Jaume Banus
email: jaume.banus-cobo@inria.fr

Marco Lorenzi

Oscar Camara

Maxime Sermesant

Large Scale Cardiovascular Model Personalisation for Mechanistic Analysis of Heart & Brain Interactions

Keywords: 0D model, Cardiovascular modelling, Personalisation, White matter damage, Brain damage, Atrial fibrillation

Cerebrovascular diseases have been associated with a variety of heart diseases like heart failure or atrial fibrillation, however the mechanistic relationship between these pathologies is largely unknown. Until now, the study of the underlying heart-brain link has been challenging due to the lack of databases containing data from both organs. Current large data collection initiatives such as the UK Biobank provide us with joint cardiac and brain imaging information for thousands of individuals, and represent a unique opportunity to gain insights about the heart and brain pathophysiology from a systems medicine point of view. Research has focused on standard statistical studies finding correlations in a phenomenological way. We propose a mechanistic analysis of the heart and brain interactions through the personalisation of the parameters of a lumped cardiovascular model under constraints provided by brain-volumetric parameters extracted from imaging, i.e: ventricles or white matter hyperintensities volumes, and clinical information such as age or body surface area. We applied this framework in a cohort of more than 3000 subjects and in a pathological subgroup of 53 subjects diagnosed with atrial fibrillation. Our results show that the use of brain feature constraints helps in improving the parameter estimation in order to identify significant differences associated to specific clinical conditions.

Introduction

Cerebrovascular diseases are related to a variety of heart diseases such as heart failure [START_REF] Ois | Heart failure in acute ischemic stroke[END_REF] or atrial fibrillation (AF) [START_REF] Benjamin | Heart disease and stroke statistics-2018 update: A report from the american heart association[END_REF], sharing several risk factors such as cholesterol, diabetes or high blood pressure. In parallel, it has been shown that stroke doubles the risk of dementia [START_REF] Azarpazhooh | Concomitant vascular and neurodegenerative pathologies double the risk of dementia[END_REF]. All these connections suggest a common underlying pathological process that links cardiac function with brain atrophy. Large scale analysis on databases combining cardiovascular and brain data from the same individuals are thus required to demonstrate and better understand the interaction between brain and heart. To this end, studies such as the UK Biobank aim at the acquisition of multi-modal databases containing both heart and brain imaging information [START_REF] Sudlow | Ukbiobank an open access resource for identifying the causes of a wide range of complex diseases of middle and old age[END_REF]. Thanks to these databases ongoing studies have focused on the study of the relationships between cardiovascular risk factors and image-derived features, such as subcortical volumes [START_REF] Simon R Cox | Associations between vascular risk factors and brain MRI indices in UK Biobank[END_REF]. However, a number of fundamental descriptors of the cardiac function are not possible to obtain in-vivo, i.e: heart contractility or fibers stiffness. Personalised modelling approaches allow us to estimate these descriptors and gain insight of the cardiac function, allowing us to obtain more reliable results and relate them to brain damage information.

Personalizing a cardiovascular model for a given subject is an ill-posed problem that implies estimating the model parameters so that the simulation behaves as close as possible to the available clinical data. In this work we will focus on a 0D model of the whole cardiovascular system. Previous studies have used multiscale models to describe the whole-body circulation and study the venous blood flow in the brain [START_REF] Müller | Enhanced global mathematical model for studying cerebral venous blood flow[END_REF]. However, their modeling of the heart chambers does not take into account the contractile and elastic properties of the heart. While other models of the whole-body circulation are available [START_REF] Safaei | Bond graph model of cerebral circulation: Toward clinically feasible systemic blood flow simulations[END_REF][START_REF] Blanco | An anatomically detailed arterial network model for onedimensional computational hemodynamics[END_REF], to our knowledge, no explicit modelling study relating neurodegeneration and cardiovascular parameters has been done.

In this paper we aim to study the relationship between cardiovascular indicators and brain volumetric features extracted from the imaging data available in UK Biobank, through the personalisation of a cardiovascular lumped model using the approach presented in [START_REF] Molléro | Populationbased priors in cardiac model personalisation for consistent parameter estimation in heterogeneous databases[END_REF]. The use of this approach allows us to tackle the ill-posedness nature of the personalisation and identify plausible and coherent solutions across the population. To achieve that, we define a regularisation term that can be extended to take into account features not present in the lumped model, allowing to explore the effect of including brain features as additional constraint. We apply this framework to a large cohort composed by more than 3 000 subjects for which cardiac and brain information was jointly available in the UK Biobank. To illustrate how to exploit the framework to identify meaningful clinical relationships, we applied it in a subset of subjects diagnosed with AF, which is considered as an independent risk factor for stroke and dementia [START_REF] Benjamin | Heart disease and stroke statistics-2018 update: A report from the american heart association[END_REF][START_REF] Alonso | Atrial Fibrillation, Cognitive Decline and Dementia[END_REF]. We identified statistically significant associations between the personalised model parameters and brain volumetric features that match findings reported in previous clinical studies.

The paper is structured as follows: in section 2.1 we detail the data preprocessing and inclusion criterion for the whole-population analysis. Following, in section 2.2 we present the lumped model and how to take into account the subject's information to constrain the solution space in the personalisation. Next, in section 2.3 we assess the impact of our model in determining significant relationships between the estimated cardiac parameters and brain damage using the AF subset. Finally, in section 3 we present the obtained results.

Methods

Data pre-processing and inclusion criterion

Our analysis includes data from UK Biobank participants for which all brain image modalities and all cardiac-image derived indicators were available, for a total of 3783 subjects. In the available cardiac images it was possible to quantify the cardiac function using indicators such as stroke volume (SV), cardiac output (CO) or ejection fraction (EF). Multi-modal brain MRI images allowed the extraction of image-derived features such as brain tissue volumes and white matter hyperintensities (WMHs), one of the most common indicators used to assess neurological damage.

Using FLAIR MR images, WMHs were segmented by the lesion prediction algorithm (LPA), available in the lesion segmentation toolbox (LST) [START_REF] Schmidt | Bayesian inference for structured additive regression models forlarge-scale problems with applications to medical imaging[END_REF] for SPM 3 . FLAIR MR images were pre-processed following the protocol described in [START_REF] Alfaro-Almagro | Image processing and quality control for the first 10,000 brain imaging datasets from uk biobank[END_REF], in which gradient distortion correction and defacing were performed. After discard subjects for which pre-processing (449) or segmentation of WMHs (250) failed, the final number of available subjects was 3 084. From the segmentations we extracted the total volume of WMHs and the number of lesions. All brainrelated volumes were normalized by head size.

Cardiovascular lumped model

The cardiovascular personalisation of the subjects was performed by using the 0D model shown in Figure 1 which is a simplification of a 3D cardiac electromechanical model [START_REF] Chapelle | Energy-preserving muscle tissue model: formulation and compatible discretizations[END_REF] derived in [START_REF] Caruel | Dimensional reductions of a cardiac model for effective validation and calibration[END_REF]. In the 0D version, which assumes spherical symmetry, the myocardial forces and motion can be described by the inner radius (R 0 ) of the ventricle. Deformation and stress tensors are also reduced to 0D forms, which allow us to characterise the heart contractile (σ 0 ) and elastic (C 1 ) properties of the heart.

The model M consists in a set of ordinary differential equations with P M parameters, e.g. maximum contraction of the heart fibers or its stiffness. The state variables of the model are denoted by O M , e.g. arterial or venous pressures, and they describe the state of the system. During the personalisation we are interested in a subset of n state variables, such that O = (O 1 , O 2 , ..., O n ), and we vary a subset θ of the P M model parameters. We consider O(θ) the set of state variables generated by the model for a given set of θ. The goal is to find θ * such that O(θ * ) best approximates the target features Ô.

