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# Laboratory Phonology

### Journal of the Association for Laboratory Phonology

Ed. by Cole, Jennifer

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Rank 85 out of 179 in category Linguistics in the 2015 Thomson Reuters Journal Citation Report/Social Sciences Edition

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Volume 6, Issue 1

# The relationship between prosodic weakening and sound change: evidence from the German tense/lax vowel contrast

Jonathan Harrington
• Corresponding author
• Institute for Phonetics and Speech Processing, Ludwig-Maximilians University, Munich, Germany
• Email
• Other articles by this author:
/ Felicitas Kleber
/ Ulrich Reubold
/ Jessica Siddins
Published Online: 2015-03-17 | DOI: https://doi.org/10.1515/lp-2015-0002

## Abstract

The study tests a model of sound change based on how prosodic weakening affects shortening in polysyllabic words. Twenty-nine L1-German speakers produced minimal pairs differing in vowel tensity in both monosyllables /zakt, zaːkt/ and disyllables /zaktə, zaːktə/. The target words were produced in accented and deaccented contexts. The duration ratio between the vowel and the following /kt/ cluster was less for lax than tense vowels and less for disyllables than monosyllables. Under deaccentuation, there was an approximation of tense and lax vowels towards each other but no influence due to the mono- vs. disyllabic difference. On the other hand, Gaussian /a/ vs. /aː/ classifications of these data showed a lesser influence due to the syllable count in deaccented words. Compatibly, when the same speakers as listeners classified synthetic sackt-sagt and sackte-sagte continua, they were shown to compensate for the syllable count differences, but to a lesser extent in a deaccented context. Deaccentuation may therefore provide the conditions for sound change to take place by which /aː/ shortens in polysyllabic words; it may do so because the association between coarticulation and the source that gives rise to it is hidden to a greater extent than in accented contexts.

## 1 Introduction

The paper is concerned with the mechanisms by which coarticulatory variation due to so-called polysyllabic shortening can give rise to sound change, but taking into account to a greater extent than before the roles of both phonological categorisation and prosodic variation. The starting point for the analysis is that the conditions for sound change can be met when the acoustic signal provides insufficient information for distinguishing between phonetic timing changes due to polysyllabic compression on the one hand and a phonological shortening of the long vowel on the other.

Following Ohala (1981, 1993), a sound change can occur when speakers and listeners parse coarticulatory relationships in different ways as a result of ambiguities in how speech production is associated with the acoustic speech signal. Thus, the presumed trigger for tonogenesis (Hombert et al. 1979) is that listeners parse the high fundamental frequency not with the source that gives rise to it, the increased tension in the vocal folds in voiceless stop production (Löfqvist et al. 1989), but instead with the following vowel, ultimately leading to a phonologization of this mis-parsed variation as tonal differences. Sound change is, according to Ohala’s model, infrequent in relation to the ubiquitous phonetic variation because listeners undo the contextual modification to phonological categories in speech production through perceptual normalisation; for example, listeners factor out the anticipatory lip-rounding that is often produced in consonants preceding rounded vowels (Bell-Berti and Harris 1979; Perkell 1990) and so reverse or undo coarticulation in speech production (see Lindblom and Studdert-Kennedy 1967; Fujisaki and Kunisaki 1976; Mann and Repp 1980 for compatible perceptual evidence).

The further development to Ohala’s model in this paper is to explore the extent to which sound change is conditioned by prosody and its association with the variation in speech production between hyper- and hypoarticulated speech. While there is extensive evidence that sound change depends on prosodic factors such as prominence and phrasing (e.g., Beckman et al. 1992; Cole and Hualde 2013), prosodic processing is rarely incorporated into models of sound change. One of the few exceptions to this is Lindblom et al. (1995), who suggest that sound change emerges from the interaction between an economy of effort in speech production combined with the need for the speaker to make the signal sufficiently clear in order that the listener’s understanding is not compromised. Lindblom et al. (1995) reason that phonetic innovation can emerge from the high degree of variation in semantically predictable parts of the speech signal. However, listeners do not typically absorb these innovations into the lexicon because they reconstruct the meaning predominantly from top-down and not bottom-up processing. It is when top-down processing becomes disengaged in such hypoarticulation contexts that phonetically innovative forms can become more salient and may be added to the lexicon.

In both Lindblom et al. (1995) and Ohala’s (1993) models, sound change can be brought about when listeners exceptionally decontextualize the speech signal (from top-down processing and from coarticulation, respectively). The models are also similar in explaining sound change as arising as a consequence of how speech production is perceptually processed. The main differences are that Lindblom et al. (1995), like Bybee (2002), give greater emphasis to the mechanisms that give rise to undershoot and hypoarticulation, and that in contrast to Ohala (1993), sound change does not come about in Lindblom et al. (1995) model because of a listener error, but emerges instead as a consequence of how speech production is adapted to the needs of the listener. Harrington et al.’s (2013) draw upon the commonalities of both models in testing whether the perception of coarticulation might be compromised in prosodically weak constituents. They showed that listeners’ compensation for coarticulation in relation to the size of anticipatory V2-on-V1 coarticulation in the production of German/pV1CV2l/non-words was less in unstressed than in stressed syllables. According to their model, sound change may be more likely in prosodically weak constituents because in such contexts listeners are less likely to attribute coarticulation to the source from which it originated than they are in prosodically strong contexts.

The present study extends that of Harrington et al. (2013) to sentence stress and to the sound change that can arise from the well-established finding of vowel shortening in polysyllabic words, which has been extensively documented for Germanic languages (e.g., Lindblom and Rapp 1973 for Swedish; Lehiste 1970; Klatt 1973; Port 1981 for English; Nooteboom 1972 for Dutch). The focus here is on the changes to segmental timing in German trochaic (sackte/sagte; ‘sagged’/‘said’) words in relation to their corresponding monosyllables (sackt, sagt; ‘sags’/‘says’).

Synchronic changes due to polysyllabic shortening can be related to two types of sound change. The first is the shortening of long vowels before two heterosyllabic consonants (Luick 1964; Bermúdez-Otero 1998), a process sometimes referred to as closed syllable shortening. This has resulted in changes such as late Old English sōfte, cēpte that were produced with long vowels in the stem to Middle English and present-day forms with an initial short vowel soft, kept (see also Hickey [1986] for an analysis based on presumed heterosyllabic geminates in late Old English fōder leading to present-day fodder with a short vowel in the stem via an intermediate diachronic stage in which the word was produced with a short vowel and geminate consonant). The second is trisyllabic shortening by which long vowels are shortened in antepenultimate syllables (Hogg 1992; Bermúdez-Otero 1998; Lahiri and Fikkert 1999) leading to vowel shortening in initial syllables between late Old English and present-day forms in words such as holiday and southerner.

There are two main hypotheses to be tested in the present study. The first is that, consistent with other studies (e.g., Fowler and Thompson 2010), there will be vowel shortening in the initial syllable of trochaic words and that listeners will compensate for this shortening. The second is that listeners will compensate less for the shortening when the words are in a prosodically weak, deaccented position. The first part of the paper is concerned with a production study designed to assess the influence of disyllables vs. monosyllables both on timing relationships in the initial syllable and on lax /a/ vs. tense /aː/ classification. The purpose of the second part of the production study was to relate the amount of information in the signal for classifying /a/ and /aː/ to listeners’ responses on a similar task in the subsequent speech perception study.

