Janine A. E. StrandbergORCID, Charlotte GooskensORCID, Anja Schüppert
Abstract
Aims and objectives:
This study examines fennicisms (i.e., Finnish loanwords and calques) in Finland-Swedish, a Swedish variety spoken in Finland. We investigate how fennicism frequency relates to speakers’ regional backgrounds and fluency in Finnish.
Methodology:
134 participants from four regions in Finland performed a picture-naming task designed to elicit fennicisms. The participants also rated their own fluency in Finnish.
Data and analysis:
A regression analysis with the outcome variable of fennicism frequency and the predictors of region, fluency in Finnish, and gender was performed.
Findings:
Results show that speakers from the more bilingual regions of Southern Finland and Helsinki used significantly more fennicisms than speakers from Ostrobothnia or Swedish-speaking Åland. The study suggests that fluency in Finnish was a strong predictor for fennicism use, as speakers with low or moderate knowledge of Finnish used fewer fennicisms than speakers with high or native(-like) fluency. No significant effect of gender was found.
Originality:
While fennicisms are considered widespread in Finland-Swedish, there is little previous research on their use and distribution.
Implications:
The results demonstrate that while many of the fennicisms are well-established in the Finland-Swedish variety, their use is limited to certain groups and communities.
Research article
Available accessResearch articleFirst published October, 2024pp. 777-798
This study investigates how the perfective/imperfective distinction is encoded across various ethnic varieties of Namibian English, with a specific focus on progressive -ing, simple verbal forms, and used to.
Design/methodology/approach:
The sample involves 158 Coloured and White Afrikaans-speaking informants, Damaras (L1: Khoekhoegowab), Hereros (L1: Otjiherero), Ovambos (L1: Oshiwambo). The sample thus includes Bantu, Khoesan, and West Germanic languages. The informants were administered an L1- into-English translation task.
Data and analysis:
The analysis follows two steps. First, the study looks at the distribution of English verbal forms across tenses, event classes, and English verb types. Second, the study looks at the distribution of English verbal forms across aspectual indicators in the informants’ respective native languages.
Findings/conclusions:
Regardless of event classes and verb types, Khoekhoegowab-speakers are the most frequent users of -ing while Afrikaans-speakers are the most frequent users of simple verbal forms. A close look at how English verbal forms are distributed across L1 aspectual indicators possibly reveals hypercorrect L1-pattern avoidance, spearheaded by women. Along with L1-pattern avoidance, one finds overreliance on positively transferable L1-patterns, manifest in the use of used to by Bantu-speakers, who grammatically mark the habitual in their L1s. Finally, one finds instances of English output features that cannot be triggered by L1-patterns. This concerns primarily -ing and used to among Afrikaans-speakers and used to among the Damaras.
Originality:
Unlike most research on aspect in postcolonial English varieties, this study systematically tracks transfers to their linguistic origin.
Significance/implications:
The paper argues that transfers (or their absence) are often better accounted for from the perspective of inter-group relations than as mere ‘interferences’.
Research article
Open accessResearch articleFirst published October, 2024pp. 799-827
Katrina Kechun LiORCID, Li NguyenORCID, Christopher BryantORCID , [...]
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Abstract
Aims and objectives:
Previous research has revealed much about the syntactic and social variables conditioning code-switching (i.e., the alternation between two or more languages in a discourse or utterance); however, little is known about the phonological effects. Our work explores this area by asking two main questions: (1) Does lexical tone affect code-switching between a tonal language and a non-tonal language? and (2) Is this effect (or lack thereof) observable cross-linguistically?
Methodology:
We examine natural code-switching production between Cantonese and English, Mandarin and English, and Vietnamese and English. We use a semi-automatic natural-language processing method to process and extract relevant variables, including tonal categories at switch points.
Data and analysis:
Data include transcribed natural speech from three bilingual corpora: the HLVC corpus (Cantonese/English, 25 speakers), the SEAME corpus (Mandarin/English, 20 speakers), and the CanVEC corpus (Vietnamese/English, 45 speakers). We use logistic mixed-effects models to examine tonal effects, taking into account other factors such as frequency and grammatical category.