Due to the high dimensionality and non-convexity of this inverse problem, we solve it with the CMA-ES optimization algorithm based on evolution strategies [START_REF] Hansen | The cma evolution strategy: A comparing review[END_REF]. CMA-ES minimizes a given error function by combining maximum likelihood principles with natural gradient descent on the ranks of the point scores (i.e: the score of each individual at every generation). The error function S(θ, Ô) is defined as the L 2 distance between O(θ) and Ô. Since each target feature has different range of values we defined a tolerance interval, T ol, for each feature i to be able to compare the different outputs. This can be formalized as shown in Equation 1:

S(θ, Ô) = n i=1 (O i (θ) -Ôi ) 2 T ol i (1)
Based on the available clinical data, we selected the following target features for the personalisation; stroke volume (SV), ejection fraction (EF), diastolic blood pressure (DBP), mean blood pressure (MBP) and end-diastolic volume (EDV). Considering the uncertainty of the measured data, the tolerance interval for each feature was set to 10 ml for the SV and the EDV, 200 Pa for the DBP and the MP, and 5% for the EF. Finally, the personalized parameters of the cardiovascular model were maximum contraction of the heart fibers σ 0 , stiffness of the heart fibers C 1 , peripheral resistance R p , venous pressure P ven , and the characteristic time τ of the aorta, which defines the time that takes for blood pressure to decrease from systolic to the systemic, or "asymptotic" value. We selected these parameters based on a sensitivity analysis in which we assessed the influence of each parameter over the selected target features.

Since the solution of equation 1 is non-unique, there is an observability difficulty in this personalisation problem. To tackle this issue, we used the iterativeupdate prior (IUP) approach presented in [START_REF] Molléro | Populationbased priors in cardiac model personalisation for consistent parameter estimation in heterogeneous databases[END_REF] to introduce constraints in the fitting process. In the IUP method a regularization term, R(θ, µ, Σ), is used to reduce the variability in the estimation of the parameters. The regularization constrains the directions in which we explore the parameter-space by using the relationships among the model parameters. Formally, the regularization term is parameterized by an expected value µ and by a covariance matrix Σ encoding the relationships across parameters.

R(θ, µ, Σ) = (θ -µ) T Σ -1 (θ -µ). (2)
Therefore, the fitting score becomes:

S(θ, Ô, µ, Σ) = S(θ, Ô) + γR(θ, µ, Σ), (3)
where γ defines the relative importance of the regularization term. This term is updated at each IUP iteration, using the obtained mean value of the fitted parameters and the estimated covariance in the previous iteration.

Accounting for brain information in the 0D model. R(θ, µ, Σ) can be extended to incorporate relationships with features not present in the cardiovascular model. In our setting, we included in the regularization term the extended feature space corresponding to the concatenation of the model parameters, θ, with the brain and clinical information, here denoted by φ. We used the total brain volume, the ventricles volume, the obtained WMHs features, age, sex and body surface area (BSA). Therefore, the problem in equation 3 becomes:

S(θ, Ô, µ, Σ, φ) = S(θ, Ô) + γR(θ, φ, µ, Σ). (4)
Equation 4 now accounts for a covariance term constraining the parameters according to the extended set of information. We have used 10 IUP iterations and assessed the results at different γ levels (0.1, 0.5, 2 and 10). The optimisation is performed over the logarithm of the parameter values.

Atrial fibrillation analysis.

Considering the dataset obtained after the pre-processing described in section 2.1 we had access to 53 subjects diagnosed with AF. Using bootstrapping we sampled 100 control groups of the same sample size of the AF group and without any significant difference in age, sex and BSA. The sampled controls came from the subset of subjects without any diagnosed cardiovascular disease (n = 2022). We applied the framework described in the previous section to each bootstrap subset composed by the AF group and sampled control group, to obtain the bootstrap distributions of the correlations between cardiac and external parameters. This approach allowed us to exploit the dataset variability for assessing the difference between cardiac and brain associations.

Results

Whole-population analysis. As expected, we observed that as the value of γ is increased (i.e. more regularisation), the fitting error increases and at the same time the number of outliers is reduced and the estimated distributions have lower variability, as can be seen in Figure 2a. We can observe that strong regularisation even shrinks some parameters close to a constant value, implying that those parameters cannot be observed from the available data. Looking at the correlation of the model parameters with the external features we note the strong correlation between the left ventricle size, R 0 , and the BSA, even for low γ values. Moreover, there is a positive correlation between peripheral resistance, the WMHs volume, and brain ventricles volume, which are at the same time negatively correlated to aorta characteristic time, τ . The number of WMHs lesions and age followed the same correlations pattern, but due to space issues they have been omitted. On the other hand, brain volume is positively correlated with τ and peripheral resistance. An increase in peripheral resistance can be associated to higher DBP, while a decrease in contractility, τ , can be interpreted as an increase in arterial stiffness leading to high SBP. Both, DBP and SBP, have been previously associated to WMHs [START_REF] Modir | Stroke blood pressure and white matter hyperintensity volume -a review of the relationship and implications for stroke prediction and prevention[END_REF]. In figure 2b we note that the significant correlations present when no regularization is applied become stronger as regularization increases, while the non-significant correlations stay close to zero. This behavior is expected since regularization is constraining the space of feasible solutions. Therefore, as we increase γ we further limit the feasible parameter-space towards the set of solutions that satisfy the existing relationships between the parameters. In figure 2c we observe the obtained correlations when the external features are not taken into account. In this case, the solutions are constrained into a different parameter-space in which the relationships between the model parameters and the external features are lost. Limiting the interpretability when assessing the parameters estimations with respect external factors not present in the mechanistic model. Atrial fibrillation. In Figure 3 we observe a statistical description of the empirical distribution of correlations obtained from the bootstrap analysis done in the AF subset. The results are obtained with trade-off γ = 0.5, which in the whole-population analysis provided a good a balance between data-fit and regularization. We assessed the difference between the controls and AF groups correlations distributions using the Wilcoxon rank-sum test with a significance level of α = 0.05. Moreover, to consider the obtained results as significant we assessed if in the AF group the obtained correlations were statistically greater or smaller than 0 with a 5% significance level.

Bootstrap distribution of correlation between cardiac parameters and brain information

In the brain volume we observe the same correlations found in the wholepopulation analysis, but it can be seen that for the AF group these correlations are stronger, which suggests that in the AF subjects brain is more susceptible to cardiovascular factors. For the BSA we found the positive correlation with the left ventricle size observed in section 3 and a negative correlation with the maximum heart contractility σ 0 . In the number of WMHs lesions we can observe a positive correlation with the left ventricle size. The associations of BSA with σ 0 and WMHs with R 0 could be related to cardiac dilation due to an increased impairment of the functioning heart in AF. Interestingly, in the AF control subjects the correlation between WMHs and left ventricle size is negative. These findings suggest an association between AF and WMHs. Moreover, they agree with previous studies reporting an association between left ventricle remodelling and AF [START_REF] Seko | Association between atrial fibrillation, atrial enlargement, and left ventricular geometric remodeling[END_REF]. No significant associations were found for brain ventricles volume, WMHs volume and age.

Conclusions

We have modeled 3 084 subjects with a 0D cardiovascular model and we constrained the available parameter-space during personalisation by incorporating external features in the regularization term, allowing us to study their influence in the estimated model parameters. The use of this approach gives access to a generative model that allows to analyze the relationships between external features and non-observable parameters such as the characteristic time of aorta, τ , which we found to be related with brain-volumetric features. Using the same framework we assessed a clinical subgroup in which we have found meaningful clinical relationships, linking AF with WMHs and heart remodelling. Our model does not currently simulate the cerebral blood flow, while previous studies [START_REF] Müller | Enhanced global mathematical model for studying cerebral venous blood flow[END_REF] suggest that WMHs are due to more localize vascular impairments. This highlights the need to obtain a local flow characterization to estimate more relevant parameters. Moreover, the presented approach can be seen as a parameter selection approach. It allows to identify which parameters cannot be estimated from the available data and find a parameter subspace of solutions in which the non-observable parameters get close to constant values. The identification of the non-observable parameters coupled with human modelling expertise can help in the selection of a reduced subset of observable cardiovascular parameters for personalisation. Future work will go towards the local blood flow characterization in the brain, as well as towards the assessment of its spatial patterns, and the modelling of more brain atrophy indicators.

