On Generalized Pseudostandard Words Over Binary Alphabets

International audienceIn this paper, we study generalized pseudostandard words over a two-letter alpha- bet, which extend the classes of standard Sturmian, standard episturmian and pseu- dostandard words, allowing different involutory antimorphisms instead of the usual palindromic closure or a fixed involutory antimorphism. We first discuss about pseu- doperiods, a useful tool for describing words obtained by iterated pseudopalindromic closure. Then, we introduce the concept of normalized directive bi-sequence (Θ,w) of a generalized pseudostandard word, that is the one that exactly describes all the pseudopalindromic prefixes of it. We show that a directive bi-sequence is normalized if and only if its set of factors does not intersect a finite set of forbidden ones. Moreover, we provide a construction to normalize any directive bi-sequence. Next, we present an explicit formula, generalizing the one for the standard episturmian words introduced by Justin, that computes recursively the next prefix of a generalized pseudostandard word in term of the previous one. Finally, we focus on generalized pseudostandard words having complexity 2n, also called Rote words. More precisely, we prove that the normalized bi-sequences describing Rote words are completely characterized by their factors of length 2

Generalized resolution for orthogonal arrays

The generalized word length pattern of an orthogonal array allows a ranking of orthogonal arrays in terms of the generalized minimum aberration criterion (Xu and Wu [Ann. Statist. 29 (2001) 1066-1077]). We provide a statistical interpretation for the number of shortest words of an orthogonal array in terms of sums of $R^2$ values (based on orthogonal coding) or sums of squared canonical correlations (based on arbitrary coding). Directly related to these results, we derive two versions of generalized resolution for qualitative factors, both of which are generalizations of the generalized resolution by Deng and Tang [Statist. Sinica 9 (1999) 1071-1082] and Tang and Deng [Ann. Statist. 27 (1999) 1914-1926]. We provide a sufficient condition for one of these to attain its upper bound, and we provide explicit upper bounds for two classes of symmetric designs. Factor-wise generalized resolution values provide useful additional detail.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1205 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Generalized trapezoidal words

The factor complexity function $C_w(n)$ of a finite or infinite word $w$ counts the number of distinct factors of $w$ of length $n$ for each $n \ge 0$. A finite word $w$ of length $|w|$ is said to be trapezoidal if the graph of its factor complexity $C_w(n)$ as a function of $n$ (for $0 \leq n \leq |w|$) is that of a regular trapezoid (or possibly an isosceles triangle); that is, $C_w(n)$ increases by 1 with each $n$ on some interval of length $r$, then $C_w(n)$ is constant on some interval of length $s$, and finally $C_w(n)$ decreases by 1 with each $n$ on an interval of the same length $r$. Necessarily $C_w(1)=2$ (since there is one factor of length $0$, namely the empty word), so any trapezoidal word is on a binary alphabet. Trapezoidal words were first introduced by de Luca (1999) when studying the behaviour of the factor complexity of finite Sturmian words, i.e., factors of infinite "cutting sequences", obtained by coding the sequence of cuts in an integer lattice over the positive quadrant of $\mathbb{R}^2$ made by a line of irrational slope. Every finite Sturmian word is trapezoidal, but not conversely. However, both families of words (trapezoidal and Sturmian) are special classes of so-called "rich words" (also known as "full words") - a wider family of finite and infinite words characterized by containing the maximal number of palindromes - studied in depth by the first author and others in 2009. In this paper, we introduce a natural generalization of trapezoidal words over an arbitrary finite alphabet $\mathcal{A}$, called generalized trapezoidal words (or GT-words for short). In particular, we study combinatorial and structural properties of this new class of words, and we show that, unlike the binary case, not all GT-words are rich in palindromes when $|\mathcal{A}| \geq 3$, but we can describe all those that are rich.Comment: Major revisio

Palindromic richness for languages invariant under more symmetries

For a given finite group $G$ consisting of morphisms and antimorphisms of a free monoid $\mathcal{A}^*$, we study infinite words with language closed under the group $G$. We focus on the notion of $G$-richness which describes words rich in generalized palindromic factors, i.e., in factors $w$ satisfying $\Theta(w) = w$ for some antimorphism $\Theta \in G$. We give several equivalent descriptions which are generalizations of know characterizations of rich words (in the terms of classical palindromes) and show two examples of $G$-rich words

Factor frequencies in generalized Thue-Morse words

We describe factor frequencies of the generalized Thue-Morse word t_{b,m} defined for integers b greater than 1, m greater than 0 as the fixed point starting in 0 of the morphism \phi_{b,m} given by \phi_{b,m}(k)=k(k+1)...(k+b-1), where k = 0,1,..., m-1 and where the letters are expressed modulo m. We use the result of A. Frid, On the frequency of factors in a D0L word, Journal of Automata, Languages and Combinatorics 3 (1998), 29-41 and the study of generalized Thue-Morse words by S. Starosta, Generalized Thue-Morse words and palindromic richness, arXiv:1104.2476v2 [math.CO].Comment: 11 page

Review: Burden on Family Caregivers Caring for Patients with Schizophrenia and Its Related Factors

