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Article Text

Attitudes and experiences with secondhand smoke and smoke-free policies among subsidised and market-rate multiunit housing residents living in six diverse communities in the USA
1. Andrea S Gentzke1,2,
2. Andrew Hyland1,
3. Marc Kiviniemi3,
4. Mark J Travers1
1. 1 Department of Health Behavior, Roswell Park Cancer Institute, Buffalo, New York, USA
2. 2 Department of Epidemiology and Environmental Health, State University of New York at Buffalo, Buffalo, New York, USA
3. 3 Department of Community Health and Health Behavior, State University of New York at Buffalo, Buffalo, New York, USA
1. Correspondence to Dr Andrea S Gentzke, Department of Health Behavior, Roswell Park Cancer Institute, Elm & Carlton Streets, Buffalo, NY 14263, USA ; aslicht01{at}gmail.com

## Abstract

Background Given that higher smoking rates persist among lower socioeconomic populations, multiunit housing (MUH) environments may result in higher secondhand smoke (SHS) exposures among subsidised MUH residents. This cross-sectional assessment compares experiences with SHS and smoke-free policies among subsidised and market-rate MUH residents living in six US communities.

Methods MUH residents (n=1565) were surveyed regarding their smoke-free rules (home and building), SHS exposures and preferences towards smoke-free policies. Binary logistic regression identified predictors of each outcome, focusing on differences by subsidised housing status (subsidised vs market rate).

Results Among residents enforcing smoke-free home rules (76%, overall), 50% reported SHS incursions into their unit. Only 23% reported living in a smoke-free building; 56% of those living in smoking-allowable buildings reported preferences towards smoke-free building policies. Among market-rate housing residents, smoke-free home (OR=4.18) and building (OR=2.26) rules were significantly higher when children were present. Smoke-free building rules reduced the odds of SHS incursions among market-rate housing residents (OR=0.50), but no association was observed among subsidised housing residents. Non-smoking subsidised housing residents exhibited stronger preferences for smoke-free policies compared with those in market-rate housing.

Discussion Smoke-free home rules may not protect MUH residents from SHS exposures, particularly in subsidised MUH. Although strong preferences towards smoke-free policies were present overall, subsidised MUH residents may have fewer alternative smoke-free housing options available. Therefore, all publicly funded housing should be smoke free to protect these vulnerable populations. However, continued efforts to encourage privately owned MUH operators to adopt smoke-free policies are also necessary.

• Secondhand Smoke
• Multiunit Housing
• Policy
• Smoke-free

## Statistics from Altmetric.com

• Secondhand smoke (SHS) can move throughout a multiunit housing (MUH) building, entering into spaces where tobacco smoking is otherwise prohibited. Although clean indoor air legislation has contributed to reductions in SHS exposure over time, the magnitude of this decline has been smaller for lower socioeconomic groups.

• To date, most of the MUH literature has focused on the attitudes and experiences with SHS and smoke-free policies among subsidised or market-rate housing residents, exclusively. The current study, however, compares the experiences with SHS and smoke-free policies between these MUH populations.

• Results show that market-rate housing residents had significantly higher odds of protection from SHS through smoke-free home and building rules, particularly when children were present in the home, while no association was observed for either outcome among subsidised housing residents. Non-smoking subsidised housing residents exhibited stronger preferences for smoke-free building policies, but may have fewer opportunities to seek out smoke-free living environments, providing support for mandated smoke-free policies in all subsidised housing.

## Introduction

The US Surgeon General has extensively documented the harmful health effects of secondhand smoke (SHS) exposure, concluding that there is no safe level of exposure to SHS.1 Smoke-free air legislation has contributed to reductions in SHS exposures among US adults and children, but gaps in protection remain particularly in private settings such as homes and vehicles.1 2 Although the prevalence of US households with voluntary smoke-free home rules has increased considerably over the last two decades,2 3 this space still represents a major source of SHS exposure in both children and adults.1 4–6 Despite efforts to reduce SHS exposure in public places, about one-third of US adults and approximately 32 million children remain involuntarily exposed to SHS.3 7 These gaps in protection are not randomly distributed in the population, as disparities in SHS exposures are also observed by socioeconomic status (SES).8 Over the last two decades, the magnitude of the decline in SHS exposures has been smaller for lower SES populations, widening the gap between low-income and high-income groups for SHS exposures.7

Multiunit housing (MUH) environments may perpetuate these disparities. Over 79 million individuals, or about 25% of the US population, currently reside in MUH,9 though a greater proportion of MUH residents live below the poverty line and identify as racial or ethnic minorities compared with the US population overall.9–11 Air circulation patterns inside MUH can facilitate the migration of tobacco smoke through shared hallways and ventilation systems, potentially entering into units and areas where smoking is otherwise restricted.5 12–15 Therefore, despite having a similar prevalence of smoke-free homes rules as the US population overall,3 it is estimated that 27.6–28.9 million MUH residents are exposed to SHS in their home.9 Given the socioeconomic disparities in MUH residency9–11 and in the prevalence of smoking,8 MUH environments may put vulnerable populations, including children, the elderly and those who are disabled, at an increased risk of health complications due to SHS exposures.

Due to differences in resident populations, attitudes and experiences with SHS and preferences towards smoke-free policies may vary between residents of subsidised (public and affordable) and market-rate MUH. However, literature to date has primarily focused on either subsidised,16–19 or market-rate housing residents,20 21 exclusively, or has not distinguished the study population according to subsidised housing status.22–25 The purpose of this paper is to describe and compare the experiences and preferences towards SHS exposures and smoke-free housing policies among both subsidised and market-rate MUH residents living in six diverse communities throughout the USA.

## Methods

This paper describes the baseline data analysis of a longitudinal community-based educational intervention study aimed at increasing the adoption of smoke-free MUH policies. This study was initiated in three community pairs across the USA: Bismarck (BM) and Grand Forks (GF), North Dakota; Fort Collins (FC) and Pueblo (PB), Colorado; and Charleston (CH) and Columbia (CO), South Carolina. These communities were selected with the assistance of Americans for Nonsmokers’ Rights (Berkley, California, USA) based on their size and the availability of an existing group of public health practitioners present in each community that were familiar with issues related to SHS but who were not actively engaged in smoke-free MUH activities at the time of selection. All communities selected were protected by comprehensive smoke-free air legislation in public places either at the state (ND and CO) or local (SC) level.

