Much research has attested to the benefits of intergenerational programs (IGPs) for older and younger participants, but there is a lack of understanding about what motivates them to participate and to persevere. We conducted structured interviews with 83 older (mean age = 77) and 96 younger (mean age = 23) participants who participated in 13 IGPs in Israel, some involving specific topics, and some providing assistance to older adults. Using a mixed-methods approach, we analyzed differences in motivation across generations and program types and compared initial and ongoing motivation to participate. We found differences regarding motivation by age group and program type: Among older participants, interest in the specific subject was a more prevalent motivation in topic-focused groups, while receiving support was more common in assistance groups. Among young persons, motivations relating to obligation, such as receipt of a financial scholarship, and the wish to help others were the most prevalent motivators. Ongoing motivation was often explained by positive intergenerational relationships and enjoyment. For older adults, offering more diverse topic-focused activities may motivate greater participation. For young adults, integrating IGPs within more and different settings, and promoting IGPs as opportunities to help others are potential motivators. Intergenerational programs [IGPs] are types of human services that involve ongoing and organized interactions between members of younger and older age groups for the benefit of all participants . They arThe negative physical and mental health impact of social isolation and loneliness on older adults has been widely studied ,6,7, yetResearch has also identified challenges, including uneasiness experienced by young people when encountering older people with dementia or frail older adults . AnotherAlthough the motivation of participants to take part in IGPs has not been studied specifically, several studies have mentioned the willingness of participants to continue participating in IGPs, e.g., . In one study, university faculty members worked in collaboration with a team of older volunteers to develop, deliver, and evaluate an undergraduate intergenerational course about aging . The oppThe limited available literature principally describes the motivation of older adults, who generally report more benefits than young adults in IGPs . Little In this paper, we aim to address the lacuna in the research regarding the motivation of younger and older individuals to participate in IGPs. Addressing this gap aims to provide organizers of IGPs with new insight into the recruitment and retention of participants. Our research questions are: (1) What motivates young and older participants to participate in IGPs? We examined both initial motivation and ongoing motivation for both groups. (2) How do the different types of motivation relate to young and older participants’ background characteristics? (3) For young participants, we also examined whether they perceived their involvement as voluntary or obligatory.We sought to include all the types of community-based IGPs offered in the greater Tel Aviv, Israel metropolitan area by contacting national and municipal offices and diverse not-for-profit organizations. We learned of additional programs through snowballing. Our data were generated from 13 programs. We included only IGP programs that were organized by community-based organizations, in contrast to school-based, and were conducted at community centers or participants’ homes, rather than in school buildings or long-term care settings.We categorized the 13 community-based IGPs found in greater Tel Aviv based on their most salient content and goals: (1) “Topic-focused programs,” which involved the performing arts such as theater and dance, the study of Jewish texts, a photography course, and a bridge club; (2) “Assistance groups” designed to provide one-on-one support with daily activities for older persons, including visits to their homes, documentation of their life stories, in-home computer assistance, or assistance with pet care. Young participants and older participants met about once weekly on average. While all programs were aimed at improving intergenerational relationships, the topic-focused programs provided meaningful recreation rather than personal assistance. Some programs recruited young persons from among specific groups, e.g., university students, soldiers. Young participants either volunteered or participated non-electively as part of their military, pre-military, national service, or academic course work. Some young participants received a financial or housing scholarship for their participation. Older participants were referred to some IGPs by social services workers, or by word of mouth, or responded to publicity about opportunities to participate in such programs.The study was approved by the ethics committee of Tel Aviv University. After informed written consent was obtained, questionnaires were either completed during personal interviews by trained interviewers (~80%) or were self-administered.The questionnaires first inquired into participants’ demographic background. The subjective health of older participants was measured using the global self-rated-health item on SF-36 , using aWhy did you choose to participate in the activity?What motivates you to continue to take part in the activity and persevere in it?Is the activity a mandatory or voluntary task for you?Motivation for participation in the IGP was queried via three questions:The third question was addressed only to young participants. While the last question requested a binary (Yes/No) response, all young participants were also asked to specify the elective or non-elective nature of their involvement. These clarifications and responses to open-ended questions enabled us to identify the organizational framework within which 91 (out of 96) young participants participated in the IGPs. This study used a mixed-methods approach, combining quantitative and qualitative analyses. The research question as to what motivated young persons and older persons to participate in IGPs was first examined qualitatively. We then used the themes reached to code the responses of each participant. We used descriptive analyses of frequencies and cross-tabulation to respond to the research questions.Qualitative analysis. We used a thematic approach to understand participants’ motivations for participating in IGPs . To adhet-tests to examine the relationships between background information and motivational factors. Because of the exploratory nature of those analyses and the use of multiple comparisons, we applied a criterion of p-value of 0.001 for presentation of significant findings. Quantitative analysis. We used chi-square and 79 = 3.34, p = 0.001).We interviewed 83 older persons and 96 young persons. For older participants, the mean age was 77.07 (SD = 7.07), and 83.1% were women. Of the older participants, 46 (55.4%) participated in the topic-focused groups and the rest in the assistance groups. Young participants reported a mean age of 23.49 (SD = 6.25), and 66.7% were women. Of the young participants, 32 (33.3%) participated in the topic-focused groups and the rest in the assistance groups. Age and gender were not significantly associated with type of IGP in either age group. Older participants in the topic-focused IGPs reported better health (mean = 3.42) than those in the assistance groups -IM) or “I really like theater and the subject is very interesting for me. When I was young, I participated in a number of plays and then I did not [further] develop this [interest]. And it intrigued me and I wanted to try and see how it would go” (#84-F-80-TF-IM) or “It is relaxing and good. This photography is relaxing and there are beautiful things to photograph in this country” (#25-F-70-TF-OM).Satisfaction and joy, OM. “I enjoy every moment, a great pleasure” (#198-F-76-TF-OM) or “The activity gives me pleasure” (#176-F-78-TF-OM). “The joy I have in my heart, I love it. It’s love” (#87-F-79-TF-OM).Intergenerational interactions and friendship, IM and OM. “They give me meaning in life, content. I am getting to know these wonderful youth” (#143-F-90-assistance-IM) or “It interests me to be among the youth” (#184-F-77-TF-IM) or “I love being close to the younger generation, I feel young in spirit and love to dance” (#197-F-78-TF-IM) or “The fact that it gives me pleasure, I feel with them … they opened up to me. I tell about my life and they tell me about their life, they are like my granddaughters. I feel like I have expanded the family” (#171-F-77-assistance-OM).Receiving support , IM and OM. “I asked for someone to come to me because I am alone. There are afternoons when I don’t even have anyone to talk to. I wanted someone to come so that I will have someone to talk to and who will help me with all sorts of things I need” (#142-M-83-assistance-IM) or “I wanted someone to teach me [about] a computer. I have a computer and did not know how to use it. When the kids were little, I avoided the computer and now I really like this device and just want to know how to develop [using] it. Bringing someone in costs a lot of money; [A volunteer organization] had a course and they stopped it … the best thing is to have someone come and sit by you and guide you, that’s the best” (#44-F-75-assistance-IM) or “It gives me a lot, I am very satisfied. I always say, [when] Ron returns he is the best. I love him, it’s not like a woman’s love, it’s like a mom and son’s love. He does everything for me when he can. He helps me a lot. I give him a score of one hundred” (#17-F-85-assistance-OM).Invited to participate, IM. “Rachel asked me to attend, so I agreed. I do not know the specific neighborhood, so it was interesting for me. It’s nice to be with young people; They instructed us nicely” (#30-F-80-TF-IM) or “I did not choose [the IGP], my friend, Rhonda, who works here at the club, teaches gymnastics or something here, told me that there is this activity and I said I would come and check it out” (#99-F-74-TF-IM).Activity/Having something to do, IM and OM. “Ever since I retired, against my will, I started looking for what to do. To my great joy, I received a gift from the job from which I had been fired, a personal computer, and I said to myself, what am I doing now?! What do I do in life? Then I started writing my life story. It took me 10 years. After that, I started writing short stories and participated in a creative writing class. And then I did research on a family that perished in the Holocaust; No one knew about the others’ existence, and finally the second generation found each other. It’s a life story of this family. So now I’m looking for activities, and each time a different activity. …Then I heard about the community theater, and I said, ‘Oh!’ Now I have something to do” (#89-M-77-TF-IM) or “A friend told me it exists; It’s something I’ve never done before. I have time now, something I have never had in my life. So, I came, I had a nice time, so I stayed. Everything was so light. Without major goals. Had it not been interesting, I would have left, but it was fine, so I stayed” (#104-F-70-TF-IM) or “Because [otherwise] it’s a day with no activities” (#81-F-79-TF-OM).Sense of commitment, OM. “It’s a responsibility, as soon as I get into something, then I take it upon myself to complete it. I take on a new challenge every year, and every time I take on such a challenge I keep going until the end. I intentionally take on all sorts of things every year that present a challenge for me “(#79-F-71-TF-OM) or “It’s a matter of character; I will not leave in the middle…” (#92-F-70-TF-OM).Philanthropy/Desire to help young persons, IM and OM. “Because it fulfills me, it’s my life. It is important for me to pass on to the young people the values with which I grew up” (#36-F-82-assistance-IM) or “It is my pleasure that I can teach young people and pass my knowledge on to them” (#36-F-82-assistance-OM).Obligation, reported both as IM and OM. “Obligation” refers to young participants’ reports that IGP participation was a non-elective activity. “[I participated] because it is mandatory to make a personal commitment and this activity was the most convenient and appropriate for my schedule” (#199-F-16-TF-IM). The obligation may have been part of national service enlistment, army service, in those units that require it, or as a condition for receipt of a financial study or housing scholarship, etc. “The military required that we participate” (#7-M-21-assistance-IM) or “I committed to the activity at the beginning of the year so I persevered with it” (#199-F-16-TF-OM).Interest in the subject matter, IM and OM. “I have been participating in community theater for three years in a row in this course. There are quite a few frustrations and challenges in this … but I do it because it is the only production in the theater department that allows all students to participate in a creative way, and writing is also one of the things that interests me and this is the only course that allows one to write. Plus, it allows me to perform on stage” (#111-F-25-TF-IM) or “I was interested in learning bridge” (#185-M-16- TF-IM) or “Both the stories themselves and the enthusiasm of the older women motivate me to continue” (#34-M-19-assistance-OM).Philanthropy/Helping others, IM and OM. “It was important for me to do something for the community before I graduate from university” (#52-M-27-assistance-IM) or “[I liked] the idea of a year of contribution to the State, in the framework of a year of national service, the understanding that I am helping and really contribute where I am needed” (#137-F-19-assistance-OM).Satisfaction with the activity and enjoyment of the activity, OM. “Personal satisfaction” (#7-M-21-assistance-OM) or “I really, really enjoy it …. love to come to meetings with Hanna” (#51-F-19-assistance-OM).Intergenerational interactions and friendship, IM and OM. “I chose to take part in the activity because I also believe in the goals of the activity; I think it is important to get the generations closer. In addition, the older generation has a lot to contribute to us, the younger generation. There is something to learn from them” (#203-F-16-TF-IM) or “I connected to that woman and that moving life story of hers, and also the need to feel the same feeling of satisfaction when I walk away knowing I made her smile and enjoy those moments” (#133-F-24-assistance-OM).Personal or professional value/experience, IM and OM. “I worked at [another volunteering organization that provides a scholarship] for three years, and I wanted to diversify a little for myself. I got a lot out of it, I do not shy away from older persons, like many people who flinch” (#154-F-26-assistance-IM) or “For me it’s part of personal development … it gives me answers and questions about my life…it connects me to the world I study, the whole subject of coaching. It gives [or] connects me to the spiritual side of what I am learning. Gives me a perspective that is more human, in the Jewish point of view and less religious … It is more a matter of a humane, philosophical outlook. This is a field that interests me very much” (#80-M-44-TF-OM) or “I think I derive something from it … I’m getting to know a population that I hadn’t had much contact with. It’s seeing different perspectives. I guess it’s also somewhere nice to have another circle of friends, both from my side and from hers” (#151-M-28-assistance-OM).Sense of commitment, OM. “I began the activity so I will finish it. It is not fair to stop in the middle, also to the old woman” (#60-F-20-assistance-OM) or “Commitment to the group, desire to learn to direct and gain experience” (#124-F-31-TF-OM).177 = 4.04, p < 0.001, see Overall, the most common motivational factors that prompted older participants to become involved in IGPs  were intTo understand the extent to which young participants’ involvement was actually voluntary or mandatory, we cross-tabulated the organizational frameworks  under which the young participants participated by their reports of the activity being voluntary or mandatory—both in the open-ended questions about their motivation to volunteer and in a question that directly addressed whether their activity was voluntary or mandatory. As shown in p ≤ 0.001 are reported.Since the examination of the relationships between reported sources of motivation and background variables was exploratory and involved multiple comparisons, only relationships that were found to be significant at a level of 79 = 3.45, p = 0.001). Similarly, older participants who reported sense of commitment as a motivational source were younger than older participants who did not report so . Older persons who reported receiving support as a motivation for participation reported worse health status than those who did not . No relationships with gender fit our criteria. For young participants, none of the relationships fit our criteria. Age, gender, and health. Older participants who reported interest in the topic as a motivational source were younger than older participants who did not report such interest . Conversely, older participants in the topic-focused IGPs were more likely to report interest in the topic as their motivation (73.9%) as compared to those in the assistance group . Among young persons, those who were in the assistance groups were more likely (56.3%) to report that the activity was motivated by philanthropy  as compared to those in the topic-focused groups .Type of activity. Among older participants, those who were in the assistance groups were more likely (70.3%) to report that the activity was motivated by need for support and companionship as compared to 2.2% in the topic-focused groups  school project, academic course, n = 23, (3) condition for financial scholarship, n = 23, and (4) purely voluntary activity not related to a formal framework, n = 10. As summarized in 23 = 27.25, p < 0.001). Philanthropy/helping others was often a motivation reported by actual volunteers (80%) and by those providing a year of military, pre-military, or national service (63%), as compared to those receiving a financial scholarship (34.8%) or those in a school-related activity . Interest in the subject was most often mentioned by actual volunteers (80%) and by those in school-related activities (60.9%), but much less so among those receiving financial scholarships (17.4%) or those enlisted in military, pre-military, or national service . We compared young persons’ reports of motivation in terms of the four organizational frameworks through which they participated: (1) pre-military, military, or national service, The most striking finding was that about 90% of the young participants were in educational or enlisted-service frameworks that mandated “voluntary” activity, either in IGPs or in other endeavors. This is especially illuminating in view of the fact that we located the young participants not through those educational or occupational frameworks but through the IGPs. This suggests that the vast majority of young persons in IGPs, at least those in the geographical area under study, are not truly volunteers. This finding corroborates previously reported complaints by organizers of IGPs concerning the difficulty of recruiting young persons . Our resThe distinction between “obligatory” and “voluntary”, however, is somewhat obscure in this context. Over half of the young participants belonging either to a pre-military, military, or national service organization, and many of those receiving a financial scholarship that required participation, described their involvement as “voluntary.” This is probably due to two types of reasoning. The first is that the choice of enrolling in the organizational framework is usually voluntary, i.e., participants choose to take the specific course or to apply for the financial scholarship that comes with conditions; the second, reported in some of the verbal responses to our open-ended questions, is that participants have the option to choose from several “voluntary” activity options. Participants who felt that they “chose” to enlist in national service, or “chose” the academic course that required participation, or “chose” the particular IGP from a number of options, considered their participation voluntary.Among older persons in the assistance programs, 70% said they were motivated by a need for support and companionship. In a sense, individuals among these older persons were compelled to participate because they required help and/or were lonely, and the IGP was the only available solution. In addition, close to 30% of all older participants said they joined because they were invited to attend, suggesting that observers perceived a need for intervention, and 10% said they enrolled because it