Fig. 1 .

 1 Fig. 1. Simplified schematic representation of the lumped model showing the parameters used in the personalisation. The 0D representation of the myocardial forces has been omitted for the sake of clarification. τ characterizes the contractility of the aorta, Rp the peripheral resistance and Pven the venous pressure.

Fig. 2 .

 2 Fig. 2. a) Estimated density distributions of the fitted parameters at different regularization levels. Initial and final distributions after 10 iterations in brown and blue respectively. The variability among the initial distributions is due to the variability in the sampling that CMA-ES performs during the optimization.Error b) Evolution of Spearman's rank correlation coefficient between the model parameters and the external parameters as the regularization level increases when external features are considered in the regularization and c) when external features are not considered. Model parameters being: maximum heart fibers contractility σ0, heart fibers stiffness, C1, left ventricle size R0, peripheral resistance Rp, aorta characteristic time τ , and venous pressure Pven. d) Error percentages with respect to the target features. Stroke volume (SV), ejection fraction (EF), diastolic blood pressure (DBP) and mean blood pressure (MBP). While end-diastolic volume (EDV) is not shown due to space issues, its error pattern was similar to the one observed in the SV

Fig. 3 .

 3 Fig. 3. Comparison of the Spearman's rank correlation coefficient bootstrap distributions obtained at γ = 0.5 between the personalised model parameters and the external features. Blue boxplots correspond to control groups and brown to AF subjects. * denotes that the correlations are significantly different according to the Wilcoxon ranksum test, and that in the AF group correlations are significantly greater or smaller than 0 (5% significance level).

https://www.fil.ion.ucl.ac.uk/spm

Acknowledgements. This work was supported by the Inria Sophia Antipolis -Méditerranée, "NEF" computation cluster. This research has been conducted using the UK Biobank Resource undder Application Number 20576 (PI Nicholas Ayache). Additional information can be found at: https://www.ukbiobank.ac.uk


Mosqueira

D Sóñora

C Carballeira

J J Ponte

F Vidal

T Grenet

J Mosqueira

Paraconductivity of granular Al films at high reduced temperatures and magnetic fields

Paraconductivity of granular Al films at high reduced temperatures and magnetic fields

I. INTRODUCTION

As summarized in Tinkham's textbook on superconductivity [START_REF] Tinkham | Introduction to Superconductivity[END_REF] (see also Ref. [START_REF] Skocpol | [END_REF]), the increase just above T c of the electrical conductivity relative to the normal-state contribution (the so-called paraconductivity, σ ) is explained in low-T c superconductors (LTSC) by taking into account two contributions: a direct (Aslamazov-Larkin, AL) contribution, due to Cooper pairs created by thermal agitation, and the indirect (Maki-Thompson, MT) contribution, associated with the quasiparticles created by the decay of these fluctuating pairs. However, this scenario does not explain the steep σ reduction at high reduced-temperatures, ε ≡ ln(T /T c ) 0.1, or magnetic fields, h ≡ H/H c2 (0) 1, H c2 (0) being the upper critical field linearly extrapolated to T = 0 K [3]. These difficulties are not completely overcome with the introduction of a momentum (or kinetic-energy) cutoff to eliminate the contribution of short-wavelength fluctuations [START_REF] Tinkham | Introduction to Superconductivity[END_REF][START_REF] Skocpol | [END_REF][START_REF] Larkin | Theory of Fluctuations in Superconductors[END_REF][START_REF] Glatz | [END_REF]. Alternative scenarios to explain this behavior have been proposed, particularly for the highly studied granular LTSC films, in terms of the reduction of the fluctuations dimensionality, in turn associated with the decoupling of superconducting grains or to a confinement due to electronic inhomogeneities [6][7][8]. Nevertheless, some central aspects, as for instance the role that may play the presence of MT contributions, still remain not well settled.

To address the long standing open problem of the σ behavior at high-ε and -h in LTSC, in this work we first present detailed σ (T, H ) measurements around T c in disordered aluminum films. Then, these data are quantitatively explained, also in the high-ε and -h regions, by introducing in the AL and the anomalous MT contributions the so-called total-energy cutoff, that takes into account the limits imposed * j.mosqueira@usc.es by the Heisenberg uncertainty principle to the shrinkage of the superconducting wavefunction [9,10]. The adequacy of this total-energy cutoff to explain the fluctuation-induced (Schmidt) diamagnetism at high-ε and h in several LTSC has been probed previously [START_REF] Mosqueira | [END_REF][12][13][14].

The choice of thin (10-100 nm thick) granular Al films is motivated by their experimental advantages: (i) They present a very well defined T c that minimizes the entanglement between the intrinsic resistivity rounding due to fluctuations and the extrinsic one associated with a possible T c distribution, mainly near the average T c [START_REF] Vidal | High-T c Superconductors and Related Materials, Material Science, Fundamental Properties, and Some Future Electronic Applications[END_REF]. (ii) For temperatures above ∼3T c the resistivity is almost T -and H-independent, allowing a reliable determination of the normal-state background. (iii) The films properties are determined by the ratio of Al evaporation rate to oxygen pressure and the substrate temperature during evaporation, both of which are easily controlled. As compared to clean systems, their properties are much less dependent on other conditions. Moreover, their superconducting properties present a remarkable correlation with the normal state resistivity [START_REF] Levy-Bertrand | [END_REF]. The interest of the σ studies in granular Al films is also enhanced by their implications in other aspects of these superconductors, including the Nernst effect above T c [17], and the understanding of the mechanisms that enhance their T c , a fundamental problem still open [8].

II. EXPERIMENTAL DETAILS

The films were grown by thermal evaporation of aluminum on 5 × 5 mm 2 thermally oxidized Si substrates at room temperature in a 1-1.5 × 10 -5 mbar oxygen pressure. The evaporation rate (0.2 nm/s) was monitored with a quartz balance. The films' microstructure was investigated with a scanning electron microscope (Zeiss FE-SEM Ultra Plus). As it may be seen in Fig. 1, due to the presence of oxygen in the chamber during the evaporation process, they present a granular structure with grain diameters in the 5-to 10-nm range. In the case of films A and B the nominal thickness is 10 nm and the substrate is essentially covered by a single layer of grains, while in the case of film C (100 nm thickness) they form three-dimensional agglomerates. A summary of the different characteristics of the samples used in this study is presented in Table I.

The resistivity, ρ, was measured with a commercial Physical Property Measurement System (Quantum Design) by using an inline contacts configuration with an excitation current of 5 μA, and under perpendicular magnetic fields up to ∼0.5 T (large enough to almost quench fluctuation effects above T c ). For the sake of clarity this section focuses on the data for film A. The results for the other studied films (B and C), that are consistent with the ones for film A, are presented in detail in the Appendix.

A. Analysis of σ for H = 0 in terms of the conventional AL and MT approaches

The ρ(T, H ) behavior around the superconducting transition is presented in Fig. 2. The T c value (2.07 K) was estimated as the temperature at which the resistivity falls to 50% the normal-state background, and the transition width, estimated as T = T c -T (ρ = 0), is only ∼0.02 K. T c is two times larger than in bulk Al [18], an effect early attributed to different finite-size effects [19][20][21][22][23][24], and that may be strongly affected by the degree of Josephson coupling between neighboring grains [8,25]. The maximum T c value observed in granular Al films is ∼3 K [8,18], so the appreciable resistivity rounding extending up to ∼6 K (see the inset of Fig. 2) can only be attributed to fluctuations. Moreover, as we will see below, the coherence length amplitudes of the films studied in this work are much larger than the grains average diameter (see Table I). This prevents the appearance of a T c distribution due to the different grain sizes, and of the associated FIG. 2. T -dependence of the resistivity of film A around T c for different perpendicular magnetic fields. Inset: Example for H = 0 and 500 mT of the procedure used to determine the background contribution by a linear fit above 8 K (indicated by an arrow). FIG. 3. ε-dependence of σ for H = 0. The dot-dashed line is the prediction of the 2D-AL approach. The agreement is improved by including the MT contribution (dashed line) but only at low ε. The introduction of a total energy cutoff (solid line) extends the applicability to the high-ε region. For comparison, the slope -2 corresponding to 0D fluctuations is indicated with a dotted line in the high-ε region. See the main text for details. effects near the average T c . The homogeneous nature of these materials is also supported by the fact that the energy gaps determined from tunneling experiments in similar granular Al films present no detectable broadening in spite of the grains size distribution [20].

