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# Conditional and unconditional genome-wide association study reveal complicate genetic architecture of human body weight and impacts of smoking

## Abstract

To reveal the impacts of smoking on genetic architecture of human body weight, we conducted a genome-wide association study on 5,336 subjects in four ethnic populations from MESA (The Multi-Ethnic Study of Atherosclerosis) data. A full genetic model was applied to association mapping for analyzing genetic effects of additive, dominance, epistasis, and their ethnicity-specific effects. Both the unconditional model (base) and conditional model including smoking as a cofactor were investigated. There were 10 SNPs involved in 96 significant genetic effects detected by the base model, which accounted for a high heritability (61.78%). Gene ontology analysis revealed that a number of genetic factors are related to the metabolic pathway of benzopyrene, a main compound in cigarettes. Smoking may play important roles in genetic effects of dominance, dominance-related epistasis, and gene-ethnicity interactions on human body weight. Gene effect prediction shows that the genetic effects of smoking cessation on body weight vary from different populations.

## Introduction

Body weight is a typical complex trait in human. Classical genetic studies have shown that the heritability of obesity is about 40% to 70%1,2,3. Compared with BMI (body mass index) and body surface, body weight can be easily and accurately measured, and thus is convenient to use for scientific researches.

Over the past two decades, more than forty candidate genes associated with human body weight have been found. Among them, many genes (FTO, MC4R, KCTD15, TMEM18, SH2B1, GNPDA2, MTCH2, NEGR1, TNNI3K, QPCTL, BDNF) have been reported several times, and genes FTO and MC4R were ones of the mostly confirmed by many studies1,4,5. The FTO gene is the first weight-related gene to be discovered with specific function and metabolic pathways being explained clearly by functional and molecular biology6. Nevertheless, the estimated heritability of these weight-related genes is still low. Even if we add all the heritability of above-mentioned genes together, the result is less than 10%2,3, and majority of heritability is missing, coined as missing heritability. Epistasis and gene-environment (e.g., life style factors) interactions are considered as one of the primary reasons for missing heritability.

Acquired behaviors may have influences on body weight by genetic effects of weight-related genes, such as FTO7. Smoking is a series social problem and has direct influences on human body weight. A consensus opinion is that the average weight of smokers is lower than nonsmokers, and quitting smoking will lead to weight gain8. One article suggested that smoking may change the sensitization potential of body fat distribution9. Moreover, mother smoking during pregnancy is a possible factor leading to childhood obesity10,11.

Cracking genetic architecture of human body weight and identification of the genes related to weight is a major task. To reveal the genetic architecture of human body weight and the impact of smoking, we have conducted a genome-wide association study (GWAS) on body weight (WT) in a multi-ethnic genetic study by a model including additive effects, dominance effects, epistasis effects as well as their ethnicity interactions. For detecting impacts of smoking on genetic effects of human body weight, pack-years of cigarette smoking (SMK) was set as a covariant by conducting conditional association mapping12 in the conditional model (WT|SMK). For analyzing genetic models, estimating genetic effects, and improving test efficiency at reasonable computational costs, a mixed linear model approach was used13. The findings will provide useful information for tailored prevention and precision treatment.

## Materials and methods

### Study sample

The Multi-Ethnic Study of Atherosclerosis (MESA) is a study of the characteristics of cardiovascular disease and its physiological and social psychological risk factors14. Several thousand participants recruited form six US communities had physical examinations, and the genotype data was obtained from gene sequencing. 5,336 individuals passed the preliminary screening (repeat of two years with missing data, contained 9,984 observations). There are 38% of European–American (E-A population), 28% of African–American (A-A population), 22% of Hispanic–American (H-A population), 12% of Chinese–American (C-A population). The gender is basically evenly distributed in each ethnic population. The MESA dataset was obtained from the dbGaP (the database of Genotypes and Phenotypes) of US National Center for Biotechnology Information website (https://www.ncbi.nlm.nih.gov/gap, accession number: phs000209.v11.p3).

### Genotype and quality control

Before the formal study began, we performed quality control on the original genotype data. Single nucleotide polymorphism (SNP) with call rate less than 95% was deleted. Polymorphic at least in one ethnic group monomorphic SNPs in each ethnic group were deleted. Due to the allelic frequency differences between ethnic groups, no SNPs were filtered based on the minor allele frequency. The uniform heterozygosity distribution is limited to the range of 0–53%, and SNPs with heterozygosity larger than 53% were removed. And we also filtered the SNP with Hardy weinberg equilibrium (HWE) P-value larger than 1 × 10–4. Finally, 866,435 SNPs from 22 autosomes met the quality control standards and were used for our analysis.

### Phenotype and the cofactor

The age, gender, ethnicity, and other basic information including smoking habits of each study participant were assessed in the form of a set of self-reporting questionnaires. Physiological index like body weight and height was obtained by physical examinations, which were conducted twice, the second time was two years later than the first time. We took body weight (the variable name is “wtlb1”, the values were recorded in pounds) as the main object. Smoking (the variable name is “pkyrslc”, which means the average number of packets of cigarettes someone smoked per year) was chosen in this study as covariant.

### Statistical analysis

In this study, mixed linear model approaches were used for genome-wide association studies. The full model includes additive and dominance effects of singular SNPs (a, d), epistasis effects between pairs of SNPs (aa, ad, da, dd), and ethnicity-specific effects (e, ae, de, aae, ade, dae, dde). The phenotypic values of the k-th individual in the h-th ethnic population ($${y}_{hk}$$) can be expressed as

\begin{aligned} y_{hk} & = \mu + c_{hk} + s_{hk} + \sum\limits_{i} {a_{i} u_{{A_{ik} }} } + \sum\limits_{i} {d_{i} u_{{D_{ik} }} } + \sum\limits_{i < j} {aa_{ij} u_{{AA_{ijk} }} } + \sum\limits_{i < j} {ad_{ij} u_{{AD_{ijk} }} } + \sum\limits_{i < j} {da_{ij} u_{{DA_{ijk} }} } + \sum\limits_{i < j} {dd_{ij} u_{{DD_{ijk} }} } \hfill \\ & \quad + e_{h} + \sum\limits_{i} {ae_{ih} u_{{AE_{ihk} }} } + \sum\limits_{i} {de_{ih} u_{{DE_{ihk} }} } + \sum\limits_{i < j} {aae_{ijh} u_{{AAE_{ijhk} }} } + \sum\limits_{i < j} {ade_{ijh} u_{{ADE_{ijhk} }} } + \sum\limits_{i < j} {dae_{ijh} u_{{DAE_{ijhk} }} } + \sum\limits_{i < j} {dde_{ijh} u_{{DDE_{ijhk} }} } + \varepsilon_{hk} \hfill \\ \end{aligned}

where $$\mu$$ is the population mean; $$c_{hk}$$ is the fixed cofactor effects; $$s_{hk}$$ is the fixed gender effects; $$a_{i}$$ is the additive effect of the i-th locus with coefficient $$u_{{A_{ik} }}$$; $$d_{i}$$ is the dominance effect of the i-th locus with coefficient $$u_{{D_{ik} }}$$; $$aa_{ij}$$, $$ad_{ij}$$, $$da_{ij}$$ and $$dd_{ij}$$ are the digenic epistasis effects with coefficients $$u_{{AA_{ijk} }}$$, $$u_{{AD_{ijk} }}$$, $$u_{{DA_{ijk} }}$$ and $$u_{{DD_{ijk} }}$$; $$e_{h}$$ is the effect of the h-th ethnic population (1 for E-A, 2 for C-A, 3 for A-A, 4 for H-A); $$ae_{ih}$$ is the additive × ethnicity interaction effect of the i-th locus in the h-th ethnic population with coefficient $$u_{{AE_{ihk} }}$$; $$de_{ih}$$ is the dominance × ethnicity interaction effect of the i-th locus in the h-th ethnic population with coefficient $$u_{{DE_{ihk} }}$$; $$aae_{ijh}$$, $$ade_{ijh}$$, $$dae_{ijh}$$ and $$dde_{ijh}$$ are the ethnicity-specific epistasis effects in the h-th ethnic population with coefficient $$u_{{AAE_{ijhk} }}$$, $$u_{{ADE_{ijhk} }}$$, $$u_{{DAE_{ijhk} }}$$ and $$u_{{DDE_{ijhk} }}$$; and $$\varepsilon_{hk}$$ is the residual effect of the k-th individual in the h-th ethnic population.