Data were collected between August and December 2012 via telephone. Individuals aged 18 and older who resided in MUH (apartment, duplex, double/multifamily home, condominium or townhouse) and lived in each specified community were eligible for this study. Data collection was initiated using random digit dial (RDD) of landline (n=18 000) and cell phone (n=60 000) numbers. The sampling frame was restricted to the zip codes that represented each community. Commercially available lists of renters in each community were also included in the sampling frame. The renter lists (n=3) were called consecutively following the exclusion of duplicate records in the previous RDD and listed samples. Calling on these samples (RDD and listed) were completed until samples were exhausted. Figure 1 provides the number of completed interviews, response rates (RDD) and participation rates (lists) for each data source. The response rate for the RDD component was 32.1% (combined cell and landline), calculated using the American Association for Public Opinion Research (AAPOR) Response Rate 3 calculator.26 The participation rates for the listed samples ranged from 7.5% to 12.1% after exclusion of ineligible records. In households with more than one adult, a random selection process determined who was interviewed. The survey took approximately 25 min to complete. All MUH residents who completed the survey were mailed a check for $10. All study procedures were approved by the Institutional Review Board at Roswell Park Cancer Institute. Figure 1 Completed interviews, response rates and participation rates by data source. AAPOR RR3 accounts for ineligible records within the denominator and estimates how many cases of unknown eligibility are likely to be actually eligible.26 Participation rates were calculated for the listed samples; all records of unknown eligibility were assumed to be eligible and are included in the denominator for these calculations. AAPOR RR3, American Association of Public Opinion Research, Response Rate calculator 3. ### Measures 1. Enforcing a voluntary smoke-free home rule was defined as a response of ‘No’ to the question: ‘Do you allow smoking inside your residence?’. 2. Experiencing an SHS incursion into the unit was defined as a response of ‘Everyday’, ‘A few times a week’, ‘A few times a month’ or ‘Rarely’ to the question: ‘In the past 12 months, how often has tobacco smoke entered your unit from somewhere else in or around your building? (vs ‘Never’)’. Analyses of SHS incursions are restricted to respondents who enforced voluntary smoke-free home rules (n=1259). 3. Two questions were combined to determine the rules about smoking inside the building: 4. Any rules about smoking in the building were defined as a response of ‘Yes’ to the question: ‘Regardless of whether you personally have set rules about smoking in your home, has your landlord or property manager set any rules regarding smoking on your property?’. 5. Among respondents with ‘any rules’ in their building, living in a smoke-free building was defined as a response of ‘prohibited inside all areas of the building, including living units’ to the question: ‘Regarding these rules about smoking in the building, which of the following most accurately describes the official smoking policy in your building?’ (vs ‘prohibited in shared areas, but allowed inside living units’). 6. Preferences towards smoke-free building policies were defined as an answer of ‘Yes’ to the question: ‘Would you prefer to have a policy in your building that prohibits smoking in all indoor areas, including individual residential units and common indoor areas?’ Analyses of preferences are restricted to MUH residents who live in smoking-allowable buildings (n=1061). ### Predictors of interest The primary predictors of interest were the respondent’s subsidised housing status and smoking status. Subsidised housing residents were defined by an answer of ‘yes’ to the question ‘Do you currently live in public, affordable or subsidized housing or participate in a voucher or low-income housing program?’ Those responding ‘no’ were defined as market-rate housing residents. Responses of ‘don’t know’ or ‘refused’ were excluded (n=38). Two questions determined the respondent’s smoking status (current smoker vs current non-smoker): (1) ‘Have you smoked at least 100 cigarettes in your lifetime?’ and, if yes, (2) ‘Do you now smoke cigarettes every day, some days, or not at all?’. Those answering ‘yes’ to question (1) and either ‘some days’ or ‘every day' to question (2) were defined as current smokers. Current non-smokers included both never smokers (‘no’ to question (1)) and former smokers (‘yes’ to question (1) and ‘not at all’ to question (2)). ### Covariates Other covariates included age (18–24 years, 25–64 years and 65+ years), sex (male and female), race/ethnicity (white, non-Hispanic, black, non-Hispanic, Hispanic, other or missing), education (<high school, high school graduate/General Educational Development (GED), some college, college graduate or graduate school), presence of 1+ children under the age of 18 in the home (no and yes), sample type (RDD and targeted lists) and knowledge of SHS (continuous, number of questions answered correctly, range: 0–10). Type of MUH was dichotomised as ‘apartment’ versus ‘other’ MUH (duplex, double/multifamily home, townhouse or condominium), given the differences in building structure and ownership status associated with these types of MUH. ### Data analysis Data were analysed using IBM SPSS V.21. Univariate and bivariate analyses are stratified by community and subsidised housing status. Significance testing for univariate analyses was calculated using the χ2 test; p values <0.05 were considered statistically significant. Binary logistic regression within generalised linear mixed modelling (GLMM) assessed sociodemographic predictors of each outcome of interest overall and according to subsidised housing status (market rate, subsidised) while accounting for community clustering. Missing data in outcomes or predictor variables were excluded from analyses. Statistical tests for multiplicative interaction between subsidised housing status and smoking status were completed to assess whether non-smokers residing in subsidised housing experienced more SHS incursions into their living unit compared with non-smokers residing in market-rate housing. ### Weighting procedures Given the demographic diversity in the communities included in this study, community-specific poststratification weights were developed from Public Use Microdata Areas (PUMAs)27 as opposed to national or state-representative data sources to allow for the comparison of data between communities. PUMAs provide population-level demographic characteristics for geographically concise areas. For each community, PUMA person data were obtained for individuals aged 18 and over who resided in MUH. The frequency distributions for age, sex and race/ethnicity of the PUMA data in each community were used to construct poststratification weights. The weights were normalised to the unweighted sample size and were applied to the data for all univariate and bivariate analyses assessing differences by community. In lieu of applying weights to the multivariate analyses (GLMM), the covariates used in the construction of the weights (age, sex and race/ethnicity) were included in the models as independent variables. ## Results Across all communities, 1565 interviews were obtained. Weighted demographic characteristics of MUH residents overall and stratified by community are reported in table 1. The proportion of current smokers ranged from 23% (GF) to 34% (CO). Between 20% (FC) and 39% (PB) of residents lived in subsidised housing and over half of residents in four communities (PB, CO, BM and GF) had annual household incomes below$30 000. Racial and ethnic characteristics varied by community but were consistent between community pairs.

Table 1

Sociodemographic characteristics of 1565 MUH residents in six communities across the USA, 2012

### Voluntary smoke-free home rules

The prevalence of each outcome stratified by community and subsidised housing status is reported in table 2. Voluntary smoke-free home rules were reported by 76% of all respondents, ranging from 64% (CH) to 95% (FC) by community. Only 10% of respondents allowed smoking inside their unit without any restrictions (range: 1.3% (FC) to 15% (CO)). A significantly higher proportion of market-rate housing residents enforced voluntary smoke-free home rules compared with subsidised housing residents overall (82% vs 69%, p<0.001) and within each community.

Table 2

Prevalence of voluntary smoke-free home rules, SHS incursions, rules about smoking in the building and preferences for smoke-free policies among MUH residents living in six communities across the USA: overall and stratified by public/affordable housing versus market-rate housing

In multivariate modelling overall, non-smokers were significantly more likely to report enforcing smoke-free home rules compared with current smokers (OR=14.73, 95% CI: 10.25 to 21.17; table 3). Residents of subsidised housing were about 30% less likely to enforce smoke-free home rules, though this was marginally non-significant (OR=0.70, 95% CI: 0.49 to 1.002; table 3).

Table 3

Demographic predictors of MUH residents reporting each outcome of interest in six communities across the USA

In analyses restricted to market-rate housing residents, those with children in the home were 4 times more likely to enforce voluntary smoke-free home policies (OR=4.18, 95% CI:1.99 to 8.79) compared with those without children. Among subsidised housing residents, higher odds of smoke-free home rules were observed for women (OR=2.18, 95% CI: 1.19 to 3.98) and those with a college education (OR=3.58, 95% CI: 1.18 to 10.87).