The paraconductivity was obtained from the data in Fig. 2 through σ (T ) = 1/ρ(T ) -1/ρ B (T ), where the almost constant background resistivity ρ B (T ) (solid line) was obtained for each applied field by a linear fit to data above 8 K. The σ dependence on the reduced temperature for H = 0 is presented in Fig. 3 in a log-log scale. The dot-dashed line is the AL prediction for two-dimensional (2D) superconductors, given by [START_REF] Tinkham | Introduction to Superconductivity[END_REF][START_REF] Skocpol | [END_REF]

σ 2D AL (ε) = e 2 16 hd ε -1 , ( 1
)
where e is the electron charge, h the reduced Planck constant, and d = 10 nm the film thickness. As it may be seen, there is a strong discrepancy with the experimental data in both amplitude and critical exponent. However, a good agreement is obtained at low reduced temperatures (dashed line) by also taking into account the anomalous MT contribution for 2D superconductors [START_REF] Skocpol | [END_REF],

σ 2D MT (ε) = e 2 8 hd 1 ε -δ ln ε δ , ( 2
)
where δ is the relative shift in the transition temperature due to pair-breaking interactions. The δ value resulting from the fit for ε 0.03 is δ = 0.05, close to the value found in Al films with a similar sheet resistance (∼160 / ) [26,27].

The disagreement in the high-ε region could be interpreted in terms of a 2D-0D dimensional transition associated to the decoupling of the Al grains [6]. In fact, in the ε-region bounded by 0.4 ε 0.7, σ approaches the 0D-AL critical exponent (see Fig. 3). However, such a behavior could also be attributed to an overestimation of the statistical weight of high-energy fluctuation modes (with energies of the order of h2 /2m * ξ 2 0 , where m * and ξ 0 are the pairs effective mass and size), that may be corrected through the introduction of a cutoff in the energy of the fluctuation modes [9]. In what follows, this procedure will be applied to the calculus of σ 2D AL and σ 2D

MT to extend its applicability to the high-ε region.

B. Generalization of σ AL to high-ε and -h

A total-energy cutoff was used in Ref. [28] to extend the Ginzburg-Landau (GL) expression for σ 3D AL to the high-ε and -h regions. The same procedure may be used to derive σ 2D AL , just by taking into account that for 2D materials the spectrum of the fluctuations is frustrated along the perpendicular direction (the corresponding component of the fluctuations wave vector, q z is bounded by -π/d and π/d, and verifies ξ (0)q z 1). When applied to Eqs. (B.17) and (B.18) of Ref. [28], this leads to [29]

σ 2D AL (ε, h, c) = e 2 32 hd 1 h ψ 1 ε + h 2h -ψ 1 c + h 2h .
(3) Here ψ n is nth derivative of the digamma function, and c is the cutoff constant, that corresponds to the ε-value at which σ vanishes. In the zero-field limit (i.e., for h ε, c), Eq. ( 3) reduces to

σ 2D AL (ε, c) = e 2 16 hd 1 ε - 1 c , (4)
that in absence of cutoff (i.e., c → ∞) leads to Eq. (1).

C. Procedure to introduce a total-energy cutoff in the anomalous MT contribution

In what concerns the anomalous MT term [START_REF]A total energy cutoff has been already applied to study the regular MT[END_REF], the quasiparticles resulting from the decay of a fluctuation of momentum q give rise to an extra contribution to the conductivity that should be proportional to the superfluid density, n s (q, ε) ∝ [ε + ξ 2 (0)q 2 ] -1 , and to a diffusion time 1/Dq 2 (where D is the diffusion constant) unrelated to the superconducting relaxation time τ s (q, ε) [START_REF] Patton | Proceedings of the 13th International Conference on Low Temperature Physics[END_REF]. A detailed calculation leads to [START_REF] Thompson | [END_REF]

σ MT = π e 2 2 hV q 1 [q 2 + δ/ξ 2 (0)][ε + ξ 2 (0)q 2 ] ( 5
)
(V is the sample volume), that corresponds to Eq. ( 21) of Ref. [START_REF] Thompson | [END_REF] with the variable change q 2 → q 2 + δ/ξ 2 (0). This transformation accounts for the low momentum cutoff, q 2 min = δ/ξ 2 (0), to remove the divergence in σ MT occurring in low dimensional systems, allowing to start the sums and integrals in momentum space at q = 0 instead of cutting their low limit. A problem now arises when attempting to apply the total energy cutoff to this equation. While the term associated to n s (q, ε) has to be cut off, no restriction has to be imposed to the relaxation time of the quasiparticles. To solve these difficulties, let us note that the total-energy cutoff forbids the existence of fluctuations above ε = c, and that σ MT (ε, c) should vanish at this reduced temperature analogously to the direct contribution. Thus, instead of directly cutting the 10) evaluated with the same parameters as in (a). If the cutoff is not introduced (dashed line) there is an appreciable disagreement in the high-h region. (c) h -ε phase diagram indicating the σ amplitude (color scale). The data points represent the low-T applicability limit for the theoretical approach used in (a). They were obtained as the temperatures at which the theory overestimates by 10% the experimental data for each H . These points agree with the onset of the critical fluctuation region as evaluated from the H -dependent Ginzburg criterion [thick solid line, Eq. ( 11)]. The dashed line is the reduced upper critical field. See the main text for details. sum over q in Eq. ( 5), we have calculated σ MT (ε, c) by subtracting to n s (q, ε) the superfluid density at ε = c, n s (q, ε) -→ n s (q, ε, c) = n s (q, ε)n s (q, c), (6) which directly leads to

σ MT (ε, c) = σ MT (ε) -σ MT (c). ( 7
)
Following this procedure we reproduce the basic features of the inclusion of a total-energy cutoff (mainly, the vanishing of the fluctuation effects at ε = c), but acting only in the superconducting part of the MT-term. It is worth noting that, when calculating direct contributions to fluctuation effects, this procedure leads to the same results as the direct limitation of the sums over q by the total-energy cutoff condition.

D. Generalization of σ MT to high-ε and -h

According to Eqs. ( 2) and (7), in absence of an external magnetic field the anomalous MT contribution under a totalenergy cutoff is given by

σ 2D MT (ε, c) = e 2 8 hd ln(ε/δ) ε -δ - ln(c/δ) c -δ . ( 8
)
As it may be seen in Fig. 3, σ 2D AL (ε, c) + σ 2D MT (ε, c) (solid line) now fits the measured σ up to reduced temperatures as high as ε ≈ 0.7, where the uncertainty associated to the background determination (error bars) is already important. The fitting parameters are δ = 0.04 (close to the value obtained by fitting the expression without a cutoff) and c = 0.85, a value near the one that may be expected for dirty superconductors [9].

