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Year : 2012 | Volume : 14 | Issue : 56 | Page : 13--20
Validity of hearing impairment calculation methods for prediction of self-reported hearing handicap
Andrew B John1, Brian M Kreisman2, Stephen Pallett3,
1 Department of Communication Sciences and Disorders, University of Oklahoma Health Sciences Center, Oklahoma City, Oklahoma, USA
2 Department of Audiology, Speech-Language Pathology and Deaf Studies, Towson University, Towson, Maryland, USA
3 Department of Audiology, Speech-Language Pathology and Deaf Studies, Towson University, Towson, Maryland; ENTAA Care, Annapolis, Maryland, USA
Andrew B John
1200 N. Stonewall Ave., Room 3080, University of Oklahoma Health Sciences Center, Oklahoma City, OK, 73117
Worker's compensation for hearing loss caused by occupational noise exposure is calculated by varying methods, from state to state within the United States (US), with many employing arithmetic formulas based on the pure-tone audiogram, to quantify hearing loss. Several assumptions unsupported or weakly supported by empirical data underlie these formulas. The present study evaluated the ability of various arithmetic hearing impairment calculations to predict a self-reported hearing handicap in a sample of presenting with sensorineural hearing loss. 204 adults (127 male, 77 female) ranging in age from 18 to 94 served as participants. The sample was selected to exclude patients who had been referred for hearing testing for a medicolegal examination or a hearing conservation appointment. A hearing handicap was measured by the Hearing Handicap Inventory for Adults/for the Elderly (HHIA/E). The covariance analysis of linear structural equations was used to assess the relative strength of correlation with the HHIA/E score among the six formulas and various forms of pure-tone average. The results revealed that all the hearing impairment calculations examined were significantly, but weakly, correlated with the self-reported hearing impairment scores. No significant differences among the predictive abilities of the impairment calculations were evident; however, the average binaural impairment assigned differed significantly among the six calculations examined. Individuals who demonstrated 0% impairment had significantly lower (i.e., better) HHIA/E scores compared to those with non-zero impairment for each formula. These results supported the idea that audiometric data provided an insufficient explanation for real-world hearing difficulties.
John AB, Kreisman BM, Pallett S. Validity of hearing impairment calculation methods for prediction of self-reported hearing handicap.Noise Health 2012;14:13-20
John AB, Kreisman BM, Pallett S. Validity of hearing impairment calculation methods for prediction of self-reported hearing handicap. Noise Health [serial online] 2012 [cited 2023 Feb 1 ];14:13-20
Available from: https://www.noiseandhealth.org/text.asp?2012/14/56/13/93321
Over the last several decades, a number of formulas have been proposed within the US to quantify adult-onset hearing loss, using audiometric data. These include calculations developed by the American Medical Association, Council on Physical Medicine; [1] the American Academy of Ophthalmology, and Otolaryngology, Committee on Conservation of Hearing; [2] the National Institute for Occupational Safety and Health; [3] the National Academy of the Science Committee on Hearing, Bioacoustics, and Biomechanics; [4] the American Medical Association/American Academy of Otolaryngology, Committee on Hearing and Equilibrium; [5] and the American Speech-Language-Hearing Association, Task Force, on the definition of a hearing handicap, duplicated by the 1997 formula revision by the National Institute for Occupational Safety and Health. [6],[7] These formulas vary in three main dimensions: (1) audiometric frequencies included in the calculation; (2) low/high fence, from which percentage-impairment-per-dB calculation is derived; and (3) better ear/worse ear weighting for the calculation of binaural impairment, or %BI [Table 1].
High fence refers to the amount of hearing loss in the decibel hearing level (dB HL) that constitutes a maximum (100%) monaural impairment for everyday speech understanding; additional losses in dB HL above the high fence do not accrue an additional handicap. Conversely, low fence refers to the maximum amount of hearing loss in dB HL that is considered to contribute no handicap (0%) to the understanding of speech. All thresholds below the low fence are considered to contribute a 0% handicap. Binaural impairment (%BI) refers to the total percentage handicap to speech understanding across both ears realized by an individual with a measurable hearing loss. All formulas evaluated here calculate %BI using unequal weighting of the two monaural impairment figures calculated for an individual, with a bias (most commonly five to one) toward the better ear. To be precise, the better ear monaural impairment is multiplied by five, added to the worse ear impairment, and the sum is divided by six to produce %BI. Notably, while a five-to-one weighting is common among hearing impairment calculations, there is no research basis for this particular proportion. [8]
The motivation for development of mathematical calculations of workplace hearing loss in the US dates back to the early twentieth century. In 1908, the Federal Employees' Compensation Act (FECA) provided America's first federally-mandated worker's insurance for occupational injury or disease. Among the conditions considered by the FECA to be a workplace injury was traumatic loss of hearing, such as tympanic membrane perforation following an explosion or head injury. [9] However, for the next 40 years, non-traumatic hearing loss due to workplace noise exposure was not compensable under United State's worker's compensation laws. In 1948, the New York Supreme Court ruled in Slawinski versus J.H. Williams and Co. that partial loss of hearing was eligible for compensation even if no wage loss could be demonstrated. That ruling led to the development of occupational hearing loss statutes, to assist in the determination of the direct effect of a hearing loss during workplace production. Later rulings by courts in Wisconsin, Maryland, and Georgia, as well as the Federal Judiciary, during the 1950s and 1960s, refined and upheld the concept of hearing loss as a compensable occupational injury even without demonstrable wage loss. [8]
Currently, all fifty states in the US and the federal government compensate workers for occupational hearing loss, either through specific hearing loss compensation statutes or the general worker's compensation law. Methods for calculating hearing impairment for the purpose of worker's compensation vary from state to state. According to recent data from the United States Department of Labor's Office on Worker's Compensation Programs, four different arithmetic calculations of hearing impairment are currently in use by state worker's compensation offices [Table 1]. In addition, 22 states report the primary use of medical evidence as a determinant of award. The state of Montana currently does not have a statute in place specifically covering compensation for hearing loss, instead it treats it under statutes covering the general occupational injury or disease. [10]
Audiometrically-derived measures of hearing impairment such as those used for Worker's Compensation awards are likely to be insensitive to individual variability in the impact of hearing loss on daily functioning. Indeed, self-report measures of hearing handicap consistently show low correlations with audiometric sensitivity. [11],[12],[13],[14],[15] The relation between hearing sensitivity and a self-reported hearing handicap is believed to be influenced by a number of non-auditory factors, including, age, gender, personality, and socioeconomic status. [16],[17],[18],[19],[20],[21] For example, Lutman and colleagues [16] noted that age, gender, and socioeconomic status are factors in hearing self-report, with males reporting more disability than females and older individuals reporting less disability than younger individuals. In another study, Lutman [17] concluded that older individuals tend to overrate disability, consistent with the findings of Gordon-Salant and colleagues, [19] who demonstrated that increased age correlated with increased disability as reported on the Hearing Handicap Inventory for the Elderly (HHIE). Gatehouse [21] also found substantial variability in the correlation between self-reported hearing handicap and hearing loss, with significant effects of age and personality.
Furthermore, several studies have noted that the specific methodology used in most audiometrically-based measures of hearing impairment limit their correlation to speech recognition. In one such investigation, Suter [22] evaluated the AAOO-1959 calculation and found that the failure of that formula to include thresholds above 2000 Hz, made it insensitive to sloping hearing losses that caused significant deterioration of speech understanding, both in quiet and with noise. This is not surprising given the critical importance of the high-frequency spectrum, in the recognition of speech sounds. [23],[24]
A subjectively-reported hearing handicap, measured using instruments such as the Hearing Handicap Inventory for Adults (HHIA; [25] ) or for the Elderly (HHIE; [26] ), may help to determine the social and communicative impact of hearing loss on daily functioning. However, the use of subjective measures of hearing impairment is clearly contraindicated for patients in cases with financial stakes, such as worker's compensation claims. Even as audiometric sensitivity measures have little face validity for the explanation of everyday listening performance and difficulty, they contain elements critical to medicolegal evaluation: They are objective, reproducible, and if performed correctly, resistant to exaggeration and opinion. [27]
Schow and Gatehouse [28] have suggested that objective hearing impairment measures may be validated by studies comparing the self-report of hearing handicap in individuals who are not seeking financial or other compensation. In one such study, Brainerd and Frankel [11] evaluated the correlation among six hearing impairment calculations, the Denver Scale of Communication Function, and the English version of the Social Hearing Handicap Index. The authors determined that no audiometrically-derived formula was a strong predictor of self-reported hearing function. In fact, the better ear pure-tone average (500, 1 k, and 2 kHz) was a stronger predictor of self-reported handicap than any formula examined. In a second study, Stewart and colleagues [29] evaluated four audiometric calculations for hearing impairment, as predictors of performance, on the screening version of the HHIA in a group of firearm users. The correlations ranged from r=.55 to r=.66, with the highest correlation seen in the formulas omitting the 500 Hz threshold as a formula element.
Few studies have found evidence for any of the several arithmetic hearing loss calculations in current or recent use in the US, as an effective measure of real-world hearing difficulty. More significantly, a literature review was unable to identify any study that has used appropriate statistical methods to evaluate the relative strength of association between these hearing impairment calculations and self-report measures. State-to-state inconsistency in the technique used to quantify compensable hearing loss suggests the need for an evidence basis in an impairment calculation. In particular there is a need to determine whether the differences in formula predictive power, in the studies cited herewith and others, are statistically significant. If so, this might suggest a real benefit for the use of one formula (or more) over others currently in use, for appropriately evaluating hearing loss as a limitation on daily activity. With these considerations in mind, we have examined the relative predictive value of seven published hearing impairment formulas for a commonly-employed self-report of hearing difficulty, the Hearing Handicap Inventory, in a compensation-ineligible clinical population.
Data were collected from adult patients seen at the speech and hearing clinics of the University of Florida, Towson University, and the ENTAA Care of Annapolis, Maryland, between 1997 and 2009. All participants were diagnosed with adult-onset sensorineural hearing loss (SNHL) and were administered the Hearing Handicap Inventory for Adults (HHIA) or Hearing Handicap Inventory for the Elderly (HHIE) at the diagnosis appointment. Patients presenting with conductive or fluctuating hearing loss were excluded from the analysis. Consistent with the recommendations of Schow and Gatehouse, [28] patients who had been referred for hearing testing for a medicolegal examination or hearing conservation appointment were not recruited. In order to minimize the varying effect of amplification use on the HHIA/E score, individuals using hearing aids or other assistive listening devices at the time of testing were excluded. A total of 204 participants (127 male, 77 female) met the criteria for inclusion. Subjects ranged from 18 to 94 years in age, with a mean of 65.6 years (SD=13.8). See [Figure 1] for a summary audiogram of this sample. For the purpose of analysis, thresholds marked as 'no response' were coded at 5 dB above the limits of the test equipment (typically 100 - 115 dBHL).{Figure 1}
Hearing impairment calculations
Six different arithmetic calculations of hearing impairment were included in our analysis. Four were selected on the basis of the current or recent (since 2006) use in worker's compensation award calculation in the US. These four calculations were designated AAOO-1959, CHABA-1975, AMA-1979, and Oregon, referring to a fourth formula, unique to the Oregon statutes. [10] In addition, we evaluated two versions of a hearing impairment calculation developed by NIOSH: NIOSH-1972 and NIOSH-1997 [Table 1]. In light of the findings of Brainerd and Frankel, [11] on the strength of the pure tone average (PTA), as a predictor of hearing impairment, we also evaluated the simple averages of monaural hearing thresholds at (1) 500, 1 k, and 2 kHz, (2) 1 k, 2 k, and 4 kHz, (3) 500, 1 k, 2 k, and 3 kHz, and (4) 500, 1 k, 2 k, and 4 kHz. Both better-ear and worse-ear PTA were included in the analysis.{Table 1}
Self-reported hearing impairment
All participants were administered the HHIA or HHIE at the appointment during which the hearing thresholds were obtained. The Hearing Handicap Inventories were the widely-used clinical measures of communicative function, containing a series of questions that attempted to quantify the social and emotional impact of hearing loss on daily communication and functioning. The HHIA and HHIE are similar, differing in three questions (out of 25) to reflect situations appropriate to individuals under age 65 (HHIA) or 65 and older (HHIE). For the purpose of analysis, we collapsed the scores across the two versions of the HHI.
A repeated-measures ANOVA was conducted, to determine whether the significant differences in the average percent binaural impairment (%BI) produced by the six calculations were evident. Post hoc paired comparisons were conducted using a Bonferroni correction for paired samples, to determine which calculations differed significantly in the assigned %BI. The Pearson product moment correlations were calculated between the HHIA/E score and %BI, calculated using the six formulas mentioned earlier, and the four variations of a pure-tone average, in the worse and better ears. As the hearing impairment calculations were used principally in the worker's compensation claims, these analyses were repeated for the subgroup of participants under age 65 (n=91).
A comparison of the strength of paired correlations among each of the impairment formulas and the HHIA/E score was conducted using a family-wise covariance analysis of linear structural equations (CALIS) in an SAS software, as described by Cheung and Chan. [30] The CALIS technique allowed for significance testing among the bivariate correlations between multiple dependent predictor variables (%BI values for the six impairment calculations, derived from the same audiometric data) and the one criterion variable (HHI score). Post hoc testing was conducted using a Holm correction. [31]
Average %BI as well as the maximum and minimum %BI for the six hearing impairment calculations can be seen in [Table 2]. Results of a repeated-measures ANOVA for the effect of calculation suggested that %BI was significantly different among the calculations, F(5,1015)=358.75, P<.001. After applying a Bonferroni correction (.05/6=.008), all paired differences among %BI of the six calculations were significant (P<.001). That is, the six calculations could be ranked discretely from the greatest to the least average %BI as follows: NIOSH-1997 (40.3%), Oregon, NIOSH-1972, AMA-1979, CHABA-1975, AAOO-1959 (15.8%) [Table 2].{Table 2}
A repeated-measures ANOVA for the effect of calculation on the subgroup of participants under age 65 also found a significant difference, F(5, 205)=86.26, P<.001. After a Bonferroni correction, all paired differences were significant (P<.001) except for the comparison between AAOO-1959 and CHABA-1975 (P=.03). For this subgroup, the six calculations could be ranked from the greatest to the least average %BI, as follows: NIOSH-1997 (35.7%), Oregon, NIOSH-1972, AMA-1979, CHABA-1975 (17.5%)/AAOO-1959 (15.7%).