gave them something to do, suggesting a need for occupation. Similarly, the findings point to significant relationships between the background variables of age and health status and the source of motivation to participate in IGPs. Thus, for some older persons, involvement in an IGP, while not mandatory, was in some way dictated by their life circumstances.Reported motivational factors differed between young and older participants in several ways. The sources of motivation of “obligation”, ”philanthropy” , and “valuable experience”  were mentioned mostly by young participants. In contrast, the factors of “being invited to participate”, “receiving support” and “having something to do” were mentioned only by older participants. Some motivational factors were reported by both young and older participants, including “interest in the subject matter”, “satisfaction and joy derived” (or anticipated) from the activity, “intergenerational interactions and friendship”, and “sense of commitment.” The differences we found partly reflect the different life experiences of the respective generations. For example, older, but not young participants, reported the need for support and the need to fill available time. The differences also reflect that a large proportion of the IGPs under study were devoted to providing assistance to older persons, which relates directly to the altruistic motivations expressed among young participants and the need for support expressed among older participants. For older participants who were involved in the topic-focused groups, interest in the subject matter was often a key motivation. We found that motivational factors change with time during the course of IGPs. The motivations of satisfaction and joy, as well as intergenerational interactions and friendship developed into ongoing motivations as the programs continued, whereas other motivational factors, such as interest in the subject matter (among older and young persons) and personal/professional value (among young persons) tended to decrease. To some extent, this suggests that the success of the IGPs in our study provided the motivation needed to promote continued participation. The implications of the motivational factors that emerged in this study should be investigated in future studies. For example, were young persons who cited enjoyment of intergenerational interactions and friendship as ongoing motivations to participate in IGPs more likely to manifest a decrease in ageist attitudes or an increase in interest in pursuing a career involving older adults? Were older persons who cited enjoyment of intergenerational interactions and friendship as ongoing motivations more likely to report a decrease in feelings of isolation and loneliness? As both young and older participants cited “interest in the topic” as a motivating factor, offering more diverse topic-focused activities may motivate greater participation, in addition to integrating IGPs within more and different settings, and promoting IGPs as opportunities to help others.A study of an American IGP, “The Experience Corps” ,30, in wWe aimed to recruit all IGPs in the greater Tel Aviv metropolitan area. Therefore, the study is limited to programs in the cultural context of Israel. We also limited the IGPs to those in the community, rather than in schools. As participants volunteered to participate in the research, the data may represent a bias towards more successful experiences in IGPs. Another limitation is the use of qualitative analysis to determine the distribution of sources of motivation, whereas future research should validate those findings via quantitative questionnaires based on our findings, which are likely to provide more precise estimates. These limitations notwithstanding, this study is innovative in examining in depth young persons’ motivation to participate in IGPs, in examining the complex meaning of “volunteering” in this context, in comparing young and older persons’ motivations to participate, in studying how those motivational factors change during the course of IGPs, and in examining community-based rather than school-based IGPs. The study is strengthened by convergent results vis a vis the few studies available on the topic of motivation to participate in IGPs and by its use of a larger sample than most studies of IGPs, thus offering valuable guidance for future planning of IGPs. The findings highlight the crucial need for motivational analysis prior to initiating an IGP. In our study, the often-proclaimed virtues of IGPs, such as promotion of intergenerational understanding and encouragement of altruistic community involvement, emerged as insufficient without the weight of additional self-interest or external requirements as motivational factors. Most topic-focused programs provided some self-interest in the form of offering an experience that would further knowledge or experience in a particular topic, but among such IGPs, most young participants were bound by additional obligatory frameworks. In the assistance groups, the self-interest of older participants was evident, and although more than half of young participants in the assistance groups reported philanthropy/altruism as a motivational factor, it is likely that many would not have participated without having been obligated to do so. These findings are important because they challenge the general perception that recruiting “volunteers” is an inexpensive means of improving intergenerational relationships and remedying social problems. Rather, the results suggest that significant funding or embedding IGPs in a compulsory framework may be needed. The motivational factors of “satisfaction”, “interpersonal relationships”, “idealism/sense of philanthropy”, and “interest in the topic” appear to need a boost from a different realm. These complexities need to be considered in planning IGPs and evaluating their outcomes. Future research should examine correlations between the motivations of dyads in one-on-one IGPs and within specific topic-based groups. Future work could also include a cost/benefit analysis of IGPs by following cohorts on a longitudinal basis, exploring participants’ ongoing and changing sources of motivation, and also the long-range impact on participants, such as the percentage of young participants who go on to pursue careers involving older adults.


Neurodevelopmental disorders (NDDs) are conditions affecting a child's cognitive, behavioural, and emotional development. Appropriate and validated outcome measures for use in children with NDDs in sub-Saharan Africa are scarce. The aim of this study was to validate the Communication Profile Adapted (CP-A), a measure developed in East Africa to assess caregivers' perception of communication among children with NDDs.We adapted the CP-A for use in Ethiopia, focusing on the communicative mode (CP-A-mode) and function (CP-A-function) scales. The CP-A was administered to a representative sample of caregivers of children with NDDs and clinical controls. We performed an exploratory factor analysis and determined the internal consistency, test-retest reliability, within-scale, known-group, and convergent validity of the identified factors.N = 300 participants . Within the CP-A-mode, we identified two factors ; the CP-A-function scale was unidimensional. Combining both scales into one summary variable  resulted in a scale with excellent internal consistency and test-retest reliability . Testing known-group validity, the CP-A-total scores were significantly higher for controls than cases . Convergent validity assessment indicated that scores were negatively and moderately correlated with clinical severity .Our analysis included The CP-A is a valid tool for the assessment of communication among children with NDDs in Ethiopia. It holds promise as a brief, quantitative, and culturally appropriate outcome measure for use in sub-Saharan Africa. The CP-A assesses caregivers' perceptions of their child's abilities and activities for communication, and participation in family and community events  and King's College London (HR-16/17–3489). In addition to validation of the CP-A, the data collection comprised further questionnaires, the validation of which are reported in Borissov et al. .Participants were 300 caregivers with long-term responsibilities for children aged 2–9 years, attending either the general paediatric or child mental health clinic of the two hospitals. The paediatric clinics consecutively recruited children with physical health conditions for the clinical control group. The mental health clinics consecutively recruited children with either NDDs alone or with a comorbid mental health condition for the case group, in accordance with the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5)  after initial test data collection. All participants provided written informed consent.Eligible families were approached by the attending clinician at either clinic and given a flyer with information about the study. Data collection and consent taking was done by clinic nurses who worked independently from the clinicians supporting the families. A subgroup of 40 caregivers was invited for a retest of an average of 19.6 days  and communicative function (CP-A-function) were selected for validation in this study, see Items' responses refer to a 0–4 Likert scale where [0] stands for ‘never’, [1] ‘rarely’, [2] ‘sometimes’, [3] ‘usually’, and [4] ‘always’. Participants were shown a visual ladder representation of this scale with 0 being the lowest and 4 the highest rung .N = 24) and ASD (N = 50). General psychiatrists working in the two hospitals rated the child's condition, based on the DSM-5 severity ratings, as [1] ‘mild’, [2] ‘moderate’, or [3] ‘severe’. A structured questionnaire was used to collect caregiver-reported demographic information.Clinician-assigned severity levels were available for a subgroup of participants with a diagnosis of ID  as cognitive interviews indicated that the direct translation of the term ‘behaviour’ was unclear to respondents. The original CP-A response scale comprised 8 rungs of ladders; this was shortened to 4 rungs  after cognitive interviewing indicated that participants could not meaningfully differentiate between the small increments of the original ladder. Within the CP-A-function, the original version of the scale did not just require a response using the ladder but also asked the extent to which the item