For the standard variety of German analysed in this study, there is a phonological opposition between lax and tense vowels. The opposition between lax /a/ and tense /aː/ in German is characterised predominantly by length (Heike 1972) and to a lesser extent quality differences (Mooshammer and Geng 2008; Harrington et al. 2011). There is also some evidence to show that German lax vowels can be modelled as similar to tense vowels but truncated by an earlier timing of the closing consonant (Vennemann 1991; Hoole and Mooshammer 2002). Domain-final stops are neutralised in German (Vennemann 1972; Wiese 1996), possibly incompletely (Port and O’Dell 1985; Piroth and Janker 2004; Kleber et al. 2010); thus the vowel tensity distinction for the present materials is within monosyllabic /zakt, za:kt/ (sackt/sagt) and disyllabic /zaktə, za:ktə/ (sackte/sagte) words.

## 2.1 Method

The subjects included 29 speakers of Standard German who produced the target words sackt, sagt, sackte, sagte in the carrier sentence Anna hatte _ verstanden (‘Anna had understood _’). For the accented context, the sentence was produced with a nuclear pitch accent (typically L+H*) on the first syllable of the target word. For the deaccented context, the L+H* nuclear accent was on the first syllable of utterance-initial Anna; since all the following words were deaccented, the pitch was low and level following the L+H* pitch peak.

Ten repetitions of the target sentences and an equal number of fillers were read in randomized order one at a time from a computer monitor at the subject’s self-selected pace. Each sentence was preceded by a question to elicit the appropriate accentual pattern: for the accented context in which the target word was nuclear accented, the sentence was preceded by the question WAS hatte Anna verstanden? (‘WHAT had Anna understood?’); for the context in which the target word was deaccented, the preceding sentence was WER hatte_ verstanden? (‘WHO had understood _?’). Three of the subjects were unable to complete the task: two of these because they consistently misread the sentences and one because of difficulty in producing the accented/deaccented contrast. Of the final 26 speakers, all were Standard German speakers with only slight regional colouring: 16 were from southern Germany (Bavarian and Baden-Württemberg) and the remaining 10 from central and northern regions of Germany. The group consisted of 17 females and 9 males aged 15 to 34 years (mean age 24 years). The final total number of analysed tokens was 10 (repetitions) × 2 (/a, aː/) × 2 (monosyllabic, disyllabic) × 2 (accented, deaccented) × 26 (speakers) = 2,080 target word tokens.

The entire corpus was automatically segmented into phoneme-sized units using the Munich automatic forced-alignment system (Schiel 2004). The boundaries of each segment were checked and manually corrected. The segments that formed part of the present analysis included the /a/ or/ aː/ vowel of the target word defined acoustically as the interval between the acoustic onset and offset of periodicity, and the consonant cluster /kt/ which marked the interval between the /a, aː/ offset and the release of the /t/-closure with no further sub-segmentation of the cluster into /k/ and /t/ components (Figure 1). The /t/-release was excluded in the analysis from the consonant cluster because of the general difficulty of segmenting reliably its offset either from the /f/ of the following word verstanden or from the final weak vowel of sackte/sagte which was often partially devoiced. We also decided not to carry out a further sub-segmentation of /kt/ because it was not possible to determine for a large proportion of the tokens whether or not the /k/ had been released (as a result of which there were often unreliable acoustic landmarks for separating the two plosives). The dependent variables were (1) V, the log of the vowel duration; (2) C, the log of the duration of the /kt/ cluster as defined above; and (3) V/C, the log of the ratio between the vowel duration and the duration of the following /kt/.

Figure 1

Waveform and spectrogram of an utterance containing nuclear accented sagte produced by one of the speakers from the corpus of this study, showing word and V = /a:/and C = /kt/segmentations.

The V/C values were classified as lax /a/ or tense /aː/ separately in the monosyllabic and disyllabic and separately in the accented and deaccented contexts following the procedure in Harrington et al. (2013). These (Gaussian) classifications were also carried out separately for each speaker (thus giving four separate classifications per speaker). The classification included a training stage in which normal distributions were fitted separately to these four contexts from which the posterior probabilities p(a|(V/C)) and p(aː|(V/C)) were derived that a given V/C value is a member of the category/a/or/aː/. The posterior probabilities were then used to derive/a-aː/classification functions (separately per speaker and for each of the four contexts). In order to do so, linear regression was used to estimate the slope, m, and intercept, k, in (1): $\text{log[}p\left(\text{a}ː|\left(\text{V/C}\right)\right]/p\left[\text{a}|\left(\text{V/C}\right)\right] = m\left(\text{V/C}\right) + k$[1] from which the corresponding sigmoid classification function was derived in eq. [2]: $p\left[\text{a}ː|\left(\text{V/C}\right)\right] = {\text{e}}^{\left[m\left(\text{V/C}\right)\text{\hspace{0.17em}}+}{{}^{k}}^{\right]}/\left[1 + {\text{e}}^{\left[m\left(\text{V/C}\right)\text{\hspace{0.17em}}+}{{}^{k}}^{\right]}\right]$[2]

using the relationship $p\left[aː|\left(V/C\right)\right] = 1–p\left[a|\left(V/C\right)\right]$. The decision boundary was given by −k/m which is the V/C value in eq. [2] for which $p\left[\text{a}ː|\left(\text{V/C}\right)\right] =$$p\left[\text{a}|\left(\text{V/C}\right)\right] = 0.5$. Four such decision boundaries were derived separately per speaker based on a total of 20 (10 /a/ and 10 /aː/) V/C values for each of the four contexts.

The influence of the syllable count on vowel tensity classifications was determined by calculating the difference between the monosyllabic and disyllabic classification functions on two measures: the 50% decision boundaries at which classifications between /a/ and /aː/ are equiprobable, and the area enclosed within the classification functions extending between these two decision boundaries in disyllabic and monosyllabic contexts (this latter measure corresponds to the shaded area between the fitted sigmoids shown in the lower panel of Figure 5). The second of these measures also takes account of the slope of the classification function which is not incorporated just by subtracting the location of the decision boundaries.

As far as classification was concerned, the hypothesis to be tested was that /a-aː/ classifications would be influenced less by the syllable count (monosyllabic vs. disyllabic) in deaccented than in accented words. If so, then the difference between the decision boundaries and the area between the classification functions should be smaller in the deaccented than in the accented context.

## 2.2 Results

The first part of the results (Figures 24) is concerned with the effect of tensity, syllable count, and sentence stress on V, C, and V/C durations; the second part is concerned with classifications of V/C. Further descriptive statistics on the former are given in the Appendix.

As far as vowel duration is concerned, Figure 2 shows the expected clear distinction between lax /a/ and tense /aː/ in both accented and deaccented words. In the deaccented context, there was a greater overlap between /a, aː/ that was primarily caused by a shift of the /aː/ distribution towards /a/. This is in line with findings from Mooshammer and Fuchs (2002) and Mooshammer and Geng (2008) that prosodic weakening shortens tense but not lax vowels. There was (to our surprise) no evidence that vowel duration was shorter in the disyllabic vs. monosyllabic contexts. The results of a repeated measures ANOVA with dependent variable V (log vowel duration) and within-subject factors Tensity (two levels /a, aː/), Syllable Count (monosyllabic, disyllabic), and Stress (accented, deaccented) showed a predictable main effect on V of Tensity (F[1, 25] = 616.3, p < 0.001), and of Stress (F[1,25] = 44.5, p < 0.001) but no significant effect of Syllable Count. There was also a significant interaction between Stress and Tensity (F[1, 25] = 92.6, p < 0.001). Post-hoc Bonferroni corrected t-tests showed, consistent with the evidence in Figure 2, a significant difference between accented and deaccented vowel duration for tense /aː/ (t[24] = 8.3, padj <0.001) but not for lax /a/. There was also a significant interaction between Stress and Syllable Count (F[1,25] = 9.0, p < 0.01). Post-hoc tests showed, consistent with Figure 2, that vowel duration in disyllables was actually significantly greater than in monosyllables but only in deaccented (t[25] = 3.1, padj <0.05) and not in accented words.