### Rules about smoking in the building

Twenty-three percent (23.4%) of all respondents reported living in a smoke-free building, but most (67.3%) reported having no rules about smoking inside the building where they lived (table 2). Smoke-free building rules ranged from 6.4% (CO) to >40% (BM, GF) in community stratified analyses. Overall, smoke-free building policies were more common among residents of market-rate housing compared with those in subsidised housing though differences were noted in community-stratified analyses.

In multivariable modelling, no differences in smoke-free building policies were observed according to subsidised housing status (table 3). Overall, non-smokers were nearly twice as likely to live in a smoke-free building compared with current smokers (OR=1.94, 95% CI: 1.32 to 2.84; table 3), and similar associations were observed in analyses stratified by subsidised housing status (table 4). Market-rate housing residents with children in the home were twice as likely to live in a smoke-free building (OR=2.26, 95% CI: 1.38 to 3.76; table 4). Non-apartment style MUH residents were significantly less likely to reside in a smoke-free building, but this association persisted only among market-rate housing residents in stratified analyses (OR=0.35 (95% CI: 0.25 to 0.51; table 4).

Table 4

Demographic predictors of MUH residents reporting each outcome in six communities across the US: stratified by subsidised housing status*

### SHS incursions

Approximately 50% of residents who enforce smoke-free home rules reported experiencing an SHS incursion into their unit from somewhere else in or around the building, ranging from 34% (CH) to over 75% (FC) in community-stratified analyses (table 2).

In multivariable modelling, no main effects of SHS incursions were observed according to smoking status or subsidised housing status (table 3). However, there is suggestive evidence (p=0.18) of an interaction between smoking and subsidised housing status on experiencing SHS incursions, in which non-smokers residing in subsidised housing had increased odds of reporting SHS incursions (OR=1.8, 95% CI: 0.8 to 4.5; not shown) compared with non-smokers residing in market-rate housing. Overall, higher odds of experiencing SHS incursions into the unit were observed among Hispanic respondents (OR=1.79, 95% CI: 1.04 to 3.09) and women (OR=1.47, 95% CI: 1.10 to 1.95). Residents of non-apartment style MUH buildings (OR=0.72, 95% CI: 0.54 to 0.96) and older individuals (65+; OR=0.46, 95% CI: 0.22 to 0.93) were both less likely to report incursions (table 3).

Respondents who lived in a smoke-free building had significantly reduced odds of reporting SHS incursions into their unit compared with those who lived in a smoke-permitted building (OR=0.59, 95% CI: 0.44 to 0.80; table 3). Although market-rate housing residents also experienced reduced odds of SHS exposure due to smoke-free building policies (OR=0.50, 95% CI: 0.35 to 0.71, table 4), there was no difference in SHS incursions according to smoke-free building status among residents of subsidised housing.

### Preferences for smoke-free building policies

Over half (56%) of respondents who reside in smoking-allowable buildings reported they would prefer to have a policy in their building that prohibited smoking in all areas (range: 45% (PB) to 68% (GF); table 2). Preferences for smoke-free policies did not differ according to subsidised housing status overall, but community-specific differences were observed (table 2). Over 90% of residents already living in a smoke-free building reported preferences towards smoke-free building policies (data not shown).

In multivariable adjusted analyses, non-smokers were significantly more likely to report preferences towards smoke-free building policies overall (OR=5.80, 95% CI: 3.99 to 8.45; table 3) and in analyses stratified by subsidised housing (table 4). Those with children in the home were about 70% more likely to prefer a smoke-free building policy (OR=1.68, 95% CI: 1.05 to 2.68; table 3). In analyses restricted to market-rate housing, non-apartment style MUH residents were 40% less likely to prefer a smoke-free building policy (OR=0.58, 95% CI: 0.40 to 0.84; table 4) compared with apartment dwellers.

## Discussion

Despite enforcing smoke-free home rules, half of MUH residents reported experiencing SHS incursions into their personal living space, indicating that personal smoke-free home rules do not confer full protection from SHS exposures in smoking-allowable MUH buildings.12–15 Although the majority of residents reported they would prefer to have a policy inside their building that prohibited smoking in all location, only about 25% actually lived in a building with a policy. Although multiple factors may play a role in choosing MUH, such as price, location and available amenities, this suggests there is a disconnect between preferences and availability of smoke-free living environments across the range of factors that are considered important in making housing decisions. Despite variation in smoke-free building rules and external influences of community-level tobacco control policies, preferences towards smoke-free policies were generally consistent across communities and with previously reported estimates.24 25

Higher smoking rates and SHS exposures persist among populations with lower socioeconomic indicators.7 8 This study found evidence to suggest that non-smokers living in subsidised housing may have higher SHS exposures in their homes compared with non-smokers living in market-rate housing. Furthermore, the relationship between smoking status and preferences towards smoke-free policies was stronger among residents of subsidised housing, suggesting that in addition to the greater need for smoke-free policies in subsidised housing, there may also be higher demand for smoke-free policies in this environment.

The clean indoor air legislation in Colorado and North Dakota prohibits smoking in commonly used areas of MUH buildings.28 29 However, a sizeable proportion of residents in these states reported that smoking was allowed ‘anywhere, or at any time’ inside their building. These findings suggest there is a lack of enforcement of these regulations and that there is a need to educate residents and building operators about these MUH-specific smoking restrictions.

The extent of state-level or community-level tobacco control policies may influence perceptions towards SHS exposures in the home environment. For example, residents living in states without comprehensive clean indoor air legislation, such as South Carolina, may become desensitised to tobacco smoke exposures occurring inside their homes due to more frequent exposures in public places. In the current study, South Carolina MUH residents reported the fewest SHS incursions into their units, despite having the lowest prevalence of smoke-free building policy coverage. Although community-level clean indoor air policies were present in both Charleston and Columbia, South Carolina, this discrepancy in SHS exposures in the home may have been influenced by the lack of a comprehensive smoke-free air policy at the state level.30

Among residents of market-rate housing, non-smokers and those with children in the home were more likely to reside in a smoke-free building, suggesting that specific populations may be seeking out smoke-free living environments. However, financial constraints and housing availability may limit the ability of subsidised housing residents to seek out smoke-free housing. In this study, market-rate housing residents were also more likely to enforce voluntary smoke-free home rules when children were present in the home, whereas no association was observed among residents of subsidised housing. Therefore, in addition to SHS exposures from in or around the building, children living in subsidised housing may also experience SHS exposures originating from active smoking inside their unit. Moreover, the accumulation of tobacco smoke pollution inside these units and buildings may contribute to thirdhand smoke exposures,31 placing children and other vulnerable populations at increased risk of health effects from these pollutants.

In late 2015, the US Department of Housing and Urban Development (HUD) proposed a rule prohibiting the use of lit tobacco products inside all public housing authority (PHA) buildings.32 According to this rule, finalised in December 2016 and effective as of February 2017, all PHAs must implement a smoke-free policy no later than 18 months from the effective date, banning the use of all prohibited tobacco products (cigarettes, cigars, pipes and waterpipes/hookahs) inside PHA living units, indoor common areas, administrative offices and buildings, and will extend to outdoor areas up to 25 feet (7.62 meters) from PHA buildings and offices.33 However, the majority of MUH residents would not be directly affected by the HUD rule, including approximately 5 million residents participating in Section 8 or other voucher programmes as well as the remaining 72 million residents living in market-rate housing.34 Although socioeconomic disparities in protection from SHS exposures underscore the importance of continued work to encourage or mandate subsidised and public housing providers to implement smoke-free building policies, efforts are also needed to promote policy adoption among market-rate housing operators.

