 360 type I transmembrane proteins in Arabidopsis. The predicted PD targeting signals in one PDLP have been experimentally verified.
Jiefu Li, Jung-Youn Lee, Li Liao

### The Agility of a Neuron: Phase Shift Between Sinusoidal Current Input and Firing Rate Curve

Abstract
The response of a neuron when receiving a periodic input current signal is a periodic spike firing rate signal. The frequency of an input sinusoidal current and the surrounding environment such as background noises are two important factors that affect the firing rate output signal of a neuron model. This study focuses on the phase shift between input and output signals, and here we present a new concept: the agility of a neuron, to describe how fast a neuron can respond to a periodic input signal. By applying the score of agility, we are capable of characterizing the surrounding environment; once the frequency of periodic input signal is given, the actual angle of phase shift can then be determined, and therefore different neuron models can be normalized and compared to others.
Chu-Yu Cheng, Chung-Chin Lu

### Efficient Sequential and Parallel Algorithms for Incremental Record Linkage

Abstract
Given a collection of records, the problem of record linkage is to cluster them such that each cluster contains all the records of one and only one individual. Existing algorithms for this important problem have large run times especially when the number of records is large. Often, a small number of new records have to be linked with a large number of existing records. Linking the old and new records together might call for large run times. We refer to any algorithm that efficiently links the new records with the existing ones as incremental record linkage (IRL) algorithms and in this paper, we offer novel IRL algorithms. Clustering is the basic approach we employ. Our algorithms use a novel random sampling technique to compute the distance between a new record and any cluster and associate the new record with the cluster with which it has the least distance. The idea is to compute the distance between the new record and only a random subset of the cluster records. We can use a sampling lemma to show that this computation is very accurate. We have developed both sequential and parallel implementations of our algorithms. They outperform the best-known prior algorithm (called RLA). For example, one of our algorithms takes 71.22 s to link 100,000 records with a database of 1,000,000 records. In comparison, the current best algorithm takes 140.91 s to link 1,100,000 records. We achieve a very nearly linear speedup in parallel. E.g., we obtain a speedup of 28.28 with 32 cores. To the best of our knowledge, we are the first to propose parallel IRL algorithms. Our algorithms offer state-of-the-art solutions to the IRL problem.
Abdullah Baihan, Reda Ammar, Robert Aseltine, Mohammed Baihan, Sanguthevar Rajasekaran

### Autoencoder Based Methods for Diagnosis of Autism Spectrum Disorder

Abstract
Autism Spectrum Disorder (ASD) is a neurological disorder that affects a person’s behavior and social interaction. Integrating machine learning algorithms with neuroimages a diagnosis method can be established to detect ASD subjects from typical control (TC) subjects. In this study, we develop autoencoder based ASD diagnosis methods. Firstly, we design an autoencoder to extract high-level features from raw features, which are defined based on eigenvalues and centralities of functional brain networks constructed with the entire Autism Brain Imaging Data Exchange 1 (ABIDE 1) dataset. Secondly, we use these high-level features to train several traditional machine learning methods (SVM, KNN, and subspace discriminant), which achieve the classification accuracy of 72.6% and the area under the receiving operating characteristic curve (AUC) of 79.0%. We also use these high-level features to train a deep neural network (DNN) which achieves the classification accuracy of 76.2% and the AUC of 79.7%. Thirdly, we combine the pre-trained autoencoder with the DNN to train it, which achieves the classification accuracy of 79.2%, and the AUC of 82.4%. Finally, we also train SVM, KNN, and subspace discriminant with the features extracted from the combination of the pre-trained autoencoder and the DNN which achieves the classification accuracy of 74.6% and the AUC of 78.7%. These results show that our proposed methods for diagnosis of ASD outperform state-of-the-art studies.
Sakib Mostafa, Wutao Yin, Fang-Xiang Wu

### FastFeatGen: Faster Parallel Feature Extraction from Genome Sequences and Efficient Prediction of DNA -Methyladenine Sites

Abstract
$$N^6$$-methyladenine is widely found in both prokaryotes and eukaryotes. It is responsible for many biological processes including prokaryotic defense system and human diseases. So, it is important to know its correct location in genome which may play a significant role in different biological functions. Few computational tools exist to serve this purpose but they are computationally expensive and still there is scope to improve accuracy. An informative feature extraction pipeline from genome sequences is the heart of these tools as well as for many other bioinformatics tools. But it becomes reasonably expensive for sequential approaches when the size of data is large. Hence, a scalable parallel approach is highly desirable. In this paper, we have developed a new tool, called FastFeatGen, emphasizing both developing a parallel feature extraction technique and improving accuracy using machine learning methods. We have implemented our feature extraction approach using shared memory parallelism which achieves around 10$$\times$$ speed over the sequential one. Then we have employed an exploratory feature selection technique which helps to find more relevant features that can be fed to machine learning methods. We have employed Extra-Tree Classifier (ETC) in FastFeatGen and performed experiments on rice and mouse genomes. Our experimental results achieve accuracy of 85.57% and 96.64%, respectively, which are better or competitive to current state-of-the-art methods. Our shared memory based tool can also serve queries much faster than sequential technique. All source codes and datasets are available at https://github.com/khaled-rahman/FastFeatGen.
Md. Khaledur Rahman

### Optimized Multiple Fluorescence Based Detection in Single Molecule Synthesis Process Under High Noise Level Environment

Abstract
Single molecule sequencing contributes to overall human advancement in the areas including but not limited to genomics, transcriptomics, clinical test, drug development, and cancer screening. Furthermore, fluorescence based sequencing is mostly employed in single molecule sequencing among other methods, specifically in the fields of DNA sequencing. Contemporary fluorescence labeling methods utilize a Charge-coupled Device camera to capture snapshots of multiple pixels on the single molecule sequencing. We propose a method for fluorescence labeling detection with a single pixel, which excels in high accuracy and low resource requirement in the low signal-to-noise ratio conditions. Such a method also benefits from higher throughput compared to others. The context in this study explores the single molecule synthesis process modeling using negative binomial distributions. Also, including the method of maximum likelihood and Viterbi algorithm in this modeling improves signal detection accuracy. The fluorescence-based model is most beneficial to simulate actual experiment processes and to facilitate in understanding the relations between fluorescence emission and signal receiving event.
Hsin-Hao Chen, Chung-Chin Lu

### Deep Learning of CTCF-Mediated Chromatin Loops in 3D Genome Organization

Abstract
The three-dimensional organization of the human genome is of crucial importance for gene regulation. Results from high-throughput chromosome conformation capture techniques show that the CCCTC-binding factor (CTCF) plays an important role in chromatin interactions, and CTCF-mediated chromatin loops mostly occur between convergent CTCF-binding sites. However, it is still unclear whether and what sequence patterns in addition to the convergent CTCF motifs contribute to the formation of chromatin loops. To discover the complex sequence patterns for chromatin loop formation, we have developed a deep learning model, called DeepCTCFLoop, to predict whether a chromatin loop can be formed between a pair of convergent CTCF motifs using only the DNA sequences of the motifs and their flanking regions. Our results suggest that DeepCTCFLoop can accurately distinguish the convergent CTCF motif pairs forming chromatin loops from the ones not forming loops. It significantly outperforms CTCF-MP, a machine learning model based on word2vec and boosted trees, when using DNA sequences only. Moreover, we show that DNA motifs binding to ASCL1, SP2 and ZNF384 may facilitate the formation of chromatin loops in addition to convergent CTCF motifs. To our knowledge, this is the first published study of using deep learning techniques to discover the sequence motif patterns underlying CTCF-mediated chromatin loop formation. Our results provide useful information for understanding the mechanism of 3D genome organization. The source code and datasets used in this study for model construction are freely available at https://github.com/BioDataLearning/DeepCTCFLoop.
Shuzhen Kuang, Liangjiang Wang

### Effects of Various Alpha-1 Antitrypsin Supplement Dosages on the Lung Microbiome and Metabolome

Abstract
Patients with Alpha-1 Antitrypsin Deficiency (A1AD) have abnormally low levels of the protein Alpha-1 Antitrypsin (AAT) in their blood, because of a double mutation that makes the protein misfold and instead collect in the liver (sometimes even causing cirrhosis). The currently accepted single dosage (SD) of AAT supplements does not produce AAT blood concentrations anywhere near normal levels; they typically only reach the effect of having a single mutation. Some have therefore advocated for a double dosage (DD) of these treatments, which generally would be enough to approach these normal concentrations. Levels of cytokines, produced by the immune system in response to an attack, have already been observed to drop dramatically when A1AD patients consuming single dosage started taking double dosage, and then either remain the same or increase again upon return to a single dosage regimen. In this study we administer the same dosage sequence to A1AD patients (SD, DD, SD) for one month each and view the effects on their lung microbiome and metabolome. We analyze both at the end of each stage, comparing and contrasting and discovering potential biomarkers for each stage, and concluding with a discussion of potential implications.
Trevor Cickovski, Astrid Manuel, Kalai Mathee, Michael Campos, Giri Narasimhan

### A Multi-hypothesis Learning Algorithm for Human and Mouse miRNA Target Prediction

Abstract
MicroRNAs (miRNAs) are small non-coding RNAs that play a key role in regulating gene expression and thus in many cellular activities. Dysfunction of cells in these tasks is correlated with the development of several kinds of cancer. As the functionality of miRNAs depends on the location of their binding on their targets, binding site prediction has received a lot of attention in the last several years. Despite its importance, the mechanisms of miRNA targeting are still unknown. In this paper, we introduce an algorithm that partitions miRNA target duplexes according to hypotheses that each represents a different mechanism of targeting. The algorithm, called multi-hypothesis learner, examines all possible hypotheses to find out the optimum data partitions according to the performance of these hypotheses for miRNA target prediction. These hypotheses were then utilized to build a superior target predictor for miRNAs. Our method exploited biologically meaningful features for recognizing targets, which enables establishment of hypotheses that can be correlated with target recognition mechanisms. Test results show that the algorithm can provide comparable performance to state-of-the-art machine learning tools such as RandomForest in predicting miRNA binding sites. Moreover, feature selection on the partitions in our method confirms that the partitioning mechanism is closely related to biological mechanisms of miRNA targeting. The resulting data partitions can potentially be used for in vivo experiments to aid in discovery of the targeting mechanisms.
Mohammad Mohebbi, Liang Ding, Russell L. Malmberg, Liming Cai

