Marie Herr

Nicolas Sirven

Hélène Grondin

Sylvain Pichetti

Catherine Sermet

Frailty, polypharmacy, and potentially inappropriate medications in old people: findings in a representative sample of the French population

Keywords: aged, anticholinergic medications, frailty, inappropriate prescribing, polypharmacy

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Frailty is defined as an increased vulnerability to stressors, resulting from a decrease in physiological reserves of multiple systems. It has been operationalized as a phenotype, determined by the presence of a critical number of impairments in physical strength, physical activity, nutrition, mobility and energy [START_REF] Fried | Frailty in older adults: evidence for a phenotype[END_REF]. Epidemiological studies have shown that frailty is associated with a higher use of healthcare resources [START_REF] Sirven | The cost of frailty in France[END_REF][START_REF] Sirven | The Dynamics of Hospital Use among Older People Evidence for Europe Using SHARE Data[END_REF] and predicts health outcomes such as occurrence or aggravation of functional limitations, falls, hospitalisations, and mortality [START_REF] Bergman | Frailty: an emerging research and clinical paradigm--issues and controversies[END_REF][START_REF] Shamliyan | Association of frailty with survival: a systematic literature review[END_REF].

The concept of frailty is now well recognized by geriatricians and prevention stakeholders.

Primary care physicians are encouraged to screen their patients for frailty and to address them when necessary to day hospitals where an evaluation of the causes of frailty will lead to a personalized care plan [START_REF] Morley | Frailty consensus: a call to action[END_REF]. The reduction of polypharmacy is part of the intervention to manage frailty.

Epidemiological studies have shown that frail people are more likely to receive polypharmacy compared to non-frail individuals [START_REF] Herr | Polypharmacy and frailty: prevalence, relationship, and impact on mortality in a French sample of 2350 old people[END_REF][START_REF] Chang | Prevalence and correlates of geriatric frailty in a northern Taiwan community[END_REF][START_REF] Gnjidic | High-risk prescribing and incidence of frailty among older community-dwelling men[END_REF][START_REF] Rosted | Frailty and polypharmacy in elderly patients are associated with a high readmission risk[END_REF][11]. Polypharmacy exposes old people to various risks [START_REF] Fried | Health outcomes associated with polypharmacy in community-dwelling older adults: a systematic review[END_REF], notably adverse drugs events [START_REF] Frazier | Health outcomes and polypharmacy in elderly individuals: an integrated literature review[END_REF], falls [START_REF] Lai | Polypharmacy correlates with increased risk for hip fracture in the elderly: a population-based study[END_REF], increased use of healthcare services, and mortality [START_REF] Gnjidic | Polypharmacy cutoff and outcomes: five or more medicines were used to identify community-dwelling older men at risk of different adverse outcomes[END_REF][START_REF] Beer | Quality use of medicines and health outcomes among a cohort of community dwelling older men: an observational study[END_REF]. Polypharmacy also increases the risk of receiving "potentially inappropriate medications" (PIMs), i.e. medications with a well-established risk of adverse effect in old people or medications with questionable efficacy. The first set of explicit criteria for PIMs was developed in 1991 by Beers et al to be used in nursing homes [START_REF] Beers | Inappropriate medication prescribing in skilled-nursing facilities[END_REF] and has since been updated several times and adapted in different countries.

In this context, this study aimed to analyse the relationship between medication use and frailty by considering the quantity of medications prescribed (polypharmacy) and the quality of medication prescribing (according to French criteria for PIMs) in people aged 65 and over participating in a nationally representative study about health and use of healthcare resources in France where data about frailty and medications were thoroughly assessed.

Methods

Study design and population

We used cross-sectional data from the 2012 French health, health care, and insurance survey

Data collection

Participants were first interviewed by telephone (or directly at home for people who did not have the telephone or for whom the telephone number was wrong) about the socio-demographic characteristics of their household. Information about their health status, access to health care services, health insurance, and the economic and social status of individuals were then collected by using self-administered questionnaires. Participants gave their informed consent and ESPS received the approval of the National Commission for Data Protection and Liberties (CNIL).

Frailty definition

Frailty was defined according to the construct derived from the Cardiovascular Health Study [START_REF] Fried | Frailty in older adults: evidence for a phenotype[END_REF] adapted to declarative data. The five frailty dimensions were defined as follows:

-Exhaustion: self-reported physical fatigue or weakness or lack of energy; -Unintentional weight loss of 5% of body weight during the past 12 months; -Muscle weakness: difficulty carrying a bag weighting 5 kg (in the absence of difficulty using hands or fingers) or difficulty bending of kneeling down without help; -Impaired mobility: difficulty walking 500 meters without help or difficulty going up or down a dozen or more steps without help; -Low level of physical activity: no practice of walk, bicycle or sport (jogging, fitness, swimming, biking, etc.).

Further details about the assessment of frailty (exact formulation of the question and coding) are given in Appendix 1. Frail individuals were those reporting three criteria or more. Previous work in ESPS 2012 showed consistency with other measures of the frailty phenotype in the general population [START_REF] Sirven | The cost of frailty in France[END_REF], as in SHARE (the Survey of Health Ageing and Retirement in Europe) where objective measurements of gait speed and grip strength are available [START_REF] Célant | Enquête sur la santé et la protection sociale 2012[END_REF].

Polypharmacy and potentially inappropriate medications

The EGB contains exhaustive information on all the medications that were reimbursed to people during the year 2012. Medications are coded using the Anatomical Therapeutic Chemical (ATC) system. We estimated the number of medications used during the year 2012 by calculating the mean of the total number of medications used per 3-month periods. Polypharmacy was defined as five or more and excessive polypharmacy as 10 medications or more [START_REF] Herr | Polypharmacy and frailty: prevalence, relationship, and impact on mortality in a French sample of 2350 old people[END_REF]. It included both regular and as required medications. PIMs were assessed over the whole year 2012 according to the Laroche list [START_REF] Laroche | Potentially inappropriate medications in the elderly: a French consensus panel list[END_REF], which results from an expert consensus and takes into account drugs marketed in France. We excluded 5 criteria that required information about underlying conditions that could not be assessed here. Concomitant use of drugs corresponded to cases where two drugs were delivered on the same day. Based on current literature and national recommendations, we also considered inappropriate duration of treatment (3 reimbursements over a 4-month period) for some a priori selected drug classes, which were non-steroidal anti-inflammatory drugs [20] and benzodiazepines [20,[START_REF]The American Geriatrics Society 2012 Beers Criteria Update Expert Panel. American Geriatrics Society Updated Beers Criteria for Potentially Inappropriate Medication Use in Older Adults[END_REF], especially hypnotics [22].

Other variables

In addition to sociodemographic characteristics (age, gender, marital status, and education), information was collected about difficulties in doing alone 5 activities of daily living (ADL: eating, dressing and undressing, getting in and out of bed, using the toilets, bathing or showering) and 7 instrumental activities of daily living (IADL: food preparation, using the telephone, shopping, managing medications, light housekeeping, heavy housekeeping, managing finances and administrative tasks). Participants were asked about their body mass index, self-perceived health on a 5-point scale, social isolation, and tobacco smoking. Chronic diseases (over the last 12 months) were assessed among a standard list of 13 diseases including asthma, chronic bronchitis/emphysema, heart attack, stroke/cerebral haemorrhage, coronary disease/angina, high blood pressure, osteoarthritis, back pain, neck pain, diabetes, allergy, liver cirrhosis, and depression. Some diseases were grouped as follows:

-Respiratory diseases: asthma, chronic bronchitis/emphysema; -Cardiovascular diseases: heart attack, stroke/cerebral haemorrhage, coronary disease/angina, high blood pressure; -Musculoskeletal diseases: osteoarthritis, back pain, neck pain.

Statistical analyses

The statistical analysis describes sociodemographic and health variables, including frailty and medication use. We used individual sampling weights (the inverse of the probability that the observation is included considering sampling design, age, gender, household size, and social security scheme) to provide representative estimates.

Complete information about the five frailty criteria was available in 70.7% of the study population. In the remaining 29.3% cases, we imputed missing data regarding frailty criteria according to age and gender (logit modelling and imputation of the variable as 1 when the probability was more than 0.5, 0 otherwise). The proportion of frail individuals did not differ between the original and the imputed dataset (p=0.11). The prevalence of the original and imputed variables is given in the descriptive statistics.

The independent and combined effects of polypharmacy and PIMs on the progression of the frailty score from 0 to 5 were assessed by using Poisson regression models with the number of frailty criteria as the dependent variable. As first step of a multi-stage approach, we modelled the effect of polypharmacy and PIMs separately. Second, we adjusted the models for confounders, corresponding to the variables associated with the number of frailty criteria with a p<0.20. The final adjustment was obtained by progressively removing variables associated with frailty with a p>0.10. Third, we entered simultaneously polypharmacy and PIMs, as well as confounders, in the model. Eventually, we added an interaction term between polypharmacy and PIMs. Results are presented in terms of Incidence Rate Ratios (IRR, i.e. exponentiated coefficients) with 95% Confidence Interval (95%CI).

Note that the analytical choices (imputation, use of sampling weights, Poisson modelling, and robust standard errors) aimed to maximise the statistical power of the analysis, which is of particular importance when introducing interaction terms in the models. Sensitivity analyses were conducted to test the influence of imputation on the frailty variables by repeating the analysis with the original variables. Analyses were performed by using Stata ® version 14.