Results of this study suggest that individuals residing in non-apartment style MUH, such as duplexes, townhouses or condominiums, may have different experiences with SHS in their homes due to varying building structures and ownership that could influence their attitudes towards smoke-free building policies. Residents of non-apartment style MUH reported fewer SHS incursions, lower preferences towards smoke-free building policies and fewer smoke-free building rules compared with apartment-style MUH. A higher proportion of residents living in townhouses or condominiums may own their units compared with apartment-style MUH residents, which may affect attitudes regarding individuals’ rights to smoke inside these personal living spaces. In addition, it may be comparatively more difficult to adopt smoke-free policies in these MUH environments because modifications to existing rules may require approvals by homeowners’ association and resident advisory boards.35 Public health personnel need to be aware of these important differences when working with non-traditional MUH communities to effectively encourage smoke-free policy adoption.

This study is not without limitations. First, as this study was confined to six US communities, the results may not be generalisable to other populations, particularly in regions not represented in this study. Second, because MUH residents are a difficult segment of the population to reach using population-based sampling methods (RDD), it was necessary to introduce targeted renter lists to increase the efficiency of data collection. Respondents identified using the targeted lists tended to be older, have lower annual household incomes and reside in subsidised housing compared with the RDD respondents. Moreover, the low participation rates obtained from these listed samples may suggest a selection bias in participation, although no information on non-responders from these samples is available to refute these concerns. To compensate for the fact that the respondents identified from these samples may not be representative of the population of MUH residents residing in each community, post-stratification survey weights were introduced to compensate for the particularly low participation among men, younger respondents and certain racial and ethnic groups in this survey. Although low, our RDD response rate (32%) was similar to that calculated for a national RDD survey of MUH residents.24 However, these issues with sample identification and potential selection biases in participation in telephone-based surveys points towards a need to adopt dual-method or non-population based data collection procedures to identify MUH populations in the future, such as the utilisation of targeted lists, address-based sampling procedures (including door-to-door surveys)23 or the integration of Internet panels.25 Third, our outcomes are based on self-report. MUH residents may overestimate the rules about smoking inside their building,21 or there may be interindividual differences in the ability to detect SHS exposures occurring inside one’s home. Moreover, the language used to assess voluntary smoke-free home rules did not elaborate on what constituted ‘smoking’. Therefore, the rules regarding the use of combustible and non-combustible products as well as electronic cigarettes inside the home may be left open to interpretation by individual respondents. Finally, because the self-reported assessment of SHS incursions was assessed over the previous 12 months, respondents living in their residence for less than 1 year (9.4% of sample) may have reported on experiences not related to their current living situation. However, our findings were largely consistent between community pairs, suggesting that any differences would be consistent in larger geographic regions (ie, at the state level). In addition, the main goal of this study overall is to assess changes over time within respondents, so any misclassification from this baseline survey is likely to continue through follow-up.

## Conclusions

Voluntary enforcement of smoke-free home rules does not adequately protect MUH residents from SHS exposures inside their homes, particularly in subsidised MUH environments. Strong preferences for smoke-free policies were observed among all MUH residents, although subsidised housing residents may have fewer housing options available to seek out alternative smoke-free living environments. Due to these disparities, all publically funded housing should be required to implement smoke-free building policies in order to protect these vulnerable populations. However, efforts to encourage privately owned MUH buildings to adopt smoke-free policies should also continue.

## Footnotes

• Contributors All authors were involved in the conceptualisation of this manuscript. ASG conducted the statistical analyses and completed the initial draft of the manuscript. All authors contributed to the interpretation of results and approved the final submitted and revised versions of this manuscript.

• Funding This project was funded by R01CA151953 (NCI; Travers, Roswell Park Cancer Institute) and R25CA113951 (NCI; Freudenheim, State University of New York at Buffalo).

• Competing interests None declared.

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

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text:
### B - Double Clique

##### Languages: MS C# .NET 4.7.2053,GNU G++11 5.1.0 ...

You are given an undirected graph $G$ with $n$ nodes and $m$ edges. The set of vertices is $V$ and the set of edges is $E$.

Let the Complement of $G$ be $G'$. The Complement of a graph is a graph with all of the same nodes, but if there’s no edge between nodes $a$ and $b$ in $G$, then there is an edge between $a$ and $b$ in $G'$, and if there is an edge between $a$ and $b$ in $G$, then there is no edge between $a$ and $b$ in $G'$.

A Clique is a subset of nodes that have an edge between every pair. A subset of nodes $S$ is called a Double Clique if $S$ forms a clique in $G$, and $V - S$ forms a clique in $G'$. Note that an empty set of nodes is considered a clique. Given a graph, count the number of double cliques in the graph modulo $10^9 + 7$.

#### Input

Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. Each test case will begin with a line with two integers $n$ and $m$ $(1 \leq n, m \leq 2 \times 10^5)$, where $n$ is the number of nodes and $m$ is the number of edges in the graph. The nodes are numbered $1..n$. Each of the next $m$ lines will contain two integers $a$ and $b$ $(1 \leq a \lt b \leq n)$, representing an edge between nodes $a$ and $b$. The edges are guaranteed to be unique.

#### Output

Output a single integer, which is the number of Double Cliques in the graph modulo $10^9 + 7$.

Input
3 3 1 3 1 2 2 3
Output
4

format: plain


text:
# Interpret change in exponential models

Given a function that models an exponential situation, find the factor by which the quantity changes per unit time interval.

### Problem

Takumi started studying how the number of branches on his tree changes over time.
The relationship between the elapsed time, t, in years, since Takumi started studying his tree, and the total number of its branches, N, left parenthesis, t, right parenthesis, is modeled by the following function:
Complete the following sentence about the yearly rate of change in the number of branches.
Every year, the number of branches
by a factor of
.


text:
# Hours

How many hours is 9 days?