Findings/conclusion:
We found a robust tonal effect in Cantonese/English, a less robust effect in Mandarin/English, and no effect in Vietnamese/English. This indicates there is a tonal effect in code-switching between a tonal and a non-tonal language, but this effect is language-dependent. We also found a specific T3 ‘step-up’ pattern at Cantonese-English switch points and offered some possible phonological explanations.
Originality:
This is the first study that systematically investigates tonal effects in code-switching across different language pairs, using comparable data and methods. Our finding of a Cantonese-English T3 ‘step-up’ pattern is also a novel discovery that hitherto has not been documented.
Significance/implications:
Theoretically, our findings support Clyne’s ‘facilitation theory’ in code-switching at a prosodic level. Empirically, we nevertheless emphasised the complexity of different prosodic features and social variables in play, thereby rejecting the idea of ‘predicting’ code-switching solely based on linguistic factors.
Research article
Open accessResearch articleFirst published October, 2024pp. 828-844
The study investigates the extent to which offline and online comprehension of reflexive possessive pronouns in L3 and L4 is modulated by previously acquired native and non-native grammars.
Design/Methodology/Approach:
Thirty-one intermediate to advanced L1 Polish learners of Swedish, who have also acquired German and English (L4 Swedish learners) or only English (L3 Swedish learners), completed a self-paced reading task and a grammaticality judgement task. The tasks targeted learners’ sensitivity to possessor and possessee agreement violations.
Data and Analysis:
Reading times and judgement accuracy were collected and analysed using generalised linear models.
Findings/Conclusions:
In online comprehension, L3/L4 Swedish learners showed sensitivity to possessor and possessee agreement, which was affected neither by their proficiency, nor by their prior native and non-native grammars. In offline comprehension, however, they had difficulty deploying possessor and possessee agreement. Performance on possessee but not possessor agreement increased with advancing proficiency, indicating the persistent influence of L1 Polish grammar. Furthermore, L4 Swedish learners achieved an overall higher judgement accuracy than L3 Swedish learners. At the same time, L4 Swedish learners read the relevant regions longer than L3 Swedish learners, suggesting a speed–accuracy trade-off for L4 Swedish learners.
Originality:
The study investigates cross-linguistic influence in the comprehension of reflexive possessive pronouns in multilingual learners, which has not been explored before. It is also the first to provide evidence for a multilingual advantage in the accuracy of reflexive possessive agreement, albeit at the cost of reading speed.
Significance/Implications:
The study demonstrates that native and non-native grammars differently affect the offline comprehension of reflexive possessive pronouns in L3 and L4. The native language seems to have a more direct effect that consists in the employment of agreement mechanisms, while the non-native language has a more indirect effect, as it can increase learners’ sensitivity to morphosyntactic violations.
Research article
Available accessResearch articleFirst published October, 2024pp. 845-862
Multiple dimensions of language dominance, such as language proficiency and demand for language use, can be reflected in bilinguals’ speech-in-speech recognition scores. This paper explores the feasibility of using a novel measure to estimate language dominance for bilinguals: relative speech-in-speech recognition thresholds (SRTs) or the within-person difference in SRTs between their two languages.
Method:
Participants were 25 Spanish/English bilingual adults (Mage = 30 years). SRTs for sentence recognition in a language-matched two-talker masker were measured in English and in Spanish using an adaptive, open-set task. Relative SRTs were calculated by subtracting the Spanish SRT from the English SRT. Language dominance and proficiency were assessed by established measures.
Analysis:
Spearman correlations were used to assess the association between measures.
Findings:
Relative SRTs were correlated with dominance scores derived from both a questionnaire (rs = −.80) and standardized testing (rs = −.80).
Originality:
We tested and validated a novel measure, relative SRTs, to predict language dominance in bilingual adults. This time-efficient method could be used by bilingual researchers interested in assessing language dominance for descriptive or experimental purposes.
Significance:
Relative SRTs show promise as a valid method to assess language dominance in Spanish/English bilingual adults.
Research article
Available accessResearch articleFirst published October, 2024pp. 863-883
The empirical goal of the paper is to document the linguistic distribution and patterning as well as the sociolinguistic conditioning of intensifiers in English spoken by educated young adults from Germany. Here, I also seek to understand how the empirical data presented here informs our understanding of the mechanisms underlying language change in those forms of English that emerge through the combined impact of naturalistic L2 acquisition and instructed foreign language learning (henceforth, EFL or Learner English).