Results

Population

The study population was composed of 1003 women and 887 men, of mean age 74.7 +/-7.4 years. They estimated their health good or very good in 39.7% of the cases. Musculoskeletal disorders were reported in more than half of the participants, followed by cardiovascular diseases, diabetes, and respiratory diseases. Frail people accounted for 16.4% of the study population, 14.8% when using the imputed variables. The characteristics of the study population are further described in table 1.

Medication use

Polypharmacy concerned 42.9% of the study population (n=799), and excessive polypharmacy 27.4% (n=474). Potentially inappropriate prescribing according to the Laroche list concerned 36.8% of the study population (n=664). When criteria assessing prolonged use of NSAIDs, benzodiazepines, and hypnotics were added, the prevalence of PIMs reached 46.7% (n=841). The most frequent PIMs involved benzodiazepines, anticholinergic drugs, NSAIDs, and cerebral vasodilators. Table 2 displays the frequency of the PIMs that concerned at least 1% of the study population. A complete description of the prevalence of PIMs is given in Appendix 2.

Relationship between frailty, polypharmacy, and PIMs

Models 1 and 2 of Table 3 show that both polypharmacy and inappropriateness of medications are associated with the number of frailty criteria in bivariate analysis. The Figure 1 illustrates the gradual increase in the prevalence of polypharmacy and PIMs with the number of frailty criteria. These associations remained significant after adjustment for confounders, including comorbidities in models 3 and 4 (IRR 5 to 9 drugs =1.16, 95%CI [1.01-1.34]; IRR 10 drugs or more =1.45, 95%CI [1.25-1.69]; and IRR PIM =1.18, 95%CI [1.07-1.30]). When polypharmacy and PIMs were both introduced in the model (model 5), excessive polypharmacy only remained significantly associated with the number of frailty criteria. Several PIMs were specifically associated with the number of frailty criteria in bivariate analysis but the only one that remained significantly associated with the number of frailty criteria after the introduction of confounders and polypharmacy in the model was the prescription of anticholinergic drugs as defined in the Laroche list. There was no significant interaction between polypharmacy and PIMs, meaning that there was no indication to stratify the analyses on the level of polypharmacy.

Sensitivity analyses

We obtained similar results concerning the association of the number of frailty criteria with polypharmacy and PIMs when analyses were replicated using the non-imputed variables for frailty (see table in Appendix 3), though the association with the prescription of anticholinergic drugs hardly remained significant.

Discussion

Main findings

By analysing the data from a nationally representative study matched with National Health

Insurance data, this study provides insights about the prevalence of polypharmacy and PIMs and about their relationships with frailty in community-dwelling people aged 65 years and over.

Polypharmacy and excessive polypharmacy were reported in 42.9% and 27.4% of the study population respectively, while 46.7% of the study population received at least one PIM, especially benzodiazepines, anticholinergic drugs, NSAIDs, and cerebral vasodilators.

Polypharmacy and PIMs were both associated with the number of frailty criteria in models adjusted for socio-demographic and health characteristics of the individuals. The prescription of anticholinergic medications was the only PIM that remained significantly associated with the number of frailty criteria after adjustment for polypharmacy.

Prevalence of polypharmacy and PIMs

Our estimates of the prevalence of polypharmacy and PIMs are relatively high compared to previous estimates [START_REF] Herr | Polypharmacy and frailty: prevalence, relationship, and impact on mortality in a French sample of 2350 old people[END_REF][START_REF] Gnjidic | High-risk prescribing and incidence of frailty among older community-dwelling men[END_REF]11,[START_REF] Tommelein | Potentially inappropriate prescribing in community-dwelling older people across Europe: a systematic literature review[END_REF], which was expected considering our methodology. Indeed, the prevalence of polypharmacy should be considered with regard to our definition that encompasses all the medications prescribed over 3-month periods. Considering PIMs, their prevalence was assessed over an entire year and not at a given time point, which obviously increased the chance of having one PIM for a given subject compared to a punctual assessment. Moreover, we added 3 criteria (assessing potentially inappropriate prolonged use of medications) to those of the Laroche list, which happened to increase by nearly 10% the prevalence of PIMs. Consistently with the review by Tommelein et al [START_REF] Tommelein | Potentially inappropriate prescribing in community-dwelling older people across Europe: a systematic literature review[END_REF], we found benzodiazepines and NSAIDs among the most reported PIMs. Conversely, inappropriate use of antidepressants was limited in our study.

PIMs and frailty

The relationship between PIMs and frailty initially observed in unadjusted and partially adjusted models became non-significant when polypharmacy was introduced in the multivariate models. This result suggests that the association between PIMs and frailty reflects the association between PIMs and polypharmacy in the one hand and between polypharmacy and frailty in the other hand.

Nevertheless, this result should be considered with caution because PIMs still tended to be associated with frailty in the model adjusted for polypharmacy and we cannot exclude a lack of power to detect a significant association. Collinearity between polypharmacy and PIMs may have increased estimates of parameter variance, hence reducing the likelihood of showing a significant association between PIMs and frailty. Nevertheless, this hypothesis is unlikely regarding the results of diagnostic tests for colinearity between polypharmacy and PIMs; both the Variance Inflation Factor (VIF) and the condition number were inferior to admitted thresholds (VIF=1.21 <10 and condition number < 15) [START_REF] Dormann | Collinearity: a review of methods to deal with it and a simulation study evaluating their performance[END_REF]. Besides, PIMs include heterogeneous situations in terms of risk. Some medications are said inappropriate because of their safety profile, whereas others are said inappropriate because of uncertainty about their efficacy. That is why we considered PIMs altogether and by criteria. Doing this, we actually showed a significant association between frailty and anticholinergic medications that persisted after adjustment for polypharmacy.

Polypharmacy, anticholinergic medications, and frailty

Our results confirm the previously reported association between excessive polypharmacy and frailty [START_REF] Herr | Polypharmacy and frailty: prevalence, relationship, and impact on mortality in a French sample of 2350 old people[END_REF], and extend to the general population the results of Moulis et al [START_REF] Moulis | Exposure to atropinic drugs and frailty status[END_REF] who showed that medications with anticholinergic properties were associated with frailty in people attending a frailty clinic in France, after adjustment for polypharmacy. Anticholinergic medications can cause peripheral (dry mouth and constipation) and central (falls, dizziness, delirium, and cognitive decline) adverse effects [START_REF] Gerretsen | Drugs with anticholinergic properties: a current perspective on use and safety[END_REF] that could participate in the development of frailty through altered nutritional intake, limitation of mobility or cognitive impairment. Gnjidic et al [START_REF] Gnjidic | Potential contribution of medications to frailty[END_REF] suggested the potential contribution of medicines to the development of frailty, consistently with the results of previous studies showing an increased risk of incident frailty in people with polypharmacy [START_REF] Gnjidic | High-risk prescribing and incidence of frailty among older community-dwelling men[END_REF][START_REF] Wang | Incidence and Effects of Polypharmacy on Clinical Outcome among Patients Aged 80+: A Five-Year Follow-Up Study[END_REF].

Strength and limitations

The strength of this study is that we used a unique dataset combining a nationally representative health survey with respondents' National Health Insurance data on medication reimbursements.

Nevertheless, this study has limitations. Though nationally representative, we had a limited sample of people aged 65 years and over. Concerning the assessment of frailty, we used selfreported variables in the absence of objective measures of grip strength and walking speed.

Another limitation is that 29.7% had missing data regarding one or more frailty variable.

Considering only people with complete information would have led to a selection bias and a loss of power, which is why we imputed missing data based on available information. Results were similar with regard to the estimated coefficients but differed somehow with regard to the standard error of the estimates. The lack of statistical power was substantial in the case of the sample with non-missing observation. However, imputation of frailty criteria conditional on age and sex helped improve the statistical power without introducing bias in the estimates because (i) age and sex are exogenous covariates (not determined by frailty or its determinants) and (ii) these two variables are included as covariates in the model, thus assigning the observations to the average individual. Though highly reliable, data about medication use only reflect medication bought by people and not those actually taken. In the case of concomitant use, defined as situations where two drugs of the same class were delivered on the same day, we miss cases where people buy their medications on different days and use medications they have left at home. Furthermore, we did not have information about the use of over-the-counter products and medications received during hospitalisation. Eventually, the cross-sectional design of this study did not enable to conclude on the causality of the relationships between frailty, polypharmacy and PIMs.

Conclusion

This study shows that polypharmacy and use of anticholinergic medications are independently associated with frailty in old people. This should increase awareness towards the overuse of medications in old people and should encourage physicians to supress the prescriptions that are known to have a poor benefit-risk ratio in their patients, especially anticholinergic medications.

Longitudinal studies are required to establish the respective role of polypharmacy and PIMs on the development of frailty.

