Ross Taylor
email: ross.taylor@vuw.ac.nz

Chris Bumby
email: chris.bumby@vuw.ac.nz

Hubertus Weijers
email: huub.weijers@vuw.ac.nz

Mark Ainslie
email: mark.ainslie@eng.cam.ac.uk

Modelling Interactions Between HTS Tapes and Permanent Magnet Fields in an HTS Dynamo

Keywords: HTS modelling, HTS dynamo, flux pump, coated conductor, numerical modelling, finite-element modelling

The interactions between a permanent magnet (PM) and a coated conductor tape lies at the heart of several types of high-temperature superconducting (HTS) devices, including the HTS dynamo. The applied magnetic field from a PM is typically highly spatially inhomogeneous with large field gradients occurring near the edges of the PM. This situation differs substantially from the well known Brandt analytical model, which describes flux penetration and current behaviour of type-II superconducting strips in a spatially-homogeneous applied magnetic field. The Brandt model also assumes the superconductor exhibits a constant critical current (J c ) throughout, and hence does not take into account the more complex behaviour of practical HTS tapes which exhibit an angular magnetic field-dependence of the critical current, J c (B, θ). In this work, a finite-element model is developed to include such considerations and describe the interaction between a permanent magnet and an HTS coated-conductor wire at small flux gap distances similar to those used in the HTS dynamo. The effect of varying key parameters is studied including: (1) changing the critical current, (2) including an anisotropic magnetic field dependence to the critical current density, (3) changing the relative width of the PM to the coated conductor tape, and (4) changing the flux gap between the magnet and the HTS tape. The effects on the flux penetration and current behaviour of the HTS tape are calculated and compared to the Brandt model. Of particular interest are the key differences in the flux penetration profile which are observed when the magnet width is much smaller than the width of the coated conductor tape.

I. INTRODUCTION

The Brandt model [START_REF] Brandt | Type-II-superconductor strip with current in a perpendicular magnetic field[END_REF], [START_REF] Brandt | Superconductors of finite thickness in a perpendicular magnetic field: Strips and slabs[END_REF] is a useful tool to describe the behaviour of superconducting tapes in the presence of magnetic fields. However key underlying assumptions (of the Bean limit (n → ∞) and a homogeneous, externally applied magnetic field) are not always valid experimentally. For exam-ple -highly inhomogeneous magnetic fields are often present in applications such as the high-temperature superconducting (HTS) dynamo, where a rotating permanent magnet (PM) passes by an HTS stator tape [START_REF] Van De Klundert | Fully superconducting rectifiers and fluxpumps Part 1: Realized methods for pumping flux[END_REF], and magnetic levitation, where strong magnetic field gradients occur perpendicular to the thrust direction [START_REF] Schultz | Superconductively levitated transport system -the SupraTrans project[END_REF].

In addition, many practical HTS devices operate at liquid nitrogen temperatures (77 K) to avoid the need for expensive liquid helium cooling. At this temperature, widely available HTS coated conductor tapes exhibit typical n-values between 20 and 40 [START_REF] Zhang | Study of Critical Current and n-Values of 2G HTS Tapes: Their Magnetic Field-Angular Dependence[END_REF] and hence are not well-described by the Bean limit. Even at 4.2 K, the n-values of many commercial tapes only increase to between 40 and 60 [START_REF] Tsuchiya | Critical current measurement of commercial REBCO conductors at 4.2 K[END_REF]. The analytical model also assumes a constant critical current density, J c , but in real HTS tapes, J c has a magnetic field-angular dependence, J c (B, θ) [START_REF] Zhang | Study of Critical Current and n-Values of 2G HTS Tapes: Their Magnetic Field-Angular Dependence[END_REF].

Consequently, modelling the magnetic field and current behaviour of real coated conductor tapes in inhomogeneous, perpendicular applied fields is of interest. In this work, such a model is constructed using the commercial finite-element modelling software COMSOL Multiphysics [7], and n-values and J c (B, θ) measured from real tapes.

II. MODELLING METHOD

The geometry of the inhomogeneous PM model is shown in Fig. 1. This assumes the 2D case for simplicity. A PM of width a, height b and remanent flux density B r , is centred stationary at a distance c beneath an HTS coated-conductor wire of width e and thickness d, such that the axis of the PM is perpendicular to the tape. The remanent flux density of the magnet is increased from 0 T to a maximum value B max at a variable rate of r mTs -1 . The parameterisation of all variables allows the model to be easily adapted for different contexts.

The model implements the coupled H-A formulation as described and applied in [START_REF] Brambilla | A Finite-Element Method Framework for Modeling Rotating Machines With Superconducting Windings[END_REF], [START_REF] Ainslie | A new benchmark problem for electromagnetic modelling of superconductors: the high-Tc superconducting dynamo[END_REF], using the H-formulation within the superconducting subdomain and the A-formulation to model the permanent magnet and the surrounding subdomain.

III. INITIAL RESULTS

To initially validate the HTS model, the shielding behaviour of an ideal HTS tape under an externally applied homogeneous magnetic field is examined by approximating to the Bean limit using n = 120. The computed results are compared to the Brandt analytical model in Fig. 2 for a range of critical currents. For each I c , the value of J c is taken to be constant as J c = Ic d•e . The value at which flux penetrates the center of the tape such that it is no longer fully shielded, B p , is calculated for different values of I c . Table I gives the parameters used for this homogeneous model. Results for the simulation agree closely with the Brandt analytical model.

The inhomogeneous PM case is then considered and also presented in Fig. 2. Parameters used for this example inhomogeneous model are shown in Table I. In this case, a more realistic n-value of n = 20 is chosen to better describe practical HTS tape. Similarly a flux gap of 3.7 mm is chosen, as this corresponds to the geometry of the HTS dynamo presented in [START_REF] Ainslie | A new benchmark problem for electromagnetic modelling of superconductors: the high-Tc superconducting dynamo[END_REF], [START_REF] Mataira | Origin of the DC output voltage from a high-Tc superconducting dynamo[END_REF]. The remanent field of the PM is increased from zero, and again B p is determined for a range of I c values. There is still linearity between B p and I c , but it is clear that magnetic shielding is present in the tape at much higher fields than the Brandt model predicts, due to the finite size of the magnet and the consequent decrease in applied field at the coated conductor.

Further results will show the effects of adding J c (B, θ) considerations, changing the flux gap and field ramp rate, and changing the PM geometry -including the presence of a fluxfocusing effect when using a PM with a much smaller width than the coated conductor tape. Ultimately, this study targets improved understanding of the flux penetration behaviour which arises from a finite-sized permanent magnet, in order to better understand the operation of the HTS dynamo and other such devices.

Fig. 1 .

 1 Fig. 1. Geometry of the inhomogeneous applied field model. A permanent magnet is stationary, centred under an HTS tape, and the remanent flux density of the magnet is increased.

Fig. 2 .

 2 Fig. 2. Applied magnetic field strength at which a superconducting strip is no longer fully shielded, Bp, for a range of critical currents, Ic. The solid lines are linear fits to the data.


Mike Beck

Mark Ainslie

Vito Cientanni

Modelling Pulsed Field Magnetization of Ring-Shaped Bulk Superconductors

Keywords: HTS modelling, bulk superconductors, ring bulks, pulsed field magnetization, flux jumps

The motion of flux into ring-shaped bulk hightemperature superconductors during pulsed-field magnetization (PFM) renders finite element modelling of the process more complicated than for disk-shaped bulks. The collision of two opposite-moving flux fronts within the bulk creates a thermal shock which, if poorly modelled, pollutes the numerical solution. In this work, the H-formulation is coupled with a thermal model to simulate the dynamics of ring-bulk PFM. The results are firstly validated against Brandt's analytical equations for a simple zero-field-cooled magnetization case, then extended to analyze PFM under more realistic material assumptions. The limitations, and ways to address these, of finite element modelling to predict the behavior of ring-shaped bulks during PFM are explored.

I. INTRODUCTION

Stacks

of ring-shaped bulk high-temperature superconductors (HTS) have been shown to be a viable source for the generation of high-strength, uniform magnetic fields for use within benchtop nuclear magnetic resonance (NMR) instruments [START_REF] Nakamura | Development of a superconducting magnet for nuclear magnetic resonance using bulk high-temperature superconducting materials[END_REF]. Experimental work to date, however, has only explored their suitability when magnetized by field cooling in an NMR-grade background field.

For such ring-bulk stacks to be used more practically, they will need to be magnetized using low-cost, portable magnetization techniques such as pulsed-field magnetization (PFM). PFM has been shown to successfully magnetize diskshaped HTS [START_REF] Ainslie | Flux jump-assisted pulsed field magnetisation of high-J c bulk high-temperature superconductors[END_REF], but is prone to inducing macroscopic breaks of the critical current in ring-shaped samples through flux jumps [START_REF] Korotkov | Flux Jumps at Pulsed Field Magnetization[END_REF]. These breaks in current lead to the trapping of negligible, or negative magnetic fields within the bore of the ring.

The behavior of disk-shaped bulk HTS is well understood and has been extensively modelled using finite element (FE) models (FEM) [START_REF] Ainslie | Modelling of bulk superconductor magnetization[END_REF], but there is little in the literature regarding the magnetization of ring-shaped devices. Brandt [START_REF] Brandt | Susceptibility of superconductor disks and rings with and without flux creep[END_REF] showed that, during zero-field-cooled (ZFC) magnetization, flux penetrates into thin rings from both the inner and outer edges of the sampleintroducing two opposite-moving flux fronts into the ringbetween which exists a shielded region of zero current and trapped field. In samples of practical height, flux preferentially propagates into the sample at the top and bottom surfaces [START_REF] Brandt | Superconductor disks and cylinders in an axial magnetic field. I. Flux penetration and magnetization curves[END_REF], thus this region of zero field and current is isolated by magnetized regions. As the fronts converge, the region of zero field and current becomes infinitely small, introducing a singularity into FE models, before vanishing. At this point, a large local dB/dt is induced, leading to significant local heating of the bulk at this point. In physical samples, this may locally induce a flux jump within the ring.

If the model is insufficiently finely meshed, this local heating will induce a thermal shockand subsequent nonphysical thermal oscillationsto the result [START_REF] Bergheau | Finite Element Simulation of Heat Transfer[END_REF]. The maximum element size, emax, is given by

𝑒 𝑚𝑎𝑥 = √ 6𝜅Δ𝑡 𝜌𝐶
where Δ𝑡, 𝜅, 𝜌 and C are the time-step size, thermal conductivity, mass density and thermal density of the material respectively [START_REF] Bergheau | Finite Element Simulation of Heat Transfer[END_REF]. As such, bulk HTS materials require a very fine mesh to avoid this issue, introducing multiple degrees of freedom (DoFs) to the model, thus rendering 3D models of PFM intractably large for many desktop PCs.

In this work, the pulsed magnetization of a finite ring is modelled using H-and mixed H-A formulations, with meshes of differing density and element order [START_REF]Understanding, and changing, the element order -Knowledge Base[END_REF]. They are compared in terms of computational requirements and performance, and the solutions obtained relative to each other. Through this, the limitations of FEM to predict the behavior of ring-shaped bulks during PFM are explored, and methods to address these identified.

II. RING MODELLING

The geometry of the analyzed ring is shown in Fig. 1, assuming for simplicity the 2D axisymmetric case. A bulk HTS ring with inner diameter (ID) 16 mm, outer diameter (OD) 40 mm, height (H) 12 mm and experimentally measured Jc(T,B) is set in an aluminum reinforcement ring of width 5 mm and height 12 mm. It is cooled from the bottom surface by a cold head at constant temperature.

The bulk is magnetized by a coil with dimensions 120/100/50 mm OD/ID/H, applying a pulse with adjustable amplitude and a 13 ms rise time. In all cases, the ring is modelled using the H-formulation with a fully coupled thermal model. The surrounding regions are modelled either using the H-formulation or the A-formulation (in the mixed formulation case).

III. INITIAL RESULTS

The data in Fig. 2 show the occurrence of a thermal shock during the PFM of a ring with a mesh size 0.25 mm radially and 0.2 mm axially. Fig. 3 shows the subsequent, nonphysical, thermal oscillations induced by this shock. By refining the mesh element size in the bulk HTS domain, this shock is damped at the expense of increased model computational time.