Figure 2

Probability density plots of the log vowel duration for all samples of the corpus in monosyllabic (black) and disyllabic (grey) words in accented (left) and deaccented (right) contexts for tense (solid) and lax (dashed) categories.

The results for C (log duration of the /kt/ cluster as defined in 2.1 above) in Figure 3 showed a greater cluster duration for lax /a/ vs. tense /aː/ and also for disyllables vs. monosyllables. The results of a repeated measures ANOVA with cluster duration as the dependent variable and with the same independent factors as above showed main effects for Tensity (F[1,25] = 178.6, p < 0.001), Stress (F[1,25] = 109.8, p < 0.001), and Syllable Count (F[1,25] = 26.4, p < 0.001) and no interaction between these factors. Thus these results show that cluster duration was greater in lax vs. tense vowels, in disyllables vs. monosyllables, and in accented vs. deaccented words.

Figure 3

Probability density plots of the log C = /kt/duration for all samples of the corpus in monosyllabic (black) and disyllabic (grey) words in accented (left) and deaccented (right) contexts for tense (solid) and lax (dashed) categories.

The third parameter to be considered was the ratio of the vowel to the /kt/ cluster duration: this measure (V/C) should magnify the difference between lax and tense vowels given that, as shown above, lax vowels had both a shorter vowel duration and longer cluster duration compared with their tense counterparts. Figure 4 shows smaller V/C ratios for lax vs. tense vowels and to some extent also for disyllables vs. monosyllables. The results of a repeated measures ANOVA with V/C as the dependent variable and with the same independent factors as above showed a significant main effect for Tensity (F[1,25] = 759.5, p < 0.001), and for Syllable Count (F[1,25] = 17.1, p < 0.001), but no main effect of Stress. There was a significant interaction between Tensity and Stress (F[1,25] = 55.7, p < 0.001). Post-hoc tests showed a larger V/C ratio for deaccented vs. accented words, but only in lax vowels (t[25] = 5.7, p < 0.001).

Figure 4

Probability density plots of the V/C ratio for all samples of the corpus in monosyllabic (black) and disyllabic (grey) words in accented (left) and deaccented (right) contexts for tense (solid) and lax (dashed) categories.

Finally, Figure 5 shows the same data as in Figure 4 together with the corresponding classification functions calculated by fitting sigmoids to the data in the manner described in Section 2.1. Three observations can be made about these averaged data. First, the decision boundaries between /a/ and /aː/ were located as expected roughly halfway between their corresponding V/C distributions. Secondly, and again compatibly with the acoustic data, the /a-aː/ classification function in the disyllabic context was left-shifted towards lower V/C values relative to that in the monosyllabic context (i.e., the vertical grey dashed lines are to the left of the vertical black dashed lines in both accented and deaccented contexts). Thirdly, the classification functions for the monosyllabic and disyllabic contexts were closer together for deaccented than for accented words. This may come about because, as the top right panel of Figure 5 shows, the distributions of lax and tense vowels on V/C were closer together. Importantly, the closer proximity of the classification functions in deaccented words did not result from the weaker influence of the syllable count. As the top row of Figure 5 shows, the difference between the monosyllabic (white) and disyllabic (grey) distributions was about the same in accented and deaccented words, which is compatible with the results in Figure 4 showing that stress had no influence on the V/C ratio and did not interact with the syllable count. In deaccented words, the closer proximity of the monosyllabic and disyllabic classification functions was instead an indirect consequence of the smaller separation between lax and tense vowels. The shallower slopes in the classification functions for deaccented words in Figure 5 are also consistent with the greater overlap and therefore greater classification uncertainty in the deaccented compared with the accented context.

Figure 5

Boxplots of the data in Figure 4 showing the interquartile range of the V/C ratio (above) for disyllables (grey boxes) and monosyllables (white boxes) in accented (left) and deaccented (right) contexts. The corresponding /a/ vs. /a:/ classification functions averaged across all speakers are shown below for sackte vs. sagte (grey) and sackt vs. sagt (black) with 50% decision boundaries superimposed as the vertical dashed lines. The shaded region is the area between the disyllabic and monosyllabic sigmoids within their respective decision boundaries.

The above trends were further quantified by analysing the speaker-specific decision boundaries, areas, and slopes. As far as the decision boundaries are concerned, there were two trends evident in Figure 6: first, the decision boundaries were lower at lower V/C ratios in the disyllabic than in the monosyllabic context; secondly, the separation between monosyllabic and disyllabic decision boundaries was slightly greater in accented than in deaccented words. A repeated measures ANOVA with the Decision Boundary as the dependent variable and with independent factors Syllable Count (monosyllabic/disyllabic) and Stress (accented/deaccented) showed a main effect for Syllable Count (F[1,25] = 15.5, p < 0.001) but no effect for Stress and no interaction between these factors. The significant effect for Syllable Count supports what is evident in Figure 5: the decision boundaries were shifted towards lower values in disyllables than in monosyllables. However, the lack of a significant interaction between the independent factors does not support the averaged data in Figure 5 showing that the decision boundaries were closer together in the deaccented context.

Figure 6

Distribution of the 50% decision boundaries resulting from classifying the V/C ratio as /a/ or /aː/ in accented monosyllables (white boxes) and disyllables (grey boxes) in accented (left) and deaccented (right) contexts. The extent of the box is the interquartile range. Each of the four distributions consists of one decision boundary per speaker.

Figure 7 shows that the areas between the mono- and disyllabic sigmoids (corresponding to the grey shaded area in Figure 5) were larger in the accented than in the deaccented context. Consistent with the observations in Figures 5 and 7, a paired (one pair per speaker) signed Wilcoxon rank test showed larger areas in an accented compared with a deaccented context (V = 266, p < 0.05).

Figure 7

Distribution resulting from calculating the area between the sigmoids in the mono- and disyllabic contexts (the shaded area in Figure 5) in accented (left) and deaccented (right) words separately per speaker. The extent of the box is the interquartile range. Each of the two distributions consists of one area per speaker.

Figure 8 shows that the slopes of the classification functions were steeper in accented than in deaccented words. These trends were confirmed by a repeated measures ANOVA with Slope as the dependent variable which showed a main effect for Stress (F[1, 25] = 22.8, p < 0.001), a non-significant effect for Syllable Count, and no interaction between these independent factors. Thus the likely reason for the discrepancy between the decision boundaries (showing a non-significant effect for stress in Figure 6) and area differences (in which the effect for stress was significant in Figure 7) is that the area between the sigmoids is determined not just by decision boundaries but also by slopes: that is, the shallower slopes are likely to have been a contributory factor in the smaller area between the mono- and disyllabic sigmoids in the deaccented context.

Figure 8

Distribution of the sigmoid slopes resulting from classifying V/C ratio as /a/ or /aː/ in monosyllables (white boxes) and disyllables (grey boxes) and in accented (left) and deaccented (right) words separately per speaker. The extent of the box is the interquartile range. Each of the four distributions consists of one slope per speaker.