Scores on the HHIA/E showed considerable variance, ranging from zero to 96 with a mean of 37.1 and standard deviation of 23.1. No significant effect of age (P=.19) or gender (P=.89) was seen for the HHIA/E score. No significant difference in inventory score was seen between subjects who were administered the HHIE (age 65+, n=113, mean=35.7, SD=23.0) and those subjects who were administered the HHIA (age 18-64, n=91, mean=38.8, SD=23.2, P=.963).
Pearson product moment correlations between the HHIA/E score and %BI, calculated by using the six formulas mentioned earlier and the four variations of a pure-tone average in the worse and better ears can be seen in [Table 2]. These coefficients were denoted as rHHI for each impairment calculation. Correlation coefficients within the whole sample ranged from .462 to .502 for the impairment formulas and from .480 to .521 for the pure-tone average variations. All correlations were highly significant (P<.001). These findings indicate that the impairment calculations predicted approximately 21% to 25% of the variance in the HHI score, while various forms of the PTA in the better and worse ear predicted approximately 23% to 27% of the variance.
Correlation coefficients for HHIA/E score and %BI in the subgroup of participants under age 65 ranged from .492 to .551 for the impairment formulas and from .548 to .596 for the pure-tone average variations. All correlations were highly significant (P<.001). These findings indicated that, for the subgroup of participants under age 65, the impairment calculations predicted approximately 24 to 30% of the variance in the HHI score, while various forms of the PTA in the better and worse ear predicted approximately 30 to 36% of the variance.
Next, a comparison of the strength of the paired correlations among each of the impairment formulas and HHIA/E score was conducted using CALIS, as described earlier. After applying a Holm correction for multiple comparisons to the P values generated, no significant differences among the predictive abilities of the impairment calculations were evident.
Finally, the likelihood that each of the six impairment calculations would return a compensable (non-zero) impairment for the individuals in the sample under study was calculated. Individuals were most likely to demonstrate impairment greater than zero under the Oregon calculation (185/204 impaired, or 90.7% in the full sample, 74/91 impaired or 81.3% in the subsample under age 65), and least likely under the AAOO-1959 calculation (147/204 impaired, or 72.1% in the full sample; 62/91 impaired or 68.1% in the subsample). As expected, for each of the six impairment calculations, individuals who demonstrated 0% impairment had significantly lower HHIA/E scores (P<.001) compared to those with nonzero impairment in both the full sample and subsample under age 65 [Table 3].{Table 3}
Large differences in calculated %BI were evident among many individuals in the sample, particularly those with steeply sloping hearing losses above 2000-3000 Hz. Calculated impairment and HHIA/E score from the selected individuals with widely varying %BI, depending on the calculation employed, are shown in [Table 4]. These data are presented to illustrate the high variance in %BI within individuals as well as the wide range in HHIA/E score across individuals with similar %BI.{Table 4}
The present study evaluated differences among the relations between six arithmetic calculations of hearing impairment and the score on a common self-report measure of hearing, the HHIA or HHIE. All the impairment calculations examined were derived from the audiometric pure-tone threshold data, but differed in the audiometric frequencies included in the calculation, low/high fence, from which the percentage-impairment-per-dB calculation was derived, as also better ear/worse ear weighting. The goals of the analysis were, (1) to determine statistically whether one or more of the calculations was significantly better in predicting self-reported hearing difficulty than others, and (2) to evaluate whether these calculations differed in the likelihood of assigning non-zero impairment to an individual with SNHL and the percentage of impairment assigned.
The results of this analysis suggest that self-reported hearing handicap as measured by the HHIA and HHIE can be estimated by calculations based on pure-tone thresholds. However, the statistical relations found were weak overall, ranging from r=.46 to r=.50 for the six impairment calculations and from r=.48 to r=.52 for forms of simple monaural pure-tone average. In the subsample of participants under age 65, these values ranged from r=.49 to r=.55 for the six impairment calculations and from r=.55 to r=.60 for forms of simple monaural pure-tone average. The strongest predictor of a handicap was the worse-ear standard PTA (500, 1000, and 2000 Hz), which had 27.1% shared variance with the HHIA/E score, in the full sample and the better-ear standard PTA, which had 35.5% shared variance with HHIA/E score, in the subsample. Our results agreed in part with Brainerd and Frankel's finding that better-ear PTA was a stronger predictor of the standard summarized handicap (r=.38) than was any impairment calculation studied (1985). It should be noted that Brainerd and Frankel did not include isolated worse-ear PTAs in their analysis. The current findings suggest that greater attention to worse-ear status may improve the understanding of the handicap in individuals with asymmetric hearing impairment.
Of the calculations currently in used in the US for Worker's Compensation claims, the strongest predictive value was seen for the AMA-1979 formula (r=.50 in the full sample and r=.55 in the subsample). The weakest predictor was the CHABA-1975 formula (r=.46 in the full sample and r=.49 in the subsample). Correlations were significant for all the formulas examined, with these impairment calculations accounting for between 21% and 25% of the variance in HHIA/E score in the full sample and between 24 and 30% of the variance in the subsample. These correlation coefficients were lower than those reported by Stewart and colleagues, [29] who reported that four audiometric formulas were more highly correlated with the HHIA screener score (r=.55 to .66) between four audiometric formulas and the screening HHIA. However, the present data are more similar to values reported by Stewart in a subset of only those individuals with a non-zero hearing handicap. For this set, the r values for the HHIA screener varied between .45 and .59 among the formulas examined. The finding that impairment of the subgroup of younger participants was better predicted by the calculations studied than the sample as a whole can likely be attributed to the greater degree of high-frequency hearing loss (to which the formulas are insensitive) resulting from presbyacusis in participants over 65.
A family-wise comparison of the strength of the correlations among the impairment formulas and HHI score found no significant differences among the calculations, suggesting that all six calculations performed equally well (or equally poorly) in predicting the HHIA/E score. This finding was the most compelling of the present study. Previously, investigators had conducted informal comparisons of these impairment calculations, but the fact that the calculations were structurally correlated (incorporating similar or identical elements) made comparisons of the strength of correlation problematic. The CALIS technique described here accounts for these structural similarities and allows for a comparison of correlation strength with the HHIA/E score among the calculations under study.
Large differences were evident in both the number of individuals in the sample classified as impaired by the six calculations and in the degree of impairment assigned by the calculations. In particular, individuals with steeply-sloping hearing losses above 2000 - 3000 Hz demonstrated much greater impairment as calculated by the NIOSH-1997 formula, which incorporated thresholds at 3000 and 4000 Hz, and the Oregon formula, which incorporated thresholds at 3000, 4000, and 6000 Hz.
Several conclusions can be drawn from these data. First, the four formulas currently in use in Worker's Compensation statutes in the US have performed equally well. No strong evidence can be found in the present analysis to support the use of one of these impairment calculations over another. Notably, each of the calculations examined predicted the HHIA/E score almost as well as the simple pure-tone average in the individual's better ear.
However, a review of the average %BI for each of the six formulas [Table 2] showed that while these formulas may have performed equally well in predicting the handicap, they differed greatly in the binaural impairment percentage assigned. The most 'conservative' of these formulas was the AAOO-1959 formula, which found an average of 15.8% binaural impairment in the full sample and 15.7% in the subsample. The NIOSH-1997 formula, by contrast, produced an average of 40.3% impairment in the full sample and 35.7% in the subsample. Although the mean %BI differed significantly among each of the six calculations examined, the NIOSH-1997 formula produced considerably higher impairment than the other five, for most of the participant samples. Two reasons for this finding were evident. First, the NIOSH-1997 formula was one of the two (along with Oregon) to consider thresholds above 3000 Hz when calculating an impairment, increasing the sensitivity of the calculation to the high-frequency-biased hearing losses consistent with noise exposure and presbycusis. Second, and probably more importantly, the NIOSH-1997 formula employed a 75 dB HL 'high fence' compared to a 92 dB HL fence for the other five formulas. This had the effect of increasing the percentage impairment per dB to 2% (compared to 1.5-1.8) and to assign 100% impairment to any individual ear with an average threshold of above 75 dB HL in the range of 1000 - 4000 Hz.
The six impairment calculations examined here also differed considerably in the number of subjects in the sample who were assigned a compensable hearing impairment (%BI>0). Of the 204 participants, 147 (72.1%) would be considered as having a hearing impairment under the AAOO-1959 calculation, 148 (72.5%) under CHABA-1975, 172 (84.3%) under NIOSH-1972, 163 (79.9%) under AMA-1979, 181 (88.7%) under NIOSH-1997, and 185 (90.7%) under the Oregon state calculation. In the subsample of 91 adults under age 65, 62 (68.1%) would be considered to have a hearing impairment under the AAOO-1959 calculation, 62 (68.1%) under CHABA-1975, 67 (73.6%) under AMA-1979, 70 (76.9%) under NIOSH-1972, 73 (80.2%) under NIOSH-1997, and 74 (81.3%) under the Oregon state calculation. By contrast, Brainerd and Frankel (1985) found that 27% of their sample was impaired under the AAOO-1959 calculation, 30% under AMA-1979, 18% under CHABA-1975, and 40% under NIOSH-1997, suggesting that the present sample had significantly greater hearing loss overall than in the previous study.
Many of these between-formula differences were substantial at the individual level as well. [Table 4] displays data from selected individuals with widely varying impairment depending on the calculation employed. Also notable here is the wide variance in the HHIA/E score among the individuals in [Table 4], illustrating the low correlation between arithmetically-derived impairment and self-reported hearing handicap. Of the six calculations examined, only the ASHA-1981 formula produced a 100% binaural impairment score, and did so for eleven individuals in the sample (5.3%). In the other six calculations, binaural impairment scores for those individuals ranged from 27.5 to 96.7%.
Ward [32] noted that each of the hearing impairment formulas in current use, in the US, is predicated on a series of arbitrary assumptions; among them are,
that some combination of threshold values at specific frequencies are related monotonically to the intelligibility of (some form of) speech in (some form of) noisethat the rate of change in impairment is linear, and equals 1.5, 1.8 or 2% per dB, and rises between the absolute cut-offs of 25 or 35 dB at the low end and 75 or 92 dB at the high endthat each frequency employed is equally important, such that a loss above the low fence at one frequency can be offset by a threshold below the low fence at another frequencythat better-ear thresholds contribute precisely five or seven times as much to overall impairment as do worse-ear thresholds.
There is little empirical data to support these assumptions. The US state and federal organizations who promulgate the calculations resulting from them offer little or no evidence basis for the use of the six formulas considered in the present study. Some criteria appear to have a pragmatic basis. For example, the apparently arbitrary choice of a 92-dB HL high fence for maximum impairment is probably a result of desire for an easy by-hand calculation of impairment (1.5% per dB) of above 25 dB HL, a reasonable low fence for compensable hearing loss. The authors are currently conducting a study to statistically evaluate the assumptions identified by Ward, and also to study whether the predictive value of impairment calculations for hearing and communication difficulty might be improved by alterations in the formula structure.
It should be noted that the hearing impairment formulas examined here are used primarily to quantify hearing loss in individuals exposed to occupational noise rather than presbyacusis, the predominant etiology of hearing loss for at least some of our sample. Noise exposure alone results in a characteristic pattern of binaural hearing loss with a significant 'notch' around 4000 - 6000 Hz. The sample used in this study includes individuals with a wide variation in the degree and configuration of hearing loss. It is possible that a sample more representative of the noise-exposed workforce might demonstrate different relations between audiometric sensitivity and self-reported hearing handicap. In particular, a more homogenous sample with this characteristic noise notch might produce different results when analyzing the role of mid-high frequency thresholds in the prediction of the hearing handicap score.
However, the aim of this investigation was to make comparisons between audiometrically-derived impairment and a self-reported handicap, in a population with no motivation to exaggerate responses on the self-report inventory. Data collection using these methods with noise-exposed workers is difficult to accomplish without introducing some bias. Participant files have been selected to exclude individuals seen for hearing conservation audiograms, Worker's Compensation evaluations, or any other medicolegal appointment. Not surprisingly, this process has resulted in an older sample whose hearing loss was not limited to noise exposure. Analysis of the subsample of participants under age 65 did reveal somewhat stronger correlations between the impairment calculations and the HHIA/E score than the sample at large; however, these correlations were still low.
It is likely that the low correlation between the objective and subjective measures of the hearing handicap seen in this study and others is related to non-auditory factors. Several studies have noted that the relation between hearing sensitivity and self-reported handicap is related to individual characteristics including age, gender, and personality, [16],[17],[18],[19],[20],[21] which may have affected the results of the present study. Considering the substantial literature demonstrating that hearing handicap by self-report is influenced by factors outside of the purview of the audiologist or physician, it is likely that any audiometrically-derived calculations would have only a limited correlation with the scores on an instrument such as the HHIA or HHIE.
It should also be noted that the present study is limited to calculations used to estimate the hearing handicap primarily in the US. Methods for quantifying hearing loss as a compensable injury vary substantially across nations, although many do employ arithmetic formulas similar to those described here. Even as the present study is limited to a sample of American adults and compares across calculations presently or recently in use within the US, examination of differences across nations may be warranted.
The principal finding of the present study is that arithmetic hearing impairment calculations currently in use predict self-reported hearing loss in a population of individuals with SNHL significantly, but weakly. No formula was able to account for more than about 25% of the variance in the HHIA/E score in the full sample or 30% in the younger subsample, and none was a more effective predictor than a simple three-frequency, pure-tone average in the better or worse ear. A CALIS technique revealed no significant differences in the correlations between the HHIA/E score and the six impairment calculations. Further research into the theoretical basis of hearing impairment calculations is needed, as few empirical data are available to support the methods employed for the allocation of impairment and for financial awards for hearing loss suffered as a result of workplace noise exposure.
The authors acknowledge the assistance of Valerie Skaggs, PhD, for her assistance in the statistical analysis. Portions of this article were presented at the Seventeenth Annual American Academy of Audiology Convention, Washington, DC, April, 2005, and at the New Zealand Audiological Society Conference, Christchurch, NZ, June, 2005.
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21 Gatehouse S. Speech tests as measures of outcome. Scand Audiol Suppl 1998;49:54-60.
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text:
\section{Introduction}