is a problem for the respondent. This second part was removed to improve consistency as well as administration ease and time. In its original format, the CP-A asked both questions on how the child communicates with the respondent, and how the respondent communicates with the child. Caregivers participating in the cognitive interviews had difficulty distinguishing between these questions. In response, the section on how the caregiver communicates with the child was removed, focusing solely on the mode of communication of the child.Within the CP-A-mode, the content of question 1j was changed to Data collection took place in Yekatit 12 and St Paul's Millennium Medical College general paediatric and child mental health clinics. To allow for the participation of non-literate caregivers, instruments were administered through face-to-face interviews by nurses who had received training in data collection procedures and questionnaire administration. Participating caregivers were reimbursed for their travel costs. Data were double entered using Epidata, version 3.1  in the CP-A were replaced by median imputation  CP-A-mode, (ii) CP-A-function, and (iii) CP-A-total were examined with the use of item exploratory factor analysis (EFA)  guidelines   was evaluated through Cronbach's alpha  included children who did not meet the criteria to be in either the case or clinical control group  and those for whom diagnostic information was incomplete (full details in online Supplementary Table S3).Our sample included s.d. = 7.1).p < 0.001), with a higher proportion of females among cases, and residence (p < 0.001), with more cases living in urban areas. Furthermore, we report significant differences in religion, (p = 0.004), observing a higher proportion of Muslims among cases compared to controls, relationship to the child (p < 0.001), where caregivers in the case group were more often mothers, and occupation (p = 0.02), with a higher proportion of housewives among cases.Statistically significant differences between cases and controls were observed for the caregivers' gender (s.d. = 2.0). Among children, comparison across cases and controls showed significant differences in age (p < 0.001), with cases being older than controls, and gender (p < 0.001), with a higher proportion of males in the case group in line with the observation that NDDs are more common in boys .62% of children were male, with an average age of 4.9 years . The correlations between items and promax-rotated common factors are displayed in a ‘Facial expression’, c ‘Gestures’, d ‘Body movements’, e ‘Looking or use of eye gaze’, f ‘Pointing’, and h ‘Showing you objects’ loaded more strongly on factor 1. Items b ‘Making noises ’, g ‘Showing you pictures’, j ‘Speaking Amharic’ and l ‘Behaviour’, instead loaded on factor 2. Factor 1 generally corresponds to physical communication, while factor 2 to verbal communication. Item g ‘Showing you pictures’ presented moderate loadings for both factors  that were maintained throughout different rotation methods. Item l had a negative loading on verbal communication, indicating that caregivers of children with poor verbal communication tended to highlight the use of behaviour as the main mode of their child's communication.The Scree-test and eigenvalues  suggested a 2-factor solution for the CP-A-mode , suggesting that the function items assessed a single construct. All items loaded positively on the factor, see For communicative function, the Scree-test and eigenvalues  suggested a 1-factor solution. The model had a good fit to the data , suggesting that all items of the analysed CP-A scales can be meaningfully subsumed under a single construct reflecting overall communication activities. All items, besides Item l, loaded positively on the factor .Similarly, for the CP-A-total the Scree-test and eigenvalues  indicated a 1-factor solution. The model had a good fit to the data (ρ) between the CP-A-mode's verbal communication and physical communication scores was significant and moderately positive . Similarly, the correlation between the CP-A-mode and CP-A-function scores was significant and moderately positive , further justifying the adoption of the CP-A-total as a unidimensional summary scale  and verbal communication . This holds true also for the CP-A-function  and CP-A-total , d estimates for CP-A-mode's factors were medium when unadjusted and small when accounting for covariates. Estimates were large for CP-A-function and CP-A-total across both scenarios  as well as verbal communication . Similar results were obtained for the CP-A-total . For the CP-A-function, the point estimate of the correlation was in the same direction and of similar magnitude (ρ = −0.23) but was not significant (p = 0.59)  ⩾ 0.78; verbal communication: 6 items, α = 0.72, 95% CI ⩾ 0.69). Excellent internal consistencies were obtained for the CP-A-function  and the CP-A-total . Assessing internal consistency separately for cases and controls suggested that levels remained acceptable for the CP-A-mode's physical communication , but were low for verbal communication . The internal consistency remained excellent for CP-A-function  and CP-A-total  , demonstrating moderate to a substantial agreement among items. For CP-A-function and CP-A-total, the agreement was moderate to near perfect for all items   and verbal communication . Excellent test-retest reliability was observed for the CP-A-function  and CP-A-total  , supported by findings of strong correlations between the identified factors. As hypothesised, children with NDDs (cases) scored lower than clinical controls. Moreover, scale scores were negatively correlated to clinical severity ratings of NDDS, indicating that children with more severe NDDs used fewer modes of communication and applied fewer functions of communication. We observed acceptable to excellent internal consistency as well as test-retest reliability. Overall, these results demonstrated the validity and reliability of the CP-A-mode, function, and communication-total scales.This paper reports the first validation study of the CP-A, a caregiver-reported tool for the assessment of children's communication. Our aim was to address the need for a psychometrically sound, brief, and culturally appropriate outcome measure for use in Ethiopia. We investigated two sections of the CP-A: communicative mode and function. Within communicative mode we identified two factors, i ‘Sign language’, j ‘Speaking English’, and m ‘Other’, most responses were equal to 0 (i.e. ‘never’), suggesting these items have little relevance in assessing communicative mode within our sample. For item i this is likely due to the fact that none of our participants was reported to have hearing loss. Furthermore, formal sign language has received very limited implementation in Ethiopia so far ’ and j ‘Speaking Amharic’. Item g ‘Showing you pictures’ showed more moderate loading, with substantial cross-loading on factor 1. Item l ‘Behaviour’ had a strong negative correlation with factor 2, suggesting that caregivers of children who did not express themselves verbally were more likely to report their child's behaviour as a form of communication. This finding is in line with theoretical notions that see verbal acts and behaviours as equivalent in function  and did not include a control group, results suggested sensitivity to change, as scores in the activities for communication sections, including communicative mode and function, were significantly higher post-intervention compared to pre-intervention  still scores higher than the case group. The clinical severity assessment was carried out by general psychiatrists, with no specialist expertise in the assessment of child NDDs. Nevertheless, diagnosis and assessment of children with NDDs part of postgraduate training in psychiatry in Ethiopia and the psychiatrists involved in the study were experienced in making these diagnostic assessments. Psychiatrists used their clinical judgement rather than standardised tools to assess the severity of impairment of the children since there are no validated standardised clinical severity assessment scales available in Ethiopia to support the assessment of severity. Furthermore, the limited educational or supportive service provision available for children with NDDs in this setting means that reports from other professionals are not available to inform severity assessments. Thus severity is based on a single report from the caregiver during the clinical encounter and observations of the psychiatrist of the child in a clinical setting. Severity scores were collected for a small sample only, covering a limited range of complex communication needs. This study was conducted as part of a larger project focusing on NDDs. Most cases included in our study had ASD or ID, rather than a wider group of complex communication needs and developmental disabilities for which the CP-A was also developed (e.g. sensory impairments). Lastly, participants were help-seeking families recruited in Addis Ababa. Our sample had an overrepresentation of urban families and may not be fully representative of the Ethiopian population.i ‘Sign language’), should be re-evaluated. Future research should further examine whether the CP-A-mode, CP-A-function, and CP-A-total are sensitive to change induced by interventions. Lastly, future studies may also wish to consider other sections of the CP-A not included in the current evaluation.Further research could test our factor structure for the CP-A-mode and CP-A-function through confirmatory factor analysis. Recruitment of cases should be extended to a wider range of complex communication needs. In such studies, the relevance of items reflecting characteristics of developmental disabilities with no representation in our sample (i.e. This work is the first investigation to explore the validity of CP-A, an open-access measure developed in and for the African context. The communication mode and function and their combined scales met the validity and reliability criteria as a measure for the assessment of caregiver-perceived activities for communication. We tested this among children with NDDs and concomitant complex communication needs in Ethiopia. We recommend the further validation of this scale. The CP-A has potential for application in intervention studies on NDDs across sub-Saharan Africa as a brief, quantitative, and culturally appropriate outcome measure.