### RiboSimR: A Tool for Simulation and Power Analysis of Ribo-seq Data

Abstract
RNA-seq and Ribo-seq are widespread quantitative methods for assessing transcription and translation. They can be used to detect differential expression, differential translation, and differential translation efficiency between conditions. The statistical power to detect differential genes is affected by multiple factors, such as the number of replicates, sequencing depth, magnitude of differential expression and translation, distribution of gene counts, and method for estimating biological variance. As power estimation of translational efficiency involves the combination of both RNA-seq measurements and Ribo-seq measurements, this task is particularly challenging. Here we propose a power assessment tool, called RiboSimR, based purely on data simulation. RiboSimR, produces semi-parametric simulations that generate data based on real RNA and Ribo-seq experiments, with customizable choices on baseline parameters and tool configurations. We demonstrate the usefulness of our tool by simulating data based on two published Ribo-seq datasets and analyzing various aspects of experimental design.
Patrick Perkins, Anna Stepanova, Jose Alonso, Steffen Heber

### Treatment Practice Analysis of Intermediate or High Risk Localized Prostate Cancer: A Multi-center Study with Veterans Health Administration Data

Abstract
Khajamoinuddin Syed, William Sleeman IV, Joseph Nalluri, Payal Soni, Michael Hagan, Jatinder Palta, Rishabh Kapoor, Preetam Ghosh

### Forecasting Model for the Annual Growth of Cryogenic Electron Microscopy Data

Abstract
In this paper, we develop a forecasting model for the growth of Cryogenic Electron Microscopy (Cryo-EM) experimental data time series using autoregressive (AR) model. We employ the optimal modeling order that maximizes the estimation accuracy while maintaining the least normalized prediction error. The proposed model has been efficiently used to forecast the growth of cryo-EM data for the next 10 years, 2019–2028. The time series for the number of released three-dimensional Electron Microscopy (3DEM) images along with the time series of the annual number of 3DEM achieving resolution 10 Å or better are used. The data was collected from the public Electron Microscopy Data Bank (EMDB). The simulation results showed that the optimal model orders to estimate both datasets are $$AR\left( 5 \right)$$ and $$AR\left( 6 \right)$$ respectively. Consequently, the optimal models obtained an estimation accuracy of $$96.8\%,$$ and $$85\%$$ for 3DEM experiments time series and 3DEM resolutions time series, respectively. Hence, the forecasting results reveal an exponential increasing behavior in the future growth of annual released of 3DEM and, similarly, for the annual number of 3DEM achieving resolution 10 Å or better.
Qasem Abu Al-Haija, Kamal Al Nasr

### Local and Global Stratification Analysis in Whole Genome Sequencing (WGS) Studies Using LocStra

Abstract
We are interested in the analysis of local and global population stratification in WGS studies. We present a new R package (locStra) that utilizes the covariance matrix, the genomic relationship matrix, and the unweighted/weighted genetic Jaccard similarity matrix in order to assess population substructure. The package allows one to use a tailored sliding window approach, for instance using user-defined window sizes and metrics, in order to compare local and global similarity matrices. A technique to select the window size is proposed. Population stratification with locStra is efficient due to its C++ implementation which fully exploits sparse matrix algebra. The runtime for the genome-wide computation of all local similarity matrices does typically not exceed one hour for realistic study sizes. This makes an unprecedented investigation of local stratification across the entire genome possible. We apply our package to the 1,000 Genomes Project.
Georg Hahn, Sharon Marie Lutz, Julian Hecker, Dmitry Prokopenko, Christoph Lange

### A New Graph Database System for Multi-omics Data Integration and Mining Complex Biological Information

Abstract
Due to the advancement in high throughput technologies and robust experimental designs, many recent studies attempt to incorporate heterogeneous data obtained from multiple technologies to improve our understanding of the molecular dynamics associated with biological processes. Currently available technologies produce wide variety of large amount of data spanning from genomics, transcriptomics, proteomics, and epigenetics. Due to the fact that such multi-omics data are very diverse and come from different biological levels, it has been a major research challenge to develop a model to properly integrate all available and relevant data to advance biomedical research. It has been argued by many researchers that the integration of multi-omics data to extract relevant biological information is currently one of the major biomedical informatics challenges. This paper proposes a new graph database model to efficiently store and mine multi-omics data. We show a working model of this graph database with transcriptomics, genomics, epigenetics and clinical data for three cancer types from the Cancer Genome Atlas. Moreover, we highlight the usefulness of graph database mining to extract relevant biological interpretations and also to find novel relationships between different data levels.
Ishwor Thapa, Hesham Ali

### SMART2: Multi-library Statistical Mitogenome Assembly with Repeats

Abstract
SMART2 is an enhanced version of the SMART pipeline for mitogenome assembly from low-coverage whole-genome sequencing (WGS) data. Novel features include automatic selection of the optimal number of read pairs used for assembly and the ability to assemble multiple sequencing libraries when available. SMART2 succeeded in generating mitochondrial sequences for 26 metazoan species with WGS data but no previously published mitogenomes in NCBI databases. The SMART2 pipeline is publicly available via a user-friendly Galaxy interface at https://neo.engr.uconn.edu/?tool_id=SMART2.


text:
Find the value of.
Question:

Find the value of $\left(a^{2}+\sqrt{a^{2}-1}\right)^{4}+\left(a^{2}-\sqrt{a^{2}-1}\right)^{4}$.

Solution:

Firstly, the expression $(x+y)^{4}+(x-y)^{4}$ is simplified by using Binomial Theorem.

This can be done as

$(x+y)^{4}={ }^{4} C_{0} x^{4}+{ }^{+} C_{1} x^{3} y+{ }^{4} C_{2} x^{2} y^{2}+{ }^{4} C_{3} x y^{3}+{ }^{4} C_{4} y^{4}$

$=x^{4}+4 x^{3} y+6 x^{2} y^{2}+4 x y^{3}+y^{4}$

$(x-y)^{4}={ }^{4} C_{0} x^{4}-{ }^{4} C_{1} x^{5} y+{ }^{4} C_{2} x^{2} y^{2}-{ }^{4} C_{3} x y^{3}+{ }^{4} C_{4} y^{4}$

$=x^{4}-4 x^{3} y+6 x^{2} y^{2}-4 x y^{3}+y^{4}$

$\therefore(x+y)^{4}+(x-y)^{4}=2\left(x^{4}+6 x^{2} y^{2}+y^{4}\right)$

Putting $x=a^{2}$ and $y=\sqrt{a^{2}}-1$, we obtain

$\left(a^{2}+\sqrt{a^{2}-1}\right)^{4}+\left(a^{2}-\sqrt{a^{2}-1}\right)^{4}=2\left[\left(a^{2}\right)^{4}+6\left(a^{2}\right)^{2}\left(\sqrt{a^{2}-1}\right)^{2}+\left(\sqrt{a^{2}-1}\right)^{4}\right]$

$=2\left[a^{8}+6 a^{4}\left(a^{2}-1\right)+\left(a^{2}-1\right)^{2}\right]$

$=2\left[a^{8}+6 a^{6}-6 a^{4}+a^{4}-2 a^{2}+1\right]$

$=2\left[a^{8}+6 a^{6}-5 a^{4}-2 a^{2}+1\right]$

$=2 a^{8}+12 a^{6}-10 a^{4}-4 a^{2}+2$

format: plain


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# Question on the construction of mapping from space of strategy profile into itself in Nash(1951)

To appeal to Brouwer fixed point theorem, Nash(1951) constructed a continuous mapping $$\operatorname{T}$$ from strategy profile space into inself:

For player $$i$$, the probability of a pure strategy $$a_{i} \in A_i$$ is

$$\operatorname{T}s_i(a_i) = \frac{s_i(a_i) + \phi_{i, a_i}(s)}{1+\sum_{b_i \in A_i}\phi_{i, b_i}(s)}$$

in which, $$A_i$$ is the player $$i$$'s pure strategy space, $$s_i$$ is a mixed strategy that assigns each pure strategy with its probability.$$s$$ is a strategy profile, a vector of every player's strategy.

My question is why we need $$\phi_{i, a_i}(s) = \max \{0, u_i(a_i, s_{-i})-u_i( s)\}$$? Why not to replace $$\phi_{i, a_i}(s)$$ with $$u_i(a_i, s_{-i})-u_i( s)$$, which is$$\operatorname{T'}s_i(a_i) = \frac{s_i(a_i) + u_i(a_i, s_{-i})-u_i( s)}{1+\sum_{b_i \in A_i}(u_i(b_i, s_{-i})-u_i( s))}$$

It seems to me, Besides being continuous, $$\operatorname{T'}$$ statisfies that $$\sum_{b_i \in A_i}\operatorname{T'}s_i(b_i)=1$$ and that a fixed point of $$\operatorname{T'}$$ necessarily constitutes a nash equilibrium.


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# How To Check If Two Strings Are Equal In Typescript

Knowing how to check if two strings are equal in Typescript will help you a lot when working with the string like you can check input from use. Using equality operator is one of those methods to do it. Let’s go into detail now.

## Check if two strings are equal in Typescript

### Use equality operator

The equality operator(==) helps you check if two strings are equal and then returns a boolean.

Example:

const str1:string = 'WooLa'
const str2:string = 'WooLa'
let result: boolean

result = str1 == str2
console.log(result)
result = (str1 == str2)
console.log(result)

Output:

[LOG]: true
[LOG]: true

Here if my string has whitespace, then the result will be false.

Example:

const str1:string = 'WooLa'
const str2:string = 'WooLa'
const str3:string = ' WooLa'
let result: boolean
let result2: boolean

result = (str1 == str2)
result2= (str1 == str3)

console.log(result)
console.log(result2)

Output:

[LOG]: true
[LOG]: false

But with the equality operator in Typescript will try to convert and compare the different types.