(

  Enquête sur la santé et la protection sociale, ESPS) matched with National Health Insurance data. The survey, coordinated by the Institute for Research and Information in Health Economics (IRDES, Paris), was designed to be representative of the French population (1 individual included in ESPS being representative of 2231 individuals on average in general population). The source population consisted of the 599,544 individuals included in the EGB (Echantillon Généraliste des Bénéficiaires) in 2012, a permanent representative sample of the population covered by the French public health insurance. A random subsample of community-dwellers was drawn from the EGB; these reference individuals together with members of their household were eligible for the survey. A total of 8413 households representing 23,047 French residents took part in the 2012 survey. Among them, 14.2 % were 65 years old or more (3271 observations remaining). Survey respondents were then matched with National Health Insurance data (in the EGB) for 1955 observations. Unmatched individuals were those household members whose public health insurance was independent from the reference individual's health insurance known in the EGB. An additional 65 observations were discarded because we did not have information about medications for these individuals. Our analysis sample eventually consisted in 1890 communitydwellers aged 65 and over.

Figure 1 .

 1 Figure 1. Prevalence of polypharmacy and PIMs according to the number of frailty criteria (N=1890)

Table 1 . Characteristics of the people aged 65+ included in ESPS 2012 (N=1890)

 1

	Variable

Table 2 . Potentially inappropriate medications received by 1% or more of the participants aged 65+ in ESPS 2012 (N=1890)

 2

	Potentially inappropriate medications	ATC	N	Prevalence
				(%)
	Laroche list criteria a
	Non-steroidal anti-inflammatory drugs		74	3.9
	(NSAIDs)
	≥ 2 NSAIDs	M01A	72	3.8
	Anticholinergic drugs		161	9.2
	Tricyclic antidepressant		47	2.4
	Amitriptyline	N06AA09	31	1.8
	Antihistamins H1		79	4.7
	Hydroxyzine	N05BB01	69	4.0
	Anticholinergic urinary		28	1.7
	antispasmodics
	Solifenacine	G04BD08	28	1.5
	Long-acting benzodiazepines		232	12.6
	Bromazepam	N05BA08	138	7.7
	Prazepam	N05BA11	45	2.5
	Clonazepam	N03AE01	25	1.2
	Antihyperthensives		98	6.2
	Centrally acting		60	4.0
	Rilmenidine	C02AC06	51	3.4
	Short-acting calcium-channel		42	2.4
	blockers
	Nicardipine	C08CA04	35	1.9

Table 3 . Poisson regression models of the number of frailty criteria according to medications among participants aged 65+ in ESPS 2012 (N=1542)

 3

	Model 1	Model 2	Model 3	Model 4	Model 5	Model 6


Colleen Ackermann

AND Peter Haïssinsky

Aimo Hinkkanen

EQUILATERAL TRIANGLE SKEW CONDITION FOR QUASICONFORMALITY

Introduction

Since quasiconformal mappings were first studied nearly a century ago, many diverse characterizations have been discovered. These have led to a wide variety of applications in many fields including Teichmüller theory, elliptic PDE's, hyperbolic geometry and complex dynamics. For an overview of these applications and the theory of quasiconformal mappings see [START_REF] Ahlfors | Lectures on quasiconformal mappings[END_REF], [START_REF] Astala | Elliptic partial differential equations and quasiconformal mappings in the plane[END_REF], and [START_REF] Hubbard | Teichmüller theory and applications to geometry, topology, and dynamics[END_REF]. In this paper we will use the metric definition of quasiconformality to obtain a new formulation for the definition of planar quasiconformal mappings. Note that f (T ) is also a topological triangle so the expression skew(f (T )) is defined. Then f : U → V is quasiconformal provided that there exists an increasing homeomorphism h : [0, ∞) → [0, ∞) such that skew(f (T )) ≤ h(skew(T ))

for all closed Euclidean triangles T ⊂ U . In fact, Hubbard showed that it is sufficient to only consider triangles with skew bounded above by 7/3. He then asked the question of whether it suffices to only consider equilateral triangles. Progress was made on this problem in a previous paper by Javier Aramayona and Peter Haïssinsky [START_REF] Aramayona | A characterisation of plane quasiconformal maps using triangles[END_REF] in which they showed that there exists a constant 0 > 0 such that if ∈ [0, 0 ) and skew(f (T )) ≤ 1 + Date: May 29, 2018. C.A. was supported in part by a gift to the Mathematics Department at the University of Illinois from Gene H. Golub. P.H. is partially supported by the ANR projects "GDSous/GSG" no. 12-BS01-0003-01 and "Lambda" no. 13-BS01-0002.

for all equilateral triangles T ⊂ U , then f is quasiconformal.

Theorem 1 of this paper answers Hubbard's question in the affirmative.

Theorem 1. Let U and V be two domains in the complex plane C, and let f : U → V be an orientationpreserving homeomorphism. For each σ ≥ 1 there exists H(σ) ≥ 1 with the following property. If skew(f (T )) ≤ σ for all equilateral triangles T ⊂ U , then, for every z ∈ U and every r < dist(z, C \ U ), the inequality M (z, r) ≤ Hm(z, r) holds where H = H(σ). In particular, the map f is quasiconformal.

Our next result, Theorem 2, serves two purposes. On the one hand, the value H from Theorem 1 gives an upper bound for the dilatation of f . However, this estimate is far from optimal. We may use the fact that quasiconformal mappings are differentiable almost everywhere to obtain a sharp bound on the dilatation in terms of σ. On the other hand, quasiconformality is a local property and demanding, as in Theorem 1, a uniform bound on every equilateral triangle is unnecessarily strong.

To show that f is quasiconformal in U , it suffices to show that f satisfies two conditions. First we must show that each z ∈ U has a neighborhood W such that skew(f (T )) is bounded for all equilateral triangles T ⊂ W . If the resulting upper bounds vary, this is not enough to get the result that f is quasiconformal in U . However, by formulating a new concept of skew that needs to be satisfied only almost everywhere and must be bounded uniformly in U , and combining it with the assumption that the skew as defined previously is bounded, not necessarily uniformly, in some neighborhood of each point of U , we obtain both quasiconformality and a better upper bound for the maximal dilatation of f . The details of how to define this new notion of skew are given below.

First note that quasiconformal maps are differentiable almost everywhere by Mori's theorem [START_REF] Mori | On quasi-conformality and pseudo-analyticity[END_REF]. Let Skew(f ) denote the supremum of skew(f (T )) over all equilateral triangles T contained in U . Next, for z ∈ U and r > 0, let skew(f, z, r) denote the least upper bound of skew(f (T )) over all equilateral triangles T ⊂ {w ∈ U : |z -w| < r}. Set skew(f, z) = lim inf r→0 skew(f, z, r) and skew(f ) = ||skew(f, z)|| ∞ . Below, the assumption that Skew(f |W ) is finite means that there is some upper bound for skew(f (T )) over all equilateral triangles T contained in the appropriate domain W .

Theorem 2. Let U be a domain in the complex plane C, and let f : U → f (U ) be an orientation-preserving homeomorphism. Suppose that each z ∈ U has a neighborhood W such that Skew(f |W ) is finite.

Suppose that σ ≥ 1. If skew(f ) ≤ σ then f is K(σ)-quasiconformal where K(σ) = σ 2 -1 + √ σ 4 + σ 2 + 1 √ 3σ .
In particular, if skew(f ) = 1 then f is a conformal mapping. The upper bound K(σ) for the maximal dilatation of f is the best possible and is attained at least for certain affine mappings.

Proof of the Main Theorem

Throughout the rest of the paper we will use the following notation and conventions:

(1) We define D(z, r) = {w ∈ C : |z -w| ≤ r} and let C(z, r) be the boundary of D(z, r).

(2) By a curve we mean the image of a not necessarily one-to-one continuous function from a closed interval into C.

(3) All triangles (without the qualifier "topological") will be closed Euclidean triangles.

(4) Let F σ denote the set of orientation-preserving homeomorphisms of any domain U ⊂ C into any domain V ⊂ C such that skew(f (T )) ≤ σ for all closed equilateral triangles T ⊂ U .

Our strategy for the proof of Theorem 1 is to show first that the image of every equilateral triangle, T , contains a disk with radius proportional to L(f (T )). To do this, we naturally need the assumption of Theorem 1 for all equilateral triangles (at least in a suitable set containing this T ) and not only for this particular T . Once it is known that the images of equilateral triangles are "thick" in this sense, it is easier to obtain the quantitative estimates that are required to prove that f is quasiconformal, whether using the metric (Definition 1) or the analytic (Definition 2 in Section 6 below) definition of quasiconformality.

The most difficult part of the proof that the image f (T ) is thick is to construct another equilateral triangle with suitable additional properties, to which the assumption of Theorem 1 can then be applied. This will be done in the proof of Proposition 2 below. That proof requires delicate geometric considerations. After these strategic comments, let us now move to the proof.

The proof of Theorem 1 relies on the following proposition.

Proposition 1. Let U be a domain containing D(0, 1), let f : U → C belong to F σ , and let T be the triangle with vertices 0, 1, and ω = 1/2 + ( √ 3/2)i. Then there exists a disk D contained in f (T ) such that

(1) D is centered at f (p) where p = 1/2 + (85

√ 3 • 2 -9
)i ≈ 0.5 + 0.29i, and (2) there exists a constant α = α(σ) such that the radius of D is at least αL(f (T )).

We note that if f is to be quasiconformal, then, certainly, the image f (T ) has to contain a disk of a definite size centered at the image of the centroid of the triangle, i.e., the point ξ = 1/2 + ( √ 3/6)i. Unfortunately, its arithmetic properties make it difficult to relate this point ξ to the vertices of T using equilateral triangles.