In our mixed linear model, we take ethnicity-specific effects into consideration, instead of detecting them separately in each ethnic population. The general effects (a, d, aa, ad, da, dd) are effects stably expressed in each ethnic population, while the ethnicity-specific effects are specifically expressed in a certain population.

Conditional genetic model is an effective method to reveal conditional effects excluding interference factors15. If we add smoking variables in the mixed linear model as cofactor, it could reveal net genetic effects after removing influence of smoking.

To reduce the computational burden in mixed model-based GWAS analysis, we first used GMDR (Generalized Multifactor Dimensionality Reduction)16 method to scan 866,435 SNPs of 9,984 observations in one dimension (single SNP), two dimensions (two SNPs epistasis), and three dimensions (three SNPs epistasis), and a set of 807 candidate SNPs were obtained. In the second step, a GPU parallel computing software QTXNetwork was used to run the mixed linear model on 807 candidate SNPs. The estimated genetic effects and the standard errors (SE) of detected SNPs were obtained by Monte Carlo Markov Chain method with 20,000 Gibbs sampler iterations. The experiment-wise P values (PEW-value) for controlling the experiment-wise type I error (PEW < 0.05) were calculated by 2,000 permutation tests17.

### Bioinformatics analysis

We also search the detected genes for biological information through the text-mining search engine BiopubInfo. BiopubInfo is a full-text biological literature search engine based on words and phrases developed by our laboratory. The literature data includes all abstract databases of PubMed and full-text databases of more than 2,500 magazines. In order to realize the identification of different biomedical concepts by BiopubInfo, we sorted out the different expressions and synonyms of all concepts, and the hierarchical and subordinate relationships between concepts using 118 ontology databases including UMLS18, OLS19, BioOntology.org20 to build a database of biomedical concepts. In this research, we use BiopubInfo to search genetic networks between weight-related genes with chemicals, diseases, and gene-gene interactions.

### Estimation of genetic effects

As is mentioned above, mixed linear model can estimate genetic effects with their standard errors12. For a person of a certain ethnicity, we calculate the sum of the general effects and the ethnicity-specific effect of this ethnicity of all the weight-related SNPs to obtain the estimated value of his body weight. For simplicity and intuitively, we showed the estimation data, a 5,336 × 15 matrix, in the form of figure. We compared estimated genetic effects of base model (WT) to conditional model (WT|SMK), and revealed impacts of smoking on gene effects of body weight.

## Results

### Detected SNPs

Eleven SNPs were identified significantly by two models (seven SNPs in the base model WT and nine SNPs in the conditional model WT|SMK). Highly significant SNPs at the experiment-wise level (PEW < 10–5) are presented in Fig. 1. We use different shapes (circle and square) and colors (green, red, blue and black) to show different kinds of genetic effects of SNPs.

The estimated genetic effects (represented by the various colors and shapes in Fig. 1) and heritabilities of two models are summarized in Tables 1, 2 and 3, revealing the genetic patterns and gene-ethnicity interactions on body weight.

### Estimated heritability of different models

Estimated heritability of the base model (WT) and conditional model (WT|SMK) are presented in Table 1. The total heritability of human body weight in the base model is up to 61.78%, within the range of heritability for human body weight discussed in the previous studies (from 40 to 70%). The heritability is mostly contributed by ethnicity-specific dominance effects ($$h_{DE}^{2} \overset{\wedge}=$$ 18.30%) and ethnicity-specific epistasis effects ($$h_{IE}^{2} \overset{\wedge}=$$ 26.21%: $$h_{AAE}^{2} \overset{\wedge}=$$ 1.67%, $$h_{ADE}^{2} \overset{\wedge}=$$ 13.66%, $$h_{DAE}^{2} \overset{\wedge}=$$ 5.49%, $$h_{DDE}^{2} \overset{\wedge}=$$ 5.39%). SNP rs2504934 in the SLC22A3 gene has the highest heritability ($$h_{{de_{3} }}^{2} \overset{\wedge}=$$ 17.93%) in A-A population, and it is the leading result of high heritability of ethnicity-specific dominance effects ($$h_{DE}^{2} \overset{\wedge}=$$ 18.30%). The C/C of rs11684785 × C/T of rs17094894 (TMEM163 × near VRK1) has the highest epistasis heritability ($$h_{{ade_{1} }}^{2} \overset{\wedge}=$$ 20.61%) in E-A population, and it leads to high heritability of ethnicity-specific additive × dominance epistasis effect ($$h_{ADE}^{2} \overset{\wedge}=$$ 13.66%) with high significance (PEW -value = 7.43 × 10−5).

The total heritability of conditional model on smoking ($$h_{T}^{2} \overset{\wedge}=$$ 66.91%) is higher than that of base model, mostly due to epistasis effects ($$h_{I}^{2} \overset{\wedge}=$$ 13.86%), and ethnicity-specific epistasis effects ($$h_{IE}^{2} \overset{\wedge}=$$ 33.91%). The highest heritability ($$h_{{dae_{3} }}^{2} \overset{\wedge}=$$ 4.42%) belongs to G/A of rs2504934 in the SLC22A3 gene × T/T of rs620175 in the SDK1 gene. This weight-decreasing epistasis was not detected in the base model, indicating that this epistasis is suppressed by smoking.

### Genetic effects of SNPs related to human body weight

There were 32 highly significant genetic effects (PEW-value < 1 × 10−5) of human body weight detected in the base model (Table 2). Among all the genetic effects, nearly half of them are ethnicity-specific effects, but C-A population has fewer (2/18) ethnicity-specific effects, suggesting that weight-related genes have minor specific effects in Chinese-American.

### Genetic effects of SNPs not affected by smoking

When removing impacts of smoking in the conditional model, we detected 13 negative effects and eight positive effects of body weight not affected by smoking (Table 2).

The homozygote genotype of rs9639575 has additive effect ($$a \overset{\wedge}=$$ – 5.12 for T/T and 5.12 for G/G), which reveals that major frequency homozygous (T/T) has negative additive effect, but minor frequency homozygous (G/G) has positive additive effect. The heterozygote genotype T/C of rs11684785 has a positive dominance effect ($$d \overset{\wedge}=$$ 6.26) and three negative epistasis effects with interaction to homozygote genotype T/T and heterozygote genotype T/C of rs620175 ($$da \overset{\wedge}=$$ – 4.02 for T/C × T/T, but 4.02 for T/C × C/C; $$dae_{1} \overset{\wedge}=$$ – 6.92 for T/C × T/T, but 6.92 for T/C × C/C. $$dd \overset{\wedge}=$$ – 7.74).