By adjusting the element-order of the HTS domain, the thermal shock behavior is damped out whilst the associated time penalty is less than that for a more finely meshed model (Table I).

The detailed findings from this work will be presented at the modelling workshop, including techniques to suppress the numerical errors introduced from the thermal shock and methods to improve the model computational time.

Fig. 1 .

 1 Fig.1. Geometry of the magnetizing fixture. A pulsed magnetic field is applied to the ring using the solenoid coil.

Fig 2 .Fig. 3 .

 23 Fig 2. Appearance of thermal shock in PFM of linear element in 0.25/0.2 mm mesh

TABLE I .

 I Impact of element order and model formulation on model solve-time.

	Formulation	Element Order	Mesh Density r/z (HTS	DoFs	Time to solve [s]
			Region) [mm]
	H	Quadratic	0.25 / 0.2	53086	11912
	H	Linear	0.25 / 0.2	19570	4816
	H	Linear	0.125 / 0.1	60446	15257
	H	Linear	0.0625 / 0.05	211339	87960
	H-A	Quadratic -Linear	0.25 / 0.2	32635	8691
	H-A	Quadratic -Quadratic	0.25 / 0.2	39774	10117


V Cientanni

M. D Ainslie

H Fujishiro
email: fujishiro@iwate-u.ac.jp

K Takahashi

Modelling the record trapped field by pulsed field magnetisation of a composite bulk MgB 2 superconducting ring

Keywords: Bulk superconductor, FEM, MgB 2, PFM

. Excellent quantitative agreement with the experimental data was achieved by making novel assumptions to the thermal physics, namely to that of the heat flow within the sample. The models are extended to investigate open & remaining questions; such as the influence of the iron yoke and the copper plates, and their interaction with the applied field. The results of the study illustrate how composite MgB 2 rings may be optimised to trap even higher fields, and how the physics of their operation affect the trapped field distribution and magnitude. This submission disseminates the physics and modelling techniques used in detail, illustrating the novel modelling methods adopted to accurately simulate bulk composite MgB 2 rings acting as trapped field magnets.

I. INTRODUCTION

Bulk superconductors fabricated from MgB 2 (magnesium diboride) materials show great promise for trapping large magnetic fields, which can be exploited in engineering applications such as magnetic separation, superconducting rotating machines, and advanced NMR/MRI [START_REF] Hull | Applications of bulk high-temperature Superconductors[END_REF]- [START_REF] Song | Review of core technologies for development of 2G HTS NMR/MRI magnet: A status report of progress in Korea University[END_REF]. MgB 2 can offer a number of advantages over materials more traditionally used for fabricating bulk superconductors, such as (RE)-BCO (RE = Rare Earth, B=Ba, C=Cu, O=oxide). The lightweight, polycrystalline structure is 'rare-earth-free', can trap highly homogenous fields due to the long coherence length below 39 K [START_REF] Kambara | High intergranular critical currents in metallic MgB2 superconductor[END_REF], and may be fabricated using a number of techniques with relative ease [START_REF] Durrell | A trapped field of >3 T in bulk MgB 2 fabricated by uniaxial hot pressing[END_REF]. Indeed, whilst the record high trapped field in an MgB 2 disk of 5.4 T at 12 K [START_REF] Fuchs | High trapped fields in bulk MgB 2 prepared by hotpressing of ball-milled precursor powder[END_REF] may be lower than (RE)-BCO [START_REF] Durrell | A trapped field of 17.6 T in melt-processed, bulk Gd-Ba-Cu-O reinforced with shrink-fit steel[END_REF], the magnetisation potential is still great; at least double the saturation field of ferromagnetic iron. Achieving these records involved using the 'Field-Cooled' (FC) magnetisation technique [START_REF] Krabbes | High temperature superconductor bulk materials Fundamentals -processing -properties control -application aspects[END_REF], which ensures full magnetisation of the bulk to characterise its performance. This requires a large, static magnetising field from bulky and expensive magnets, that are usually superconducting. Pulsed-field magnetisation (PFM) meanwhile, has no such constraints, and has greater commercial and practical applications due to its portable and inexpensive setup, despite an often lower trapped field [START_REF] Fujishiro | Pulsed Field Magnetization for GdBaCuO Bulk With Stronger Pinning Characteristics[END_REF]. Adopting favourable magnetising coil fixtures, and optimising the magnetisation technique with multiple pulses has yielded promising results for PFM [START_REF] Mizutani | Pulsed-field magnetization applied to high-Tc superconductors[END_REF]- [START_REF] Wu | Stress analysis in high-temperature superconductors under pulsed field magnetization[END_REF].

Hirano et al. [START_REF] Hirano | A record-high trapped field of 1.61 T in MgB 2 bulk comprised of copper plates and soft iron yoke cylinder using pulsed-field magnetization[END_REF] achieved the PFM record high trapped field, by embedding bulk MgB 2 superconducting rings between layers of copper and inserting a soft-iron yoke. When pulsed with a magnetic field, the copper plates induce eddy-currents, which in turn increase the effective pulse rise time. This should reduce the heat generated in the bulk; thought to be beneficial to trapping higher fields using the PFM technique [START_REF] Fujishiro | Simulation of temperature and magnetic field distribution in superconducting bulk during pulsed field magnetization[END_REF]. Further, the high thermal conductivity and heat capacity of copper may be favourable in transferring heat from the bulk generated during the PFM process. Meanwhile, the iron yoke inserted into the bore of the sample acts to retain flux during the pulse, and concentrate the trapped field. We therefore have a large number of variables which may be controlled; from the number of copper layers adopted and their thickness, to the effect of a soft iron yoke insert compared to no insert. This high variable number heeds well for finite-element modelling [START_REF] Ainslie | Modelling of bulk superconductor magnetization[END_REF] [START_REF] Ainslie | Numerical Modelling of Bulk Superconductor Magnetisation[END_REF], where parameters such as these can be readily and simply altered. This is the impetus behind the presented work; to leverage the advantage of numerical modelling for this large variable space problem. In this submission we illustrate the key outcomes of this modelling based investigation, and disseminate the novel modelling methods adopted to achieve a successful and accurate replication of the experiment of Hirano et al. [START_REF] Hirano | A record-high trapped field of 1.61 T in MgB 2 bulk comprised of copper plates and soft iron yoke cylinder using pulsed-field magnetization[END_REF] II. MODELLING DETAILS Figure 1 illustrates the geometry of the 2D axisymmetric models implemented with COMSOL Multiphysics version 5.4. As per the experiment, three sample configurations were modelled [START_REF] Hirano | A record-high trapped field of 1.61 T in MgB 2 bulk comprised of copper plates and soft iron yoke cylinder using pulsed-field magnetization[END_REF]. These included; a 'Single Bulk' consisting of MgB 2 only, a 'Composite w/o Yoke' bulk, consisting of three copper layers sandwiching two MgB 2 bulks, and finally a 'Composite w. Yoke' bulk: which is the 'Composite w/o Yoke' configuration but with an iron yoke inserted to the bore of the ring. Utilising the H-formulation of Maxwell's equations, we are able to capture the electrodynamics of the experiment. The E-J Power law was applied to the MgB 2 domains to model the highly non-linear I-V relationship this material exhibits in the superconducting state. The magnetic field-dependence of the critical current density, J c (B), was interpolated using reference data [START_REF] Ogino | Optimization of Infiltration and Reaction Process for the Production of Strong MgB 2 Bulk Magnets[END_REF]. Meanwhile, the thermal transient equation was adopted to model the thermal physics of the bulk and sample holder; allowing heat to flow towards a cold-stage, as per the experiment [START_REF] Hirano | A record-high trapped field of 1.61 T in MgB 2 bulk comprised of copper plates and soft iron yoke cylinder using pulsed-field magnetization[END_REF].

Split Coil

Figure 2 illustrates the validation of the model by the experimental data; comparing the magnitudes of the applied pulses and the maximum temperature observed in the experiments. Excellent quantitative agreement is obtained, which gives us confidence in the accuracy of the later explored 'extension studies'.

III. RESULTS

Whilst above we present results illustrating the validation of the model, the full results will be presented at the HTS Modelling Workshop. Here, we will present in full the 'extension studies' of this investigation. This involved investigating the influence of the copper plates, and the iron yoke in magnetising the composite MgB 2 bulks. We first investigated how varying the number of copper layers affects the pulse magnitude, duration, and magnetisation potential of the composite bulks. We also investigated the influence of the iron yoke in concentrating flux during magnetisation, and how this action affects the heating of the bulks.

IV. CONCLUSION

In conclusion, we have successfully validated our numerical modelling framework by replicating the experimental data presented by Hirano et al. [START_REF] Hirano | A record-high trapped field of 1.61 T in MgB 2 bulk comprised of copper plates and soft iron yoke cylinder using pulsed-field magnetization[END_REF]. We present an accurate modelling of the pulsed-field magnetisation of a superconducting composite MgB 2 bulk, which led Hirano et al. to achieve a record-high trapped field. We extend our studies to investigate elements of the experiment which are difficult or impractical to investigate experimentally, and shall present these results in full at the HTS Modelling Workshop.

Fig. 1 :

 1 Fig. 1: Geometry of the 2D axisymmetric model implemented in COMSOL Multiphysics; complete with split coil magnetisation fixture with iron yokes.

Fig. 2 :

 2 Fig. 2: TOP: Applied field (B ex (shunt)) vs the measured field at the centre of the top surface of the sample (B ex (Hall)), at 40 K. BOTTOM: Experimental data compared to the modelled results, for the maximum temperature recorded at the temperature probe (r = 48 mm, z = 0). Prior to pulse; samples at 20 K.

  B ex (Hall) and B ex (shunt) at T = 40 K

	1.5 2.0 2.5 3.0 Sing. Bulk Comp. w/o y. Comp. w. y. ex (Hall) [T] B	}	Exp. Data
	1.0				Sing. Bulk Comp. w/o y Comp. w. y.	}	Model
	0.5
	0.8	1.2	1.6	2.0	2.4	2.8	3.2
			B ex (shunt) [T]
	32
	30	Sing. Bulk Comp. w/o y. Comp. w. y.	Exp.
	28	Sing. Bulk Comp. w/o y.
	26	Comp. w. y.
	24
	22
	20
	0.5		1.0		1.5			2.0


Marcin Kowalik
email: kowalik.marcin@interia.pl

Marek Giebultowski

Ryszard Zalecki
email: zalecki@agh.edu.pl

Ryszard Zalecki The

Marek Giebułtowski

The application of unsupervised learning to a dataset

Keywords: High-temperature superconductors, AC susceptibility, unsupervised learning, clustering

I. INTRODUCTION

High-temperature superconductors (HTS) are materials which exhibit properties like zero electrical resistance and expulsion of an external magnetic field from the interior of a superconductor [START_REF] Tinkham | Introduction to superconductivity[END_REF]. These unique properties are used in the state of the art applications for the medicine (MRI/NMR machines), science (particle accelerators), transportation (Maglev) and electrical industry (electric power transmission, fault current limiters). The greatest drawback of superconductors is that they only function in low temperatures up to 134 K for HgBa 2 Ca 2 Cu 3 O 9 at ambient pressure [START_REF] Schilling | Superconductivity above 130K in the Hg-Ba-Ca-Cu-O system[END_REF]. Scientists are still pursuing the discovery of room-temperature superconductor. The phenomenon of hightemperature superconductivity is still not fully understood.

Machine learning (ML) is the study of computer algorithms that improve automatically through experience [START_REF] Mitchell | Machine Learning[END_REF]. Machine learning algorithms build a mathematical model based on data, order to make predictions or decisions without being explicitly programmed to do so [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF]. Unsupervised learning (UL) is a subfield of ML. UL algorithms look for previously undetected patterns in a dataset with no preexisting labels and with a minimum of human supervision [START_REF] Hinton | Unsupervised Learning: Foundations of Neural Computation[END_REF]. The great progress has been made in a quest to discover, develop or refine various machine learning algorithms in recent years. Also, new ways of data analysis (Artificial Neural Network autoencoders [START_REF] Kramer | Nonlinear principal component analysis using autoassociative neural networks[END_REF], t-SNE [START_REF] Van Der Maaten | Visualizing Data Using t-SNE[END_REF]) have been shown. ML application to the analysis of datasets is a state of the art technique and allows to gain new knowledge in various areas of science and engineering.

