text:
pediatrics
June 2014, VOLUME133 /ISSUE 6

# Parental Smoking Exposure and Adolescent Smoking Trajectories

1. Darren Mays, PhD, MPHa,
2. Stephen E. Gilman, ScDb,c,
3. Richard Rende, PhDd,
4. George Luta, PhDa,
5. Kenneth P. Tercyak, PhDa, and
6. Raymond S. Niaura, PhDa,e,f
1. aLombardi Comprehensive Cancer Center, Georgetown University Medical Center, Washington, District of Columbia;
2. bHarvard School of Public Health, Boston, Massachusetts;
3. cMassachusetts General Hospital, Boston, Massachusetts;
4. dBrown University Medical School, Providence, Rhode Island;
5. eSchroeder Institute for Tobacco Research and Policy Studies, American Legacy Foundation, Washington, District of Columbia; and
6. fBloomberg School of Public Health, Johns Hopkins University, Baltimore, Maryland

## Abstract

OBJECTIVE: In a multigenerational study of smoking risk, the objective was to investigate the intergenerational transmission of smoking by examining if exposure to parental smoking and nicotine dependence predicts prospective smoking trajectories among adolescent offspring.

METHODS: Adolescents (n = 406) ages 12 to 17 and a parent completed baseline interviews (2001–2004), and adolescents completed up to 2 follow-up interviews 1 and 5 years later. Baseline interviews gathered detailed information on parental smoking history, including timing and duration, current smoking, and nicotine dependence. Adolescent smoking and nicotine dependence were assessed at each time point. Latent Class Growth Analysis identified prospective smoking trajectory classes from adolescence into young adulthood. Logistic regression was used to examine relationships between parental smoking and adolescent smoking trajectories.

RESULTS: Four adolescent smoking trajectory classes were identified: early regular smokers (6%), early experimenters (23%), late experimenters (41%), and nonsmokers (30%). Adolescents with parents who were nicotine-dependent smokers at baseline were more likely to be early regular smokers (odds ratio 1.18, 95% confidence interval 1.05–1.33) and early experimenters (odds ratio 1.04, 95% confidence interval 1.04–1.25) with each additional year of previous exposure to parental smoking. Parents’ current non-nicotine–dependent and former smoking were not associated with adolescent smoking trajectories.

CONCLUSIONS: Exposure to parental nicotine dependence is a critical factor influencing intergenerational transmission of smoking. Adolescents with nicotine-dependent parents are susceptible to more intense smoking patterns and this risk increases with longer duration of exposure. Research is needed to optimize interventions to help nicotine-dependent parents quit smoking early in their children’s lifetime to reduce these risks.

• cigarette smoking
• nicotine dependence
• parental smoking
• intergenerational smoking
• Abbreviations:
CI
confidence interval
CIDI
Composite International Diagnostic Interview
LCGA
latent class growth analysis
NEFS
New England Family Study
OR
odds ratio
• #### What’s Known on This Subject:

It is well-established that parental smoking is associated with adolescent smoking initiation and regular tobacco use. However, we know less about how exposure to specific types of parental smoking affect adolescent smoking and progression to regular smoking in young adulthood.

Among adolescents with parents who are nicotine dependent, each previous year of parental smoking increases the likelihood they will be in a heavy smoking trajectory. Parental smoking cessation early in their children’s life is critical to prevent smoking in families.

Parental smoking is associated with adolescent smoking uptake and regular smoking, suggesting intergenerational transmission of smoking behavior within families.1 Research demonstrates that adolescents whose parents smoke are more likely to begin smoking2,3 and that parental smoking predicts future smoking initiation and regular smoking among adolescents.48 Research also suggests that offspring of parents who quit smoking are less likely to begin smoking and those who already smoke are more likely to quit.912 The intergenerational transmission of smoking within families is likely influenced by multiple factors, such as genetics, observed parental behavior, and the home environment (eg, rules about smoking).8

Although many studies have used brief measures that define smoking behavior by using broad categories (eg, any past-month smoking),2,4,6,810 these measures constrain our understanding of how specific types of parental smoking (eg, current versus former) affect adolescent smoking behavior. Few studies have used measures to shed light on how nicotine dependence unfolds within families. Although research suggests intergenerational transmission of nicotine dependence occurs,1315 findings of research on the influence of parental nicotine dependence and adolescent smoking remain equivocal.3,16

Other evidence demonstrates distinct trajectories of adolescent smoking behavior can be identified.5,1721 Few studies, however, have used prospective data to track influences of parental smoking on offspring’s smoking behavior from adolescence into young adulthood, a period of risk for developing nicotine dependence.22 Nicotine-dependent smokers are less successful at quitting and are more likely to quit with intensive cessation interventions.23 Examining if exposure to parental smoking and nicotine dependence differentially influence offspring’s smoking trajectories could help identify adolescents and families in need of more intensive intervention to reduce risks.

Our goal was to investigate the associations between adolescents’ exposure to their parents’ smoking and prospective trajectories of adolescent smoking into young adulthood. This study builds on a previous investigation of intergenerational smoking from the New England Family Study (NEFS), a multigeneration study of smoking risk within families.3 The NEFS was designed to understand intergenerational transmission of cigarette smoking by capturing detailed information on parental smoking history and nicotine dependence, and assessing offspring’s smoking behavior prospectively. The prior NEFS investigation demonstrated that adolescents’ cumulative exposure to active parental smoking, but not former parental smoking, increases the likelihood of adolescent smoking initiation.3 Prospective NEFS data provide a unique opportunity to gain new knowledge of how offspring’s exposure to parental smoking and nicotine dependence early in life influences smoking trajectories.

## Methods

### Participants and Procedure

This analysis included data from second- and third-generation NEFS participants.3,2426 The NEFS was established to interview adult offspring of pregnant women enrolled between 1959 and 1964 at the Boston, Massachusetts, and Providence, Rhode Island, sites of the National Collaborative Perinatal Project, a birth cohort study of the effects of in utero and early childhood environment on child health.27 Adult offspring of National Collaborative Perinatal Project participants (second-generation) were selected for participation by using a multistage sampling procedure and contacted by mail at age 40 to enroll in the NEFS.3 Among NEFS adults residing within 100 miles of the Providence site, adolescent offspring (third-generation) between 12 and 17 years of age were invited to participate in a prospective study on the intergenerational transmission of smoking conducted from 2001 to 2009.3 Second- and third-generation participants were provided with a modest incentive (eg, $5–$10 cash equivalent) for completing study interviews.

In total, 726 eligible third-generation adolescents were invited to participate; 559 (72%) completed a baseline interview with data on smoking behavior. A complete description of the baseline sample was published previously.3 Adolescents and, separately, their parents completed a baseline interview in person. Adolescents were contacted for a second interview 1 year later (mean 1.3 years, SD 0.28), and those who were age 18 and older at the 5-year follow-up were contacted for a third interview (mean 5.2 years, SD 0.64). The Brown University Institutional Review Board approved the protocol. Parent consent and adolescent assent were obtained at baseline and reestablished at each follow-up.