The point p was chosen, because it is both close to the centroid (|ξ

-p| = √ 3/(2 9 • 3)),
and it is a vertex of a tiling of the plane by equilateral triangles whose vertices include the vertices of T . Indeed, we have

p = 1/2 -85 • 2 -9 + 85 • 2 -8 ω, cf. Lemma 1.
We finally observe that p is closer to the horizontal side of T than ξ.

We first derive the proof of Theorem 1 from Proposition 1. We will then focus on the proof of the latter.

Proof of Theorem 1. Fix z ∈ U and r > 0. If D(z, r) ⊂ U , let M (z, r) = max{|f (z) -f (w)| : w ∈ C(z, r)} and m(z, r) = min{|f (z) -f (w)| : w ∈ C(z, r)}. Denote by z M a point in C(z, r) such that |f (z M ) -f (z)| = M (z, r).
Since F σ is invariant under pre-and post-composition by affine maps of the form z → az +b where a, b ∈ C and a = 0, we assume that z = 0, r = 1, and z M = 1. Thus |f (0) -f (1)| = M (0, 1). Let T 1 be the equilateral triangle with vertices 0, 1, and ω. Since 0 and 1 are vertices of T 1 , we have |f (0

) -f (1)| ≤ L(f (T 1 )). Hence M (0, 1) ≤ L(f (T 1 )).
By Proposition 1, the image of T 1 must contain a disk D centered at f (p) and of radius at least αL(f (T 1 )). 1). The triangle T 2 is contained in the unit disk, and A maps p to 0 and 1 to p. Since the vertices of T 2 other than p lie outside of T 1 , their images lie outside of D, and so we have

Let us consider the isometry

A(z) = z -p. Let T 2 = A(T 1 ) (see Figure
L(f (T 2 )) ≥ αL(f (T 1 )).
Moreover, another application of Proposition 1 implies that f (T 2 ) contains the disk

D(f (0), αL(f (T 2 ))).
Since all the points of f (C(0, 1)) are outside the interior of f (T 2 ) and thus also outside the interior of

D(f (0), αL(f (T 2 ))), it follows that m(0, 1) ≥ αL(f (T 2 )).
Summing up these estimates, we obtain

m(0, 1) ≥ αL(f (T 2 )) ≥ α 2 L(f (T 1 )) ≥ α 2 M (0, 1) . T 1 T 2 z M -1.0 -0.5 0.5 1.0 -1.0 -0.5 0.5 1.0 Figure 1. Configuration of C(0, 1), T 1 and T 2

Construction of Certain Triangles

Proposition 1 is a consequence of the following proposition.

Proposition 2. Let U be a neighborhood of D(0, 1), and let f : U → C be a homeomorphism onto its image such that f ∈ F σ . Let T be the closed triangle with vertices 0, 1, and ω = 1/2 + ( √ 3/2)i. Let q = p + 2 -9 . Then there exist points t 1 , t 2 ∈ T such that the points q, t 1 , t 2 form the vertices of an equilateral triangle and the inequalities |f (t j ) -f (p)| ≤ Cµ, and |f (p) -f (q)| ≥ cL(f (T )) hold for some constants c = c(σ) and

C = C(σ) where µ = dist(f (p), C \ f (T )).
We permit the trivial triangle where we have t 1 = t 2 = q.

Proof of Proposition 1 assuming Proposition 2. If t 1 = t 2 = q, then we have

cL(f (T )) ≤ |f (p) -f (q)| ≤ Cµ.
Otherwise, by the triangle inequality we have [START_REF] Ackermann | Quasiconformal mappings on planar surfaces[END_REF] |f

(p) -f (q)| ≤ |f (t 1 ) -f (p)| + |f (t 1 ) -f (q)|.
Since f ∈ F σ and t 1 , t 2 , q are the vertices of an equilateral triangle, we obtain

(2) |f (t 1 ) -f (q)| ≤ σ|f (t 1 ) -f (t 2 )|.
By Proposition 2, we have |f (t 1 ) -f (p)| ≤ Cµ and |f (t 2 ) -f (p)| ≤ Cµ, so that by this and the triangle inequality we get

|f (t 1 ) -f (t 2 )| ≤ |f (t 1 ) -f (p)| + |f (t 2 ) -f (p)| ≤ 2Cµ.
By Proposition 2, we have |f (p) -f (q)| ≥ cL(f (T )). Using this and appealing to |f (t 1 ) -f (p)| ≤ Cµ again, we obtain from ( 1) and (2) that

cL(f (T )) ≤ |f (p) -f (q)| ≤ Cµ + σ(2Cµ), whence µ ≥ c (2σ + 1)C L(f (T )) .

Proof of Proposition 2

Let the assumptions of Proposition 2 be satisfied. Recall that we write µ = dist(f (p), C \ f (T )). The idea of our proof of Proposition 2 is to define a curve γ in T going through p such that

(1) for all t ∈ γ we have |f (t) -f (p)| ≤ σµ(1 + 2σ 3 );

(2) there are two points t 1 , t 2 ∈ γ , such that q, t 1 , t 2 form the vertices of an equilateral triangle.

The proof of Proposition 2 results from Lemma 1 and Lemma 3.

We first prove the following result. Recall that ω = 1/2 + ( √ 3/2)i.

Lemma 1. Let T be the closed triangle with vertices 0, 1, and ω. Let p = 1/2+(85

√ 3•2 -9 )i and q = p+2 -9 . Then |f (q) -f (p)| ≥ cL(f (T )) for some positive constant c = c(σ).
Proof. Let us first consider the tiling of the plane by equilateral triangles with vertices in Λ = Z⊕ωZ. Define a chain of triangles (T j ) 0≤j≤J as a sequence of triangles with vertices in Λ such that T j ∩ T j+1 is an edge for all j with 0 ≤ j < J. Given two edges (v, w) and (v , w ), we may connect them by a chain of minimal length n ≥ 0. A simple induction argument implies

(3) |f (v) -f (w)| ≤ σ n |f (v ) -f (w )| if f ∈ F σ is defined in a neighborhood of the chain.
Let T be as defined in our hypotheses: it is tiled by N = 2 18 triangles of 2 -9 Λ, and [p, q] is an edge of this tiling. Therefore, for every other edge [v, w], it follows that

|f (v) -f (w)| ≤ σ N |f (p) -f (q)|.
But each side of T is the union of less than N edges of our tiling, therefore, the triangle inequality implies

L(f (T )) ≤ N σ N |f (p) -f (q)| .
We now prove a geometric lemma which will be used in the proof of Lemma 3. Proof. We note that cos θ ± ≥ 1/2 -1/4 > 1/8 ≥ |z| so that ϕ is less than π.

For the other inequality, we will estimate tan | arg(e iθ± -z)| to obtain a lower bound of both angles with the horizontal line:

tan | arg(e iθ± -z)| ≥ √ 3/2 -(|z| + 1/8) 1/2 + (|z| + 1/8) ≥ √ 3/2 -1/4 1/2 + 1/4 ≥ 2 √ 3 -1 3 ≥ 2 3 > tan(π/6).
Therefore ϕ is at least π/3. Now we demonstrate how to find the curve γ mentioned above.

Lemma 3. Under the assumptions of Proposition 2, there exists a curve γ in T going through p such that for all t ∈ γ we have

|f (t) -f (p)| ≤ σµ(1 + 2σ 3 )
and there are two points t 1 , t 2 ∈ γ , such that q, t 1 , t 2 form the vertices of an equilateral triangle. We permit the trivial triangle where we have t 1 = t 2 = q.

Proof. We will do this in several steps. We first define a curve that will join two points of the boundary of a disk contained in T (Step 1). To make sure that we will be able to find two points that form an equilateral triangle with q, we will extend this curve so that it has end points in a slightly larger disk, and is only close to the boundary of the larger disk when it is also close to its end points (Step 2). Then we will use Lemma 2 to find our triangle (Step 3).

Since √ 3 ≥ 8/5, it follows that dist(p, ∂T ) = 85 √ 3 • 2 -9 > 1/4 + 2 -6 , so that D(p, 1/4 + 2 -6
) is contained in the interior of T .

Throughout the proof, for x ∈ C, R x will denote the rotation centered at x by π/3 radians, defined by R x (z) = x + (z -x)ω and Rx the rotation centered at x by -π/3 radians, defined by Rx (z) = x + (z -x)ω. Recall that we set ω = 1/2 + ( √ 3/2)i.

Step 1: There exists a curve γ 2 that satisfies the following:

(1) We take w to be the first point of C(p, 1/4) encountered when moving along γ starting from p.

γ 2 ⊂ D(p, 1/4), (2)

Now define

γ 2 = R p (γ 1 ) ∪ Rp (γ 1 ) .
Note that, for every s ∈ γ 1 , R p (s) and Rp (s) are two points in γ 2 which make an angle of 2π/3 seen from p.

Since f ∈ F σ , for all t ∈ γ 2 , we have

|f (t) -f (p)| ≤ σ|f (s) -f (p)| ≤ σµ
where s ∈ γ 1 denotes a point such that either t = R p (s) or t = Rp (s).