## 2.3 Discussion

The first part of the analysis was concerned with the way in which the syllable count influenced vowel and cluster duration in accented and deaccented words. Syllable count was found (somewhat surprisingly) not to affect vowel duration (V). On the other hand, the duration of the post-vocalic cluster (C) was greater; as a result, the V/C ratio decreased in a disyllabic vs. monosyllabic context. Stress predictably caused a lengthening of both the vowel and of the following cluster. However, it did not affect the V/C ratio. Consequently, the syllable count’s influence was not different in accented and deaccented words, at least on the V/C ratio parameter.

The second part of the analysis was concerned with the influence of the syllable count on tensity (/a/ vs. /aː/) classifications. The results of this analysis showed that the classification functions were shifted towards lower values on the V/C parameter (resulting in a greater likelihood of /aː/ classifications) in a disyllabic compared with a monosyllabic context. However, this shift from a disyllabic to a monosyllabic context was not as great for deaccented compared with accented words; the likely cause of this diminished shift in the deaccented context (in which the vertical dashed lines are closer together for the deaccented than in the accented context in Figure 5) was that the distributions of lax and tense vowels on V/C overlapped to a greater extent in the deaccented than in the accented context.

Thus the overall conclusion from all of these findings is that, whereas the syllable count’s influence on the V/C parameter was much the same in an accented and deaccented context, its influence on /a/ vs. /aː/ categorisations was greater in accented than in deaccented words.

From the point of view of speech perception, the implication of these results is that listeners may find it more difficult to factor out the influence of the syllable count from /a-aː/ classifications in a deaccented context. Suppose, for example, that the sigmoids in Figure 5 had been psychometric functions resulting from forced-choice classification of /a/ vs. /aː/. In this case, the evidence in Figure 5 would show that listeners compensate for coarticulation because they would be shifting their categorisations towards lower values in the disyllabic context, commensurate with the lowering effect of disyllabic words on V/C seen in the top panel of the same figure. But the same data would also show that listeners compensate less for the coarticulatory effect of syllable count in deaccented words; this is because the monosyllabic and disyllabic classification functions are evidently closer together in deaccented words, even though the size of the monosyllabic vs. disyllabic effect on V/C in speech production (Figure 5, top panel) was much the same in accented vs. deaccented words.

The question of whether listeners do compensate less for the effects of syllable count in a deaccented context is the main concern of the next section.

## 3.1 Method

Four continua were created by embedding the same 11-step continuum between tense /aː/ and lax /a/ in monosyllabic /z_kt/ (sagt/sackt) and disyllabic /z_ktə/ (sagte/sackte) contexts in the carrier Anna hatte _ verstanden (the same sentence that had formed part of the materials for the speech production experiment). Just as in speech production, the target word was either nuclear accented or deaccented, with the nuclear accent occurring on the initial word in the second case.

In order to create these continua, a trained phonetician and female speaker of Standard German with slight South German regional characteristics produced 10 repetitions each of the target sentences in the two sentence-stress contexts. The mean durations of /z/ and /kt/ of the target word were calculated separately in the accented and deaccented context; then, two productions of /zVkt/ were selected, one from each of the speaker’s accented and deaccented productions which were closest in duration to these means. The vowel was spliced out leaving an accented /z_kt/ and a deaccented /z_kt/. In both cases, the /k/ was produced with a release.

An /aː/ vowel from one of the speaker’s accented productions of sagte which had F1 values closest to the F1 median of this speaker’s /a/ and /aː/ tokens was manipulated in duration to create 11 equidistant steps using Praat’s (Boersma and Weenink 2012) implementation of the PSOLA algorithm (Moulines and Charpentier 1990). The 11 steps ranged from 40 ms to 112 ms; these selected durations were based on the model speaker’s range of target vowel durations in two small pilot studies. The vowel duration and quality were identical in both accented and deaccented contexts, but the deaccented vowels were subsequently manipulated to contain a slightly falling F0 contour from 215 to 195 Hz and an intensity of 7 dB lower than that of the accented vowels. These vowel continua were then spliced into the aforementioned two (accented, deaccented) /z_kt/ contexts. The closure duration of the /kt/ cluster between the acoustic vowel offset and onset of the /t/ release was 81 ms for all accented stimuli and 62 ms for all deaccented stimuli. Finally, the sackte-sagte continua were derived by post-pending one of the speaker’s /ə/ tokens to accented /zVkt/ and to deaccented /zVkt/ (where V denotes the 11-step continuum).

The four (monosyllabic/disyllabic × accented/deaccented) continua created in this way were spliced into the carrier sentence Anna hatte _ verstanden, with one carrier taken from an accented utterance and the other from a deaccented utterance. As such, the final stimuli contained natural intonation contours without the need for pitch and intensity resynthesis (see Figure 9).

Figure 9

Fundamental frequency contour, waveform, and segmentation of two of the synthetic utterances in the accented and deaccented contexts. The target word is shown for sagt at the /aː/ endpoint of the sackt-sagt continuum.

The 44 synthetic utterances were each repeated 10 times, and presented in the same randomised order over headphones in a quiet room in a two-alternative forced-choice identification test to 32 subjects, 29 of whom were the same subjects as in the speech production experiment. The listeners, who were paid for their participation, classified each word by clicking on one of four possible orthographic forms (sackt, sagt, sackte, sagte) displayed on a computer monitor. The two alternatives were presented counter-balanced so that half of the time the orthographic representation of the tense vowel was on the right-hand side of the screen and half of the time on the left. Participants were requested to respond as quickly as possible. Repeated listening of the same stimulus was not allowed, but listeners could change their choice before clicking the “OK” button to confirm their response. There were 11 (steps) × 2 (mono/disyllabic) × 2 (accented/deaccented) × 10 (repetitions) × 32 (listeners) = 14,080 responses in total.

Classification (sigmoid) functions explaining the relationship between the response (/a/ or /aː/) and the V/C ratio were obtained using binary logistic regression separately per listener in each of the four contexts (monosyllabic/disyllabic × accented/deaccented) using the relationship: ${p}_{\text{a}ː} ={\text{\hspace{0.17em}e}}^{\left[m\left(\text{V}/\text{C}\right)\text{\hspace{0.17em}}+k\right]} /\text{\hspace{0.17em}}\left[\text{1\hspace{0.17em}}+{\text{\hspace{0.17em}e}}^{\left[m\left(\text{V}/\text{C}\right)\text{\hspace{0.17em}}+k\right]}\right]$[3]in which p was the proportion of /aː/ responses (pa: + pa = 1), V/C the values on the V/C ratio in the synthetic continuum, and m and k respectively the listener-specific slope and intercept that were fitted using logistic regression. Decision boundaries at 50% cross-over points were obtained, one for each of the four contexts (i.e., four per listener) from −k/m in eq. [3]. In addition, and analogously to the procedure for classifying the production data, the areas between the classification functions in the disyllabic and monosyllabic contexts were calculated (two areas per listener, one for the accented, one for the deaccented context).

The hypothesis to be tested was that the decision boundaries would be closer together and that the area between the classification functions would be smaller in the deaccented than in the accented context.

## 3.2 Results

The aggregated results in Figure 10 suggest that listeners’ decisions were influenced both by syllable count and by stress. Consistent with the production data (Figure 5), listeners’ decision boundaries were, on average, shifted towards lower values of the V/C ratio in disyllabic compared with monosyllabic words. The much greater positive V/C values in the deaccented context are, however, not at all consistent with the production data which showed no main effect of stress, neither in duration (Figure 4 and Figure 5, top panel) nor classification (Figure 5, lower panel). The rightwards shift in the deaccented context observed in these perception data is likely to have come about because the range of V/C values of the 11-step continuum on which responses were obtained was itself right-shifted for deaccented (between −0.44 and 0.58) compared with accented (between −0.70 to 0.32) by an amount (approximately 0.25) which corresponds to the size of the rightward displacement for deaccented observed in Figure 10. That is, the unexpected rightwards displacement of the responses in the deaccented relative to the accented context is most likely an artefact of the different V/C values for the two stress contexts on which responses were obtained.