Example:

const str1:string = '1'
const str2:number = 1
let result: boolean

result = (str1 == str2) // This condition will always return 'false' since the types' string' and 'number' have no overlap
console.log(result)

Output:

[LOG]: true

Typescript also warned us, but it still works if we try to run the code.

### Use strict equality operator

To solve the problem with type, we can use strict equality operator(===). The strict equality operator is like the equality operator, but it will never deal with different types. So I recommend you always use a strict equality operator.

Example:

const str1:string = '1'
const str2:number = 1
let result: boolean

result = (str1 === str2) //This condition will always return 'false' since the types' string' and 'number' have no overlap.
console.log(result)

Output:

[LOG]: false

The strict equality operators also provides us with the IsStrictlyEqual semantic(!==) to check if two strings are not equal.

Example:

const str1:string = 'Hello'
const str2:string = 'World'
let result: boolean

result1 = (str1 !== str2)
console.log(result1)

result2 = (str1 === str2)
console.log(result2)

Output:

[LOG]: true
[LOG]: false

You can also apply the equality operator inside conditional logic code.

Example:

const str1:string = 'Hello'
const str2:string = 'Hello'

if (str1===str2){
console.log("Hello World")
}

Output:

[LOG]: "Hello World"

## Summary

In this tutorial, we showed and explained how to check if two strings are equal in Typescript. You can use the equality operator or strict equality operator. You can also apply it to your conditional logic code. Good luck for you!

Maybe you are interested:


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# How to interpret correlations with negative numbers in SPSS

I'm trying to establish a bivariate Pearson correlation between two groups of variables in SPSS, however one of the groups has positive decimal numbers and the other negative decimal numbers.

The results show a significant negative correlation between the two groups. If the negative numbers were positive instead this analysis would show a significant positive correlation.

What Im wondering is, how do I interpret these results? Can I accurately say that there's a negative correlation?

• Note that use of SPSS by poster is immaterial to the question. I've removed the tag. – Nick Cox May 5 '13 at 14:56

Yes, there is a negative correlation. The positive correlation means there is a positive relationship between the variables; as one variable increases or decreases, the other tends to increase or decrease with it. The negative correlation means that as one of the variables increases, the other tends to decrease, and vice versa.

If the negative numbers were positive instead this analysis would show a significant positive correlation.

Not necessarily! For example, if you add a large enough constant to all the negative numbers so that they're all positive - i.e. add something a little larger than the absolute value of the smallest (most negative) number - then the correlation would still be negative.

You interpret the negative correlation as a negative correlation between the sets of numbers you have. The fact that if you transformed some of the numbers to different numbers (such as by multiplying them by $-1$) can yield a different correlation is immaterial -- if you had wanted to know that correlation with a transformed variable, you would have transformed the numbers before computing the correlation.

Correlations are between two variables; if you have groups of variables, then you will many correlations. For example, if there are five variables in group A and 3 in group B you will have 15 correlations.

Whether the values of the variables are positive, negative or a mixture, the correlation can be positive or negative and can be anywhere from -1 to +1. Correlation is a measure of the linear relationship between 2 variables. For example,

set.seed(1929100) #Pick random seed
x <- rnorm(1000) #Standard normal, mean = 0, sd = 1
y <- x^2 # Square
cor(x,y) #Correlation
plot(x,y)

there is a perfect relationship between x and y, but the correlation with this seed is -0.10


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ROUGH DRAFT authorea.com/99858

# TYC 8241 2652 1 and the case of the disappearing disk: no smoking gun yet $\star$$\star$footnotetext: Based on observations made with ESO telescopes at the Paranal Observatory (ESO program IDs 090.C-0697(A), 090.C-0904(A), and 095.C-0438(A)) and on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.

TYC 8241 2652 1 is a young star that showed a strong mid-infrared (mid-IR, 8-25 $$\mu$$m) excess in all observations before 2008 consistent with a dusty disk. Between 2008 and 2010 the mid-IR luminosity of this system dropped dramatically by at least a factor of 30 suggesting a loss of dust mass of an order of magnitude or more. So far there is no conclusive explanation for this observational fact; possibilities include removal of disk material by stellar activity processes, a collisional cascade that rapidly grinds dust of all sizes down to where radiative blowout is effective, or a run-away accretion event spurred by the presence of gaseous material in the disk. We present new X-ray observations, optical spectroscopy, near-IR interferometry, and mid-IR photometry of this system to constrain its parameters and identify the cause of the dust mass loss.

In X-rays TYC 8241 2652 1 has all properties expected from a young star: Its luminosity is in the saturation regime and the abundance pattern shows enhancement of O/Fe. The photospheric H$$\alpha$$ line is filled with a weak emission feature, indicating chromospheric activity consistent the observed level of coronal emission. Interferometry does not detect a companion and sets upper limits on the companion mass of 0.2, 0.35, 0.1 and 0.05 $$M_{\odot}$$ at a distance of 0.1-4 AU, 4-6 AU, 6-11 AU, and 11-34 AU, respectively. Our mid-IR measurements, the first of the system since 2012, are consistent with the depleted dust level seen after 2009.

The new data confirms that stellar activity is unlikely to destroy the dust in the disk and shows that scenarios where either TYC 8241 2652 1 heats the disk of a binary companion or a potential companion heats the disk of TYC 8241 2652 1 are higly unlikely.


text:
Browse Questions

# Find the equation of parabola if : Vertex $(1 , 4 );$ focus :$(-2 , 4 ).$

This is the fourth part of the multi-part question Q1

Toolbox:
• Standard parabolas :
• (i)$y^2=4ax$
• In this case the axis of symmetry is $x$-axis.
• The focus is $F(a,0)$ and the equation of the directrix is $x=-a$
• (ii)$y^2=-4ax$
• In this case the focus is $F(-a,0)$ and the equation of directrix is $x=a$
• (iii)$x^2=4ay$
• In this case the axis of symmetry is $y$-axis.
• The focus is $F(0,a)$ and the equation of directrix is $y=-a$
• (iv)$x^2=-4ay$
• In this case the focus is $F(0,-a)$ and the equation of directrix is $y=a$
• Shifting of origin from (0,0) to $(h,k)$ then if the new axes form $X-Y$ coordinate system then
• $x=X+a$
• $y=Y+k$
• The new coordinates of a point $P(x,y)$ become $X=x-a$,$Y=y-k$.
Step 1:
$VF=1+2=3$units=$a$
$VF$ is the axis whose equation is $y=4$(parallel to the $x$-axis).
Step 2:
Focus is on the -ve side of the vertex.
Therefore equation is of the form $y^2=-4ax$
Where $X=(x-1),Y=(y-4),a=3$
$(y-4)^2=-4\times 3(x-1)$
$\Rightarrow -12(x-1)$
$(y-4)^2=-12(x-1)$
This is the required equation.
edited Jun 23, 2013


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# Example for a basis of $\mathbb{R}^4$ which consists of several conditions

Let V be a vector space over $\mathbb{R}$ and let $B=\{v_1,v_2,v_3,v_4\}$ be its basis.
Let $T : V \to V$ be a linear transformation which satisfies the following condition:

1. $T(v_1) = T(v_2)$
2. $T(v_3) = T(v_4)$
3. $kerT \subseteq ImT$

Can you find an example for a basis which satisfies the above conditions?
(The question actually asks to prove that $kerT = ImT$, but the conditions which are presented here seem unreasonable.)

Take for example the standard basis:
$$T(e_1) = T(e_2) = e_1 \\ T(e_3) = T(e_4) = e_3$$ Thus the image is spanned by $\{e_1,e_3\}$, but if $kerT = ImT$, then $T(e_1) = 0$ !
We conclude that $ImT = \{0\}$ which isn't true according to the data of the question.
What am I missing?

Let $w_1=v_1-v_2,w_2=v_3-v_4$. Then $w_1,w_2\in\ker T$ and they are linearly independent. Hence $\dim\ker T\ge2$. By the rank-nullity theorem, $\dim\text{im}T=4-\dim\ker T\le2$. However $\ker T\subseteq\text{im}T$ implies $2\le\dim\ker T\le\dim\text{im}T\le2$ so we must have $\ker T=\text{im}T$.

To find an example of such an operator, note that you cannot have $Tv=v$ for any $v\ne0$, since this would imply $v\in\text{im}T=\ker T$ so $v=Tv=0$, a contradiction. Try instead $$Tv_1=Tv_2=v_1-v_2,\quad Tv_3=Tv_4=v_3-v_4.$$

• In the last example - Amongst $v1,v2,v3,v4$, who is in $kerT$ and who is in $ImT$? – Dor Feb 9 '15 at 7:39
• None. We have $\ker T=\text{im}T=\text{span}\{v_1-v_2,v_3-v_4\}$. – Jason Feb 9 '15 at 8:03

what about a transformation $T$ defined by $Te_1 = Te_2 = 0, Te_3 = e_1, Te_4 = e_2$ then a basis for both $ker T$ and $im T$ is $\{e_1, e_2\}$

we can fix $v$'s by $v_1 = e_1 + e_3, v_2 = -e_1 + e_3, v_3 = e_2+e_4, v_4=-e_2 + e_4.$

showing $\{v_1, v_2, v_3, v_4\}$ is linearly independent. suppose $av_1+ bv_2 + cv_3 + dv_4 = 0.$ then $a - b= 0, c+d = 0, a + b= 0, c-d = 0$. only solution is $a = b = c = d = 0.$ that shows the independence of the $v$'s.

we can verify that $Tv_1 = Tv_2 = Te_3 = e_1$ and $Tv_3 = Tv_4 = Te_4 = e_2.$

• I'm not sure that I understand the example with the $v$'s. Why does that "fix" them? (I understood why they are linearly independent). – Dor Feb 9 '15 at 7:41
• $e$'s satisfy your third condition. you need to satisfy the first two. $v_1, v_2$ have one element from the $ker$ and one from $im$ so that $Tv_1 = Tv_2$. same for $v_3, v_4.$ – abel Feb 9 '15 at 11:07