We investigate the analogues, in  Being interested in the maximum order of d(n), he defined highly composite integers n to be those for which In , Ramanujmentclass2pt{minimf, and observe that a generic polynomial f in We consider the question of maximising the divisor function in the function field setting. Let highly composite polynomial of degree n if We call Highly composite polynomials of a given degree are not necessarily unique. For example, all linear polynomials in f be a highly composite polynomial of degree n and suppose otherwise, so that There is (at least) one new highly composite polynomial at each degree. Indeed, let If See “Appendix” for an illustrative table of highly composite polynomials in We first investigate a family of highly composite polynomials nujan in ), which x-superior highly composite polynomial, or just x-SHC, if for all h is an x-semi-superior highly composite polynomial, or x-SSHC, if for all x-SHC or x-SSHC for some superior highly composite or semi-superior highly composite, respectively.Let x-SHC, then it is x-SSHC. Moreover, any polynomial h which is x-SSHC is highly composite, since if Clearly, if n by Eq.  we have x of an x-SSHC,After defining a particular set for the parameter lettingwe are able to determine the structure of the superior highly composite polynomials.x-SHC polynomial, namely There is one, and only one, S, then S and the set of superior highly composite polynomials, given by If Let and that of the semi-superior highly composite polynomialsx-SSHC polynomial, namely the polynomial If d in Eq. .x-SSHC polynomials of the form h given by If equation , \documeh(x) is x-SSHC and g is also x-SSHC.If Let This family of semi-superior highly composite polynomials is not too sparse, so we can use it to construct polynomials at every degree which make the divisor function close to its maximum. In particular, if we let h be an x-SSHC polynomial of degree Let s in Eq. , and h bN in Theorem N in the form presented in Theorem From the final sentence of part 2 of Theorem We begin by showing, contingent on some auxiliary lemmas proven subsequently, thath is unique.If h.Else, if For each From , we can j, so by Lemma If s, r) such that j, so by Lemma Otherwise, if h is in fact a finite product.Notice in both cases that This leads us first to the proof of Theorem x-SHC. Moreover, if x-SHC, and moreover, if If S, and let S and the set of superior highly composite polynomials, given by Let For Then we move to the proof of Theorem If x-SSHC polynomials are precisely the E, which we can rewrite as x and y, we have y, we have for h(x) is the (distinct) superior highly composite polynomial immediately preceding If mentclass2pt{minimh(x) be x-SSHC and g is also x-SSHC.Let entclass1pt{minimaFor Finally we conclude by proving the auxiliary lemmas used in the proof of Proposition j, then If j, then Else, if Let If Else, if Let ands, r) such that Let r, s) and  distinct. If Suppose otherwise, so that Now Since the family of semi-superior highly composite polynomials is not too sparse, we can use it to construct polynomials at every degree which make the divisor function close to its maximum, and thus prove Theorem h is highly composite, so h is x-SSHC, and so for any If entclass1pt{minimaentclass1pt{minima


Obtaining valuable clues for noncoding RNA (ribonucleic acid) subsequences remains a significant challenge, acknowledging that most of the human genome transcribes into noncoding RNA parts related to unknown biological operations. Capturing these clues relies on accurate “base pairing” prediction, also known as “RNA secondary structure prediction”. As COVID-19 is considered a severe global threat, the single-stranded SARS-CoV-2 virus reveals the importance of establishing an efficient RNA analysis toolkit. This work aimed to contribute to that by introducing a novel system committed to predicting RNA secondary structure patterns  that leverage syntactic pattern-recognition strategies. Having focused on the pseudoknot predictions, we formalized the secondary structure prediction of the RNA to be primarily a parsing and, secondly, an optimization problem. The proposed methodology addresses the problem of predicting pseudoknots of the first order (H-type). We introduce a context-free grammar (CFG) that affords enough expression power to recognize potential pseudoknot pattern. In addition, an alternative methodology of detecting possible pseudoknots is also implemented as well, using a brute-force algorithm. Any input sequence may highlight multiple potential folding patterns requiring a strict methodology to determine the single biologically realistic one. We conscripted a novel heuristic over the widely accepted notion of free-energy minimization to tackle such ambiguity in a performant way by utilizing each pattern’s context to unveil the most prominent pseudoknot pattern. The overall process features polynomial-time complexity, while its parallel implementation enhances the end performance, as proportional to the deployed hardware. The proposed methodology does succeed in predicting the core stems of any RNA pseudoknot of the test dataset by performing a 76.4% recall ratio. The methodology achieved a F1-score equal to 0.774 and MCC equal 0.543 in discovering all the stems of an RNA sequence, outperforming the particular task. Measurements were taken using a dataset of 262 RNA sequences establishing a performance speed of 1.31, 3.45, and 7.76 compared to three well-known platforms. The implementation source code is publicly available under knotify github repo. The RNA molecule, being the intermediate representation of the information flowing from DNA to proteins, holds a crucial role in many biological processes. Recent studies reveal that its detailed structural analysis is of utmost importance. The RNA structure is often depicted by a 2-D representation of the base-pairing nucleotide  known as the “secondary structure”, which conduces to the more-complex construction 3-D representation, termed the tertiary structure. The substantial role of RNA in the expression of proteins, along with its contribution to the gene expression regulation, catalysis, and site recognition, requires the enlightenment of its territorial structure, which is associated with considerable biological functionalities.More specifically, non-coding RNAs (ncRNAs) are functional RNA molecules transcripted from DNA but not translated into proteins. The latter must not be misinterpreted as not enclosing important information or contributing to any biological operation. Current evidence implies that ncRNAs transcribe most of the genomes of mammals and other complex bions, to contradict the widespread assumption that proteins transcribe most genetic information. Their purpose is to fulfill diverse catalytic and structural functions, along with regulating gene expressions at the transcriptional and post-transcriptional level.The literature contains a considerable number of publications discussing the prediction of RNA secondary structure. The employed methodologies are mainly based on dynamic programming algorithms, thermodynamic models, stochastic methods, and syntactic pattern-recognition techniques. A thorough analysis of the related work is presented in The article is organized at the following sections. In Non-coding RNAs (ncRNAs) are functional RNA molecules that are transcripted from DNA but not translated into proteins. This is not to say that ncRNAs do not provide knowledge or serve a purpose. Although it has been generally assumed that proteins transcribe most genetic information, recent evidence suggests that ncRNAs transcribe most of the genomes of mammals and other complex bions. Their purpose is to fulfill diverse catalytic and structural functions and regulate gene expressions at the transcriptional and post-transcriptional level.RNA is the cornerstone of various biological processes; it carries genetic information that is encoded into four distinct nitrogenous bases, i.e., adenine (A), cytosine (C), guanine (G), and uracil (U). As a messenger , it actsContrary to DNA, RNA features a single-stranded molecule resulting in a solid inclination of its bases to interact in two principal ways, either perpendicular to their planes (stacking) or hydrogen-bonded within the base planes (pairing). Those interactions form base pairs via hydrogen bonds between the corresponding nucleotide, i.e., the standard set of RNA base-pairs  known as Watson–Crick base-pairs  and the According to the literature, there is a strong correlation between the structure of a RNA molecule and its function ,11,12 toTurnip Yellow Mosaic virus  ← EMPTY_SET  endforfunction EARLEY_PARSER  INITIALIZE(input_string)  n ← LENGTH(input_string)Start → •S, 0), Sets[0])  ADD_TO_SET((  for i ← from 0 to n    for each state in Sets[i]      if (state is not completed)        if (RIGHT_TO_DOT(state) is a nonterminal)          PREDICTOR        else          SCANNER        endif      else        COMPLETER      endif    endfor  endfor  return Setsα• Cβ, j), i, grammar)function PREDICTOR((B → δ) in GRAMMAR_RULES  for each , Sets[i])    ADD_STATE_TO_SET, i, input_string)function SCANNER((B →   if (a is input_string[i])γ a •δ, j), Sets[i+1])    ADD_STATE_TO_SET, i)function COMPLETER((A → γ• A β), j) in Sets[x]  for each , Sets[i])    ADD_STATE_TO_SET RNA sequence is parsed using a CFG parser in order all trees that include a pseudoknot to be produced; (ii) all derived trees are then traversed to identify additional base pairs around the pseudoknot; and (iii) the optimal tree is selected via the criteria of minimum energy and the maximum number of base pairs of the pseudoknot. These three main tasks of the proposed methodology see  are thorAs exhibited in The proposed methodology of detecting pseudoknots in sequences of nitrogenous bases representing RNA rests on syntactic-pattern-recognition techniques and specifically on an efficient CFG parser. Consequently, it is vital to select the right primitive patterns. In the case of RNA recognition, the most-typical option is to represent the nitrogenous bases adenine, cytosine, guanine, and uracil as single characters “A”, “C”, “G”, and “U”, respectively. These characters in sequence constitute an RNA representation. Hence, in