The lightest Dark Matter (DM) particle ought to be stable
(\textit{i.e.}\ unable to decay),
or at least it should have a lifetime of order the age of the Universe.
If it is stable,
then there is an unbroken cyclic $\mathbbm{Z}_N$ symmetry which is non-trivial
(\textit{i.e.}\ it has $N \ge 2$),
such that standard matter is invariant under $\mathbbm{Z}_N$ while DM is not;
the $\mathbbm{Z}_N$ charge different from $1$ of the lightest DM particle
prevents it from decaying to standard matter,
which has $\mathbbm{Z}_N$ charge $1$.

The most usual option in Model Building is $N = 2$.
However,
some authors have considered possibilities $N > 2$.
For DM stabilized by a $\mathbbm{Z}_3$ symmetry,
see Ref.~\cite{z3}.
Larger cyclic groups have been used to stabilize DM,
like $\mathbbm{Z}_4$ and $\mathbbm{Z}_6$~\cite{z4z6},
$\mathbbm{Z}_5$~\cite{z5},
or a general $\mathbbm{Z}_N$~\cite{zn}.

The $\mathbbm{Z}_N$ that stabilizes DM may be the center of
a larger internal-symmetry group $G$.\footnote{The center of a group $G$
is its Abelian subgroup formed by the elements of $G$
that commute with all the elements of $G$.}
The simplest possibility consists in $G$ being a discrete group
of order\footnote{The order of a discrete group is the number
of its elements.}
$O$ that is isomorphic to the direct product $\mathbbm{Z}_N \times G^\prime$,
where $G^\prime$ is a group of order $O / N$.
In that case,
all the irreducible representations (`irreps') of $G$
consist of the product of an irrep of $\mathbbm{Z}_N$
(which is one-dimensional,
because $\mathbbm{Z}_N$ is Abelian and Abelian groups have one-dimensional irreps)
and an irrep of $G^\prime$;
standard matter must be placed
in the trivial representation\footnote{The trivial representation of a group
is the one where all the group elements are mapped onto the unit matrix.}
of $\mathbbm{Z}_N$ while DM is placed in non-trivial representations of $\mathbbm{Z}_N$.

However,
also discrete groups $G$ that cannot be written
as the direct product of a cyclic group and a smaller group
may have a non-trivial $\mathbbm{Z}_N$ center.
If that happens,
then once again an irrep of $G$ may represent $\mathbbm{Z}_N$ either trivially
or in non-trivial fashion
(\textit{viz.}\ when some elements of $\mathbbm{Z}_N$ are represented
by a phase $f$ with $f \neq 1$ but $f^N = 1$).
If $\mathbbm{Z}_N$ remains unbroken when $G$ is
(either softly or spontaneously)
broken,
and if there are particles with $\mathbbm{Z}_N$ charge different from $1$,
then those particles play the role of DM,
while the particles with $\mathbbm{Z}_N$ value $1$ are standard matter.

This possibility was recently called to our attention by Ref.~\cite{chulia},
where a group $G$ of order 81,
named $\Sigma(81)$,\footnote{The group $\Sigma(81)$ can \emph{not}
be written as a direct product $\mathbbm{Z}_3 \times G^\prime$,
$G^\prime$ being a group of order 27.
Rather,
$\Sigma (81)$
(which has {\tt SmallGroups} identifier $\left[ 81,\, 7 \right]$)
is of the form $\left( \mathbbm{Z}_3 \times \mathbbm{Z}_3 \times \mathbbm{Z}_3 \right) \rtimes \mathbbm{Z}_3$,
\textit{i.e.}\ it is a semi-direct product.}\footnote{The group $\Sigma(81)$
was used in Model Building by E.~Ma\cite{ma}.
See also Ref.~\cite{ishimori}.}
was used as internal symmetry of a model.
The authors of Ref.~\cite{chulia} rightly pointed out that
``[some] irreducible representations [of $\Sigma(81)$]
form a closed set under tensor products,
implying that if every Standard Model field transforms as
[one of those representations],
then any field transforming as
[a representation that is not in that closed set]
will belong to the dark sector.
The lightest among them will then be a dark matter candidate.''