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 P: 5,462 Confused about thermodynamic quantities and their minimums. Not quite sure of your symbols here. E is not normally used for enthalpy. Assuming you are using F for what is normally called the Helmoltz free energy (or work function) and H for enthalpy, equation 38 should read. G = H - TS = F + PV. since there is some doubt as to the meaning of E I shall avoid it and use U for internal energy. So for a change at constant temperature along a reversible path dU = TdS + dwrev (Gibbs equation) At constant temp TdS = d(TS) so we can rewrite the above as d(U-TS) = dwrev (at constant T) Let F = A-TS then dF = dwrev (at const T) Now the expression wrev refers to all possible kinds of work. If only pressure/volume work is envisaged dwrev = PdV At const volume dV = 0 so dF = 0 so F is a minimum, at equilibrium. So the condition for equilibrium at constant temp and volume is that the Helmholtz free energy is a minimum and dF = 0. Now to develop a similar discussion for constant pressure Note that H = U + PV dH = dU + PdV + VdP Substitute from the above dH = TdS + dwrev + PdV + VdP or at constant T and P d(H-TS) = PdV + dwrev This new function is called the Gibbs free energy and is G = H - TS = F+PV So dG = PdV + dwrev Again restricting this to PV work dwrev, dwrev = -PdV So at const T and P dG = 0, Thus G is a minimum, at equilibrium. Now applying all that to my melting ice example. dG= 0 = d(H-TS) = dH - d(TS) = dH - TdS dS = dH / T The entropy of melting (fusion) can be calculated by dividing the enthalpy (=latent heat) by the melting point. You might like to look at this thread and in particular post#11 (Which no longer contains a typo) http://www.physicsforums.com/showthr...lmholtz&page=2 does this help?


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## anonymous 5 years ago solve the equation x+1=sqrt/19-x

1. anonymous

So I think you are trying to solve the equation: $x+1=\sqrt{19-x}$ Right? If not, let me know... To solve this equation you need to square both sides... the square cancels out the square root but you have to be careful! Squaring both sides of an equation can give solutions that aren't of the original equation. These are called extraneous solutions... $(x+1)^2=(\sqrt{19-x})^2$

2. anonymous

i dont kno how to write it step by step i dont get it

3. anonymous

O.k., $(x+1)^2= (\sqrt{19-x})^2$ $(x+1)(x+1)=19-x$ $x^2+x+x+1=19-x$ $x^2+2x+1=19-x$ -19 +x -19 +x $x^2+3x-18=0$ (x-3)(x+6)=0 so either x = 3 or x = -6 Now you have to check your answer in the original equation

4. anonymous

if x = 3, $x+1 = \sqrt{19-x}$ becomes $(3)+1 = \sqrt{19-(3)}$ which reduces to $4 = \sqrt{16}$ which is true so x = 3 is a solution. if x = -6, $x+1 = \sqrt{19-x}$ becomes $(-6)+1 = \sqrt{19-(-6)}$ which reduces to $x+1 = \sqrt{19-x}$ which becomes $-5 = \sqrt{25}$ which is -5 = 5 which isn't true so x is not equal to -6 Therefore your only solution is x = 3

5. anonymous

Is that enough steps? :)


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# Finding Limits and Determining Uniform Convergence

Define $f_n: \mathbb{R} \to \mathbb{R}$ by $f_n(x)= \dfrac{n+\cos(x)}{10n+\cos^2(x)}$.

(a) Find the limit $f(x) = \lim\limits_{n \to \infty} f_n(x)$

(b) Does $f_n\to f$ converge uniformly on $\mathbb R$?

Not quite sure how to start this? I can figure out the limit but not sure how to show whether it converges uniformly...

• What do you have for $f$? Then what is $\sup_{x \in \mathbb{R}} \left| f_n(x) - f(x) \right| = \|f_n - f\|_\infty$? – Jon Warneke Dec 2 '15 at 2:04
• For all $x$ we have $(n-1)/(10n+1)\leq f_n(x)\leq (n+1)/10n.$ – DanielWainfleet Dec 2 '15 at 16:21

When , $x=(2n-1)\frac{\pi}{2}$ then $\cos x=0$ and then $f_n(x)\to \frac{1}{10}$.
When , $x\not=(2n-1)\frac{\pi}{2}$ then $0<|\cos x|\le 1$. Then , $f_n(x)\to \frac{1}{10}$.
Hence , $\displaystyle f(x)=\lim_n f_n(x)=\frac{1}{10}$.
Now , $\displaystyle \sup_n|f_n(x)-f(x)|\to 0$ as $n\to \infty$. So , the sequence of function $\{f_n(x)\}$ converges uniformly.


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# Who is Mr. T and where does he work?

Part 1:

Write each answer as ONEWORD, without spaces, punctuation or abbreviating. Some questions may have more than one answer when taken alone, but only certain answers will work with respect to the whole puzzle.

1: Where can you find the easternmost traffic light in the United States, including territories and possessions?

2: What is the longest irregular verb?

3: Based on what you saw last month (December 2020), which is the most visible planet in our solar system?

4: Which four consecutive letters in the alphabet can be rearranged to form a familiar word, with each letter appearing at least once?

5: What’s a lung disease that comes from breathing in very fine ash and sand dust?

6: Name an 11-letter monosyllabic word.

7: Which two letters from the English alphabet are the least frequently used in the Spanish language?

8: What word means “to exalt”, and has an anagram that means “to instruct”?

Part 2:

145 46 365 415 32 43 82 58 81 48 24 22 105 96 58 27 21 275 17 16

Part 3:

Who is Mr. T and where does he work?

Notes:

Partial answers are encouraged, as this may be tricky to solve alone, but the checkmark will go to whomever answers all parts correctly. When you’ve got it all figured out, copy from others if needed (and give credit) to form a complete answer.

• How do i format an answer to make it so people dont have to see any questions i answer if they dont want to Jan 15 at 20:32
• @Benja You can put the content inside spoilers. The syntax can in some cases be annoying, but generally it's >! your text. If you need a multi-line spoiler, you either need to add two spaces or a <br> at the end of the line. Jan 15 at 20:34

## 3 Answers

I believe that the theme is related to being extreme (easternmost, longest, most visible, etc.) so the answers that i came up with for some of the questions are: 5.pneumonoultramicroscopicsilicovolcanoconiosis(the longest word in any major dictionary) 6. squirrelled

1: Probably ALASKA; it extends westward into the eastern hemisphere
3: From my experiences, EARTH
4: rstu => RUST

1. The longest irregular verb I can find has 13 letters:

Misunderstand

1. The letters are:

K and W


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Article Text

Original research
Preterm birth and stillbirth rates associated with socioeconomic disparities during COVID-19 pandemic: a population-based cross-sectional study
1. Laila Aboulatta1,
2. Kaarina Kowalec1,2,
3. Christine Leong1,3,
4. Joseph A Delaney1,4,
5. Jamie Falk1,
6. Silvia Alessi-Severini1,5,
7. Dan Chateau6,
8. Qier Tan5,
9. Katherine Kearns7,
10. Christina Raimondi7,
11. Christine Vaccaro1,
12. Alekhya Lavu1,
13. Lara Haidar1,
14. Payam Peymani1,
15. Sherif Eltonsy1,8
1. 1College of Pharmacy, Rady Faculty of Health Sciences, University of Manitoba, Winnipeg, Manitoba, Canada
2. 2Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden
3. 3Department of Psychiatry, Rady Faculty of Health Sciences, University of Manitoba, Winnipeg, Manitoba, Canada
4. 4Department of Epidemiology, University of Washington, Seattle, Washington, USA
5. 5Manitoba Centre for Health Policy, University of Manitoba, Winnipeg, Manitoba, Canada
6. 6College of Health and Medicine, Australian National University, Canberra, Australian Capital Territory, Australia
7. 7Department of Family Medicine, College of Medicine, University of Manitoba, Winnipeg, Manitoba, Canada
8. 8Children's Hospital Research Institute of Manitoba, Winnipeg, Manitoba, Canada
1. Correspondence to Dr Sherif Eltonsy; Sherif.Eltonsy{at}umanitoba.ca

## Abstract

Background Conflicting evidence exists on the impact of the COVID-19 pandemic restrictions on preterm birth (PTB) and stillbirth rates. We aimed to evaluate changes in PTB and stillbirth rates before and during the pandemic period and assess the potential effect modification of socioeconomic status (SES).

Methods Using the linked administrative health databases from Manitoba, Canada, we conducted a cross-sectional study among all pregnant women, comparing 3.5 years pre-pandemic (1 October 2016 to 29 February 2020) to the first year of the pandemic (1 March 2020 to 31 March 2021). We used generalised linear models to assess the quarterly rates of PTB (<37 weeks) and stillbirths. We calculated the predicted trends based on pre-pandemic period data. Finally, we evaluated the PTB and stillbirth rates among lower and higher SES pregnant women (average annual household income) using subgroup analysis and interaction models.

Results We examined 70 931 pregnancies in Manitoba during the study period. The risk of PTB increased by 7.7% (95%CI 1.01 to 1.13) and stillbirths by 33% (95% CI 1.08 to 1.64) during the pandemic period. Following COVID-19 restrictions implemented in March 2020, there were increases in the quarterly rates of both PTB (immediate increase (β2)=1.37; p=0.0247) and stillbirths (immediate increase (β2)=0.12; p=0.4434). Among the lower income groups, the pandemic restrictions resulted in an immediate relative increase in PTB and stillbirth rates by 20.12% (immediate increase (β2)=3.17; p=0.0057) and 27.19% (immediate increase (β2)=0.48; p=0.0852). However, over the pandemic, the overall PTB rate significantly decreased as a rebound effect by 0.85% per quarter (p=0.0004), whereas the overall stillbirth rate did not decrease significantly (slope decrease (β3) =−0.01; p=0.8296) compared with the pre-pandemic period. The quarterly rates during the pandemic among the higher income group decreased by 0.39% (p=0.1296) for PTB and increased by 0.07% (p=0.1565) for stillbirth. We observed an effect modification by SES for PTB rates (p=0.047).