our case, where a grammar parser is proposed to recognize pseudoknots in RNA, the proposed grammar vocabulary contains only the four terminal symbols T = {“A”, “C”, “G”, “U”}, with each one representing a distinct base: adenine, cytosine, guanine, and uracil, respectively. Therefore, every RNA sequence may linguistically be represented as a string containing the terminal symbols, e.g., UAGGC or AUGGCCGUACG.The task to syntactically recognize a given pattern may actually be converted into using an appropriate pattern grammar, in order to parse the linguistic representation of the original patterns. The design of the pattern grammar may have a significant impact on the recognition’s result. Therefore, the formation of the CFG to be used is an important subtask in implementation having as the underlying model syntactic-pattern-recognition techniques. Hence, the design of an efficient grammar is indispensable in order to describe the syntax of the pseudoknot within any arbitrary RNA sequence. It is well known that CFGs are adequate to represent structural features. The core stems. The second column of ∅ is the empty set. Hence, L may produce strings of length greater than zero as A, UA, CCGGAU, etc.L is the non-terminal symbol that will produce sequences of bases forming the two interior loops of the pseudoknot, i.e., sequences of bases between C and U as well as between G and A. Non-terminal L may produce strings belonging to set D is the non-terminal symbol that will produce sequences of bases between the two crossing base pairs, i.e., between bases U and G in this example. Using non-terminal symbols K and N, L may recognize substrings of terminal symbols of length zero to two, i.e., hub repo .C …U G ……A. Then, rules 19, 18, and 22 were used to recognize the bases between C and U, i.e., C G C C U G ……A. After that, rules 24, 29, and 34 were used to recognize the empty string between the U G bases of the core stems. Finally, rules 16, 17, 17, 17, and 23 were used to recognize the bases between C and U, i.e., C G C C U G A U U U G A. The integration of this substring in the initial RNA sequence and the process of decorating the pseudoknot with additional base pairs is explained in The parse tree produced by the parsing of substring “C G C C U G A U U U G A” is shown in The proposed methodology parses all substrings, beginning with the one that starts with the first sequence symbol and features the minimum potential length. Iteratively, the length is extended by one symbol to include the entire initial RNA sequence finally. In the same iterative fashion, string starting points are augmented to exclude the previous set starting symbol. The parsing is over when the substring’s length to be parsed deteriorates further than a predefined threshold . This methodology, considering that Context-free grammar was selected with the view of augmenting it with attributes (forming an attribute grammar) in order to store probabilities and to manage to prune parse trees during the parse-tree construction process, in future work. In order to enhance the performance of the proposed system, an alternative implementation of the first task was proposed using a brute-force algorithm. This approach traverses the input string in order to spot all possible base pairs and then traverses all base pairs so as to identify couples of base pairs, which potentially form the core stems of a pseudoknot, i.e., consisting of two helical segments that are bound by two single-stranded sections or loops. The brute-force approach achieved a performance speed of 2.55 compared to the implementation that makes use of the context-free grammar at the first task. The speedup, though, comes at the cost of the overall extensibility that the grammar approach provides. In any case, both proposed implementations are faster than two well-known platforms . Performdecoration. Having detected the core stems U–A and C–G at positions 9 and 11 and 5 and 10, respectively, all bases in both pseudoknot loops,  and 11 to 15 (right loop)) were examined to identify against forming base pairs with bases outside the pseudoknot loops. Hence, base pairs belonging in the left loop were checked out if they may form base pairs with bases at positions 17 to 19, while bases belonging in the right loop were examined to match the ones at positions 1 to 4. In both the left and the right loops of the pseudoknot, the base pairs at positions 8–17, 7–18 and 4–11, and 3–12 were sequentially detected, respectively. Once the parse trees are generated following the methodology described in the previous subsection, all trees are traversed to decorate the pseudoknot with additional base pairs. The proposed system permits the user to optionally select the existence of base pairs U–G in the pseudoknot’s loops. At the same time, the existence of bulges or interior loops in the loops of the pseudoknot is part of our future work.According to the literature, multiple methodologies have been presented that tackle the problem of RNA base-pair prediction, with the prevalent ones being the (i) method of minimum free energy , which dThe proposed system employs a hybrid model of optimal tree selection, combining principles originating from the two most-prevalent techniques, i.e., the method of maximum pairing and that of MFE, to predict the pseudoknot pattern of any RNA secondary structure accurately yet in an efficient way. MFE is cost-effective in terms of performance; initially, all trees are sorted by the number of base pairs around the detected pseudoknot, and MFE applies only to the ones ranking the top score of the base-pairs count. This is a heuristic that outperforms the MFE original approach.H-In order to select the best candidate from the set of secondary structures, our method chooses the one with the minimum free energy. To carry out this important task, a module from HotKnots  was incotally in , and refAccording to the literature, the prediction of pseudoknots of any arbitrary RNA sequence is a NP-complete problem. On the one hand, free-energy-minimization algorithms proposed to provide pseudoknot predictions invoke dynamic programming to rank high regarding their computational cost, while their precision decreases proportionally to the length of the input sequence. On the other hand, existing heuristic approaches lack generalization capabilities when being tested under different datasets. In this setting, we introduced a novel hybrid strategy to pick the RNA sub-sequence that is the most-probable pseudoknot expression. According to The proposed implementation is hybrid itself as well. By invoking Python and C code routines, a performant, scalable yet easy-to-use, and extended software package was implemented. Python was used to provide high-level flexibility and out-of-the box features such as flexible parallelization capabilities, sub-process supervision, and file management, while C nailed the parsing task by minimizing the memory footprint and maximizing CPU utilization.The input sequence was sliced into multiple sub-sequences  the accuracy of the pseudoknot’s core stems prediction, (ii) the capability to predict the base pairs existing in the ground truth dot-bracket (confusion matrix), and (iii) the execution time.i, j) is equivalent to , , , and , as proposed in [posed in .The measurements divided per RNA sequences’ length are presented in The percentage of pseudoknot’s core stems exact prediction per platform is also shown in The percentage of pseudoknot’s core stems exact prediction per platform for all groups of length of RNA sequences is also shown in We benchmarked our framework on the sequences of To evaluate the overall performance, we focused on the precision, the MCC, and the F1-score. The latter is the harmonic mean of the PPV and the recall. Our methodology outperformed on average all methods in regards to precision metric having 0.784, while Knotty was 0.729, IPknot 0.718, Hotknots 0.706, and IHFold 0.608. Regardingthe F1-score and the MCC metrics, Knotty outperformed on average all methods with a F1-score equal to 0.807 and a MCC equal to 0.569. The proposed methodology had an accuracy very close to Knotty . In addition, HotKnots scored F1-score = 0.738 and MCC = 0.452, while IPknot  and IHFold  had lower accuracy on both the F1-score and the MCC.The above-mentioned results are also shown in In The last metric that was used in order to compare the proposed methodology with other platforms is that of execution time. In The execution time required per platform is also shown in In Total execution time required per platform for the four groups of different RNA sequences’ length is also shown in Recently, COVID-19 evolved into a severe global threat. This virus’ connection to RNA is one of the most-prominent artifacts justifying the requirement of concrete and accurate RNA-analysis toolkits. This study introduced an innovative methodology to detect H-type pseudoknots in RNA secondary structures accurately and performantly. The method is based on Early’s parser, which, given an RNA sequence in the form of a string, produces a space of all possible parse-trees, each one expressing a potential pseudoknot structure. The optimal tree is picked through a hybrid model combining pairing maximization and free-energy minimization for each context–tree pair. The evaluation of the proposed algorithm demonstrated its outperformance to the task of pseudoknot prediction, paving the way for our future work endeavours: (i) discovering even more complex patterns such as pseudoknots enclosing bulges or hairpins, (ii) building advanced searching algorithms leveraging common patterns of the secondary structure of RNA, and (iii) creating an open web platform to make our work accessible to all researchers. The work in ,91 may b