As a matter of fact,
this mechanism had already been suggested before,
\textit{viz.}\ in Ref.~\cite{valle}.
There,
it was noted that some discrete subgroups of $SU(2)$
have subsets of irreps that are closed under tensor products,
and this fact might be used to stabilize DM.\footnote{In the course of
the present investigation we have found that this indeed happens
for \emph{all} the discrete subgroups of $SU(2)$.}

In this paper we make a survey of all the discrete groups
of order $O \le 2000$,
except groups with either $O = 512$,
$O = 1024$,
or $O = 1536$.
We select the groups that cannot be written as the direct product
of a non-trivial cyclic symmetry and a smaller group,
and that moreover have at least one \emph{faithful} irreducible representation
(`firrep').
We identify the center $\mathbbm{Z}_N$ of each of those groups,
and also the dimensions $D$ of their firreps.
We construct various tables with the integers $O$,
$N$,
and $D$.
We find that very many discrete groups,
especially those that are not subgroups of any continuous group $SU(D)$,
have centers $\mathbbm{Z}_N$ with $N \ge 2$,
and $N$ is sometimes quite large.

This paper is organized as follows.
In Sec.~2 we explain,
through the well-known cases of $SU(2)$ and $SU(3)$,
that some groups have a center $\mathbbm{Z}_N$ with $N \ge 2$
and some irreps of those groups represent $\mathbbm{Z}_N$ trivially
while other irreps do not.\footnote{Hurried readers may skip Section~2.}
In Sec.~3 we make a systematic survey of the centers
of all the discrete groups $G$ of order up to 2000
that cannot be written as the direct product
of a cyclic group and another group and that
have some faithful irreducible representation.\footnote{We do not survey
groups of order either~512,
1024,
or~1536,
because there are unpractically very many groups of those orders.}
In Sec.~4 we briefly state our conclusions.
As an Appendix to this paper,
comprehensive listings of the groups that we have studied
are available online at
\href{https://github.com/jurciukonis/GAP-group-search}{{\tt
    https://github.com/jurciukonis/GAP-group-search}}.

\section{$SU(2)$ and $SU(3)$}

\subsection{$SU(2)$}

The defining representation of $SU(2)$
consists of the $2 \times 2$ unitary matrices with determinant $1$.
One such matrix is
\begin{equation}
\label{a2}
A_2 = \left( \begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array} \right).
\end{equation}
This is proportional to the unit matrix
and therefore it commutes with all the $2 \times 2$ matrices;
in particular,
it commutes with all the matrices in the defining representation of $SU(2)$.
Hence,
in any irrep of $SU(2)$,
$A_2$ must be represented by a matrix that commutes
with all the matrices in that irrep.
Schur's first lemma~\cite{elliott} states that
any matrix that commutes with all the matrices in an irrep of a group
must be proportional to the unit matrix.
Therefore,
in the $D$-dimensional irrep of $SU(2)$,\footnote{As is well known,
$SU(2)$ has one and only one $D$-dimensional irrep for each integer $D$.}
$A_2$ must be represented by a multiple of the $D \times D$ unit matrix
$\mathbbm{1}_D$.
But,
as is clear from Eq.~\eqref{a2},
$\left( A_2 \right)^2 = \mathbbm{1}_2$
is the unit element of $SU(2)$ in the defining representation.
This property must be reproduced in the $D$-dimensional irrep of $SU(2)$.
One hence concludes that,
in that irrep,
\begin{equation}
A_2 \mapsto \left( - 1 \right)^{q_D} \times \mathbbm{1}_D,
\end{equation}
where $q_D$ is an integer that is either 0 or 1 modulo 2.
The integer $q_D$ depends on the irrep.

The irreps of $SU(2)$ wherein $A_2$ is mapped onto the unit matrix,
\textit{i.e.}\ the ones for which $q_D$ is $0$ modulo 2,
are unfaithful.\footnote{An unfaithful representation of a group
represents two or more distinct elements by the same matrix.}
Those are the integer-spin irreps.
They have odd $D$ and are faithful irreps of the quotient group
\begin{equation}
\label{so3}
SU(2) \left/ \ \mathbbm{Z}_2 \right. \cong SO(3).
\end{equation}
In Eq.~\eqref{so3},
\begin{equation}
\label{z2}
\mathbbm{Z}_2 = \left\{ \mathbbm{1}_2,\ A_2 \right\}
\end{equation}
is the center of $SU(2)$,
\textit{i.e.}\ it is the subset of $SU(2)$ elements
(in the defining representation)
that commute with \emph{all} the elements of $SU(2)$;
it is a $\mathbbm{Z}_2$ subgroup of $SU(2)$.

The $D$-dimensional irreps of $SU(2)$ with even $D$
are the half-integer-spin representations and represent $SU(2)$ faithfully,
\textit{viz.}\ they map $A_2 \mapsto - \mathbbm{1}_D$.

Let us consider the tensor product of the irreps of $SU(2)$
with dimensions $D_1$ and $D_2$.
Clearly,
$A_2 \mapsto \left( -1 \right)^{q_{D_1}} \times \mathbbm{1}_{D_1}$
in the irrep with dimension $D_1$
and $A_2 \mapsto \left( -1 \right)^{q_{D_2}} \times \mathbbm{1}_{D_2}$
in the irrep with dimension $D_2$.
In the product representation,
which is in general reducible,
\begin{equation}
\label{su2product}
A_2 \mapsto \left( -1 \right)^{q_{D_1} + q_{D_2}} \times \mathbbm{1}_{D_1 + D_2}.
\end{equation}
Therefore,
the subset of the irreps of $SU(2)$ that have $q_D = 0$ modulo 2
is closed under tensor products.
This property of the irreps of $SU(2)$
also holds for the irreps of discrete subgroups of $SU(2)$.
If one such subgroup contains $A_2$ in its defining representation,
then a $D$-dimensional irrep of that subgroup
must represent $A_2$ either by $\mathbbm{1}_D$ or by $- \mathbbm{1}_D$,
and the subset of irreps that represent $A_2$ by unit matrices
is closed under tensor products.
It was suggested in Ref.~\cite{valle} that
this property may be used to stabilize DM.
In that suggestion,
Nature possesses an internal symmetry under a discrete subgroup of $SU(2)$
that contains in its defining irrep the matrix $A_2$;
standard matter sits in a (in general, reducible) representation
of that internal symmetry where $A_2$ is mapped onto the unit matrix,
while DM is in a representation of the internal symmetry
in which $A_2$ is mapped onto minus the unit matrix.
Then,
any collection of standard-matter particles will be invariant
under the transformation represented in the defining representation by $A_2$,
which implies that the lightest DM particle,
which changes sign under that transformation,
is stable.
Dark matter is stabilized by the $\mathbbm{Z}_2$ symmetry in Eq.~\eqref{z2};
that $\mathbbm{Z}_2$ symmetry is the center of the internal-symmetry group of the model.

\paragraph{Example:} The quaternion group $Q_8$
is the order-eight subgroup of $SU(2)$ formed,
in its defining two-dimensional irrep,
by the matrices\footnote{In Eq.~\eqref{quat} and below,
we separate the classes of each group through semicolons.}
\begin{subequations}
\label{quat}
\begin{eqnarray}
& &
A = \left( \begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right),
\qquad
A^3 = \left( \begin{array}{cc} -i & 0 \\ 0 & i \end{array} \right);
\\
& &
B = \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right),
\qquad
B^3 = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right);
\\
& &
A B = \left( \begin{array}{cc} 0 & i \\ i & 0 \end{array} \right),
\qquad
B A = \left( \begin{array}{cc} 0 & -i \\ -i & 0 \end{array} \right);
\\
& &
A_2 = A^2 = B^2 = \left( \begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array} \right);
\qquad
\left( A_2 \right)^2 = A^4 = B^4 =
\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right).
\label{ud}
\end{eqnarray}
\end{subequations}
The center of this group consists of the $\mathbbm{Z}_2$ in Eq.~\eqref{z2},
\textit{cf.}\ line~\eqref{ud};
yet,
$Q_8$ is not the direct product of that $\mathbbm{Z}_2$
and any order-four group.
The quaternion group has five irreps:
the two-dimensional one in Eqs.~\eqref{quat}
and the four one-dimensional ones
\begin{equation}
\mathbf{1}_{rs}: \quad A \mapsto r,\ B \mapsto s,
\end{equation}
where both $r$ and $s$ may be either $1$ or $-1$.
Clearly,
in all the one-dimensional irreps $A_2 = A^2 = B^2$
is mapped onto $1$,
while in the two-dimensional irrep $A_2$ is mapped onto $- \mathbbm{1}_2$.
In Model Building,
standard matter might sit in singlet irreps of $Q_8$
while DM would be placed in doublets of $Q_8$.

\subsection{$SU(3)$}

The defining representation of $SU(3)$
consists of the $3 \times 3$ unitary matrices with determinant $1$
and includes the matrix
\begin{equation}
\label{a3}
A_3 = \left( \begin{array}{ccc}
  \omega & 0 & 0 \\ 0 & \omega & 0 \\ 0 & 0 & \omega \end{array} \right)
= \omega \times \mathbbm{1}_3,
\end{equation}
where $\omega = \exp{\left( 2 i \pi / 3 \right)}$.
The Abelian group
\begin{equation}
\label{z3}
\mathbbm{Z}_3 = \left\{ \mathbbm{1}_3,\ A_3,\ \left( A_3 \right)^2 \right\}.
\end{equation}
forms the center of $SU(3)$ in the defining representation.
The matrix $A_3$ commutes with all the matrices in the defining representation
of $SU(3)$ and satisfies $\left( A_3 \right)^3 = \mathbbm{1}_3$.
Therefore,
in a $D$-dimensional irrep of $SU(3)$
\begin{equation}
\label{qvalue}
A_3 \mapsto \omega^{q_D} \times \mathbbm{1}_D,
\end{equation}
where $q_D$ is an integer that depends on the irrep and may be either $0$,
$1$,
or $2$ modulo $3$.\footnote{The value of $q_D$
is the `triality' of the irrep~\cite{khanna}.}
Irreps with $q_D = 0$
(like the octet and the decaplet)
have $A_3$ represented by $\mathbbm{1}_D$ and are unfaithful
representations of $SU(3)$.
Irreps with either $q_D = 1$
(like the triplet)
or $q_D = 2$
(like the sextet and the anti-triplet)
are faithful.
Clearly,
if $A_3 \mapsto \omega^{q_{D_1}} \times \mathbbm{1}_{D_1}$
in an irrep with dimension $D_1$
and $A_3 \mapsto \omega^{q_{D_2}} \times \mathbbm{1}_{D_2}$
in an irrep with dimension $D_2$,
then in the product representation
\begin{equation}
\label{su3product}
A_3 \mapsto \omega^{q_{D_1} + q_{D_2}} \times \mathbbm{1}_{D_1 + D_2}.
\end{equation}
Therefore,
the irreps with $q_D = 0$ form a closed set under tensor products.
There is a selection rule in tensor products of irreps of $SU(3)$,
similar to the selection rule in tensor products of irreps of $SU(2)$,
but with the group $\mathbbm{Z}_3$ of Eq.~\eqref{z3} in $SU(3)$
instead of the group $\mathbbm{Z}_2$ of Eq.~\eqref{z2} in $SU(2)$.