In what follows we obtain an expression for σ 2D MT to analyze the paraconductivity in presence of a perpendicular magnetic field (Fig. 4). Our starting point is Eq. (5) in the 2D limit, i.e., for ξ (0)q z 1. The magnetic field transforms the in-plane spectrum of the fluctuations into the one of a charged particle in a magnetic field [33]. Subsequently, one must replace q 2 xy by (n + 1/2)4eμ 0 H/ h (here μ 0 is the vacuum magnetic permeability and n the Landau-level index) and introduce the Landau degeneracy factor eμ 0 HS/π h, where S is the film surface. An example of the application of this procedure to the calculus of the AL term in the presence of a magnetic field may be seen, for instance, in Ref. [28]. This leads to

σ 2D MT = e 2 4 hd ∞ n=0 1 (2n + δ/2h + 1)[ε + h(2n + 1)] . (9)
After summing and taking into account Eq. ( 7) to introduce the total-energy cutoff, it results

σ 2D MT (ε, h, c) = e 2 8 hd 1 ε -δ ψ 0 ε + h 2h -ψ 0 δ + h 2h - 1 c -δ ψ 0 c + h 2h -ψ 0 δ + h 2h . ( 10
)
This expression reduces to Eq. ( 8) for h ε, δ, and includes only one additional free parameter, H c2 (0), that is implicit in h. Note also that for ε, δ, h c, Eq. (10) reduces to the σ 2D MT (ε, h) expression without cutoff (see Ref. [START_REF] Glatz | [END_REF]).

E. Analysis of σ for finite H in terms of the generalized AL and MT approaches

The solid lines in Fig. 4(a) are a fit of Eq. (3)+Eq. (10) by using the above δ and c values, letting H c2 (0) as the only free parameter. The data included in the fit correspond to magnetic fields up to 0.2 T and reduced temperatures down to ε = 0, although an excellent agreement is observed well beyond those limits (see below). The resulting μ 0 H c2 (0) is 0.82 T, which corresponds to a coherence length amplitude of ξ (0) = [φ 0 /2πμ 0 H c2 (0)] 1/2 = 20 nm. This value is larger than the average grains' diameter, indicating that they are strongly coupled. It is also larger than the film thickness (∼10 nm), consistently with the observed 2D behavior. As ξ (T ) is always larger than ξ (0), it is expected that no dimensional transition will take place upon increasing the temperature above T c , in particular the 2D-0D transition that would be associated to a decoupling of the grains [although this effect could still be present in films with a smaller ξ (0)].

A detail of the field dependence of σ for two selected ε values is presented in Fig. 4(b). The solid lines are Eq. ( 3)+Eq. (10) evaluated with the same parameters as in Fig. 4(a). An excellent agreement is observed up to h ≈ 0.4 (i.e., μ 0 H ≈ 0.3 T). For comparison, the dashed line for ε = 0.015 was obtained by using the same approach but without cutoff (i.e., by setting c → ∞). While a good agreement with the data is obtained for H → 0 (as expected after the analysis presented in Fig. 3), a notable disagreement is obtained in the finite field region. This shows that the introduction of a total-energy cutoff is not only needed to explain the data at high reduced temperatures, but also at high reduced magnetic fields, where the theory also overestimates the contribution of the high-energy modes [START_REF] Tinkham | Introduction to Superconductivity[END_REF][START_REF] Skocpol | [END_REF].

The theory agrees with the observed σ down to the data points in the h -ε phase diagram presented in Fig. 4(c). This limit is consistent with the H-dependent Ginzburg criterion for the onset of the critical fluctuation region in 2D superconductors (solid line) [34],

T /T c = 1 -h -2k B h/ξ 2 (0)d c, ( 11
)
as evaluated by using the ξ (0) value obtained above, and a specific heat jump at T c of c = 345 J/m 3 K, a value in reasonable agreement with the one obtained in similar granular Al films [35].

IV. CONCLUSIONS

The present results show that the seemingly anomalous σ behavior of granular Al thin films at high-ε, attributed in similar films to dimensional transitions, may be explained by introducing a total energy cutoff in the fluctuation spectrum in both the direct AL term and the anomalous MT contribution. Such cutoff is also crucial to describe at a quantitative level the high-h behavior, where the theoretical approaches also overestimate the contribution of the short wavelength fluctuations. These results could thus help to elucidate long standing but still open questions concerning the Al thin films, including the mechanism that enhance their T c . In particular, it has been recently proposed that the Josephson coupling between neighboring grains weakens the quantum confinement in each grain, and should also weaken the associated T c enhancement. In our films we find ξ (0) ∼ 20-48 nm, much larger than the grains size (5-10 nm), indicating a strong intergrain coupling. In spite of that, T c ∼ 1.9-2.1 K, still as high as two times the bulk value and almost independent on ξ (0). Our results suggest that the superconductivity in these materials is homogeneous in nature, and that the grains size distribution does not seem to play a relevant role. This questions a purely confinement mechanism for the T c enhancement, which still remains an open issue. It would be interesting to extend the present study to other superconductors and experimental conditions (e.g., other grain sizes and/or dimensionalities).

ACKNOWLEDGMENTS

This work was supported by FEDER/Ministerio de Ciencia, Innovación y Universidades-Agencia Estatal de Investigación (Project No. FIS2016-79109-P), Xunta de Galicia, Consellería de Educación, Universidades e FP (Grant No. ED431C 2018/11), Xunta de Galicia/FEDER (Network No. ED431D 2017/06 and Strategic Group No. ED431E 2018/08), and by the CA16218 Nanocohybri COST Action. Authors appreciate the use of RIAIDT-USC analytical facilities.

APPENDIX: EXPERIMENTAL RESULTS AND ANALYSIS OF THE OTHER STUDIED FILMS

To complement the results on film A described in detail in the main text, here we summarize the experimental results and analysis of films B (grown under a different oxygen pressure), and C (with a larger thickness), that confirm the applicability of our theoretical approach and support the results obtained in film A.

Film B

This film was grown under a oxygen pressure 33% smaller than film A (10 -8 bar). The temperature dependence of its resistivity around T c under different perpendicular magnetic fields is presented in Fig. 5. The normal state resistivity and the T c value (1.9 K) are slightly smaller than for film A (see Table I), probably as a consequence of a better coupling between the Al grains. The background contribution was obtained for each applied field by a linear fit above T onset = 5 K, the temperature above which fluctuation effects are below the noise level. Examples for H = 0 and 200 mT are presented as solid lines in the inset of Fig. 5 (both lines are indistinguishable in this scale).

The paraconductivity for H = 0 is presented in Fig. 6(a). The solid line is a fit of σ 2D AL (ε, c) + σ 2D MT (ε, c) [Eq. (4)+Eq. ( 8)], with c = ln(T onset /T c ) ≈ 1 and δ as the only free parameter. The agreement is good in all the accessible ε-region above ε ≈ 10 -2 , and the resulting δ value (∼0.1, compiled in Table I), is within the values found in Al films with a similar sheet resistance, see Refs. [26,27]. Just for comparison, the same theoretical approach but without a cutoff (i.e., with c → ∞) is presented as a dashed line. As it may be seen, the agreement in this case extends only up to ε ≈ 0.1. The ε-dependence of σ under different applied magnetic fields is presented in Fig. 6(b). The solid lines are the best fit of σ 2D AL (ε, h, c) + σ 2D MT (ε, h, c) [Eq. ( 3)+Eq. (10)], with the above T c , c and δ values, and with H c2 (0) (that is implicit in h) as the only free parameter. The agreement is good down to ε ≈ 0, near the onset of the critical fluctuation region. The resulting μ 0 H c2 (0) value (0.14 T) leads to a coherence length amplitude of ξ (0) = 48 nm, significantly larger than the superconducting grains. This suggests that the grains are strongly coupled, which is consistent with the observed 2D behavior. The h dependence of σ is illustrated in the inset for some ε values. The lines are the 2D AL + MT approach under a cutoff, evaluated with the same parameters as in the main figure.

Film C

This film was grown under the same oxygen pressure (10 -8 bar) as film A but is 10 times thicker (100 nm). The temperature dependence of its resistivity around T c under different perpendicular magnetic fields is presented in Fig. 7. As expected, the T c value (2.07 K) is the same as for film A (see Table I), but the normal state resistivity is slightly smaller, probably as a consequence of its three-dimensional microstructure. The background contribution was obtained for each applied field by a linear fit above T onset = 4.4 K, the temperature above which fluctuation effects vanish. Examples for H = 0 and 200 mT are presented as solid lines in the inset of Fig. 7 (both lines are indistinguishable in this scale).