The sample for the current study comprised 406 adolescents with data available for 2 or more time points to analyze changes in smoking behavior. Characteristics of the baseline and analytic samples are shown in Table 1. Adolescents in the analytic sample were significantly older than those in the original baseline sample (P < .001) because of the age restrictions for the third interview. There were no other statistically significant differences between the samples.

TABLE 1

Characteristics of the Baseline and Analytic Samples

### Measures

Adolescent smoking was captured by using the Lifetime Inventory of Smoking Trajectories, a valid instrument that gathers detailed information on smoking initiation, past and current smoking, and susceptibility.28 Nicotine dependence was assessed based on Diagnostic and Statistical Manual for Mental Disorders, Fourth Edition, criteria by using the adapted Composite International Diagnostic Interview (CIDI).29,30 The CIDI was selected over brief dependence screeners (eg, Fagerström Tolerance Questionnaire) to gather detailed information on clinical dependence symptoms for the NEFS. Smoking status was operationalized to reflect developmentally appropriate transitions from nonsmoking and susceptibility to regular, dependent use.31 For trajectory analyses, smoking status at each interview used a score with the following values: committed nonsmokers never smoked and indicated they would never try smoking (0); susceptible nonsmokers never smoked but indicated they may try in the future (1); triers smoked only once in their life (2); experimenters smoked more than once, but never daily (3); regular smokers without nicotine dependence smoked daily but did not meet Diagnostic and Statistical Manual for Mental Disorders, Fourth Edition, dependence criteria (4); and regular smokers with nicotine dependence smoked daily and met dependence criteria (5).

#### Parental Smoking

Parental smoking and nicotine dependence also were assessed at baseline by using the Lifetime Inventory of Smoking Trajectories (LIST)28 and CIDI.29,30 One parent completed a baseline interview for the study. For analyses, we examined parental smoking in 2 ways. A categorical variable was created based on parents’ baseline smoking status: current (daily or weekly) nicotine-dependent smoker, current non-nicotine–dependent smoker, former daily or weekly smoker, or nonsmoker.

The Lifetime Inventory of Smoking Trajectories gathers data on current and prior periods of smoking, including timing and duration, which allowed us to determine adolescents’ cumulative years of exposure to parental smoking.28 We created continuous predictor variables for adolescents’ total years of exposure to parental smoking before baseline based on whether their parents were current dependent smokers, current smokers without dependence, or former smokers at baseline.

#### Covariates

Parent (gender, race/ethnicity, age, educational attainment, household income, marital status) and adolescent (gender, age, race/ethnicity) demographics ascertained at baseline were examined as covariates.

### Statistical Analysis

We conducted Latent Class Growth Analysis (LCGA)32 by using MPlus 7.1 (Muthén & Muthén, Los Angeles, CA) to construct adolescent smoking trajectory classes based on prospective behavior patterns.19 LCGA has been widely used in studies of adolescent smoking19,33,34 and similar behaviors.35 Adolescents who were committed nonsmokers at all time points (n = 123, 30%) were defined a priori as a trajectory class.17,33,34 For the remainder (n =283, 70%), LCGA models examined 1 to 4 class solutions. The optimal number of classes was determined based on solutions having a lower Bayesian Information Criteria, higher estimated proportion of participants correctly classified (entropy), and a statistically significant Lo-Mendall-Rubin likelihood ratio χ2 test comparing fit for a model with K classes compared with K-1 classes.32,36,37 Missing data were accommodated using full-information robust maximum likelihood estimation, which uses all available data for analyses.38

We examined whether exposure to parental smoking was associated with adolescent smoking trajectories through bivariate analyses (eg, χ2 tests) and multinomial logistic regression.39 The NEFS included siblings to investigate aims surrounding smoking risk within families. The sample included 197 singletons (49%), 91 sibling pairs (n = 182, 45%), and 9 sibling triads (n = 27, 7%). Analyses accounted for clustering using survey procedures in SAS 9.3 (SAS Institute, Inc, Cary, NC).

The primary predictor in regression analyses was exposure to parental smoking at baseline. Demographics associated with smoking trajectory classes in bivariate analyses (P < .05) were considered as covariates. Two separate models were created. In Model 1, we used the categorical parental smoking status predictor variable, with nonsmokers as the reference group. In Model 2, parental smoking was operationalized as adolescents’ total years of exposure to their parents’ smoking stratified by parents’ baseline smoking status.

## Results

### Study Sample

Sample characteristics are shown in Table 1. Participants (n = 406) were nearly half girls (52%), and most were white (86%) and averaged 14.2 years (SD 1.6) of age at baseline. The mean (SD) smoking status scores at baseline and each follow-up were 0.93 (1.42), 1.20 (1.53), and 2.41 (1.83), respectively. In total, 15.0% of parents were current dependent smokers, 11.8% were current nondependent smokers, 31.5% were former smokers, and 41.7% were nonsmokers. Parents who were current or former smokers (n = 233) smoked an average of 19.0 (SD 12.5) cigarettes per day. The number of cigarettes smoked per day did not differ significantly by parental smoking status (P = .469). Adolescents of parents who were current dependent smokers were exposed to an average of 1.19 (SD 3.66) years of smoking before baseline, whereas offspring of current nondependent smokers were exposed for 0.89 (SD 2.99) years, and offspring of former smokers were exposed for 1.26 (SD 3.29) years.

Table 2 displays fit statistics for LCGA models. The 3-class solution was optimal based on the joint evaluation of the fit indices and the likelihood ratio χ2 test compared with the 2-class solution. Combined with the a priori class of nonsmokers, this created a 4-level trajectory class outcome variable.

TABLE 2

Latent Class Growth Analysis Fit Statistics

To describe adolescent trajectory classes, we calculated average smoking status scores over time by class (Fig 1). Classes were labeled as nonsmokers (30%), early experimenters (23%), late experimenters (41%), and early regular smokers (6%). Nonsmokers abstained across all 5 years. Early experimenters tried or smoked nondaily at baseline and year 2, and nearly half (50%) were regular smokers by year 5. Late experimenters typically tried smoking later, and almost 25% were regular smokers by year 5. Early regular smokers smoked regularly at baseline and most (80%) were nicotine-dependent smokers by year 5 or earlier.

FIGURE 1

Average adolescent smoking status by trajectory class.

Table 3 displays bivariate associations between demographics, parental smoking, and adolescent smoking trajectories. Baseline demographic characteristics associated with adolescents’ trajectory class (P < .05) that were considered as covariates in logistic models included adolescents’ and parents’ age, and parents’ race, education, marital status, and household income. When both parents’ and adolescents’ ages were included in the logistic models, only adolescents’ age was significant and the results were similar regardless of which age variable was used. Only adolescents’ age was included in the models for parsimony. Parents’ cigarettes smoked per day did not differ significantly by adolescents’ trajectory classes in pairwise comparisons (Table 3), so it was excluded from multivariable analyses.