This also holds for many---but not all---the discrete subgroups of $SU(3)$.
The three matrices in Eq.~\eqref{z3}
may all belong to the defining representation
of a discrete subgroup of $SU(3)$;
when that happens,
a $D$-dimensional irrep of that subgroup possesses a $q_D$-value,
defined by Eq.~\eqref{qvalue}.
The $q_D$-values help determine the tensor products of irreps of the subgroup.
This may be used to explain the stability of DM:
if Nature had an internal symmetry that was a discrete subgroup of $SU(3)$,
that contained the matrix $A_3$ in its defining representation,
and that stayed unbroken,
then standard matter would sit in irreps of that subgroup with $q_D = 0$
while DM would be in irreps with either $q_D = 1$ or $q_D = 2$;
the lightest DM particle would then automatically be stable.

\paragraph{Example:} The group $A_4$ is the order-12 subgroup of $SU(3)$
formed,
in its defining representation,
by the matrices
\begin{subequations}
\begin{eqnarray}
& &
B = \left( \begin{array}{ccc}
  1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array} \right),
\quad
A^2 B A = \left( \begin{array}{ccc}
  -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{array} \right),
\quad
A B A^2 = \left( \begin{array}{ccc}
  -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{array} \right);
\hspace*{7mm} \\
& &
A = \left( \begin{array}{ccc}
  0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} \right),
\quad
A B = \left( \begin{array}{ccc}
  0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0 \end{array} \right),
\nonumber \\
& &
B A = \left( \begin{array}{ccc}
  0 & 1 & 0 \\ 0 & 0 & -1 \\ -1 & 0 & 0 \end{array} \right),
\quad
B A B = \left( \begin{array}{ccc}
  0 & -1 & 0 \\ 0 & 0 & 1 \\ -1 & 0 & 0 \end{array} \right);
\\
& &
A^2 = \left( \begin{array}{ccc}
  0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right),
\quad
A^2 B = \left( \begin{array}{ccc}
  0 & 0 & -1 \\ 1 & 0 & 0 \\ 0 & -1 & 0 \end{array} \right),
\nonumber \\
& &
B A^2 = \left( \begin{array}{ccc}
  0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0 \end{array} \right),
\quad
A B A = \left( \begin{array}{ccc}
  0 & 0 & -1 \\ -1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right);
\quad
A^3 = B^2 = \left( \begin{array}{ccc}
  1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right).
\end{eqnarray}
\end{subequations}
Neither the matrix $A_3$ nor $\left( A_3 \right)^2$
belong to the defining representation of $A_4$;
the center of $A_4$ is trivial,
\textit{i.e.}\ it is formed just by the unit element.
The four irreps $\mathbf{3}$,
$\mathbf{1}$,
$\mathbf{1}^\prime$,
and $\mathbf{1}^{\prime \prime}$ of $A_4$ do not have any selection rule
in their tensor products.
Thus,
the group $A_4$ is of no use to explain the stability of DM.

\section{Group search}

\subsection{Motivation}

The defining representation of $SU(D)$ consists of
the $D \times D$ unitary matrices with determinant $1$.
It is obvious that,
in this representation,
the center of $SU(D)$ is formed by the $D$ diagonal matrices
\begin{equation}
\label{4jvio}
\Delta \times \mathbbm{1}_D, \
\Delta^2 \times \mathbbm{1}_D, \
\Delta^3 \times \mathbbm{1}_D, \
\ldots, \ \Delta^D \times \mathbbm{1}_D = \mathbbm{1}_D,
\end{equation}
where $\Delta = \exp{\left( 2 i \pi / D \right)}$.
Thus,
the center of $SU(D)$ is a $\mathbbm{Z}_D$ group.
Any discrete group that has a firrep
formed by matrices that belong to $SU(D)$
may contain in that representation either
\begin{description}
  %
\item all the matrices in Eq.~\eqref{4jvio},
  %
\item only the last one of them,
  %
\item or---if $D$ is not a prime number and may be divided
  by an integer $m$ different from both $1$ and $D$---the
  $m^\mathrm{th}, 2 m^\mathrm{th}, \ldots, D^\mathrm{th}$
  matrices in Eq.~\eqref{4jvio}.
  %
\end{description}
In general,
if $m$ is an integer that divides $D$
and $\mu = \exp{\left( 2 i \pi / m \right)}$,
then there is a cyclic symmetry $\mathbbm{Z}_m$ given,
in the defining representation of $SU(D)$,
by
\begin{equation}
\mathbbm{Z}_m = \left\{ \mu \times \mathbbm{1}_D,\
\mu^2 \times \mathbbm{1}_D,\
\mu^3 \times \mathbbm{1}_D,\ \ldots,\
\mu^m \times \mathbbm{1}_D = \mathbbm{1}_D \right\}.
\end{equation}
One or more discrete subgroups of $SU(D)$ may then have $\mathbbm{Z}_m$ as their center.

Thus,
discrete subgroups of $SU(D)$
that are not subgroups of any $U(D^\prime)$ with $D^\prime < D$
may have very few centers.
For instance,
a discrete subgroup of $SU(10)$
that is not a subgroup of any $U(D^\prime)$ with $D^\prime < 10$
may only have center $\mathbbm{Z}_2$,
$\mathbbm{Z}_5$,
$\mathbbm{Z}_{10}$,
or the trivial group;
and a discrete subgroup of $SU(11)$
that is not a subgroup of any $U(D^\prime)$ with $D^\prime < 11$
may only have center $\mathbbm{Z}_{11}$ or the trivial group.

On the other hand,
discrete subgroups of $U(D)$ do not bear the constraint that
the determinants of the matrices in their defining representations
should be $1$.
As a consequence,
if
\begin{equation}
\mathbbm{Z}_t = \left\{ \theta \times \mathbbm{1}_D,\
\theta^2 \times \mathbbm{1}_D,\
\theta^3 \times \mathbbm{1}_D,\ \ldots,\
\theta^t \times \mathbbm{1}_D = \mathbbm{1}_D \right\},
\end{equation}
where $\theta = \exp{\left( 2 i \pi / t \right)}$,
is the center of a discrete subgroup of $U(D)$,
then there appears to be \textit{a priori}\/ no restriction on $t$.

\paragraph{Example:} The discrete group $\mathbbm{Z}_8 \rtimes \mathbbm{Z}_2$ has order 16
and {\tt SmallGroups}\ identifier $\left[ 16,\, 6 \right]$.
In its defining representation it is formed by the matrices
\begin{subequations}
\label{v8r90}
\begin{eqnarray}
& &
\left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right),
\quad
\left( \begin{array}{cc} -1 & 0 \\ 0 & 1 \end{array} \right);
\quad
\left( \begin{array}{cc} i & 0 \\ 0 & -i \end{array} \right),
\quad
\left( \begin{array}{cc} -i & 0 \\ 0 & i \end{array} \right);
\\
& &
\left( \begin{array}{cc} 0 & -1 \\ i & 0 \end{array} \right),
\quad
\left( \begin{array}{cc} 0 & 1 \\ -i & 0 \end{array} \right);
\quad
\left( \begin{array}{cc} 0 & i \\ 1 & 0 \end{array} \right),
\quad
\left( \begin{array}{cc} 0 & -i \\ -1 & 0 \end{array} \right);
\\
& &
\left( \begin{array}{cc} 0 & i \\ -1 & 0 \end{array} \right),
\quad
\left( \begin{array}{cc} 0 & -i \\ 1 & 0 \end{array} \right);
\quad
\left( \begin{array}{cc} 0 & -1 \\ -i & 0 \end{array} \right),
\quad
\left( \begin{array}{cc} 0 & 1 \\ i & 0 \end{array} \right);
\\
& &
\left( \begin{array}{cc} i & 0 \\ 0 & i \end{array} \right);
\quad
\left( \begin{array}{cc} -i & 0 \\ 0 & -i \end{array} \right);
\quad
\left( \begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array} \right);
\quad
\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right).
\label{8tg9u}
\end{eqnarray}
\end{subequations}
This is the firrep $\mathbf{2}$ of $\mathbbm{Z}_8 \rtimes \mathbbm{Z}_2$.
The other inequivalent irreps of that group are the $\mathbf{2}^\ast$
(wherein each matrix of the $\mathbf{2}$
is mapped onto its complex-conjugate matrix)
and eight inequivalent unfaithful singlet irreps.
Most of the $2 \times 2$ unitary matrices~\eqref{v8r90}
do not have determinant $1$;
therefore,
$\mathbbm{Z}_8 \rtimes \mathbbm{Z}_2$ is a subgroup of $U(2)$ but not of $SU(2)$.
One sees in line~\eqref{8tg9u} that the center of $\mathbbm{Z}_8 \rtimes \mathbbm{Z}_2$ is
\begin{equation}
\mathbbm{Z}_4 = \left\{
\left( \begin{array}{cc} i & 0 \\ 0 & i \end{array} \right),\
\left( \begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array} \right),\
\left( \begin{array}{cc} -i & 0 \\ 0 & -i \end{array} \right),\
\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) \right\}.
\end{equation}
Thus,
while discrete subgroups of $SU(2)$ may have center either $\mathbbm{Z}_1$ or $\mathbbm{Z}_2$,
discrete subgroups of $U(2)$ enjoy further possibilities,
for instance $\mathbbm{Z}_4$.

Motivated by this observation that discrete subgroups of $U(D)$
may in general have diverse centers,
in our work we have surveyed many discrete groups
in order to find out their centers
and also which groups $U(D)$ they are subgroups of.

\subsection{{\tt GAP} and {\tt SmallGroups}}

{\tt GAP}~\cite{gap} is a computer algebra
that provides a programming language and
includes many functions that implement various algebraic algorithms.
It is supplemented by libraries
containing a large amount of data on algebraic objects.
With {\tt GAP} it is possible to study groups and their representations,
to display the character tables,
to find the subgroups of larger groups,
to identify groups given through their generating matrices,
and so on.

{\tt GAP} allows access to the {\tt SmallGroups} library~\cite{sg}.
This library contains all the finite groups of order less than 2001,
but for order 1024---because there are many thousands of millions of groups
of order 1024.
{\tt SmallGroups} also contains some groups for some specific orders
larger than 2000.
In {\tt SmallGroups} the groups are ordered by their orders;
for each order,
the complete list of nonisomorphic groups is given.
Each discrete group of order smaller than $2001$
is labeled $\left[ O,\, n \right]$ by {\tt SmallGroups},
where $O \le 2000$ is the order of the group
and $n \in \mathbb{N}$ is an integer that distinguishes
among the non-isomorphic groups of the same order.