Result

h =  216 h

#### Solution:

$h = 24 \cdot \ 9 = 216 = 216 \ \text{ h }$

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

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Result of the product of the numbers 1, 2, 3, 1, 2, 0 is:
16. Icecreams
Karthik bought 30 icecreams for his birthday .2/5 of them are chocolate icecreams. Find the number of chocolate icecreams he bought?
17. Decimal expansion
Calculate: 2 . 1 + 0 . 10 + 7 . 10000 + 4 . 1000 + 6 . 100 + 0 . 100000 =


text:
Learn the difference between rational and irrational numbers, and watch a video about ratios and rates Rational Numbers. Mathematics. Rationale is a noun. log10 x is 2 when x = 100. All of the following numbers are rational. Start studying Rational or Irrational Numbers. Since rationale and reason both contain the letter E, you should have little trouble remembering that rationale is a set of reasons that support an argument. Rationales can be formal documents or an informal defense of a perspective. not governed by or according to reason. Someone who is rational is able to think clearly about something and reach a sound conclusion. The spelling and pronunciation of these two words are similar, but they differ ever so slight. Rational and Irrational Numbers: A rational number is any number that can be expressed in the form of {eq}\dfrac{p}{q} {/eq} where both p and q are integers and q is not equal to zero. Quiz Now: How Well Aware Are You of HIV AIDS. 8th grade. 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Since these words are actually different parts of speech, your choice between them comes down to how you use them in a sentence. Edit. Rational and Irrational numbers both are real numbers but different with respect to their properties. 1. a. The square root of 2 is irrational.How do I know? A rational number can be written as a ratio of integers. Many people are surprised to know that a repeating decimal is a rational number. Irrational beliefs include words like should, must, and ought to, while rational beliefs contain phrases like, I would like, it would be nice, or I would appreciate. Irrational Numbers. Most of the numbers that people use in everyday life are rational. 0 times. Preview this quiz on Quizizz. But an irrational number cannot be written in the form of simple fractions. The difference between a preference (rational belief) and demand (irrational belief) is substantial. Here, we will try to explain the difference between rational and irrational numbers with the … Khan Academy is a 501(c)(3) nonprofit organization. This smart calculator is provided by wolfram alpha. 2 can be written as 2/1. Depending on your profession, you may be required to provide a rationale for many or most of your decisions, especially if those decisions are expensive or time-consuming. If you're a straight-A student and still you worry about failing all of your classes, you're being irrational. The √4 is 2. 0. This quiz will evaluate students' ability to identify rational numbers and differentiate them from irrational numbers. $0.58\overline{3}$ Is the number rational or irrational? 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I will use each of these words in at least one example sentence, so you can see how they appear in context. a number which can be expressed as quotient or fraction of two integers is known as rational number and on other hand an irrational number is a number which cannot be expressed as a ratio. Rational Numbers. Save. Let me explain ... Squaring a Rational Number. Is 4 plus the square root of 2 rational or irrational? It is a rational decimal. Let’s summarize a method we can use to determine whether a number is rational or irrational. DRAFT. mama_g. Only the square roots of the square numbers; that is, the square roots of the perfect squares. What is the Difference Between Rational and Rationale? irrational synonyms, irrational pronunciation, irrational translation, English dictionary definition of irrational. A rational idea is realistic, achievable, and useful. Mathematics. Hainley’s rationale for murdering her physics professor was that if she didn’t stop him from inventing a time machine, he could inadvertently alter the course of all human history. Literally, the answer is "yes, 1 is rational or irrational." Is it rationale or rational? Any real number that cannot be expressed as a ratio of integers, i.e., any real number that cannot be expressed as simple fraction is called an irrational number. Wellbeing or Well-Being – Which is Correct? First category is known as rational numbers and the second category is known as irrational numbers. Rationale vs. Rational numbers are all real numbers, and can be positive or negative.A number that is not rational is called irrational.. It means sensible or logical. About the calculator: This super useful calculator is a product of wolfram alpha, one of the leading breakthrough technology & knowledgebases to date. A rational number is the one which can be represented in the form of P/Q where P and Q are integers and Q ≠ 0. A rational number is a number that can be expressed as a fraction (ratio) in the form where p and q are integers and q is not zero. Rationale is a noun that refers to a set of reasons that support a claim. Type your name beside your answer. Some of the worksheets below are Rational and Irrational Numbers Worksheets, Identifying Rational and Irrational Numbers, Determine if the given number is rational or irrational, Classifying Numbers, Distinguishing between rational and irrational numbers and tons of exercises. It is a rational square root number. containing such a syllable. 2 is an integer, it is rational. After your class has gained a basic grasp of rational and irrational numbers, use this worksheet to push them a little further in their understanding. An Irrational Number is a real number that cannot be written as a simple fraction.. Irrational means not Rational. 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## rational or irrational

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## *USER SECTION

Keyword type: model definition

This option is used to assign material properties to user element sets. The parameters ELSET, MATERIAL and CONSTANTS are required. The parameter ELSET defines the user element set to which the material specified by the parameter MATERIAL applies. Finally, with the parameter CONSTANTS the number of parameters needed to describe the user element (e.g. the thickness for a shell-type user element) is defined.

First line:

• *USER SECTION
• Enter any needed parameters.

Following line:

• First constant
• Second constant
• etc (maximum eight constants on this line)
Repeat this line if more than eight constants are needed to describe the user section.

Example:

*USER SECTION,MATERIAL=EL,ELSET=Eall,CONSTANTS=1
0.01

assigns material EL to all elements in (user element) set Eall. There is one constant with value 0.01.

Example files: .


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# Why a ball doesn't stop when it collides with a wall?

If a ball collides with a ball of same mass the first ball stops and the second ball gets the velocity of first ball.The first ball stops due to the reaction force acting on it. But when a ball collides with a wall why doesn't it stop due to the reaction force?

• The mass of the ball and the wall aren't the same. Also in real-life situations, the collision isn't elastic. That explains why the ball doesn't stop. The one you've described where the velocities get swapped is only true if the collision is between two equal masses and is elastic. – Pratham Hullamballi Apr 3 at 17:47
• Can you explain why you think it should stop? – BioPhysicist Apr 3 at 17:52
• When a ball collides with another ball of same mass it stops due to reaction force.When it collides with a wall there is also a reaction force.Then why doesnt it stop? – Abdullah Al Zami Apr 3 at 18:08
• oh ok, so your thought process is just "Reaction forces should stop things"? – BioPhysicist Apr 3 at 18:21
• But the action and the reaction forces aren't on the same object, therefore, don't expect them to always cancel out. You've misinterpreted Newton's 3rd Law. – Pratham Hullamballi Apr 3 at 18:26

I'll let the animation speak for itself.

The blue arrow shows the force on the object.

For this scenario to happen it is important that the collision is elastic (all energy is conserved). I used a force that is proportional to penetration depth. This way the balls feel a force that is the same during the deceleration and recoil. In inelastic collisions it is possible for two balls to stick together after the collision. Imagine two pieces of clay colliding in mid air. In inelastic collisions the force is smaller (or zero) during the recoil than during the deceleration.

• Is the force larger at second scenario? – Abdullah Al Zami Apr 4 at 12:56
• @AbdullahAlZami Both cases have the same spring constant so for a given penetration depth they experience the same force. The maximal force is larger though for the second scenario. This is because in the first case the second ball moves starts moving during the collision, decreasing the duration of the deceleration. – AccidentalTaylorExpansion Apr 4 at 13:09
• In the second case if the ball hits the wall with a velocity of 1 ms^-1(assume the mass is 1 kg).Then what will be amount of the force for reflecting the ball?Will it be (2xForce at first case)? – Abdullah Al Zami Apr 4 at 13:12
• @AbdullahAlZami For a general collision you can't know this. The change in momentum is $\Delta p=2$kgm/s though. You can calculate the force for the case where the force is like a spring (like I did) but this would be too long for a comment. Do you really want to know this? – AccidentalTaylorExpansion Apr 4 at 13:19
• Is the force at second case = 2x(force at first case)? – Abdullah Al Zami Apr 4 at 13:23

Because the reaction force is larger.

The force needed to make an object reach a certain speed is the same as the force needed to slow it down from that speed to zero over the same time. This is Newton's 2nd law.