Figure 10

The points are proportional /aː/ responses averaged across listeners responding to the 11-step monosyllabic (black) and disyllabic (grey) continua in the accented (solid) and deaccented (dashed) contexts. The decision boundaries averaged across listeners in these four contexts are shown as vertical lines.

As far as the main hypothesis is concerned, there is some marginal evidence from Figure 10 that the difference between responses in a disyllabic and monosyllabic context was greater in the accented vs. deaccented context: this is shown both by the slightly greater separation between the decision boundaries derived from logistic regression and by the generally greater leftward displacement of the disyllabic responses in the accented context. The distribution of the listener-specific decision boundaries in Figure 11 shows marginally lower decision boundaries for disyllables vs. monosyllables in the accented context. A repeated measures ANOVA with the 50% decision boundary (the data in Figure 11) as the dependent variable and with independent factors Syllable Count (mono/disyllabic) and Stress (accented/deaccented) showed significant effects for Syllable Count (F[1,31] = 12.4, p < 0.01) and for Stress (F[1,31] = 483.2, p < 0.001) and a significant interaction between these factors (F[1,31] = 5.9, p < 0.05). The significant effect of Syllable Count shows that listeners compensated for the coarticulatory effects of V/C reduction in a disyllabic context. The significant effect of stress is likely to be an artefact of the different range of V/C ratios on which responses were obtained in the two stress contexts, as discussed above. The significant interaction is consistent with the evidence in Figure 10 showing a greater separation between disyllabic and monosyllabic responses in the accented than in the deaccented context. Compatibly, post-hoc Bonferroni corrected t-tests showed a significant difference in the location of the decision boundaries between monosyllabic and disyllabic contexts in the accented (t = 4.4, p < 0.001) but not in the deaccented context.

Figure 11

Distribution of the 50% decision boundaries resulting from perceptual classification of the continua in accented monosyllables (white boxes) and disyllables (grey boxes) in accented (left) and deaccented (right) contexts. The extent of the box is the interquartile range. Each of the four distributions consists of one decision boundary per speaker (compare with Figure 6).

The other dependent variable used to test the hypothesis was the area between the monosyllabic and disyllabic classification functions (between the decision boundaries) which was predicted to be greater for the accented than for the deaccented context. Although, as Figure 12 shows, there is a very general tendency for this to be so, the results of a paired Wilcoxon signed rank test with Area as the dependent variable (one pair per listener) and with Stress as the independent factor showed no significant differences between accented and deaccented words on this parameter.

Figure 12

Distribution resulting from calculating the area between the sigmoids in the mono- and disyllabic contexts (corresponding to the shaded area in Figure 5) in accented (left) and deaccented (right) words. The extent of the box is the interquartile range. Each of the two distributions consists of one area per listener (compare with Figure 7).

Based on the results of the speech production data, the prediction about slopes in perception was that these should be flatter for the deaccented than for the accented context (Figure 8). The statistical quantification of slope for the perception data was, however, problematic because of data from six listeners who had almost infinite slopes, i.e., who showed no ambiguity in their responses at the 50% decision boundary. After removing the data from these listeners, a similar tendency (although not as marked as for speech production) emerged for steeper slopes in the accented than in the deaccented context (Figure 13). Consistent with this trend in Figure 13, the results of a repeated-measures ANOVA with Slope as the dependent variable and the same independent factors as above showed steeper Slopes for accented vs. deaccented (F[1,25] = 8.1, p < 0.01), no effect of Syllable Count, and no interaction between these factors. 1 Thus, there is some evidence to show that, at least for those 26/32 listeners who showed some kind of ambiguity in the vicinity of the decision boundary, the slope was steeper for accented than deaccented words.

Figure 13

Distribution of the sigmoid slopes resulting from perceptual classifications in monosyllables (white boxes) and disyllables (grey boxes) in accented (left) and deaccented (right) contexts for 26/32 listeners. The extent of the box is the interquartile range. Each of the four distributions consists of one slope per listener (compare with Figure 8).

## 3.3 Discussion

There are two main findings from this perception experiment. The first is that listeners adjusted their responses due to the syllable count such that their /a/ vs. /aː/ classifications were shifted towards lower V/C values in disyllabic vs. monosyllabic words. This result is compatible with their speech production data, which showed both that V/C ratios were smaller (Figure 4) and that the binary /a/ vs. /aː/ classification of the production data was shifted towards lower V/C ratios (Figure 5). Independently of this finding, the perception experiment also demonstrates that listeners must have responded to the V/C ratio (or at least to some combination of V and C) in making their responses. If they had been responding just to synthetic vowel duration, then their classifications should have been at about the same point on the V/C continuum in the accented vs. deaccented context; yet the results show that their responses in the deaccented context were shifted towards higher V/C values by an amount corresponding to the difference in the synthetic /kt/ cluster duration between the two stress contexts. Listeners must therefore have taken the cluster duration into account in making /a/ vs. /aː/ categorisations.

The second main finding is that the influence of the syllable count on classifications was less in the deaccented than in the accented context: that is, consistent with the main hypothesis that was tested, there was some evidence to suggest that listeners did not compensate as much for the phonetic influence of a disyllabic vs. monosyllabic context in deaccented words. The speech perception and production classifications were generally in the same direction, although not always significantly so: thus in deaccented compared with accented words, the decision boundaries were closer together (significantly for perception), the areas between the classification functions were smaller (significantly for production), and the slopes were flatter (significantly in production and perception, but only after removing 6 listeners from the latter due to near-infinite slopes).

## 4 General discussion

The results from speech production showed that the syllable count – whether the word was monosyllabic or disyllabic – had an influence on V/C ratios, where V is the vowel duration and C the duration of the following cluster. The size of this phonetic effect was no different in accented and deaccented words. When the same speech production data were classified for tensity as /a/ or /aː/ on the same V/C parameter, then categorisations were affected by the syllable count such that, compatibly with this phonetic effect, classifications were shifted towards lower V/C values in disyllabic words: that is, the probability of classifying the signal as /aː/ was slightly greater in disyllabic words. However, the extent of the syllable count’s influence on classification was less in the deaccented context, even though the size of the V/C lowering in disyllables was no different in deaccented than in accented words. Thus, although deaccentuation did not diminish the influence of the syllable count on speech production, it did weaken its influence on /a/ vs. /aː/ categorisation. The weakening in categorization was an indirect consequence of the closer proximity (less differentiation) of lax vs. tense vowels and the greater speech production variability in deaccented speech. From another point of view: the contextual effect of the syllable count was not obliterated in speech production but was instead hidden to a greater extent from tensity categorisations due to the greater /a/ vs. /aː/ overlap and variability in deaccented words. That is, when speech production was categorised for tensity, there was less information available in a deaccented context about the confounding influence of the syllable count.