Design/methodology/approach:
The study reports intensification data from 53 advanced learners of English from Germany and compares it with that reported for L1 English vernaculars. The data were elicited with the help of sociolinguistic interviews tapping into learners’ natural linguistic behaviour. The processes of language change are explored against the backdrop of grammaticalisation theory, as espoused by variationist sociolinguists.
Data and analysis:
The study employs two methods of analysis: distributional percentages and fixed-effects logistic regression analyses. These allow pinpoint rates and patterns of intensification as well as its language-internal and language-external conditioning.
Findings/conclusions:
Systematic comparisons of language-internal conditioning underlying the occurrence of linguistic variants pinpoint ‘functional specialisation’ (also ‘entrenchment’) as a driving mechanism of and a potent constraint on language change.
Originality:
While demonstrating how functional specialisation is operative in the system of German Learner English intensification, the present study highlights the relevance of EFL data to empirical testing and advancement of (socio-)linguistic theory. The study also highlights the relevance of learners’ linguistic identity and the sociopsychological construct of ‘relatedness’ in the adoption of the globally available features of the English language.
Significance/implications:
The study pleads for more research at the intersection of historical/general linguistics and SLA as such an approach adds to knowledge about language acquisition, language variation and, ultimately, language change.
Research article
Open accessResearch articleFirst published October, 2024pp. 884-906
Nana LehtinenORCID, Anna KauttoORCID, Kati Renvall
Abstract
Purpose:
Verbal fluency (VF) tasks are an efficient tool for exploring lexical retrieval. Attriters generally produce fewer words in semantic categories than monolinguals, but the processes underlying attriters’ lexical retrieval remain unclear. Phonemic tasks are scarcely applied in language attrition studies. We aim to identify processes underlying attriters’ phonemic verbal fluency (PVF) and semantic verbal fluency (SVF) performance by extending the analysis beyond total scores in the first (L1) and second (L2) language.
Design:
We modeled total scores and temporal aspects of task performance in L1 and L2 for the attriters, with L1 use and length of residence (LoR) as predictors. We analyzed the number and types of errors between languages and compared L1 task performance with a matched monolingual group.
Data and analysis:
Attriters’ (N = 38) phonemic and semantic task performance in L1 (Finnish) and L2 (English) were modeled, and results were contrasted to a matched monolingual control group (N = 50).
Findings:
Attriters demonstrated strong proficiency in L1 and similar lexical retrieval processes in L1 and L2 after 20 years of immersive L2 exposure. Frequent L1 use supported overall performance but slowed performance down in both languages. Compared with monolinguals, attriters show a disadvantage in SVF, but not due to slower initiation. Instead, attriters rely more on rapid L1 retrieval than monolinguals.
Originality:
Prior research on VF tasks in language attrition populations has mainly focused on SVF total scores in L1. Our investigation explores underlying mechanisms of L1 and L2 task performance in PVF and SVF within a language attrition population and contrasts L1 performance to a matched monolingual group performance.
Implications:
This study demonstrates the importance of investigating L2 alongside L1 in language attrition studies for a holistic approach to language processing strategies. Our findings enhance understanding of the processes underlying VF task performance, emphasizing the significance of VF task early-stage performance.
Research article
Available accessResearch articleFirst published October, 2024pp. 907-925
As there remains a paucity of research into the use of communication strategies in Polish transnational communities, this paper seeks to delineate the code-switching practices of Polish-speaking families residing in Australia. The tripartite framework of Family Language Policy has been applied for the analysis of affordances of code-switching patterns.
Methodology:
Based on the data collected through the online questionnaire supplemented with in-depth interviews and observations, a qualitative analysis has been conducted to obtain the sociolinguistic picture of the code-switching practices.
Data and analysis:
The interviews and discussions during the observations were recorded, transcribed verbatim and coded according to the Grounded Theory Approach. The recurrent themes were noted. The existing patterns were conceptualised through the process of constant comparison.
Findings and conclusions:
The conducted study draws our attention to the fact that Polish speakers in Australia differ from speakers of other heritage languages. Language management efforts through a great exposure to Polish translates to HL proficiency, maintenance and the degree of bilingualism, as well as the rate of code-switching. All joint family and extracurricular activities organised by Saturday schools and cultural centres augment the HL retention despite the constant attempts to code-switch among the young family members.