So, when two equal and movable objects collide, the (action) force that is required to make one object speed up to a certain speed is exactly the same as the (reaction) force required to slow the impacting object down to precisely zero.

When two non-equal objects collide, this is not necessarily so, and thus you will not see any of the objects stopping. Instead you might see that when

• a lighter object hits a heavier object, the action/reaction forces between them are larger than they would be between equal objects. And thus, the reaction force that stops the lighter object is larger than what is needed to precisely slow it down to zero. Thus, the lighter object slows down to zero and then start moving backwards, as if it bounces off.
• When an object hits a wall, you can consider this a collision between a light object and a very heavy object, and so, this bouncing-back effect is much stronger (the reaction force is much bigger). It might actually be so strong that the impacting light object bounces backwards with exactly the speed it came with, just directed oppositely.
• Doesn't this beg the question as to why the reaction force is different in different scenarios? – BioPhysicist Apr 3 at 18:06
• Yeah!Are they different or not? – Abdullah Al Zami Apr 3 at 18:12
• The action-reaction terminology here is very misleading. I strongly recommend using different words. Action-reaction is almost universally associated with Newton’s 3rd law, and there the action-reaction forces are always equal and opposite – Dale Apr 3 at 19:07
• Action-reaction forces are always equal and opposite. But Newton's 2nd law says that a body's acceleration is inversely proportional to its mass, so during the time while the forces are applied, the lighter mass is accelerated to a higher speed (or change of speed to be precise) than the heavier one. – Cuspy Code Apr 3 at 19:58

Try considering momentum conservation. The wall won’t be moving before or after the collision with the ball, and presuming the collision is elastic (no energy lost to heat, sound etc.) then the sum of the momenta will be conserved before and after the collision and you should find that the ball must have the same momentum before and after the collision. So it must be in moving too.


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# Normal Form Games¶

Game theory is the study of interactive decision making. Consider the following situation:

Two friends must decide what movie to watch at the cinema. Alice would like to watch a sport movie and Bob would like to watch a comedy. Importantly they would both rather spend their evening together then apart.

To represent this mathematically we will associate utilities to the 4 possible outcomes:

1. Alice watches a sport movie, Bob watches a comedy: Alice receives a utility of $1$ and Bob a utility of $1$.
2. Alice watches a comedy, Bob watches a sport movice: Alice receives a utility of $0$ and Bob a utility of $0$.
3. Alice and Bob both watch a sport movie: Alice receives a utility of $3$ and Bob a utility of $2$.
4. Alice and Bob both watch a comedy: Alice receives a utility of $2$ and Bob a utility of $3$.

This is referred to as the "battle of the sexes" and we will represent it using two matrices, $A$ will represent the utilities of Alice:

$$A = \begin{pmatrix} 3 & 1\\ 0 & 2 \end{pmatrix}$$

and matrix $B$ will represent the utilities of Bob:

$$B = \begin{pmatrix} 2 & 1\\ 0 & 3 \end{pmatrix}$$

We refer to Alice as the row player and Bob as the column player:

• The row player chooses which row of the matrices the players will gain their utilities;
• The column player chooses which column of the matrices the player will gain their utilities.

Thus if the row player (Alice) chooses the first row (this corresponds to a sport movie) and the column player (Bob) chooses the second column (this corresponds to a comedy):

• The row player receives a utility of $A_{12}=1$
• The column player receives a utility of $B_{12}=1$

This representation of the stategic interaction between Alice and Bob is called a Normal Form Game.

## Definition of Normal Form Game¶

An $N$ player normal form game consists of:

• A finite set of $N$ players
• Strategy spaces for the players: $\{S_1,S_2,S_3,\dots,S_N\}$;
• Payoff functions for the players: $u_i:S_1\times S_2\dots\times S_N\to\mathbb{R}$

In this course we will only consider the case of $N=2$.

For the battle of the sexes:

• We have $N=2$ players (Alice and Bob)
• The strategy spaces: $S_1=S_2=\{\text{comedy}, \text{sport movie}\}$ or equivalently $S_1=S_2=\{1, 2\}$
• The payoff functions mapping an element of $\tilde s \in S_1\times S_2=\{(1, 1), (1, 2), (2, 1), (2, 2)\}$ to $\mathbb{R}$:

$$u_1(\tilde s)=A_{\tilde s},$$

$$u_2(\tilde s)=B_{\tilde s}.$$

We can use Python to represent these games, we will use the nashpy library to do so and we start by building our two matrices:

In [1]:
import nash
A = [[3, 1], [0, 2]]
B = [[2, 1], [0, 3]]

We then create a nash.Game instance:

In [2]:
battle_of_the_sexes = nash.Game(A, B)
battle_of_the_sexes

Out[2]:
Bi matrix game with payoff matrices:

Row player:
[[3 1]
[0 2]]

Column player:
[[2 1]
[0 3]]

In the next chapter we will start to see how to use that for further calculations.

# Examples of other common games¶

## Prisoners Dilemma¶

Assume two thieves have been caught by the police and separated for questioning. If both thieves cooperate and don’t divulge any information they will each get a short sentence. If one defects he/she is offered a deal while the other thief will get a long sentence. If they both defect they both get a medium length sentence.

This corresponds to:

$$A = \begin{pmatrix} 3 & 0\\ 5 & 1 \end{pmatrix}\qquad B = \begin{pmatrix} 3 & 5\\ 0 & 1 \end{pmatrix}$$
In [3]:
A = [[3, 0], [5, 1]]
B = [[3, 5], [0, 1]]
prisoners_dilemma = nash.Game(A, B)
prisoners_dilemma

Out[3]:
Bi matrix game with payoff matrices:

Row player:
[[3 0]
[5 1]]

Column player:
[[3 5]
[0 1]]

## Hawk Dove game¶

Suppose two birds of prey must share a limited resource. The birds can act like a hawk or a dove. Hawks always fight over the resource to the point of exterminating a fellow hawk and/or take a majority of the resource from a dove. Two doves can share the resource.

This corresponds to:

$$A = \begin{pmatrix} 0 & 3\\ 1 & 2 \end{pmatrix}\qquad B = \begin{pmatrix} 0 & 1\\ 3 & 2 \end{pmatrix}$$
In [4]:
A = [[0, 3], [1, 2]]
B = [[0, 1], [3, 2]]
hawk_dove = nash.Game(A, B)
hawk_dove

Out[4]:
Bi matrix game with payoff matrices:

Row player:
[[0 3]
[1 2]]

Column player:
[[0 1]
[3 2]]

## Pigs¶

Consider two pigs. One dominant pig and one subservient pig. These pigs share a pen. There is a lever in the pen that delivers food but if either pig pushes the lever it will take them a little while to get to the food. If the dominant pig pushes the lever, the subservient pig has some time to eat most of the food before being pushed out of the way. If the subservient pig push the lever, the dominant pig will eat all the food. Finally if both pigs go to push the lever the subservient pig will be able to eat a third of the food.