Our work aims to provide first insights if clustering technique applied to the dataset consisting of about 1000 measurements of HTS using the AC susceptibility method, will allow recovering known relationships (features) between different types of HTS and their superconducting properties, which depend on sample preparation conditions like sintering and annealing temperatures, etc.

II. THEORY AND EXPERIMENTAL DETAILS

The AC magnetic susceptibility can be written as a complex number by the formula χ=χ'+iχ", where χ' is the dispersion and χ" is the absorption part of the AC susceptibility. The value of the dispersion part corresponds to the diamagnetic nature -a negative magnetization of the HTS sample when an external magnetic field is applied [START_REF] Gömöry | Characterization of high-temperature superconductors by AC susceptibility measurements[END_REF]. The values of χ' and χ" for HTS change with temperature. Above a certain temperature, called the critical temperature T c , both parts of AC susceptibility are equal to zero. On the other hand, below critical temperature T c , the χ' part has negative values and the χ" part is positive or equal to zero. The shapes of χ'(T) and χ"(T) curves strongly depend on superconductor properties, so measuring the temperature dependence of the complex AC susceptibility χ is the most common procedure for characterizing the properties of a superconductor. On

III. MAIN METHODOLOGICAL ASSUMPTIONS

(1) A single AC susceptibility measurement χ(T) can be treated as a sequence of several hundred data points (sentence), therefore it can be represented as a collection of smaller subsequences (words) (Fig. 2). Data point is a 3D vector consisting of values of sample temperature, χ' and χ" IV. THE MOST IMPORTANT DETAILS ON COMPUTATION All χ'(T) measurements were normalised to [-1,0] range and then were divided into subsequences of the size of about several dozen data points. Values of temperatures were dropped. A data point is a single value of χ'. Resulting subsequences were normalised to [0,1] range. Next, further reduction of dimensionality of subsequences to 9 dimensions (9D) was performed by using an 1D convolutional autoencoder, which task was to learn the most efficient representation for a given subsequence. Then, all subsequences with reduced dimensionality were analysed by K-means and DBSCAN clustering algorithms to find subsequences (words) with unique features (Fig. 2) and create word dictionary. The results of clustering were evaluated by visualisation of the words dataset on 2D plane using the t-SNE algorithm. The most reasonable results were achieved by the k-Means with a number of classes set to 5. In the final step, the autoencoder and word dictionary were used to transform every single measurement in χ'(T) dataset into 5D vector. The vector coordinates correspond to word occurrence in single χ'(T) measurement.

V. RESULTS AND DISCUSSION

It is possible to represent the most significant features of a single χ'(T) measurement of HTS sample as 5 numerical values by using Convolutional 1D Autoencoder and Bag Of Words model. The 5D representation of χ'(T) dataset preserves the most important features of measured HTS, because the most distant 5D representations of χ'(T) are for samples, which have the most different superconducting properties i.e. thin layer HTS (very high value of critical current density) and grinded and pressed polycrystalline HTS (current is equal to zero) (Fig. 3). However the cluster analysis of 5D χ'(T) dataset by K-Means and DBSCAN did not reveal the existence of clearly distinct classes of χ'(T) measurements. Though a t-SNE visualisation (Fig. 4) shows that some clustering exists and some of the measurements are mainly arranged on some sort of cluster boundary. In our opinion, these results are interesting and more advanced analysis should be tried in future.

  Fig. 1 are shown four selected examples of χ(T) for different types of High-Temperature Superconductors.

Fig. 2 .

 2 Fig. 2. An idea of Bag Of Words approach in the analysis of χ'(T) measurement. Small figures marked as A, B, C, D and E represent the words with unique features of a sequence shown on middle plot.

Fig 4 .

 4 Fig 4. Visualisation of about 1000 measurements of χ'(T) in 3D feature space by t-SNE. Single measurement is 5D vector and is represented as single circle.

Fig 3 .

 3 Fig 3. Upper row shows χ'(T) measurements for thin layer HTS (left) and grinded and pressed polycrystalline HTS powder (right). Bottom row shows representations of these measurements in 5D space.

ACKNOWLEDGMENT

This work is supported by The National Science Center Poland under Project No. 2018/02/X/ST3/01741.


Jan Srpčič

John Durrell
email: john.durrell@eng.cam.ac.uk

Mark Ainslie
email: mark.ainslie@eng.cam.ac.uk

Yunhua Shi

The Campbell penetration depth in type-II superconductors

Keywords: Bean model, Campbell model, penetration depth, AC magnetic fields

The flux-line-lattice response to AC magnetic fields in type-II superconductors is determined by the shape of the flux pinning potential in the material. This response is normally well described within the Bean critical state model, provided the amplitude of the AC magnetic field is sufficient to unpin and move the vortices from their pinning sites, leading to hysteresis. Conversely, if the vortices remain within their potential wells, their movement remains reversible, and cannot be described by the Bean critical state model. Instead, the mixed state can be viewed as an ensemble of coupled linear harmonic oscillators, each oscillating within its potential well. This is explained by the Campbell model, which extends upon the Bean model to include the finite size of the pinning potential, and a corresponding linear and elastic pinning force. Here, the dynamic equations for flux movementgiven a linear pinning forceare described, linearized, and solved analytically. Subsequently, the solution is used to extract the value of the Campbell penetration depth of an applied AC magnetic field from experimental data obtained with a pick-up method.

I. INTRODUCTION

The flux pinning force in the Bean critical state model [START_REF] Bean | Magnetization of hard superconductors[END_REF] is assumed equal and opposite the Lorentz force, regardless of the vortex position with respect to its pinning centre. Essentially, the pinning force is treated as a constant frictional force opposing vortex movement. This assumption is sufficient to lead to the prediction of the hysteretic behaviour, as observed in type-II superconductors.

In reality, the pinning force will depend on the vortex position with respect to its pinning centreclose to the equilibrium position within the pinning centre the force will be linearly proportional to the displacement from equilibrium. This is true because the minimum of the pinning potential can always be approximated locally by a harmonic potential, which is quadratic in displacement (thus, the force is linear). Conversely, for large displacements the pinning force becomes frictional, as in the Bean model, hence its value approaches a constant independent of displacement. This was first observed and explained by Campbell [START_REF] Campbell | The response of pinned flux vortices to lowfrequency fields[END_REF] in pick-up measurements of magnetic flux due to an applied AC magnetic field.

A candidate form of the pinning force, proposed by Campbell, taking into account the above assumptions, can be written as

FP(y)=B0JC(1-exp(-y/d), ()
and is shown in Figure 1. Here, y denotes the vortex displacement from its equilibrium in the pinning potential well. B0 and JC are the background DC magnetic field and critical current density, respectively. The parameter d is the effective size of the pinning potential and marks the displacement, at which the pinning force, FP, transitions from the linear (Campbell) to the constant (Bean) regime.

For instance, for small displacements y<d the pinning force can be written as

FP(y)≈(B0JC/d)y, ()

whereas for y>d the force becomes

FP(y)≈B0JC. ()
Thus, the pinning force, as it is defined in equation (1), will describe the transition from the Campbell regime to the Bean regime, depending on the magnitude of the vortex displacement y. In this work, the above pinning force is combined with the general flux conservation equation, leading to a system of two partial differential equations dy(x)/dx=-b(x)/B0, () and

db(x)/dx=µ0JC(1-exp(-y(x)/d), ()

where µ0 is the permeability of free space. The solutions of equations ( 4) and (5) are y(x), the vortex displacement as a function of position in the superconductor x, and b(x), the local magnetic field change due to y(x). In general, the system has no analytical solutions and must be solved by numerical integration. However, it is possible to linearize the pinning force expression at a given value of x simply by substituting its linear approximation (i. e. tangent) into the equations ( 4) and ( 5). This leads, subsequently, to an analytical solution for vortex displacement, y(x), which can be shown to decay exponentially with distance into the superconductor,

y(x) ∝ exp(-x/λC), ()
where λC is the Campbell penetration depth, and is defined as

λC  = (B0d)(µ0JC). ()
Since its value directly determines the vortex displacement, y(x), and, in turn, the local magnetic field change, b(x), it can be determined experimentally via magnetic flux measurements using pick-up methods. Hence, the effective size of the pinning potential, the parameter d, which is typically on the nanometre scale, can be determined. In this talk, we present such measurements in a bulk GdBa2Cu3O7-δ superconductor.

II. RESULTS

By measuring the changing magnetic flux in a 5 mm wide and 10 mm long GdBa2Cu3O7-δ sample due to an applied AC magnetic field, the shape of b(x) in the sample can be deduced, since its integral over the cross-section of the sample will correspond to the magnetic flux. In turn, the shape of y(x) can be deduced from b(x) by using equation (4), from which the value of the Campbell penetration can be extracted. Here, this is done by measuring the pick-up voltage in a coil, wound round the superconducting sample, exposed to and AC magnetic field of amplitude 1 mT at a frequency of 300 Hz. The sample temperature is 70 K, with helium gas used as coolant. The resulting values of Campbell penetration depth, at different values of background DC magnetic field, B0, are shown in Figure 2 (date presented originally in [START_REF] Srpčič | Flux vortex dynamics in type-II superconductors[END_REF]).

Figure 1 .

 1 Figure 1. The pinning force transition from the linear Campbell regime to the constant Bean regime.

Figure 2 .

 2 Figure 2. The Campbell penetration depth for background DC magnetic fields up to B0 = 6 T.


Ilhom Rahmonov
email: rahmonov@theor.jinr.ru

Nikolay Chtchelkatchev
email: n.chtchelkatchev@gmail.com

Yury Sjukrinov
email: shukrinv@theor.jinr.ru

The resonance properties of Superconductor/Ferroelectric/Superconductor heterostructure

Keywords: Josephson effect, Josephson junction, resonance branch, Ferroelectric layer, Landau-Devonshire theory

The phase dynamics of the Superconductor/Ferroelectric/Superconductor hetero structure, which can be considered as the Josephson junction with the ferroelectric barrier, is investigated. The dynamics of the Josephson junction is described based on the Resistively and Capacitively Shunted Junction model, and the Landau-Devonshire theory is used to describe the dynamics of polarization in ferroelectric layer. The currentvoltage characteristic and the voltage dependence of polarization amplitude are calculated. It is shown that, when the Josephson frequency reaches the polarization eigenfrequency as a result of the resonance an increase in the polarization amplitude is observed and an additional resonance branch appears on the currentvoltage characteristic.

I. INTRODUCTION

The Josephson effect, which consists in the tunneling of Cooper pairs between superconducting layers separated by a barrier (Josephson junction), is of great interest in connection with the prospect of using it in superconducting electronics [START_REF] Likharev | Introduction in dynamics of Josephson junctions[END_REF][START_REF] Golubov | [END_REF]. As a barrier, we consider dielectrics, normal metals, ferromagnets, topological insulators, nanowires, etc., which lead to various physical phenomena, in particular, the shift of the phase difference and the manifestation of the Majorana mode [START_REF] Likharev | Introduction in dynamics of Josephson junctions[END_REF][START_REF] Golubov | [END_REF][3][4][5].

In this work we investigate the Superconductor/Ferroelectric/Superconductor hetero structure, which can be considered as the Josephson junction with the ferroelectric barrier. The dynamics of Josephson junction is described by the RCSJ (Resistively and Capacitively Shunted Junction) model, and the Landau-Devonshire theory is used to describe the dynamics of polarization in ferroelectric layer. We have calculated the currentvoltage characteristic of the Josephson junction and the voltage dependence of polarization amplitude in ferroelectric layer.