Step 2: Let a, b be the end points of γ 2 . There exists a curve γ 3 such that Clearly γ 2a also has an end point on the boundary of D a . Let a denote an end point of γ 2a on the boundary of D a . Use the tangent line to D(p, 1/4) at a to divide D a in half, and then divide each half into thirds. So we have divided D a into closed sectors of π/3 radians with three such sectors lying entirely outside of D(p, 1/4). Let S a denote the middle sector lying completely outside of D(p, 1/4). Then there exists n ∈ {2, 3} such that when γ 2a is rotated nπ/3 radians in an appropriate direction about a, the image of a under the rotation will lie in S a . Let the image of γ 2a under this rotation be denoted by γ 3a . Now we will bound the quantity |f (t) -f (p)| where t ∈ γ 3a . Fix t ∈ γ 3a . Let t 0 be the point on γ 2a whose image under the rotation is t. Without loss of generality we will assume this rotation was clockwise. Let t i denote the image of t 0 under a clockwise rotation of iπ/3 radians where i = 1, . . . , n (t = t n ). Then since a, t i-1 , t i (1 ≤ i ≤ n) form an equilateral triangle, we have

|f (t i ) -f (a)| ≤ σ|f (t i-1 ) -f (a)|. Since a, t 0 ∈ γ 2 we have |f (a) -f (t 0 )| ≤ |f (a) -f (p)| + |f (p) -f (t 0 )| ≤ 2σµ and |f (a) -f (p)| ≤ σµ.
Thus since n is at most 3 we may apply (3) with n ≤ 3, which together with the triangle inequality yields

|f (t) -f (p)| ≤ |f (a) -f (p)| + |f (a) -f (t)| ≤ σµ + σ n |f (a) -f (t 0 )| ≤ σµ(1 + 2σ n ) ≤ σµ(1 + 2σ 3 ). Furthermore γ 3a must intersect the circle C(p, 1/4 + √ 3 • 2 -7
). This is because γ 3a has an end point in S a and therefore the distance of the end point of γ 3a from D(p, 1/4) must be at least cos(π/6)

• 2 -6 = √ 3 • 2 -7
. This is depicted in Figure 2. ). Let γ 3 be the connected component of (γ

2 ∪ γ 3a ∪ γ 3b ) ∩ D(p, 1/4 + √ 3 • 2 -7
) which includes points in both γ 3a and γ 3b . Then for all points t ∈ γ 3 ,

|f (t) -f (p)| ≤ σµ(1 + 2σ 3 ).
The curve γ in Lemma 3 can be chosen as γ = γ 3 .

Step 3: Let q = p + 2 -9 . There exist t 1 , t 2 ∈ γ 3 such that {q, t 1 , t 2 } form an equilateral triangle.

Let D q be the smallest disk centered at q which contains D(p, 1/4).

Then D q ⊂ D(p, 1/4 + √ 3 • 2 -7 ) since |p -q| = 2 -9 ≤ √ 3 • 2 -8 . Let γ 4 be the connected component of γ 3 ∩ D q which has end points A ∈ D a ∩ D q and B ∈ D b ∩ D q .
Note that, if we write a = p + |a -p|e iθa and A = p + |A -p|e iθ A , where |θ A -θ a | is chosen to be as small as possible modulo 2π, then Therefore, by Lemma 2 applied in D(p, 1/4), the angle between A -q and B -q lies in (π/3, π). Hence,

|θ A -θ a | ≤ 2|a -A|/(1/4) ≤ 2 • 2 -6 /(1/4) ≤ 1/
D a S a D(p, 1/4 + √ 3/2 7 ) D(p, 1/4)
Figure 2 the images A r and B r of A and B respectively under Rq will separate A and B on ∂D q . Thus the image Rq (γ 4 ) must intersect γ 4 at a point q , say. This gives us our desired equilateral triangle since q, q , and the pre-image q of q form an equilateral triangle. This completes the proof of Lemma 3.

To complete the proof of Proposition 2, we note that the points t 1 and t 2 required will be the points q and q . It remains to be shown that |f (q ) -f (p)| ≤ Cµ and |f (q ) -f (p)| ≤ Cµ for some constant C. Since q ∈ γ 4 ⊂ γ 3 , we have |f (q ) -f (p)| ≤ σ(1 + 2σ 3 )µ by part (3) in Step 2. But also q ∈ γ 4 , so that for the same reason |f (q ) -f (p)| ≤ σ(1 + 2σ 3 )µ. This completes the proof of Proposition 2.

Proof of Theorem 2

We prove Theorem 2 by approximating f by linear mappings at points where f is differentiable.

Therefore

cos x = - 2 ν cos π 3 = - 1 ν .
It follows that sin 2 x = 1 -1/ν 2 so κ = 0 for

z = 1 ν -1 + iε ν 2 -1
with ε ∈ {±1}. For these values of z, one gets

κ = ν + 2Re(αz) ν + 2Re(ᾱz) = ν 2 -1 -ε 3(ν 2 -1) ν 2 -1 + ε 3(ν 2 -1)
which is maximal for ε = -1. So we obtain

skew( f ) 2 = ν 2 -1 + 3(ν 2 -1) ν 2 -1 -3(ν 2 -1) = (ν 2 -1)/3 + 1 (ν 2 -1)/3 -1 > 1 . Note that ν 2 -1 > 3 since ν > 2. Let us write τ = skew( f ) > 1 so that (ν 2 -1)/3 = τ 2 + 1 τ 2 -1 , ν 2 = 3 τ 2 + 1 τ 2 -1 2 + 1 = 4(τ 4 + τ 2 + 1) (τ 2 -1) 2
and thus

µ + µ -1 = ν = 2 √ τ 4 + τ 2 + 1 τ 2 -1 . Hence µ 2 -2µ √ τ 4 + τ 2 + 1 τ 2 -1 + 1 = 0 .
We compute the reduced discriminant

∆ = τ 4 + τ 2 + 1 (τ 2 -1) 2 -1 = 3τ 2 (τ 2 -1) 2
and we deduce from 0 < µ < 1 that

µ = √ τ 4 + τ 2 + 1 - √ 3τ τ 2 -1 . Thus K( f ) = 1 + µ 1 -µ = τ 2 -1 + √ τ 4 + τ 2 + 1 - √ 3τ τ 2 -1 - √ τ 4 + τ 2 + 1 + √ 3τ = ϕ(τ ), say. Write t = √ τ 4 + τ 2 + 1 and note that t 2 = (τ 2 -1) 2 + 3τ 2 so that ϕ(τ ) = [(τ 2 -1 + t) - √ 3τ ][(τ 2 -1 + t) + √ 3τ ] [(τ 2 -1) + √ 3τ ] 2 -t 2 = [(τ 2 -1) + t] 2 -3τ 2 2 √ 3τ (τ 2 -1) = 2[(τ 2 -1) 2 + t(τ 2 -1)] 2 √ 3τ (τ 2 -1) = τ 2 -1 + √ τ 4 + τ 2 + 1 √ 3τ .
Differentiation shows that ϕ(τ ) is an increasing function of τ , so that since τ = skew( f ) = skew(f, z 0 ) ≤ σ, we have

K( f ) = ϕ(τ ) ≤ ϕ(σ) = σ 2 -1 + √ σ 4 + σ 2 + 1 √ 3σ .
Hence K(f ) ≤ ϕ(σ) = K(σ) as defined in Theorem 2.

If µ ∈ (0, 1) is given and if f (z) = z + µz, then we may take f = f and σ = τ in the above argument, and we see that K(f ) = K(σ). Thus the upper bound in Theorem 2 is best possible.

6. An Alternative Proof of the Quasiconformality of Mappings Satisfying the

Hypotheses of Theorem 1

From Proposition 1, there are several ways of establishing that a mapping f satisfying the hypotheses of Theorem 1 satisfies the analytic definition of quasiconformality which is equivalent to Definition 1. For

|ξ| = 1, we denote by ∂ ξ f (z) the directional derivative lim t↓0 (f (z + tξ) -f (z))/(tξ).
Definition 2. We say that a homeomorphism f : U → V is absolutely continuous on lines if for every rectangle R = {(x, y) : a < x < b, c < y < d} with R ⊂ U , f is absolutely continuous on a.e. interval I x = {(x, y) : c < y < d} and a.e. interval I y = {(x, y) : a < x < b}. An orientation-preserving homeomorphism f is called quasiconformal if f is absolutely continuous on lines and there exists K ≥ 1 such that

max ξ |∂ ξ f (z)| ≤ K min ξ |∂ ξ f (z)| a.e.
Proposition 1 tells us that the image of every equilateral triangle, T , contains a disk with radius proportional to L(f (T )). In [6, Section 4.5], Hubbard uses this to prove that the map belongs to the Sobolev space W 1,2 loc by an approximation argument. We propose another approach which shows directly that the map satisfies the ACL property.

