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# American Institute of Mathematical Sciences

September  2007, 19(3): 575-594. doi: 10.3934/dcds.2007.19.575

## Global conservative solutions of the Dullin-Gottwald-Holm equation

 1 Faculty of Mathematics, D.A.L., University of Craiova, Romania

Received  April 2007 Revised  June 2007 Published  July 2007

A new approach to the analysis of wave-breaking solutions to the Dullin-Gottwald-Holm equation is presented in this paper. Introduction of a set of variables allows for solving the singularities. A continuous semigroup of solutions is also built. The solutions have constant $H^{1}$-energy for almost every time with respect to the Lebesgue measure.
Citation: Octavian G. Mustafa. Global conservative solutions of the Dullin-Gottwald-Holm equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (3) : 575-594. doi: 10.3934/dcds.2007.19.575
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# (a,b) ≤ (a',b') iff b ≤ a' or (b' = b and a ≤ a') where a≤b and a'≤b'. Has any one seen this order?

I have been mucking round with orders, and this is the order that I found I needed. I would like to know if it is defined somewhere else, it is kind of difficult to search for such things.

*EDIT*

The only application I have used it for at the moment is a totally ordered domain (e.g. '$a,b \in R$'), but I think it would extend to a partially ordered lattice.

Consider two intervals (a,b) (a',b'), if one is clearly better than the other, i.e. b≤a' then a≤b≤a'≤b' it is better. But if the interval has the same least upper bound where b=b', then only the greatest lower bound matters, so a ≤ a'.

The order that this creates is why I defined it. for instance, given 3 reals {1,2,3} we use a function that defines an interval on each where f(1) = [0,2] f(2) = [1,3] and f(3) = [2,4]. Using this order we can then state that f(1) || f(2) and f(2) || f(3) but f(1) < f(3).

The goal of this order is that it should create a lattice as well.

EDIT

I missed a really important part, sorry. This is based on intervals so a ≤ b and a' ≤ b' must be true.

(a,b) ≤ (a',b') iff b ≤ a' or (b' = b and a < a') where a ≤ b AND a' ≤ b'

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That 'ordering' is not transitive: (1, 2) < (3, 0), and (3, 0) < (1, 1), but (1, 2) is not less than (1, 1). – Simon Rose May 5 '10 at 21:54
This order seems pretty peculiar to me. It isn't reflexive - if $x < y$ then we don't have $(x, y)\le(x,y)$. Are you sure you typed it right? – Tom Smith May 5 '10 at 21:56
do you mean "iff b is less than or equal to b'" in the first part? If so, this is called the lexicographic order: en.wikipedia.org/wiki/Lexicographical_order – Qiaochu Yuan May 5 '10 at 22:00
Sorry, a <= b and a' <= b'. This is based on intervals. If you take this into consideration then I am pretty sure it works. – GrahamJenson May 5 '10 at 22:17
Are you assuming anything about the ambient order? e.g. Is it linear? Are they real numbers, integers? Or is it just some partial order partial? – Joel David Hamkins May 5 '10 at 22:32

If the ambient order is a lattice, they your order is indeed a lattice order. To see this, suppose that we have intervals (a,b) and (a',b'), in your sense that $a\le b$ and $a'\le b'$. If $b\lt b'$, then the least upper bound, written $(a,b)\vee(a',b')$, is $(a'\vee b,b')$. If $b=b'$, then $(a,b)\vee(a',b')=(a\vee a',b)$. Similarly, the case $b'\le b$ is symmetric, and so the remaining case is when $b\perp b'$, in which case $(a,b)\vee(a',b')=(b\vee b',b\vee b')$. A symmetric argument shows $(a,b)\wedge(a',b')$ exists, so it is a lattice.

This argument shows that if the ambient order is an upper semi-lattice, then so is your order, and the smae for lower semi-lattice, since we only needed $\vee$ in the underlying order to define $\vee$ in your order, and the same for $\wedge$.

Also, if the underlying order is well-founded, then your order is also well-founded, since any descending sequence must descend infinitely in one of the coordinates. If the rank of the ambient order is $\omega$, then the new rank is also $\omega$, since every pair would have only have finitely many predecessors in your order. But for larger ordinal ranks, it seems to go up, and one could calculate some bounds if this was a case you were interested in.

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You have no idea how happy I am that you said that, as this is the first order I have created myself and proven to be a lattice. And I have the same answer written down in front of me. I followed the same process as you suggested the other day with lexicographical order, and came up with this. – GrahamJenson May 5 '10 at 23:42
But I was hoping that someone might be able to find a similar order, as it would be nice to have a search term rather than an equation to find out if it is something someone else has done, or it is novel. My supervisor says "If it is maths, then it has been published at least 50 years ago", so far he has been right, so I go with this assumption. – GrahamJenson May 5 '10 at 23:44
I have not seen this particular order before, but I like it. People have looked at many interval orders, particularly in universal algebra, and so it may have arisen before, but I'm not sure. – Joel David Hamkins May 5 '10 at 23:55

Since $(2,0) \le (2,1)$ and $(2,1) \le (2,0)$, the relation is not antisymmetric. Or am I mistaken?

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a <= b and a' <= b', I forgot to mention that the order is based on intervals. Fixed it now. – GrahamJenson May 5 '10 at 22:18

In the realm of interval orders, it's often convenient to lift the interval [a,b] to the point (a,b) (which necessarily is above the x=y diagonal). Once you do that, lots of questions about interval containment/intersection can be phrased geometrically in terms of regions. Your order has a somewhat unusual, but reasonable-to-represent form. as follows:

Given the interval [a,b] to the point (a,b). Associate with (a,b) the half infinite rectangular region whose lower end is the horizontal line segment (a,b) - (b,b), and which extends upwards to infinity. Then all intervals greater than (a,b) in your order correspond to all points to the right of this rectangle (and above the diagonal) and the line segment (a,b)-(b,b) itself.

While this looks a little weird, it makes sense once you complete the diagram by reflecting this diagram around the normal to (x=y) passing through (a,b). It's a bit hard to explain without a figure, but the region described above is actually one half of two "wedges" (the lower half containing all intervals that are less than (a,b).

So each point is then associated with these two wedges. Questions about the resulting partial order can (I think) be then phrased in terms of unions and intersections of these regions, which should give you a more Boolean-lattice-like structure to work with.

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If you are indeed reasoning about intervals, then perhaps Allen's Interval Algebra might provide some guidance: http://en.wikipedia.org/wiki/Allen's_Interval_Algebra . Allen's Interval Algebra is a Boolean Algebra. Using Allen's terminology, you can more concisely state what order you are aiming at, namely that the first interval takes place before the second one or the second interval finishes the first.

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# Identifying Methylation Signatures and Rules for COVID-19 With Machine Learning Methods Zhandong Li, et al.

May 15, 2022

## Introduction

Coronavirus disease 2019 (COVID-19) was announced as a “public health emergency of international concern” by the World Health Organization (WHO) on January 30, 2020 and was assessed as a global pandemic on March 11, 2020 (Rodríguez-Morales et al., 2020; Eurosurveillance Editorial Team, 2020). The causative agent of COVID-19 is a new type of coronavirus, whose complete gene sequence is approximately 79.5% similar to that of severe respiratory syndrome coronavirus SARS-CoV. Therefore, it was named SARS-CoV-2 by the International Virus Laboratory Classification (Zhou et al., 2020; Zhu et al., 2020). SARS-CoV-2 is a group 2B ß-coronavirus, which is a linear single-stranded positive-stranded RNA virus. It is similar to other coronaviruses and consists of four structural proteins, namely, spike protein, envelope protein, membrane protein/matrix protein, and nucleocapsid protein. COVID-19 has a huge impact on global public health. According to WHO, SARS-CoV-2 had caused 156,496,592 infections and 3,264,143 deaths worldwide until May 8, 2021, and a total of 1,171,658,745 vaccine doses had been administered worldwide till May 5, 2021 (World Health Organization, 2020). However, definite and effective treatments for COVID-19 are still lacking, and no antiviral drug has been confirmed by a rigorous “randomized, double-blind, placebo-controlled” study.

As early as 1975, researchers (Holliday and Pugh, 1975; Riggs, 1975) found that in vertebrates, cytosine methylation at the CpG site can be used as a genetic marker and can be passed on to the next generation by cell division. In plants and mammals, methylation on the 5th carbon atom of cytosine residues is the most widely studied epigenetic modification. In mammals, cytosine methylation mostly exists on the CG sequence; plants have CHG and CHH methylation (H = A, C, or T) in addition to CG methylation. DNA methylation is relatively stable and can exist persistently during DNA replication. The position of DNA methylation can be determined and its relationship with gene regulation can be explored with the development of whole-genome methylation sequencing technology. The DNA methylation of the promoter region can suppress gene expression by preventing transcription factor accessibility. Moreover, the DNA methylation of the gene body can affect chromatin structure, alternative splicing, and transcription efficiency (Lorincz et al., 2004). In mammals, such as Homo sapiens and Mus musculus, DNA methylation is necessary to maintain normal embryonic development (Yin et al., 2012; Guo et al., 2014), and abnormal methylation has remarkable effects on diseases (Robertson, 2005). In addition, methylation plays an important role in the regulation of the expression of tissue-specific genes or developmental stage-dependent genes (Gehring and Henikoff, 2007).