Conclusion While the onset of COVID-19 pandemic restrictions was not associated with significant effects on stillbirth rates, we observed an immediate and rebound effect on PTB rates. The impact of COVID-19 on preterm birth was dependent on SES, with higher influence on families with lower SES. Further studies are needed to detect future trend changes during pandemic waves after 2021 and assess potential underlying mechanisms.

• COVID-19
• epidemiology
• neonatology

## Data availability statement

Data cannot be shared publicly because of privacy restrictions. Data are available from the University of Manitoba and MCHP Institutional Data Access/Ethics Committee (contact via https://umanitoba.ca/manitoba-centre-for-health-policy/) for researchers who meet the criteria for access to confidential data.

This is an open access article distributed in accordance with the Creative Commons Attribution Non Commercial (CC BY-NC 4.0) license, which permits others to distribute, remix, adapt, build upon this work non-commercially, and license their derivative works on different terms, provided the original work is properly cited, appropriate credit is given, any changes made indicated, and the use is non-commercial. See: http://creativecommons.org/licenses/by-nc/4.0/.

## Request Permissions

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#### WHAT IS ALREADY KNOWN ON THIS TOPIC

• COVID-19 restriction measures have been associated with either a decrease, increase or no change in the rate of preterm birth (PTB) and stillbirth.

• Scarce evidence exists on the association between these changes and maternal socioeconomic status during the pandemic.

• This study highlights that the impact of the restrictions on PTB differed by maternal socioeconomic status.

• We observed an increase in PTB rates among the lower income group followed by return to pre-pandemic averages within the following three quarters.

#### HOW THIS STUDY MIGHT AFFECT RESEARCH, PRACTICE OR POLICY

• Our findings provide an additional insight into the complex aetiologies that can be attributed to the increased incidence of preterm and stillbirth deliveries, revealing the differential impacts of pandemic restrictions.

## Introduction

Provincial and local governments across Canada have cooperated to impose measures to reduce the COVID-19 burden.17 Studies in two Canadian provinces, Ontario and Alberta, examined the influence of the COVID-19 first wave (~March 2020) lockdown measures on perinatal health outcomes and found non-significant changes in both PTB and stillbirth rates compared with pre-pandemic period.18–21 In Manitoba, different pandemic strategies were implemented affecting antenatal care services which could have influenced the health of pregnant women and their infants.22–27 The policy responses adopted in Manitoba and the low rates of COVID-19 infection provide a unique opportunity to serve as a natural experiment to determine their effects on perinatal outcomes.28 We, therefore, aimed to assess the impact of the pandemic restrictions implemented during the first and second waves in Manitoba on the incidence of PTB and stillbirth. In addition, we aimed to evaluate if the effect of the COVID-19 restrictions was modified by the maternal SES.

## Methods

### Data source

We used data from the Manitoba Population Research Data Repository at the Manitoba Centre of Health Policy (MCHP) to conduct the study. The Repository is a secure data-rich environment containing person-level health information on virtually the entire population of Manitoba.29 All records are de-identified and linkable at individual and family levels using a scrambled health number. The validity and reliability of the MCHP Repository for epidemiological studies has been previously reported.30–32 We used the following databases: (1) the Manitoba Health Insurance Registry (date of birth, sex); (2) hospital abstracts and physician visits (to identify the perinatal outcomes of interest); (3) Drug Program Information Network (prescription drug data to identify comorbidities); (4) Manitoba laboratories (for COVID-19 testing data); (5) postal codes from the Canada Census (to distinguish urban from rural regions); (6) census data for income quintiles based on ranges of mean household income, and grouped into five categories with each quintile assigned to approximately 20% of the population (quintile 1 (mean income=$C17 910) to quintile 5 (mean income=$46 230)).

### Study design and population

We conducted a population-based cross-sectional study using the linked data from Manitoba to create a large, pregnancy cohort pre-COVID-19 and during COVID-19. All pregnancies (live birth or stillbirth) in Manitoba were included in the final cohort during the pre-pandemic period (Q4-2016 to Q1-2020) and during the pandemic period (Q2-2020 to Q1-2021). The delivery date was defined as the date that a procedure code of delivery was recorded in the database. Maternal age at delivery, pre-existing comorbid diseases (pre-validated definitions of asthma,33 diabetes34 and hypertension35), income (quintiles),36 parity (primipara and multipara) and residence (urban and rural) were extracted from the database. As a proxy for the maternal SES, we used the pre-validated classification based on the median neighbourhood income quintile: lower income (individuals in the lowest and second lowest median neighbourhood income quintile), higher income (individuals residing in the neighbourhoods with the three highest median neighbourhood income quintiles) and income unknown (individuals who cannot be assigned a neighbourhood income from the census data—this category includes individuals residing in facilities such as personal care homes, psychiatric facilities, prisons or wards of the Public Trustee and Child and Family services).37

### Exposure and outcomes

Pregnant women exposed to the pandemic restrictions during the first and second waves between 1 March 2020 and 31 March 2021 were compared with those who were pregnant before the pandemic period. A strict mitigation strategy was implemented in Manitoba from 13 March 2020 (first wave) followed by a subsequent ease of some restrictions between June and July 2020. More restrictions were applied in August 2020, with Manitoba meeting their peak lockdown measures during the second wave.25 38 However, we did not investigate the differences in the effect of exposure to pandemic restrictions by trimester. Preterm birth was defined as a live birth where an infant is born at less than 37 weeks’ gestation,39 while stillbirth was identified as a fetal death with a gestation of 20 weeks or greater.40

### Statistical analysis

We examined pregnant women demographics such as age (≤19 years, 20–34 years and >35 years), area of residence, income and comorbidities that were described before and during the pandemic. We estimated the quarterly rates of both preterm birth which was defined as the number of preterm deliveries within each quarter during the study period with the total number of births during the quarter used as denominator. Stillbirth rate was defined as the number of stillbirths within each quarter during the study period with the total number of births during the quarter used as denominator. We chose Q2-2020 as the intervention point for the COVID-19 period since the restriction measures started to impact clinical practice in Manitoba by the end of Q1-2020 (March 2020). We calculated the relative risk (RR) and 95% CI for the preterm births and stillbirth rates for the overall pregnancies and by subgroups.

The effect of restriction measures on the preterm birth and stillbirth rates was evaluated using interrupted time series analysis through generalised linear models, to assess both the immediate impact and rebound effect of the COVID-19 pandemic on the quarterly outcome rates within three subgroups: (1) overall incidence, (2) among lower income group and (3) among higher income group.41 The immediate impact was estimated by the immediate change in level from pre-COVID-19 to post-COVID-19 interruption. Relative percentage change was examined, which is calculated as the relative change in the percentage of rates between Q1-2020 and Q2-2020. The rebound effect was estimated using the change in slope from pre-interruption to post-interruption in the quarterly PTB and stillbirth rates. If an interruption or intervention (COVID-19) has an effect on the outcomes, then a change in level (β2) and/or slope (β3) should be detected between pre-phase and post-phase.42 In addition, we used generalised linear models to test for the potential effect modification by maternal SES using multiplicative interactions.43 Sensitivity analysis was conducted, including monthly rates of preterm birth and stillbirth. To assess bias due to seasonal variability that is known to affect pregnancy outcomes, we conducted the analysis using ARIMA models, accounting for autocorrelation between consecutive quarterly observations.44 45 We also generated forecast models using the autoregressive moving average analysis to compare the observed quarterly PTB and stillbirth rates during the pandemic with the counterfactual expected rates in the absence of the pandemic. We opted to use this method as it provides superior prediction accuracy (using autocorrelations and moving averages over residual errors) and comprehension of the data patterns. Analyses were conducted using SAS V.9.4 (SAS Institute).

There was no patient or public involvement in this study

## Results

During the study period, we included 70 931 pregnant women: 54 306 in the pre-pandemic period and 16 625 during the pandemic period. Of the pregnant women included, the mean age was 29.4 years (SD 5.61) pre-pandemic and 29.7 years (SD 5.56) during the pandemic and over half lived in urban areas in both groups. Twenty five per cent of the pregnancies were among women in the lowest income quintile, while 15% of pregnancies were among women in the highest quintile. Asthma (24%), hypertension (8%) and diabetes (6%) were the most common comorbidities in the pregnant women (table 1).

Table 1

Maternal characteristics before and during COVID-19 periods in the Canadian province of Manitoba

### Preterm birth

Figure 1 illustrates the trends of the observed PTB rates during the study period. The average rates of PTB were 8.77% and 9.45% before and during the pandemic period, respectively, with a relative increase of 7.7% (95% CI 1.01 to 1.13). The implementation of mitigation measures was associated with a 4.5% relative increase in PTB rates (immediate increase (β2)=1.37; p=0.0247). Furthermore, we observed a rebound decrease by 0.85% (p=0.0004) among the quarterly PTB rates (table 2).

Figure 1

Observed preterm birth (PTB) rates among lower income, higher income and all pregnant women in Manitoba from Q4-2016 until Q1-2021.

Table 2

Preterm birth and stillbirth outcomes before and during the pandemic periods in the Canadian province of Manitoba

The risk of PTB deliveries during the pandemic have increased with a RR of 1.08 (95% CI 1.00 to 1.16) among the lower income group and RR of 1.05 (95% CI 0.97 to 1.15) among the higher income group. Before the pandemic period, we found a 30.2% relative difference between lower and higher income groups (10.05% vs 7.72%). In the lower income group, 10.86% of women had preterm delivery compared with 8.13% in the higher income group during the pandemic, with a relative difference of 33.6% (table 2).

Among pregnant women with lower income, the onset of pandemic measures was associated with a 20.2% relative increase in PTB rates (immediate increase (β2)=3.17; p=0.0057), while a 12.2% relative reduction was observed among higher income women (immediate decrease (β2)=−0.33; p=0.6535). Moreover, we observed a rebound decrease in the preterm deliveries among the lower income group (1.33% decrease quarterly; p=0.0013) (table 2). Online supplemental figure S1 shows the relative percentage change of preterm birth during the study period.