Proof. First we show that f is absolutely continuous on lines. This part of the proof parallels Pfluger's proof that a mapping satisfying the geometric definition of quasiconformality is absolutely continuous on lines. His proof can be found in [START_REF] Pfluger | Über die Äquivalenz der geometrischen und der analytischen Definition quasikonformer Abbildungen[END_REF] and is reproduced in English in [START_REF] Lehto | Quasiconformal mappings in the plane[END_REF], p. 162. We fix a rectangle R = {(x, y) : a < x < b, c < y < d} and let I y = {(x, y) : a < x < b} for y between c and d. Define A(y) to be the area in f (R) beneath the image of I y . Since A is an increasing function of y, it is differentiable almost everywhere. We will show that f | Iy is absolutely continuous for all y at which A is differentiable; a similar argument applies to vertical line segments. We select an arbitrary collection {(z * k , z k )} n k=1 of disjoint sub-intervals of I y where z k = (x k , y) and z * k = (x * k , y). We consider the collection of rectangles {R k } n k=1 where each R k has height δ and has its bottom side contained in the kth sub-interval. More precisely, fix k and draw a rectangle of height δ above (z * k , z k ). Set N k equal to the smallest integer greater than or equal to

|x * k -x k | δ . We draw N k equilateral triangles in R k .
The triangles all have one side on the interval [z * k , z k ] and they overlap only at their vertices. The first N k -1 triangles have width δ and the last triangle has width x * k -x k -δ(N k -1). Let ∆ k,i denote the side length of the ith triangle for

1 ≤ i ≤ N k and set ∆ k,0 = 0. Set N = n k=1 N k . Then N ≤ n k=1 |x * k -x k | δ + n.
Let α be as in Proposition 1. By Proposition 1, the image of each of our triangles must contain a disk of radius comparable to the greatest distance between the images of the vertices of the triangle. Thus the total area of all the images of our rectangles of height δ and width x * k -x k is greater than or equal to (4)

n k=1 N k j=1 πα 2 |f (x k + j i=1 ∆ k,i , y) -f (x k + j i=1 ∆ k,i-1 , y)| 2 .
Then by the Cauchy-Schwarz inequality, ( 4) is greater than or equal to

πα 2 N   n k=1 N k j=1 |f (x k + j i=1 ∆ k,i , y) -f (x k + j i=1 ∆ k,i-1 , y)|   2 ≥ πα 2 N n k=1 |f (z * k ) -f (z k )| 2 ≥ πα 2 n k=1 |x * k -x k | δ + n n k=1 |f (z * k ) -f (z k )| 2 = πα 2 n k=1 |x * k -x k |+δn δ n k=1 |f (z * k ) -f (z k )| 2 .
Recall that A(y) is defined to be the area in f (R) beneath the image of the line segment I y . Since the sum of the areas of the images of our rectangles of height δ and width x * k -x k is less than or equal to A(y + δ) -A(y), we have

πα 2 n k=1 |f (z * k ) -f (z k )| 2 ≤ A(y + δ) -A(y) δ n k=1 |x * k -x k | + δn .
Since we chose y at which A is differentiable, letting δ → 0 gives

πα 2 n k=1 |f (z * k ) -f (z k )| 2 ≤ A (y) n k=1 |x * k -x k | .
Since A (y) exists almost everywhere this gives absolute continuity on almost every horizontal line segment.

The proof is analogous for vertical line segments.

To then conclude that f is quasiconformal we note that since f is open and absolutely continuous on lines, f is differentiable almost everywhere by a theorem of Gehring and Lehto [START_REF] Astala | Elliptic partial differential equations and quasiconformal mappings in the plane[END_REF]. Now the computations in the proof of Theorem 2 show that

max ξ |∂ ξ f (z)| ≤ K(σ) min ξ |∂ ξ f (z)|
at points where f is differentiable and hence almost everywhere. According to the analytic definition of quasiconformality (Definition 2), the mapping f is quasiconformal.

7. Appendix by Colleen Ackermann: An Analogue of the Main Theorem in Hilbert Spaces of Dimension at Least Three

In dimensions three and higher the proof of an analogue of Theorem 1 is surprisingly simpler than the proof of Theorem 1. Furthermore the proof itself gives an elegant bound on K(σ).

Theorem 3. Let H 1 and H 2 be Hilbert spaces with dim(H 1 ) = dim(H 2 ) ≥ 3 and let U ⊂ H 1 , V ⊂ H 2 be domains. Suppose that f : U → V is a homeomorphism and that for all closed equilateral triangles T ⊂ U , skew(f (T )) ≤ σ. Then f is σ 3 -quasiconformal when using the metric definition of quasiconformality.

Note that Definition 1 can be used to define quasiconformal mappings also between Hilbert spaces, whether finite-dimensional or infinite-dimensional. The same discussion on the locality of quasiconformality made in the introduction applies here as well and enables us to weaken the hypothesis that skew(f (T )) ≤ σ holds for all equilateral triangles.

Proof. Fix a point p ∈ U , a positive number r with r < dist(p, ∂U ) and points a 1 , a 2 ∈ ∂B(p, r). We will prove |f (a 1 ) -f (p)| ≤ σ 3 |f (p) -f (a 2 )|. Let P ⊂ H 1 be the affine plane containing {p, a 1 , a 2 } and let us fix a point q ∈ H 1 so that → pq is orthogonal to P and let us restrict ourselves to the three-dimensional space containing {p, a 1 , a 2 , q} that we may identify with R 3 . For the sake of convenience, we will consider spherical coordinates (r, θ, ϕ) with origin p so that (r, θ, π/2) reduces to polar coordinates in P and so that a 1 and a 2 have coordinates (r, ±θ/2, π/2) where θ ∈ (0, π] denotes the non-oriented angle between → pa 1 and → pa 2 . Case 1: If θ ≤ 2π/3, then we may find ϕ ∈ [0, π/2] such that 2 sin ϕ cos(θ/2) = 1 and we set b = (r, 0, ϕ).

Both triangles T j with vertices {p, a j , b}, j = 1, 2, are equilateral by construction and they share a common side with end points at b and p.

Acknowledgments

We would like to thank the referee for several very helpful comments that improved the clarity of the exposition. The authors would also like to thank Jeremy Tyson for many productive discussions particularly with regard to the Appendix. We would further like to thank Kari Astala for helpful remarks related to the formulation of Theorem 2.

The key point of both of our proofs of Theorem 1 given here is contained in Proposition 1, where it is proved that the image of a triangle contains a disk of a definite size, exhibiting a certain length-area estimate. This approach goes back to Pfluger and was pushed forward by P. Koskela where |f (D(x, r))| denotes the Lebesgue measure of f (D(x, r)); see [START_REF] Koskela | Linear dilation and absolute continuity[END_REF] for details. The authors would like to thank Pekka Koskela for mentioning this similarity.

Definition 1 .

 1 Let f : U → V be a homeomorphism between planar domains. When D(z, r) = {w ∈ C :|z -w| ≤ r} ⊂ U , define M (z, r) = sup{|f (z) -f (w)| : |z -w| = r}, and m(z, r) = inf{|f (z) -f (w)| : |z -w| = r}. Suppose that K ≥ 1. The mapping f is said to be K-quasiconformal if H(z) = lim supr→0 M (z, r) m(z, r) ≤ K for a.e. z ∈ U , and if H(z) is bounded in U . In his book [6] John Hubbard obtained a new characterization of quasiconformal mappings. Let T be a closed topological triangle with specified vertices, L(T ) = max{|a -b| : a, b are vertices of T } and l(T ) = min{|a -b| : a, b are vertices of T }. We define skew(T ) = L(T ) l(T ) .

Lemma 2 .

 2 Let |z| ≤ 1/8 and suppose that |θ ± -(±π/3)| ≤ 1/8. Then the angle ϕ between e iθ+ -z and e iθ--z which crosses the positive real axis belongs to (π/3, π).

  γ 2 has end points on C(p, 1/4) which are exactly 2π/3 radians apart, and (3) for all points t ∈ γ 2 we have |f (t) -f (p)| ≤ σµ. Let p ∈ ∂T be such that d(f (p), f (p )) = µ and let γ = f -1 ([f (p), f (p )]). Since D(p, 1/4) is contained in the interior of T , we may consider the component γ 1 of γ ∩ D(p, 1/4) that contains p, and we denote by w ∈ C(p, 1/4) the other end point of γ 1 .

( 1 )

 1 γ 3 ⊂ D(p, 1/4) ∪ D(a, 2 -6 ) ∪ D(b, 2 -6 ); (2) γ 3 has both end points on C(p, 1/4 + √ 3 • 2 -7 ); (3) for all points t ∈ γ 3 , |f (t) -f (p)| ≤ σµ(1 + 2σ 3 ).Let D a = D(a, 2 -6 ) and D b = D(b, 2 -6 ). Let γ 2a and γ 2b be the components of γ 2 ∩ D a and γ 2 ∩ D b that have end points at a and b respectively.

  We proceed similarly near b and define a curve γ 3b contained in D b with end points at b and at some point on the intersection of the boundary of D b and S b (defined analogously to S a ) such that for all t ∈ γ 3b we have |f (t) -f (p)| ≤ σµ(1 + 2σ 3 ); as above, γ 3b intersects the circle C(p, 1/4 + √ 3 • 2 -7

8

 8 and similarly for b and B. Note that | arg(a -p)| + | arg(b -p)| = 2π/3 and |p -q|/(1/4) = 2 -7 ≤ 1/8.