\subsection{Procedure}

We have surveyed all the discrete groups of order $O \le 2000$
in the {\tt SmallGroups} library,
except the groups of order either 512,
1024,
or 1536.\footnote{Rather exceptionally,
we have included in our search four groups
of order 1536 that are known to have three-dimensional firreps,
according to our previous paper~\cite{jurciukonis}.}
We have discarded all the groups that are isomorphic to the direct product
of a smaller
(\textit{i.e.}\ of lower order) group and a cyclic group.\footnote{{\tt SmallGroups}
itself informs us about the structure of each group,
\textit{viz.}\ whether it is isomorphic to the direct product of smaller groups.
We have found that there are,
however,
at least two exceptions.
One of them is the group with {\tt SmallGroups} identifier
$\left[ 180,\, 19 \right]$;
{\tt SmallGroups} informs us that
this is the group $\mathrm{GL} \left( 2,\, 4 \right)$
but omits the well-known fact that $\mathrm{GL} \left( 2,\, 4 \right)$
is isomorphic to $\mathbbm{Z}_3 \times A_5$,
where $A_5$ is the group of the even permutations of five objects.
(Thus,
$\left[ 180,\, 19 \right]$ is discarded in our search,
because it is the direct product of $A_5$
and the cyclic group $\mathbbm{Z}_3$.)
The other exception are the dihedral groups $D_O$ of order $O = 12 + 8 p$,
where $p$ is an integer,
\textit{viz.}\ the groups $D_{12}$,
$D_{20}$,
$D_{28}$,
and so on.
({\tt SmallGroups} instead uses the notation $D_{O/2}$ for these groups,
\textit{viz.}\ it uses $D_6$,
$D_{10}$,
$D_{14}$,
and so on.)
It is easy to check analytically that these specific $D_O$ groups
are isomorphic to $\mathbbm{Z}_2 \times D_{O/2}$,
but {\tt SmallGroups} omits this fact.
We have used a method,
suggested to us by Horvath G\'abor,
to check whether \emph{any} group $\left[ O,\, n \right]$
is a direct product of smaller groups,
namely the succession of {\tt GAP} commands
{\tt G := SmallGroup(O, n)}
and {\tt ListX(DirectFactorsOfGroup(G), StructureDescription)}.}
We have used {\tt GAP} to find out all the irreps of each remaining group,
and then to ascertain whether those irreps are faithful or not.
We have discarded all the groups
that do not have any firrep.\footnote{There are groups
for which all the faithful representations are reducible.
An example is the group formed by the 32 matrices
\begin{equation}
\left( \begin{array}{cccc}
  a & 0 & 0 & 0 \\ 0 & b & 0 & 0 \\ 0 & 0 & c & 0 \\ 0 & 0 & 0 & d
\end{array} \right)
\quad \mbox{and}\, \quad
\left( \begin{array}{cccc}
  0 & a & 0 & 0 \\ b & 0 & 0 & 0 \\ 0 & 0 & 0 & c \\ 0 & 0 & d & 0
\end{array} \right),
\label{uit3}
\end{equation}
where $a$,
$b$,
$c$,
and $d$ may be either $1$ or $-1$.
This group---with {\tt SmallGroups}\ identifier $\left[ 32,\, 27 \right]$
and structure $\left(
\mathbbm{Z}_2 \times \mathbbm{Z}_2 \times \mathbbm{Z}_2 \times \mathbbm{Z}_2 \right) \rtimes \mathbbm{Z}_2$---has
eight inequivalent singlet irreps and six inequivalent doublet irreps,
but all of them are unfaithful.
The defining representation in Eq.~\eqref{uit3} is,
by definition,
faithful,
but it is reducible.}
We have thus obtained 87,349 non-isomorphic groups,
that are all listed in our tables available at the site
\href{https://github.com/jurciukonis/GAP-group-search}{{\tt
    https://github.com/jurciukonis/GAP-group-search}}.
We have looked only at the firreps of each group;
non-faithful irreps,
and all reducible representations,
were neglected.
We have computed the determinants of the matrices of each firrep
in order to find out whether all those determinants are $1$ or not.
We have also looked for all the matrices in the firreps
that are proportional to the unit matrix
and we have checked that those matrices form a group $\mathbbm{Z}_N$
for some integer $N$
(which for some groups is just $1$).

We have tried to answer the questions
on how the following three integers are related:
\begin{enumerate}
  %
\item The integer $O$ that is the order of the discrete group $G$.
  %
\item The integer $N$ corresponding to the group $\mathbbm{Z}_N$
  that is the center of $G$.
  %
\item The integer $D$ such that $G$ has one or more firreps of dimension $D$.
  %
\end{enumerate}
We have also examined the question whether each $D$-dimensional firrep
is equivalent to a representation through matrices
of $SU(D)$.\footnote{All the irreps of discrete groups are equivalent
to representations through unitary matrices,
and therefore we know that all the generators that {\tt GAP} provides to us
are equivalent to unitary generators,
even though {\tt GAP} often gives them in non-unitary version.
In order to know whether the generators belong to $SU(D)$
we just compute their determinants.}

There are relatively few groups that have firreps
with different dimensions.
(For instance:
$A_5$ has firreps of dimensions three,
four,
and five.
On the other hand,
the group $\Sigma \left( 36 \times 3 \right)$,
that has {\tt SmallGroups}\ identifier $\left[108,\, 15 \right]$,
has irreps of dimensions $1$,
$3$,
and $4$,
but the $1$- and $4$-dimensional irreps are unfaithful---all the firreps
have dimension $3$.)
We have found just 2787 such discrete groups,
out of the total 87,349 groups that we have surveyed;
they are collected in table Intersections at
\href{https://github.com/jurciukonis/GAP-group-search}{{\tt
    https://github.com/jurciukonis/GAP-group-search}}.

\paragraph{Computing time:}
The scan over the {\tt SmallGroups}\ library
to find the firreps of all possible dimensions
constituted a computationally very expensive task.
Our computations with {\tt GAP} took about three months.
It is difficult to estimate the total number of CPU hours (CPUH) spent
in the computations,
because various computers with different CPUs were used.
Most of the time was consumed in the computation of the irreps of the groups.
For example,
the computation for group $\left[ 1320,\, 15 \right]$,
\textit{viz.}\ $\mathrm{SL} \left( 2, 11 \right)$,
took about 320~CPUH running on Intel Xeon~CPU~@~1.60GHz
or about 46~CPUH in the newer Intel i9-10850K~CPU @~3.60GHz.
Also,
some groups of orders 1728 and 1920 require quite a few CPUH
to find the irreps.
Orders 768,
1280,
and 1792 have more than one million non-isomorphic groups of each order
and therefore require many CPUH to scan over all of them.

\paragraph{Example:} The discrete group $\mathrm{GL} \left( 2,\, 3 \right)$
has order 48 and {\tt SmallGroups} identifier $\left[ 48,\, 29 \right]$.
By definition,
it is the group generated by three transformations $a$,
$c$,
and $d$ that satisfy~\cite{internet}
\begin{subequations}
\begin{eqnarray}
& & a^4 = c^3 = d^2 = \left( c d \right)^2 = e, \label{1} \\
& & b^2 = a^2, \\
& & b^3 = d a d, \\
& & b a b^{-1} = d b d = a^{-1}, \\
& & b = a^{-1} c a c^{-1}, \label{5}
\end{eqnarray}
\end{subequations}
where $e$ is the identity transformation and $b \equiv c^{-1} a c$.
There is a faithful representation of these transformations
through $2 \times 2$ unitary matrices:
\begin{equation}
\label{jgf999}
a \mapsto \frac{1}{3} \left( \begin{array}{cc} i \sqrt{3} & \sqrt{6} \omega \\
  - \sqrt{6} \omega^2 & - i \sqrt{3}
\end{array} \right),
\quad
c \mapsto \left( \begin{array}{cc} \omega & 0 \\ 0 & \omega^2
\end{array} \right),
\quad
d \mapsto \left( \begin{array}{cc} 0 & 1 \\ 1 & 0
\end{array} \right).
\end{equation}
The first two matrices~\eqref{jgf999} have determinant $1$
while the third one has determinant $-1$;
hence,
we classify $\mathrm{GL} \left( 2,\, 3 \right)$ as a subgroup of $U(2)$,
but it is not a subgroup of $SU(2)$.
On the other hand,
there is another faithful irrep of $\mathrm{GL} \left( 2,\, 3 \right)$,
through $4 \times 4$ unitary matrices,
all of them with determinant $1$:
\begin{equation}
a \mapsto \frac{1}{9} \left( \begin{array}{cccc}
  - 3 \sqrt{3} i & 0 & 6 i & - 3 \sqrt{2} \\
  0 & 3 \sqrt{3} i & 3 \sqrt{2} & - 6 i \\
  6 i & - 3 \sqrt{2} & i \sqrt{3} & - 2 \sqrt{6} \\
  3 \sqrt{2} & - 6 i & 2 \sqrt{6} & - i \sqrt{3}
\end{array} \right),
\quad
d \mapsto \left( \begin{array}{cccc}
  0 & - \omega^2 & 0 & 0 \\ - \omega & 0 & 0 & 0 \\
  0 & 0 & 0 & - 1 \\ 0 & 0 & - 1 & 0
\end{array} \right),
\end{equation}
and $c \mapsto \mathrm{diag} \left( \omega,\ \omega^2,\ 1,\ 1 \right)$.
Therefore,
we classify $\mathrm{GL} \left( 2,\, 3 \right)$
as a subgroup of both $U(2)$ and $SU(4)$,
but $\mathrm{GL} \left( 2,\, 3 \right)$ earns these two classifications
\emph{through different irreps}.

\subsection{The discrete subgroups of $U(3)$ and $SU(3)$}

In Ref.~\cite{jurciukonis}
a classification of the discrete subgroups of $SU(3)$
and of the discrete subgroups of $U(3)$ that are not subgroups of $SU(3)$
has been provided.
All those subgroups were classified
according to their generators and structures.
In this subsection we give relations between the integer $N$
characterizing the center $\mathbbm{Z}_N$ of each group
and the identifiers of the group series defined in Ref.~\cite{jurciukonis}.