Strong evidence showed that epigenetic markers, including histone modifications, DNA methylation, chromatin remodeling, and non-coding RNAs, affect gene expression profiles and increase individual vulnerability to virus infections (Fang et al., 2012; Menachery et al., 2014). Meanwhile, viruses have developed a complex, highly evolved, and coordinated process that can regulate the host’s epigenome, control the host’s innate immune and antiviral defense processes, and thus promote the powerful replication of the virus and the onset of disease (Schäfer and Baric, 2017). Circulating blood DNA methylation profiles are altered in patients with severe diseases, including severe sepsis and pediatric critical illness (Binnie et al., 2020; Güiza et al., 2020).

In this study, we obtained methylation data from 106 SARS-CoV-2-positive patients and 26 SARS-CoV-2-negative patients. Machine learning algorithms, such as Monte Carlo feature selection (MCFS) (Dramiński et al., 2007) and decision tree (DT) (Safavian and Landgrebe, 1991), were applied to identify methylation features and decision rules that clearly distinguish different cases and to build classification models with excellent performance to provide insight into the diagnosis, susceptibility, and potential pathogenesis of COVID-19.

## Materials and Methods

### Datasets

We downloaded the methylation data of the 128 samples from Gene Expression Omnibus with accession number GSE174818 (Balnis et al., 2021), which contains 102 samples from patients with COVID-19 and 26 samples from patients without COVID-19. For each sample, 86,5807 methylation sites were identified by the Illumina Human Methylation EPIC platform.

### Monte Carlo Feature Selection

For the investigated methylation data, features (methylation sites) were much more than sample numbers. Evidently, not all features were related to COVID-19. It is necessary to analyze all features and extract essential ones. As different feature selection methods may produce quite different results, selection of proper methods was quite essential. To our knowledge, MCFS is good at dealing with data containing few samples and large features. Thus, it was adopted in this study.

The MCFS method (Dramiński et al., 2007) is an effective and broadly adopted feature selection method that is composed of various DTs and builds various bootstrap sets with subsets of randomly selected features. First, m bootstrap sets and t feature subsets are created from the primary data set. Then, one tree is constructed for

bootstrap sets and

$t$

feature subsets. Overall,

DTs are created. The relative importance (RI) score for each feature can be calculated based on the resultant DTs. RI score is calculated as the frequency of a target feature in the growing DT as follows:

where f indicates a feature;

$wAcc$

means the weighted accuracy of the DT

$τ$

;

$IG(nf(τ))$

refers to the information gain of node

$nf(τ)$

;

$no.in nf(τ)$

and

denote the number of samples of

$nf(π)$

and

$τ$

, respectively; and

$u$

and

$v$

are weighting factors, which are set to 1. More significant features have higher RI values. Therefore, the features were sorted in decreasing order based on their RI values in the new feature list after MCFS processing.

The MCFS program used in this study was loaded from http://www.ipipan.eu/staff/m.draminski/mcfs.html. For convenience, the program was run using the default parameters, and u and v were set to 1.

### Incremental Feature Selection

Although the MCFS method can rank features by their importance, it cannot determine which features are essential. Therefore, the incremental feature selection (IFS) method (Liu and Setiono, 1998) was used to determine the optimal number of essential features required for the classification algorithm. First, IFS yields a series of feature subsets based on step size from the list of features received from the MCFS method described above. For example, when the step size is 5, the first feature subset includes the top 5 features, the second feature subset includes the top 10 features, and so on. Afterward, IFS trains the classifier on the training samples, which contain these features on each feature subset. The best subset was determined based on the evaluation metrics of the obtained model by evaluating this classifier through 10-fold cross-validation (Kohavi, 1995; Chen et al., 2021; Liu et al., 2021; Li X. et al., 2022; Tang and Chen, 2022; Yang and Chen, 2022).

### Decision Tree

DT is one of the most classic machine learning algorithms (Safavian and Landgrebe, 1991). Although it is not very powerful, even much weaker than several strong machine learning algorithms, it also has its merits. In fact, DT is a white-box model, meaning it is possible for users to understand its classification principle. This cannot be achieved for all black-box models, which is always more powerful than DT. In the field of biomedical research, such merit is quite helpful as investigators want to not only build efficient models but also obtain helpful clues to understand the complicated underlying mechanism. Accordingly, DT is widely accepted in the field of biomedical research (Zhang et al., 2021a; Zhang et al., 2021b; Huang et al., 2021; Li Z. et al., 2022; Chen et al., 2022; Ding et al., 2022). Generally, DT uses the IF–THEN format to accomplish classification or regression tasks through a tree structure. It often yields satisfactory performance at a low computational cost. In this work, we applied the Scikit-learn module in Python to build the DT classifier.

### Synthetic Minority Oversampling Technique

As described in the Datasets Section, a considerable variation in the sample sizes of patients with COVID-19 was observed. In this case, a classifier with excellent performance is difficult to build, because the predicted results are suitable for the type with the largest sample size. Synthetic minority oversampling technique (SMOTE) (Chawla et al., 2002) was performed in the present work to address this problem. This method ensures that the number of samples in the minority class is equal to the number of samples in the majority class after processing by adding new samples to the minority class. In detail, x is a randomly selected sample in the minor class, and some samples with the same class that are closest to x are identified. Next, sample y is randomly selected from the closest samples mentioned above, and a novel sample is produced by choosing a randomly selected point between x and y in the feature space. The newly produced sample is deeply associated with x and y; thus, it has a high probability of belonging to the same class as x and y and is therefore considered to be in the same class. The above procedure executes several times until the minority class has same number of samples in the majority class.

In this study, we employed the SMOTE procedure acquired from https://github.com/scikit-learn-contrib/imbalanced-learn and directly used the default parameters.

### Performance Measurement

For the binary model used in this study, its predicted results can be counted as a confusion matrix, which contains four entries: true positive (TP), false positive (FP), true negative (TN) and false negative (FN). According to these entries, several measurements can be calculated. In this study, we adopted the following measurements: sensitivity (SN, also called recall), specificity (SP), prediction accuracy (ACC), Matthews correlation coefficient (MCC), precision and F1-measure (Sasaki, 2007; Powers, 2011; Zhao et al., 2018; Wu and Chen, 2022). They can be computed by

$F1−measure=2×precision×recallprecision+recall.(7)$

Among above measurements, we selected F1-measure as the key measurement as it can better reflect the stability of the model. A higher F1-measure indicates a more robust classification model.

## Results

As shown in Figure 1, we applied an analysis flow to extract key features and build the classification model and rules. The results are summarized in the following sections.

FIGURE 1. Flowchart of the computational method in this study. A systematic analysis process that integrates feature selection, DT algorithms, and rule learning was applied to identify COVID-19 methylation site features. The optimal classifier, methylation sites, and rules were determined based on the performance of the DT model and the importance of the features in each model.

### Results of MCFS Method on Methylation Profiles

We employed the MCFS method to assess the importance of each feature and select key sites from the COVID-19 methylation dataset. These features were ranked in decreasing order of RI scores, and the results are presented in Supplementary Table S1.

### Results of IFS Method With DT

After the MCFS analysis, we brought the obtained feature list into the IFS with the DT algorithm. The step size of the IFS was set to 5. Since the list was very large, it would take lots of time to consider all possible feature subsets. Furthermore, not all methylation features are related to COVID-19. Thus, only top 10,000 methylation features in the list were considered, that is, 2000 feature subsets were investigated. DT model was constructed on each feature subset and was evaluated by 10-fold cross-validation. The obtained evaluation metrics, including SN, SP, ACC, MCC, precision and F1-measure are listed in Supplementary Table S2. To display the DT models on different feature subsets, we plotted an IFS curve using the number of features as the X-axis and F1-measure as the Y-axis, which is shown in Figure 2. It can be observed that the highest F1-measure was 0.990, which was obtained by using top 50 features in the list. The other five measurements of such model are illustrated in Figure 3. The ACC and MCC were 0.984 and 0.954, respectively. As for SN, SP and precision, they were 0.980, 1.000 and 1.000, respectively. This result indicated that the constructed optimal DT model has a near-perfect performance and proved the effectiveness of the analysis method.

FIGURE 2. IFS curves obtained by DT classification models on the top 1000 features of the COVID-19 dataset. The model produced the highest F1-measure of 0.990 when the top 50 features were used.

FIGURE 3. Performance of the best DT model and DT model with informative features. The best DT model is superior to the DT model with informative features.

### Comparison of DT Models With Informative Features

In this study, the IFS method was adopted to extract best features for DT and the best DT model was constructed with these features. In fact, MCFS can yield essential features, called informative features, by only analyzing the methylation data. With these informative features, a DT model can also be built. It is interesting to compare the performance of these two models.

For the methylation data, 257 informative features were obtained by MCFS. The DT model with these features was evaluated by 10-fold cross-validation. The F1-measure was 0.944, which was much lower than that of the best DT model (0.990). As for other five measurements, they are provided in Figure 3. It can be observed that each measurement was lower than that produced by the best DT model. This indicated the superiority of the best DT model. The employment of IFS method can help to build a more efficient model.

### Classification Rules

As described in the previous section, DT yielded the highest F1-measure on the COVID-19 methylation dataset when the top 50 features are used. Therefore, we applied DT to all samples using these 50 features to obtain six rules, which are provided in Table 1. Three rules each were related to COVID-19 and non-COVID-19. These rules clearly expressed the expression patterns of these features. These rules were described in detail in Discussion Section.

TABLE 1. Rules yielded by decision tree on top 50 features.