Case 2 :

 2 Thus |f (p) -f (a 1 )| ≤ σ|f (p) -f (b)| ≤ σ 2 |f (p) -f (a 2 )|. If θ > 2π/3, consider the equilateral triangle T 0 with vertices p, a 1 and b where b is the image of a 1 under a rotation in P of angle π/3 so that the smaller angle between b and a 2 is less than or equal to 2π/3. Thus by Case 1|f (p) -f (b )| ≤ σ 2 |f (p) -f (a 2 )|.Then since the triangle T 0 has sides with endpoints at p and a 1 , and p and b we have|f (p) -f (a 1 )| ≤ σ|f (p) -f (b )| ≤ σ 3 |f (p) -f (a 2 )|.

  and S. Rogovin who proved that the ACL property of a homeomorphism f between open sets of R n , n ≥ 2, could be established from anL 1 -control of (5) k f = lim inf r→0 diam(f (D(x, r))) n |f (D(x, r))| 1/(n-1)

Proof. Let the assumptions of Theorem 2 be satisfied. Theorem 1 implies that f is locally quasiconformal and hence differentiable with f z = 0 at almost every point of U . Let z 0 be a point of differentiability such that skew(f, z 0 ) ≤ σ. We compute the maximum possible value for H(z 0 ), which yields the same upper bound for the dilatation of f at z 0 . Since H(z 0 ) is invariant under Möbius transformations we may compose with translations, a dilation and a rotation to assume that

Then f (z) = z + f z (z 0 )z + (z) where (z)/|z| tends to 0 as z tends to z 0 , and thus skew(f, z 0 ) = skew( f , z 0 ) where f (z) = z + f z (z 0 )z. It now suffices to prove that for all affine mappings f that can arise in this way, under the assumptions of Theorem 2, the maximal dilatation of f is at most K(σ). This then shows that also the maximal dilatation of our original mapping f is at most K(σ), as required. Thus from now on we only consider f . We will first compute the skew of f , and then express K( f ) in terms of skew( f ).

and all r > 0. This implies skew( f (T )) where T is an equilateral triangle is invariant under translations, complex conjugation and dilations of T . Thus for all equilateral triangles T ,

Indeed, suppose T has vertices A, B and C, and skew(T

First we translate A to the origin, and then we dilate T so its side lengths are equal to 1. If AB is π/3 radians clockwise from AC, it is clear that our statement is true; otherwise we take the complex conjugate of T to change the orientation of T and then, since skew( f (T )) is invariant under complex conjugation of T , our claim is true.

Hence we have skew

Let µ = f z (z 0 ). If µ = 0, then skew( f ) = 1, so we assume that 0 < µ < 1. Write ν = µ + µ -1 > 2 and β = e iπ/6 . Let w ∈ C with |w| = 1. We have


Rabab Bouziane

Erven Rohou

Abdoulaye Gamatié

Partial Worst-Case Execution Time Analysis

Keywords:

Computing the worst-case execution time (WCET) of tasks is important for real-time system design. The industry and research communities have developed a wealth of techniques to compute relevant WCET approximations. Traditionally, WCETs are estimated at the granularity of a function (or task). We propose an approach to estimate partial WCET (pWCET), i.e., the worst-case execution time between two locations in a function, such as basic blocks or instructions. Our technique is derived from the well-known implicit path enumeration technique. It takes into account both the control flow graph and the architecture (pipeline and cache hierarchy). Some useful applications of such pWCETs are motivated in this paper.

Motivation for Partial WCET Estimates and Positioning

Real-time systems are composed of tasks that must deliver their results within a well-defined time-frame. Designers of such systems compute an upper bound of the worst-case execution time of the tasks of a system such that any execution of the task takes less time than the estimate (the bound is said to be safe). In addition, to be useful, the bound shall be as close as possible to the actual worst-case (the bound is tight). Traditional WCET estimates are computed at the granularity of a function (or task). This is convenient for both computation and exploitation of the results. Functions are well-defined code fragments with a single entry and few exits, a few parameters, and one or no return value. Compilers and analysis tools have developed extensive theory and a wealth of tools based on basic blocks, control flow graphs, and graph theory to deal with functions. Yet, designing computing systems requires careful attention to a number of aspects such as performance, energy consumption, or security concerns. Computing partial worst case execution time (pWCET) estimates proves useful in many contexts as discussed below.

Example 1: Performance debugging. Many tools exist to measure the actual (average) execution time of regions of interest in code and identify performance bottlenecks. Pinpointing fragments with high WCET is more difficult because the worst-case may occur in rare circumstances and may not be easy to actually expose, hence usual profiling techniques do not apply. pWCET is useful for developers to identify potential (worst-case) performance bottlenecks in their applications and focus their effort in the relevant code fragments. In multicore systems, performance can be limited by contention on shared resources (caches, bus...), and interactions between tasks must be taken into account when estimating WCET. Without precise knowledge about the occurrence of competing events in the different cores, pessimistic values must be considered. pWCETs are useful to split a task in finer grain fragments, whose executions can be proven to not overlap. This results in fewer contentions and tighter overall WCET. Execution time also matters for security: some software attacks consist in injecting new code in a target application. A protection may consist in computing the WCET of a task and verifying at runtime that the actual time does not exceed the computed value. Any timing anomaly suggests an intrusion. Hence, if we consider the example of Figure 1(a) (taken from the Mälardalen benchmarks [START_REF] Gustafsson | The Mälardalen WCET benchmarks: Past, present and future[END_REF]) which consists of an unbalanced if-statement: the left branch, starting at block 26, has a pWCET of 78 900 cycles and the right branch, starting at block 33, has 629 524 cycles, then bounding the execution time of the function to the latter value would be overly pessimistic when the left branch executes. pWCET for each branch is much tighter.

Example 2: Energy-efficiency. Energy consumption is another major concern driving the design of a computing system. It led to the development of many techniques such as power gating, DVFS, non-volatile memories... Knowing the WCET of the remainder of a computation makes it possible to apply optimizations. Consider Figure 1(b), and assume that the execution has reached block 38 in advance with respect to its (pessimistic) bound. This slack time combined with the knowledge of the rest of the computation makes it possible to apply energy saving techniques, such as reducing the clock frequency or switching the task to as less power hungry core (as in Arm's big.LITTLE architecture). In the context of Internet of Things (IoT), many systems do not have any exchangeable battery. They harvest energy from physical phenomena (light, vibration...) into a capacitor, and run as long as there is energy left. For these intermittently powered systems, it is crucial to guarantee that the system stops at predefined locations. In other words, when a fragment starts executing, we must guarantee that execution will reach the next checkpoint where the state can be safely stored. Combining pWCET estimates with an energy model of the system is a promising way to ensure that the system makes forward progress and never runs out of power at unwanted locations. Designers also stated incorporating non-volatile memories in their products. On standard STT-RAM, non-volatility refers to a 10-year retention period. However, whenever shorter retention is acceptable, cheaper designs are possible [START_REF] Sun | Multi retention level STT-RAM cache designs with a dynamic refresh scheme[END_REF]. We envisioned a system with several memory banks designed at various energy/retention points [START_REF] Bouziane | How Could Compile-Time Program Analysis help Leveraging Emerging NVM Features?[END_REF]. By computing the WCET between a memory write and its subsequent reads, we can assign writes to the most appropriate bank. The loop is marked to iterate a maximum of ten times. ANNOT_MAXITER is a macro that stores this information in a dedicated ELF section of the binary, to be retrieved by Heptane and attached to the CFG. Some related work on WCET estimation. Wilhelm et al. [START_REF] Wilhelm | The worst-case execution-time problem -overview of methods and survey of tools[END_REF] presented an overview of methods and existing WCET estimation tools. Two classes of methods are distinguished: static versus measurement-based. Static methods do not rely on real hardware executions. They analyze the code itself, combine the control flow graph with a model of the hardware architecture, and produce an upper bound of this combination. On the other hand, measurement-based methods execute the code on real hardware or a simulator for certain inputs. Then, based on the measured times, the minimal and maximal execution times are derived. Therefore, static methods are safer and guarantee that the execution time will not be higher than the obtained bound. Recently, Jacobs et al. [START_REF] Jacobs | WCET analysis for multi-core processors with shared buses and event-driven bus arbitration[END_REF] considered a special case of pWCET. They focus on interference of concurrent tasks sharing a bus, and they compute for how many cycles concurrent cores may be granted access to the resource in any time interval of a given length.

i n t main ( void ) { i n t i ; i n t a =10 , b =20 , c = 0 ; a = 6 ; i f ( a>b ) { b=a ; a = 2 0 ; c = 1 0 ; f o r ( i = 0 ; i <c ; i ++) { ANNOT_MAXITER( 1

Background: the Heptane WCET Estimation Tool

We consider Heptane [START_REF] Hardy | The Heptane Static Worst-Case Execution Time Estimation Tool[END_REF], a static WCET estimation tool, divided in two parts: HeptaneExtract and HeptaneAnalysis. HeptaneExtract generates the control flow graph G from a program written in C language. Then, it identifies the different loops, attaches the loop bounds information provided in the source file and attaches the instruction addresses based on the binary file. Heptane does not include the analysis of maximum number of loops iterations, which are not always statically computable in the general case. Thus, loops must be annotated by the user with their maximum number of iterations (maxiter) 1 . Afterwards, HeptaneAnalysis implements IPET (Implicit Path Enumeration Technique) along with cache analysis techniques for several cache architectures. Static WCET estimation methods are divided into two steps: high level analysis and low level analysis (see Figure 3). The high level analysis consists of determining the longest execution path. The low level analysis takes into consideration the micro architecture. 1 External tools such as oRange [START_REF] De Michiel | Partial flow analysis with oRange[END_REF] are able to provide loops upper bounds of C programs in some cases. For the high level analysis, Heptane performs an IPET analysis, based on Integer Linear Programming (ILP) formulation of the WCET estimation problem. The program flow is mapped into a set of graph flow constraints. An upper bound of the program's WCET is then obtained by maximizing the following objective function : max i n i × w i where w i is the duration of the basic block i (constant in the ILP problem) determined by the low-level analysis, and n i is the number of times the basic block i is executed (variable in the ILP problem). For the low level analysis, Heptane performs data address analysis, a cache analysis and pipeline analysis. Note that Heptane performs context-sensitive analysis, which means that every call path of a function is analyzed separately.