One dominance effect and three ethnicity-specific dominance effects of the heterozygote genotype G/A of rs2504934 will increase body weight by 4.22 lb, but decrease body weight by – 7.59 lb in E-A population and – 13.67 in H-A population, increase body weight by 22.27 lb in A-A population, respectively. Interestingly, this SNP not only has the largest positive effect but also the smallest negative effect among all genetic effects with highly significance ($$de_{3} \overset{\wedge}=$$ 22.27, PEW-value < 1 × 10−47; $$de_{4} \overset{\wedge}=$$ – 13.67, PEW-value < 1 × 10−32). This means SNPs can have totally different effects in different ethnic populations. The homozygote genotype G/G of this SNP also has five different epistasis effects with interaction to rs620175 in the SDK1 gene (additive × additive effect and additive × additive epistasis effects in all four populations, increases body weight in E-A and H-A populations, decreases body weight in C-A and A-A population), indicating that this epistasis regulates body weight through different metabolic networks in different populations and has widespread effects in different populations. These results show that SNP rs2504934 in the SLC22A3 gene has major effects on body weight, and it is not affected by smoking.

Genetic effects of rs620175 in the SDK1 gene have large parts of total effects (11/21). In addition to single effects ($$ae_{3} \overset{\wedge}=$$ 8.74 for T/T but – 8.74 for C/C, $$ae_{4} \overset{\wedge}=$$ – 5.68 for T/T but 5.68 for C/C), SNP rs620175 also has many epistasis effects with interaction to three SNPs (rs11684785 in the TMEM163 gene, rs2504934 in the SLC22A3 gene, and rs1277840 in the CACNB2 gene). Epistasis rs620175 × rs1277840 has positive dominance × dominance effect ($$dd \overset{\wedge}=$$ 6.46), while all the effect values of rs11684785 × rs620175 are negative ($$da \overset{\wedge}=$$ – 4.02 for T/C × T/T, $$dd \overset{\wedge}=$$ – 7.74, $$dae_{1} \overset{\wedge}=$$ – 6.92 for T/C × T/T).

### Genetic effects of SNPs caused by smoking

Eleven highly significant genetic effects were detected due to smoking (Table 2). All five single effects and ethnicity-specific effects in E-A population are positive, but ethnicity-specific effects in other three ethnic populations are negative. This indicates that smoking not only has overall effects of increasing body weight in all populations, but also can increase weight in E-A population and decrease weight in C-A, A-A, and H-A populations.

Compared to effects of SNPs not affected by smoking, genetic effects caused by smoking are smaller. The largest effect is 6.94 with heritability 1.74% of rs17094894 near the VRK1 gene. This shows that effect of smoking on weight-related genes is not so significant as compared to their own effects on body weight. This SNP also has ethnicity-specific additive × dominance epistasis effect with interaction to rs11684785 in the TMEM163 gene in E-A population ($$dae_{1} \overset{\wedge}=$$ 4.51 for T/C × C/C and – 4.51 for T/C × T/T), indicating that this SNP is a key factor in the mechanism of weight gain contributed by smoking in all populations, but is also a smoking-related factor in the mechanism of weight gain in E-A population.

SNP rs620175 has many additive-related effects, including ethnicity-specific additive effects in A-A population and three epistasis effects with rs2504934, rs9639575, and rs1277840. The only additive effect due to homozygote genotype A/A of rs1415210 in the CRB1 gene is highly significant ($$a \overset{\wedge}=$$ 4.13 for A/A but – 4.13 for T/T, PEW-value < 1 × 10−22), and this SNP also has another effect in E-A population ($$de_{1} \overset{\wedge}=$$ – 4.83). SNP rs1277840 is involved in two effects, one single effect ($$de_{3} \overset{\wedge}=$$ – 5.11) and one epistasis effect ($$dde_{1} \overset{\wedge}=$$ 8.34) with interaction to rs620175 in the SDK1 gene.

### SNPs suppressed by smoking

There were seven highly significant SNPs (PEW-value < 1 × 10−5) detected only in the conditional model with 25 genetic effects (Table 3).

SNP rs1467194 could only be detected in the conditional model; whereas SNP rs11684785 could not be detected in this model. However, these two SNPs are located close to each other in the same gene TMEM163. The homozygote genotype of rs1467194 takes part in two epistasis effects: additive × additive effect with interaction to rs9639575 ($$aa \overset{\wedge}=$$ – 3.51 for G/G × T/T and A/A × G/G, and 3.51 for G/G × G/G and T/T × A/A), ethnicity-specific additive × additive effect with rs17094894 in E-A population ($$aae_{1} \overset{\wedge}=$$ 3.45 for G/G × C/C and A/A × T/T, − 3.45 for G/G × A/A and C/C × T/T), and ethnicity-specific additive × dominance effect with rs17094894 in C-A population ($$ade_{2} \overset{\wedge}=$$ 8.44 for G/G × C/T but – 8.44 for A/A × C/T).

The dominance × additive effect ($$dae_{3} \overset{\wedge}=$$ – 11.76 for G/A × T/T but 11.76 for G/A × C/C) of rs2504934 × rs620175 is the negatively largest effect suppressed by smoking with the heritability of 4.42%, which contributes to a large part of the total heritability of epistasis effects in the conditional model. SNP rs2656825 was not detected in the base model, but has highly significant additive effect ($$d \overset{\wedge}=$$ 9.06, PEW-value < 1 × 10−64) in the conditional model. Apart from this single effect, six epistasis effects were detected due to interactions with three SNPs (rs620175, rs1277840 and rs2145965), implying that this SNP is involved in body weight regulation and could be suppressed by smoking.

Smoking also suppresses the ways in which SNPs regulates the body weight via other two important SNPs (rs1415210 and rs2145965). SNP rs1415210 has epistasis effects with rs17094894 ($$aa \overset{\wedge}=$$ – 6.44 for A/A × C/C and T/T × T/T, 6.44 for A/A × T/T and C/C × T/T; $$ade_{4} \overset{\wedge}=$$ – 9.97 for A/A × C/T but 9.97 for T/T × C/T). The heterozygote genotype G/C of rs2145965 near the KIZ gene has a positive dominance effect ($$d \overset{\wedge}=$$ 6.42) and three negative effects with interaction to rs2656825 ($$ad \overset{\wedge}=$$ – 3.17 for A/A × G/C but 3.17 for T/T × G/C, $$dd \overset{\wedge}=$$ – 7.77, $$ade_{1} \overset{\wedge}=$$ – 5.17 for A/A × G/C but 5.17 for T/T × G/C).

### Gene ontology analysis

Using BioPubInfo (https://ibi.zju.edu.cn/biopubinfo/), we also assessed the structural and functional connectivity for the detected candidate genes.

Figure 2a shows the network of two SNP-related genes (CREB5 and SLC22A3) not caused or suppressed by smoking. The CREB5 gene is connected to the SLC22A3 gene via three kinds of chemicals (estradiol, polyestradiol phosphate, and estradiol sulfate), which are all related to estrogen. Gonadal hormones can effectively control body weight, the activation of estradiol will affect the body's homeostasis for a long time in female animals, it has similar effects in humans21.

Figure 2b shows three SNP-related genes with the largest effects caused by smoking (CRB1, SDK1, SLC22A3). The SDK1 gene is connected to the CRB1 gene through valproic acid and to the SLC22A3 gene via tetrachlorodibenzo-p-dioxin (TCDD). These two chemicals were both proved to be related to body weight. A common adverse reaction of valproic acid is weight gain22. TCDD is a main compound in cigarettes, and animals treated with TCDD will develop overeating and become obese23. SNP rs2504934 in the SLC22A3 gene has seven kinds of epistasis effects with interaction to rs620175 in SDK1 gene in all populations (Table 3), indicating that this epistasis is important in body weight regulation, and that these effects may be caused by smoking through TCDD. From gene ontology analysis, the relationship between these three genes became clearer for smoking and body weight.

Figure 2c shows three SNP-related genes (TMEM163, VRK1, SLC22A3) with the largest effects suppressed by smoking. All the three genes are connected to each other by benzopyrene. Long-term exposure to benzopyrene may cause obesity24, and overweight/obesity can be experimentally induced by benzopyrene25, which is a main compound in cigarettes26, and all these three genes have effects contributed or suppressed by smoking. This gene ontology plot proves the reliability of our results in some ways.