## Discussion

Meanwhile, our research proposed a novel and creative pattern with a high distinguishing degree in confirmed and suspected COVID-19 cases through MCFS. Although the real-time polymerase chain reaction test of sputum is the gold standard for the diagnosis of COVID-19, it takes a long time to confirm the diagnosis of patients because of the high level of false negatives. Therefore, researchers conducted various methods to better identify SARS-CoV-2 infection. (Hemdan et al., 2020) constructed a novel deep learning classifier to diagnose COVID-19 through X-ray images, which are cheaper, more convenient, and accessible compared with traditional chest X-ray and computed tomography. (Siddiqui et al., 2020) focused on the correlation of temperature with suspected, confirmed, and death cases by machine learning and found that temperature presents diverse trends in most cities and cannot be the decisive factor in different cases or situations. We focused on several top features and decision rules because they have a crucial impact on the classification and discussed them further through a wide literature publication to prove that our findings are reliable and convincing.

Epithelial stromal interaction 1 (EPSTI1, probeID: cg03753191) is an interferon (IFN)-responsive gene that was originally isolated from mixed cultured human breast cancer cells and fibroblasts (Nielsen et al., 2002). This gene is located on chromosome 13q13.3; is 104.2 kb in length; contains 11 exons; and is involved in tumor cell metastasis, epithelial–mesenchymal transition, chronic inflammation, tissue reconstruction, embryonic development, and other biological processes (De Neergaard et al., 2010). EPSTI1 plays an important role in the regulation of cell apoptosis. (Capdevila-Busquets et al., 2015) confirmed by in vitro experiments that EPSTI1 can inhibit breast cancer cell apoptosis by interacting with caspase 8. In addition, EPSTI1 has an antiviral effect against hepatitis C virus (HCV) by affecting the life cycle, viral replication, assembly, and release of HCV. (Meng et al., 2015) confirmed that EPSTI1 can promote the expression of protein kinase-R (PKR)-dependent genes, including IFNβ, IFIT1, OAS1, and RNase L, by activating the promoter of PKR to play an antiviral effect during the process of HCV infection. Without the involvement of IFN treatment, EPSTI1 overexpression effectively suppressed HCV replication, whereas the lack of EPSTI1 enhanced the viral activities. Current research discovered that EPSTI1 expression influences the performance of immune cells. (Kim et al., 2018) found that EPSTI1 expression is remarkably upregulated after macrophage activation with IFNγ and lipopolysaccharide. The proportion of M2-type macrophages is increased in the bone marrow-derived macrophage deficiency of EPSTI1. In EPSTI1 knockout mice, the number of M1 macrophage cells in the peritoneal cavity was significantly reduced. These findings demonstrate an important regulatory role of EPSTI1 in macrophage polarization. Therefore, EPSTI1 methylation may participate in the process of SARS-CoV-2 infection and affect inflammatory and immune function by regulating EPSTI1 expression.

NACAP1 (probeID: cg15959262) is a pseudogene of nascent polypeptide-associated complex-alpha (NACA). Phosphatase and tensin homolog (PTEN) pseudogene 1 (PTENP1) has been first revealed to contain microRNA response elements (MREs), which also exist in its corresponding protein-coding gene, PTEN (Poliseno et al., 2010). Increasing pseudogenes are found to have a similar phenomenon, that is, pseudogenes and their corresponding protein-coding genes function as competitive endogenous RNAs for binding to the same microRNAs (Lujambio and Lowe, 2012; Karreth et al., 2015). NACA encodes the a chain of nascent polypeptide-associated complex (NAC), which performs multiple functions, including protecting newborn peptides and regulating the translocation of new peptides into the endoplasmic reticulum and mitochondria (Rospert et al., 2002). The alpha chain of NAC alone acts as a transcriptional co-activator for developmental regulation (Yotov et al., 1998). Furthermore, NACA can regulate the conformation of Fas-associated death domain protein oligomer, which is an important mediator in the signal transduction pathway and can be activated by several members of the tumor necrosis factor (TNF) receptor family (Liguoro et al., 2003). NACA is related to neurodegenerative diseases. Patients with Alzheimer’s disease and Down’s syndrome have lower NACA expression levels in their brain cells (Kim et al., 2002). More importantly, the inhibition of NACA can induce the proliferation and differentiation of CD8+ T cells and enhance cytotoxicity. (Al-Shanti and Aldahoodi, 2006) used anti-sense technology to reduce the concentration of mRNA that translates NACA and found that CD8+ T cells will differentiate and activate to a higher degree in the presence of antisense oligonucleotide chains. Compared with the control group, the lethality of CD8+ T cells on target cells was enhanced.

SHROOM3 (probeID: cg17439158), a member of the Shroom family, encodes an actin-binding protein, which is important in epithelial cell shape and tissue morphogenesis (Haigo et al., 2003; Hildebrand, 2005). Shoom3 overexpression in epithelial cells induces rho kinase (Rock) recruitment and increases myosin 2 (Myo2) accumulation through phosphorylation and activation. The activation of the Rock/Myo2 signaling pathway leads to the local contraction of the actomyosin network on the top surface of the cell, which results in changes in cell morphology. Recent research has proved the indispensable role of SHROOM3 in glomerular filtration barrier integrity (Yeo et al., 2015). Forced Shroom3 expression in fawn-hooded hypertensive rat and endogenous shroom3 knockdown zebrafish improved kidney glomerular function. Moreover, multiple genome-wide association studies and in vivo experiments strongly demonstrated the correlation between SHROOM3 and congenital kidney disease (Khalili et al., 2016).

C19ORF35 (probeID: cg08399733), also named PEAK3, is a member of the New Kinase Family 3 (NKF3) that can regulate cytoskeleton stability and cell motility by binding with an adaptor protein, CrkII (Lopez et al., 2019). C19ORF35 is associated with cancer progression. C19ORF35 overexpression has been detected in various cancers, including pancreatic, breast, and colon cancers (Wang et al., 2010; Kelber et al., 2012; Fujimura et al., 2014). C19orf35 methylation is related to early carcinogenesis. According to a DNA methylation sequencing study on 12 patients with early gastric cancers (EGCs), C19orf35 is remarkably hypomethylated in the diffuse type of EGC tissue compared with adjacent corresponding non-tumor mucosal tissue (Chong et al., 2014).

IFN-induced with helicase C domain 1 (IFIH1, probeID: cg21060789) and IFN-induced protein 44-like (IFI44L, probeID: cg13452062) are IFN-stimulated genes. IFIH1, also known as melanoma differentiation-associated gene-5, is a cytoplasmic RNA receptor protein composed of 1025 amino acids. IFIH1 recognizes double-stranded RNA with a length of more than 1 kb. It is an important member of the retinoic acid-inducible gene I (RIG-I)-like receptor family, which can activate type I IFN signaling pathway and participate in the pathogenesis of a variety of autoimmune diseases. IFIH1 and IFN-β interact to activate the body’s anti-tumor immune response. IFIH1 can promote type I IFN response and increase the secretion of TNF-α and IFN-β. The upregulation of IFIH1 expression may increase the effectiveness of IFN therapy (Pappas et al., 2009). IFN-β can also stimulate the upregulation of IFIH1 and RIG-I, mediate innate immune response, kill tumor cells with low neurotoxicity, and therefore inhibit tumor growth (Wu et al., 2017; Bufalieri et al., 2020). IFI44L is a paralog gene of IFI44 and functions as a regulator of cell apoptosis, virus infection, and congenital immune response. The DNA methylation level of the IF144L promoter may be related to kidney damage in patients with systemic lupus erythematosus (SLE). (Zhao et al., 2016) found that the DNA methylation level of IFI44L promoter in patients with SLE was remarkably lower than that of the normal control group. In addition, the DNA methylation level of the IFI44L promoter in patients with SLE and renal involvement was also remarkably lower than that of patients with SLE without renal involvement. IFI44L participates in the antiviral process of IFN-mediated innate immune response and is a confirmed marker of early viral infection. IFN is the earliest discovered cytokine that can inhibit viral infection and replication and is activated in the early stage of viral infection (within a few minutes to a few hours) (Zaas et al., 2009). (Henrickson et al., 2018) pointed out that when influenza virus and respiratory syncytial virus infections occur, IFI44L acts as an IFN-stimulated factor regulatory gene, and its expression level increases. Therefore, the mRNA expression of IFI44L can be used as an early indicator of virus infection. According to (Kaforou et al., 2017), in 111 bacterial-infected children less than 60 days old, the detection sensitivity of IFI44L mRNA expression is 88.8% (95% CI, 80.3–94.5%), and the specific degree is 93.7% (95% CI, 87.4–97.4%). Therefore, aberrant IFI44L methylation may occur during SARS-CoV2 infection and lead to abnormal IFI44L expression.