As explained in subsections~2.2 and~3.1,
the finite subgroups of $SU(3)$
can only have either trivial center or center $\mathbbm{Z}_3$;
thus,
they have either $N = 1$ or $N = 3$.
Explicitly,
they have the following values of $N$:
\begin{itemize}
  %
\item The groups $\Delta \left( 3 n^2 \right)$,
  $\Delta \left( 6 n^2 \right)$,
  and $C_{rn,n}^{(k)}$ have $N = 1$ when $n$ cannot be divided by three,
  and $N = 3$ when $n$ is a multiple of three.
\item The groups $D_{3l,l}^{(1)}$ have $N =3$.
\item The exceptional groups
  $\Sigma \left( 60 \right)$ and $\Sigma \left( 168 \right)$ have $N = 1$.
\item The exceptional groups $\Sigma \left( 36 \times 3 \right)$,
  $\Sigma \left( 72 \times 3 \right)$,
  $\Sigma \left( 216 \times 3 \right)$,
  and $\Sigma \left( 360 \times 3 \right)$ have $N = 3$.
  %
\end{itemize}
 %

The series of finite subgroups of $U(3)$ that are not subgroups of $SU(3)$
constructed in Ref.~\cite{jurciukonis} have the following values of $N$:
\begin{itemize}
  %
\item The groups $Y \left( m, j \right)$,
  $L \left( m \right)$,
  $J \left( m \right)$,
  and $\Theta \left( m \right)$ have $N = 3^m$.
  %
\item The groups $T_r^{(k)} \left( m \right)$,
  $\Delta \left( 3 n^2, m \right)$,
  $L_r^{(k)} \left( n, m \right)$,
  $P^{(k)}_r \left( m \right)$,
  $Q^{(k)}_r \left( m \right)$,
  $Q^{(k)\prime}_r \left( m \right)$,
  $X \left( m \right)$,
  $S_r^{(k)} \left( m \right)$,
  $S_r^{(k)\prime} \left( m \right)$,
  $Y_r^{(k)} \left( m \right)$,
  $V_r^{(k)} \left( m \right)$,
  $W \left( n, m \right)$,
  $Z \left( n, m \right)$,
  $Z^\prime \left( n, m \right)$,
  $Z^{\prime \prime} \left( n, m \right)$,
  $\Upsilon \left( m \right)$,
  $\Upsilon^\prime \left( m \right)$,
  and $\Omega \left( m \right)$ have $N = 3^{m-1}$.
  %
\item The groups $M_r^{(k)}$,
  $M_r^{(k)\prime}$,
  $J_r^{(k)}$,
  $Y \left( j \right)$,
  $\tilde Y \left( j \right)$,
  $V \left( j \right)$,
  and $D \left( j \right)$ have $N = 3$.
  %
\item The groups $U \left( n, m, j \right)$ have $N = 3^{j}$.
  %
\item The groups $S_4 \left( j \right)$ have $N = 2^{j-1}$.
  %
\item The groups $\Delta \left( 6 n^2, j \right)$ have $N = 2^{j-1}$
  when $n$ cannot be divided by three,
  and $N = 3 \times 2^{j-1}$ when $n$ is a multiple of three.
  %
\item The groups $\Delta^\prime \left( 6 n^2, m, j \right)$,
  $H \left( n, m, j \right)$,
  $G \left( m, j \right)$,
  $\hat \Xi \left( m, j \right)$,
  and $\Pi \left( m, j \right)$ have $N = 3^{m} 2^{j-1}$.
  %
\item The groups $Z \left( n, m, j \right)$ and $Z' \left( n, m, j \right)$
  have $N = 3^{m-1} 2^{j-1}$.
  %
\item The groups $\Xi \left( m, j \right)$ have $N = 3^{m} 2^{j-2}$.
  %
\end{itemize}
In Ref.~\cite{jurciukonis} a few more subgroups of $U(3)$
that are not subgroups of $SU(3)$ are mentioned,
which could not be classified into any series.
Their values of $N$ are the following:
\begin{itemize}
  %
\item The groups $\left[ 729,\, 96 \right]$,
  $\left[ 729,\, 97 \right]$,
  $\left[ 729,\, 98 \right]$,
  $\left[ 1458,\, 663 \right]$,
  $\left[ 1458,\, 666 \right]$,
  $\left[ 1701,\, 130 \right]$,
  and $\left[ 1701,\, 131 \right]$ have $N = 3$.
  %
\item The group $\left[ 1296,\, 699 \right]$ has $N = 6$.
  %
\item The groups $\left[ 972,\, 170 \right]$,
  $\left[ 1701,\, 102 \right]$,
  and $\left[ 1701,\, 112 \right]$ have $N = 9$.
  %
\end{itemize}

\section{Conclusions}

In this paper we have pointed out that Dark Matter
may be stabilized by a $\mathbbm{Z}_N$ cyclic group
under which it has a non-trivial charge---contrary to standard matter,
which is invariant under that $\mathbbm{Z}_N$---and that that $\mathbbm{Z}_N$
may be the center of the larger internal symmetry group $G$ of Nature,
while $G$ is not a direct product $\mathbbm{Z}_N \times G^\prime$.
Thereafter we have performed an extensive
and computationally very time-consuming search for the centers
of discrete groups that cannot be written in the form $\mathbbm{Z}_N \times G^\prime$
and that have faithful irreducible representations.
The following are our conclusions:
\begin{itemize}
  %
\item We have found groups with centers $\mathbbm{Z}_N$ for $N \le 162$.
  %
\item We have found groups with $N = 2^p \times 3^q$
  for \emph{all} the integers $p$ and $q$ such that $N \le 162$.
  %
\item We have found groups with $N = 2^p \times 5$
  for $0 \le p \le 3$.
  %
\item We have also found groups with $N = 7$,
  $N = 11$,
  $N = 14$,
  $N = 15$,
  and $N = 25$.
  %
\item The number $N$ always divides the order $O$ of the group.
  The integer $O/N$ always has at least two prime factors;
  we have found groups with $O / N = 4$,
  $6$,
  $8$,
  $9$,
  $10$,
  $12$,
  $14$, and so on.
  %
\end{itemize}

In the cases of some smallish groups,
we have explicitly computed the way an element $g$
of order\footnote{The order of an element $g$
is the smallest integer $o$ such that $g^o$ is the unit element of the group.}
$N$ belonging to the $\mathbbm{Z}_N$ center of the group
is represented in the various irreps.
We have found that
the sum of the squares of the dimensions of the irreps
where $g$ is represented by any $N^\mathrm{th}$ root of unity
times the unit matrix is always equal to $O/N$.

\paragraph{Acknowledgements:} L.L.\ thanks Salvador Centelles Chuli\'a
for a discussion on his paper~\cite{chulia}.
We thank the {\tt GAP} support team,
specifically Bill Alombert,
Stefan Kohl,
and Horvath G\'abor for helping us solve a problem with that software.
D.J.\ thanks the Lithuanian Academy of Sciences
for financial support through project DaFi2021.
L.L.\ thanks the Portuguese Foundation for Science and Technology
for support through projects UIDB/00777/2020 and UIDP/00777/2020,
and also CERN/FIS-PAR/0004/2019,
CERN/FIS-PAR/0008/2019,
and CERN/FIS-PAR/0002/2021.

redpajama_set_name: RedPajamaArXiv


text:
Home Literature Book House On Mango Street
House On Mango Street
I Know Why The Caged Bird...
Nickel and Dimed
Drawing The Color Line
The Pearl
A Worn Path
Angela' Ashes
Young Goodman Brown
To Build A Fire
House On Mango Street Essay Prompts
Esperanza is torn between deciding whether she wants to escape MangoStreet.
She is embarrassed by the superficial appearance of her identity,but appreciates her roots. Her house is a wreck and the neighborhood,probably not much better off. However, she has loving family and friends. Although marriage has caused the suffering of many of the women in herneighborhood, she realizes that she needs men to fulfill the new desiresshe attains as she hits adolescence. Through the novel, Esperanza matures both physically and mentally.
Thefirst thing that struck me about this novel was that the chapters werevery short. I realized that the narrator is young and has a shortattention span, judging from her fragmented observations. However,Esperanza begins to mature and to develop a desire for men. While shesenses that many women are caged by men, they cannot be truly freewithout them. Most of the women Esperanza knows on Mango Street are either trapped intheir marriages or tied down by their children. For example,Esperanza's grandmother.
Esperanza does not want to "inherit her placeby the window. " She neither likes what she has already inherited fromher grandmother – her name. Esperanza plays with words when she firstexpresses her dissatisfaction with her name. She says that in Spanish,her name means "too many letters. It means sadness from the oppositeof esperar, which is desesperarse, it means waiting from the verbesperar. " She settles on changing her name to "Zeze the X".
AsEsperanza observes, the Mexicans and the Chinese do not want theirwomen to be strong like horses. Esperanza hopes for a different future. Although she likes to sleep near her mother's hair, the noveleventually reveals that she wants to escape Mango Street. Clearly,Esperanza's name suits her; she has hope. In House on Mango Street, Cisneros constantly reminds the reader not tojudge a book by its cover.
The idea of a dirty outside but appealinginside is prevalent at many levels – the neighborhood, the householdand the individual. Cathy, Esperanza's first friend in theneighborhood, tells Esperanza that her family is moving because "theneighborhood is getting bad", because of the many immigrants likeEsperanza's family beginning to move in. Cathy says that Lucy andRachel, who Esperanza eventually befriends, "smell like a broom. " Hermentioning her distant relation to the queen of France makes her seemvery pretentious. In reality, she is not much better off economicallyfrom the rest of the neighborhood. In her house, "The floors slant"(21).
"There are no closets" and the steps are "all lopsided andjutting like crooked teeth" (22). At the household level, Esperanza is ashamed by her house that has"crumbling bricks", "only one washroom" and "paint peeling" (4). However, in the second chapter, "Hairs", Esperanza writes about whatis inside the house on Mango Street: "her mother holding her",making her "feel safe" and the "warm smell of bread before you bakeit". Similarly, first impressions of individuals in the novel are based onexternal factors: race, gender, and perhaps their name.
Meme and hisdog each having two names highlights the neighborhood's two cultures:Hispanic and American, and two languages: Spanish and English. Esperanza points out that everyone in the neighborhood is "all brownall around", suggesting that what makes people feel safe is beingaround others who are of the same race. Esperanza is afraid to talk tothe owner, perhaps because "he is a black man". The second factor that causes Esperanza to experience difficulty indeciding whether she wants to leave Mango Street is her physically andmentally maturing simultaneously. She is old enough to realize thatthere is much in the world to explore other than one, her house andtwo, her neighborhood.
Esperanza points out that "you don't pick yoursisters" (8). She then proceeds to say that "someday she will have abest friend" who she "can tell her secrets to" (9) and "who willunderstand her jokes without her having to explain them. Until then,she is "a red balloon, a balloon tied to an anchor" (9). Later, sherealizes that Nenny, her sister, and her are "more alike than youwould know" (19).
In "Laughter", Esperanza points out that a house"looks like Mexico". Rachel and Lucy "look at her like she iscrazy", but Nenny was thinking the exact same thing. Her own sistersometimes understands Esperanza better than her friends do. Most of the wives in the neighborhood are discontent with theirmarriages.
While many want to escape their husbands, Marin issearching for one. Marin waits for "a car to stop, a star to fall,someone to change her life". Esperanza faces a dilemma: she cannothave relationships with men unless she is married (and marriage hascaused most of the problems the women on Mango Street face). She mustchoose between sex and freedom.
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House On Mango Street Essay Prompts. (2019, Jan 29). Retrieved from https://happyessays.com/house-on-mango-street-essay-prompts/
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redpajama_set_name: RedPajamaCommonCrawl