Originality:
This paper delves into how Polish is maintained as a HL among Polish-speaking immigrants to Australia and their offspring. It explores one of the well-established yet understudied communities that makes up multicultural Australia. It unfolds an account of the dynamics of code-switching, illustrating how its affordances are utilised to foster communication.
Research article
Available accessResearch articleFirst published October, 2024pp. 926-943
This study examines language accommodation in cross-dialectal encounters. It demonstrates how speakers of different Arabic varieties use (non)accommodation strategies that include both verbal and embodied actions.
Approach:
The study uses qualitative data from Arabic cross-dialectal communication. In the course of their interactions, participants draw on linguistic and nonlinguistic resources to achieve either convergent or divergent accommodation. Using the theoretical frameworks of discourse analysis (DA) and communication accommodation theory (CAT), the study investigates natural conversations for insight into qualities of Arabic cross-dialectal communication.
Data and analysis:
The data were collected in the United States. In this study, the participants originate from four major geographical areas in the Arab world, including the Maghreb, Egypt and Sudan, the Levant, and the Gulf.
Findings/conclusions:
The analysis demonstrates that successful accommodation requires a verbal and nonverbal collaborative construction of meaning to achieve co-participation. Participants linguistically converge while they display a shared Arab identity. The data also show that when a sense of shared Arabness is deemphasized and the positive social image to a group’s variety is violated, divergence as a nonaccommodative strategy is used.
Originality:
This is the first study that demonstrates how different modes of accommodation are expressed, not only through speech but also through embodied actions. The study shows that accommodation as a process is not complete at the verbal level; it must be orchestrated at multiple semiotic levels, including nonverbal ones.
Significance/implications:
Scholars have paid less attention to nonverbal accommodation. This study seeks to bridge this gap by analyzing both verbal and nonverbal accommodation, whether convergent or divergent, in Arabic cross-dialectal communication. It shows how speakers of different dialects use accommodation to signal intergroup relations and negotiate different social identities.
Research article
Available accessResearch articleFirst published October, 2024pp. 944-961
Profiles of bilinguals vary among studies due to the diversity of intrinsic and extrinsic factors, bilingualism classification and to the discrepancies between measures. Thus, a generalizable index with a defined threshold is needed to capture the linguistic dominance of bilinguals and facilitate comparisons between studies. This study’s objective was to define and examine the validity of a new bilingualism index.
Methodology:
This index was derived from the Language Experience and Proficiency Questionnaire (LEAP-Q) conducted on 100 bilingual Lebanese participants (age = 68.2 ± 9.4 years; education = 12.8 ± 5.2 years).
Data and analysis:
The bilingualism index was based on selected items of the LEAP-Q, optimizing the explained variance on a linear regression using the differential Arabic-French score on the Boston Naming Test as the dependent variable. The validity of the classification was examined using other linguistic (articulatory rate, shortened Token test, Stroop reading subtest) and cognitive screening tests (mini-mental state examination [MMSE]).
Findings and conclusion:
LEAP speaking and oral comprehension scores provided a parsimonious index that accounted for naming variance (R2 = .435, p = .0001) and subdivide our population into three bilingualism subgroups (prominent Arabic, balanced, prominent French). A prominent language advantage was found in the expected direction on linguistic (articulatory rate: p = .03; shortened Token test: p = .026; Stroop reading subtest: p = .0001) and cognitive screening tests (MMSE: p = .08).
Originality and implications:
These results show that a simple index can accurately characterize adult bilingualism subtypes and offers clinicians an easy and fast tool compared with the usual procedure used to determine individuals and patient’s bilingualism subtype. More broadly, the index’s validity in other bilingual populations is warranted for generalizability of the present findings.
Research article
Available accessResearch articleFirst published October, 2024pp. 962-979
Dina Abdel Salam El-DakhsORCID, Mervat M. Ahmed, Jeanette Altarriba , [...]
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Abstract
Aims and Objectives:
This study aimed to examine how bilinguals express their emotions in their first language (L1) versus their second language (L2) in autobiographical narratives.
Methodology:
A total of 64 Egyptian learners of English as a foreign language (EFL) took an English language proficiency test, and wrote about the happiest or saddest experiences they recalled in Arabic and in English.
Data and Analysis:
The narratives were analysed for the frequency and the valence of emotion words.