Background: Family caregiver is the most important person who cares for patient with schizophrenia. However when care is provided for long time, he/she may experiences the burden.Purpose: The purpose was to review concept and factors related to burden on family caregivers caring for patients with schizophrenia.Method: A literatures were searched from databases: Pubmed, CINAHL, and Science Direct. Key words used to retrieve literature include caregiver burden and schizophrenia. Searching was limited in English language, full text, and the year of publication from 2000 to 2009 was used.Results: Twenty two studies were reviewed in this paper. The result showed that the caregivers caring for patients with schizophrenia experience burden. Burden was defined as a negative impact of caring for the impaired person experienced by caregiver on their activity (objective burden) or feeling (subjective burden) that involves emotional, physical health, social life, and financial status. Factors related to burden on family caregiver were grouped into: 1) caregiver‟s factors included age, gender, educational level, income, health status, and spent time per day, knowledge of schizophrenia, culture, and coping; 2) patient‟s factors included age, clinical symptoms, and disability in daily life; 3) environmental factors included mental health service and social support.Conclusion: Definition of burden have quite same meaning and mostly factors focus on the patient‟s symptoms, demographic factors of caregiver, and time spent per day. Most of studies cannot be generalized due to small sample used in the study and that too conducted in western countries. For further research, the correlation between burden and resources of family caregiver should be investigated particularly in eastern country

Generating functions for Wilf equivalence under generalized factor order

Kitaev, Liese, Remmel, and Sagan recently defined generalized factor order on words comprised of letters from a partially ordered set $(P, \leq_P)$ by setting $u \leq_P w$ if there is a subword $v$ of $w$ of the same length as $u$ such that the $i$-th character of $v$ is greater than or equal to the $i$-th character of $u$ for all $i$. This subword $v$ is called an embedding of $u$ into $w$. For the case where $P$ is the positive integers with the usual ordering, they defined the weight of a word $w = w_1\ldots w_n$ to be $\text{wt}(w) = x^{\sum_{i=1}^n w_i} t^{n}$, and the corresponding weight generating function $F(u;t,x) = \sum_{w \geq_P u} \text{wt}(w)$. They then defined two words $u$ and $v$ to be Wilf equivalent, denoted $u \backsim v$, if and only if $F(u;t,x) = F(v;t,x)$. They also defined the related generating function $S(u;t,x) = \sum_{w \in \mathcal{S}(u)} \text{wt}(w)$ where $\mathcal{S}(u)$ is the set of all words $w$ such that the only embedding of $u$ into $w$ is a suffix of $w$, and showed that $u \backsim v$ if and only if $S(u;t,x) = S(v;t,x)$. We continue this study by giving an explicit formula for $S(u;t,x)$ if $u$ factors into a weakly increasing word followed by a weakly decreasing word. We use this formula as an aid to classify Wilf equivalence for all words of length 3. We also show that coefficients of related generating functions are well-known sequences in several special cases. Finally, we discuss a conjecture that if $u \backsim v$ then $u$ and $v$ must be rearrangements, and the stronger conjecture that there also must be a weight-preserving bijection $f: \mathcal{S}(u) \rightarrow \mathcal{S}(v)$ such that $f(u)$ is a rearrangement of $u$ for all $u$.Comment: 23 page

Дослідження впливу середовища на реакцію оксосинтезу

Узагальнено дані по константах швидкості реакції оксосинтезу з метилакрилатом у різних розчинниках за допомогою лінійного багатопараметрового рівняння, причому сольватаційні параметри сприяють реакції, а енергія когезії та мольний об’єм розчинників її сповільнюють. Подібно можна узагальнити дані по каталізуючому впливу додатку третинних амінів, але тут вплив сольватаційних характеристик протилежний. Ключові слова: оксосинтез, метилакрилат, багатопараметрові рівнянняValues of the rate constants of oxosynthese reaction with the methylmetacrylate in various solvents have been generalized by means of a linear multiparameter equation in which the solvation factors favour the reaction. On the contrary the cohesion energy density and molar volume of solvents decelerate it. The data of catalysing influence of tertiary amines addition can be similarly generalized too, however in such case the influence of solvation characteristics is opposite. Key words: oxosynthesis, methylacrylate, multiparametric equations

Quantitative Social Dialectology: Explaining Linguistic Variation Geographically and Socially

In this study we examine linguistic variation and its dependence on both social and geographic factors. We follow dialectometry in applying a quantitative methodology and focusing on dialect distances, and social dialectology in the choice of factors we examine in building a model to predict word pronunciation distances from the standard Dutch language to 424 Dutch dialects. We combine linear mixed-effects regression modeling with generalized additive modeling to predict the pronunciation distance of 559 words. Although geographical position is the dominant predictor, several other factors emerged as significant. The model predicts a greater distance from the standard for smaller communities, for communities with a higher average age, for nouns (as contrasted with verbs and adjectives), for more frequent words, and for words with relatively many vowels. The impact of the demographic variables, however, varied from word to word. For a majority of words, larger, richer and younger communities are moving towards the standard. For a smaller minority of words, larger, richer and younger communities emerge as driving a change away from the standard. Similarly, the strength of the effects of word frequency and word category varied geographically. The peripheral areas of the Netherlands showed a greater distance from the standard for nouns (as opposed to verbs and adjectives) as well as for high-frequency words, compared to the more central areas. Our findings indicate that changes in pronunciation have been spreading (in particular for low-frequency words) from the Hollandic center of economic power to the peripheral areas of the country, meeting resistance that is stronger wherever, for well-documented historical reasons, the political influence of Holland was reduced. Our results are also consistent with the theory of lexical diffusion, in that distances from the Hollandic norm vary systematically and predictably on a word by word basis