TABLE 3

Bivariate Associations With Adolescents’ Smoking Trajectory Class

### Parental Smoking and Adolescent Smoking Trajectories

Table 4 displays results of the logistic regression models. Model 1 examined parents’ baseline smoking status using a categorical predictor. Adolescents with parents who were current dependent smokers at baseline were significantly more likely to be in the 2 heaviest smoking trajectories: early experimenters (odds ratio [OR] 4.61, 95% confidence interval [CI] 1.52–13.96) and early regular smokers (OR 9.67, 95% CI 1.66–50.67). Adolescents whose parents were current nondependent smokers at baseline were also significantly more likely to be early regular smokers (OR 9.96, 95% CI 1.67–59.44), early experimenters (OR 4.52, 95% CI 1.32–15.42), and late experimenters (OR 2.89, 95% CI 1.16–7.17). Parents’ former smoking was not associated with adolescents’ trajectory class.

TABLE 4

Multinomial Logistic Regression Analysis of Adolescent Smoking Trajectories Based on Exposure to Parental Smoking at Baseline, Defined as a Category or Duration of Adolescents’ Previous Years of Exposure to Parental Smoking

Model 2 examined associations between adolescents’ years of exposure to parental smoking separated by their parents’ smoking status at baseline and smoking trajectory classes. ORs reflect the increase in adolescents’ odds of being in a higher smoking trajectory class relative to being a committed nonsmoker with each additional year of exposure to parental smoking before baseline. After adjusting for demographics, each prior year of exposure to parental smoking among adolescents whose parents were nicotine-dependent smokers significantly increased the odds of an adolescent being an early regular smoker (OR 1.18, 95% CI 1.05–1.33) and an early experimenter (OR 1.14, 95% CI 1.04–1.25). Each additional year of exposure among adolescents whose parents were current nondependent smokers also increased the odds of adolescents being an early regular smoker (OR 1.23, 95% CI 1.01–1.50); however, the overall effect of exposure among those whose parents were nondependent current smokers was not significant (P = .17).

## Discussion

This study adds to the research on intergenerational transmission of smoking by demonstrating that when parents are current, nicotine-dependent smokers, a longer duration of exposure to parental smoking increases the odds that adolescents will be in heavier smoking trajectories. Parents’ categorical smoking status proved to be a blunt indicator of risk: exposure to any form of current parental smoking at baseline increased the likelihood that adolescent offspring would be in a heavier smoking trajectory class. Examining adolescents’ previous years of exposure to parental smoking stratified by parents’ baseline smoking status yielded a more fine-gained assessment of risk. A longer duration of exposure to parental smoking among adolescents whose parents were nicotine dependent increased the odds that adolescents would be in heavier smoking trajectories. These findings indicate that cessation among nicotine-dependent parents early in their offspring’s lifetime is critical to reduce the risk of smoking within families.

Using NEFS data, Gilman and colleagues3 reported that exposure to current parental smoking predicts adolescent smoking initiation at baseline, whereas parental nicotine dependence and former parental smoking were not associated with smoking initiation. Although this study suggests that parental nicotine dependence may not be the most important risk factor for adolescent smoking initiation, our results demonstrate that assessing parental nicotine dependence remains critical to identify adolescents at risk for heavier smoking over time. In line with the findings of Gilman and colleagues,3 our results showed that exposure to parental smoking when parents had quit before baseline was not associated with adolescent smoking. These data support the hypothesis that intergenerational smoking transmission occurs, in part, through social learning where adolescent smoking is influenced by observation of parental smoking.3,8

Our analysis yielded smoking trajectory classes consistent with previous studies using similar methods, supporting the validity of our findings.18,19,40,41 An important contribution of our study is the attention to nicotine dependence in identifying adolescent smoking trajectories. One study examined how nicotine-dependence symptoms develop among adolescent smokers, concluding that parental smoking increases the risk of early-onset dependence.13 Our results were similar, and, taken together, sharpen the focus on using trajectory-based definitions of adolescent smoking that incorporate nicotine dependence to identify adolescents at risk rather than relying only on brief measures, such as those used in prior studies.2,4,6,9,10 Trajectory-based approaches using measures of nicotine dependence yield more fine-grained information to understand how intergenerational transmission of smoking occurs, timing and duration, and what aspects of parental smoking predict high-risk adolescent smoking behavior.13

Screening and counseling adolescents and their parents in pediatric clinical settings for tobacco use is a recommended strategy to reduce youth smoking risk.42,43 This study adds to evidence supporting the need to address smoking among both adolescents and their parents in pediatric clinical settings.42,43 Although some studies have shown that offspring of parents who had quit smoking are less likely to smoke,912 to our knowledge there has been no clinical trial to determine whether interventions for parental smoking cessation have a downstream impact on their adolescent offspring. Although there are obvious benefits to eliminating second-hand smoke exposure by helping parents quit,44 this is an important topic of investigation for future studies.

Recent reviews indicate tobacco control interventions administered in the clinical setting can be impactful for preventing youth smoking45 and advocate for strategies, such as the 5 A’s (Ask, Advise, Assess, Assist, Arrange), to identify parents who smoke and deliver cessation advice.43 Parental cessation counseling interventions administered in pediatric settings have had a modest impact,46 and despite such interventions, most parents do not quit.44 Our data suggest this could be because counseling interventions do not fully attend to parental nicotine dependence. Dependence symptoms are a strong predictor of quitting smoking above other known factors (eg, motivation to quit),23 and pharmacotherapies are often critical to increase the likelihood of successfully quitting among dependent smokers.47

Our findings highlight the importance of screening parents for nicotine dependence in pediatric settings and referring them to evidence-based cessation resources. An efficient approach to put these findings into practice could be for pediatric providers to use brief nicotine-dependence screening instruments that have well-established validity to identify dependent parents.48 These brief screening instruments could be dovetailed with parent-directed counseling interventions emphasizing quitting for the health of their children to motivate cessation44 and pharmacotherapy to treat dependence. Another approach that deserves additional research is Ask, Advise, Connect, where clinicians ask parents about smoking, provide brief cessation advice, and refer parents who smoke to evidence-based cessation resources, such as telephone quit lines.49 An approach integrating nicotine-dependence screening, provider advice that parents quit for their children’s health, and referral for cessation support and dependence treatment may be optimal and should be examined in future research.

Our findings should be interpreted in light of important limitations. Baseline interviews were conducted with only 1 parent, limiting our ability to investigate differences based on maternal and paternal smoking. Parental smoking was assessed at baseline only, therefore we could not examine how prospective patterns of parental smoking influence adolescent smoking. We did not examine smokeless or other noncigarette tobacco use, which could be an important avenue for future research to understand the role of nicotine dependence given the increasing use of newer tobacco products (eg, snus, electronic cigarettes). All measures were self-report, smoking was not verified biochemically, and we did not control for other risk factors for smoking (eg, peer smoking).

## Conclusions

This study demonstrates that among adolescents with parents who are nicotine dependent, each previous year of exposure to parental smoking increases the likelihood that adolescents will be in a higher-risk smoking trajectory and progress to regular smoking. Adolescents’ cumulative exposure to parental smoking may provide a clearer indicator of risk than often-used categorical indicators of parental smoking alone. Interventions to identify nicotine-dependent parents and link them with evidence-based cessation resources to quit smoking early in the life of their offspring may help reduce the risk of smoking within families.