In this sample it is possible to appreciate the T c shift with H. This allows to estimate the coherence length from the GL relation

ξ 2 (0) = φ 0 2π T c |dH c2 /dT | . (A1)
By using a 10% (90%) criterion on the normal-state resistivity to determine T c (H ) it is obtained ξ (0) = 18 (21) nm. As this value is ∼5 times smaller than the film thickness, it is expected FIG. 6. Comparison of the experimental σ for film B with the 2D AL + MT approach under a total-energy cutoff (solid lines). The dashed line in (a) is the same approach without a cutoff (with c → ∞). The inset in (b) illustrates the h-dependence of σ near T c . See the main text for details.

that the fluctuation effects are not 2D in this case. However, the ε-dependence of σ for H = 0 [Fig. 8(a)] cannot be explained even at low-ε by the conventional AL + MT 3D approach (see, e.g., Ref. [START_REF] Thompson | [END_REF]) evaluated with ξ (0) = 20 nm and δ = 0 (dot-dashed line). The use of a finite δ value increases the disagreement. Thus, we have compared the data with a generalization of the AL + MT approach to superconductors with an intermediate 2D-3D dimensionality. This may be easily obtained from the 2D approach presented above by following the procedure described in Ref. [START_REF] Thompson | [END_REF]. The result for the AL contribution under a total-energy cutoff is and the total-energy cutoff may be introduced by using the prescription described in Sec. III C, σ MT (ε, h, c) = σ MT (ε, h) -σ MT (c, h). (A4)

σ 2D-3D AL (ε, h, c) = e 2 32 hd 1 h n max n=0 ψ 1 ε + h + ξ 2 (0)(nπ/d ) 2 2h -ψ 1 c + h + ξ 2 (0)(nπ/d ) 2 2h , ( A2
A detailed calculation of these expressions will be presented elsewhere. It is worth noting that they reduce to the 2D and 3D limits when d ξ (0) and d ξ (0), respectively. The solid line in Fig. 8(a) corresponds to the 2D-3D AL + MT approach under the cutoff, evaluated with c = ln(T onset /T c ) ≈ 0.75, δ ∼ 0.02 (within the values found in Al films with a similar sheet resistance, see Refs. [26,27]), and ξ (0) = 21 nm (within the values obtained above from the T c (H ) shift). These values are compiled in Table I. The agreement is now good in all the accessible temperature region above ε ≈ 10 -2 . For comparison, the same theoretical approach but without a cutoff (i.e., with c → ∞) is presented as a dashed line. As expected the agreement in this case extends only up to ε ≈ 0.1. As for the other samples, the coherence length is larger than the superconducting grains, FIG. 8. Comparison of the experimental σ for film C with the 2D-3D AL + MT approach under a total-energy cutoff (solid lines). In (a), the dashed line is the same approach without a cutoff (with c → ∞), that illustrates the disagreement at high-ε, and the dot-dashed line is the best fit of the conventional 3D AL + MT approach. The inset in (b) illustrates the h-dependence of σ near T c . See the main text for details.

which is consistent with a strong intergrain coupling and with the observed non-0D behavior.

The ε-dependence of σ under different applied magnetic fields is presented in Fig. 8(b). The solid lines are the 2D-3D AL + MT approach with a total-energy cutoff, evaluated with the same parameters as in Fig. 8(a). As it may be seen the agreement is excellent down to temperatures slightly below ε = 0, where critical fluctuations are expected to be appreciable. The h-dependence of σ is illustrated in the inset of Fig. 8(b) for some ε values. The lines are the 2D-3D AL + MT approach under a cutoff, evaluated with the same parameters as in the main figure.

FIG. 1 .

 1 FIG. 1. Scanning electron micrographs of the samples studied. The detailed views presented in the insets are 100 × 100 nm 2 in size.

FIG. 4 .

 4 FIG. 4. (a)T -dependence of σ for film A under different perpendicular magnetic fields. The lines are the best fit of the AL 2D + MT 2D approach under a total-energy cutoff, Eq. (3)+Eq. (10), with H c2 (0) as the only free parameter. (b) h dependence of σ for selected ε values. The solid lines are Eq. (3)+Eq. (10) evaluated with the same parameters as in (a). If the cutoff is not introduced (dashed line) there is an appreciable disagreement in the high-h region. (c) h -ε phase diagram indicating the σ amplitude (color scale). The data points represent the low-T applicability limit for the theoretical approach used in (a). They were obtained as the temperatures at which the theory overestimates by 10% the experimental data for each H . These points agree with the onset of the critical fluctuation region as evaluated from the H -dependent Ginzburg criterion [thick solid line, Eq. (11)]. The dashed line is the reduced upper critical field. See the main text for details.

5 .

 5 T -dependence of the resistivity of film B around T c for different H . Inset: Example for H = 0 and 200 mT of the procedure used to determine the background contribution (solid lines) by a linear fit above 5 K.

)FIG. 7 .

 7 FIG. 7. T -dependence of the resistivity of film C around T c for different H . Inset: Example for H = 0 and 200 mT of the procedure used to determine the background contribution (solid lines) by a linear fit above 4.4 K.

TABLE I .

 I Growth conditions and superconducting parameters of the studied films, as results from the σ analysis (see the main text for details).

	d Film (nm) (10 -8 bar) (K) (μ cm) (nm) T c ρ n ξ (0) μ 0 H c2 (0) P O 2 (T)	δ
	A	10	1.5	2.07	163	20	0.82	0.04
	B	10	1.0	1.90	75.2	48	0.14	0.1
	C	100	1.5	2.07	117	21	0.75	0.02
				III. DATA ANALYSIS


Lei Fu
email: lei.fu2@mail.mcgill.ca

Gary Scavone

Claudia Fritz

PLAYER EVALUATION OF PERFORMANCE AND STUDENT VIOLINS

Keywords:

Several playing experiments have been reported in recent years to assess violin qualities but the results have shown large inter-subject inconsistencies. A new perceptual playing experiment was conducted to examine how violinists might differentiate between "good" and "bad" violins, and to what extent they might agree with each other. A pool of six violins of was assembled: three performance violins and three student violins. Nine violinists participated: 3 professional violinists, 1 graduate student, 4 undergraduate students in music performance and 1 had a conservatory degree. The experiment was organized in two phases. During the first phase, subjects were asked to rate and rank the six violins from least preferred to most preferred on a continuous scale. During the second phase, subjects were asked to rate and rank the violins according to five criteria: responsiveness, resonance, clarity, richness, and balance. The results showed that the three performance violins were on average rated significantly higher than the three student violins in terms of preference and all five criteria except responsiveness. We also found that the violinists preferred violins with richer and clearer sound. Three professional violinists rated performance violins much higher than student violins.

Introduction

It has been a long-standing goal for scientists to find correlations between measurable physical properties, acoustic characteristics and the perceived sound qualities of music instruments. The fundamental premise of "goodness" of the violins however had not been investigated scientifically until recent years. In [START_REF] Fritz | Player preferences among new and old violins[END_REF] and [START_REF] Fritz | Soloist evaluations of six Old Italian and six new violins[END_REF], the researchers designed two perceptual experiments to examine violinists' preference between old and distinguished Italian violins and new violins made by professional violin makers. The studies found that the violinists could not tell old violins from the new ones at better than chance levels. A general preference for new violins was shown within their studies. To investigate how players evaluate the violins and whether there is agreement between them, Saitis et al. [START_REF] Saitis | Perceptual evaluation of violins: A quantitative analysis of preference judgments by experienced players[END_REF] conducted a series of experiments. They found that players were self-consistent evaluating violins in different trials and different days but that there was a significant lack of agreement between different individuals. The participants were also asked to rank the violins according to several specific criteria. It was shown that the violinists tended to prefer instruments that they judged to have high "richness" and "dynamic range," though there was no agreement in terms of which instruments were most "rich" or had the largest "dynamic range".