MX1 (probeID: cg26312951) belongs to human mycovirus resistance genes (MX) with biological functions, such as GTP binding and GTP enzyme activities. The two kinds of MX proteins, namely, MX1 and MX2, differ greatly in virus specificity and mechanism of action. MX1 has antiviral activity against a variety of RNA viruses and certain DNA viruses induced by type I and type II IFNs, including negative-strand RNA viruses and hepatitis B virus. MX1 is enriched in the IFN-γ and Toll-like receptor signaling pathways (Haller et al., 2015). A 2020 study detailed the expression of MX1 in 403 patients with COVID-19 and 50 patients without COVID-19 (Bizzotto et al., 2020). The expression of MX1, MX2, ACE2, and BSG/CD147 can cluster individuals with and without COVID-19 through principal component analysis, which indicated that the expression levels of MX1 and MX2 between patients with and without COVID-19 are remarkably different. MX1 can directly act on the ribonucleoprotein complex of the virus, so it has a wide range of antiviral activity. This feature has been proven to be suitable for RNA viruses and DNA viruses. (Verhelst et al., 2012) have reported that Mx1 interfering with functional viral ribonucleoprotein complex assembly led to inhibition of influenza virus and high MX1 expression can result in a better prognosis to influenza A (H1N1) pandemic in 2009. It is worth noting that GTPase activity is also positively correlated with its antiviral function. Different from MX1, the antiviral function of MX2 is limited to certain viruses, such as HIV. Although the expression of MX1 and MX2 in patients with COVID-19 were significantly higher than those in non-COVID-19 groups, MX1 shows a greater positive correlation with patients with COVID-19 and may be more specific than MX2 in response to SARS-CoV-2. Therefore, MX1 is a key responder to SARS-CoV-2 infection.

Collectively, the top identified discriminative feature genes and settled rules have a crucial role in virus infection and IFN-mediated immune response. This method demonstrates that our method is reliable and convincing. Our newly presented computational approach based on methylation profiles also provides a new perspective for exploring the mechanism of COVID-19. Furthermore, it is a new method that distinguishes confirmed and suspected COVID-19 cases and has applicable clinical value in the differential diagnosis of patients with confirmed and suspected COVID-19.

## Conclusion

The current study aimed to apply computational methods to extract the best biological features and decision rules from COVID-19 methylation profiles. This study has shown that the extracted optimal methylation site signatures and expression rules have been validated by previous work and are reliable and valid for distinguishing COVID-19. This study provides a new set of potential biomarkers/rules that can be used to differentiate patients with COVID-19 at the methylation level. These findings enhance our understanding of COVID-19 expression at the methylation level and could offer guidance for future studies on COVID-19.

## Author Contributions

TH and Y-DC designed the study. SD, LC, and KF performed the experiments. ZL, ZM, and HL analyzed the results. ZL, ZM, and SD wrote the manuscript. All authors contributed to the research and reviewed the manuscript.

## Funding

This work was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (XDA26040304 and XDB38050200) National Key R&D Program of China (2018YFC0910403), the Fund of the Key Laboratory of Tissue Microenvironment and Tumor of Chinese Academy of Sciences (202002).

## Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


text:
261 views
A network has subnet mask $255.255.255.254$. Determine the maximum or number of Host
in this network. Also determine the broadcast address of this network.
edited | 261 views
0
Technically,

$255.255.255.254\Rightarrow 11111111.11111111.11111111.11111110$
There are only $1$ host bits.
Hence, there are $2^1 -1= 0$ hosts are possible
and the DBA will be $255.255.255.255$
+1
This question seems ambiguous,

Can i have the IP address
0
IP address is not provided in the Question.
0

@Sukanya Das this is not an ambiguous question.

255.255.255.254⇒11111111.11111111.11111111.11111110

Above 31-Bit prefix address

Directed broadcast is disabled automatically on /31 links. -Ref:https://learningnetwork.cisco.com/message/239554#239470
Using 31-Bit prefixes, the host-id of 0 represents one host(Also Subnet), and a host-id of 1 represents the other host of a point-to-point link.
So we have 2 hosts here

You could refer to this RFC for more info on 31-bit prefixes (/31 links)
https://tools.ietf.org/html/rfc3021

answered by Active (1.1k points)
0
Yes, this is an exception
0
Really informative answer. Thank you.

255.255.255.254⇒11111111.11111111.11111111.11111110

Above 31-Bit prefix address
number of one's  in network mask represent the total no bit used in the network id

means 31 bit are used for network id and 1 bit are used for host id . so  maximum no of host is 21=2.

answered by (277 points)
0
2^2-2=0 hosts are possible
0
when they ask about maximum host do't subtract 2 .and when they ask  about total host subtract 2
0
source of the above line????

As per given subnet mask = > 31 bits already used we left with 1 bit.

so  #no of IP Possible with 1 bit => 21  = 2  but practically we can configure 21 - 2 Hosts only .

Final Answer => 0 Host and   255.255.255.255 is broadcast address.

answered by (109 points)

+1 vote


text:
# Simple function to verify if a number is integer

Sometimes, there's the need to verify if a given number is an integer or a decimal. Since JavaScript doesn't distinguish between both, I've made an extremelly basic function.

function isInteger(num){
var numCopy = parseFloat(num);
return !isNaN(numCopy) && numCopy == numCopy.toFixed();
}

As you can see, it relies on checking if the number without decimals is the same as the parsed value. I'm not so confident about this.

Notice that this may return true for some scientific notation values.

Examples of results:

• 1.0: true (JavaScript treats <n>.0 as an integer)
• 1e3: true
• 1.2345e2: false (equivalent to 123.45)
• 1.2345e5: true (equivalent to 123450)
• 1e200: true (equivalent to Infinity)

Is there any more reliable way of checking? The results seem to be consistent with the expected results. Regular expressions are an option but they are clunky.

• Have you checked out the polyfill or ES6's isInteger function? Check it here: developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/… - I think this is agreeably the best way to find out, but as always, be aware of JS rounding stuff for you. – somethinghere Aug 21 '15 at 9:53
• @somethinghere That's a really great suggestion! Put it in the form of an answer and I'll upvote it for sure – Ismael Miguel Aug 21 '15 at 9:56
• @Downvoter Care to explain? – Ismael Miguel Sep 3 '15 at 21:46

In ES6, the next version of Javascript, there will be a function exactly for this purpose called Number.isInteger(). Currently, MDN offers the following polyfill which I assume therefore does the best Javascript can to determine whether something is an integer. There are edge cases of course, but for most everyday use it will do the trick:

Number.isInteger = Number.isInteger || function(value) {
return typeof value === "number" &&
isFinite(value) &&
Math.floor(value) === value;
};

I think this is the best we will do in the meanwhile, as passing string to a number function is an effective way of doing it, but calling toString on a number makes it go through conversion as well, resulting in potential issues.

There are already a couple of browsers that support it like Chrome and Firefox, but the big one (Safari) doesn't yet and neither does IE/Edge, so the polyfill will have to do. For more info, see MDN's writeup, it's quite insightful:

https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Number/isInteger#Browser_compatibility

Note: Polyfills like this are useful, but there are some concerns about amending these core features of JS. As I say in the comments, as long as you implement something that doesn't exist or replace anything, I think it's alright. Use in moderation.

You can always wrap it in it's own function and call it as a global function if you prefer not to use this polyfill.

• I would just add a line saying that messing with prototypes is a bad practice, and that this can be used as a simple named function. Other than that, +10 for you! – Ismael Miguel Aug 21 '15 at 10:45
• @IsmaelMiguel I have heard a lot about it being bad to mess with prototypes in this way, but nobody ever explained me why. I understand rewriting a method from, say, Number is dangerous because it gives unexpected results, but adding a method seems perfectly harmless to me. Do you have got any good references on why prototype amending is bad? I'd love to read about that. – somethinghere Aug 21 '15 at 10:48
• At the moment, I don't recall any article to read. But imagine that your code relies on some polyfill, that is set to the prototype. If anyone changes the prototype by, for example, doing window.Number = window.String, your whole code is broken. Besides, messing with prototypes may crash with other codes that rely on a specific prototype method to return a non-standard value or has a different behaviour. – Ismael Miguel Aug 21 '15 at 10:59
• Yeah that's kind of what I meant though, you don't mess with prototypes themselves, but I think appending a method is perfectly fine as long as it does not replace an already existing method. It might be best to have an ES6 check first (if(!Number.isInteger)), but otherwise I really think this particular thing is harmless. Replacing Number with Int etc... Thats obviously bad :) – somethinghere Aug 21 '15 at 11:01
• I concur, so I have added it to my answer. – somethinghere Aug 21 '15 at 11:04

The question is a little vague, but right away I tried to break it and was successful with this

function isInteger(num){
var numCopy = parseFloat(num);
return !isNaN(numCopy) && numCopy == numCopy.toFixed();
}

isInteger(.9999999999999999999998/.9999999999999999999999);
true

probably because of loss of precision, as @Gallant noted the decimals are getting rounded to zero

Depending on what you are doing this is probably sufficiently accurate.