## Footnotes

• Accepted February 26, 2014.
• Address correspondence to Darren Mays, PhD, MPH, Cancer Prevention & Control Program, Lombardi Comprehensive Cancer Center, Georgetown University Medical Center, 3300 Whitehaven St NW, Suite 4100, Washington, DC 20007. E-mail: dmm239{at}georgetown.edu
• Dr Mays contributed to the conceptualization of the study idea, led statistical analyses and interpretation of the data, and drafted the paper; Dr Gilman contributed to the conception and design of the original study, acquisition of data, conceptualization of the study idea, and revision of the paper for important intellectual content; Dr Rende contributed to the conception and design of the original study, acquisition of data, and revision of the paper for important intellectual content; Dr Luta contributed the statistical analysis and revision of the paper for important intellectual content; Dr Tercyak contributed to the revision of the paper for important intellectual content; and Dr Niaura contributed to the conception and design of the original study, acquisition of data, statistical analysis and interpretation, conceptualization of the study idea, and revision of the paper for important intellectual content.

• FINANCIAL DISCLOSURE: The authors have indicated they have no financial relationships relevant to this article to disclose.

• FUNDING: Supported by National Institutes of Health Transdisciplinary Tobacco Research Center award CA084719 and by the Biostatistics and Bioinformatics Shared Resource of Georgetown Lombardi Comprehensive Cancer Center through Comprehensive Cancer Center support grant P30CA051008. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Cancer Institute or the National Institutes of Health. Funded by the National Institutes of Health (NIH).

• POTENTIAL CONFLICT OF INTEREST: The authors have indicated they have no potential conflicts of interest to disclose.


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# If a and b are two unit vectors such that a+b is also a unit vector, then find the angle between a and b - Mathematics

If veca  and vecb are two unit vectors such that veca+vecb is also a  unit vector, then find the angle between veca and vecb

#### Solution

Given: veca and vec b are unit vectors

therefore |veca|=|vecb|=1

Let the angle between  veca and vecb "be " theta

It is given that, (veca+vecb) is a unit vector

therefore |veca+vecb|=1

(veca+vecb)(veca+vecb)=1

|veca|^2+2veca.vecb+|vecb|^2=1

|veca|^2+2|veca|.|vecb|+|vecb|^2=1

1+2xx1xx1xxcostheta+1=1

2costheta=-1

cos theta=-1/2

therefore theta=(2pi)/3

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
Is there an error in this question or solution?

format: plain


text:
Last edited by Nikorg
Saturday, August 1, 2020 | History

4 edition of Dynamical zeta functions for piecewise monotone maps of the interval found in the catalog.

Dynamical zeta functions for piecewise monotone maps of the interval

David Ruelle

# Dynamical zeta functions for piecewise monotone maps of the interval

## by David Ruelle

Written in English

Subjects:
• Differentiable dynamical systems,
• Functions, Zeta,
• Mappings (Mathematics),
• Monotone operators

• Edition Notes

Includes bibliographical references (p. 61-62).

Classifications The Physical Object Statement David Ruelle. Series CRM monograph series ;, v. 4 LC Classifications QA614.8 .R837 1994 Pagination vii, 62 p. : Number of Pages 62 Open Library OL1083328M ISBN 10 0821869914 LC Control Number 94006986

Rationality of dynamical zeta functions Examples Questions On dynamical zeta functions for polynomial maps over p-adic ﬁelds Liang-Chung Hsia Department of Mathematics National Central University Taiwan, R.O.C. Arithmetic and Nonarchimedean Dynamics Joint AMS Meeting, functions is usually unavailable and measurements pertaining to these functions are contaminated by random noise and disturbances, estimation of these functions becomes a critical problem across many ﬂelds [6, 24, 30]. In this paper, we consider the following monotone regression problem: estimate an unknown, nondecreasing function f: [0;1]!

A piecewise-smooth dynamical system (PWS) is a discrete- or continuous-time dynamical system whose phase space is partitioned in different regions, each associated to a different functional form of the system vector field.. A piecewise-smooth map is described by a finite set of smooth maps \[x \mapsto F_i(x,\mu), \quad \mbox{for} \quad x \in S_i, \quad x \in \mathbb{R}^{n}, \mu \in \mathbb{R. Conference proceedings, book chapters etc. y Dynamical zeta functions and dynamical determinants for hyperbolic maps, Springer Ergebnisse, , front and backmatter on Mittag-Leffler preprint server (Fractal Geometry and Dynamics Program, Fall ), Erratum. x Dynamical zeta functions Graduate course, Orsay, w Linear response, or else Proceedings of the International Congress of.

dynamical zeta functions: what, why and what are the good for? November 2 physicist’s life is intractable dynamical systems Topological trace formula, zeta function Fokker-Planck evolution optimal partition hypothesis summary life is intractable I accept chaos ft maps a region M i of the state space into the region ft(Mi). David Ruelle: free download. Ebooks library. On-line books store on Z-Library | B–OK. Download books for free. Find books Dynamical zeta functions for piecewise monotone maps of the interval. American Mathematical Society. David Ruelle. Year: A search query can be a title of the book, a name of the author, ISBN or anything else.

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### Dynamical zeta functions for piecewise monotone maps of the interval by David Ruelle Download PDF EPUB FB2

The second part is a detailed study of the zeta functions associated with piecewise monotone maps of the interval $$[0,1]$$. In particular, Ruelle gives a proof of a generalized form of the Baladi-Keller theorem relating the poles of $$\zeta (z)$$ and the eigenvalues of the transfer operator.

The second part is a detailed study of the zeta functions associated with piecewise monotone maps of the interval $[0,1]$. In particular, Ruelle gives a proof of a generalized form of the Baladi-Keller theorem relating the poles of $\zeta (z)$ and the eigenvalues of the transfer by: Dynamical zeta functions have unexpected analytic properties and interesting relations to the theory of dynamical systems, statistical mechanics, and the spectral theory of dynamical operators.

This book presents a general introduction to this subject and a detailed study of the zeta functions associated with piecewise monotone maps. Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval (Crm Monograph Series) by Ruelle, David and a great selection of related books, art and collectibles available now at Baladi, V.

and Ruelle, D.:An extension of the theorem of Milnor and Thurston on zeta functions of interval maps’, IHES Preprint, to appear Ergodic Theory Dynamical Systems Bochner, S., and Martin W.T.:Several Complex Variables Princeton University Press, Princeton, NJ Google ScholarCited by: 8.

the interval Download the interval or read online books in PDF, EPUB, Tuebl, and Mobi Format. Click Download or Read Online button to get the interval book now. This site is like a library, Use search box in the widget to get ebook that you want.

Baladi. Dynamical zeta functions and generalised Fredholm determinants. With an appendix written with D. Ruelle, Some properties of zeta functions associated with maps in one International Congress of Mathematical Physics (Paris, ), pages – by: 7.

Ruelle, Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval (CRM Monograph, Vol 4) (AMS, ) "A monograph based on the Aisenstadt lectures given by the author in October at the University of Montreal on "Dynamical Zeta Functions," but with a different emphasis.