Contribution: Computing pWCET Estimates

We first present the general principle of the pWCET Estimation, described through an algorithm. Then we validate it on the Mälardalen Benchmarks [4]2 .

pWCET Estimation Algorithm

A given program is given as a regular executable (both ARM and MIPS instruction sets are supported), in binary format, and an entry point (function main). Considering two basic blocks A and B, we are interested in estimating the worst-case execution time from block A to block B. We first compute the WCET estimate for the entire program, which consists in computing for each basic block its WCET estimate, and its worst case execution frequency. Through this step, we obtain the system constraints of the original ILP that we want to use later for pWCET estimations, as well as the contextual analysis. Secondly, we compute the set of blocks and edges that can be traversed in any path from A to B. Consider the example on Figure 2, where we selected block 23 for A and 26 for B. All blocks colored in light blue must be considered, which means that all the blocks in the different paths leading to B must be considered. We achieve this by a Depth First Traversal -like (DFS-like) walk on the CFG, starting from node A, and we reach B or a block that reaches B. Note that the WCET path from A to B is not necessarily on the overall WCET, i.e, the path from A to B may not be a part of the longest path in the program, thus, the WCET from A to B is not a sub-WCET of the program's WCET. Then, we compose a new ILP problem for the sub-graph G obtained from DFS-like, to compute the WCET from A to B, see Algorithm 1 for the details. Therefore, the objective function will include the nodes in G along with the callee nodes (with the callee context) if there is a function call. Moreover, in order to tighten the WCET (make it less pessimistic), we analyze the maxiter annotations. The potential for improvement comes from the fact that the annotation applies to the execution of the entire function, while we consider only a subgraph. We take into account the following cases, as illustrated in Figure 4.

1. The backedge is part of the subgraph, the loop may execute its maximum number of iterations on a path from A to B (as in Figure 4(a)). We do not modify the value of maxiter.

2. The backedge is not part of subgraph, as in Figure 4(b). B is necessarily reached in the same iteration as A. We set maxiter=0 so that the ILP formulation for the subgraph does not consider pessimistic frequencies due to the loop structure.

3. The backedge is part of subgraph, but it does not contribute to any cycle, as in Figure 4(c). Reaching B requires taking the backedge just once, hence, we set maxiter=1.

So, the new ILP problem has a new system constraints, slightly different from the original one to consider the above cases. Note that the maxiter modification is applied to all backedges that are not part of the subgraph. The whole implementation of the pWCET algorithm has been done inside Heptane, as shown in Figure 3. Given a start node A and an end node B, the high-level analysis of Heptane performs IPET analysis along with the contextual analysis to estimate WCET and then performs the DFS-like search and creates a new ILP problem using the contextual analysis related to the output of the DFS-like and the maxiter analysis.

Application to a benchmark-suite

We experimented with the Mälardalen Benchmarks, a typical suite for WCET-related experiments. Benchmarks were compiled for ARM using GCC. As customary with real-time systems, no optimization is applied (optimization level -O0). The reason for this is the need to Modify w i based on whether A and/or B are inside a loop or not Maxiter analysis (see Figure 4) keep source-level annotations consistent with the binary representation. Compiler optimizations heavily restructure the program representation, to the point that the CFG representation at binary level cannot be matched with the source level, making annotations invalid 3 . The ILP problems are solved by lp_solve or cplex. For the sake of brevity, we present two Mälardalen benchmarks: expint and crc, see Figure 1. For expint, we observe that the WCET of the subgraph from the block 33 to 49 is much higher that the WCET of the subgraph from the block 26 to 49. For crc, we see that the CFG can be divided to two parts: from block 29 to block 38 and from block 38 to block 47. This kind of graphs can actually be illustrated in security where attacks like code injection may occur. Thus, by computing the pWCETs of the two sub-graphs, we can verify at run-time that the real execution time is not higher than the estimated WCET. We are not aware of any previous work computing estimations of partial WCET, which makes it impossible to compare with prior techniques. To illustrate the potential of our technique, we show that we can encompass previous work, proposed by Avila et al. [START_REF] Avila | Impact of automatic gain time identification on tree-based static WCET analysis[END_REF]. The difference between the estimated WCET and the actual execution time is known as gain time. Early identification of gain time requires to obtain pWCETs of the code instead of considering the code as a whole. We successfully applied our algorithm on their toy program. In Figure 5, the example is taken from [START_REF] Avila | Impact of automatic gain time identification on tree-based static WCET analysis[END_REF] where the authors placed three gain points: gp1, gp2 and gp3, in which the time actually consumed by the program is measured. These measurements are used to identify all sources of pessimism of WCET analysis. In Figure 6, different pWCETs are computed to cover possible paths between gain points. 1 reports several pWCET computed for some nodes within the CFGs of expint and crc, presented in Figure 1. In the case of expint, the function consists in a conditional statement whose branches are heavily unbalanced: the right branch is almost 9× longer than the left one in the worst case. This is mostly due to the presence of a doubly nested loop, and a more complex computation. Recalling the discussion on performance debugging in Section 1 (Example 1), the much tighter bound for the WCET on the left branch makes it possible to make more informed decisions regarding performance analysis or security assessment. Regarding crc, the function consists in two computations, scheduled one after the other. Again, we observe a large unbalance between the two subgraphs, the first one requiring 27× more cycles in the worst case, due to a larger tripcount, and the presence of a function call in the loop body. Recalling Example 2 from Section 1, the slack time when reaching block 38 may well be in the order of the remaining of the computation, making it possible to apply aggressive energy saving techniques, or, in the IoT context, skipping a checkpoint. Note that the pWCET estimates depend on the structure of the CFG and the number and type of instructions inside each block, but also on the capability of Heptane to apply address and cache analysis to bound the number of cache misses.

i n t main ( void ) { i n t i , j = 0 ; f o r ( i = 0 ; i < 3 ; i ++) { ANNOT_MAXITER ( 3 )

Complexity of pWCET computation.

Concluding remarks

In this work, we presented many uses of partial worst case execution time (pWCET) estimates. We showed how to use Heptane, a static WCET estimation tool, to compute pWCETs, based on ILP formulations. Different applications of such an approach are possible as it can be linked to other WCET estimation problems like gain time identification. Hence, pWCET opens many opportunities as discussed in the introductory section of this paper. Future work will consist in leveraging this information in order to deal with the energy efficiency of programs. In particular, we consider exploiting pWCET for reducing energy consumption.

Figure 1 :

 1 Figure 1: Motivation for pWCET on sample control-flow graphs.

Figure 2 :

 2 Figure2: A sample program with its associated Control Flow Graph (CFG). The loop is marked to iterate a maximum of ten times. ANNOT_MAXITER is a macro that stores this information in a dedicated ELF section of the binary, to be retrieved by Heptane and attached to the CFG.

Figure 3 :

 3 Figure 3: Heptane implementation enhanced with the proposed pWCET analysis (additions shown in yellow).

Figure 4 :

 4 Figure 4: Handling of maxiter

Algorithm 1 pWCET algorithm 1 : 3 :list of nodes encountered in possible paths 6 : 7 : 8 :

 13678 procedure DFS-LIKE(A,B)for all nodes N visited in the DFS and not included in the output do procedure GENERATION OF THE ILP PROBLEM FROM A TO B max i n i × w i where i ∈ L the rest of the basic blocks are excluded by setting their w i to 0 for all calls in L do9: add to the objective function the callee nodes with their callee context 10:

Figure 5 :

 5 Figure 5: Gain time computation on example from Avila et al. [1].

Figure 6 :

 6 Figure 6: pWCET estimates (cycles) between gain points

Table 1 :

 1 Computation time is hardly noticeable on a modern PC for the Mälardalen benchmarks. For larger problems, solving the ILP problem can require large amounts of time. We add invocations to the solver for each pWCET, but these problems are also much smaller than the overall problem. Since solving ILP problem has exponential complexity, we expect that smaller problems are much faster to solve.

	crc	expint

pWCET Estimation for crc and expint benchmarks Table

http://www.mrtc.mdh.se/projects/wcet/benchmarks.html

Li et al.[START_REF] Li | Traceability of Flow Information: Reconciling Compiler Optimizations and WCET Estimation[END_REF] have successfully traced annotations throughout compiler optimizations, but this requires heavy compiler machinery and it is not the focus of this work.

Acknowledgments

We would like to thank Loïc Besnard, Damien Hardy, and Isabelle Puaut for fruitful discussion and feedback on this work. This work has been funded by the ANR Continuum project, under the grant ANR-15-CE25-0007-01.

Bibliographie