Figure 2d shows other three key genes (CACNB2, VRK1, CREB5) that have interesting stories about body weight. The VRK1 and CREB5 genes are connected via 7,8-dihydro-7,8-dihydroxybenzo(a)pyrene-9,10-oxide, which is an analogue of benzopyrene. Cyclosporine is connected to genes CACNB2, VRK1 and CREB5. Cyclosporine, which is widely used in organ transplantation, was found to cause weight gain after transplantation27. The CREB5 gene and the VRK1 gene are connected through the JUN gene and the ATF2 gene. CRE (cAMP response element) can bind to the product of the CREB5 gene with c-Jun (another name of the JUN gene). The protein encoded by the ATF2 gene is closely related to CRE and c-Jun. An essay clearly described the regulatory principles and the complex metabolic interaction of CRE, c-Jun, the ATF2 gene, and the VRK1 gene28. This network may be a way of controlling body weight and the reasons are as follows. CREB (CRE binding protein) and ATF2 could affect the expression of UCP1, a protein essential for the thermogenic function of brown adipose tissue. The brown adipose tissue is related with heat production in human body, and there is a weight loss therapy for the purpose of activating the tissue29. Thus, we may have discovered a novel mechanism involving CRE of these four genes that has impacts on body weight and weight-related diseases through a complex metabolic network, the specific reason remained to be revealed.

### Gene effect prediction

There were 15 SNP/epistasis detected in the base model and 22 SNP/epistasis detected in the conditional model. All the 15 SNP/epistasis were also detected in the conditional model. The patterns of gene effects for 15 SNP/epistasis in the conditional model are almost identical with those in the base model (some new effects were detected when setting smoking as a covariant, indicating that these effects are suppressed by smoking). This shows the stability and reasonableness of our conditional method. Although some effects of the 15 SNP/epistasis were not detected in the conditional model (some effects disappeared when setting smoking as a covariant, indicating that these effects are caused by smoking), these effects have little effects on the population and are not large in number. Therefore, we believed that the overall impact of smoking on body weight is produced by suppressing the expression of certain weight-related SNP/epistasis.

In Fig. 3, we present genetic effects of the rest seven SNP/epistasis in the conditional model, which are suppressed by smoking. We arranged individuals of same population together, did clustering on the effects in each population, and then visualized the result by R software.

The most important result of Fig. 3 is that smoking has different effects on different populations. The effects of these seven SNP/epistasis were smaller in effect size and less in effects number compared to other three ethnic populations (represented in Fig. 3 as the region of C-A population has large area of blank or light color), showing that smoking has limited effects on C-A population. However, A-A population has large area of dark color, and the color is the darkest in all effects. From the point of single SNP/epistasis, rs620175 × rs2656825 has negative effects (purple color) on some people in A-A population but not in other populations; rs1277840 × rs2656825 has positive effects (orange color) on some people in H-A population but not in other populations. Some other SNP/epistasis have the same phenomenon.

The first column is the sum of all columns, from which we can infer when people quit smoking, he or she will gain or lose how much body weight. In E-A and H-A populations, a small number of people will gain weight heavily (people have red color in the first column), but also a small number of people will lose weight heavily (people have blue color in the first column), and major part of people will have slight weight change. In C-A population, smoking cessation has limited effects on body weight, but in A-A population, a majority of people will lose weight when quitting smoking.

## Discussion

Body weight is a typical complex trait in human that is affected by genetic and environment factors as well as their interactions. In our study, we assessed the genetic effects of individual genes, epistasis effects between pairs of genes, and ethnic-specific interaction effects by mixed linear model approaches. We identified eleven significant SNPs for body weight with PEW-value less than 1 × 10–5, and seven of them were found to be within the introns of identified genes. Among all the genetic loci detected, one gene (CRB1) has been reported by recent GWAS studies30, and three genes (TMEM163, SLC22A3, CACNB2) have been shown to have associations with human body weight.

Gene ontology analysis indicates that there may be new mechanism involving cAMP response element, which may affect body weight. Also, three chemicals (cyclosprine, estradiol, and benzopyrene) have a vast number of associations with genes detected through our gene ontology analysis, which have been reported to be related to body weight.

Smoking behavior as well as body-weight both may have correlative association with some SNPs, but direct biological evidences are insufficient. We have conducted a conditional GWAS to detect the impact of smoking on body weight by comparing the different results between two models (WT and WT|SMK), rather than directly studying the interaction with them. GWAS is hypothesis-free and does not require prior knowledge of a SNP’s biological impact on body weight.

Setting the first few principal components of genetic relationship matrix as covariates in the model is a common method for controlling population stratification in GWAS. The ethnicity-specific effect in our model is not a covariant, but an effect with estimate value, standard error, P-value and heritability. The population stratification due to different ethnic populations can be effectively controlled by ethnicity-specific effect, and there is no need to set principal components in the model as covariates.

Conditional genetic model is an effective method to reveal conditional effects excluding interference factors. When we added smoking variables in the mixed linear model as cofactor, it could reveal net genetic effects after removing influence of smoking. In this way, we can detect SNPs associated with body weight which are affected by smoking. It is worth mentioning that, a SNP will not be detected in the conditional model if it affect body weight only through smoking.

Gene effect prediction shows that smoking has different impacts on different human populations. The genetic effects of smoking cessation on body weight vary from different populations. Body weight of C-A population is not sensitive to smoking, while A-A population has opposite situation. In E-A and H-A populations, people can be divided into several subgroups, and precision medicine and intervention of living habits can be applied on each subgroup for smoking people to keep good fit.

Comparing with low heritability (less than 10%) of weight-related genes estimated by most of the previous GWASs2,3, the total heritability of human body weight in this study is up to 61.78% (base model). Majority of the heritability is contribute by ethnicity-specific epistasis $$h_{IE}^{2} \overset{\wedge}=$$ 26.21% ($$h_{AAE}^{2} \overset{\wedge}=$$ 1.67%, $$h_{ADE}^{2} \overset{\wedge}=$$ 13.66%, $$h_{DAE}^{2} \overset{\wedge}=$$ 5.49%, $$h_{DDE}^{2} \overset{\wedge}=$$ 5.39%), indicating that epistasis and ethnicity-specific effects are important parts of genetic structure of human body weight.

The heritability of epistasis × ethnicity effects accounts for the largest proportion of total heritability and more than half of the effects are ethnicity-specific effects. The same singular gene could have different impacts on different ethnic groups, therefore ethnicity-specific effects should be considered when designing personalized medicine therapies for weight related diseases.

Over the past decade, dozens of genes associated with body weight have been identified, but biologists have not confirmed most of them. A highly significant gene associated with human body weight detected in our study have been found to be significant in the previous GWAS studies on traits related with body weight. One GWAS assessed the relationship between the CRB1 gene and human body weight in Wight American30. The SNP rs1415210 in the first intron of CRB1 gene has single dominance effects and also epistasis effects with interaction to many other SNPs (rs7006789, rs2504934, rs17094894), indicating that rs1415210 regulates body weight through different metabolic networks. This gene plays a role in photoreceptor morphogenesis in the retina and may maintain cell polarization and adhesion. However, no evidence has been revealed to support the relationship between this gene and weight.