Surely the chance to ditch your servers and outsource to someone who is steeped in server management seems like a gift from the universe.
The arguments in favor of cloud computing are easy to make, especially to someone frustrated by the intellectual overhead and raw cost of maintaining their own servers.
The promises of the cloud include the following.
You pay only for what you use, so it's incredibly flexible; you can scale up or down at will.
Security, upgrading, and server configuration are in the hands of experts.
In these days of everything being "as-a-service," the idea of owning anything like a server seems downright old-fashioned. If Uber can run the world's largest taxi service without owning any taxis, why on earth would you need to own servers?
There are a few things you need to factor in to make sure you're comfortable with any potential compromises.
Power: Cloud providers can't match the power of a dedicated server that's properly configured.
Speed: The scalability of the cloud has to do with getting more or less storage, not faster storage, which might be a concern when another customer is flogging the server you're on.
The promise of cloud computing is most clearly seen in companies meeting one or more of the following criteria.
A company that has a business-to-business model or has well-established usage needs and predictable growth will likely find running its own servers cheaper and more efficient. This is something you can quickly run the numbers on, and the results might surprise you, considering that "cheaper" is a clarion call of the cloud industry.
It's also worth running the decision through the filter of security. Hackers fish where the fish are, which makes cloud hosts attractive targets. You're not just outsourcing server configuration and the like. You're trusting another company with your security. If security is a concern, you're probably better off keeping your servers in-house, where you can tailor security to your needs.


text:
Q: Is this function the Gamma function I have a function here that I'm not sure if it is the Gamma function, or perhaps the Gamma function by some periodic function. Or if I am just making a series of mistakes.
$$f(z) = \sum_{n=0}^\infty \frac{(-1)^n}{n!(z+n)} - \frac{1}{e} \sum_{n=0}^\infty \frac{1}{z(z+1)\cdots(z+n)}$$
We can show, really fast:
$$
\begin{eqnarray}
zf(z) &=& \sum_{n=0}^\infty \frac{z (-1)^n}{n!(z+n)} - \frac{1}{e} - \frac{1}{e}\sum_{n=0}^\infty \frac{1}{(z+1)(z+2)\cdots(z+n)}\\
&=&\sum_{n=0}^\infty (\frac{(-1)^n z}{n!(z+n)} - \frac{(-1)^n}{n!}) - \frac{1}{e}\sum_{n=0}^\infty \frac{1}{(z+1)(z+2)\cdots(z+n)}\\
&=&\sum_{n=0}^\infty \frac{(-1)^{n+1} n}{n!(z+n)} - \frac{1}{e}\sum_{n=0}^\infty \frac{1}{(z+1)(z+2)\cdots(z+n)}\\
&=& f(z+1)
\end{eqnarray}
$$
However, the residues at the poles come out to zero and $f(1)=0$ as well, unlike the Gamma function. Is it not the Gamma function or am I miscalculating something..? Because I think it should be the Gamma function, just not certain.
Any help on clarifying what this function may or may not be, or if you've seen this expression somewhere before, that would be amazing.
Thanks a lot.

redpajama_set_name: RedPajamaStackExchange


International students considered for the following awards must have a temporary resident visa as a member of the student class under the Immigration and Refugee Protection Act of Canada on the first day of classes.
All students who are required to pay the international differential fee including university transfers, second degree students, college diploma and college transfer students are eligible for the Entrance Scholarship.
Under no circumstances will international scholarships be deferred. Students must accept the terms and conditions of admission and scholarship awarded for the year the offer was made.
US residents are not eligible for the scholarship.
The Chancellor's Award is a scholarship valued at $5,000 annually for international students entering their first year of study at Algoma University. To qualify, candidates must be studying on a full-time basis (30 credits over two consecutive terms of study). Students must complete all their qualifying requirements in the academic year prior to commencing studies in any program of study at Algoma University.
Student must achieve a minimum A+ average (or country equivalent) on their previous academic record.
International students studying in an Ontario secondary school and meeting the OSSD requirements must achieve 90 – 100 per cent on their top 6 grade 12 U/M courses presented for admission.
This award is renewable for up to three additional years. Candidates must maintain an average of 80 per cent each year on previous fall/winter courses with no failures. Students must enrol in and maintain 24 credits over two consecutive fall/winter terms of study. In the event the student fails to meet the criteria in any given year, the scholarship will not be awarded and the student will no longer be eligible for renewal in any subsequent years. Recipients of the international scholarship transitioning to permanent residency will continue to be eligible for the balance of their renewable scholarship for a total of four years. Transitioning does not infringe upon the funding of the scholarship award.
Transfer students entering Algoma U during the winter term of any academic year will be eligible for one-half of the award (minimum 15 credits).
The President's Award is an entrance scholarship valued at $2,500 for international students entering their first year of study at Algoma University. To qualify, candidates must be studying on a full-time basis (24-30 credits over two consecutive terms of study). Registrations in course loads less than 30 credits but greater than or equal to 24 credits will be prorated. Students must complete all their qualifying requirements in the academic year prior to commencing studies in any program of study at Algoma University.
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'Life changes in the instant. The ordinary instant'
BOOKS FOR THE INDEPENDENT READER
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Marti Leimbach
A powerful, compelling, topical novel about a relationship between a radio DJ and a teenage girl from the author of DYING YOUNG and DANIEL ISN'T TALKING
Radio DJ Craig Kirtz took a shine to Bobbie when she was only thirteen years old, embroiling her in a secret life of sex and drugs.
He remembers their relationship as a love affair. She remembers it as abuse. Thirty years later, she
returns to her hometown to testify against him.
He denies the charges. And he has a witness Bobbie could not have anticipated – her own mother, who is now Craig's wife.
Reviews of Age of Consent
'An account of estrangement between mother and daughter and the toll abuse can take on a family … woven with pain and fractured relationships … The story will keep readers turning pages until the bitter end … Fans of Jodi Picoult may enjoy' Library Journal
'A nuanced portrayal of a mother and daughter at once linked and divided by a ferociously exploitative man … The novel brings memorable depth to issues often oversimplified' Publishers Weekly
'Leimbach is known for tackling tough subjects in an unflinching manner, and this novel is no exception … The descriptions of abuse don't come across as gratuitous or overdone … Readers who enjoy issue-driven women's fiction—and who can handle the dark subject matter—will be moved by Bobbie's story' Booklist
'A gripping if uncomfortable novel' Mail on Sunday
'An unflinching look at sexual abuse from an author who isn't afraid of difficult subjects … Leimbach offers a horribly believable depiction of a child ensnared by a predator. In giving a voice to Bobbie's mother as well as Bobbie, she foregoes the urge to simply blame a woman who failed to protect her daughter … Devastatingly powerful' Kirkus Reviews
'What makes Marti Leimbach's new novel so chilling is not only its tale of predator and prey, but the story of the mother and daughter, entangled in its web. It is so terrifying to see a mother in denial, a daughter who can't and won't forget. As Leimbach weaves her story, moving between past and present, this thrilling novel forces us to ask to whom do we owe our loyalties – to those we love, or to ourselves' Mary Morris
'Three people: a young girl, her mother, and the man who has seduced them both. Leimbach explores the ambiguities of loss, love, and desire, and plumbs the dark places where they meet. At the center of the story is an illicit, illegal affair, the consequences of which never really stop reverberating. Age of Consent is spellbinding' Whitney Otto
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#EAA – Accessibility: Making products and services in the EU easier to use
EU Reporter Correspondent | March 6, 2019
Parliament and EU Council negotiators have reached an agreement on a European Accessibility Act (EAA). The new rules are a step towards a fairer and more inclusive Europe and should improve the daily lives of the elderly and people with disabilities across the EU.
The agreement will still need to be approved by MEPs and the Council before the legislation can enter into force.
More accessible products and services
More than 80 million people live with disabilities in the EU and many have difficulties using everyday products, such as smartphones, computers, e-books, and encounter problems in accessing key services via ticket machines or ATMs.
The UN Convention on the Rights of Persons with Disabilities (UNCRPD) requires the EU and member states to ensure accessibility. Measures at EU level are needed to set common accessibility requirements for key products and services.
The European Accessibility Act sets standards for key products and services:
Ticketing and check-in machines;
ATMs and other payment terminals;
PCs and operating systems;
smartphones, tablets and TV equipment;
access to audio-visual media services, e-books;
e-commerce;
some elements of passenger transport services, and;
electronic communications, including the 112 emergency number.
Opportunities for businesses and consumers
Having common standards at EU level will prevent member states from developing different laws? This will make it easier and more attractive for businesses to sell accessible products and services in the EU and abroad.
New rules would encourage competition between economic operators and promote the free movement of accessible products and services. It is expected to give consumers more choice of accessible products and service and reduce their cost.
Exemptions for micro-enterprises
Because of their size and limited resources, exemptions would apply to some micro-enterprises, which are small companies with fewer than 10 employees and an annual turnover or balance sheet of less than €2 million.
However, these firms will be encouraged to manufacture and distribute products and provide services that comply with the accessibility requirements of the new rules.
EU countries will have to provide guidelines to these micro-enterprises to facilitate the implementation of the legislation.
MEPs will vote on the draft directive during a plenary session in March. It will also need to be approved by the Council of Ministers before it can enter into force.
Text of the provisional agreement
Check progress on the legislative proposal
Category: A Frontpage, EU, European Commission
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it will be nice for you to give us a picture of the reverse of each coin and give us their weight of each of them with their dimensions and separate photos.
merci Rol pour ton complément d 'info sur ces monnaies ..
My weight doesn't calculate the decimals. Is that a problem?
Celle ci appartient à la série Belvès.
Pouvez vous montrer l'avers svp ?
Belles monnaies gauloises "aux meches ondulées"
very nice coins! well done.


text:
Q: SSL handshake failure due to java update My client java -version is 1.7.0_85 and the server version is lower than that. the communication takes place fine when the client was a version lower. But since the update i get the handshake_failure error. Upon firing openssl command for the server it said that the cipher was RC4-SHA in the latest update RC4 were disabled by default. Cant update the server not in my control, how should i establish the communication using properties in the java.security
 file

A: You can look at the value for the jdk.tls.disabledAlgorithms parameter in java.security and remove RC4 from that values list.
Rather than retain this change and make clients insecure, it would be a better option if a parallel thread is opened with the server operator(s) to have them fix their SSL setup.
I'd like to repeat what the JRE release notes states about the usage of RC4 (emphasis mine).

Area: security-libs/javax.net.ssl
Synopsis: Prohibit RC4 cipher suites
RC4 is now considered as a compromised cipher. RC4 cipher suites have
  been removed from both client and server default enabled cipher suite
  list in Oracle JSSE implementation. These cipher suites can still be
  enabled by SSLEngine.setEnabledCipherSuites() and
  SSLSocket.setEnabledCipherSuites() methods.
See JDK-8077110 (not public).