II. MODEL AND SIMULATION METHOD

In order to describe the dynamics of the polarization P in ferroelectric layer we can use the Landau-Devonshire theory. The free energy in the framework of this theory can be written as

2 2 ] [ 4 2 0 P B P A F P F    , (1)
where 0 F is the free energy in Curie temperature, A and B are the standard parameters of the Landau-Devonshire theory. The dynamics of the polarization in ferroelectric layer can be considered us the oscillator dynamics under the external field

PV P F  ] [
, where V is the voltage in the junction [6]. In this case the dynamical equation of polarization can be written as

V P F dt dP dt P d m       P 2 2 eff  , ( 2
)
where eff m is the parameter of the inertia, P  -is the damping parameter in the ferroelectric. In this case the total current through the system consists of the superconducting current, 

sin c s I I  , quasiparticle current, J qp R V I / 
, and displacement current.

                 ] [ sin 1 1 1 2 1 2 1 q C q dt dq V dt q d L V dt d dt dq V I dt dV P     , ( 3
)
where

I is bias current, ) /( 1 0 C R J p   
is the dissipation parameter of the JJ, p  is plasma frequency of JJ, 0 1 / q P q  -is the polarization charge,

p c I q  / 0  , ) /( 0 0 q V p P    
-is the damping parameter of the ferroelectric layer,

0 2 0 / q V m L p P   is the normalized inductance, ) /( 1 ] [ 2 1 0 1 q A C q C P   
-is the nonlinear capacitance, with nonlinear parameter 2 0 0 Bq C   . In the system of equations (3) time is normalized to the plasma frequency. We note that the considered system has its eigenfrequency, which is determined by the expression

P P P C L C / ) 1 (    . (4)
In this work we have considered the simples case, i.e. the dissipation and nonlinearity of the ferroelectric are neglected. The system of equation ( 3) is solved numerically using fourth order Runge-Kutta method. All calculations are performed for The calculated CVC shows the branches at voltages 2  V and 4  V

, which are the result of resonance of Josephson oscillations and oscillations of the polarization charge, which leads to the appearance of an average superconducting current and a decrease in the differential resistance. At these voltage values, the Josephson frequency coincides with the eigenfrequency of the oscillations of the polarization charge.

On the other hand, as a result of this resonance, the amplitude of the oscillations of the polarization charge increases, as shown in Figure 2 . We note that due to the small dissipation of the JJ, the I -V characteristic shows hysteresis, and therefore, only one side of the resonance peak is observed in the dependence of the maximum amplitude on voltage, the other side is shown by a dashed line. In conclusion we note that, we have demonstrated the interaction of ferroelectricity and superconductivity through the Josephson effect in the Superconductor/Ferroelectric/Superconductor hetero structure. It is shown that when the Josephson frequency and the eigenfrequency of polarization oscillations coincide, an increase in the amplitude of the polarization charge is observed as a result of the resonance and a resonant branch appears on the I -V characteristic of the Josephson junction.

Fig. 1 .

 1 Fig. 1. Equialent scheme of superconductor/Ferroelectric/Superconductor structure. Here P C , P L and P R are the capacitance inductance and resitance of oscillator, which describes the behavior of ferroelectric layer.

  C are critical current, resistance and capacitance of the junction, respectively. The final system of equation takes a form

Figure 2

 2 Figure 2 (a) shows the currentvoltage characteristic (CVC) of the Josephson junction. The calculation of this CVC was carried out in three stages of changing the bias current: a) increased from 0  I to 5 . 1 max  I and decreased zero; b) increased from 0  I to 5 . 1 max  I and decreased to 27 . 0  I and then increased to 5 . 1 max  I ; c) increased from 0  I to 5 . 1 max  I and decreased to 79 . 0  I and then increased again to 5 . 1 max  I .

  (b), where is presented the dependence of the maximum amplitude of the polarization charge on the voltage in the JJ. It can be seen the increasing of the amplitude of the polarization charge at 

Fig. 2 .

 2 Fig. 2. (a) Current-voltage characteristic of a JJ with a ferroelectric barrier. Shaded arrows indicate the direction of change of the base current, hollow arrows indicate the resonant branches of the I -V characteristic; (b) Dependence of the maximum polarization charge on voltage (Josephson frequency) in the Josephson junction.

ACKNOWLEDGMENT

The study was carried out with the financial support of the Russian Science Foundation in the framework of scientific project № 18-71-10095.


Min Zhang
email: min.zhang@strath.ac.uk

Muhammad Zulfiqar Ali

Felix Huber

3D modelling of HTS ring magnets

Keywords: HTS ring, 3D, H formulation

We have developed a 3D model for HTS ring shape magnets using homogenised 3D H formulation. Every ten HTS rings are homogenised into one domain in order to speed up the calculation. We have compared the simulation results to experiments performed using field cooling. The model agrees well with experimental results. We then use the model to guide the design of large HTS ring magnets for MRI applications.

I. INTRODUCTION (HEADING 1)

A HTS ring magnet is made from YBCO coated conductor 1 (Superpower M4 AP tape with 20 um copper stabilizer). Each sample conductor is 88 mm in length. Laser cutting was used to cut the tape in the center for 72 mm and each 8mm tape was left intact on both ends as shown in Fig 2(a). Fig 2(b) shows the critical current of one HTS tape after splitting. The critical current of a tape measured by the manufacturer was 520 A at 77 K. After splitting, the critical current reduces to 490 A. 100 of HTS rings are stacked together to form a ring stack. The former inside the HTS ring is 20 mm in diameter. A doublelayered ring stack is used in this experiment as shown in Fig. 1 (a) and (b), by placing two ring stacks together. Due to the asymmetric geometry, the closed side of the magnet sees two HTS layers closely stacked together while the opened side of the magnet sees a 12 mm gap between two HTS layers as shown in Fig. 3a and3b. This geometry gives a freedom in the selection of the inner diameter of the magnet. In addition, each ring works independently so any local degradation will not degrade the performance of whole magnet. The stacked ring magnet was placed in a brass holder and impregnated with paraffin wax mixed with Aluminium nitride. Fig 1 . (c) shows the sample holder after impregnation. The outer diameter of the magnet holder is 90 mm. Three hall sensors are positioned respectively in the centre (H3), 4.5mm from the center towards the closed side (H4) and 4.5 mm from the center towards the opened side (H2), as shown in Fig. 1 (d). These sensors are mounted on a double sided PCB board: H2 and H4 are on same side and H3 is on the opposite side.

II. MODELLING

To help understand the physics of HTS ring magnets, a threedimensional finite element modelling was performed using the homogenised H formulation 4,[START_REF] Cruz | Study of the Voltage Behavior of Jointless Superconducting 2G Loops during Pulse Magnetization[END_REF] . The 3D governing equation can be represented by the general form: The HTS ring magnet has 200 rings with a 3D asymmetric geometry, it is not possible to model all the rings in detail due to the high aspect ratio of 2G HTS. The previous established H formulation homogenisation 19 takes two assumptions into account; first the thickness of HTS has been artificially increased from 1 µm to 74 µm, which is the total thickness of the HTS tape; secondly, it is assumed that every ten HTS rings has the same critical current penetration depth. Therefore, every ten HTS rings is homogenised as a single HTS domain, which means the ten HTS rings have the same current distribution. One layer of HTS rings with 100 tapes is therefore represented by ten HTS domains, separated by nine air domains. There is an air domain between each two layers of HTS domains. Fig. 2 shows the comparison between experiment and modelling results. The model employs a much fast ramping current of external magnetic fields due to calculation time limits so Fig. 2 is shown in a normalised time plot. Solid lines are for experiment results and the dotted lines are for the modelling results. The difference between the modelling result and the experimental result is less than 7%. Fig. 3 illustrates the current distributions in the HTS ring during a FC cycle, corresponding to five different time steps. The current distributions in the rings are not uniform, affecting by the critical current density distribution and the magnetic field distribution. The current is induced simultaneously from the innermost and the outermost HTS domains when the magnetisation starts. When the applied field is further reduced, more current is induced in middle HTS domains and the penetration depth of each HTS domain is increasing. When the applied field reduced, the inner and outer domains start to induce persistent current loops to compensate the field reduction from the applied field. The current in the inner and outer domain reaches its peak value at 20% of applied field reduction. This saturation of inner and outer rings' current leads to a weakened shielding effects for the middle domains and forces the current to be induced in the middle rings. This process continues until all the domains saturates. The middle domains with ring 41-60 have the highest current flowing in them after the magnetisation. The changing magnetic field decreases the critical current of the outer and inner rings, so maximum induced current value is also limited by the reduction in critical current. According to Fig. 7 (e) and (e'), the outermost domain is not fully penetrated even when the magnetisation is finished. This is because it has the highest critical current density.

FIG 1 .

 1 FIG 1. (a) and (b) show the double layered ring magnet sitting in a brass holder. The opened side and the closed side of the ring are indicated respectively. (c) after paraffin impregnation. (d) Hall sensor mounted on double sided PCB board.

( 3 )

 3 where H = [Hx, Hy, Hz] T , ρ = [ρx , ρy , ρz] T is the resistivity. For HTS domains, ρ is governed by the E-J power law21 . The magnetic field dependence of critical current density Jc(B) is considered in the model using direct interpolation of measurements of Jc(B) at 25 K.

Fig. 2 .

 2 Fig.2. Simulation results. Points a, b, c, d, e and e' show the time steps for normalised current distribution shown in Fig.9. The calculation uses a 10sramping rate for the magnetic field.

Fig. 3 .

 3 Fig.3. The distribution of the normalised current density (J/Jc ) in the ring magnet during the ramping down process. (a) applied field = 4 T, (b) applied field = 3 T, (c) applied field = 2 T, (d) applied field = 1 T, (e) applied field = 0 T for the closed side and (e') applied field =0 T for the opened side.

Fig. 4

 4 Fig.4 shows the current distribution in individual HTS domains and the total current induced in the rings during the field cooling process. Ring 1-10 indicate the innermost domain while ring 91-100 indicate the outermost domain. When the applied field reduced, the inner and outer domains

Fig. 10 .

 10 Fig.10. Calculated induced currents in 10 HTS domains during the 5T field cooling process.


V Flammang

THE ABSOLUTE TRACE OF TOTALLY POSITIVE ALGEBRAIC INTEGERS

published or not. The documents may come

The absolute trace of totally positive algebraic integers

Introduction

Let α be a totally positive algebraic integer of degree d ≥ 2 (i.e. its conjugates α 1 = α, ..., α d are all positive real numbers). The trace of α is defined by tr(α) = The Schur-Siegel-Smyth trace problem (so called by P. Borwein in his book [START_REF] Borwein | Computational excursions in analysis and number theory[END_REF]) is the following:

Fix ρ < 2. Then show that all but finitely many totally positive algebraic integers α satisfy tr(α) d > ρ.

Remark: Solving this problem is equivalent to prove that 2 is the smallest limit point of T .