In physics, communication theory, engineering, statistics, and other areas, one of the methods of deriving distributions is the optimization of an appropriate measure of entropy under relevant constraints. In this paper, it is shown that by optimizing a measure of entropy introduced by the second author, one can derive densities of univariate, multivariate, and matrix-variate distributions in the real, as well as complex, domain. Several such scalar, multivariate, and matrix-variate distributions are derived. These include multivariate and matrix-variate Maxwell–Boltzmann and Rayleigh densities in the real and complex domains, multivariate Student-t, Cauchy, matrix-variate type-1 beta, type-2 beta, and gamma densities and their generalizations. A,B, etc. No tilde will be used on constants. If A is written as x and y with differentials X, where X may be scalar real variable x, scalar complex variable X, or vector/matrix complex variable X and The following notations will be used in this paper: Real scalar variables, whether mathematical variables or random variables, will be denoted by lower-case letters, such as x,  and Ex are fixeussed by . A modifussed by , and it X may be real scalar x, complex scalar X, or complex vector/matrix a is a fixed real scalar anchoring point, a is measured in In early 2000, the second author introduced a generalized entropy of the following form:plied by  in time-plied by  used a pplied by  discusseplied by  studied In the present article, the term “entropy” is used as a mathematical measure of uncertainty or information characterized by some basic axioms, as illustrated by . Thus, iShannon entropy x be a real scalar variable and let x and a is a fixed real scalar constant, x as a mathematical model or as a statistical model. The model x is replaced by m is the relocation parameter. For Let ation of  under thd, this gives the Fermi–Dirac density, and for Fermi–Dirac and Bose–Einstein densities are also available from the same procedure. In this case, the second factor Equation  becomes In model-building situations, if X be a X be X from the point of location X from u. Consider the following constraints:X is a Let he modelf3(X)=C3[Lemma 1.Let For the proof of this result, as well as for other similar results, see . We willLemma 2.Let X be real where, for example, where .s by using a real type-1 beta integral. Hence, for s by using a real type-2 beta integral, we have the normalizing constant Consider Observe that the model in (9) is a multivariate generalized real type-1 beta model, (10) is a multivariate generalized real type-2 beta model, and (11) is a multivariate generalized real gamma model. For sis; see . The modons; see .h-th moment of the ellipsoid of concentration for an arbitrary h. That is, for From the normalizing constants ion, see . Then, [Theorem 1.For the Note 1.We can relax the condition Note 2.Consider a function over all p matrix with distinct real scalar variables u is an important quantity in statistical literature. Hence, we will impose restrictions in terms of moments of u. Consider the optimization of Mathai’s entropy in (4) over all densities X is a Y, and, hence, Z is a s-integral by using a real scalar type-1 beta integral; for s-integral by using a real scalar type-2 beta integral; for s-integral is evaluated by using a real scalar gamma integral. Then, the normalizing constants are the following:Let Observe that (21) and (22) are available from (20). Similarly, (20) and (22) are available from (21). In other words, all densities in (20)–(22) are available through the pathway parameter Note 3.If a location parameter p matrix with distinct elements p matrix-variate densities a is a fixed scalar constant. In order to avoid having too many symbols, we will use the same notations of e, we have S-integral by using a real matrix-variate type-1 beta integral for Let S-integral by using real matrix-variate type-1 beta, type-2 beta, and gamma integrals. The final expressions will be the following:Let us consider the matrices Note 4.If a location parameter is needed, then replace X with h-th moment of this determinant u for an arbitrary h is needed, then this moment can be written down by looking at the normalizing constants in (32)–(34). For h-th moment is the following: Let Both sides have the same distribution. Similarly, for Note 5.Note that Note 6.If A, its transpose will be written as A is called Hermitian. Any complex matrix A can be written as A is Hermitian, then A is A is written as For a matrix ain, see .Lemma 3.Let Lemma 4.Let where p-dimensional complex domain. This ellipsoid is an important quantity in statistical analysis, as well as in various other situations. When u is the generalized distance of u. Consider the following constraints: For a is a fixed location, and s is s-integral by using a real scalar type-1 beta integral for As a first problem, let For een from . Observep matrix with distinct complex scalar variables as elements. Let A. Consider the optimization of Mathai’s entropy in (4) for the density p. Then, proceeding as in the real case, we end up with the following densities: For Let q, then we can take p and q interchanged and A and B interchanged. We obtain results parallel to the ones above for the case of Note that (50) can be considered as a multivariate version of the complex Maxwell–Boltzmann and Rayleigh densities for p with distinct complex scalar variables as elements. Let p. Then, proceeding as in the real case, we have the following densities:Let tE[|detAX˜BX˜*A12given by . The stah-th moment of the absolute value of the determinant h. Then, from (54), we have the following: For q, then we can obtain results for p with q and A with B.As in the real case, we can consider the In this paper, it is shown that a large number of statistical densities belonging to the pathway family  of densi


Typically, n is an integer and bounded from above by the dimension of the local (or single particle) Hilbert space, the most prominent example being spin-1/2 systems with n restricted to 2. Here, we show that a clean spin-1/2 system in the presence of long-range interactions and transverse field can sustain a huge variety of different ‘higher-order’ discrete time crystals with integer and, surprisingly, even fractional n > 2. We characterize these  non-equilibrium phases of matter thoroughly using a combination of exact diagonalization, semiclassical methods, and spin-wave approximations, which enable us to establish their stability in the presence of competing long- and short-range interactions. Remarkably, these phases emerge in a model with continous driving and time-independent interactions, convenient for experimental implementations with ultracold atoms or trapped ions.Discrete time crystals are periodically driven systems characterized by a response with periodicity  Discrete time crystals are typically characterized by a period doubled response with respect to an external drive. Here, the authors predict the emergence of rich dynamical phases with higher-order and fractional periods in clean spin-1/2 chains with long-range interactions. A DTC is a nonequilibrium phase of matter breaking the discrete time translational symmetry of a periodic  drive. In the thermodynamic limit, the defining feature of an n-DTC is a subharmonic response at 1/nth of the drive frequency (n > 1), which is robust to perturbations and which persists up to infinite time12. Following the first seminal proposals8, DTCs have been widely investigated both theoretically and experimentally23.Due to its foundational and technological relevance, the study of condensed matter systems out of equilibrium has attracted growing interest in recent years, accounting among others for the discovery of dynamical phase transitionsn = 2). This fact naturally emerges from the dimension 2 of the local Hilbert space of the spins6, and can be generalized to n-DTCs in models of n-dimensional clocks25. Another well-studied setting is that of bosons in a gravitational field bouncing on an oscillating mirror9, where the single-particle Hilbert space dimension is infinite (as the particle’s position is continuous) and where n-DTCs with arbitrary integer26 and fractional27n have been shown.In this context, most work has focused on spin-1/2 systems, which have largely been shown to exhibit a 2-DTC where at every Floquet period each spin (approximately) oscillates between the states 8. Alternatively, in clean  systems, heating can be escaped with all-to-all interactions28, or significantly slowed down with long-range interactions32. Very recently, ref. 32 has provided the theoretical framework to study Floquet, clean, long-range interacting systems, in which novel prethermal phases of matter are expected. While their framework allows for the possibility of n-DTCs with n larger than the size of the local (or single-particle) Hilbert space, their concrete examples are limited to n = 2. From our analysis below, we see that part of the difficulty in numerically observing what we call “higher-order” DTCs may lie in their emergence at system sizes that are typically beyond the reach of exact diagonalization.In these systems, heating to a featureless “infinite temperature” state is typically avoided by introducing disorder, which leads to a (Floquet) many-body-localized (MBL) phasen-DTCs with integer and, remarkably, even fractional n > 2 . These novel dynamical phases give rise to a peculiar fragmentation of the magnetization spectrum, which is intriguingly reminiscent of the plateau structure of the fractional quantum Hall effect.Here, we overcome this limitation by considering a system amenable to a set of complimentary methods, which enable us to discover an unusually rich dynamical phase diagram hosting a zoo of novel, exotic  nonequilibrium phases of matter. More specifically, we show that a clean spin-1/2 chain in the presence of long-range interactions  limit, and finally show that the physics observed extends far beyond the fine-tuned limit. We note that our work is distinct from traditional MBL DTCs, which do not have such a semiclassical limit. On a conceptual level, our analysis is closer to that of equilibrium statistical physics, where, for example, the ferromagnetic phase in the Ising model is best understood in a mean-field description, which is exact in the limit of all-to-all interactions. In our out-of-equilibrium and clean setting, the existence of a conceptually simple mean-field limit is particularly valuable, and highlights profound differences between the clean DTCs considered here and the pioneering works on MBL DTCs.N spins in the thermodynamic limit (N → ∞), driven according to the following time-periodic Hamiltonianjth spin, periodic boundary