On ϕ \phi -amicable pairs (with appendix)

Let $\phi(n)$ denote Euler's totient function, i.e., the number of positive integers~$<n$ and prime to $n$. We study pairs of positive integers $(a_0,a_1)$ with $a_0\le a_1$ such that $\phi(a_0)=\phi(a_1)=(a_0+a_1)/k$ for some integer $k\ge1$. We call these numbers $\phi$--{\it amicable pairs with multiplier\/} $k$, analogously to Carmichael's multiply amicable pairs for the $\sigma$--function (which sums all the divisors of $n$). We have computed all the $\phi$--amicable pairs with larger member $\le10\^9$ and found 812 pairs for which the greatest common divisor is squarefree. With any such pair infinitely many other $\phi$--amicable pairs can be associated. We present a table of the $58$ so-called primitive $\phi$--amicable pairs for which the larger member does not exceed $10^6$. Next, $\phi$--amicable pairs with a given prime structure are studied. It is proved (a.o.) that a relatively prime $\phi$--amicable pair has at least twelve distinct prime factors and that, with the exception of the pair $(2^2,2\cdot 3)$, if one member of a $\phi$--amicable pair has two distinct prime factors, then the other has at least four distinct prime factors. Analogies with construction methods for the classical amicable numbers are shown; application of these methods yields another 79 primitive $\phi$--amicable pairs with larger member $>10^9$, the largest pair consisting of two 46-digit numbers

Spartan Daily, May 3, 1948

Volume 36, Issue 128https://scholarworks.sjsu.edu/spartandaily/11086/thumbnail.jp

Affection and Conflict in Intergenerational Relationships of Women in Sixteen Areas in Asia, Africa, Europe, and America

Studies of intergenerational solidarity in affluent societies suggest that relationships between generations consist simultaneously of both emotional closeness and conflicts. This analysis extends the standard model of intergenerational relationships, which until now has been applied only to countries with bilineal kinship systems, to culturally and economically diverse areas with varying kinship systems. Latent class analysis was applied to measure affection and conflict in the ongoing relationships of young and middle-aged women with their mothers (7,522 relationship pairs) and fathers (5,338 relationship pairs). The empirical analysis was based on standardised oral interviews with mothers from areas in China, Indonesia, North and South India, South Africa, Palestine, Israel, Turkey, Russia, Estonia, Poland, East and West Germany, France, Jamaica, and the United States (n = 8,756). The best fitting model of relationship differences consisted of four latent classes: „amicable“ (45 percent), “detached“ (28 percent), “ambivalent” (22 percent), and “disharmonious“ (5 percent). Based on a cross-culturally largely invariant measurement model, results revealed significantly different distributions for fathers and mothers and across areas. Multinomial three-level regression analysis was used to analyse the complex cross-level interdependence of area effects, individual characteristics, and the respective relationship on class membership. In patrilineal societies, relationships of women with their biological parents are more likely to be ambivalent, less likely to be detached, and very likely to become disharmonious in case of spatial proximity. In affluent societies, the relationships are less likely to be disharmonious and most likely to be harmonious. Whereas the frequency of contact decreases the likelihood of detached or disharmonious relationships in affluent societies, functional exchange with the parents increases the likelihood of disharmonious relationships.Dieser Beitrag liegt nur in englischer Sprache vor.Studies of intergenerational solidarity in affluent societies suggest that relationships between generations consist simultaneously of both emotional closeness and conflicts. This analysis extends the standard model of intergenerational relationships, which until now has been applied only to countries with bilineal kinship systems, to culturally and economically diverse areas with varying kinship systems. Latent class analysis was applied to measure affection and conflict in the ongoing relationships of young and middle-aged women with their mothers (7,522 relationship pairs) and fathers (5,338 relationship pairs). The empirical analysis was based on standardised oral interviews with mothers from areas in China, Indonesia, North and South India, South Africa, Palestine, Israel, Turkey, Russia, Estonia, Poland, East and West Germany, France, Jamaica, and the United States (n = 8,756). The best fitting model of relationship differences consisted of four latent classes: „amicable“ (45 percent), “detached“ (28 percent), “ambivalent” (22 percent), and “disharmonious“ (5 percent). Based on a cross-culturally largely invariant measurement model, results revealed significantly different distributions for fathers and mothers and across areas. Multinomial three-level regression analysis was used to analyse the complex cross-level interdependence of area effects, individual characteristics, and the respective relationship on class membership. In patrilineal societies, relationships of women with their biological parents are more likely to be ambivalent, less likely to be detached, and very likely to become disharmonious in case of spatial proximity. In affluent societies, the relationships are less likely to be disharmonious and most likely to be harmonious. Whereas the frequency of contact decreases the likelihood of detached or disharmonious relationships in affluent societies, functional exchange with the parents increases the likelihood of disharmonious relationships

Spartan Daily, March 11, 1963

Volume 50, Issue 84https://scholarworks.sjsu.edu/spartandaily/4431/thumbnail.jp

The New Hampshire, Vol. 23, No. 8 (Nov 17, 1932)

An independent student produced newspaper from the University of New Hampshire

Influence of Noise on Scattering-Parameter Measurements

We present a general model of noisy scattering-parameter (S-parameter) measurements performed by a vector network analyzer (VNA). The residual error of the S-parameter due to the noise is examined to appear like a complex Gaussian quotient. The statistical analysis of the residual error is given, and relevant statistical quantities are derived and discussed. Experiments were conducted on a two-port VNA to validate the noise-influenced S-parameter model. We show that the uncertainty due to the noise is often critical in S-parameter measurements, in particular for S-parameters of a small magnitude

Bulletin of the Center for Children's Books 38 (07) 1985

published or submitted for publicatio

Ariel - Volume 11 Number 5

Executive Editors Ellen Feldman Leonardo S. Nasca, Jr. Business Managers Alex Macones Martin B. Getzow News Editor Hugh A. Gelabert Features Editor Aaron D. Bleznak CAHS Editor Joan M. Greco Editorial Page Editor Samuel Markind Photography Editor Todd L. Demmy Sports Editor Paul F. Mansfield Commons Editor Saul I. Helfin