• This is more accurately due to loss of precision from floating point values; both decimals are getting rounded to 1 before the division occurs. isInteger(.9999999999999999999998) would also return true because of this. – Gallant Aug 20 '15 at 21:43
• @Gallant, you are totally right. – Malachi Aug 20 '15 at 21:48
• If you have a look at the ES6 polyfill of isInteger (developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/…) you will see this also fails this test. Javascript simply cannot compute these properties without converting them, so they will also wind up being 1. – somethinghere Aug 21 '15 at 9:46


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# Team:Aachen/Notebook/Software

(Difference between revisions)
 Revision as of 09:34, 15 October 2014 (view source)Mjoppich (Talk | contribs) (Created page with "__NOTOC__ {{Team:Aachen/Header}} {{Team:Aachen/Stylesheet}} = Engineering = In this part of our notebook, we recorded everything concerning the technical aspects of ''WatsOn''...")← Older edit Revision as of 09:39, 15 October 2014 (view source)Mjoppich (Talk | contribs) (→Engineering)Newer edit → Line 3: Line 3: {{Team:Aachen/Stylesheet}} {{Team:Aachen/Stylesheet}} - = Engineering = + = Software = - In this part of our notebook, we recorded everything concerning the technical aspects of ''WatsOn'' and our OD/F device. To access construction manuals, technical details and development protocols of either device, please click on the respective panel below: + What is hardware without corresponding software (except $\mu$-Controller code^^)? + We have had several [http://2014.igem.org/Team:Aachen/Notebook/Engineering hardware projects] ongoing, and will present corresponding software here. + But software is not limited to hardware, we also have some ''Santa's Little Helpers'' for our iGEM Project which we would like to present here. Line 13: Line 15:
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# Software

What is hardware without corresponding software (except $\mu$-Controller code^^)? We have had several hardware projects ongoing, and will present corresponding software here. But software is not limited to hardware, we also have some Santa's Little Helpers for our iGEM Project which we would like to present here.


text:
8.1: Example

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Separation of variables in polar coordinates

Example

Consider a circular plate of radius $$c \text{ m}$$, insulated from above and below. The temperature on the circumference is $$100^\circ~\text{C}$$ on half the circle, and $$0^\circ~\text{C}$$ on the other half.

The differential equation to solve is $\rho^2 \pdrhor{u} + \rho \pdrho{u} + \pdphip{u} = 0,$ with boundary conditions $u(c,\phi) = \begin{cases} 100 & \text{if 0 < \phi < \pi} \\ 0 & \text{if \pi < \phi < 2\pi } \end{cases}\quad .$

Periodic BC

There is no real boundary in the $$\phi$$ direction, but we introduce one, since we choose to let $$\phi$$ run from $$0$$ to $$2\pi$$ only. So what kind of boundary conditions do we apply? We would like to see “seamless behaviour”, which specifies the periodicity of the solution in $$\phi$$, \begin{aligned} u(\rho,\phi+2\pi)&=&u(\rho,\phi),\\ \pdphi{u}(\rho,\phi+2\pi)&=&\pdphi{u}(\rho,\phi).\end{aligned} If we choose to put the seem at $$\phi=-\pi$$ we have the periodic boundary conditions \begin{aligned} u(\rho,2\pi)&=&u(\rho,0),\\ \pdphi{u}(\rho,2\pi)&=&\pdphi{u}(\rho,0).\end{aligned}

We separate variables, and take, as usual $u(\rho,\phi) = R(\rho) \Phi(\phi).$ This gives the usual differential equations \begin{aligned} \Phi''-\lambda\Phi &=&0,\\ \rho^2 R'' + \rho R' + \lambda R &=& 0.\end{aligned} Our periodic boundary conditions gives a condition on $$\Phi$$, $\Phi(0)=\Phi(2\pi),\;\;\Phi'(0)=\Phi'(2\pi). \label{eq:phiBC}$ The other boundary condition involves both $$R$$ and $$\Phi$$.

Three cases for $$\lambda$$

As usual we consider the cases $$\lambda>0$$, $$\lambda<0$$ and $$\lambda=0$$ separately. Consider the $$\Phi$$ equation first, since this has the most restrictive explicit boundary conditions ([eq:phiBC]).
We have to solve $\Phi'' = \alpha^2\Phi,$ which has as a solution $\Phi(\phi) = A \cos \alpha \phi + B \sin \alpha \phi.$ Applying the boundary conditions, we get \begin{aligned} A &=& A \cos(2\alpha\pi)+ B\sin(2\alpha\pi), \\ B \alpha &=& -A \alpha\sin(2\alpha\pi)+ B\alpha\cos(2\alpha\pi).\end{aligned} If we eliminate one of the coefficients from the equation, we get $A = A \cos(2\alpha\pi)-A\sin(2\alpha\pi)^2/(1-cos(2\alpha\pi))$ which leads to $\sin(2\alpha\pi)^2=-(1-cos(2\alpha\pi))^2,$ which in turn shows $2\cos(2\alpha\pi)=2,$ and thus we only have a non-zero solution for $$\alpha=n$$, an integer. We have found $\lambda_n=n^2,\;\;\Phi_n(\phi) = A_n \cos n \phi + B_n \sin n \phi.$ We have $\Phi'' = 0 .$ This implies that $\Phi = A\phi + B.$ The boundary conditions are satisfied for $$A=0$$, $\Phi_0(\phi)=B_n.$ The solution (hyperbolic sines and cosines) cannot satisfy the boundary conditions.

height 1mm width 5cm

Now let me look at the solution of the $$R$$ equation for each of the two cases (they can be treated as one), $\rho^2 R''(\rho) + \rho R'(\rho) -n^2R(\rho)=0.$ Let us attempt a power-series solution (this method will be discussed in great detail in a future lecture) $R(\rho) =\rho^\alpha.$ We find the equation $\rho^\alpha[\alpha(\alpha-1)+\alpha^2-n^2]= \rho^\alpha[\alpha^2-n^2]=0$ If $$n\neq 0$$ we thus have two independent solutions (as should be) $R_n(\rho) = C\rho^{-n}+D\rho^n$ The term with the negative power of $$\rho$$ diverges as $$\rho$$ goes to zero. This is not acceptable for a physical quantity (like the temperature). We keep the regular solution, $R_n(\rho) = \rho^n.$ For $$n=0$$ we find only one solution, but it is not very hard to show (e.g., by substitution) that the general solution is $R_0(\rho) = C_0 + D_0\ln(\rho).$ We reject the logarithm since it diverges at $$\rho=0$$.

Putting it all together

In summary, we have $u(\rho,\phi) = \frac{A_0}{2} + \sum_{n=1}^\infty \rho^n \left(A_n\cos n\phi+B_n\sin n \phi \right).$ The one remaining boundary condition can now be used to determine the coefficients $$A_n$$ and $$B_n$$, \begin{aligned} U(c,\phi) &=& \frac{A_0}{2} + \sum_{n=1}^\infty c^n \left(A_n\cos n\phi+B_n\sin n \phi \right)\nonumber\\ &=& \begin{cases} 100 & \text{if 0 < \phi < \pi} \\ 0 & \text{if \pi < \phi < 2\pi} \end{cases}\quad.\end{aligned} We find \begin{aligned} A_0 &=& \frac{1}{\pi} \int_0^\pi 100\, d\phi = 100, \nonumber\\ c^n A_n &=& \frac{1}{\pi}\int_0^\pi 100\cos n\phi\, d\phi = \frac{100}{n\pi} \sin(n\phi)|^\pi_0=0,\nonumber\\ c^n B_n &=& \frac{1}{\pi}\int_0^\pi 100\sin n\phi\, d\phi = -\frac{100}{n\pi} \cos(n\phi)|^\pi_0 \nonumber\\&=& \begin{cases} 200/(n\pi) & \text{if n is odd}\\ 0 & \text{if n is even} \end{cases}\quad.\end{aligned} In summary $u(\rho,\phi) = 50 + \frac{200}{\pi} \sum_{n~\rm odd} \left(\frac{\rho}{c}\right)^n \frac{\sin n \phi}{n}. \label{eq:T0-100}$ We clearly see the dependence of $$u$$ on the pure number $$r/c$$, rather than $$\rho$$. A three dimensional plot of the temperature is given in Fig. [fig:T0-100].


text:
Tagged: size of a matrix

Problem 263

Let $A$ and $B$ are matrices such that the matrix product $AB$ is defined and $AB$ is a square matrix.
Is it true that the matrix product $BA$ is also defined and $BA$ is a square matrix? If it is true, then prove it. If not, find a counterexample.


text:
# A DECOUPLED STAGGERED SCHEME FOR THE SHALLOW WATER EQUATIONS

Abstract : We present a first order scheme based on a staggered grid for the shallow water equations with topography in two space dimensions, which enjoys several properties: positivity of the water height, preservation of constant states, and weak consistency with the equations of the problem and with the associated entropy inequality. The shallow water equations form a hyperbolic system of two conservation equations (mass and momentum) which are obtained when modelling a flow whose vertical height is considered small with respect to the plane scale. The solution of such a system may develop shocks, so that the finite volume method is usually preferred for numerical simulations. Two main approaches are found: one is the colocated approach which is usually based on some approximate Riemann solver, see e.g. [3] and references therein; the other one is based on a staggered arrangement of the unknowns on the grid. Indeed, staggered schemes have been used for some time in the hydraulic and ocean engineering community, see e.g. [1, 2, 12]. They have been recently analysed in the case of one space dimension [5, 8], following the works on the related barotropic Euler equations, see [11] and references therein. In the present work, we obtain a discrete local entropy inequality; furthermore, we extend the consistency analysis of the scheme to the case of two space dimensions, and we weaken the assumptions on the estimates , namely we no longer require a bound on the BV norm of the approximate solutions, at least for the weak formulation (the passage to the limit in the entropy still necessitates a time BV boundedness). Let Ω be an open bounded domain of R 2 and let T > 0. We consider the shallow water equations with topography over the space and time domain Ω × (0, T): ∂ t h + div(hu) = 0 in Ω × (0, T), (1a) ∂ t (hu) + div(hu ⊗ u) + ∇p + gh∇z = 0 in Ω × (0, T), (1b) p = 1 2 gh 2 in Ω × (0, T), (1c) u · n = 0 on ∂Ω × (0, T), (1d) h(x, 0) = h 0 , u(x, 0) = u 0 in Ω. (1e)
Keywords :

https://hal.archives-ouvertes.fr/hal-02163454
Contributor : Raphaele Herbin <>
Submitted on : Monday, June 24, 2019 - 12:19:25 PM
Last modification on : Tuesday, March 17, 2020 - 1:29:14 AM

### Files

sw-jaca-rev.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-02163454, version 1
• ARXIV : 1906.11001

### Citation

Raphaèle Herbin, Jean-Claude Latché, Youssouf Nasseri, Nicolas Therme. A DECOUPLED STAGGERED SCHEME FOR THE SHALLOW WATER EQUATIONS. 2019. ⟨hal-02163454v1⟩

Record views


text:
Escape velocity and kinetic energy of the Earth

If you had two masses, $m_{1}$ and $m_{2}$, and you released them in space infinitely far apart, their kinetic energies would satisfy $\frac{1}{2}m_{1}v_{1}^2+\frac{1}{2}m_{2}v_{2}^2=\frac{Gm_{1}m_{2}}{r}$ if they met with a distance r between their centres of mass. This equation therefore tells you the velocities needed for the two bodies to escape the gravitational pull of each other, i.e. the escape velocities. So, why does the formula for the escape velocity of an object on earth only include the kinetic energy of the object, and not the earth itself? Is the kinetic energy of the earth just negligibly small (because the conservation of momentum means its velocity is pretty much zero) and can therefore be ignored? Would it only be necessary if the two objects had similar mass?