The problem was solved in 1918 by I. Schur for ρ < √ e [START_REF] Schur | Uber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten[END_REF] then in 1945 by C. L. Siegel for ρ < 1.7337 [START_REF] Siegel | The trace of totally positive and real algebraic integers[END_REF]. The results of Schur and of Siegel involved inequalities on the discriminant of an algebraic integer which is the quantity:

Disc(α) = 1≤i<j≤d (α i -α j ) 2 .
In 1984, C.J. Smyth solved the problem for ρ < 1.7719 [START_REF] Smyth | Totally positive algebraic integers of small trace[END_REF] by using for the first time the principle of auxiliary functions. It lies on the fact that the resultant of two polynomials with integer coefficients and without common factors is a nonzero integer. His idea was at the origin of many works on the subject. In 1997, the author, M. Grandcolas and G. Rhin solved the problem for ρ < 1.7735 [START_REF] Flammang | Small Salem numbers[END_REF]. In 2004, J. McKee and C.J. Smyth for ρ < 1.7783786 [START_REF] Mckee | Salem numbers of trace -2 and traces of totally positive algebraic integers[END_REF], in 2006, J. Aguirre, M. Bilbao and J. C. Peral for ρ < 1.7800 [START_REF] Aguirre | The trace of totally positive algebraic integers[END_REF]. In 2007, J. Aguirre and J. C. Peral solved the problem for ρ < 1.7836 [START_REF] Aguirre | The integer Chebyshev constant of Farey intervals[END_REF] then in 2008 for ρ < 1.784109 [START_REF] Aguirre | The trace problem for totally positive algebraic integers. With an appendix by Jean-Pierre Serre[END_REF]. In all these works, the polynomials involved in the auxiliary functions were found heuristically and have all positive roots. In 2003, Q. Wu [START_REF] Wu | On the linear independence measure of logarithms of rational numbers[END_REF] linked the auxiliary functions with the integer transfinite diameter ( see Section 3 below) and developed an algorithm which allows a systematic search for relevant polynomials. In 2009, we improved the previous algorithm as described in Section 4 and the polynomials are now found by induction. Hence, we call this new algorithm the recursive algorithm. This led us to solve the problem for ρ < 1.78702 [START_REF] Flammang | Trace of totally positive algebraic integers and integer transfinite diameter[END_REF]. J. McKee [START_REF] Mckee | Computing totally positive algebraic integers of small trace[END_REF] solved the problem in 2011 for ρ < 1.78839 using several of our polynomials with complex roots appearing in the auxiliary function. In 2011 also, Y. Liang and Q. Wu [START_REF] Liang | The trace problem for totally positive algebraic integers[END_REF] solved the problem for ρ < 1.79193. Finally, in 2016 [START_REF] Flammang | Une nouvelle minoration pour la trace absolue des entiers algébriques totalement positifs[END_REF], a slight variant in the use of our recursive algorithm allowed us to improve the known lower bounds and we get : if α is a totally positive algebraic integer of degree d whose

minimal polynomial is different from x -1, x 2 -3x + 1, x 3 -5x 2 + 6x -1, x 4 -7x 3 + 13x 2 -7x + 1 and x 4 -7x 3 + 14x 2 -8x + 1 then tr(α) d ≥ 1.792812.
Besides, J.P. Serre (see Appendix B in [START_REF] Aguirre | The Trace Problem for Totally Positive Algebraic Integers, Number Theory and Polynomials[END_REF]) showed that this method does not give such an inequality for any ρ larger than 1.898302... Nevertheless, it is interesting to try to get lower bounds for tr(α) d . For instance, it was used for the search of Salem numbers of smallest degree with trace equal to -2 by J.F. McKee and C.J. Smyth [START_REF] Mckee | Salem numbers of trace -2 and traces of totally positive algebraic integers[END_REF].

Another kind of inequality was established by the author, G. Rhin and C.J. Smyth in 1997 [START_REF] Flammang | The integer transfinite diameter of intervals and totally real algebraic integers[END_REF]. We got : if α is a totally positive algebraic integer of degree d ≥ 2 with minimum conjugate α 1 then, with a finite number of explicit exceptions, tr(α)

d ≥ 1.6 + α 1 .
We used the method of auxiliary functions with heuristic search of "good "polynomials. In 2006, J. Aguirre, M. Bilbao and J C Peral [START_REF] Aguirre | The trace of totally positive algebraic integers[END_REF] using the same method but adding new polynomials to ours improved the constant 1.6 to the constant 1.66. Now, thanks to our recursive algorithm, we prove the following result:

Theorem 1. Let α be a totally positive algebraic integer of degree d ≥ 2 with least conjugate α 1 . Then,

tr(α) d ≥ 1.68 + α 1
unless α is a zero of one of the polynomials listed in Table 1.

Table 1: List of all monic irreducible polynomials with positive roots, least root α 1 in (0,1) and Trace/Degreeα 1 at most 1.68

Polynomials P tr(P)/deg(P) -α 1 x 2 -3x + 1 1.1180340 x 2 -4x + 2 1.4142136 x 2 -5x + 5 1.1180340 x 3 -5x 2 + 6x -1 1.4686044 x 3 -6x 2 + 8x -2 1.6751309 x 3 -6x 2 + 9x -3 1.5320889 x 3 -7x 2 + 14x -7 1.5803129 x 3 -8x 2 + 19x -13 1.4686044 x 3 -9x 2 + 24x -19 1.5320889 x 4 -7x 3 + 13x 2 -7x + 1 1.5222229 x 4 -7x 3 + 14x 2 -8x + 1 1.5770909 x 4 -9x 3 + 27x 2 -31x + 11 1.6056743 x 5 -9x 4 + 26x 3 -29x 2 + 11x -1 1.6688217 x 6 -11x 5 + 42x 4 -68x 3 + 46x 2 -12x + 1 1.6702846 x 6 -13x 5 + 64x 4 -151x 3 + 177x 2 -96x + 19 1.6703513
This result implies that the only totally positive algebraic integers that satisfy tr(α) d < 1.68 + α 1 are those whose minimal polynomial belongs to the list above. The proof of this appears in [START_REF] Flammang | The integer transfinite diameter of intervals and totally real algebraic integers[END_REF] for the constant 1.6.

Proof of Theorem 1

The proof of the theorem is based on the following result: The b j are found recursively. We start with b 0 equals to 0 and P 0 is the polynomial-power used in [START_REF] Flammang | On the absolute trace of polynomials having all zeros in a sector[END_REF] to prove that tr(α) d ≥ 1.792812 = m(0) for all totally positive algebraic integers not a zero of the polynomials making up P 0 . The polynomials involved in P 0 and their coefficients are available on [10]. Thus, we take b 1 = 0.1138. Our recursive algorithm allows us to find a polynomial-power P b 1 such that m(b 1 ) is as big as possible. Then the process is iterated. We explain in the following sections how the P b j and the minima m(b j ) are calculated.

The proof of Theorem 1 (originally used in [START_REF] Flammang | The integer transfinite diameter of intervals and totally real algebraic integers[END_REF]) now follows easily. Let α and α 1 be as in the statement of the theorem. By replacing α by α -α 1 we can assume that α 1 ∈ (0, 1). Then we take b = α 1 in Proposition 1 and we obtain tr(α)

d ≥ 1.68 + α 1 + log | d i=1 P α 1 (α i )| 1/d i.e., tr(α) d ≥ 1.68 + α 1 + 1≤j≤J c j log | d i=1 |Q j (α i )| 1/d where P α 1 = J j=1 Q c j j .
The polynomials Q j and the exponents c j are available on [10]

Since the minimal polynomial P of α does not divide any Q j then

d i=1 Q j (α i ) is a nonzero integer because it is the resultant of P and Q j . Hence, if α is not a root of Q j , we have tr(α) d ≥ 1.68 + α 1 .

Remark:

The following sections reproduce the corresponding sections of [START_REF] Flammang | Une nouvelle minoration pour la trace absolue des entiers algébriques totalement positifs[END_REF].

3 Auxiliary function and generalized integer transfinite diameter

Generalized integer transfinite diameter

Let K be a compact subset of C. The transfinite diameter of K is defined by

t(K) = lim inf inf |P | 1 n ∞,K n ≥ 1 P ∈ C[X] n → ∞ P monic deg(P ) = n where |P | ∞,K = sup z∈K |P (z)| for P ∈ C[X].
We define the integer transfinite diameter of K by

t Z Z (K) = lim inf inf |P | 1 n ∞,K n ≥ 1 P ∈ Z Z[X] n → ∞ deg(P ) = n
Finally, if ϕ is a positive function defined on K, the ϕ-generalized integer transfinite diameter of K is defined by

t Z,ϕ (K) = lim inf inf sup |P (z)| 1 n ϕ(z) . n ≥ 1 P ∈ Z[X] z ∈ K n → ∞ deg(P ) = n
This version of weighted integer transfinite diameter was introduced by F. Amoroso [START_REF] Amoroso | f-transfinite diameter and number theoretic applications[END_REF] and is an important tool in the study of rational approximations of logarithms of rational numbers.

Link with the auxiliary functions

Let b be in [0,1). The auxiliary function involved here is of the following type:

for x > b, f (x) = x - 1≤j≤J c j log |Q j (x)| (1)
where the c j are positive real numbers and the polynomials

Q j are nonzero polynomials in Z Z[x].
In (1), we replace the coefficients c j by rational numbers a j /q where q is a positive integer such that q.c j is an integer for all 1 ≤ j ≤ J. Then we can write:

for x > b, f (x) = x - t r log |Q(x)| ≥ m (2)
where

Q = J j=1 Q a j j ∈ Z Z[X] is of degree r = J j=1 a j deg Q j and t = J j=1 c j deg Q j (this formulation
was introduced by J. P. Serre). Note that we have

t r = 1 q . We seek a polynomial Q ∈ Z Z[X] such that sup x>b |Q(x)| t/r e -x ≤ e -m .
If we suppose that t is fixed, it is equivalent to find an effective upper bound for the weighted integer transfinite diameter over the interval [b, ∞) with the weight ϕ(x) = e -x :

t Z Z,ϕ ([b, ∞)) = lim inf inf sup |P (x)| t r ϕ(x) r ≥ 1 P ∈ Z Z[X] x > b r → ∞ deg(P ) = r
Remark: Even if we have replaced the compact K by the infinite interval [b, ∞), the weight ϕ ensures that the quantity t Z Z,ϕ ([b, ∞)) is finite.

Construction of an auxiliary function

The main point is to find a set of "good "polynomials Q j , i.e., which gives the best possible value for m. Until 2003, the polynomials were found heuristically. For example, in [START_REF] Smyth | Totally positive algebraic integers of small trace[END_REF] and [START_REF] Aguirre | The integer Chebyshev constant of Farey intervals[END_REF], the authors have searched a collection of polynomials with small absolute trace all of whose roots are positive real numbers. In 2003, Q. Wu [START_REF] Wu | On the linear independence measure of logarithms of rational numbers[END_REF] has developed an algorithm that allows a systematic search of "good "polynomials. His method was the following. We consider an auxiliary function as defined by [START_REF] Aguirre | The trace of totally positive algebraic integers[END_REF]. We fix a set E 0 of control points, uniformly distributed on the real interval I = [b, A] where A is "sufficiently large ". Thanks to the LLL algorithm, we find a polynomial Q small on E 0 within the meaning of the quadratic norm. We test this polynomial in the auxiliary function and we keep only the factors of Q which have a nonzero exponent. The convergence of this new function gives local minima that we add to the set of points E 0 to get a new set of control points E 1 . We use again the LLL algorithm with the set E 1 and the process is repeated.

In 2009 [START_REF] Flammang | Trace of totally positive algebraic integers and integer transfinite diameter[END_REF], we made two improvements to this previous algorithm in the use of the LLL algorithm. The first one is, at each step, to take into account not only the new control points but also the new polynomials of the best auxiliary function. The second one is the introduction of a corrective coefficient t. The idea is to get good polynomials Q j by induction. Thus, we call this algorithm the recursive algorithm. We detail it, always for the trace. The first step consists in the optimization of the auxiliary function f 1 = x -t log x. We have t = c 1 where c 1 is the value that gives the best function f 1 , i.e, whose minimum is as large as possible . We suppose that we have some polynomials Q 1 , Q 2 , . . . , Q J and a function f as good as possible for this set of polynomials in the form [START_REF] Amoroso | f-transfinite diameter and number theoretic applications[END_REF]. We seek a polynomial R ∈ Z Z[x] of degree k (here, k is varying from 15 up to 22) such that sup

x∈I |Q(x)R(x)| t r+k e -x ≤ e -m
where Q = J j=1 Q j . We want the quantity

sup x∈I |Q(x)R(x)| exp -x(r + k) t
to be as small as possible. We apply the LLL algorithm to the linear forms

Q(x i )R(x i ) exp -x i (r + k) t .
The x i are control points which are points uniformly distributed on the interval I to which we have added points where f has local minima. Thus we find a polynomial R whose irreducible factors R j are good candidates to enlarge the set {Q 1 , . . . , Q J }. We only keep the factors R j that have a nonzero coefficient in the newly optimized auxiliary function f . After optimization, some previous polynomials Q j may have a zero exponent and so are removed.

Optimization of the c j

We have to solve a problem of the following form: find max

C min x∈X f (x, C)
where f (x, C) is a linear form with respect to C = (c 0 , c 1 , . . . , c k ) (c 0 is the coefficient of x and is equal to 1) and X is a compact domain of C, the maximum is taken over c j ≥ 0 for j = 0, . . . , k.

A classical solution consists in taking very many control points (x i ) 1≤i≤N and in solving the standard problem of linear programming:

max C min 1≤j≤N f (x i , C).
The result depends then on the choice of the control points.