When the same speakers classified synthetic stimuli that varied only in vowel duration (and therefore also in the V/C ratio), their responses were related to the speech production classifications in two ways. First, their responses were affected by the syllable count: just as in classifications of the speech production data, they were more likely to classify a given V/C value as /aː/ in a disyllabic context. Secondly, and also entirely compatibly with the speech production classifications, their responses were affected by the syllable count to a lesser degree in a deaccented context, even though the (synthetic) changes to the V/C ratio were the same in mono- and disyllables. The speech perception results showed that listeners parsed the V/C ratio into at least two components: the component that is responsible for differences in vowel tensity and the component that is due to the influence of the syllable count. This perceptual factoring (Fowler and Smith 1986; Fowler 2005) of the signal is manifested as a shift in the classification function in the disyllabic relative to the monosyllabic context. But listeners were less successful at factoring the signal perceptually in a deaccented context, even though acoustically the same information was there for them to do so, and even though in speech production the shift in the V/C ratio due to the syllable count was significant and largely unaffected by whether the words were accented or not. The question as to why listeners should have compensated less for coarticulation in deaccented speech is more difficult to answer. But one possibility is suggested by the results from classifying the speech production data in which the greater proximity of the /a, aː/ distributions in deaccented words also pushed the decision boundaries for monosyllabic and disyllabic words closer together. Perhaps for this reason, and also because there is less certainty in /a, aː/ decisions in deaccented words, as shown by the shallower classification slopes, the information about how the signal is factored into contributions from tensity on the one hand and syllable count on the other is noisier and therefore not as learnable as in accented speech.

Deaccented speech may be a source of sound change precisely because the effect of context on categorisation – i.e., what has often been termed compensation for coarticulation – is weakened. If sentence stress has no effect on the size of the phonetic effect in distinguishing sackt from sackte or sagt from sagte but diminishes compensation for coarticulation, then there is likely to be variation in V/C that is not parsed with the syllable count. The conditions for sound change to occur by which /aː/ changes to /a/ would then be met if the decrease in the V/C ratio towards/a/caused by disyllables were instead parsed with vowel tensity. This type of parsing error is more likely in any condition such as deaccentuation in which the parsing or factoring of the signal is itself compromised.

The further issue raised by the present study is how sound change interacts with prosodic prominence. On the one hand, the results are consistent with the finding of an association between sound change and prosodic weakening (e.g., Beckman et al. 1992), but on the other hand, they do not seem to be compatible with the evidence that, as recently discussed in Cole and Hualde (2013), there are numerous sound changes such as umlaut and diphthongization that are often confined to stressed syllables. There is, however, no contradiction between our results and the occurrence of sound change in lexically stressed syllables because we are not proposing an association between word stress and sound change; our model suggests instead that it is a hypoarticulation context brought about by semantic redundancy (of which deaccenting is an example) that can provide the conditions for sound change to occur. These conditions are met in lexically stressed syllables when they are hypoarticulated as they have been shown to be in deaccented words (de Jong 1995; Harrington et al. 2000) or perhaps more generally in semantically redundant contexts (Lindblom 1988, Lindblom1998).

From another point of view, stressed syllables that are accented at the level of the utterance (i.e., marked for a pitch-accent in Germanic languages) are processed more rapidly by listeners, because prominent syllables direct the listener’s attention towards semantically salient parts of the utterance (Cutler 1976; Mehta and Cutler 1988) and/or because metrical expectations cause listeners to focus their attention on metrically prominent syllables (Zheng and Pierrehumbert 2010). Analogously, speech signals that are low in prominence are processed more slowly, perhaps because listeners can deploy semantic context to a greater extent in such contexts, which are often hypoarticulated (Lindblom 1988). Our suggestion is that this processing advantage in prominent syllables also extends to the listener’s compensation for coarticulation. That is, in hyperarticulated speech in which the listener’s attention is closely focussed on the signal content (Lindblom et al. 1995), listeners also normalise or compensate for coarticulation. But they may do so to a lesser extent in hypoarticulated and weakly prominent speech in which speech processing is less rapid and in which, following Lindblom et al.’s (1995) model, they shift their attention away from the detailed composition of the signal as they engage their ‘what-mode’ of perception (i.e., top-down processing) to a greater extent.

The more general conclusion is that it is hypoarticulation which is more likely to cause an error in parsing coarticulation. Although this proposed generalisation to hypoarticulation is not directly inferable from the results of the present study, hypoarticulation may be the common cause linking the findings of diminished perceptual compensation for coarticulation in deaccentuation (this paper) and in rhythmically weak syllables (Harrington et al. 2013). Both these studies suggest that the conditions for sound change to take place are likely to be met when, first, category boundaries become more blurred, either through greater variation and/or because they are shifted closer together; and, secondly, when the magnitude of the coarticulatory effect that a source gives rise to is maintained or increased. In Harrington et al. (2013), this argument was applied to anticipatory V1CV2 coarticulation in which the high vowels in V1 position were found to shift closer together when V1 was prosodically weak, but the coarticulatory influence of V2 on V1 was maintained.

Our assumption has been that the shortening effect from sackt to sackte or sagt to sagte is attributable to polysyllabic shortening of the kind in which there is an association between a compression to segment duration and the number of syllables in the word (Klatt 1973; Lindblom and Rapp 1973), leading to a progressive stressed vowel shortening in, e.g., triplets such as speed, speedy, and speedily (Lehiste 1972; Port 1981; see Turk and Shattuck-Hufnagel 2000; White and Turk 2010 for more recent studies). However, there are at least two reasons why polysyllabic shortening may not be the mechanism responsible for the observed duration differences in the present data. First, in contrast to those studies, we found no evidence of a shortening of the lexically stressed vowel in disyllables compared with monosyllables, which, for accented and deaccented words in our study, were instead associated with differences in the ratio of the lexically stressed vowel to that of the following consonant cluster. Second, the greater duration of the /kt/ cluster in disyllables than in their monosyllabic counterparts may instead be due to the presence of a syllable boundary within the /kt/ cluster in sack.te and sag.te (as a result of which, according to some accounts, an underlying voiced /ɡ/ in the stem of sag becomes voiceless by a rule of syllable or word-final devoicing – see, e.g., Röttger et al. [2014] for a recent review and analysis). Whether the mechanism is polysyllabic shortening or presence vs. absence of a syllable boundary within the cluster or some form of both, the conclusion is the same: classifications of lax /a/ vs. tense /aː/ are influenced by the syllable count (monosyllables vs. disyllables) but less so in deaccented compared with accented words.

The prediction from these results is that the conditions for sound change to take place are likely to be met when, due to a hypoarticulated speaking style, coarticulation is maintained (or increased) but categorisation strength is weakened. For example, there is some, albeit weak, empirical evidence that the well-documented sound change of velar palatalization by which sequences of /k g/ + /i j e/ change diachronically into alveolar or palatal affricates (Grammont 1933; Bhat 1978; Guion 1998; see Ohala and Solé 2008 for numerous examples) arises from a perceptually conditioned re-analysis of faster speech (Guion 1998). The explanation for such a change in terms of the model derived from the present set of results is first that the magnitude of the coarticulatory influence of front vowels on velar stops would be the same or even increase in a hypoarticulated speaking style (the hypoarticulation in the case of Guion’s [1998] analysis being due to fast speech); and second, that /k/ and /t/ would shift closer together in this context, possibly as a result of the degradation of the mid-frequency spectral burst peak in /ki/ (Winitz et al. 1972; Chang et al. 2001) in a hypoarticulated speaking style. That is, the maintenance or increase of anticipatory coarticulation coupled with a weakening of the /k-t/ boundary could diminish the available information in a hypoarticulated speaking style for factoring the fricative noise into a /k/-release on the one hand and the coarticulatory influence due to the high vowel on the other.