PeroK
Homework Helper
Gold Member
2020 Award
The equation you are used to for the gravitational potential assumes a small mass and a large mass, where the large mass is assumed not to move. If you have two masses of a similar relative size, then you have a different equation for the gravitational potential of the system. You might like to derive it as a useful exercise.

Alexander350
sophiecentaur
Gold Member
2020 Award
Is the kinetic energy of the earth just negligibly small (because the conservation of momentum means its velocity is pretty much zero) and can therefore be ignored?
The momentum is proportional to the velocity but the KE is proportional to the velocity squared. So the sharing of the available energy into the KE of each object is proportional to the square of the ratio of the velocities and inversely with the ratio of the masses. Velocity wins.

Alexander350
The equation you are used to for the gravitational potential assumes a small mass and a large mass, where the large mass is assumed not to move. If you have two masses of a similar relative size, then you have a different equation for the gravitational potential of the system. You might like to derive it as a useful exercise.
I have been thinking about this for a while and am still not sure how to derive it. The problem is that both the objects move different distances, so how can you calculate the work done on them?

PeroK
Homework Helper
Gold Member
2020 Award
I have been thinking about this for a while and am still not sure how to derive it. The problem is that both the objects move different distances, so how can you calculate the work done on them?

Look at how the force acts over a small distance. Don't forget conservation of momentum, which relates the velocities of the two masses.

Look at how the force acts over a small distance. Don't forget conservation of momentum, which relates the velocities of the two masses.
Even trying to use conservation of momentum, I always end up with the kinetic energy of one of the masses equalling the potential energy of the system. This cannot be right as then there would be twice as much energy as there actually is. How do I set up the equations as I am obviously approaching it wrong.

PeroK


text:
# Epoch: Ancient World (from 4000 BC to 1 BC)

## Description

Different civilizations and forms of government form on the banks of big rivers in Afrika (Egypt), Asia (including Mesopotamia, India, China), later also in the Mediterranean Area (Greece, Roman Empire). Although different in regions and time of origin, all these state forms have still a lot in commona hierarchical system of social classes, with aristocracy on the top and many specialized classes below, including priests, scribes, soldiers, craftsmen, farmers, and slaves. Most of these state forms lasted for millennia and centuries. This allowed the melting of the religion with the state forms.

The mathematics in these different state forms was developed to serve practical means, for instance, administration or collection of taxes. But over the centuries, this practical respect of mathematics became more and more independent from the practical application. Scribes and priests began to bring more abstraction into the practical calculations, to think about the concepts behind the pure practical meaning of arithmetics and geometry.

Unfortunately, the knowledge of mathematics of this time is very limited, because only a little evidence is left. It did not survive, either because the material used by scribes to write what they knew was not so resistant (e.g. bamboo, papyrus, tree bark, clay), or because of wars, fire, or intentional destruction of generations which came after.

## Chronology

• The Ishango Bone as a possibly the earliest known counting evidence

• Megalithic stone and wood settings
• Found mainly on the British Isles; following astronomic and geometric principles.
• The most famous example is Stonehenge. The settings form circles, ellipses, egg-shaped rings or more complicated figures.

• Babylonian Mathematics
• Solving linear, quadratic and some cubic equations, square roots, interpolation of logarithms * First occurrence of theorem, later known as Pythagoras' theorem * Babylonians did not formulate any proofs for the theorems they stated.

### about 630 BC to 580 BC

• Thales Of Miletus formulates first theorems in geometry ("father of geometry")
• Probably first to predict an eclipse of the Sun in 585 BC.
• The first natural philosopher in the Milesian School.
• Probably the teacher of Anaximander

### about 530 BC to 510 BC

• The Pythagoreans formulate interconnections between numbers, geometry, astronomy, and music.
• Influenced by travels to Egypt, and southern Italy.

### -480 BC

• Anaxagoras is first to introduce philosophy to the Athenians.
• In about 450 BC imprisoned for claiming that the Sun was not a god and that the Moon reflected the Sun's light.

• Empedocles Of Acragas formulates a four element theory of the worldfire, air, water, and earth. He also conducts on of the first empiric science experiments showing that air exists and is not just empty space by observing that water did not enter a vessel when placed underwater.
• First estimation of the period after which the motions of the sun and moon came to repeat themselves to 59 years by Oenopides Of Chios. Development of a theory for the Nile floods.
• Zeno Of Elea formulates paradoxes concerning the continuum ("paradoxes of motion"), some of which had an influence on the later development of mathematics.
• Influenced by the arguments of Parmenides and Plato he met in Athens.

• Beginnings of atomic theory by Leucippus Of Miletus, i.e. the theory that matter and space are not infinitely divisible.
• First proposal of the method of exhaustion by Antiphon, i.e. calculating an area by approximating it by the areas of a sequence of polygons.
• Finding the areas of "lunes" by Hippocrates Of Chios, i.e. crescent-shaped figures, using his theorem that the ratio of the areas of two circles is the same as the ratio of the squares of their radii.

• Contribution to the development of irrational numbers, Theodorus Of Cyrene proved that $\sqrt 3, \sqrt 5, \ldots, \sqrt {17}$ were not commensurable in length with the unit length.
• Further development of the atomic theory by Democritus Of Abdera i.e. the theory that matter and space are not infinitely divisible.
• Invention of "quadratrix" by Hippias Of Elis which may have been used by him for trisecting an angle and squaring the circle.

• First ideas towards the infinitesimal calculus: Bryson Of Heraclea claimed that the circle was greater than all inscribed polygons and less than all circumscribed polygons.

### about 400 - 350 BC

• Plato creates a new cosmology: he's name is attached to the Platonic solids representing the "elements of the universe" i.e. cube (=earth), tetrahedron (=fire), octahedron (=air), icosahedron (=water). Plato associated the dodecahedron with the whole universe.
• Archytas Of Tarentum finds
• two mean proportionals between two line segments,
• a solution to the problem of duplicating the cube
• a proof that there can be no number which is a geometric mean between two numbers in the ratio $\frac{n+1}n.$.

• Further developments in the theory of proportion, astronomy, exhaustion method by Eudoxus Of Cnidus.
• Anaxagoras Of Clazomenae formulates a first understanding of centrifugal force. He also formulates first known trials of squaring the circle with ruler and compasses (which was proven impossible not before 1882).

• Beginnings of propositional logic by Aristotle
• Plato's nephew Speusippus becomes head of the Academy on Plato's death, but in 340 BC he sent for Xenocrates to return to Athens to prepare to become his successor.

• Aristotle founds his own school: the Lyceum in Athens.
• Callippus
• Mades accurate determinations of the lengths of the seasons
• Constructs a 76 year cycle comprising 940 months to harmonise the solar and lunar years
• This calendar adopted in 330 BC and used by many later astronomers.

• Eudemus writes "History of Arithmetic" (two or more books), "History of Geometry" (two or more books), and "History of Astronomy" (two or more books).

### 250 BC

• Philon writes his treatise Mechanics.
• Euclid Euclid of Alexandria.
• The 'complete works of Euclid' were written by a team of mathematicians at Alexandria who took the name Euclid from the historical character Euclid of Megara who had lived about 100 years earlier.
• Eudoxus writes a book on geography called "Tour of the Earth".

### 232 BC

• Chrysippus to become the third head of the Stoa Poikile following the death of Cleanthes.

### 212 BC

• Archimedes killed during the capture of Syracuse by the Romans in the Second Punic War after all his efforts to keep the Romans at bay with his machines of war had failed.

### 194 BC

• Nicomedes criticises the method that Eratosthenes used to duplicate the cube.

• Hipparchus produces a star catalogue containing about 850 stars, probably not listed in a systematic coordinate system but using various different ways to designate the position of a star.

### 100 BC

• Posidonius to become the head of the Stoic School in Rhodes.
• Using a distance of 5000 stadia between Rhodes and Alexandria, Posidonius estimates a value of 240000 stadia for the circumference of the Earth.

• Zeno of Sidon first to show about 250 years after Euclid that Euclid's propositions were not deduced from the postulates and axioms alone, and Euclid does make other subtle assumptions.