### Stillbirth

Figure 2 demonstrates the trends of the observed stillbirth rates pre-pandemic and during the pandemic periods. The average rates of stillbirth were 0.58% and 0.77% before and during the pandemic period, respectively, with a relative increase of 33% (95% CI 1.08 to 1.64). Immediately after the onset of the pandemic, there was a relative 18.6% increase in the overall stillbirth rates (immediate increase (β2)=0.12; p=0.4434). A minor non-significant rebound decrease among the quarterly stillbirth rates was observed (slope decrease (β3)=−0.01, p=0.8296) (table 2).

Figure 2

Observed stillbirth rates among lower income, higher income and all pregnant women in Manitoba from Q4-2016 until Q1-2021.

The risk of stillbirths during the pandemic have increased with a RR of 1.46 (95% CI 1.11 to 1.93) among the lower income group and RR of 1.18 (95% CI 0.85 to 1.63) among the higher income group. Before the pandemic period, we found a 27.5% relative difference between lower and higher income groups, 0.65% versus 0.51%, respectively. Among pregnant women with lower income, the rate of stillbirth was 0.95% compared with 0.60% in the higher income group during the pandemic, with a relative difference of 58.3%. Within the income subgroups, the rates of stillbirth deliveries did not differ significantly between the pandemic and pre-pandemic periods (table 2). Online supplemental figure S2 shows the relative percentage change of stillbirth during the study period.

### SES effect modification

We assessed the interaction between COVID-19 restrictions and income (as a proxy for SES) on the overall incidence of PTB and stillbirth. Our models suggested an effect modification by income for the overall incidence of PTB (p=0.047). Although there was an increase in stillbirth rates among the lower income group, the income effect modification was not significant (p=0.643, online supplemental table S1).

### Sensitivity analysis

Examining the effect of restrictions using the monthly rates of PTB and stillbirth was conducted, which provided larger number of observation points to analyse. However, given the small numbers of monthly stillbirth rates in Manitoba, the analysis was not conducted. Monthly PTB rates analysis provided similar results to the quarterly rates (online supplemental table S2). Online supplemental figures S3 and S4 illustrate the trends of the observed versus expected PTB and stillbirth rates during the study period. During the first quarter of 2021, the absolute differences in the observed and expected overall PTB rate was 2.05% compared with 1.89% and 1.64% among the lower and higher income groups, respectively. However, the absolute difference in the observed and expected stillbirth rates were 0.04%, 0.69% and 0.17% among the overall, lower income and higher income groups, respectively.

## Discussion

In this Canadian province-wide study, we observed a significant increase in the overall rates of PTB immediately after the onset of COVID-19 restrictions. The observed increase in the PTB rate could be considered clinically significant, followed by a return to pre-pandemic averages within the following three quarters. Such a trend was observed among stillbirths but was hardly detected due to the low sample size. We also observed a 20% relative increase in PTB rates among the lower income group immediately after the measures were applied. Such an impact was not observed within the higher SES group. Our data suggest that the pandemic restrictions had a rebound effect on the overall PTB rates, mostly attributed to the impact within the lower income group.

While the exact reasons for the observed increase followed by reversal to pre-pandemic level have yet to be determined, various social or behavioural changes during the pandemic lockdown may have contributed to the change in the overall PTB rates.46 Our study shows that changes in PTB rates are inconsistent among pregnant women with different SES. Our results support the hypothesis that the higher income group has some social privilege as increased social support, reduced work-related or social stress, decreased anxiety, reduced exposure to environmental pollutants and better hygiene measures which can minimise the impact of the pandemic measures.11 47–50 On the other hand, pandemic restrictions among the lower SES group resulted in financial insecurities, increased stress and mental health concerns, and changes in maternity care practice that affected maternal health leading to higher PTB rates.7 19 51 52

Evidence indicates that the COVID-19 pandemic and associated lockdown measures exacerbated existing social inequalities.7 53 However, in a study published in the Netherlands,47 no association between COVID-19 restrictions and adverse perinatal outcomes were observed while accounting for SES as an effect modifier. Caniglia et al showed a significant reduction in PTB rates post-lockdown period compared with during lockdown, highlighting that lockdown restrictions may have a delayed effect on adverse pregnancy outcomes.54 In the largest city in the Canadian province of Alberta (Calgary), Alshaikh et al reported that during the first-wave lockdown, stillbirth and PTB rates remained unchanged; however, very preterm birth rate (<32 weeks of gestation) dropped significantly.19 Three retrospective cohort studies, in the Canadian province of Ontario, examined the association between the pandemic first-wave and perinatal outcomes, and reported non-significant changes in PTB and stillbirth rates.18 20 21 Differences between these studies and the current study may be attributable to differences in the analytical methods performed, the pandemic time frame examined and the pandemic measures implemented among the provinces. Our study assessed 12 months during the pandemic period (up to March 2021), whereas these studies examined the period until October and December 2020. In addition, previous studies used databases which covered only maternal and neonatal data until discharge, whereas the current study provides more generalisable results due to the use of a province-wide database that includes the entire Manitoban pregnant population without restrictions of insurance coverage. In addition, the effect modification by SES was not assessed in the aforementioned studies.18–21

In a meta-analysis of 36 studies, Yang et al found a non-significant reduction in the odds of PTB during the pandemic period in national studies (unadjusted OR 0.99, 95% CI 0.94 to 1.03); however, a significant reduction was observed in single-centre studies (unadjusted OR 0.99, 95% CI 0.86 to 0.94).55 Moreover, there was no difference in stillbirth rates between pre-pandemic and pandemic periods in both single-centre and national studies.55 In another meta-analysis of 17 studies, Vaccaro et al observed a significant association between lockdown restrictions and an increased risk of stillbirth (RR=1.33, 95% CI 1.04 to 1.69), but not preterm birth (RR=0.93, 95% CI 0.84 to 1.03).56

Our study has major strengths to mention. We used province-wide administrative health data which included the entire pregnant population in the province of Manitoba. Furthermore, we included 1 year of the pandemic period, thus facilitating the investigation of the association between COVID-19 restrictions and the examined outcomes. In addition, we used different model specifications to account for autocorrelation that could influence the study outcomes and we used interrupted time series analysis which minimised the impact of the measured and unmeasured confounding factors. The limitations of this study should also be acknowledged. We did not examine the influence of COVID-19 infection on the study outcomes; however, the number of reported COVID-19 positive pregnant women was small in relation to the total pregnancies in our cohort, and it is unlikely that viral infection would change our results.46 Although we did not investigate variations across the different regions of Manitoba, the changes to maternal healthcare were implemented across the provincial healthcare system minimising any differential variability within the data. However, some rural regions and those of lower SES may have been differentially impacted during the pandemic compared with urban areas and larger cities. Given that pandemic restrictions disrupted the health systems and lifestyle of pregnant women, we used interrupted time series analysis to assess the immediate and long-term (rebound) effect of the pandemic restrictions. We interpreted the immediate change (β2) as representing a homogeneous group of newborns of deliveries in a span of 3 months (one quarter). As restrictions continue to impact perinatal healthcare, this might lead to additional cases of preterm/stillbirth, represented in a group of newborns experiencing those restrictions at varied pregnancy stages and periods, creating a rather heterogeneous group. However, we observed a rebound to pre-pandemic baseline levels through the change in slope results. Importantly, causal interpretation of restrictions cannot be inferred from this study, as women were not only exposed to restrictions but also to the virus and other factors—so results should be interpreted with caution. Moreover, we could not consider the following: births occurring outside of hospitals, maternal smoking and alcohol/substance use, and vaccination rates among pregnant women in Manitoba. We used quarterly rates for our main analysis, and monthly stillbirth rates analysis was not feasible. In a sensitivity analysis, monthly PTB rates showed similar results, although additional noise was introduced. The study power was limited at instances due to the short follow-up period; however, our sensitivity analysis supported the primary model’s results.57

## Conclusion

The initial implementation of COVID-19 restrictions was associated with an increase in the rates of preterm births, followed by reversion to the pre-COVID-19 level. The pandemic has revealed important maternal socioeconomic disparities, with significantly higher rates of PTB among lower income pregnant women. The restrictions did not coincide with significant changes in the stillbirth rates in Manitoba. Further studies are needed to monitor trend changes during subsequent pandemic waves and assess potential underlying mechanisms.

## Data availability statement

Data cannot be shared publicly because of privacy restrictions. Data are available from the University of Manitoba and MCHP Institutional Data Access/Ethics Committee (contact via https://umanitoba.ca/manitoba-centre-for-health-policy/) for researchers who meet the criteria for access to confidential data.

## Acknowledgments

The authors acknowledge the Manitoba Centre for Health Policy for use of data contained in the Manitoba Population Research Data Repository under project (HIPC#2020/2021-33). Data used in this study are from the Manitoba Population Research Data Repository housed at the Manitoba Centre for Health Policy, University of Manitoba and were derived from data provided by Manitoba Health.

• ## Supplementary Data

This web only file has been produced by the BMJ Publishing Group from an electronic file supplied by the author(s) and has not been edited for content.

## Footnotes

• Contributors SE, KK and LA conceptualised and designed the study, drafted the manuscript, carried out the initial analyses and interpretation, and reviewed and revised the manuscript. JAD, SA-S, JF, CL, DC, KK and CR contributed to study conception and design, and reviewed and critically revised the manuscript. QT assisted in data acquisition and data analyses, and critically revised the manuscript. PP, AL, LH and CV contributed to the study design, and reviewed and critically revised the manuscript. All authors approved the final manuscript as submitted and agree to be accountable for all aspects of the work. SE is responsible for the overall content as the guarantor.

• Funding This work was supported by a research grant from Research Manitoba.

• Disclaimer The results and conclusions are those of the authors and no official endorsement by the Manitoba Centre for Health Policy, Manitoba Health or other data providers is intended or should be inferred.

• Competing interests None declared.

• Patient and public involvement Patients and/or the public were not involved in the design, or conduct, or reporting, or dissemination plans of this research.