This corresponds to:

$$A = \begin{pmatrix} 4 & 2\\ 6 & 0 \end{pmatrix}\qquad B = \begin{pmatrix} 2 & 3\\ -1 & 0 \end{pmatrix}$$
In [5]:
A = [[4, 2], [6, 0]]
B = [[2, 3], [-1, 0]]
pigs = nash.Game(A, B)
pigs

Out[5]:
Bi matrix game with payoff matrices:

Row player:
[[4 2]
[6 0]]

Column player:
[[ 2  3]
[-1  0]]

## Matching pennies¶

Consider two players who can choose to display a coin either Heads facing up or Tails facing up. If both players show the same face then player 1 wins, if not then player 2 wins.

This corresponds to:

$$A = \begin{pmatrix} 1 & -1\\ -1 & 1 \end{pmatrix}\qquad B = \begin{pmatrix} -1 & 1\\ 1 & -1 \end{pmatrix}$$
In [6]:
A = [[1, -1], [-1, 1]]
B = [[-1, 1], [1, -1]]
matching_pennies = nash.Game(A, B)
matching_pennies

Out[6]:
Zero sum game with payoff matrices:

Row player:
[[ 1 -1]
[-1  1]]

Column player:
[[-1  1]
[ 1 -1]]

As indicated by nashpy, this is a Zero sum game:

$$A + B = 0$$

## Definition of a zero sum game¶

A two player normal form game with payoff matrices $A, B$ is called zero sum iff:

$$A = -B$$

To define a zero sum game using nashpy we can pass a single payoff matrix (it infers what the other will be):

In [7]:
A = [[1, -1], [-1, 1]]
matching_pennies = nash.Game(A)
matching_pennies

Out[7]:
Zero sum game with payoff matrices:

Row player:
[[ 1 -1]
[-1  1]]

Column player:
[[-1  1]
[ 1 -1]]


text:
# How to cut one hundred apples into pieces

In a basket there are 100 apples. Every apple weighs at least 25 gram, and the total weight of all apples is 10 kilogram. The teacher wants to cut these apples into pieces and assign the pieces to 100 children, so that each child receives 100 gram.

Question: Does there always exist a cutting so that each apple piece weighs at least 25 gram?

• can there be leftovers pieces (of like 8g or so)? – RiddlerNewComer Feb 16 '16 at 7:38
• In other words: is it always possible to cut pieces so that no child receives more than four pieces? – Fimpellizieri Feb 16 '16 at 8:05
• Is there limit for no of pieces for each child.? – Anbu.Sankar Feb 16 '16 at 8:35
• @RiddlerNewComer: no, there cannot be any leftover pieces (the total weight of the apples equals the total weight that the children should receive). – Gamow Feb 16 '16 at 13:14
• @Fimpellizieri: No. If the conditions of the puzzle are satisfied, then yes, every child receives at most four pieces. But if every child receives at most four pieces, then no, this does not imply that the conditions of the puzzle are satisfied. – Gamow Feb 16 '16 at 13:16

Suppose that we have $N+1$ apples (each at least $25$ grams) with a total weight of $100N$ grams, and we want to cut them into pieces of at least $25$ grams so that we can make $N$ groups weighing $100$ grams each.

If $N=1$, this is trivial: we have two pieces totalling $100$ grams, and we just put them together.

If $N>1$, I will show that we can always reduce the situation to make $N$ smaller.

There are three possibilities:

• The smallest apple and the largest apple together weigh at least $125$ grams, and the smallest apple weighs no more than $75$ grams;
• The smallest apple and the largest apple together weigh less than $125$ grams;
• Every apple weighs more than $75$ grams.

### Case 1: The smallest apple and the largest apple together weigh at least $125$ grams, and the smallest apple weighs no more than $75$ grams.

In this case, we can cut a piece off the larger apple so that the piece and the smallest apple weigh $100$ grams together. We have gotten rid of one apple and one set of $100$ grams, so we have reduced $N$ by $1$.

### Case 2: The smallest apple and the largest apple together weigh less than $125$ grams.

Because all the apples' weights sum to $100N$, the rest of the apples must have a total weight of at least $100N-125$ grams. The average weight of these apples is at least $\frac{100N-125}{N-1}=100-\frac{25}{N-1}$, which is at least $75$. Therefore the second-largest and largest apples must each weigh at least $75$ grams, and the smallest apple must weigh less than $50$ grams. We can cut a piece off the larger apple so that the piece and the smallest apple weigh $75$ grams together, then cut $25$ grams from the second-largest apple, for a total of $100$ grams. Again, we have gotten rid of one apple and one set of $100$ grams, so we have reduced $N$ by $1$.

### Case 3: Every apple weighs more than $75$ grams.

Let the weight of the largest apple be $x$. For each $m\le N$, let $W_m$ be the weight of the $m$-th smallest apple, let $S_m$ be the sum of the weights of the $m$ smallest apples, and let $D_m=100m-S_m$. $D_m$ represents the size of the piece that would need to be added to the first $m$ apples to make them total $100m$ grams. In particular, $D_0=0$ and $D_N=x$.

$D_m-D_{m-1}=100+S_{m-1}-S_m=100-W_m<25$, so $D_m$ cannot increase by more than $25$ grams at a time. Because $x>75$, the difference between $25$ and $x-25$ is more than $25$ grams. Because $D_0<25$ and $D_N>x-25$, some $D_m$ will necessarily have a value between $25$ and $x-25$. Then we can cut off a piece of size $D_m$ from the largest apple and combine it with the smallest $m$ apples to make $m+1$ pieces totalling $100m$ grams. The remaining $N-m+1$ pieces total $100(N-m)$ grams. This reduces our situation to two smaller situations.

Therefore, for all $N$, if we have $N+1$ apples with a total weight of $100N$ grams, we can divide them so that we have $N$ groups of $100$ grams.

We start with $100$ apples. The largest one is at least $100$ grams, so we can safely cut it in half. Then we have $101$ pieces totalling $10000$ grams. This corresponds to $N=100$. As just shown, we can cut these pieces to make $100$ groups of $100$ grams.

• @f'' proved that "101 apples (total 10000 grams) can be cut into >25-gram pieces and share to 100 children". But the question asks 100 apples. Are they the same thing? Am I missing something? – wilson Feb 17 '16 at 7:11
• @wilson Note the last paragraph "The largest one is at least 100 grams, so we can safely cut it in half". so you create a 101st apple this way – Ivo Beckers Feb 17 '16 at 14:38
• @Ivo fully understand now, thanks :) – wilson Feb 18 '16 at 2:01

I think the answer is:

Yes

Proof:

Use the following strategy. Take the smallest apple. If it is <= 75 g take the largest apple and cut a piece from it to match the first one to 100g. This is always possible because the largest apple is always >= 100 g. If the smallest apple is > 75 g you cut it in half and match each piece with pieces from the two largest apples. After doing this once you end up with essentially 99 apples that are at least 25g with an average of 100g, so exactly the original problem but with less apples. So repeat this strategy until there are no apples left and your done. This also actually proofs that you can do it in a way that every kid only gets 2 pieces.

• ugh... I think I made a mistake. In the case of >75g I actually take 200g away but reducing the apple count only by 1. But I think I'm close – Ivo Beckers Feb 16 '16 at 14:40

This is basically Ivo Beckers' solution. I'm not sure if the etiquette is to edit his answer or to post mine and credit him. I'm choosing the latter. Ivo, the glory goes to you!


text:
# Number of factors in $972$

Question: How many factors does $972$ have?

I listen them all out and got $18$. Which is a notourisly inefficient method.