Thank you to the contributors under CC BY-SA 4.0!

Github:

### References

#### Bibliography

1. Struik, D.J.: "Abriss der Geschichte der Mathematik", Studienbücherei, 1976

#### Adapted from other CC BY-SA 4.0 Sources:

1. O’Connor, John J; Robertson, Edmund F: MacTutor History of Mathematics Archive


text:
2 e-. 2. Half-reaction method depends on the division of the redox reactions into oxidation half and reduction half. Example #4: Sometimes, the "fake acid" method can be skipped. Cr 2O 7 2 - → Cr3+ 5. First, divide the equation into two halves; one will be an oxidation half-reaction If you have properly learned how to assign oxidation numbers (previous section), then you can balance redox equations using the oxidation number method. The half-reaction method works better than the oxidation-number method when the substances in the reaction are in aqueous solution. Cr2O72-  +  C2H6O     Simplify the equation by subtracting out water molecules, to obtain the Chemists have developed an alternative method (in addition to the oxidation number method) that is called the ion-electron (half-reaction) method. Sixth, equalize the number of electrons lost with the number of electrons You establish your two half reactions by looking for changes in oxidation numbers. The following reaction, written in … First of all balance the atoms other than H and O. 2) Here are the correct half-reactions: 4e¯ + 4H + … Hello, Slight road-bump while doing my … Redox Reactions: It is the combination oxidation and reduction reactions. By adding one electron to the product side of the oxidation half-reaction, there is a $$2+$$ total charge on both sides. records this change. You then use some arrows to show your half-reactions. not have to do anything. This page will show you how to write balanced equations for such reactions even when you do not know whether the H 2 O(l) and H + (aq) are reactants or products. $6 \ce{e^-} + 14 \ce{H^+} \left( aq \right) + \ce{Cr_2O_7^{2-}} \left( aq \right) \rightarrow 2 \ce{Cr^{3+}} \left( aq \right) + 7 \ce{H_2O} \left( l \right)$. For the reduction half-reaction above, seven H 2 O molecules will be added to the product side, \begin{align} &\text{Oxidation:} \: \ce{Fe^{2+}} \left( aq \right) \rightarrow \ce{Fe^{3+}} \left( aq \right) \\ &\text{Reduction:} \: \overset{+6}{\ce{Cr_2}} \ce{O_7^{2-}} \left( aq \right) \rightarrow \ce{Cr^{3+}} \left( aq \right) \end{align}. (There are other ways of balancing redox reactions, but this is the only one that will be used in this text. To determine the number and the ethanol/acetaldehyde as the oxidation half-reaction. Balancing it directly in basic seems fairly easy: Fe + 3OH¯ ---> Fe(OH) 3 + 3e¯ And yet another comment: there is an old-school method of balancing in basic solution, one that the ChemTeam learned in high school, lo these many years ago. listed in order to identify the species that are oxidized and reduced, Cr 2O 7 2 - → Cr3+ 5. In the oxidation half-reaction above, the iron atoms are already balanced. and the other a reduction half- reaction, by grouping appropriate species. First, divide the equation into two halves by grouping appropriate species. ion. Balancing Redox Reactions: The Half-Reaction Method Balanced chemical equations accurately describe the quantities of reactants and products in chemical reactions. one must now add one (OH-) unit for every (H+) present in the equation. Step 1: Write the unbalanced ionic equation. Balancing Redox Reactions with Half-Reaction Method? of dichromate ions. The Half-Reaction Method . Each half-reaction is balanced separately and then the equations are added together to give a balanced overall reaction. Here are the steps for balancing redox reactions using the oxidation state method (also known as the half-equation method): Identify the pair of elements undergoing oxidation and reduction by checking oxidation states; Write two ionic half-equations (one of the oxidation, one for the reduction) Now equalize the electrons by multiplying everything in one or both equations by a coefficient. Chemists have developed an alternative method (in addition to the oxidation number method) that is called the ion-electron (half-reaction) method. Each of these half-reactions is balanced separately and then combined to give the balanced redox equation. You cannot have electrons appear in the final answer of a redox reaction. You cannot have electrons appear in the final answer of a redox reaction. The chromium reaction can now be identified as the reduction half-reaction the following steps: The electrons must always be added to that side which has the greater By following this guideline in the example Organic compounds, called alcohols, are readily oxidized by acidic solutions MnO 2 → Mn 2O 3 Balance each redox reaction in acid solution using the half reaction method. give the 6 electrons required by the reduction half-reaction. Recall that a half-reaction is either the oxidation or reduction that occurs, treated separately. Balancing Redox Reactions: The Half-Reaction Method Balanced chemical equations accurately describe the quantities of reactants and products in chemical reactions. 4. Balance the atoms other than H and O in each half reaction individually. Oxidation half reaction: l (aq) → l 2(s) +7 +4. Balance the atoms in each half reaction separately according to the following steps: 1. this guideline, the oxidation half reaction must be multiplied by "3" to The half-reaction method for balancing redox equations provides a systematic approach. Since And that is wrong because there is an electron in the final answer. Here, you do all the electron balancing on one line. Recall that a half-reaction is either the oxidation or reduction that occurs, treated separately. \begin{align} 6 \ce{Fe^{2+}} \left( aq \right) &\rightarrow 6 \ce{Fe^{3+}} \left( aq \right) + \cancel{ 6 \ce{e^-}} \\ \cancel{6 \ce{e^-}} + 14 \ce{H^+} \left( aq \right) + \ce{Cr_2O_7^{2-}} \left( aq \right) &\rightarrow 2 \ce{Cr^{3+}} \left( aq \right) + 7 \ce{H_2O} \left( l \right) \\ \hline 14 \ce{H^+} \left( aq \right) + 6 \ce{Fe^{2+}} \left( aq \right) + \ce{Cr_2O_7^{2-}} \left( aq \right) &\rightarrow 6 \ce{Fe^{3+}} \left( aq \right) + 2 \ce{Cr^{3+}} \left( aq \right) + 7 \ce{H_2O} \left( l \right) \end{align}. Organic compounds, called alcohols, are readily oxidized by acidic solutions of dichromate ions. Another method for balancing redox reactions uses half-reactions. Balancing Redox Reactions. They serve as the basis of stoichiometry by showing how atoms and mass are conserved during reactions. gained by multiplying by an appropriate small whole number. (There are other ways of balancing redox reactions, but this is the only one that will be used in this text. Calculator of Balancing Redox Reactions When balancing redox reactions, the overall electronic charge must be balanced in addition to the usual molar ratios of the component reactants and products. When balancing redox reactions we have always - apart from all the rules pertaining to balancing chemical equations - additional information about electrons moving. 8. To do this, add water Is there a species that is being reduced and a species that is being oxidized? The example is the oxidation of $$\ce{Fe^{2+}}$$ ions to $$\ce{Fe^{3+}}$$ ions by dichromate $$\left( \ce{Cr_2O_7^{2-}} \right)$$ in acidic solution. The reduction half-reaction needs to be balanced with the chromium atoms, Step 4: Balance oxygen atoms by adding water molecules to the appropriate side of the equation. Separate the reaction into two half-reactions, one for Zn and one for N in this case. First, separate the equation into two half-reactions: the oxidation portion, and the reduction portion. For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. An unbalanced redox reaction can be balanced using this calculator. Oxidation and reduction reactions need to be bal- Electrons are included in the half-reactions. Check to make sure the main atoms, Zn and N are balanced. The sixth step involves multiplying each half-reaction by the smallest above, only the, The third step involves balancing oxygen atoms. Each half-reaction is then balanced individually, and then the half-reactions are added back together to form a new, balanced redox equation. by reduction with the number of electrons produced by oxidation. The OH- ions, Then, on that side of the equation which contains both (OH. It can be done via the following systematic steps. The half-reaction method works better than the oxidation-number method when the substances in the reaction are in aqueous solution. The reduction half-reaction needs to be balanced with the chromium atoms, Step 4: Balance oxygen atoms by adding water molecules to the appropriate side of the equation. It depends on the individual which method to choose and use. In the ion-electron method, the unbalanced redox equation is converted to the ionic equation and then broken […] Balance the unbalanced redox reaction without any complications by using this online balancing redox reactions calculator. For reactions occurring in acidic medium, add H2O to balance O atoms and H+ to balance H atoms. $14 \ce{H^+} \left( aq \right) + \ce{Cr_2O_7^{2-}} \left( aq \right) \rightarrow 2 \ce{Cr^{3+}} \left( aq \right) + 7 \ce{H_2O} \left( l \right)$. They already are in this case. Determine the oxidation numbers first, if necessary. Now the hydrogen atoms need to be balanced. Below is the modified procedure for balancing redox reactions using the oxidation number method. For the oxidation half-reaction, the electrons will need to be added to the product side. BALANCING REDOX REACTIONS. Balancing Redox Reactions via the Half-Reaction Method Redox reactions that take place in aqueous media often involve water, hydronium ions (or protons), and hydroxide ions as reactants or products. Balancing Redox Equations for Reactions in Acidic Conditions Using the Half-reaction Method. It happens when a transfer of electrons between two species takes place. Step 7: Check the balancing. Balancing it directly in basic seems fairly easy: Fe + 3OH¯ ---> Fe(OH) 3 + 3e¯ And yet another comment: there is an old-school method of balancing in basic solution, one that the ChemTeam learned in high school, lo these many years ago. Another method for balancing redox reactions uses half-reactions. H 2O 2 + Cr 2O 7 2- → O 2 + Cr 3+ 9. The reduction reducing to the smallest whole number by cancelling species which on both and non-oxygen atoms only. Step 3: Balance the atoms in the half-reactions other than hydrogen and oxygen. Balancing Redox Equations: Half-Reaction Method. The nature of each will become evident in subsequent steps. Step 5: Balance the charges by adding electrons to each half-reaction. C2H4O oxidized, and chromium, is being reduced. (e-). Redox equations are often so complex that fiddling with coefficients to balance chemical equations doesn’t always work well. whole number that is required to equalize the number of electrons gained Let’s take a look at a simple reaction WITHOUT HYDROGEN OR OXYGEN to balance: K (s) + Br 2 (l) ! SO 4 2- → SO 2 7. under basic conditions. Example #4: Sometimes, the "fake acid" method can be skipped. 22.10: Balancing Redox Reactions- Half-Reaction Method, [ "article:topic", "showtoc:no", "license:ccbync", "program:ck12" ], https://chem.libretexts.org/@app/auth/2/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FIntroductory_Chemistry%2FBook%253A_Introductory_Chemistry_(CK-12)%2F22%253A_Oxidation-Reduction_Reactions%2F22.10%253A_Balancing_Redox_Reactions-_Half-Reaction_Method, 22.9: Balancing Redox Reactions- Oxidation Number Change Method, 22.11: Half-Reaction Method in Basic Solution, Balancing Redox Equations: Half-Reaction Method, information contact us at info@libretexts.org, status page at https://status.libretexts.org. Bewick, Richard Parsons, Therese Forsythe, Shonna Robinson, and then combined to give a balanced overall is... } \ ) ions as shown below in net ionic balancing redox reactions half reaction method, records this change hydroxide ions reduced... 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Such as photosynthesis and respiration are dependent upon the redox reaction can be with! Reaction worked example ( OCl- ) ion Cr^ { 3- } } \ ) ions will be used this... Will need to be added to the following steps: 1 simplify equation! Numbers 1246120, 1525057, and the reduction processes are the correct half-reactions: the oxidation portion, then... Solution is typically either acidic or basic, so hydrogen ions to balance the atoms than. Half-Reactions ” method to choose and use half-reactions of the equation into two half-reactions, one for in. Ions will be multiplied by six acidic medium, add water ( H the! Then balanced individually, and 1413739 all the electron balancing on one line lectures » half reaction.! \ ) molecules will be multiplied by six the hydrogen atoms Foundation under. Added together to form a new, balanced redox equation are other of! Chromium reaction can be done via the following steps: 1 is powerful... H+ and H2O ) problem illustrates how to use the “ half-reactions ” method to the. Are in aqueous solution N in this method of balancing redox equations provides a systematic approach do not occur,! Represents a balancing redox reactions half reaction method equal to ( -1 ) ” method to balance the reaction are in aqueous.... Of life such as this redox reaction in a half-reaction, the overall reaction is its with! Oxidation states, indicates that carbon is being reduced fifth, use electrons ( ). They serve as strong oxidizing agents in chemical reactions method is an integral part understanding... Be added to the ionic equation and then broken [ … ] balancing redox reactions: it is modified! Thunder Dolphin amusement ride trains listed in order to identify the species that are oxidized and reduced,.. Of balancing redox equations is described iron atoms are already balanced Slight road-bump while doing my half! Following systematic steps charge is \ ( 24+\ ) on both sides of the equation by subtracting out water,. Be added to the oxidation or reduction that occurs, treated separately atoms! In scenarios such as this redox reaction in acid solution ( adding H+ and )...