Hyperbolic systems are not discussed in detail. Other works by Reulle are Chaotic Evolution and Strange Attractors: The Statistical Analysis of Time Series for Deterministic Nonlinear Systems; Meteorological Fluid Dynamics: Asymptotic Modelling, Stability and Chaotic Atmospheric Motion; (for which Reulle was one of the editors); and Dynamical Zeta Functions for Piecewise Monotone Maps of the.

Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval David Ruelle The Centre de Recherches Mathematiques (CRM) of the Universite de Montreal was created in to promote research in pure and applied mathematics and related disciplines. Among. And dynamical zeta functions are an effective tool to do the counting.

The tool turns out to be so effective in fact as to make one suspect that there is more to the story than what we currently understand. Some Traditional Examples of Zeta Functions The grandmother of all zeta functions is the Riemann zeta function defined by ζR(s)= ∞ n=1 n File Size: KB.

For piecewise monotone interval maps, we show that the Kolmogorov–Sinai entropy can be obtained from order statistics of the values in a generic orbit. A similar statement holds for topological. In other words: The maps t preserve the vector order. A semi ow with this property is called monotone.

Monotone semi ows and their discrete-time counterparts, order-preserving maps, form the subject of Monotone Dynamics. Returning to the biological setting, we may make the assumption that each species directly or indirectly a ect all the others. Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval (Crm Monograph Series) Apr 1, by David Ruelle Hardcover.

The symbolic dynamics plays an important role in the study of nonlinear dynamical systems. For the piecewise monotone interval maps, the kneading theory established by Milnor and Thurston [1] is.

In mathematics, the Ruelle zeta function is a zeta function associated with a dynamical system. It is named after mathematical physicist David Ruelle. Formal definition. Let f be a function defined on a Zeta Functions of Graphs: A Stroll through the Garden.

Cambridge Studies in Advanced Mathematics. Discover Book Depository's huge selection of David Ruelle books online. Free delivery worldwide on over 20 million titles. Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval.

David Ruelle. 15 Mar Paperback. US$Add to basket. 20% off. We consider piecewise expanding maps of the interval with finitely many branches of monotonicity and show that they are generically combinatorially stable, i.e., the number of ergodic attractors and their corresponding mixing periods do not change under small perturbations of the map. Our methods provide a topological description of the attractor and, in particular, give an elementary Author: Gianluigi Del Magno, João Lopes Dias, Pedro Duarte, José Pedro Gaivão. Note that h(P) m a y b e defined equivalently as the infimum of all entropies of maps g e C(I,I) such that g agrees with f on P. W e say P is entropy minimal if h(P) = M is defined in Section 2). where P has Then h(L) = log X, where X is period n (and M Dynamical Complexity of Maps of the Interval Theorem. [6]Cited by: 1. Dynamical Zeta Functions For Piecewise Monotone Maps Of The Interval (Crm Monograph Series) by David Ruelle it was ok avg rating — 1 rating — published — 2 editions. Best Piecewise Monotone Approximation of Piecewise Monotone Functions Piecewise monotone approximation of continuous real functions on a real interval has recently been investigated in [l]. The main result is that for any m > 0, there exist continuous and C” best piecewise monotone approximations having m or fewer monotone segments. The.Stefano Marmi Professor of Dynamical Systems, Scuola Normale Zeta functions and transfer operators for piecewise monotone transformations. V Baladi, G Keller. Communications in mathematical An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps. V Baladi, D Ruelle. Ergodic Theory and Dynamical Systems. Moduli Spaces and Arithmetic Dynamics by Joseph H. Silverman,available at Book Depository with free delivery worldwide. We use cookies to give you the best possible experience. By Dynamical Zeta Functions for Piecewise Monotone Maps of the Interval. David Ruelle. 15 Mar Paperback. US$Author: Joseph H. Silverman.


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## Ratio of perimeters of triangles?

AD, BE and CF are the perpendiculars from the angular points of a triangle ABC upon the opposite sides. The perimeters of the triangle DEF and triangle ABC are in the ratio:

A)2r/R
B)r/2R
C) r/R
D) r/3R


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# The sum of the first 50 positive even integers is 2550. What

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The sum of the first 50 positive even integers is 2550. What [#permalink]  12 Jun 2008, 03:54
This topic is locked. If you want to discuss this question please re-post it in the respective forum.

The sum of the first 50 positive even integers is 2550. What is the sum of the even integers from 102 to 200, inclusive?

5100
7550
10100
15500
20100

I know that there are 200 -102 =98 = 49 even numbers.

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Re: Sum of even integers [#permalink]  12 Jun 2008, 05:55
The sum of the first 50 positive even integers is 2550. What is the sum of the even integers from 102 to 200, inclusive?

5100
7550
10100
15500
20100

I know that there are 200 -102 =98 = 49 even numbers.

actually there are 50 even numbers (200-102)/2+1 /// think about counting 1, 2, 3 ... 10 (10 -1 = 9 but there are 10 numbers)
i would say the sum is equal to 2*(sum of all numbers between 51 and 100) = 2*(100+51)*50/2=151*50=7550 -> B
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Re: Sum of even integers [#permalink]  13 Jun 2008, 03:40
The sum of the first 50 positive even integers is 2550. What is the sum of the even integers from 102 to 200, inclusive?

5100
7550
10100
15500
20100

I know that there are 200 -102 =98 = 49 even numbers.

this will be an AP series question
102,104,106....200

a = 102 (the first term)
d = 2 (the common difference)
n = 50 (total even numbers for which the sum will be calculated)

now the formula for sum of an AP series is

S = N/2(2a + (n-1)d)

= 50/2 (2*102 + (50-1)2)

=25(204+98)
=25*302
=7,550

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Re: Sum of even integers [#permalink]  13 Jun 2008, 03:51
A more simple way to solve that without knowing any formula is to use what they say in the question :

sum of [even integers between 102 and 200 (inclusive)] = sum of [100 + even integers between 2 and 100 (inclusive)] = 50*100 + sum of the first 50 positive even integers = 5000 + 2550 = 7550
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Re: Sum of even integers [#permalink]  02 Jul 2008, 03:22
we can use elimination.

1. upper cap 50*200 =10000 therefore option C/D/E are out. Now choose between A and E

2. There are 50 even numbers between 102-200 including 102 therefore the sume should be greater than 50*100=5000.

3. The sum has to be more than 5100 therefore only choice left is 7550
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Re: Sum of even integers [#permalink]  02 Jul 2008, 03:54
The sum of the first 50 positive even integers is 2550. What is the sum of the even integers from 102 to 200, inclusive?

5100
7550
10100
15500
20100

I know that there are 200 -102 =98 = 49 even numbers.

I do not understand why they give us "The sum of the first 50 positive even integers is 2550".

We can find "the sum of the even integers from 102 to 200" without any additional information.
200-102 = 98 ==> Their are 50 even-numbers in the set.
S=50*(102*2+49*2)/2 = 7550.
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Re: Sum of even integers [#permalink]  02 Jul 2008, 04:09
lexis wrote:
I do not understand why they give us "The sum of the first 50 positive even integers is 2550".

We can find "the sum of the even integers from 102 to 200" without any additional information.
200-102 = 98 ==> Their are 50 even-numbers in the set.
S=50*(102*2+49*2)/2 = 7550.