So I'm wondering, if there is an algorithm for finding all the factors of a number, and what is the proof.

• Why the downvote? Nov 6, 2016 at 23:44

.Prime factorize $$972$$ first. This has to be done, otherwise there are extremely complicated algorithms, such as the elliptic curve factorization algorithm that do the job for ordinary numbers. Since $$972$$ is small, we can do our work by hand.

$$972 = 2\cdot 486 = 2\cdot 2\cdot 243 = 2\cdot 2\cdot 3\cdot 81 = 2\cdot 2\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3 = 2^23^5$$.

Now, any factor of $$972$$ comes from choosing numbers $$0 \leq i \leq 2$$, $$0 \leq j \leq 5$$, because then $$2^i3^j$$ is a factor of $$972$$.

Now, there are $$3$$ choices of $$i$$ and $$6$$ choices of $$j$$, hence the number of factors is $$3\cdot 6=18$$. This way, we do not need to write down the factors as well.

• Better method than mine! Nice! :) Nov 5, 2016 at 15:46
• @Frank Have I answered your question? Or is there something else you want answered? Nov 5, 2016 at 15:48
• Nah, I'm fine. The second half is answered in another answer. Nov 5, 2016 at 15:49
• Great. That's very good. Nov 5, 2016 at 15:50

If there were an (efficient enough) algorithm for that, then the RSA cryptosystem, which depends on factoring being hard, would be broken. So no, there's no easy algorithm yet known.

But for small numbers like yours, the Sieve of Eratosthenes is a not-bad place to start, and you only need to work up to about 35 or so, since if you find a factor $f$ bigger than $\sqrt{N}$, then $N/f$, which is less than $\sqrt{N}$, is also a factor, so you would have found it already.

• What's an RSA cryptosystem? I'm not too familiar with math programs such as Mathematica, Pari, PLSQ etc. Could you please explain? Nov 5, 2016 at 15:46
• See google.com/…. Nov 5, 2016 at 15:46

$$972=2^2\times3^5$$ Therefore, total number of factors must be

$$(2+1)\times(5+1)=18$$

• Please don't answer old questions which have already been answered if you have nothing new to add - even more so in the case of really trivial questions like this. Dec 6, 2016 at 16:17


text:
# ABM11 PDFs and the cross section benchmarks in NNLO

@inproceedings{Alekhin2013ABM11PA,
title={ABM11 PDFs and the cross section benchmarks in NNLO},
author={Sergey I. Alekhin and Johannes Bluemlein and S. Moch},
year={2013}
}
• Published 6 February 2013
• Physics
We report an updated version of the ABKM09 NNLO PDF fit, which includes the most recent HERA collider data on the inclusive cross sections and an improved treatment of the heavy-quark contribution to deep-inelastic scattering using advantag es of the running-mass definition for the heavy quarks. The ABM11 PDFs obtained from the updated fit are in a good agreement with the recent LHC data on the W- and Z-production within the experimental and PDF uncertainties. We also perform a determination of…
14 Citations

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text:
# How to create a multi column multi row table without borders in LaTeX?

I just want to create a table like this

but without the in-between line among rows. I also don't need the line between columns like this

but below the Existing Methods CFNeT multi-column i need a line which doesn't span the full table. I mean that the line must fit into the content.

• Your question leaves all the effort to our community, even typing the essentials of a TeX document such as \documentclass{}...\begin{document} etc. As it is, most of our users will be very reluctant to touch your question, and you are left to the mercy of our procrastination team who are very few in number and very picky about selecting questions. You can improve your question by adding a minimal working example (MWE) that more users can copy/paste onto their systems to work on. If no hero takes the challenge we might have to close your question. – Stefan Pinnow Oct 8 at 16:40
• You might want to look at the booktabs package. – barbara beeton Oct 8 at 17:40

\documentclass[11pt]{article}
\usepackage{graphicx}

\begin{document}
This a multirow and multicolumn table
\begin{tabular}{|c|c|c|}
\hline
a & b & g\\
\hline
b & c & h\\
\hline
g & v & a\\
\hline
\end{tabular}

\end{document}

The | represent the vertical lines separating each element of the cell and c is the option to centre the cell's content (there are others like r -> right and l -> left).

Finally \hline stands for horizontal line. I hope I was clear enough.

• THe OP clearly states in their question that they want a table "without the in-between line among rows. I also don't need the line between columns like this enter image description here". The example code in your answer produces a table with a lot of horizontal and vertical rows. Also your table does not make use of \multirow or \multicolumn. Could you please comment on that? – leandriis Oct 8 at 18:49

You have to compile at least twice with every change in the percentage (last) column (since it uses eqparbox for getting the appropriate alignment).

\documentclass{book}

\usepackage[
margin=1in,
landscape
]{geometry} % Just for this example

\usepackage{booktabs,collcell,eqparbox,makecell}

\begin{document}

\begin{table}
\centering
\newcommand{\pa}[1]{\eqmakebox[CFNetvsNeuMF][r]{#1}}% Percentage alignment
\newcommand{\hlc}{\textbf}% Highlight cell
\caption{Comparison results of different methods in terms of NDCG@10 and HR@10.}
\begin{tabular}{ *{9}{c} >{\collectcell\pa}c<{\endcollectcell} }
\toprule
&
&
\multicolumn{4}{c}{Existing methods} &
\multicolumn{3}{c}{CFNet} &
\\
\cmidrule(lr){3-6}\cmidrule(lr){7-9}
Dataset & % \raisebox{.7\normalbaselineskip}[0pt][0pt]{Dataset}
Measures & % \raisebox{.7\normalbaselineskip}[0pt][0pt]{Measures}
ItemPop & eALS & DMF & NeuMF &
CFNet-rl & CFNet-ml & CFNet &
\multicolumn{1}{c}{\smash{\makecell[b]{Improvement of \\ CFNet \emph{vs.} NeuMF}}} \\
\midrule
ml-1m &  HR  & 0.4535 & 0.7018 & 0.6565 & \hlc{0.7210} &
0.7127 & 0.7073 & \hlc{0.7253} &
0.6\% \\
& NDCG & 0.2542 & 0.4280 & 0.3761 & \hlc{0.4387} &
0.4336 & 0.4264 & \hlc{0.4416} &
0.7\% \\
lastfm &  HR  & 0.6628 & 0.8265 & 0.8840 & \hlc{0.8868} &
0.8840 & 0.8834 & \hlc{0.8995} &
1.4\% \\
& NDCG & 0.3862 & 0.5162 & 0.5804 & \hlc{0.6007} &
0.6001 & 0.5919 & \hlc{0.6186} &
3.0\% \\
AMusic &  HR  & 0.2483 & 0.3711 & 0.3744 & 0.3891 &
0.3947 & \hlc{0.4071} & \hlc{0.4116} &
5.8\% \\
& NDCG & 0.1304 & 0.2352 & 0.2149 & 0.2391 &
\hlc{0.2504} & 0.2420 & \hlc{0.2601} &
8.8\% \\


text:
# 2 A set of 4 vectors in RSis linearly independent cannot span RS spans a 4-dimensional subspace of RS

###### Question:

2 A set of 4 vectors in RS is linearly independent cannot span RS spans a 4-dimensional subspace of RS

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