In this study, we found eight new significant SNPs associated with human body weight. The most important SNP is rs2504934. This SNP not only has the largest positive effect, but also has the largest negative effects that are highly significant ($$de_{3} \overset{\wedge}=$$ 22.27, PEW-value < 1 × 10−47; $$de_{4} \overset{\wedge}=$$ – 13.67, PEW-value < 1 × 10−32), indicating that this SNP is a key factor on body weight regulation. SNP rs2504934 of the first intron of gene SLC22A3 was detected to be highly significant in both base and conditional models, and has large positive effects. This gene is one of three similar cation transporter genes located in a cluster on chromosome 6, and this cluster was identified as a strong susceptibility locus for coronary artery disease through a genome-wide haplotype association study31. Coronary artery disease is one of the obesity-related diseases32,33.

SNP rs11684785 and SNP rs1467194 are located in the fourth intron of the TMEM163 gene. This gene encodes a protein of the same name that is associated with cellular zinc homeostasis34. Zinc is of great importance on metabolism, and seriously affects weight and the incidence of type II diabetes35,36. In a GWAS of Indian population for type II diabetes, TMEM163 is the most significant gene associated with type II diabetes37.

SNP rs17094894 is near the VRK1 gene, a gene has been identified to be associated with childhood obesity38. SNP rs17094894 has the largest effect with high heritability among effects caused by smoking. In a GWAS that studied the growth of boars, researchers found the association between the VRK1 gene and weight of boars, and this gene also related to cell growth and division39.

SNP rs620175 in the SDK1 gene has most effects among all SNPs. It has epistasis effects with interactions to other five SNPs, indicating that this gene regulates body weight by lots of mechanisms. Food conversation, which is closely connected to weight growth, is partly contributed by extracellular matrix and cell adhesion protein as SDK140.

SNP rs1277840 is in the fourth intron of the CACNB2 gene, and most of the effects involving rs1277840 are general effects, indicating that this gene can be expressed stably in all the ethnic groups. This gene is significantly associated with obesity-related diseases involving blood pressure41,42. A study on agouti showed that agouti regulates fatty acid synthase and fat storage via calcium channel, which is related to the protein encoded by this gene43. This gene was also found in human, and this illustrated that this gene may regulate human body weight via similar ways. In addition, increasing adipocyte intracellular calcium resulted in fat production, and increasing dietary calcium suppressed adipocyte intracellular calcium and thereby modulated energy metabolism and attenuates obesity risk44.

Fewer genetic effects were identified in C-A population, compared to the other populations. One possible reason is that the sample population consisted of only 12% Asian Americans. Perhaps a larger sample might help reveal more genetic structure of C-A population. Another problem is that human body weight may be affected by rare alleles, but we used a relatively conservative threshold in the first screen of SNPs to ensure the confidence; some rare alleles with large effects may be filtered out because of low P-value. Nonetheless, we still obtained a high total heritability and detected a number of SNPs in four populations with smoking as a covariant. Our results provide new insight in dissecting and understanding the sophisticated genetic architecture of human body weight, which will be useful for designing personalized medicine therapies for complex diseases.

## Data availability

Raw data is available on request.

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## Acknowledgements

The study was supported in part by grants from National Key Research and Development Program (2016YFC1303300), National Natural Science Foundation of China (31671570, 31871707), 111 Project (BP2018021), and National Science Foundation grant DMS2002865. The funders had no role in study design, data analysis. The authors thank the investigators, the staff, and the participants of the MESA study for their valuable contributions. A full list of participating MESA investigators and institutions can be found at https://www.mesa-nhlbi.org.

## Author information

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### Contributions

T.X. designed the study and wrote manuscript. M.M. checked and improved manuscript. X.L., J.Z. and H.X. developed the methods, analyzed the data, and revised the manuscript. All authors read and approved the final manuscript.

### Corresponding authors

Correspondence to Haiming Xu or Jun Zhu.

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Xu, T., Monir, M.M., Lou, X. et al. Conditional and unconditional genome-wide association study reveal complicate genetic architecture of human body weight and impacts of smoking. Sci Rep 10, 12136 (2020). https://doi.org/10.1038/s41598-020-68935-x

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text:
# The General Binomial Theorem: Part 2

In the previous post we established the general binomial theorem using Taylor's theorem which uses derivatives in a crucial manner. In this post we present another approach to the general binomial theorem by studying more about the properties of the binomial series itself. Needless to say, this approach requires some basic understanding about infinite series and we will assume that the reader is familiar with ideas of convergence/divergence of an infinite series and some of the tests for convergence of a series.

# The General Binomial Theorem: Part 1

### Introduction

One of most basic algebraic formulas which a student encounters in high school curriculum is the following $$(a + b)^{2} = a^{2} + 2ab + b^{2}$$ and its variant for $(a - b)^{2}$. And after many exercises and problems later one encounters another formula of similar nature namely $$(a + b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}$$ and one wonders if there are similar formulas for higher powers of $(a + b)$.

# Theories of Circular Functions: Part 3

Continuing our journey from last two posts we present some more approaches to the development of the theory of circular functions. One approach is based on the use of infinite series and requires basic knowledge of theory of infinite series. This approach is particularly well suited for treating circular functions as functions of a complex variable, but we will limit ourselves to the case of real variables only.

format: plain


text:
### Theory:

Consider plotting the points $$(0,4)$$, $$(1,5)$$, $$(2,6)$$, $$(3,7)$$ and $$(4,8)$$ in the graph.

From the graph, we can see that all these points lie on a line and also, there is a pattern in the coordinates.

Consider the $$y$$ - coordinate in every pair.

(I) $$4 = 0 + 4$$

(II) $$5 = 1 + 4$$

(III) $$6 = 2 + 4$$

(IV) $$7 = 3 + 4$$

(V) $$8 = 4 + 4$$

If all the points lie on the line, it is called as "a linear pattern".

In every pair, we can see that, the $$y$$ - coordinate is $$4$$ more than the $$x$$ - coordinate.

Also, we can write that $$y$$ - value $$=$$ $$x$$ - value $$+ 4$$

That is, $$y = x + 4$$.

This equation is called a linear equation, and the graph of the linear equation is called a linear graph.

In a linear equation there can be one or more variables where one variable is dependent on the other.
Example:
Let us consider if we buy $$1 \ kg$$ of tomato for $$₹25$$. If we buy more $$kgs$$ of tomato, the more we have to pay to buy.

Hence, if the quantity of tomato increases, then its cost also increases with it. Here, tomato is the independent variable whereas the cost depends on the quantity of tomato we buy. Hence, cost is a dependent variable.


text:
## 命令行工具

### scontrol

scontrol show node [nodename]

## Slurm节点状态

Slurm中节点的状态非常多，而且节点状态由状态和特殊符号两部分构成。节点状态中可能会有一个特殊的字符后缀用于标识节点关联的状态标志。组合起来的作业状态就更复杂了，经验不足的使用者想要理解还是有一些难度的。这里我以官方文件加上我自己的理解逐一解释各个状态。

### 特殊字符

“*”:

The node is presently not responding and will not be allocated any new work. If the node remains non-responsive, it will be placed in the DOWN state (except in the case of COMPLETING, DRAINED, DRAINING, FAIL, FAILING nodes).

“~”:

The node is presently in a power saving mode (typically running at reduced frequency).

“#”:

The node is presently being powered up or configured.

• 未知： UNKNOWN


text:
# Amplitude of a Wave

## What is called Amplitude of a Wave?

In a wave propagation, particles of the medium oscillates / vibrates to and fro about their mean position. The mean position is called equilibrium position. The displacement of the particle is measured by the distance from its mean position. When the particle is at mean position, its displacement is zero. The maximum displacement is at the extreme positions of the motion. Amplitude of a wave is the maximum distance covered on either side of the mean position by the medium particles.

Therefore, amplitude of a wave motion is defined as the maximum displacement made by the vibrating medium particles about their mean position.

Amplitude of wave is an important parameter by which properties of wave motions are specified. It is denoted by ( A ) . A medium particle covers a total distance of 4 / A , during one complete cycle of vibration.