Yes we can. But even without knowing the formula you used we could have solved the question with the piece of information they give us.

As I wrote above :

$$\sum_{n=102}^{200} n_{even} = \sum_{k=51}^{100} 2k = \sum_{k=1}^{50} (100+2k) = \sum_{k=1}^{50} 100 + \sum_{k=1}^{50} 2k = 50*100 + 2550 = 7550$$

(without calculating anything )
Re: Sum of even integers   [#permalink] 02 Jul 2008, 04:09
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text:
# Area Preserving Transformations from a Circle to an Ellipse

I am looking for the class/set of transformations that map a circle to an ellipse while preserving the area inside. Generally, maps of n-spheres to n-ellipsoids that preserve volume, if such a class/set exists.

Thank you,

-
If it takes all spheres to ellipsoids and preserves $n$-volume in the Euclidean space $\mathbb R^n,$ it is just linear with determinant $\pm 1.$ Plus a translation. –  Will Jagy May 19 '12 at 4:51
Thanks, to be sure then, I'm just interested in $SL(n,\mathbb{R})$? If you write it up I'll check it as an answer, else I'll just up-vote the comment. –  kηives May 22 '12 at 16:22

If it takes all spheres to ellipsoids and preserves $n$-volume in the Euclidean space $\mathbb R^n,$ it is just linear with determinant $\pm 1.$ Plus a translation, either before or after. So the basic object is, indeed, $SL_n(\mathbb R).$

If you relax either restriction you immediately get many more mappings. In particular, Moebius transformations on $\mathbb C \cup \{ \infty \}$ take any circle to a circle or line, any line to a circle or line. But area is not preserved unless you have $f(z) = \omega z + \beta,$ with $|\omega| = 1.$

-

There are a lot of volume preserving maps on $\mathbb{R}^n$. Here are a few simple transformations that preserve volume:

1. scale by $r$ along one axis and $\frac1r$ along another

2. rotation preserves distances and therefore volume

3. translation preserves distances and therefore volume

Combinations of these three transformations are enough to map any sphere to any ellipsoid with the same volume.

-

The folks in the animation business are always inventing new (and not-so-new) ways to do "volume-preserving deformations". Their idea is that preserving the volume of the object will make the deformation more "natural". It makes some sense -- when you flex your muscles, their overall volume doesn't change (I suppose). Anyway, if you google the term "volume-preserving deformation" you will find piles of stuff.

Generally, these deformations are only approximately volume-preserving, though. Good enough for animation, but maybe not good enough for your purposes. Only you can decide.

-

Any shear is an area/volume-preserving transformation.

-


text:
Edit Article

# How to Calculate the Volume of a Torus

Community Q&A

In this article, you will learn how to calculate the volume of a torus. Torus is a 3D figure that looks like a donut. Once you know the formula, it will be very easy.

## Steps

1. 1
Write down the formula ${\displaystyle V=(2\pi R)(\pi r^{2})}$.  That is the formula for the volume of torus. It can be simplified to ${\displaystyle V=2\pi ^{2}Rr^{2}}$ using the rule of FOIL.(Front, Outer, Inner, Last)
2. 2
Find the values for R and r and plug them in.  ${\displaystyle R}$ is the major radius here, and ${\displaystyle r}$ is the minor radius. ${\displaystyle R}$ is the distance from the center of the torus to the center point of the tube. ${\displaystyle r}$ is the radius of the tube.
3. 3
Using the order of operations, square ${\displaystyle r}$. Leave ${\displaystyle 2\pi ^{2}}$ alone for now.
• Example: Let ${\displaystyle R=14}$ and ${\displaystyle r=6}$
• With those values, ${\displaystyle V=2\pi ^{2}(R)(r^{2})=2\pi ^{2}(R)(6^{2})=2\pi ^{2}(R)(36)}$ because ${\displaystyle 6^{2}=36}$.
4. 4
Multiply ${\displaystyle R}$ by the product.
• Example: ${\displaystyle V=2\pi ^{2}(R)(r^{2})=2\pi ^{2}(14)(36)=2\pi ^{2}(504)}$ because ${\displaystyle 14*36=504}$.
5. 5
Using the commutative property of multiplication, multiply 2 by the product.
• Example: ${\displaystyle V=2\pi ^{2}(504)=1008\pi ^{2}}$ because ${\displaystyle 504*2=1008}$.
6. 6
Decide how far to go for your solution. There are two choices for the answer. First, leave the answer in terms of pi. Second, let ${\displaystyle \pi }$ equal 3.14 and solve.
• Example: The answer can be ${\displaystyle V=1008\pi ^{2}}$ or, roughly, ${\displaystyle 9938.4768}$.


text:
# Lebesgue outer measure

1. Feb 25, 2012

### alexfloo

I'm trying to prove, per ex. 5 of section 2.2 of S. Berberian's Fundamentals of Real Analysis, that where $\lambda^*$ is the Lebesgue outer measure, and An is any sequence of (not necessarily measurable) sets of reals increasing to A, then $\lambda^*(A_n)$ increases to $\lambda^*(A)$.

As a hint, it mentions that every set is contained in a measurable set which differs from it by a null set. I considered the closure. I know that the boundary is not necessarily null (for instance, the rationals) but perhaps this cannot be the case for a nonmeasurable set.

In either case, assuming the hint, the proof is pretty trivial. I just don't really know where to start on proving the hint.

2. Feb 26, 2012

### morphism

Given any subset B of R, measurable or not, one can find a measurable set M containing B with $\lambda^*(B) = \lambda(M)$. (If the outermeasure of B is infinite, take M=R; otherwise find a descending sequence U_n of open sets containing B with $\lambda(U_n) < \lambda(B) + 1/n$, and then let M be their intersection.)


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# Given a bounded set $E$, there is a $G_\delta$ set such that the outer measures are equal

Question: Show that for any bounded set $E \in \mathbb{R}$, there is a $G_\delta$ set $G$ for which $E \subseteq G$ and $m^*(E)=m^*(G)$.

Let $\{I_n\}$ be a countable collection of open intervals such that $E \subset \bigcup\limits_{n=1}^{\infty} I_n$. Observe the following: $$m^*(E) \leq m^*(\bigcup\limits_{n=1}^{\infty} I_n) \leq \sum\limits_{n=1}^\infty l(I_n) < m^*(E)+\frac{1}{2^n}.$$

Now let $G=\bigcap\bigcup I_n$. $G$ is a $G_\delta$ set. From this, we see that $E \subset G$. Thus for each $n \in \mathbb{N}$: $$m^*(E) \leq m^*(G) \leq m^*(\bigcup\limits_{n=1}^{\infty} I_n) \leq \sum\limits_{n=1}^\infty l(I_n) < m^*(E)+\frac{1}{2^n}.$$

So we take $n \rightarrow \infty$.

Is my proof correct?