• Provenance and peer review Not commissioned; externally peer reviewed.

• Supplemental material This content has been supplied by the author(s). It has not been vetted by BMJ Publishing Group Limited (BMJ) and may not have been peer-reviewed. Any opinions or recommendations discussed are solely those of the author(s) and are not endorsed by BMJ. BMJ disclaims all liability and responsibility arising from any reliance placed on the content. Where the content includes any translated material, BMJ does not warrant the accuracy and reliability of the translations (including but not limited to local regulations, clinical guidelines, terminology, drug names and drug dosages), and is not responsible for any error and/or omissions arising from translation and adaptation or otherwise.


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## 27.2 Hypotheses and notation: One mean

The decision making process begins by assuming that the population mean internal body temperature is $$37.0^\circ\text{C}$$.

The sample mean $$\bar{x}$$ is likely to be different for every sample (sampling variation). The sampling distribution of $$\bar{x}$$ describes how the value of $$\bar{x}$$ varies from sample to sample. Because $$\bar{x}$$ varies, the sample mean $$\bar{x}$$ probably won’t be exactly $$37.0^\circ\text{C}$$, even if $$\mu$$ is $$37.0^\circ\text{C}$$.

If $$\bar{x}$$ is not $$37.0^\circ\text{C}$$, two broad reasons could explain why:

1. The population mean body temperature is $$37.0^\circ\text{C}$$, but $$\bar{x}$$ isn’t exactly $$37.0^\circ\text{C}$$ due to sampling variation (that is, the sample mean varies and is likely to be different in every sample); or
2. The population mean body temperature is not $$37.0^\circ\text{C}$$, and the sample mean body temperature reflects this.

These two possible explanations are called hypotheses. More formally, the two hypotheses above are:

1. The null hypothesis ($$H_0$$): $$\mu=37.0^\circ\text{C}$$; the population mean body temperature is $$37.0^\circ\text{C}$$; and
2. The alternative hypothesis ($$H_1$$): $$\mu \ne 37.0^\circ\text{C}$$; the population mean body temperature is not $$37.0^\circ\text{C}$$.

Since the null hypothesis is assumed true, the evidence is evaluated to determine if it is supported by the data, or not.

Note that the alternative hypothesis asks if $$\mu$$ is $$37.0^\circ\text{C}$$ or not: the value of $$\mu$$ may be smaller or larger than $$37.0^\circ\text{C}$$. Two possibilities are considered: for this reason, this alternative hypothesis is called a two-tailed alternative hypothesis.


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# The Flying Colours Maths Blog: Latest posts

## How the Mathematical Ninja approximates $\ln(5)$

"Isn't it somewhere around $\phi$?" asked the student, brightly. "That number sure crops up in a lot of places!" The Mathematical Ninja's eyes narrowed. "Like shells! And body proportions! And arrawk!" Hands dusted. The Mathematical Ninja stood back. "The Vitruvian student!" The student arrawked again as the circular machine he

## Ask Uncle Colin: A Separable Difficulty

Dear Uncle Colin, I have an equation $3y, \dydx =x$. When I separate and integrate both sides, I end up with $\frac{3}{2}y^2 = \frac{1}{2}x^2$, which reduces to $y = x\sqrt{\frac{1}{3}}+c$. With the initial condition $y(3) = 11$, I get $y = x\sqrt{\frac{1}{3}}+11-3\sqrt{\frac{1}{3}}$, but apparently this is incorrect. What am I

## A common problem: decimal division

I'm a big advocate of error logs: notebooks in which students analyse their mistakes. I recommend a three-column approach: in the first, write the question, in the second, what went wrong, and in the last, how to do it correctly. Oddly, that's the format for this post, too. The question

## Ask Uncle Colin: Are normals… normal?

Dear Uncle Colin, I don't understand why the normal gradient is the negative reciprocal of the tangent gradient. What's the logic there? — Pythagoras Is Blinding You To What's Obvious Hi, PIBYTWO, and thanks for your message! My favourite way to think about perpendicular gradients is to imagine a line

## From Euclid to Cantor

One of my favourite quotes is from Stefan Banach: "A good mathematician sees analogies between theorems. A great mathematician sees analogies between analogies." This post is clearly in the former camp. I'm fairly sure it's a trivial thing, but it's not something I'd noticed before. One of the first serious


text: A spherical balloon with a diameter of $10 m$, shown in the figure below is used for advertisements. The balloon is filled with helium ($R_{He}$ = $2.08 \: kJ/kg.K$) at ambient conditions of $15^\circ C$ and $100 kPa$. Assuming no disturbances due to wind, the maximum allowable weight (in newton) of balloon material and rope required to avoid the fall of the balloon ($R_{air} = 0.289$ $kJ/kg.K$) is _______


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Unlimited Plugins, WordPress themes, videos & courses! Unlimited asset downloads! From \$16.50/m

# Ruby for Newbies: Working with DataMapper

Difficulty:BeginnerLength:MediumLanguages:
This post is part of a series called Ruby for Newbies.
Ruby for Newbies: Working with Gems
Ruby for Newbies: Regular Expressions

Ruby is a one of the most popular languages used on the web. We've started a new Session here on Nettuts+ that will introduce you to Ruby, as well as the great frameworks and tools that go along with Ruby development. Today, we'll look at the DataMapper gems to get up and running with a database in Ruby.

## Step 0: Introducing DataMapper

DataMapper is an ORM: an Object-Relational Mapping. Basically, it’s a library that lets you work with your database from object-oriented code. There’s absolutely no SQL in this tutorial at all. However, an ORM uses a regular database under the covers; we’ll be using sqlite3 today, but you could just use a different adapter to work with a mysql, postgresql or other database.

In Singing with Sinatra - The Recall App, Dan Harper introduced you to DataMapper. In this tutorial, we’re going to take a deeper dive in to working with the library.

## Step 1: Installing the Right Gems

The first step is installing the required gems. The DataMapper functionality is broken into many different gems, so you’ll have to install several different parts. Of course, we’re not going to work with it all; but these are the gems you’ll have to install.

• sqlite3: This is the database gem.
• dm-core: This is the core functionality of DataMapper.
• dm-migrations: This gem does the database migration.
• dm-validations: As you’ll guess, this offers data validation functionality.
• dm-timestamps: Helps with timestamping database records.
• dm-sqlite-adapter: This is the adapter that connects DataMapper to your database; we’ll be using sqlite here, but you can use the dm-postgres-adapter, dm-mysql-adapter, or whatever suits your fancy.

Once you’ve got all those gems installed (see the last chapter if you need to know how to install gems), we’re ready to go.

## Step 2: Creating a Basic Model

Let’s start by creating a basic model. Models are defined in classes. However, we first have to connect to our database.

Actually, the very first thing is requiring our libraries at the top of our file.

So now that we have DataMapper in the environment, let’s connect to the database.

The first parameter tells DataMapper to use the default adapter for the database type. The second is the link / URL for the database. Since we’re using sqlite, we’re just linking to a database file. Note that we don’t have to create this file; DataMapper will create it for us.

Now we’re ready to create the model. As you know, this is a class.

The first step is to include the DataMapper::Resource module. This gives you the custom methods you’ll use in your class. The most important method here is property. Here, we’re using it to create three different properties: an id, a username, and an email. As you see, the first parameter in property is a symbol that’s the name of the property. The second is the type. You understand String, of course, but what’s serial. Actually, property :id, Serial is DataMapper’s shorthand for the primary key; ‘serial’ is an auto-incrementing integer. That’s your primary key!

## Step 3: Migrating the Database

Now that we’ve created our model, we need to migrate the database. If you’re not familiar with migrating a database, it’s the process of changing the schema of the database. This could be adding a column, renaming a column, or changing properties of a column. DataMapper offers two ways to do this:

The difference here is that auto_migrate! will clear all the data from the database; the auto_upgrade! methods tries to reconcile what’s in the database already with the changes you want to make. The way this works is that after your model class, you’ll call one of these methods. You don’t want to be running auto_migrate! every time you load the model, of course, but you might want to run auto_upgrade! on every reload in development. I’ve done it this way in Sinatra:

You’ll notice that so far, we haven’t had to touch a single SQL query; that’s the point of using on ORM is that you can write normal code and have that work with relational databases.

Now that we have our feet wet with DataMapper, let’s take our model to another level. Let’s start with timestamps.

### Timestamps

We’re requiring the dm-timestamps gem, so why not use it? If we add ‘created_at’ and ‘updated_at’ properties to the model, this gem will automatically update those fields.

Of course, you don’t need to add both, if you don’t want them.

### Options

There are several options that you can add to each field. For example, if you want a field to be required, or unique, or have a default value, you can do that there. Let’s create a post model to showcase some of this:

We’re mixing things up a bit here; our ‘title’ and ‘body’ are required fields. We’re defining the ‘slug’ property as the primary key, and saying that it must be a unique index. Don’t get scared off by the default value of ‘slug.’ Of course, you can just use a raw value of whatever type your property is, but we’re doing something more. Ruby (and other languages) has lambdas, which you could think of as a small function. It’s something that can take “parameters” and return a value, just like a function. If we use a lambda as the value of the ‘default’ property, DataMapper will pass it the resource (or database record you’re working with) and the property itself (in this case, ‘slug’). So here, what we’re doing is taking the value in resource.title (the title property), putting it in lowercase, and using gsub method (think global substitution) to switch every space to a dash. This way, something like this:

Will become this:

Note: Don’t get confused with how we’re using options here. First of all, remember that when a hash is the last parameter of a method, we don’t need to add the curly braces. Also, with Ruby 1.9, there’s a new hash syntax. Previously, hashes looked like this:

You can still do this in 1.9, and you have to if you you’re not using symbols as your keys. But, if you are using symbols as keys, you can do this instead:

Basically, you just move the colon to the end of the symbol (no space!) and remove the rocket.

### Validations

There’s a lot you can do with validation in DataMapper, and you can read all about it here. However, let’s take a look at the basics.