Before the formal experiments, Saitis et al. conducted a pilot study to select instruments for the experiments. They found that the musicians could easily discriminate the poorly maintained Suzuki instruments. Thus, those violins were omitted from consideration as it was felt they would skew the consistency of the results. On hindsight, we began to wonder why the musicians could consistently distinguish the poorly maintained Suzuki violins from "good" violins. Are there some specific poor qualities of violins agreed upon by most violinists in the Suzuki violins? If the answer is yes, it might be possible to correlate the "bad" qualities to acoustical characteristics and physical measurements of the violins. Thus, this experiment sought to repeat the pilot study of Saitis et al. [START_REF] Saitis | Perceptual evaluation of violins: A quantitative analysis of preference judgments by experienced players[END_REF] to assess whether the lower quality Suzuki violins would be consistently distinguished from the better quality violins under more controlled conditions and whether there would be agreement regarding the qualities of those instruments that the subjects found less desirable.

Materials and Methods

This section describes the details of this experiment. It includes general design, test violins, characteristics of participants, controls and detailed procedure.

General Design

The goal of this experiment is to examine whether there is agreement on less desirable features of violins among violinists, and whether they agree on what the less desirable features are. This experiment consisted of two phases. The first phase allowed the violinists to rate all violins on a continuous scale from 0 to 5 based on their own preference. After the preference rating, several open questions were given to the subjects to answer in order to determine how different violinists evaluate violins. During the second phase, the subjects were asked to rate each violin on a continuous scale from 0 to 5 for responsiveness, resonance, clarity, richness, and balance.

Test violins

A pool of 3 performance violins (labeled P1, P2 and P3) and 3 student violins (labeled S1, S2 and S3) from Schulich School of McGill (SSM) was assembled. The performance violins were from a set of higher quality instruments donated to the SSM over the years while the student violins came from a collection of Suzuki violins used by music education students. They were not played on a regular basis, especially the performance violins. While scientific studies [START_REF] Fritz | Player preferences among new and old violins[END_REF][START_REF] Fritz | Soloist evaluations of six Old Italian and six new violins[END_REF][START_REF] Inta | Measurement of the effect on violins of ageing and playing[END_REF] may suggest that this should not influence the individual evaluations, players may argue that this could lower the perceived quality of these instruments. However, it should certainly not influence inter-individual agreement. Violin P1 was adjusted and setup with new strings before the experiment. The violinists were given the option to either use a provided shoulder rest (Kun Original model), use their own shoulder rest, or not use a shoulder rest at all.

Participants

Nine violinists took part in this experiment (6 females, 3 males; 6 native English speakers, 2 native Chinese speakers and 1 native Catalan speaker; average age=30 yrs, SD=14 yrs, range= 20-55 yrs). They had at least 12 years of violin experience (average years of violin playing = 22 yrs, SD=11 yrs, range = 12-40 years; average years of violin training = 14 years, SD=4 yrs, range = 8 -23 yrs; average hours of violin practice per week = 19 hrs, SD= 13 hrs, range= 0 -35 hrs). The estimated prices of their own violins range from $10K to $20K, and they were paid for their participation. Three violinists described themselves as professional violinists, and 1 had a master's degree in music performance (MMus), 3 had bachelor degrees (BMus, B.A.), 1 had a conservatory degree, and 4 were undergraduate students in music performance. They reported playing various musical styles [classical (100%), folk (22%), baroque (22%), jazz/pop (44%), contemporary (22%) and electronic (11%)], and in various types of ensembles [chamber music (67%), symphonic orchestra (89%), solo (67%), private violin teacher (11%), electronic (11%), indie (11%), and R&B (11%)].

Controls

The possible effect of visual information, such as the style of the violin, the colour of the varnish, identifying marks of the violin, may cause preference biases in the evaluation process. In order to eliminate this possible influence and also ensure the safety of the players and instruments, the subjects were provided dark sunglasses and the light level in the room was significantly reduced.

Considering, like in previous studies [e.g. 1-3] the bow as an extension of the player, we asked violinists to use their own bow. The violinists were also asked to bring their own violins with them, in case they wanted to use it as a reference during the tests.

This experiment took place in a diffusive sound space (walls treated with diffusive panels) in order to minimize the effects of room reflections on the direct sound from the instruments. The area of the room was approximately 27 m 2 , and the reverberation time was approximately 0.18s.

Detailed procedure

This experiment was organized in two phases and lasted around 1 hour. Subjects were scheduled individually. The experimenter was constantly present in the room for instructing and taking notes for the subjects. Before the experiment, the subjects answered a questionnaire and signed the consent form. Then they were given instructions about the experiment. Before the experiment, the six violins were assigned a letter from A to F randomly, to avoid presentation order effects; the letter was written on a small piece of paper, which was then stuck on the scroll of each violin. The violins were ordered from A to F and placed on a table along with the subject's own violin. During the first phase, the subjects were given up to 25 minutes to play all six violins, and compare and rate the violins from least preferred to most preferred on a continuous scale from 0 to 5. Subjects were free to play the instruments in any manner and any order. They were also encouraged to comment out loud when assessing the violins, and the experimenter took notes of the subjects' comments. They were instructed to follow their own strategy imagining that they were choosing violins for themselves at a violin shop. They were allowed to play their own violins whenever it seemed useful. Upon completing the first phase of the experiment, subjects were asked to provide written responses to a set of very general open-ended (in order to avoid confining the answers into pre-existing categories) questions as follows:

A1. How and based on which criteria did you make your rankings/ratings? A2. Why did you choose the violin ranked as the most-preferred? A3. Why did you choose the violin ranked as the least-preferred? A4. In general, what distinguished the less-preferred violins from the more-preferred violins?

A5. Do you have any comments or remarks about the task you were involved in? To what extent was wearing sunglasses disturbing?

After finishing the first phase of this experiment, subjects were given five criteria for assessment of each violin: responsiveness, resonance, clarity, richness and balance. The criteria were carefully chosen from previous publications [START_REF] Fritz | Player preferences among new and old violins[END_REF][START_REF] Fritz | Soloist evaluations of six Old Italian and six new violins[END_REF][START_REF] Saitis | Perceptual evaluation of violins: A quantitative analysis of preference judgments by experienced players[END_REF]. Subjects were given 5 minutes to evaluate each criterion and rate the six violins on a continuous scale from 0 to 5. To ensure all subjects had a common interpretation of the rating scales, each criterion was presented with a descriptive phrase and an explanatory text.

After the rating of each criterion, subjects were given a question to answer in written form: B1. Do you have specific comments or remarks about the "balance (each criterion)" of the violins? Was there a particular behavior in the violin rated as least balanced or the one rated as most balanced?

After rating all five criteria, subjects were asked to answer two optional questions:

C1. Do you have any other further comments or remarks about the violins? C2. Would you like to change the preference ranking after rating these criteria?

Detailed analyses and results of phase 1

This section provides the subjects' preference ratings of the violins and verbal responses from phase 1. This section also studies the rating difference between performance violins and student violins. The relationship between the rating difference and subject characteristics are also analyzed.

Overall preference ratings of the violins

The across-subjects average preference score for each violin is shown in Fig. 1. Error bars of twosided 95% confidence interval (CI; all CIs are two-sided 95% intervals through this chapter) of the means are also displayed. Violin P2 has the highest mean rating in terms of preference, and the 95% CI for the population mean is [2.15,4.78]. The mean rating of violin S2 is markedly below the other violins, and the 95% CI for the population mean is [-0.04, 2.88]. Independent-Samples Kolmogorov-Smirnow test showed that the null hypothesis that all six violins had the same population means could not be rejected, i.e., the overall preference ratings between the six violins were not significantly different.