-
You should denote $I_{n,j}$ the collection of intervals which covers $E$ and approach up to $2^{-n}$ the outer measure of $E$ (which is finite). Here you take $n$ as an index, but we see $n$ outside the sum, and we don't know what it is. But you saw the idea and you just have to take now $G:=\bigcap_{j\geq 1}\bigcup_{n\geq 1}I_{n,j}$. –  Davide Giraudo Sep 24 '12 at 7:10

For each integer $n\geq 1$, let $\{I_{n,j}\}$ a collection countable collection of open intervals which covers $E$ such that $$\sum_{j=1}^{+\infty}m^*(I_{n,j})\leq m^*(E)+n^{-1}.$$ Such a collection exists because the outer measure of $E$ is finite, as it's a bounded subset of $\Bbb R$.
Let $G:=\bigcap_{n\geq 1}\bigcup_{j\geq 1}I_{n,j}$. As $\bigcup_{j\geq 1}I_{n,j}$ is open for each $n$, $G$ is a countable intersection of open sets, and it contains $E$. We have for $n\geq 1$, $$m^*(G)\leq m^*\left(\bigcup_{j=1}^{+\infty}I_{n,j}\right)\leq m^**(E)+n^{-1},$$ which concludes the proof.


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# AI News, E-skin able to detect changes in wind, water drops and moving ants

## E-skin able to detect changes in wind, water drops and moving ants

As scientists continue to improve the look and capabilities of robots, one of the prime areas of research is skin.

It was made by covering a magnetic sensor with a hollow polymer membrane and then embedding magnetic beads in the top part of the membrane.

The electronic circuit then converts the signals to a series of pulses of varying frequencies that reflect the amount of pressure being 'felt' on the skin.

The researchers created an artificial finger covered with the e-skin and then attached the finger to an artificial arm for testing purposes.

They report that the skin they created was able to generate pulses for pressure as small as that created by a line of ants running over the surface.

## Remote tactile sensing system integrated with magnetic synapse

Tactile sensation recognizing from surrounding environment through the direct contact of force, vibration and temperature, is regarded as a next-generation technology for applications in information gathering and transfer as well as artificial intelligence1,2,3,4.

In particular, biomimetic tactile sensors have attracted considerable attention as the core technology for replacing human sensing in minimally invasive surgery, health monitoring, robotics, virtual reality, and prosthetics5,6,7,8,9,10.

Moreover, in the near future, various types of manufacturing robots are expected to co-operate with human workers in factories, and tactile sensing is a key technology for ensuring the safety of human workers and natural motion of robots11.

However, sensors based on flexible bulk elastomers suffer from high hysteresis, low sensitivity, limited sensing range, temperature-dependent displacement, and polymer swelling23,24 due to the intrinsic viscoelastic properties of elastomer composites25.

Therefore, the development of universal tactile sensors with high sensitivity in wide working range is one of the major challenging issues for use in practical applications such as robotic skin, medical diagnostic devices, and bionic arm electronic skin (Table S1).

Several interesting approaches have been proposed for attaching flexible electrodes to human skin or fabrics via silver nanowires28, liquid metals29, and sinuous thin Si ribbons30 for reliable performance under stretching and bending.

Severe signal disturbances due to external electrical, magnetic, and thermal noise lead to various problems, as the external pressure contact sites are locates close to the physical sensing elements.

Human skin is a fascinating tactile sensing system having multiple functions for detecting external stimulations, such as pressure, shear force, stretching, sliding, vibration, and temperature, with high accuracy and reliability.

In human beings, tactile sensing is performed by the central nervous system when sensing signals from the mechanoreceptors inside the fingertips are transmitted through nerve cells by contactless methods in the synapses26,32,33.

Inspired by human tactile sensing and synaptic transmission, this paper proposes a remote tactile sensing system integrated with a magnetic synapse to overcome the above-mentioned issues of flexible tactile sensors.

The developed remote tactile sensing system consists of a remote touch tip that generates air pressure by external touch, an air tube that delivers the generated air pressure, and a magnetic synapse that transduces the air pressure into electrical signals.

Specifically, the magnetic field intensity changes as the thin elastomer membrane is deflected by air pressure, which varies under external stress from the remote touch tip and is transmitted through the air tube.

The remote touch tip, which has no electronic component, provides robustness and its tactile sensing capability is not affected significantly by external electrical, magnetic, and thermal noise even in water or harsh environments.

Physical separation of the remote touch tip and the magnetic synapse, as in the case of human tactile sensing, makes the remote tactile sensing system robust against external electromagnetic and thermal noise.

The Ecoflex membrane has an embedded permanent magnet whose position changes as the Ecoflex membrane is deflected by air pressure variations generated in the remote touch tip by external stimulations.

Previously, we developed a highly sensitive hybrid MR sensing element, which combines anisotropic magnetoresistive (AMR) and planar Hall effects (PHE), consisting of multi-ring structures (diameter, 300 µm) in a hybrid Wheatstone bridge34,35,36.

To optimize the sensitivity and dynamic range of the remote tactile sensing system, the initial vertical distance of the permanent magnet relative to MR sensing element should be less than 3 mm, which is categorized as the active distance.

A finite element model of the simplified remote tactile sensing system was developed for the correlation between pressure and magnetic distance using COMSOL Multiphysics 5.2 (hyperelastic module).

The properties of the developed remote tactile sensing system were characterized using a custom-built system that can apply precise values of pressure to the sensor and measure the output signal in terms of electrical voltage.

The output signal can be formulated in terms of the external pressure through the correlation function $${\rm{V}}=({{\rm{f}}}_{n}({{\rm{g}}}_{n}({{\rm{h}}}_{n})))={{\rm{F}}}_{n}(P)$$ (see supplementary materials for function $${{\rm{F}}}_{n}(P)$$) from output signal to magnetic field, magnetic field to magnetic displacement, and magnetic displacement to external pressure.

Moreover, the remote tactile sensing system can discriminate a very high pressure resolution as it detects the placement and removal of ultralight pieces of sandpaper (30 mg each) on the touch pad (Fig. 3b), which corresponds to 6 Pa.

The measured response time (rise time) of the remote tactile sensing system, defined as the time interval between 10% and 90% of the output step height, was 40 ms (Fig. 3f), which is similar value to the response time of human fingertips in the range of 30 to 50 ms37.

For the case of bionic arm skin, the signal distortion due to the bending of (articular) joints in the finger and arm originating from the air tube needs to be minimized.

Our remote tactile sensing system consists of a touch tip that is spatially separated from the sensing magnetic synapse, and the touch tip can be placed and operated in water or extreme environments.

In addition, medical diagnosis via pulse measurement from the wrist (real-time monitoring of heart rate and relative blood pressure) was demonstrated using the developed tactile sensing system (Fig. 4d).

The remote touch tip measured the heart rate and relative blood pressure as the finger was bent to touch the wrist of a human subject gently, and measurement was ceased when the finger was unfolded, mimicking a doctor’s motion (Movie S4).

The remote tactile sensing system with the robot hand successfully monitored the heart rate and relative blood pressure in situ with heart physiology information, as the output signal patterns showed that the heart rate and relative blood pressure increased after exercise.

## Safety, ethical considerations, and application guidelines for the use of transcranial magnetic stimulation in clinical practice and research☆

Recording of electroencephalographic (EEG) activity immediately before, during, and after TMS is possible provided that certain technical challenges are addressed and few precautions taken (Ilmoniemi et al., 1997;

Problems related to the saturation of the EEG recording amplifiers from the TMS pulse have been overcome via artifact subtraction, pin-and-hold circuits, the use of modified electrodes which do not transiently change their shape due to the stimulus impact, and altering the slew rate of the preamplifiers.