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# What is better in Monte Carlo integration: product of means or mean of products?

Let $$X$$ and $$Y$$ be two independent continuous random variables with pdfs $$f_X$$ and $$f_Y$$, respectively. Let $$\varphi_1$$ and $$\varphi_2$$ be two continuous functions from $${\mathbb R}$$ to $${\mathbb R}$$. I want to calculate $$E[\varphi_1(X)\varphi_2(Y)]< \infty$$ numerically. This is,

$$I = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \varphi_1(x)\varphi_2(y) f_X(x)f_Y(y)dxdy\\ =\int_{-\infty}^{\infty} \varphi_1(x)f_X(x)dx\int_{-\infty}^{\infty}\varphi_2(y)f_Y(y) dy< \infty.$$

Using Monte Carlo integration, I can either approximate $$I$$ using $$I \approx \frac{1}{n}\sum_{j=1}^{n} \varphi_1(x_j)\varphi_2(y_j),$$ where $$(x_j,y_j)$$ is an independent sample from the joint distribution of $$(X,Y)$$. Alternatively, I can use the approximation: $$I \approx \left[\frac{1}{n}\sum_{j=1}^{n} \varphi_1(x_j)\right]\left[\frac{1}{n}\sum_{j=1}^{n} \varphi_2(y_j)\right] .$$ In my case, and minding potential implementation errors, I am obtaining different results.

Question. Is there any reasons to prefer one approximation over the other?

Simulating from the pdfs is straightforward in my case.

• A variant of the second approximation is better, replacing the first factor by $$\frac1m \sum_{i=1}^m \varphi_1(x_i).$$ Then you can get an answer accurate to within (e.g.) 1% by choosing $m$ and $n$ separately so each factor is accurate to within 0.5%. Nov 11 '19 at 22:20
• You must have particular distributions and functions in mind because you have done the simulations. Depending on the choices there might be various difficulties with simulation. Please show your work and results so far so someone can answer based on context. Nov 11 '19 at 22:39
• @BruceET The model I am working on is relatively complex. I have tried to summarize my main question as other aspects of the model are irrelevant. Simulating for the pdfs is not a problem at all, in fact, it is straightforward. Nov 11 '19 at 22:42
• If you're in doubt, maybe you should let us decide what may be surprisingly problematic and what may seem irrelevant and actually be important. As it stands, I'll stick with my Answer. Nov 11 '19 at 23:27

Both approaches lead to unbiased estimators, hence comparing their variances amounts to comparing their second moments.

Without loss of generality, since the distributions of $$X$$ and $$Y$$ are arbitrary, we can take both $$\varphi_1$$ and $$\varphi_2$$ to be the identity transform. Then $$\mathbb E[(\bar X\bar Y)^2]=\mathbb E[\bar X^2]\mathbb E[\bar Y^2]=\frac{n\mathbb E[X]+\sigma^2_X}{n}\frac{n\mathbb E[Y]+\sigma^2_Y}{n}$$ while $$\mathbb E[\overline {XY}^2]=\frac{n\mathbb E[X]\mathbb E[X]+\sigma^2_{XY}}{n}$$ Therefore if both $$\mathbb E[X]$$ and $$\mathbb E[Y]$$ are close to zero $$\mathbb E[(\bar X\bar Y)^2]=\text O(n^{-2})\quad\text{and}\quad\mathbb E[\overline {XY}^2]=\text O(n^{-1})$$ while otherwise mileage may vary $$\mathbb E[(\bar X\bar Y)^2]=\text O(n^{-1})\quad\text{and}\quad\mathbb E[\overline {XY}^2]=\text O(n^{-1})$$

Using the same example as in the earlier answer shows that the variability is slightly lower for the product of the averages (left) than for the average of the products (right), obtained over 10³ replications of 10⁴ simulations (the 10³ curves are the cumulated means):

If we compare directly the variances estimated from the 10⁷ simulations in this example $$\text{var}(\bar X\bar Y)=\text{var}(\bar X)\mathbb E[\bar Y^2]+ \text{var}(\bar Y)\mathbb E[\bar X]^2\qquad\qquad\qquad\\=\frac{\text{var}(X)}{n}\frac{\mathbb E[Y^2]+(n-1)\mathbb E[Y]^2}{n}+\frac{\text{var}(Y)\mathbb E[X]^2}{n}$$ is estimated by

[1] 8.990575e-07

while the variance of $$\overline{XY}$$ is estimated by

[1] 1.097814e-06

If anything, I would favor the $$\bar X\times\bar Y$$ solution as it also writes as$$\frac{1}{n^2} \sum_{i=1}^n\sum_{j=1}^n X_iY_j$$giving the impression it exploits the independence between both samples in a more systematic manner.

Here is a trivial example in which both distributions are standard normal, the first function is $$\phi_1(x) = x^\prime = x+2,$$ and the second is $$\phi_2(y) = y^\prime = y^2.$$ So that $$E(X^\prime Y^\prime) = E(X^\prime)E(Y^\prime) = 2.$$

Letting $$n = 10^7,$$ simulations in R give essentially the same answer for both methods.

set.seed(1234)
x = rnorm(10^7);  y = rnorm(10^7)
x1 = x + 2;  y1 = y^2

mean(x1*y1)
[1] 1.998185

mean(x1); mean(y1); mean(x1)*mean(y1)
[1] 2.000025
[1] 0.9993708
[1] 1.998766


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## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

RECALL: The graph of an equation of the form $x=h$, where $h\ne0$, is a vertical line whose x-intercept is $(h, 0)$. All points on the line have an x-coordinate of $h$. Thus, the graph of $x=3$ is a vertical line whose x-intercept is $(3, 0)$. All points on the line have an x-coordinate of $3$ so the following points are on the line: $(3, 2)$ $(3, 0)$ $(3, -2)$ Plot the three points then connect them using a line to complete the graph. (Refer to the graph in the answer part above.)