The idea of the semi infinite linear programming (introduced into Number Theory by C. J. Smyth [START_REF] Smyth | Totally positive algebraic integers of small trace[END_REF]) consists in repeating the previous process adding at each step new control points and verifying that this process converges to m, the value of the linear form for an optimum choice of C . The algorithm is the following: We stop when there is a good enough convergence, when m i -m i ≤ 10 -4 . Suppose that p iterations are sufficient then we take m = m p .

  by T the set of all such tr(α) d .

Proposition 1 .

 1 For every b ∈ [0, 1) there is a polynomial-power P b = J j=1 Q c j j where the Q j s are in Z Z[X] and the c j s are positive real numbers such that for x ≥ b, x -log |P b (x)| ≥ 1.68 + b. Let m(b) denote the minimum of the function x -log |P b (x)| for x ≥ b. We construct a partition b 0 < b 1 < . . . < b 36 = 1 of [0,1) into 35 subintervals such that m(b j ) -b j+1 ≥ 1.68 (j=0,. . .,35).

( 1 )

 1 We choose an initial value for C i.e., C 0 and we calculatem 0 = min x∈X f (x, C 0 ).(2) We choose a finite set X 0 of control points belonging to X and we havem 0 ≤ m ≤ m 0 = min x∈X 0 f (x, C 0 ).

( 3 )min x∈X 1 f 1 f( 5 )

 3115 We add to X 0 the points where f (x, C 0 ) has local minima to get a new set X 1 of control points. (4) We solve the usual linear programming problem: max C (x, C) We get a new value for C denoted by C 1 and a result of the linear programming equal to m 1 = min x∈X f (x, C 1 ). Then we have m 0 ≤ m 1 ≤ m ≤ m 1 = min x∈X (x, C 1 ) ≤ m 0 , We repeat the steps from (2) to (4) and thus we get two sequences (m i ) and (m i ) which satisfy m 0 ≤ m 1 ≤ . . . ≤ m i ≤ m ≤ m i ≤ . . . ≤ m 1 ≤ m 0 ,

Table 2 :

 2 The values of b used in the proof of Proposition 1 and required functions of m(b)

	Run j	b j	m(b j )
	0	0.0000	1.792812
	1	0.112812 1.801901
	2	0.121901 1.805919
	3	0.125919 1.809730
	4	0.12973	1.812613
	5	0.132613 1.812854
	6	0.132854 1.812870
	7	0.13287	1.815242
	8	0.135242 1.817070
	9	0.13707	1.821673
	10	0.141673 1.827923
	11	0.147923 1.833573
	12	0.153573 1.842692
	13	0.162692 1.853991
	14	0.173991 1.875294
	15	0.195294 1.900870
	16	0.22087	1.925140
	17	0.24514	1.943162
	18	0.263162 1.957081
	19	0.277081 1.992052
	20	0.312052 2.033790
	21	0.35379	2.088989
	22	0.408989 2.105574
	23	0.425574 2.146377
	24	0.466377 2.162143
	25	0.482143 2.190407
	26	0.510407 2.223393
	27	0.543393 2.246183
	28	0.566183 2.297433
	29	0.617433 2.345141
	30	0.665141 2.367420
	31	0.68742	2.432224
	32	0.752224 2.485535
	33	0.805535 2.552683
	34	0.872683 2.604163
	35	0.924163 2.708093


V Flammang

ON THE INTEGER TRANSFINITE DIAMETER OF INTERVALS OF THE FORM r s , u OR [0, ( √ a -√ b) 2 ] AND OF FAREY INTERVALS

Firstly, we consider intervals of the form r s , u where r, s are positive integers with gcd(r, s)=1 and u is a real number, or of the form [0, ( √ a-√ b) 2 ] where a, b are positive integers. Thanks to a lemma of Chudnovsky, we give first a lower bound of the integer transfinite diameter of such intervals. Then, using the method of explicit auxiliary functions and our recursive algorithm, we explain how to get an upper bound for this quantity. We finish with some numerical examples. Secondly, we prove inequalities on the integer transfinite diameter of Farey intervals, i.e., intervals of the type a q , b s where |as -bq| = 1.

worth including in this paper. For example, we can obtain t Z Z 0, 1 64 ≤ 0.01520467, whereas J. Aguirre and J.C. Peral had t Z Z 0, 1 64 ≤ 0.01520616. Thus, we turn our attention to the integer transfinite diameter of intervals of the type r s , u , where r, s are positive integers with

Introduction

Let K be a compact subset of C. The transfinite diameter of K is defined by

t(K) = lim inf inf |P | 1 n ∞,K , n ≥ 1 P ∈ C[X] n → ∞ P monic deg(P ) = n
where |P | ∞,K = sup z∈K |P (z)| for P ∈ C[X], while the integer transfinite diameter of K is defined by

t Z Z (K) = lim inf inf |P | 1 n ∞,K n ≥ 1 P ∈ Z Z[X] n → ∞ deg(P ) = n
If I is a real interval of length |I|, we have the well-known properties t(I) = |I| 4 and, if |I| ≥ 4, t Z Z (I) = t(I) = |I| 4 (for more details, see [F1]). Therefore, we restrict our attention to intervals I with |I| < 4. In this case, we know that t(I) ≤ t Z Z (I) ≤ t(I) = 1 2 |I|. The first inequality came from the definition and the second one is due to D. Hilbert [H]. Many authors have been interested in the study of t Z Z (I). The case I = [0, 1] was particularly studied because of a theoretical result of Gelfond and Schnirelman [L] showing that, if t Z Z ([0, 1]) = 1/e = 0.3678 were to hold, then a simple proof of the prime number theorem would follow. However, we know that 0.4213 ≤ t Z Z ([0, 1] ≤ 0.422685. The lower bound is due to I. Pritsker [Pr] while the author proved the upper bound [F2]. The exact value of t Z Z (I) remains unknown for I with |I| < 4.

In [F1], we gave lower and upper bounds for several Farey intervals i.e., I = r s , p q with |ps -qr| = 1. Using more polynomials than us (polynomials found heuristically), J. Aguirre and J.C. Peral [AP] have improved the upper bound of each our intervals. Thanks to our recursive algorithm described in Section 3, we can improve their bounds slightly, but not by enough to be gcd(r, s)=1 and u is a real number or of the type [0, ( √ a -√ b) 2 ] where a, b are positive integers. Using methods developed in [F1] and our recursive algorithm, we obtain lower and upper bounds for them. To illustrate the method, we give some numerical examples.

Then we consider an interval I = a q , b s , where |as-bq| = 1, and an interval J = c qs , d (q + s) 2 with |c(q + s) 2 -dqs| = 1. We prove the following inequalities:

Theorem 1. t Z Z (J) ≤ t Z Z (I) 2 ≤ max( q s , s q )t Z Z (J).

The lower bounds

In [F1], we proved the following result:

Theorem 2.

t Z Z r s , p q ≥ 1 q + s ∞ i=0 (1 + λ i ) -1 2 i+1 ,
where λ 0 = qs (q + s) 2 and λ i+1 =

λ i (1 + λ i ) 2 .
The essential tool of the proof is a lemma of Chudnovsky [C] that is applied to a particular family of polynomials introduced by C. J. Smyth [S1]. Now, consider I = r s , u where r, s are positive integers with gcd(r, s)=1 and u is a real number. Euclid's algorithm allows us to find the largest fraction p q such that r s , p q ⊂ I, so that t Z Z r s , p q ≤ t Z Z (I). Then, we apply Theorem 2 to the interval r s , p q and we deduce a lower bound for I.

Let I be the interval [0, ( √ a - √ b) 2 ]
where a, b are positive integers. We search for the smallest integer q > 0 such that 0, 1 q ⊂ I. Again, we apply Theorem 2 to the interval 0, 1 q to obtain a lower bound for I.

3 The upper bounds 3.1 The case I = r s , u

Let I be an interval of the form r s , u , where r, s are integers and u is a real number. Here, we will use the principle of explicit auxiliary functions and our recursive algorithm to get an upper bound for I.

First, we choose two integers p, q such that x(t) = qt -p r -ts maps r s , u into [b, ∞) with the conditions x(u) = b and 0 < b < 1, and x r s → ∞. We know that our recursive algorithm is very efficient on [0, ∞) , which is why we impose the latter condition. The first condition implies that qr -ps = -1. Now, suppose that we can find a sequence of polynomials (Q j ) 1≤j≤J in Z[X], a sequence of positive real numbers (c j ) 1≤j≤J and a real m such that

(3.1) ∀x ∈ [b, ∞), f (x) = log |q + sx| - 1≤j≤J c j log |Q j (x)| ≥ m,
which can also be written

∀x ∈ [b, ∞), J j=1 |Q j (x)| b j ≤ e -N m |q + sx| N ,
where (b j ) 1≤j≤J is a sequence of integers and N is an integer such that

1≤j≤J b j deg(Q j ) ≤ N .
We deduce from the previous inequality

∀x ∈ [b, ∞), |R(x)| ≤ e -mN |q + sx| N ,
where we put R(x)

= J j=1 |Q j (x)| b j and note that deg(R) ≤ N .
Here we make a change of variable to get back to the interval I. We define y = p + rx q + sx . Since

x The polynomial P is explicit and we obtain t Z Z (I) ≤ e -m . Now, we have to find the polynomials Q j , the real numbers c j and the minimum m of the function f which is called an auxiliary function. This notion was introduced into number theory by C. J. Smyth in [S2].

We need the following definition: let K be a compact subset of C, if ϕ is a positive function defined on K, the ϕ-generalized integer transfinite diameter of K is defined by

t Z,ϕ (K) = lim inf inf sup |P (z)| 1 n ϕ(z) . n ≥ 1 P ∈ Z[X] z ∈ K n → ∞ deg(P ) = n
This version of weighted integer transfinite diameter was introduced by F. Amoroso [A] and is an important tool in the study of rational approximations of logarithms of rational numbers.

In the auxiliary function (3.1), we replace the coefficients c j by rational numbers a j /q, where q is a positive integer such that q.c j is an integer for all 1 ≤ j ≤ J. Then we can write:

(3.2) ∀x > b, f (x) = log |q + sx| - t r log |Q(x)| ≥ m.
where

Q = J j=1 Q a j j ∈ Z Z[X] is of degree r = J j=1 a j deg Q j and t = J j=1 c j deg Q j (this formulation
was introduced by J. P. Serre). Thus we seek a polynomial Q

∈ Z Z[X] such that sup x>b |Q(x)| t/r |q + sx| -1 ≤ e -m .
If we suppose that t is fixed, it is equivalent to find an effective upper bound for the weighted integer transfinite diameter over the interval [b, ∞) with the weight ϕ(x) = |q + sx| -1 :

t Z Z,ϕ ([b, ∞)) = lim inf inf sup |P (x)| t r ϕ(x) . r ≥ 1 P ∈ Z Z[X] x > b r → ∞ deg(P ) = r
Remark: Even though we have replaced the compact set K by the infinite interval [b, ∞), the weight ϕ ensures that the quantity

t Z Z,ϕ ([b, ∞)) is finite.
The main challenge is to find a set of "good "polynomials Q j , i.e., which gives the best possible value for m. Until 2003, the polynomials were found heuristically. This was the case in [F1] when we have bounded the integer transfinite diameter of Farey intervals. In 2003, Q. Wu [Wu] developed an algorithm to systematically search for "good "polynomials. His method was the following. We consider an auxiliary function as defined by (3.1). We fix a set E 0 of control points, uniformly distributed on the real interval L = [b, A] where A is "sufficiently large ". Thanks to the LLL algorithm, we find a polynomial Q small on E 0 withrespect to the quadratic norm. We test this polynomial in the auxiliary function and we keep only the factors of Q which have a nonzero exponent. The convergence of this new function gives local minima that we add to the set of points E 0 to get a new set of control points E 1 . We use again the LLL algorithm with the set E 1 and the process is repeated.