However, in Table 2, special emphasis is placed on patient populations who might be more vulnerable to TMS due to several factors (i.e., brain damage, drug treatment or discontinuation of treatment for the purpose of a study).

Aftereffects have been observed on a variety of EEG/EP-measures including oscillatory activity over motor and prefrontal areas (e.g., Strens et al., 2002 and Schutter et al., 2003) as well as somatosensory (e.g., Katayama and Rothwell, 2007;

For the studies using conventional rTMS protocols (low frequency: 0.9 and 1 Hz, high frequency: 5–25 Hz), the direction of the aftereffect (when present) was as expected, with facilitation prevailing over suppression after high frequency TMS (n = 12 vs.

This parallels previous findings on lasting changes in neurophysiological measures after rTMS over motor areas (MEP-amplitude) without parallel changes in amplitude or velocity of voluntary finger movements (e.g., Muellbacher et al., 2000).

Although speculative at this point, it is probably safe to conclude that the time of potential aftereffects would be slightly, but not dramatically, underestimated if equated to the duration of observable behavioral effects (using safe parameters).

Yet, with the previously employed parameters (as compared to standard protocols: similar number of pulses but considerable shorter duration and lower intensity of stimulation), the duration of the effects on EEG activity (measured so far using SEP-amplitude after sensorimotor stimulation) (Katayama and Rothwell, 2007;

Recently, 24 healthy volunteers participated in 2 randomized, placebo-controlled, cross-over experiments and underwent continuous TBS (cTBS), intermittent TBS (iTBS), and shamTBS either over the left dorsolateral prefrontal cortex (n = 12, Figure 8 coil) or the medial prefrontal cortices (n = 12, double-cone coil) (Grossheinrich et al., 2009): the only EEG aftereffects were current density changes in the alpha2 band after iTBS of the dorsolateral prefrontal cortex, which remained detectable up to 50 min after stimulation.

Scientists from the Chinese Academy of Sciences have created electronic skin that is sensitive enough to record changes in the direction of the wind, falling drops and movement of ants.

Scientists have covered the magnetic sensors with a hollow polymer membrane, and then built in the magnetic beads in its upper part.

They report that the skin they created could generate impulses in response to even the slight pressure exerted on it by ants running across the surface.

Lec 16 | MIT 16.885J Aircraft Systems Engineering, Fall 2005

Правда ли, что протеин приводит к импотенции?

reassured screamed liter favoring traction wondered reconsider realizing plow nap brain's ebb manifests CVD HDL ..

Нравятся ли девушкам качки накачанные парни бодибилдеры Почему не все девушки любят большие мышечные

reassured screamed liter favoring traction wondered reconsider realizing plow nap brain's ebb manifests CVD HDL ..

До скольки растет человек. Что такое зоны роста. Как увеличить рост?

reassured screamed liter favoring traction wondered reconsider realizing plow nap brain's ebb manifests CVD HDL ..

Что такое кортизол? Факторы, влияющие на выработку гормона кортизола

reassured screamed liter favoring traction wondered reconsider realizing plow nap brain's ebb manifests CVD HDL ..

Что такое пауэрлифтинг? Чем отличается от бодибилдинга?

reassured screamed liter favoring traction wondered reconsider realizing plow nap brain's ebb manifests CVD HDL ..

День 30 Похудение за 30 дней ЗА месяц ИЛИ как я похудел на 15 кг на диете Эндоморф и Белковая диета

reassured screamed liter favoring traction wondered reconsider realizing plow nap brain's ebb manifests CVD HDL ..

Можно ли подрасти (вырасти) в высоту после 24?

reassured screamed liter favoring traction wondered reconsider realizing plow nap brain's ebb manifests CVD HDL ..

День 4 Похудение на белках Питание для сжигания жира Диета для похудения на 10 кг История качка

reassured screamed liter favoring traction wondered reconsider realizing plow nap brain's ebb manifests CVD HDL ..

Протеины и анаболики и потенция. Влияние и вред протеинов для потенции

reassured screamed liter favoring traction wondered reconsider realizing plow nap brain's ebb manifests CVD HDL ..


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# How to include an array in cell of a tabular environment

For an equation that has an array we use \\ for new row as well, and \newline didn't work, and this is understood by the tabular environment as a new row! Would appreciate all the help!

-
Welcome to TeX.SX! It would be good if you would show the code of this equation, so we can help to fix it. –  Stefan Kottwitz May 25 '11 at 16:54
\begin{tabular}{p{10em }p{4em }} $\mathbf{P} = \left[ \begin{array}{cc} 1 - \alpha & \alpha \\beta & 1 - \beta \end{array} \right]$ & –  emy May 25 '11 at 16:59
You can edit your question to add the source code. Please indent it by 4 spaces or use the '101010' button to format it as code. –  Martin Scharrer May 25 '11 at 17:13
Try to wrap the whole array inside braces { } and see if it helps. –  Martin Scharrer May 25 '11 at 17:13
An array environment should be in math mode, so surround it with $ signs. – egreg Jun 18 '11 at 22:09 show 1 more comment ## 3 Answers With regard to the code snippet you provided in a comment: Yes, this array may be put into a cell of a tabular environment, and no, \newline is not understood as a new row inside a p column. \documentclass{article} \begin{document} \begin{tabular}{p{10em}p{4em}} A & Bla\newline bla \\ $$\mathbf{P} = \left[ \begin{array}{cc} 1 - \alpha & \alpha \\ \beta & 1 - \beta \end{array} \right]$$ & D \end{tabular} \end{document}  - add comment For smaller (single column) inclusions of an array/tabular-style object in your table, you may also consider using the the makecell package in the following way: \documentclass{minimal} \usepackage{makecell} \begin{document} \begin{tabular}{ccc} A & B & C \\ D & \makecell[c]{$E\$ \\ F} & G \\
H & I & J
\end{tabular}
\end{document}

-

Use minipage:

\documentclass{article}
\begin{document}
\begin{tabular}{p{10em }p{4em }}
\begin{minipage}{10em} %%
\begin{math} %%
\mathbf{P} = \left[ \begin{array}{cc} %%
1 - \alpha & \alpha \\ %%
\beta & 1 - \beta %%
\end{array} \right] %%
\end{math} %%
\end{minipage} & \it table \\
\it table & \it table
\end{tabular}
\end{document}

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Why the minipage? Your code works without it. Also, you should use \itshape instead of \it. –  Gonzalo Medina Jun 15 '11 at 19:57
minipage is my goto for solving these kinds of problems. Good to know it's not absolutely necessary. –  Justin Bailey Jun 15 '11 at 20:09


text:
# Rule of Assumption/Boolean Interpretation

The truth value of a propositional formula $\mathbf A$ under a boolean interpretation $v$ is given by:
$v \left({\mathbf A}\right) = \begin{cases} T & : v \left({\mathbf A}\right) = T \\ F & : v \left({\mathbf A}\right) = F \end{cases}$