Consider about the displacement-distance curve of a wave pulse as shown in figure. In the figure, ( O ) represents the mean position of vibrating medium particles. Therefore, amplitude of wave is the maximum displacement of vibrating medium particles. It is from ( O ) \ \text {to} \ ( + A ) in positive direction and from ( O ) \ \text {to} \ ( - A ) in negative direction of displacements from mean position.

## Properties of Wave Motion

Along with amplitude of wave, some other parameters of wave motion are also important by which the properties of wave are specified. These are –

1. Time period of wave motion.
2. Frequency of wave motion.
3. Wavelength of wave motion.
4. Wave number.
5. Propagation constant.
6. Wave velocity etc.

### Time Period of Wave Pulse

Consider about the displacement-distance curve of a wave pulse as shown in figure. In the figure, two complete cycles of wave pulses are shown.

Time period is the time taken to complete one complete cycle of oscillation or vibration by the medium particles.

It is denoted by ( T ) .

### Frequency of Wave Pulse

Frequency is the number of waves produced per unit time. It is the reciprocal of time period and denoted by ( \nu )

Therefore, \quad \nu = \left ( \frac { 1 }{ T } \right )

### Angular Frequency

The rate of change of phase with time is called angular frequency of the wave motion. It is denoted by ( \omega )

We know that, change in phase in time ( T ) is ( 2 \pi ) . Therefore, angular frequency will be –

\omega = \left ( \frac { 2 \pi }{ T } \right ) = 2 \pi \nu

When a wave propagates through a medium, all the medium particles oscillate about their mean position identically but they remain in differentPhase of one oscillating particle is different than the phase of its adjacent particles.

### Wavelength

Wavelength is defined as the distance covered by a wave during the time in which the medium particle completes one cycle of vibration. It is denoted by ( \lambda )

It may also be defined as the distance between two nearest particles of the medium which are vibrating in the same phase.

### Wave number

It is the count of number of waves present in unit distance of medium. It is denoted by ( \bar { \nu } ) .

Wave number is equal to the reciprocal of wavelengthTherefore, \quad \bar { \nu } = \left ( \frac { 1 }{ \lambda } \right ) .

### Propagation constant

The quantity \left ( \frac { 2 \pi }{ \lambda } \right ) is called angular wave number or propagation constant of a wave. It is denoted by ( k ) .

Therefore, \quad k = \left ( \frac {2 \pi }{ \lambda } \right )

### Wave Velocity

The distance covered by a wave per unit time is called wave velocity or phase velocity. It is denoted by ( v ) .

We know that, medium particle completes one cycle in time ( T ) . In this time interval, wave pulse travels a distance equal to the wavelength  ( \lambda ) .

Therefore, wave velocity \quad v = \frac {\text {Distance}}{\text {Time}}

Or, \quad v = \left ( \frac { \lambda }{ T } \right )

But, \quad \nu = \left ( \frac { 1 }{ T } \right )

Therefore, \quad v = \nu \lambda

\text {Wave velocity} = \text {Frequency} \ \times \ \text {Wavelength}

## Transverse Wave Motion in a Stretched String

Wave motion in a stretched string or rope is a transverse wave.

Hence, it is expressed by two factors –

1. Tension ( T ) in the string which is the measure of elasticity in the string. Wave cannot propagate in a string without tension.
2. Linear mass density ( m ) i.e. mass per unit length of the string which can store kinetic energy of wave propagation.

The speed of transverse wave through a stretched string is given by –

v = \sqrt { \frac { T }{ m }}

Ripples in a rope, vibration of guitar string etc. are examples of wave motion through a stretched string.

DIMENSIONAL CHECK

Dimension of tension ( T ) is [M L T ^ { - 2 }] . Dimension of mass per unit length ( m) is \left ( \frac { Mass }{ Length } \right ) = [M L ^ { - 1 }] .

Therefore, dimension of wave speed will be –

\sqrt { \frac { Dimension \ of \ T }{ Dimension \ of \ m }}

= \left ( \frac { M L T ^ { -2 } }{ M L ^ { - 1 } } \right ) ^ { \frac { 1 }{ 2 } }

= [ L  T ^ { - 1 } ] = Dimension of linear speed.

## Transverse Wave Motion through Solids

In solid bodies, a propagating wave can be a form of transverse wave.

The speed of transverse wave through a solid is expressed by two factors –

1. Modulus of rigidity ( \eta ) of solid body.
2. Mass per unit volume or density ( \rho ) .

The speed of transverse wave through a solid is given by –

v = \sqrt { \frac { \eta }{ \rho }}

Secondary waves or seismic waves during earthquake are the examples of transverse waves through solids.

DIMENSIONAL CHECK

Dimension of ( \eta ) is [M L ^ { - 1 } T ^ { - 2 }] . Dimension of ( \rho ) is \left ( \frac { Mass }{ Volume } \right ) = [M L ^ { - 3 }]

Therefore, dimension of wave speed will be –

\sqrt { \frac { \eta }{ \rho }} = \left ( \frac { M L ^ { - 1 } T ^ { - 2 } }{ M L ^ { - 3 } } \right ) ^ { \frac { 1 }{ 2 } }

= [ L  T ^ { - 1 } ] = Dimension of linear speed.

## Longitudinal Wave Motion through Solids

In solid bodies, a propagating wave can be a form of longitudinal wave.

The speed of longitudinal wave through a solid is expressed by two factors –

1. Modulus of compression or elongation i.e., Young’s modulus of elasticity ( \gamma ) of the solid.
2. Mass per unit volume or density ( \rho ) .

The speed of longitudinal wave through a solid is given by –

v = \sqrt { \frac { \gamma }{ \rho }}

Propagation of sound waves through solid materials, waves in helical springs etc. are examples of longitudinal waves through solids.

DIMENSIONAL CHECK

Dimension of ( \gamma ) is [M L ^ { - 1 } T ^ { - 2 }] . Dimension of ( \rho ) is \left ( \frac { Mass }{ Volume } \right ) = [M L ^ { - 3 }]

Therefore, dimension of wave speed will be –

\sqrt { \frac { \gamma }{ \rho }} = \left ( \frac { M L ^ { - 1 } T ^ { - 2 } }{ M L ^ { - 3 } } \right ) ^ { \frac { 1 }{ 2 } }

= [ L  T ^ { - 1 } ] = Dimension of linear speed.

## Longitudinal Wave Motion through Fluids

In fluid bodies, the propagating wave is a longitudinal wave.

The speed of longitudinal wave through liquids and gases are expressed by two factors –

1. Volume elasticity or Bulk modulus ( k ) of the solid.
2. Mass per unit volume or density ( \rho ) .

The speed of longitudinal waves through liquids and gases are given by –

v = \sqrt { \frac { k }{ \rho }}

Sound waves through liquids and gases are examples of longitudinal waves through fluids.

DIMENSIONAL CHECK

Dimension of Bulk modulus ( k ) is [M L ^ { - 1 } T ^ { - 2 }] . Dimension of density ( \rho ) is \left ( \frac { Mass }{ Volume } \right ) = [M L ^ { - 3 }]

Therefore, dimension of wave speed will be –

\sqrt { \frac { k }{ \rho }} = \left ( \frac { M L ^ { - 1 } T ^ { - 2 } }{ M L ^ { - 3 } } \right ) ^ { \frac { 1 }{ 2 } }

= [ L  T ^ { - 1 } ] = Dimension of linear speed.


text:
# Relation between CS/WZW and AdS/CFT

One precise example of realization of the holographic principle is the CS/WZW correspondence, which relates 3d Chern-Simons theory with the 2d Wess-Zumino-Witten model. As explained for example in this article in nlab there is a relation between this correspondence and the AdS3/CFT2 correspondence (which in turn relates 3d gravity and 2d conformal field theory). In particular CS/WZW appears to be some part (nlab says "sector") of the full AdS/CFT correspondence.