In 2006, we made two improvements to this previous algorithm in the use of the LLL algorithm. The first one is, at each step, to take into account not only the new control point but also the new polynomials of the best auxiliary function. The second one is the introduction of a corrective coefficient t. The idea is to get good polynomials Q j by induction. Thus, we call this algorithm the recursive algorithm. The first step consists in the optimization of the auxiliary function f 1 = log(q + sx) -t log x. We have t = c 1 where c 1 is the value that gives the best function f 1 . We suppose that we have some polynomials Q 1 , Q 2 , . . . , Q J and a function f as good as possible for this set of polynomials in the form (??)E2. We seek a polynomial R ∈

Z Z[x] of degree k (k = 10 for instance) such that sup x∈L |Q(x)R(x)| t r+k |q + sx| -1 ≤ e -m , where Q = J j=1 Q j . We want the quantity sup x∈L |Q(x)R(x)||q + sx| (r+k) t
to be as small as possible. We apply the LLL algorithm to the linear forms

Q(x i )R(x i )|q + sx i | (r+k) t .
The x i are control points, which are uniformly distributed points on the interval L to which we have added points where f has local minima. Thus we find a polynomial R whose irreducible factors R j are good candidates to enlarge the set {Q 1 , . . . , Q J }. We only keep the factors R j that have a nonzero coefficient in the newly optimized auxiliary function f . After optimization, some previous polynomials Q j may have a zero exponent and so are removed. We stop when two consecutive steps do not give any new polynomial.

Besides, we have to optimize the c j , that is to solve a problem of the following form: find max

C min x∈X f (x, C)
where f (x, C) is a linear form with respect to C = (c 0 , c 1 , . . . , c k ) (c 0 is the coefficient of x and is equal to 1) and X is a compact domain of C, the maximum is taken over c j ≥ 0 for j = 0, . . . , k.

A classical solution consists in taking very many control points (x i ) 1≤i≤N and in solving the standard problem of linear programming:

max C min 1≤j≤N f (x i , C).
The result depends then on the choice of the control points.

The idea of the semi infinite linear programming (introduced into Number Theory by C. J. Smyth [S2]) consists in repeating the previous process adding at each step new control points and verifying that this process converges to m, the value of the linear form for an optimum choice of C . The algorithm is the following:

(1) We choose an initial value for C i.e., C 0 and we calculate

m 0 = min x∈X f (x, C 0 ).
(2) We choose a finite set X 0 of control points belonging to X, and we have

m 0 ≤ m ≤ m 0 = min x∈X 0 f (x, C 0 ).
(3) We add to X 0 the points where f (x, C 0 ) has local minima to get a new set X 1 of control points. (4) We solve the usual linear programming problem:

max C min x∈X 1 f (x, C).
We get a new value for C denoted by C 1 and a result of the linear programming equal to m 1 = min x∈X f (x, C 1 ). Then we have

m 0 ≤ m 1 ≤ m ≤ m 1 = min x∈X 1 f (x, C 1 ) ≤ m 0 ,
(5) We repeat the steps from (2) to (4) and thus we get two sequences

(m i ) et (m i ) which satisfy m 0 ≤ m 1 ≤ . . . ≤ m i ≤ m ≤ m i ≤ . . . ≤ m 1 ≤ m 0 ,
We stop when there is a good enough convergence, for example when m i -m i ≤ 10 -6 . Supposing that p iterations are sufficient, we then take m = m p .

The case

I = [0, ( √ a - √ b) 2 ] Let I be an interval [0, ( √ a- √ b) 2 ]
where a, b are positive integers. We define α to be the integer part of

√ a + √ b a -b 2 . Then, the change of variable x(t) = αt -1 -t maps I to [b, ∞) where b is the fractional part of √ a + √ b a -b 2 .
Here, we consider the auxiliary function defined on [b, ∞) by:

f (x) = log |α + x| - 1≤j≤J c j log |Q j (x)|.
Then we proceed in a similar way to the method described in the paragraph above to get an upper bound for t Z Z (I).

Numerical examples

Applying the method describe above, we prove that:

Results 1.

0.097580 ≤ t Z Z 1 2 , 1 √ 3 ≤ 0.104355 0.039854 ≤ t Z Z 7 5 , √ 2 ≤ 0.047285 0.0496984 ≤ t Z Z 27 10 , e ≤ 0.054964 0.102371 ≤ t Z Z ([3, π]) ≤ 0.115497 0.1295 ≤ t Z Z ([0, ( √ 2 -1) 2 ]) ≤ 0.133449 0.284642 ≤ t Z Z ([0, ( √ 3 -1) 2 ]) ≤ 0.295697 Proof:
We detail the proof for the first interval. We search (p 0 , q 0 ) ∈ Z Z 2 such that q 0 -2p 0 = -1 ( ) with q 0 > 0 as small as possible: p 0 = q 0 = 1 is convenient. Then, the general solution of the equation ( ) is of the type:

(p k = 1 + k, q k = 1 + 2k), k ∈ Z Z. For k = 3, we have: 1 2 , 4 7 ⊂ 1 2 , 1 √
3 so that we apply Theorem 2 to this interval to get the lower bound. We are in the case where

1 √ 3 < p 0 q 0 thus b = q 2 / √ 3 -p 2 1 -2/ √ 3 = √ 3 -1.
We optimize the auxiliary function

f (x) = log |5 + 2x| - 1 leqj≤J c j log |Q j (x)| on [ √ 3 -1, ∞).
For the third interval, we have (p 0 , q 0 ) = (19, 7) then here, p 0 q 0 < e. In this case, we put b = q -1 e -p -1 27 -10e .

The polynomials involved in the previous inequalities can be read off from Table 1 to Table 6 below.

Proof of Theorem 1

We put I = a q , b s where |as -bq| = 1 and J = c qs , d (q + s) 2 with |c(q + s) 2 -dqs| = 1.

The left hand-side inequality

We suppose that t Z Z (I) = e -m I . Thus, for all > 0, sufficiently small for m = m I -> 0, there exists a polynomial

H 1 ∈ Z Z[t], deg(H 1 ) = D, satisfying (5.1) max t∈I |H 1 (t)| 1/D ≤ e -m .
We can write H 1 (t) = (ts -b) D-k K(t) where deg(K) = k ≤ D and ts -b does not divide K.

Now, put x = a -tq ts -b then x ≥ 0. H 1 becomes H 1 a + bx q + sx = ±1 q + sx D (q + sx) k K a + bx q + sx H 2 (x)
where deg(H 2 ) = deg(K) ≤ D. From (5.1), we deduce that ∀x ≥ 0, |H 2 (x)| 1/D ≤ e -m (q + sx).

If we put f (x) = log(q + sx) -

1 D log |H 2 (x)| for all x ≥ 0, we have inf x≥0 f (x) ≥ m.
We define the function g by : ∀x > 0, g

(x) = 1 2 [f (x) + f ( 1 x )].
There exists a polynomial

H 3 ∈ Z Z[x], deg(H 3 ) ≤ deg(H 2 ) such that g(x) = 1 2 log (q + s) 2 + qs(x + 1 x -2) - 1 D log |H 3 (x + 1 x -2) . Now, put y = x + 1 x -2 and h(y) = log (q + s) 2 + qsy -1 D log |H 3 (y)|. Then (5.2) inf y≥0 h(y) ≥ 2m.
Next, put z = cy + d qsy + (q + s) 2 , so that z belongs to the interval J. From (5.2), we deduce that

∀z ∈ J, |qsz -c||H 3 d -z(q + s) 2 qsz -c | 1/D ≤ e -2m , i.e., ∀z ∈ J, |(qsz -c) D-deg(H 3 ) H 3 d -z(q + s) 2 qsz -c (qsz -c) deg(H 3 ) H 4 (z) | 1/D ≤ e -2m ,
where deg(H 4 ) = deg(H 3 ).

Hence, there exists a polynomial

H 5 (z) = (qsz -c) D-deg(H 4 ) H 4 (z) ∈ Z Z[z], deg(H 5 ) = D such that ∀z ∈ J, |H 5 (z)| 1/D ≤ e -2m . It means that t Z Z (J) ≤ |H 5 (z)| 1/D ≤ (e -(m I -) ) 2 .
Finally, for that tends towards 0, we obtain

t Z Z (J) ≤ t Z Z (I) 2 .

The right-hand side inequality

We suppose that t Z Z (J) = e -m J . Thus, for all > 0, sufficiently small for m = m J -> 0, there exists a polynomial

H 1 ∈ Z Z[t], deg(H 1 ) = D, satisfying (5.3) max t∈J |H 1 (t)| 1/D ≤ e -m .
We can write H 1 (t) = (qst -c) D-k K(t) where deg(K) = k ≤ D and qst -c does not divide K.

Now, put x = d -t(q + s) 2 qst -c then x ≥ 0. then H 1 becomes H 1 cx + d qsx + (q + s) 2 = ±1 qsx + (q + s) 2 D (qsx + (q + s) 2 ) k K cx + d qsx + (q + s) 2 H 2 (x)
, where deg(H 2 ) = deg(K) ≤ D. From (5.3), we deduce that

(5.4) ∀x ≥ 0, |H 2 (x)| 1/D ≤ e -m (qsx + (q + s) 2 ).
For all x ≥ 0, it is possible to find y ≥ 0 such that x = y + 1 y -2. Thus, (5.4) becomes:

(5.5)

y D H 2 y + 1 y -2 1/D
≤ e -m (q + sy)(qy + s).

At last, put z = a + by q + sy so that z belongs to the interval I. From (5.5), we deduce

(a -qz) D (sz -b) D H 2 a -qz sz -b + sz -b a -qz -2 1/D ≤ e -m |(s 2 -q 2 )z + aq -bs|,
i.e., there exists a polynomial

H 3 ∈ Z Z[z], deg(H 3 ) = 2 deg(H 2 ) such that | (a -qz) D-deg(H 2 ) (sz -b) D-deg(H 2 ) H 3 (z) H 4 (z) | 1/D ≤ e -m |(s 2 -q 2 )z + aq -bs|, with deg(H 4 ) = 2D. Thus, we have |H 4 (z)| 1/2D ≤ e -m |(s 2 -q 2 )z + aq -bs| 1/2 . As the func- tion z → (s 2 -q 2 )z +aq -bs is monotonic on the interval I, it follows that ∀z ∈ I, ||H 4 (z)| 1/2D ≤ e -m/2 max q s , s q
. This means that t Z Z (I) ≤ e -(m J -)/2 max q s , s q .

Now, letting → 0, we get

t Z Z (I) ≤ t Z Z (I) 1/2 max q s , s q . Table 1: t Z Z 1 2 , 1 √ 3 j c j Q j 1 0.194694226959 x -1 2 0.035494760974 x -2 3 0.002816285615 x -3 4 0.004358713735 2x 3 -12x 2 + 20x -9 5 0.018800399043 x 3 -7x 2 + 14x -7 6 0.008005832569 x 4 -10x 3 + 33x 2 -41x + 16 7 0.001938639658 x 4 -10x 3 + 32x 2 -39x + 15 8 0.004174431074 x 4 -10x 3 + 33x 2 -42x + 17 9 0.001311916586
x 5 -11x 4 + 47x 3 -95x 2 + 88x -29 10 0.000845050740