conditions are assumed, and both ℏ and the drive frequency have been set to 1. J measures the strength of a power-law interaction with characteristic exponent α, λ is the strength of a nearest-neighbor interaction, and πh is the average over one drive period of the monochromatic transverse magnetic field. The Kac normalization α = λ = 0), in which the underlying complex physics is reduced to its essence and most easily interpreted.We consider a one-dimensional chain of z-polarized state J = λ = 0, for which the magnetization t = 0, 1, 2, … reads h is the system’s characteristic frequency. The essential question to diagnose a n-DTC is whether, upon switching on the interactions, there exists a finite range of h for which the system’s characteristic frequency ν remains instead locked to a constant value 1/n < 1, signaling the stability of the DTC to perturbations of the drive. In the following, we answer this question affirmatively not only for the well-known n = 2 case, but, if the interactions are sufficiently long range, also for integer and even fractional n > 2, corresponding to the higher-order DTCs. Note, the 33. This issue, however, disappears for the higher-order n-DTCs that, because the Hamiltonian lacks any n > 2), must indeed be a “genuine” manifestation of time-symmetry breaking.The dynamics from an initially mentclass2pt{minimα = λ = 0), which allows for a conceptually simple semiclassical interpretation of the various dynamical phases. The dynamics of the system is in this case described by a semiclassical Gross–Pitaevskii equation (GPE) for the complex fields ψ↑ and ψ↓  = 1, ψ↓(0) = 0), and the corresponding Fourier transform h is plotted in Fig. 12, the 2-DTC results in the system characteristic frequency ν being locked to 1/2 for h ≈ 1/2. Surprisingly, the same locking occurs at frequencies 1/n with integer and fractional n > 2 , giving rise to a fragmentation of the spectral line of h ≈ 1/n. Each of these plateaus signals a higher-order  DTC, the width of the plateau being a signature of the DTC’s robustness to drive perturbations. The “halos” surrounding the plateaus in Fig. The dynamics of the magnetization  the GPE  from an ν = 0 for h ≈ 0 signals the tendency of the spins to remain aligned along z in a dynamical ferromagnetic phase (F). This corresponds to macroscopic quantum self-trapping of weakly driven bosons in a double well34, which can in fact be exactly mapped to the LMG limit .The plateau at N. This allows a scaling analysis extended up to large system sizes, showing a progressive emergence of the spectral line plateaus for an increasing number of spins N. For the standard 2-DTC, the plateau is clearly visible already for N ⪆ 10, whereas, crucially, for the 4-DTC it appears only for N ⪆ 100  is highlighted with red markers. For a weak drive h ≈ 0, the spins tend to remain aligned along z in a dynamical ferromagnetic phase (a), giving rise to a Poincaré map which closely resembles the phase portrait of undriven bosons in a double well34. For h ≈ 1, the micromotion consists of approximately an entire revolution of the spins around the Bloch sphere per period, with a preferential z alignment restored at stroboscopic times despite the detuning in the magnetic field strength (b). For h ≈ 1/n and n = 2, 4, 8 in (c), (d), and (e), respectively, the n-DTC results in the presence of n “islands” in the phase space, which the system visits sequentially jumping from one to the next at each drive period. In n drive periods, the system visits all the n islands once, and the magnetization m completes one oscillation.The stroboscopic dynamics generated by the GPE  can be ch ≈ 3/8 the system behaves as a n-DTC with fractional n = q/p = 8/3 (f). In this case, the system cyclically visits q islands of the phase space in q drive periods. Differently from a q-DTC, however, during this time the magnetization m completes p oscillations, resulting in a characteristic frequency p/q. Finally, for larger interactions J the Poincaré maps become chaotic (as in ref. 12), signaling thermalization35.Furthermore, for Since the island-to-island hopping that underpins the subharmonic response holds for any point of any island, the islands themselves can be interpreted as stability regions of the DTCs with respect to perturbations of the initial state. The Poincaré maps also provide an interpretation of the subdominant frequencies visible as halos around the plateaus in Fig. 10. It becomes thus of primary importance to assess the role of the interactions also for the higher-order DTCs. To this end, as a concrete example, in Fig. J on the 4-DTC. If the interaction is weak, a slightly mistaken magnetic field strength h = 1/4 + ϵ, with ϵ ≪ 1, originates in envelopes  with period ~1/ϵ, resulting in the Fourier transform ν ≈ h and in trivial dynamics (a). Crucially, stronger interactions can compensate the mistake in the flipping field (b): the envelopes in m(t) disappear, the peak in ν = 1/n, and the discrete time symmetry is broken. The time-glassy character of the DTC is observed in a small aperiodic modulation of the magnetization on top of the subharmonic response.It is well established for the standard 2-DTC that the robust subharmonic response hinges on the interaction being sufficiently strong. The fact that interactions are necessary for the robustness of DTCs is critical, as it underpins the many-body nature of the DTCs and it justifies their classification as nonequilibrium phases of matterJ, h) plane (c). The 4-DTC phase opens up from the integrable point J = 0, h = 1/4 for increasing interactions, in analogy with the opening of the standard 2-DTC from J = 0, h = 1/216. This opening, which in dynamical system theory would be referred to as Arnold’s tongue, confirms that larger interactions J allow the higher-order DTCs to bear larger detunings in the field h. However, at even larger J ⪆ 0.8 semiclassical chaos sets in and the time crystalline order is broken irrespectively of h. To see this, we introduce a decorrelator 35.The subharmonic peak magnitude entclass1pt{minimaα > 0) and nearest-neighbor (λ > 0) interactions. In this case, the system is no longer described as a collective spin, and spin-wave excitations are rather generated. To account for them, we adopt a spin-wave approximation (see “Methods”), in which the central dynamical variable is the density of spin-wave excitations ϵ(t)bk are bosonic creation and annihilation operators for the spin-wave excitations with momentum k.As shown, the DTCs rely on the interactions being sufficiently (but not too) strong. Crucially, in contrast to the standard 2-DTC, higher-order DTCs also necessitate the interactions to be sufficiently long range. We now probe the robustness of the higher-order DTCs along yet a different direction in the drive space, exploring the effects of non-all-to-all interaction on higher-order DTCs, particularly assessing their stability upon breaking the mean-field solvability of the dynamics with power-law (λ = α = 0), no spin-wave excitation is generated and ϵ = 0 at all times. When departing from such a limit, two scenarios are possible , for which we consider the spin-wave approximation consistent, or (ii) ϵ rapidly grows to values ⪆ 1, for which the spin-wave approximation breaks down. Although the method is not exact and may fail to capture the very long-time physics, it suggests that (i) and (ii) correspond to prethermalization and thermalization, respectively37.In the LMG limit  ϵ λ and α), whereas thermalization quickly sets in for shorter-range interactions  plane plotting the amplitude of the subharmonic peak ϵ〉t in Fig. n-DTC is stable for a whole region of the parameter space surrounding the LMG point (α = λ = 0), that is, if the interactions are sufficiently long range. The DTC is also robust to arbitrary perturbations to the initial state, as long as the initial spin-wave density ϵ(0) is sufficiently small, as we have checked by injecting a small amount of spin-wave excitations to the initial state. For completeness, in the Supplementary Note hz. We found in this case that the expected prethermal nature of the DTC is only changed quantitatively, with the heating time scaling exponentially with the ratio of the driving frequency  and the energy scale hz. Finally, we note that the time-glassy character of the DTCs is maintained throughout their whole stability region. In particular, we find small peaks at incommensurate frequencies that do (do not) vary when slightly perturbing the drive .We observe that the higher-order DTCs are stable  for sufficiently long-range interactions  Hilbert space. The dynamical phase space fragments to host many higher-order d2(t)44m = Δθ = 10−6. The decorrelator time average T is the total simulation time, for example, T = 103 in Figs. We quantify the sensitivity to the initial conditions of Eq.  with a d37 and the Supplementary Material therein, to which we redirect the reader for further details. In a DTC evolving from an initially z-polarized state Z-axis tracks the orientation of the collective spin at all times. This tracking is encoded in the condition 〈SX〉 = 〈SY〉 = 0, from which the dynamics of the rotating frame is obtained self-consistently. The Holstein–Primakoff transformation from spin degrees of freedom to bosonic degrees of freedom is then performed as N ordinary differential equations similar to Eqs. (26) and (29) in the Supplemental Material of ref. 36, describing the rotation of the new reference frame and the dynamics of the spin waves at the various momenta.Here, we briefly summarize the idea behind the spin-wave approximation, which is thoroughly explained in refs. 36. First, ref. 36 considers a constant-in-time Hamiltonian, whereas the parameters of the Hamiltonian in the present work are time-dependent. As a consequence, the parameters in the system of ordinary differential equations become time-dependent. Second, ref. 36 considers a nearest-neighbor interaction on top of an all-to-all one, whereas we consider the more general case of nearest-neighbor interaction on top of a power-law one. The 36 is therefore substituted by a more generic Finally, we remark the main differences between the implementation of the spin-wave approximation in our work and in ref. Supplementary Information