How does CS/WZW appear inside AdS3/CFT2? I'm asking, if possible, for a rough explanation of how this works. It seems that this happens in other dimensions too. Is there any similar TFT/CFT correspondence that can be seen as a sector of the usual AdS/CFT setting with $\rm{AdS}_5\times S^5$ type II supergravity / 4d $\mathcal{N}=4$ super Yang-Mills?

Holographic duality doesn't necessarily require $AdS$ space. See, for example, the famous CS/Topological String duality. In particular, I don't think that it is possible to consider CS/WZW as $AdS_3/CFT_2$.


text: With the add-in you can open a dialog, enter something like $y = \sum_{i=0}^n x_i$ and you will get an equation in your slide. This is done by embedding your Latex code in a template. Then, latex.exe is called in order to create a DVI which is afterwards converted in a PNG file by Ghostscript. The PNG is then inserted in your slide. Your Latex code is stored in the genertated PowerPoint object. Hence, you can edit your code later.


text: For a diatomic molecule the energy difference between rotational levels (J to J+1) is given by: For a certain diatomic molecule, the lowest-energy photon observed in the vibrational spectrum is 0.29 eV. This line occurs in the infrared, typically around 1000 cm-1, giving force constants k of the ... vibrational energy to be that of the ground state, and the other is to take the zero to be the bottom of Then we will use the BornOppenheimer approximation, to separate the nuclear and - electronic wavefunctions . A way to estimate the dissociation energy of a diatomic molecule is to determine the value of the vibrational quantum number, imax, at which the vibrational energy stops increasing. Seminar of atomic and molecular physics Presented by DINESH KUMAR KASHYAP. Vibrational motion of atoms bound in a molecule can be taken to be nearly simple harmonic. maximum value n max , i.e. Here’s an example that involves finding the rotational energy spectrum of a diatomic molecule. Another way a diatomic molecule can move is to have each atom oscillate—or vibrate—along a line (the bond) connecting the two atoms. The vibrational level spacing in the diatomic molecule N2 is 2330 cm^-1. In accordance with common practice, the bond axis is taken along the z-direction.There are six degrees of freedom, three of which are translations and two of which are rotations (about the x- and y-axes), leaving a single vibrational mode, which is a bond stretching “breather” mode. Rotational energy levels of diatomic molecules A molecule rotating about an axis with an angular velocity C=O (carbon monoxide) is an example. diatomic molecule . The selection rule for a rotational transition is, ∆ J = ± 1 (13.10) In addition to this requirement, the molecule has … The rotational energy levels of a diatomic molecule are shown in Fig. Within the harmonic and rigid rotor approximations, the rotational-vibrational energy levels (in wavenumbers) of a diatomic molecule are given by , where , are the vibrational and rotational quantum numbers, respectively, is the harmonic vibrational constant, and is the rotational constant. The vibration is associated with the two atoms moving in and out relative to one another's positions. The lowest vibrational transitions of diatomic molecules approximate the quantum harmonic oscillator and can be used to imply the bond force constants for small oscillations. the value for the vibrational quantum number where dissociation occurs, which allows us to determine the dissociation energy of the diatomic molecule. Equation (6-13) predicts that the vibrational spectrum of a diatomic molecule will consist of just one line. The frequency of molecular vibrations are in the order of 10-12 to 10-14 Hz. The lowest rotational energy level of a diatomic molecule occurs for l … Energy expended for one vibrational degree of freedom = 1/2RT + 1/2RT (1/2RT is kinetic energy and 1/2RT is potential energy) now, let us do the calculations for total energy being expended by the diatomic linear molecule. At high temperatures, they may also have vibrational energy. In case of a diatomic molecule, translational, rotational and vibrational movements are involved. Hence, each vibrational mode will contribute two degrees of freedom. $$3N-5=3(3)-5=4$$ And it would have 8 energy degrees of freedom associated with it When a molecule is irradiated with photons of light it may absorb the radiation and undergo an energy transition. What is the energy of a photon emitted in a transition from the fourth excited vibrational energy level to the second excited vibrational energy level, assuming no change in the rotational energy? The vibrational energy relaxation (VER) of a homonuclear diatomic molecule (X 2) in a 4 He superfluid nanodroplet (HeND; T = 0.37 K) was studied adapting appropriately a hybrid theoretical quantum approach recently proposed by us. The figure shows the setup: A rotating diatomic molecule is composed of two atoms with masses m 1 and m 2. This is the maximum possible value of the vibrational quantum number i in the anharmonic approximation. The term dissociation energy may be appreciated by reference to potential energy internuclear distance curves. At room temperature, what fraction of the N2 molecules are vibrationally excited (meaning not in the vibrational ground state)? The spectroscopic constants can be found in: Demtröder, Kapitel 9.5 Atome, Moleküle und Festkörper; CRC Handbook of Chemistry and Physics; K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure IV.Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979., Van Nostrand Reinhold, New York, 1979. 13.2. w1 & w2 are angular speeds} Thus, diatomic molecules are in the v = 0 vibrational level. For a general diatomic molecule, the vibrational motion is modelled by an infinite ladder of energy levels with energy spacing Δε = 252 J/mol. At ordinary temperatures, the molecules of a diatomic gas have only translational and rotational kinetic energies. Diatomic molecules are molecules composed of only two atoms, of the same or different chemical elements.The prefix di-is of Greek origin, meaning "two". We will derive the eigen energy values to understand the rotational and vibrational 22. Hence the Energy component of translational motion= 1/2 mv x 2 + 1/2 mv y 2 + 1/2 mv z 2. The vibrational energy is approximately that of a quantum harmonic oscillator: where n is an integer h is Planck's constant and f is the frequency of the vibration. The following is a sampling of transition frequencies from the n=0 to n=1 vibrational level for diatomic … Rotational motion of a diatomic molecule 2 Mtf itiI h R i th ilib i I R 0 Moment o inertia, , w ere 0 s e equilibrium internuclear separation, and is the reduced mass. We will start with the Hamiltonian for the diatomic molecule that depends on the nuclear and electronic coordinate. Fig. Vibrational energies. [SOUND] Now let's move on, to look at the last place where energy can be stored in a diatomic molecule, and that is, in the vibrations. The vibrational energy states of a heteronuclear diatomic molecule may be modeled using a potential energy function U(R) = 91.2.V (R – 0.115nm)", where R is the bond length of the molecule. 23. We have seen that the energy levels of a diatomic molecule in a state may be written as where the three terms are the energies of the electron cloud, of nuclear vibration along the internuclear axis , and rotation of the nuclei about an axis normal to . By Steven Holzner . In this simple molecule, the only vibration mode available is along the bond. The energy of the transition must be equivalent to the energy of the photon of light absorbed given by: $$E=h\nu$$. The vibrational state of the diatomic molecule refers to the frequency at which the atoms oscillate. At about 0 K all molecules have no rotational energy but are merely vibrating with their zero-point energy. Let us also note that the function d E ( 2 a ) n d n versus ( n + 1 2 ) decreases as a linear function of the variable ( n + 1 2 ) . More complicated molecules have many types of vibration and stretching modes. So, we'll look at the vibrational energy levels. The right panel shows the ground and first excited vibrational states, labeled and , respectively, with thei Since the reduced mass m r of the diatomic molecule is easily worked out, the vibrational frequency enables us to find a value for the force constant k. Together with the bond length, which we find from the rotational spectrum, we can thus obtain a fairly detailed picture of the diatomic chemical bond. A linear triatomic molecule would have 4 normal modes. As a result of this compared to lower temperatures, a diatomic gas at higher temperatures will have- Derive the Formula for the Rotational Energy of a Diatomic Molecule. The vibrational energy relaxation (VER) of a homonuclear diatomic molecule (X2) in a 4He superfluid nanodroplet (HeND; T = 0.37 K) was studied adapting appropriately a hybrid theoretical quantum approach recently proposed by us. 13.2 Rotational energy levels of a rigid diatomic molecule and the allowed transitions. Whats an equation I can use to be able to solve for this problem? 4.4 illustrates the vibrational energy level diagram for a diatomic molecule with a stiff bond (nitrogen N 2; left) and one with a looser bond (fluorine F 2; right). Fig. The vibrational energy level, which is the energy level associated with the vibrational energy of a molecule, is more difficult to estimate than the rotational energy level. Vibrational and Rotational Spectroscopy of Diatomic Molecules Spectroscopy is an important tool in the study of atoms and molecules, giving us an understanding of their quantized energy levels. energy curves associated with distinctive vibrational states, each with a range of differently spaced vibrational levels, indexed by sets of quantum numbers v¼0, 1, 2,…. Show that imax =Hn è e +xe n è eLêH2 xe n è eL. Diatomic molecules provide a convenient starting point for the discussion of molecular vibrations. The time scale for VER is found to decrease markedly with the increasing solute dipole moment, consonant with many previous studies in polar solvents. The difference is mostly due to the difference in force constants (a factor of 5), and not from the difference in reduced mass (9.5 u vs. 7 u). Energy component of rotational motion= 1/2 I 1 w 1 2 + 1/2 I 2 w 2 2 {I1 & I2 moments of inertia. 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View Poll Results: vote one norton 5 50.00% mcafee 4 40.00% pccillin 0 0% AVG 1 10.00% Voters: 10. You may not vote on this poll