Findings/Conclusions:
The results showed no difference between the narratives of the two languages in terms of the frequency of words. However, the L2 narratives were generally more negatively valenced than the L1 narratives, and more emotional polarity was noted in the L2 narratives, particularly in sad situations. The results failed to show any effect for the participants’ age, gender or L2 proficiency on the frequency or valence of the emotion words.
Originality:
This study compared two languages that are distant at several levels, including typography and culture. In addition, the study examined some variables that were not investigated in earlier emotion studies of bilingualism.
Significance/Implications:
The study presents several implications in relation to the emotional contexts of learning theory and the pedagogy of emotion words.
Research article
Available accessResearch articleFirst published October, 2024pp. 980-1001
This paper investigates the diffusion of English into the linguistic ecologies of Tunisia, an Expanding Outer Circle society in North Africa. It analyzes the language practices of Tunisian business operators in five commercial localities in metropolitan Tunis. The paper focuses, in particular, on the uses of English and its interaction with Tunisia’s dominant languages, the ways the resources of English are locally understood and deployed and the intersection of the emerging “language regime” (Kroskrity, 2000) with the official linguistic landscapes policies and the wider sociohistorical, political, and economic conditions of the country.
Methodology and data:
Detailed discourse-ethnographic analysis of 363 linguistic landscape signs collected from 5 commercial districts in metropolitan Tunis.
Findings and conclusions:
The findings indicated the emergence of a bottom-up English-led public signage communicating a global corporate ideology circumventing the official language policies mandating the use and visibility of Modern Standard Arabic in public signage. However, this English-led public posting is intertwined in complex ways with Tunisia’s dominant languages, resulting in translanguaging, linguistic puns, the commodification of vernacular forms of communication as well as (supra)national and aesthetic identity markers.
Originality:
Tunisia, a traditionally constructed Francophone country moving toward English, is very little explored in the major Global English(es) paradigms. The paper focuses on the possible ideological, economic, and cultural changes engendered by English and connects these changes to the global socioeconomic, political, and cultural transformations undergone by the country over the last decades.
Research article
Available accessResearch articleFirst published October, 2024pp. 1002-1015
Wojciech Milczarski, Anna Borkowska, Mariola Paruzel-Czachura , [...]
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Abstract
Aims and objectives/purpose/research questions:
Using a foreign language is suggested to affect how we think, even reducing bias. However, the exact mechanisms of its influence are still unknown. In this project, we tested two variants of the “thinking more” mechanism driving the foreign language effect: increased cognitive reflection and greater task engagement.
Design/methodology/approach:
We analyzed previously collected data from five experiments employing the Cognitive Reflection Test (CRT) and/or the Berlin Numeracy Test (BNT), each administered in the native or foreign language. If the increased reflection explanation is true, the CRT score should be higher in participants using a foreign language. If the greater task engagement explanation is true, the BNT score should be higher in participants using a foreign language.
Data and analysis:
We report data from N = 1,288 participants who filled the CRT and N = 413 who filled the BNT. Data were analyzed using the NHST approach and its Bayesian alternative over a spectrum of liberal, default, and conservative priors. Data are publicly available at https://osf.io/m8tn9
Findings/conclusions:
All the test results were nonsignificant and showed anecdotal to strong evidence that neither the CRT nor the BNT scores improved in the foreign language condition, also when controlling for proficiency. The proportion of intuitive to computational errors was also unaffected.
Originality:
We are the first to comprehensively test the thinking-more explanation of the foreign language effect.
Significance/Implications:
Our results undermine the claim that using a foreign language promotes more cognitive reflection or engagement. The foreign language effect likely has a different cause.
Research article
Open accessResearch articleFirst published October, 2024pp. 1016-1032
Alex CairncrossORCID, Margreet Vogelzang, Ianthi Tsimpli
Abstract
Aims:
While research on syntactic L1 attrition has largely focused on interface phenomena (e.g., overt pronouns in null-subject languages), attrition has also been reported to affect syntactic parsing. This paper extends previous work by looking at the attrition of (pseudo)relative clause parser biases in L1-Italian L2-English speakers. This was done to investigate whether previous findings of attrition in the interpretation of ambiguous relative clauses can be accounted for by changes to the parsing of pseudorelatives.