The model that is being proposed here is inspired by Ohala’s (1993) idea that coarticulatory misparsing contributes to sound change, but with the difference that speaker-hearer relationships are not quite as prominent and also that, following Lindblom et al. (1995) and Bybee (2002), there is a much greater role for the contributory effects of reduction and undershoot in sound change. Thus, whereas in Ohala’s (1993) model there is a mismatch between how the speaker interleaves and how the listener parses coarticulation, the mismatch in the model being presented here is instead between coarticulation and categorisation. More specifically, context affects speech production (coarticulation), but it also affects categorisation and not necessarily always in the same way. It is especially in hypoarticulated speech that the contextual influences that give rise to coarticulation in speech production on the one hand and to shifts in category boundaries on the other may diverge, providing a mismatch that can provide the conditions for sound change to take place. The listener is of course likely to have a role in this divergence, but the mismatch between categories and coarticulation should be in evidence in both speech production (as in Figure 5 of the present study) and in speech perception (Figure 10); thus, what is being proposed here is that under hypoarticulation there can be a change in the relationship between categorisation and coarticulation in both modalities. This proposed model is also consistent with earlier (Phillips 1984; Pierrehumbert 2001, Pierrehumbert2003; Bybee 2002) and more recent findings (Lin et al. 2014) that sound change may be more likely in words of high frequency because such words, being prone to reduction (Wright 2004), should make the association of coarticulation with the source that gives rise to it more opaque.

In summary, the main conclusion from the study is that there is greater noise in hypoarticulated speech such that the information for separating coarticulation from categorisation is weakened, which in turn creates an ambiguity that can provide the conditions for sound change to take place.

## Acknowledgments

Our thanks to Associate Editor Jonathan Barnes and to two anonymous reviewers for very helpful comments on an earlier draft of this paper. This research was supported by European Research Council grant number 295573 ‘Sound change and the acquisition of speech’ (2012–2017).

## Appendix

Some descriptive statistics of the three duration parameters analysed in Section 2 are included in Table 1 and Figure 14 below.

Figure 14

Mean durations of the vowel (left), of the following /kt/ cluster (center), and the ratio between the two (right) shown separately for stress (accented/deaccented), tensity (/a, aː/), and syllable count (white/grey rectangles for monosyllabic/disyllabic). The data were aggregated by speaker so that each boxplot includes one data point per speaker. The extent of the rectangle is the interquartile range.

Table 1

Mean (m) and standard deviation (s) calculated across all tokens produced by all speakers for the duration of the stressed vowel (V), the consonant cluster C = /kt/, and the ratio V/C between the two shown separately for stress (accented vs. deaccented), vowel tensity /a, aː/, and mono/disyllables. The V and C durations are in milliseconds.

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## Footnotes

• 1

When the repeated measures ANOVA is re-run on these data with only 26 listeners to test for an effect of decision-boundaries, then the interaction between the independent factors of Syllable Count and Stress is still significant (F[1,25] = 4.8, p < 0.05).

Published Online: 2015-03-17

Published in Print: 2015-02-01

Citation Information: Laboratory Phonology, Volume 6, Issue 1, Pages 87–117, ISSN (Online) 1868-6354, ISSN (Print) 1868-6346,

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text:
Homework Help: Green's Function

1. Nov 27, 2012

the_kid

1. The problem statement, all variables and given/known data

I'm trying to show that the Green's function for the Laplace operator $-\nabla^2$ is badly behaved at infinity. I.e.

$$\int d^dx|G(x,y)|^2=\infty$$ for d=1,2,3. What happens when d>4?

I know the 1D Green's function is given by

$$G(x,y)=-\frac{|x-y|}{2}$$

2. Relevant equations

3. The attempt at a solution

$$G(x,y)\propto|x-y|^{-(d-2)}$$ for d>2

I need to be able to show that the above integral diverges...

format: plain


text:
# Thread: Exponents and Logarithms-- 3 questions.

1. ## Exponents and Logarithms-- 3 questions.

Hi there i dont know if i should make separate threads maybe some one could let me know if so but here are three question i wanted to check if i was right or wrong without further adieu

1) Graph the following function: $\displaystyle f(x)=-\log(x+2)-1$ when i usually see a log function graph question i change to exponential form and when i do i get $\displaystyle 10^1=-(x+2)$ should it be $\displaystyle y+1=-(x+2)$ ?

2)Solve the following question leave in exact form: $\displaystyle (6^{3x+1})=(2^{2x-3})$ which i turn into $\displaystyle 3x+1\log(6)=2x-3\log(2)$ then i factor it out and bring x's together and solve from there i get $\displaystyle x= 1/24$

2. ## Re: Exponents and Logarithms-- 3 questions.

Let us see what have you tried so far.

3. ## Re: Exponents and Logarithms-- 3 questions.

Originally Posted by Gurp925
2)Solve the following question leave in exact form: $\displaystyle (6^{3x+1})=(2^{2x-3})$ which i turn into $\displaystyle 3x+1\log(6)=2x-3\log(2)$ then i factor it out and bring x's together and solve from there i get $\displaystyle x= 1/24$
No, this should be \displaystyle \displaystyle \begin{align*} \left( 3x + 1 \right) \log{(6)} = \left( 2x - 3 \right) \log{(2)} \end{align*}.

4. ## Re: Exponents and Logarithms-- 3 questions.

Originally Posted by Gurp925
Hi there i dont know if i should make separate threads maybe some one could let me know if so but here are three question i wanted to check if i was right or wrong without further adieu

1) Graph the following function: $\displaystyle f(x)=-\log(x+2)-1$ when i usually see a log function graph question i change to exponential form and when i do i get $\displaystyle 10^1=-(x+2)$ should it be $\displaystyle y+1=-(x+2)$ ?
No "logarith" is not linear and neither is an exponent.

2)Solve the following question leave in exact form: $\displaystyle (6^{3x+1})=(2^{2x-3})$ which i turn into $\displaystyle 3x+1\log(6)=2x-3\log(2)$ then i factor it out and bring x's together and solve from there i get $\displaystyle x= 1/24$
Did you try just putting 1/24 into the equation? Even without using a calculator it should be easy to see that (3/24)+ 1 is positive (so 6 to that power is greater than 1) while (2/24)- 1 is negative (so 2 to that power is less than 1).

5. ## Re: Exponents and Logarithms-- 3 questions.

Here is the graph


text:
## What is Spring Potential Energy?

The potential energy is the energy possessed by a body by virtue of its position or the energy held by the object due to its position relative to other objects, its force like stresses, charge, and other factors affecting the particles of the body. The potential energy is the measure of the net work done by or on the body. The spring potential energy is the potential energy stored due to the deformation of an elastic spring and this elasticity is due to the stretched spring. The elasticity is the ability of a solid-state matter to come back to its equilibrium position or original position after any kind of external force is acted on it. It is also known as Elastic potential energy. The potential energy here is caused due to the displacement of the spring from its equilibrium position to another position and this potential energy is equal to the net work done due to the stretching of the spring. It also resembles the same mechanics of the oscillation of a simple pendulum, where due to the external force, the object moves from its equilibrium position to its maximum ends and the periodic motion continues till it comes back to the equilibrium position to rest.

An ideal spring may undergo compression and expansion, so the force applied will be in its opposite direction. The spring is generally considered to be ideal, which means masses in its equilibrium position and when it is made to stretch, there will be a displacement from its equilibrium. So the stretched distance's value will be negative.