## Which Anti virus software is the best ?

1) Norton
2) Mcafee
3) Panda
4) Norman
5) Bullguard
6) Pccillin
7) AVG grisoft--------FREE of charge
8) you name it

 PhysOrg.com science news on PhysOrg.com >> Heat-related deaths in Manhattan projected to rise>> Dire outlook despite global warming 'pause': study>> Sea level influenced tropical climate during the last ice age
 I gave up on anti virus a long time ago. It is too expensive and viruses are always a step ahead of the software. I just back up my computer every once and a while and I don't download from kaza or anything like that.
 Recognitions: Staff Emeritus Who needs antivirus protection when you got Linux.

Recognitions:
Gold Member

## Which Anti virus software is the best ?

 Originally posted by DarkAnt I gave up on anti virus a long time ago. It is too expensive and viruses are always a step ahead of the software. I just back up my computer every once and a while and I don't download from kaza or anything like that.
That's your choice of course but not very safe. AVG is free as mentioned above. Backing up your computer is in no way a protection against viruses. Any viruses that are on your system will be backed up as well. P2P programs like Kazaa are not the only source for viruses.

Certainly there are going to be new viruses that software won't be aware of, that still leaves the thousands of known viruses that can infect your system. Statistically, you are more likely to get a known virus than an unknown one.

 Thanks for the info, I guess I didn't read all the options. I didn't know there was a fairly good free one. ty [:D]
 There's about 2 Linux viruses I've seen so far. Bliss and Staog.
 I have had the best luck with Mcafee version 7 enterprise. It is a pretty good program as long as the repositories are up. The only drawback that I can see is that it adds a few more service processes. They don't use up CPU time, but they are there. The reason that I don't like Norton is that they keep getting bought out and they keep changing suites on their AV software. HTH M2k
 F-prot is the best one I've used to date. AVG is OK. Norton sucks. Haven't used McAfee long enough to know if it's any good. F-prot hasn't created any conflict problems for me. I *know* that Norton causes some problems while running simultaneously with other programs. And the real-time protector that comes with F-prot is great!! [:D]
 it keeps my computer from getting infected no matter how long it takes for companies to up date their definitions..... Linux and Mac OS X :-D...... ok, ok, I use windows XP as well, but I read my e-mail on OS X, and do most of my work in Linux and OS X, so my XP machine does not really have much of a chance on my network to get infected.
 The best antivirus I own is a Powermac G4 1.25 using OS X v. 10.2.8. :) Not a single virus for it. :)
 when talking about viruses, windows sucks i used almost every anti virus software (no matter if it was free, share or trial) and i find that mc afee is the best. with all mcafee (antivirus+firewall) ive got the best protection for my PC. at work we dont use any antivirus. but from time to time i scan my computer on line. on line scanning its a solution too...

before talking about wich anit virus is the best we shoult talk a lil bit about viruses and trojans.

 Originally posted by DarkAnt I gave up on anti virus a long time ago. It is too expensive and viruses are always a step ahead of the software. I just back up my computer every once and a while and I don't download from kaza or anything like that.
virus=a computer program that is able to replicate
(this is a "simple" definition of a virus)
a virus can be found in other parts of the computer (boot sectors, memory, etc) not only in your OS. if u make a back up with a virus inside u`ll do it for nothing... u will remain infected.


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## Functiones et Approximatio Commentarii Mathematici

### Fourier coefficients of Hecke eigenforms

Ronald Evans

#### Abstract

We provide systematic evaluations, in terms of binary quadratic representations of $4p$, for the $p$-th Fourier coefficients of each member $f$ of an infinite class $\mathcal{C}$ of CM eigenforms. As an application, previously conjectured evaluations of three algebro-geometric character sums can now be formulated explicitly without reference to eigenforms. There are several non-CM newforms that appear to share some properties with the eigenforms in $\mathcal{C}$, and we pose some conjectures about their Fourier coefficients.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 46, Number 2 (2012), 147-159.

Dates
First available in Project Euclid: 25 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.facm/1340628398

Digital Object Identifier
doi:10.7169/facm/2012.46.2.1

Mathematical Reviews number (MathSciNet)
MR2931662

Zentralblatt MATH identifier
1368.11038

#### Citation

Evans, Ronald. Fourier coefficients of Hecke eigenforms. Funct. Approx. Comment. Math. 46 (2012), no. 2, 147--159. doi:10.7169/facm/2012.46.2.1. https://projecteuclid.org/euclid.facm/1340628398

#### References

• B. C. Berndt, R. J. Evans, and K. S. Williams, Gauss and Jacobi sums, Wiley-Interscience, New York, 1998.
• J. Booher, A. Etropolski, and A. Hittson, Evaluations of cubic twisted Kloosterman sheaf sums, Int. J. Number Theory 6 (2010), 1349–1365.
• R. J. Evans, Seventh power moments of Kloosterman sums, Israel J. Math. 175 (2010), 349–362.
• R. J. Evans, Hypergeometric $\ \ABFF3F2(1/4)\$ evaluations over finite fields and Hecke eigenforms, Proc. Amer. Math. Soc. 138 (2010), 517–531.
• R. J. Evans and M. Van Veen, Rational representations of primes by binary quadratic forms, Int. J. Number Theory 3 (2007), 513–528.
• H. Iwaniec, Topics in Classical Automorphic Forms, Amer. Math. Soc., Providence, RI, 1997.
• H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc., Providence, RI, 2004.
• I. Miyada, On imaginary abelian number fields of type $(2,2,\dots,2)$ with one class in each genus, Manuscripta Math. 88 (1995), 535–540.
• K. Ono, The web of modularity: Arithmetic of the coefficients of modular forms and $q$-series, CBMS No. 102, Amer. Math. Soc., Providence, RI, 2004.


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# Jason leans a 12 foot ladder against the side of his house with the base of the ladder 3.5 feet from the house. To the nearest tenth, how high up the side of the house does the ladder reach?

$11.5$
${12}^{2} - {3.5}^{2} = 144 - 12.25 = 131.75$
$\sqrt{131.75} \approx 11.5$