Method:
To tap parser biases, we conducted a sentence interpretation task in Italian with ambiguous items. Within these items, pseudorelative availability was manipulated. Participants consisted of two groups of adult native Italian speakers: one living in Italy and one living in an English-speaking country.
Data analysis:
Responses from the sentence interpretation task were subjected to a mixed-effects logistic regression.
Findings:
Results indicated a global effect of attrition resulting in more low attachment interpretations. This did not interact with pseudorelative availability.
Originality:
We take this to be a novel contribution to the literature for two reasons. First, our results extend previous results from Spanish–English bilinguals to a new language pair. Second, we believe that this is the first study to investigate attrition of relative clause attachment biases in light of Grillo and Costa’s, Pseudorelative-First Hypothesis. Once this is taken into consideration, we argue that attrition of parser biases is not due to a loss of a global high attachment bias under pressure from the L2, but a strengthening of a low attachment bias already present in the L1.
Implications:
Results suggest that future work on attrition should look beyond conflicts in parser biases between a bilingual’s languages and raise the possibility that attrition affects attachment biases in other contexts.
Research article
Available accessResearch articleFirst published October, 2024pp. 1033-1052
This study aims to explore the segmental and suprasegmental acoustic adjustments in the clear speech of Korean vowels produced by English-dominant Korean-heritage speakers and native Korean speakers.
Methodology:
Native Korean students at a university in the United States and Korean-American students enrolled in an intermediate Korean language course at the same university read isolated sentences featuring three representative Korean vowels (i.e., /i, a, u/), once in casual speech, and once in clear speech. They repeated this sequence five times.
Data and analysis:
The vowel space of the first and second formants (F1 and F2) in the clear speech was expanded for both heritage and native speakers, and the heritage vowel space was more fronted than the native vowel space, suggesting the influence of English among heritage speakers. However, the peripheral extent and direction in the clear speech and the dominant language effect in heritage speakers were inconsistent across the vowels. Also, pitch dynamics were highly significant in clear speech, exhibiting the salient fundamental frequency (F0) rising at the syllable offset for both heritage and native speakers. The vowel length enhancement in the clear speech was also confirmed in both native and heritage Korean vowels.
Conclusions:
The results confirmed the cross-language tendency in clear speech to induce the modification of vowel quality, pitch, and duration, which also applies to Korean heritage vowel production. Furthermore, the asymmetrical acoustic modifications across the vowels in heritage clear speech support the Speech Learning Model (SLM) prediction.
Originality and implications:
The findings present new data for the clear speech effects of Korean heritage vowel production and a further enhancement strategy of syllable offset pitch raising in clear speech for both native and heritage groups. The results contribute to the theoretical implication of bilingualism by connecting the SLM premise and heritage clear speech.
Research article
Available accessResearch articleFirst published October, 2024pp. 1053-1064
Andrea M. PhilippORCID, Marc Syndicus, Mathieu DeclerckORCID
Abstract
Aims and objectives/purpose/research questions:
Prior research has put forward that asymmetrical switch costs in language switching do not occur when bilinguals are in a voluntary language-switching context. While several studies have provided evidence along these lines, two very recent studies have put this into question. This study further investigates the possibility that voluntary language switching abolishes asymmetrical switch costs.
Design/methodology/approach:
We presented German-English bilinguals (n = 79) with pictures that had to be named in either German or English, whichever they chose, by typing.
Data and analysis:
Reaction times (both the latency of the first keystroke and interkeystroke intervals) were analyzed using linear mixed-effects regression modeling with the factors Trial type (switch vs. repetition trial) and Language (German vs. English), whereas the error rates were analyzed using logistic mixed-effects regression modeling.
Findings/conclusions:
The onset latencies and inter-keystroke intervals both showed larger second-language (English) than first-language (German) switch costs (i.e., reversed asymmetrical switch costs). As switch costs were modulated by language dominance in a voluntary language-switching context, this study provides evidence that voluntary language switching is not a boundary condition for asymmetrical switch costs.
Originality:
In contrast to most previous language-switching studies, we used typed rather than vocal responses in a voluntary language-switching paradigm. In addition, our findings further support and generalize very recent findings of (reversed) asymmetrical switch costs in voluntary language switching.
Significance/implications:
Since asymmetrical switch costs, an important measure of bilingual inhibitory control, are not consistently observed throughout the language-switching literature, it is essential to understand its boundary conditions.