x 5 -14x 4 + 72x 3 -168x 2 + 174x -62 11 0.000018382517

x 5 -13x 4 + 60x 3 -123x 2 + 113x -37 12 0.000659169385

x 6 -15x 5 + 87x 4 -245x 3 + 347x 2 -232x + 58 13 0.000522369339

x 6 -15x 5 + 84x 4 -224x 3 + 297x 2 -185x + 43 14 0.000661169596

x 6 -16x 5 + 96x 4 -274x 3 + 389x 2 -260x + 65 15 0.000060014755

x 7 -18x 6 + 129x 5 -474x 4 + 958x 3 -1059x 2 + 592x -130 16 0.000046641048 2x 7 -33x 6 + 217x 5 -736x 4 + 1388x 3 -1453x 2 + 781x -167 17 0.000968791114 x 7 -18x 6 + 129x 5 -475x 4 + 966x 3 -1080x 2 + 613x -137 18 0.000283740742 x 7 -19x 6 + 145x 5 -571x 4 + 1240x 3 -1469x 2 + 873x -202 19 0.001171298002 x 8 -21x 7 + 180x 6 -820x 5 + 2170x 4 -3416x 3 + 3124x 2 -1518x + 301 20 0.000599474735 x 8 -20x 7 + 165x 6 -731x 5 + 1896x 4 -2938x 3 + 2648x 2 -1267x + 247 21 0.000513081115 x 12 -31x 11 + 422x 10 -3333x 9 + 17000x 8 -58960x 7 + 142509x 6 -241765x 5 + 285610x 4 -229089x 3 + 118442x 2 -35464x + 4657 22 0.000266864041 x 12 -31x 11 + 422x 10 -3333x 9 + 17001x 8 -58975x 7 + 142601x 6 -242068x 5 + 286200x 4 -229789x 3 + 118939x 2 -35658x + 4689 Table 2: t Z Z 7 5 , √ 2 j c j Q j 1 0.162145981678 x -1 2 0.016639017547 2x -1 3 0.047659908308 x 2 -3x + 1 4 0.008735631418 x 2 -4x + 2 5 0.006909486855 x 2 -5x + 2 6 0.008890960380 x 3 -6x 2 + 9x -3 7 0.007016724017 x 4 -9x 3 + 20x 2 -14x + 3 8 0.000210255230 x 4 -7x 3 + 13x 2 -7x + 1 9 0.001167637570 x 5 -10x 4 + 33x 3 -46x 2 + 26x -5 10 0.005671721498 x 6 -13x 5 + 60x 4 -121x 3 + 111x 2 -46x + 7 11 0.000866481352 x 6 -13x 5 + 61x 4 -132x 3 + 138x 2 -65x + 11 12 0.000552782161 x 7 -15x 6 + 86x 5 -239x 4 + 344x 3 -256x 2 + 93x -13 13 0.001261761511 x 7 -14x 6 + 77x 5 -209x 4 + 297x 3 -219x 2 + 79x -11 14 0.001985253256 x 7 -15x 6 + 85x 5 -229x 4 + 315x 3 -222x 2 + 76x -10 15 0.000621304236 2x 7 -27x 6 + 138x 5 -343x 4 + 446x 3 -303x 2 + 101x -13 16 0.001330105312 x 9 -20x 8 + 159x 7 -657x 6 + 1551x 5 -2168x 4 + 1799x 3 -863x 2 + 220x -23 17 0.000656024029 x 9 -19x 8 + 147x 7 -597x 6 + 1388x 5 -1908x 4 + 1555x 3 -733x 2 + 184x -19 18 0.000658635577 x 10 -23x 9 + 217x 8 -1104x 7 + 3347x 6 -6301x 5 + 7439x 4 -5436x 3 + 2366x 2 -559x +55 19 0.000367168471 x 10 -22x 9 + 198x 8 -959x 7 + 2763x 6 -4941x 5 + 5550x 4 -3873x 3 + 1617x 2 -368x +35 20 0.000486211368
x 10 -23x 9 + 217x 8 -1106x 7 + 3364x 6 -6357x 5 + 7531x 4 -5518x 3 + 2406x 2 -569x +56 21 0.000294675856

x 10 -22x 9 + 199x 8 -968x 7 + 2795x 6 -4999x 5 + 5608x 4 -3905x 3 + 1626x 2 -369x +35 22 0.000155171828

x 12 -27x 11 + 309x 10 -1978x 9 + 7867x 8 -20424x 7 + 35410x 6 -41270x 5 + 32141x 4 -16392x 3 + 5234x 2 -947x + 74 23 0.000751307129

x 12 -26x 11 + 288x 10 -1791x 9 + 6945x 8 -17655x 7 + 30136x 6 -34807x 5 + 27052x 4 -13857x 3 + 4467x 2 -819x + 65 24 0.000018796797

x 12 -28x 11 + 332x 10 -2201x 9 + 9074x 8 -24482x 7 + 44308x 6 -54232x 5 + 44648x 4 -24206x 3 + 8246x 2 -1594x + 133 25 0.000014657665

x 13 -31x 12 + 416x 11 -3191x 10 + 15570x 9 -50902x 8 + 114432x 7 -178754x 6 +193820x 5 -144150x 4 + 71665x 3 -22651x 2 + 4101x -323 x 5 -13x 4 + 62x 3 -136x 2 + 136x -49 14 0.002838151941

x 5 -14x 4 + 73x 3 -175x 2 + 190x -73 15 0.003090337490

x 5 -14x 4 + 69x 3 -153x 2 + 153x -55 16 0.000877501362

x 5 -14x 4 + 70x 3 -157x 2 + 158x -57 17 0.000783458036

x 7 -21x 6 + 174x 5 -732x 4 + 1678x 3 -2091x 2 + 1315x -325 18 0.000718382052

x 8 -23x 7 + 216x 6 -1080x 5 + 3143x 4 -5451x 3 + 5503x 2 -2959x + 651 19 0.000776048203

x 9 -25x 8 + 264x 7 -1542x 6 + 5481x 5 -12283x 4 + 17346x 3 -14882x 2 + 7040x -1401 20 0.000669081482

x 13 -39x 12 + 673x 11 -6800x 10 + 44870x 9 -204169x 8 + 659163x 7 -1529181x 6 +2549851x 5 -3019341x 4 + 2468843x 3 -1320786x 2 + 414697x -57780 21 0.000505379075

x 13 -39x 12 + 673x 11 -6801x 10 + 44892x 9 -204377x 8 + 660280x 7 -1532961x 6 +2558282x 5 -3031903x

4 + 2481202x 3 -1328481x 2 + 417440x -58206 Table 4: t Z Z ([3, π]) j c j Q j 1 0.069920470860 x 2 0.054008763288 x -1 3 0.004496196393 x -2 4 0.019096822557 x 2 -3x + 1 5 0.001274142158 x 2 -4x + 1 6 0.000255157554 x 2 -4x + 2 7 0.000316881818 x 3 -6x 2 + 9x -3 8 0.005641656329 x 3 -5x 2 + 6x -1 Table 5: t Z Z ([0, ( √ 2 -1) 2 ]) j c j Q j 1 0.115071100342 x -1 2 0.042666343105 x -2 3 0.006877529069 x -3 4 0.008229041311
x 2 -5x + 5 5 0.005431600967 x 3 -8x 2 + 18x -10 6 0.000180696893 x 4 -11x 3 + 40x 2 -55x + 24 7 0.000156842585 x 4 -11x 3 + 39x 2 -53x + 23 8 0.000204367141 2x 4 -18x 3 + 56x 2 -70x + 29 9 0.001513115862 x 4 -10x 3 + 34x 2 -45x + 19 10 0.000644176350

x 5 -12x 4 + 55x 3 -119x 2 + 119x -43 11 0.001573791306

x 5 -13x 4 + 63x 3 -141x 2 + 144x -53 12 0.000719661252

x 5 -13x 4 + 62x 3 -135x 2 + 133x -47 13 0.000865530666

x 5 -13x 4 + 62x 3 -136x 2 + 136x -49 14 0.001236127494

x 5 -13x 4 + 63x 3 -140x 2 + 141x -51 15 0.000391029270

x 5 -14x 4 + 73x 3 -175x 2 + 190x -73 16 0.000918506387

x 6 -16x 5 + 99x 4 -302x 3 + 479x 2 -375x + 113 17 0.000697517226

x 6 -16x 5 + 98x 4 -296x 3 + 466x 2 -363x + 109 18 0.000463327673

x 7 -18x 6 + 132x 5 -510x 4 + 1119x 3 -1392x 2 + 908x -239 19 0.000367368001

x 7 -18x 6 + 133x 5 -520x 4 + 1156x 3 -1455x 2 + 957x -253 20 0.000732687113

x 7 -18x 6 + 132x 5 -511x 4 + 1125x 3 -1404x 2 + 917x -241 21 0.000025639888 2x 11 -55x 10 + 667x 9 -4704x 8 + 21414x 7 -66001x 6 + 140395x 5 -205912x 4 + 203889x 3 -129715x 2 + 47698x -7679 22 0.000716635362

x 11 -29x 10 + 370x 9 -2738x 8 + 13041x 7 -41928x 6 + 92748x 5 -141021x 4 + 144313x 3 -94600x 2 + 35735x -5893 23 0.000397521950

x 11 -29x 10 + 369x 9 -2717x 8 + 12853x 7 -40986x 6 + 89835x 5 -135252x 4 + 136992x 3 -88857x 2 + 33207x -5417 24 0.000524876998

x 11 -29x 10 + 370x 9 -2738x 8 + 13040x 7 -41913x 6 + 92655x 5 -140713x 4 + 143727x 3 -93962x 2 + 35368x -5807 25 0.000612892287

x 11 -29x 10 + 369x 9 -2717x 8 + 12852x 7 -40973x 6 + 89766x 5 -135058x 4 + 136679x 3 -88566x 2 + 33062x -5387 26 0.000148377159

x 12 -33x 11 + 484x 10 -4167x 9 + 23428x 8 -90515x 7 + 246146x 6 -474230x 5 + 641859x 4 -594750x 3 + 357931x 2 -125575x + 19423 27 0.000103216390

x 12 -33x 11 + 485x 10 -4192x 9 + 23699x 8 -92192x 7 + 252704x 6 -491157x 5 + 671055x 4 -627975x 3 + 381809x 2 -135365x + 21163 28 0.000133150862

x 12 -32x 11 + 455x 10 -3799x 9 + 20732x 8 -77852x 7 + 206129x 6 -387456x 5 + 512803x 4 -465781x 3 + 275478x 2 -95228x + 14551 29 0.000246669901

x 13 -35x 12 + 552x 11 -5191x 10 + 32453x 9 -142327x 8 + 450113x 7 -1038385x 6 +1745715x 5 -2110202x 4 + 1780812x 3 -992518x 2 + 327128x -48119 30 0.000039809568

x 14 -38x 13 + 655x 12 -6783x 11 + 47114x 10 -232039x 9 + 835104x 8 -2229770x 7 +4435843x 6 -6539179x 5 + 7027795x 4 -5336585x 3 + 2705220x 2 -819024x + 111689 0.000528294702 4x 5 -44x 4 + 180x 3 -343x 2 + 305x -101 9 0.000696152224 x 5 -14x 4 + 72x 3 -167x 2 + 173x -64 10 0.009486871917 x 5 -13x 4 + 63x 3 -141x 2 + 144x -53 11 0.000692925074 x 5 -14x 4 + 70x 3 -158x 2 + 161x -59 12 0.001337857384 x 5 -14x 4 + 71x 3 -163x 2 + 168x -62 13 0.000171441668 x 5 -15x 4 + 82x 3 -203x 2 + 225x -88 14 0.000196328975 x 6 -17x 5 + 111x 4 -355x 3 + 586x 2 -475x + 148 15 0.000542120909 x 6 -17x 5 + 111x 4 -355x 3 + 586x 2 -474x + 147 16 0.005326448972 x 6 -16x 5 + 100x 4 -311x 3 + 506x 2 -408x + 127 17 0.000722607689 x 7 -19x 6 + 146x 5 -587x 4 + 1332x 3 -1705x 2 + 1140x -307 18 0.001404371641 x 7 -19x 6 + 146x 5 -587x 4 + 1333x 3 -1709x 2 + 1145x -309 19 0.000116819748 x 7 -19x 6 + 146x 5 -588x 4 + 1340x 3 -1727x 2 + 1165x -317 20 0.000502577474 x 7 -19x 6 + 146x 5 -588x 4 + 1339x 3 -1722x 2 + 1157x -313 21 0.000262250259 x 7 -20x 6 + 161x 5 -677x 4 + 1606x 3 -2148x 2 + 1498x -419 22 0.000701618498 x 7 -18x 6 + 133x 5 -522x 4 + 1171x 3 -1495x 2 + 1002x -271

  > b we have y ∈ I. We define P (y) = R qy -p r -sy (r -sy) N . Thus deg(P ) = deg(R) + (Ndeg(R)) = N and the previous inequality becomes ∀y ∈ I, |P (y)| ≤ e -mN , because qr -ps = -1.

Table 3 :

 3 t Z Z

	27 10	, e

Table 6

 6

	: t Z Z ([0, (	√	3 -1) 2 ])