Difference between performance violins and student violins

The ratings of performance violins and student violins were compared in this section. The mean rating of performance violins is 3.30, 95% CI = [2.70; 3.89]; the mean rating of student violins is 1.82, 95% CI = [1.18; 2.47]. An independent-samples Mann-Whitney U test showed that the null hypothesis that the two types of violins had the same population mean could be rejected.

Influence of participant characteristics

The association between preference rating difference (performance violins and student violins) on the one hand, and the self-reported age, degree in music performance, the years of violin experience, weekly hours of violin practice and price of the owned violin, on the other was assessed. This analysis was carried out by calculating the Spearman rank correlation 𝜌 " between preference rating difference and participant characteristics. The correlations between age, degree and rating difference were significant: 𝜌 " =0.835 (p=0.005) and 𝜌 " =0.766 (p=0.016), respectively. Participants who were older, those who were professional musicians, and/or with a higher educational degree in music performance rated performance violins much higher than student violins.

Conclusion

The results of phase 1 of this experiment showed that performance violins were on average rated significantly higher than student violins in terms of preference even though there was no significant difference between the ratings of the six violins. And it was found that the subjects who were older, those who were professional musicians, and/or with a higher educational degree in music performance rated the performance violins much higher than the student violins.

Detailed analyses and results of phase 2

In this section, the results of phase 2 were analyzed. The analysis was conducted in the following aspects: comparison of the attribute criteria ratings between the six violins, between performance violins and student violins, the relationship between preference and criteria ratings, and the verbal descriptions of each attribute by subjects.

Criteria ratings

Across-subjects average ratings on specific criterion of each violin are shown in Fig. 2. Violin P2, the violin which had the highest mean preference rating in phase 1, was rated as most rich and most resonant, but the second least responsive and medium clear and balanced. Violin S1, the violin which had the lowest mean preference rating in phase 1 was rated the worst in every criterion. Violin P1 was rated as most clear. Violin P3 was rated as most balanced. The responsiveness of violin P1, P3, S2 and S3 were rated similarly.

An individual sample Kruskal-Wallis test was carried out to test the equality of the mean specific criterion ratings of the six violins. The results showed that only richness and balance lead to significantly different ratings between the violins. Further investigation showed that this is due to significant differences in richness between violins P1 and S1 (p=0.028) and P2 and S1 (p=0.006), and a significant difference in balance between violins P3 and S1 (p = 0.003).

Difference between performance violins and student violins

Across-subjects average attributes ratings on performance violins and student violins were also analyzed. Performance violins were on average rated higher than student violins for each criterion except responsiveness. An independent-samples Mann-Whitney U test showed that the null hypothesis that the two types of violins had the same population means could be rejected for all the criteria except responsiveness.

Relationship between preference and attribute ratings

The violinists may have employed a highly economic strategy in the evaluation process, which might lead to similar ratings to all rating scales. One of the subjects had mentioned at the end of this experiment: "each violin had very different personalities but when broken down into categories it is hard to list them without taking other aspects into account".

To avoid this interpretation when analyzing the relationship between preference ratings of phase 1 and attribute ratings of phase 2, partial correlation coefficients 𝜌 # was employed. 𝜌 # (𝐴, 𝐵 • 𝐶) measures the correlation between A and B removing the effect of variable C. For example, in order to measure the correlation between preference and resonance, the effect of responsiveness, clarity, richness and balance were eliminated by the calculation of 𝜌 # (𝑅𝑒𝑠𝑜𝑛𝑎𝑛𝑐𝑒, 𝑃𝑟𝑒𝑓𝑒𝑟𝑒𝑛𝑐𝑒 • 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑖𝑣𝑒𝑛𝑒𝑠𝑠, 𝑐𝑙𝑎𝑟𝑖𝑡𝑦, 𝑟𝑖𝑐ℎ𝑛𝑒𝑠𝑠, 𝑏𝑎𝑙𝑎𝑛𝑐𝑒).

Partial correlation coefficients 𝜌 # were computed between each of the attribute scale ratings and the preference ratings of all subjects. The results are shown in Fig. 3. All criteria ratings correlated to preference ratings positively except the responsiveness ratings. Richness and clarity correlated to preference significantly: 𝜌 # =0.507 (p=0.0002) and 𝜌 # = 0.282 (p=0.047), respectively. The results thus indicated that subjects preferred violins with a richer and clearer sound. None of the other partial correlation coefficients between attributes ratings and preference ratings was significant (p>0.05).

Verbal descriptions of violin attributes

The subjects were asked to give comments or remarks about the specific criterion that they had evaluated and describe the particular behavior they noticed of the violin that was rated as best or worst for that criterion.

For responsiveness, some subjects thought the most responsive violin had very clear sound at the attack of string and required less effort to create the sound. Some subjects thought responsiveness meant ringing and sound coming out. Subjects thought the sound of the least responsive violin was small, noisy, and less resonant. One subject mentioned that the violin with a bad and dry sound responded quickly, while the violin with a thick sound needed more time to response. Another subject commented that the violin with a lower bridge responded fast, but the sound wasn't solid.

For resonance, subjects considered that the most resonant violin had a bright sound and rang very open, overtones came out, and which could be powerful; the least resonant violin had a muted, stiff sound with no ringing/brilliant qualities. Two subjects thought that resonance was different from responsiveness, as not the more resonance the better: good resonance seems to imply ringing very well, but dark sounding instruments usually have less resonance, e.g. muffled, so the challenge is to find a good "dark" violin: "ringing one still dark".

For clarity, three of the subjects considered that clarity was related to responsiveness. They commented that the clearest violin was easy to obtain a brilliant, pure, concentrated and clean sound, which didn't change with bow force. Each note boomed nicely, and the ringing tones around it helped with the transition between notes, making them connect well with each other. The least clear violin had a buzzing, muddy sound that lacked purity, and it became more serious as the bow force increased. There was one subject however who thought the clearest violin was brilliant but lacked flavor. So, his most preferred violin had good flavor, but was not the clearest violin.

For richness, subjects generally thought the richest violin had lots of colour and undertones in the sound and good expression. The sound was deep, dark, sweet, thick, big and fat. Two subjects considered that richness was partly related to resonance. They considered that the least rich violin did not have many colors and lacked depth, the sound was hollow, open, narrow and flat.

For balance, the subjects considered the most balanced violin had an even, consistent sound and stable, balanced playing across strings. The least balanced violin sound was considered either not thick in the lower register, not bright in the higher register or good at one side but bad at the other side. Some subjects considered the overall bad sounding violins as balanced. Finally, some subjects thought that the bridge mattered a lot. They thought that the most balanced violin resonated and rang well over all strings; the least balanced violin had a sloped bridge that did not allow the strings to resonate separately.

Conclusion

Performance violins were rated significantly higher than student violins in all attributes rating scales except responsiveness. The analysis of relationship between preference ratings and attributes ratings showed that violinists preferred violins with rich and to a lesser extent clear sound.

From verbal collections, the violinists stated that some rating criteria of the violins were correlated despite that we have given them definitions of each criterion, e.g., resonance and richness, clarity and responsiveness. Resonance and responsiveness were anti-correlated to some extent.

Conclusion

This study investigated players evaluation between performance and student violins through a perceptual playing test. The results showed that performance violins were on average rated significantly higher than student violins in terms of performance, and all five attribute criteria except responsiveness. Through the analysis of the relationship between preference ratings and attribute ratings, we found that violinists preferred violins with rich and to a lesser extent clear sound.

And it was found that the subjects who were older, those who were professional musicians, and/or with a higher educational degree in music performance rated the performance violins much higher than the student violins in terms of preference.

Figure 1 :

 1 Figure 1: Across-subjects average of the overall preference score for each violin (error-bar = 95% confidence interval of the mean).

Figure 2 :

 2 Figure 2: Across-subjects average ratings on specific criterion of each violin (error-bar = 95% confidence interval of the mean).

Figure 3 :

 3 Figure 3: Partial correlation coefficient 𝝆 𝒑 between ratings of each attribute scale and preference.