## Hooke's Law

Hooke's law states that the force required to stretch the spring will be directly proportional to the amount of stretch or the displacement of the string from its equilibrium. The mathematical equation that explains Hookes' law is

F= k·x

Here F is the force applied to stretch the spring, k is the constant known as spring constant and x is the displacement. The law can also be stated as F is the restoring force that the spring exerts to come back to its equilibrium position. In that case, we have a negative sign :

F= -kx

## The Equation Formulation of the Work done by the Spring

The work done in stretching the aspring from equilibrium to a position x, gets stored in the spring as spring potential energy. It can be derived as below, using the force equation from Hooke’s Law,

$\begin{array}{c}W=\int F.dx\\ =\underset{0}{\overset{x}{\int }}kx.dx\\ =k\underset{0}{\overset{x}{\int }}x.dx\\ =k.\frac{{x}^{2}}{2}\\ =\frac{1}{2}k{x}^{2}\end{array}$

Thus the spring potential energy is Us = ½ kx2 .(Let the spring potential energy be Us). Now if the graph is plotted to understand the force and displacement relation, they both are perpendicular to each other and the spring constant will be at 45 degrees to both the components, hence in the graph, it forms like a right angle triangle. So the base of the right angle triangle is the product of displacement and spring constant and the height is also the displacement. If x is the displacement and k is the spring constant, then by calculating the equation it comes to  Us = 1/2·k·x2

Hence the work done by the spring will be equal to the difference between the final and initial potential energies. So it can be denoted by DU. The potential energy of the spring will be zero when it is in the equilibrium position. When the spring is stretched the potential energy increases. Then at maximum potential energy points, its kinetic energy would be zero.

NOTE: The value of the spring's potential energy cannot be negative irrespective of the restoring spring force and its negative sign.

## Application of Spring's Potential Energy

1) The spring potential energy is used in various applications in plenty of mechanical devices. The feature of the compression and expansion of the spring adds an advantage to it. The springs are used in plenty of automobiles as a shock absorber and in regenerative braking systems.

2) The potential energy of the springs is also used in control system application and its capacity to store makes it more efficient.

3) It is also used in manufacturing of certain guns where due to the compression of the spring and its ability to store the elastic potential energy, when the gun's trigger is pulled, then these springs apply a force and hence there is recoil.

4) The real life application of elasticity is also seen in using the rubber band, pole vault poles and in trampolines.

## Practice Problems

1) If a spring extends by x on loading, then calculate the  energy stored in the spring. Let T be the spring force and K is a force constant

a) 2x/T2  b) T2/2k c)2k/T2  d) T2/2x

Answer: When the spring extends a restoring force is developed in it, which depends on the deflection in the spring. The magnitude of this force is given as

$T=kx$

The energy stored in the spring is given as

$E=\frac{1}{2}k{x}^{2}$

Rearrange the equation of the force and substitute the expression of the deflection in the formula of the energy to calculate its value.

$\begin{array}{l}E=\frac{1}{2}k{\left(\frac{T}{k}\right)}^{2}\\ E=\frac{{T}^{2}}{2K}\end{array}$

The correct option is (b).

2) Determine the potential energy of a spring having a spring constant as 200 Nm-1 and the displacement is 0.8 m

Answer: The formula to calculate the potential energy of the given spring is

$E=\frac{1}{2}k{x}^{2}$

The given values of the spring constant and the deflection are 200 Nm-1 and 0.8 m respectively. Substitute the values to calculate potential energy.

The energy is 64 Joules.

3) An engineer attaches a spring to a sterdy wooden board. Let the board be placed horizontally on a table. The spring uses an energy value of three joules to stretch and the displacement moved by the spring is 99 cm. What will be the value of the spring constant using the equation of the spring's potential energy ?

Answer: The formula to calculate the potential energy of the given spring is

$E=\frac{1}{2}k{x}^{2}$

The displacement and energy in the spring is given as 99 cm and 3 Joules. Substitute the values in the formula and calculate the value of the spring’s constant.

This is the given value of the spring constant.

4) A person wants to do an experiment with the spring and analyze the potential energy of the spring. So the person attaches one end of the spring to a metallic board and the other end is free, so the spring can be expanded or compressed. The person pulls the string having a spring constant of 100 Netwon per meter and stretches it from a distance of 0.20 meters to 0.40 meters. Determine the value of the spring's potential energy

The formula to calculate the potential energy of the given spring is

$E=\frac{1}{2}k{\left({x}_{f}-{x}_{i}\right)}^{2}$

It is given that the value of the spring constant is 100 Nm-1, final displacement is 0.40 m and initial displacement is 0.20 m. Substitute the values to calculate the value of the energy of the spring.

The energy is 2 J.

## Context and Applications

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

• Bachelors in Science (Physics)
• Masters in Science (Physics)

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text:
Anachor's blog

By Anachor, history, 3 years ago,

Suppose you have a set of lattice points each of whose coordinates lie in the interval [0, N]. I've heard that the number of points on the convex hull is O(N2 / 3). However, I cannot seem to be able to prove this. How can this be proven?

Also, how to generate testcases such that the convex hull contains as many points as possible?

• +73

 » 3 years ago, # |   -15 I might have misunderstood, but if we take the 4 corner points (and have the remaining N-4 points inside), then their convex hull is a square with O(N) lattice points on every edge...
•  » » 3 years ago, # ^ |   +8 No, I mean the number of vertices of the convex hull. So, a square would only have four vertices.
 » 3 years ago, # |   +14 A heuristic approach: take all the irreducible lattice vectors in [0, a]2 for some a, and sort them by the angle. Now construct the lower right envelope by taking prefix sums of the sorted list of those vectors. Expected gain in both x and y coordinate per point is Θ(a), and the number of vectors should be Θ(a2). If you plug in , the last prefix sum is (Θ(N), Θ(N)), and the lower right envelope contains points.As for the other direction, it seems hard. I couldn't find a coherent proof for the 2-dimensional case on the Internet.
•  » » 3 years ago, # ^ | ← Rev. 2 →   +8 Nice, this allows us to construct testcases with O(N2 / 3) points on the hull.
 » 3 years ago, # | ← Rev. 2 →   +23 There's a paper in German ("Über die Gitterpunke auf konvexen Kurven" by Vojtěch Jarník) that proves a Θ(l2 / 3) bound for convex C1 curves F of length l with 0 ≤ F' ≤ 1.If we have a convex polygon, we can draw a convex C1 curve through all its vertices and apply his result to each octant. (The length of the curve is O(N).) This gives a O(N2 / 3) bound for convex polygons.Edit: The paper also talks about convex polygons. One result is that the maximum number of vertices a convex polygon with integer coordinates and circumference x can have is
•  » » 3 years ago, # ^ |   +16 Nice to see a paper by Jarník cited, as someone whose head managed to collide with his books.
 » 3 years ago, # | ← Rev. 2 →   +57 A simplified proof for the fact that a convex lattice polygon with coordinates in [0, N] has at most O(N2 / 3) vertices:Let (x0, y0), ..., (xn, yn) be the vertices with (x0, y0) = (xn, yn). Let (ai, bi) = (xi + 1 - xi, yi + 1 - yi) be the i-th line segment. W.l.o.g. let (a0, b0), ..., (ak, bk) be the line segments with 0 ≤ ai ≤ bi (the ones in one octant). We'll only look at these segments. Note that so there are at most N2 / 3 indices i with bi ≥ N1 / 3. Let's call those segments long and the other ones short. Let m + 1 be the number of short segments and (asj, bsj) be the j-th short segment. Then so On the other hand, by restriction on the octant and convexity of the polygon, the sequence is sorted, so using ai ≤ bi we get so m ≤ 2N2 / 3 hence k ≤ 3N2 / 3. Summing up over all 8 octants, we get an upper bound of 24N2 / 3 for the number of vertices.
•  » » 3 years ago, # ^ |   +2 Just FYI, in your first summation you probably want yk + 1 - yk.
•  » » » 3 years ago, # ^ |   0 I assume you meant replacing bk by bi and made the same error in your comment. That's correct. Thanks.