There are two ways to do validations; we’re going to use the method that adds your validations to the options hash. For the email property in the User model, we’ll set the format validation:

In this case, we’re using a built-in regex that DataMapper offers; we could put a custom regex there if we wanted something else.

Let’s require a certain length on the password:

If you’re not familiar with the 10..255 notation, that’s a Ruby range. We're saying that the password must be between 10 and 255 characters long.

### Associations

How about foreign keys? DataMapper makes this real easy. Let’s associate our User and Post models. We want a user to be able to have many posts, and a post to belong to a user.

In the User model, add this line

Then, in the Post model, do this:

In the database, this adds a user_id property to a post table. In practice, it’s really easy; we’ll see this soon.

### Custom Property Accessors

If you want to customize the input for a given property, you can add custom property accessors. For example, let’s say we want to make sure a user’s username is always stored in lowercase. We can add property accessor methods similar to the way you would in a normal class. This way, we take the value the user is trying to store and fix it up. Let’s do this:

We’re defining the username=, so when the username is assigned, it will be lowercased. The super part just passes our value to this method's super method, which is the one we are overriding.

Note: According to the documentation (see both here and here), we should be able to do @username = new_username.downcase in the method above. This is what I did in the screencast, and as you know, it didn't work as expected. Since recording the screencast I've discovered that the documentation is wrong, and that super is the way to do this.

## Step 5: Creating and Finding Records

Well, now that our models are created, let’s add a few records to test them out. We can do this a few ways. First, we can create a record with the new method, passing a hash of attributes, or assigning them individually.

When using User#new, you have to call the save method to actually put the record in the database. If there’s an error (remember those validations?), the save method will return false. Then, you can go to the errors property to see the errors; it’s a DataMapper::Validations::ValidationsErrors object, but you can iterate over the errors with the each method.

If you want to make and save a record in one fell swoop, use the create method, of course passing it an attributes hash.

Boom: created and saved!

How about finding a record in the database? If you know the key of the record you’re looking for, just use the get method:

Yes, you’re seeing that this works with both normal integer keys and other types of keys. Since we said the slug was the key in the Post model, we can get by slug.

What about those fields that could be the same for multiple records? You’ve got three options for that: first, last, and all. Just pass them a hash, and they get the records for you

## Conclusion: Learning More

There’s a lot more you can do with DataMapper; check out the documentation for more! Hit the question box if you have any questions.


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# ultrarelativistic gas

Consider the ideal ultrarelativistic gas Hamiltonian $\mathbf{H = }\sum_{i = 1}^N \mathbf{c |\vec{p_{i}}|}$, now if we let molecules to interact with a potential term like $\mathbf{d|\vec{q_{i}}|}$; is it still possible to find density of states $\omega(E)$?

-
What is this 'ultra-relativistic gas Hamiltonian', have you got a reference for this? –  Killercam Feb 16 '13 at 15:02
@Killercam $E^2=m^2c^4+p^2c^2$. Ultrarelativistic means $cp >> mc^2$ so that $E\approx cp$ –  Jorge Feb 17 '13 at 19:18
Thanks for that, but that was not my question... –  Killercam Feb 17 '13 at 19:24
If the system is at equilibrium then $w$ is proportional to $\exp(-\beta H)$. This result is independent of the specific form of $H$.


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# How the Sobel Operator Works

In this post, I will explain how the Sobel Operator edge detection algorithm works.

In order to understand how the Sobel Operator detects edges, it is important to first understand how to model an edge. Let’s take a look at that now before we examine how the Sobel Operator works.

# How to Model Edges Mathematically

Consider a grayscale image that is 10 x 1 pixels. Such an image would look something like the image below:

One can clearly see in the image that there is a sudden change in the intensity in the middle of the image. This sudden change is an edge.

Now, let us draw the corresponding intensity graph of each pixel, going from left to right. We will assume that black has an intensity of -1 (the lowest intensity), and white has an intensity of 1 (the highest intensity).

Mathematically, we can detect an abrupt change in the curve above by graphing the slope of the curve (i.e. taking the first derivative at each point along the curve).

The orange curve below is the graph of the slope at each point along the pixel intensity curve. Note how the curve reaches a maximum in the middle. This peak corresponds to an edge in the original image. It is an area where the intensity (brightness) of the image changes abruptly.

However, we have run into a problem here because I have drawn a continuous curve, but pixel intensity values are discrete. Therefore, in image processing, we cannot calculate the first derivatives of an image (known as “spatial derivatives”) directly using standard calculus methods; instead we have to estimate the derivative of the image using a mathematical operation known as convolution.

Professor Andrew Ng of Stanford University has a great video on how to perform the convolution operation given an input image:

I will explain the process below.

## How to Perform the Convolution Operation Given an Image as Input

To do the convolution operation, we need to use a mathematical tool called a kernel (or filter). A kernel is just a fancy name for a small matrix.

Below is an example of a kernel. This small matrix is 3×3 (3 rows and 3 columns). It happens to be the kernel used in the Sobel algorithm to calculate estimates of the derivatives in the vertical direction of an image. I will explain the Sobel algorithm later in this section.

The middle cell in a kernel is known as an anchor. In this case, the anchor is 0.

The Sobel Operator, a popular edge detection algorithm, involves estimating the first derivative of an image by doing a convolution between an image (i.e. the input) and two special kernels, one to detect vertical edges and one to detect horizontal edges.

Let’s see how to estimate the first derivative of an image by doing an example convolution calculation…

Suppose we have a 6×6 pixel color image.

A color image has 3 channels (red, green and blue). To perform the convolution operation to detect edges, we only need one channel, so we first convert the image to grayscale. Converting the color image to grayscale is the first step of most of the edge detection algorithms, including the Sobel Operator algorithm.

Here are what the intensity values might look like after the grayscale conversion.

Now, consider the kernel below (this happens to be the Sobel Vertical Kernel used to detect vertical edges).

To convolve matrix A with matrix B, we first overlay Matrix B on top of the upper-left portion of Matrix A and then take the element-wise product:

After completing the 9 multiplications, that upper-left portion becomes…

To complete the convolution operation, we add up all the numbers in the matrix above to get:

-150 + -300 + -150 + 0 + 0 + 0 + 150 + 510 + 255 = 315

The output matrix for convolving matrix A with matrix B, will be a 4×4 matrix. The value above, 315, becomes the first entry into this output matrix. You can see that 315 is a relatively large number.

We keep shifting the 3×3 kernel over the image, pixel by pixel, until we have filled up the output matrix above. That is the convolution operation.

# How the Sobel Operator Works

The Sobel Operator uses kernels and the convolution operation (described above) to detect edges in an image. The algorithm works with two kernels

1. A kernel to approximate intensity change in the x-direction (horizontal)
2. A kernel to approximate intensity change at a pixel in the y-direction (vertical).

In mathematical jargon, “approximating the intensity change” means approximating the gradient of the intensity values at a pixel. The gradient is the multivariable generalization of the derivative or slope mentioned earlier in this discussion.

Here are the two special kernels used in the Sobel algorithm:

The two kernels above are convolved with each pixel in the original image to identify the regions where the change (gradient) is maximized in magnitude in the x and y directions.

## Mathematical Formulation of the Sobel Operator

More formally,

• Let gradient approximations in the x-direction be denoted as Gx
• Let gradient approximations in the y-direction be denoted as Gy.

Suppose we have a source image Ii.

We want to calculate the magnitude of the gradient at a pixel (x, y), so we extract from image Ii a 3×3 matrix with pixel (x,y) as the center pixel (i.e. anchor).

The gradient approximations at pixel (x,y) given a 3×3 portion of the source image Ii are calculated as follows:

Gx = x-direction kernel * (3×3 portion of image A with (x,y) as the center cell)

Gy = y-direction kernel * (3×3 portion of image A with (x,y) as the center cell)

* above is not normal matrix multiplication. * denotes the convolution operation I described earlier in this discussion.

We then combine the values above to calculate the magnitude of the gradient at pixel (x,y):

magnitude(G) = square_root(Gx2 + Gy2)

The direction of the gradient Ɵ at pixel (x, y) is:

Ɵ = atan(Gy / Gx)

where atan is the arctangent operator.

So, in short, the algorithm goes through each pixel in an image and defines 3×3 matrices that have that pixel as the center cell (anchor pixel). Convolving these matrices with both the x and y-direction kernels yields two values, gradient approximations in the x-direction (Gx) and y-direction (Gy). We square both of these values, add them, and then take the square root of the sum to calculate the magnitude of the gradient at that pixel. Pixels that have gradients of large magnitude are likely part of an edge in an image.

## Example Calculation

Here is an example calculation showing how to calculate the gradient approximation at a single pixel in the Sobel algorithm. We have already converted the input image into grayscale:

We start the calculation of the gradient in the y-direction (Gy) at the pixel in the second column and second row (marked in yellow).

First, we convolve the selection in the original image with the y-direction kernel. The result is as follows:

Which becomes…

Then we add up all the numbers in the matrix above:

Gy for the pixel in row 2, column 2 is 315.

We now do a similar calculation to calculate Gx at that same pixel.The only thing that changes is the kernel that we multiply our selection by.

We start the calculation of the gradient in the x-direction (Gx) at the pixel in the second column and second row (marked in yellow).

We convolve the selection in the original image with the x-direction kernel. The result is:

Which becomes…

Then we add up all the numbers in the matrix above:

Gx for the pixel in row 2, column 2 is 315.

Thus, at the pixel at row 2, column 2…

Gx = 315

Gy = 315

We then combine the values above to calculate the magnitude of the gradient at pixel (x,y):

magnitude(G) = square_root(Gx2 + Gy2)

magnitude(G) = square_root(3152 + 3152)

magnitude(G) ≈ 445

This magnitude above is just our calculation for a single 3×3 region. We have to do the same thing for the other 3×3 regions in the image. And once we have done those calculations, we know that the pixels with the highest magnitude(G) are likely part of an edge.

This tutorial on GitHub has a really nice animated demonstration of the Sobel Edge Detection algorithm I explained above.

# Sobel Operator Code

This tutorial on image gradients has the Python code for the Sobel Operator.