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This study introduces an innovative local statistical moment approach for estimating Kramers-Moyal coefficients, effectively bridging the gap between nonparametric and parametric methodologies. These coefficients play a crucial role in characterizing stochastic processes. Our proposed approach provides a versatile framework for localized coefficient estimation, combining the flexibility of nonparametric methods with the interpretability of global parametric approaches. We showcase the efficacy of our approach through use cases involving both stationary and non-stationary time series analysis. Additionally, we demonstrate its applicability to real-world complex systems, specifically in the energy conversion process analysis of a wind turbine.

Using the well-known realisation of the root system $ H_3 $ as a folding of $ D_6 $, one can construct examples of $ H_3 $-graded groups from Chevalley groups of type $ D_6 $. Such Chevalley groups are defined over a commutative ring $ R $, and the root groups of the resulting $ H_3 $-grading are coordinatised by $ R \times R $. We show that every $ H_3 $-graded group arises as the folding of a $ D_6 $-graded group, or in other words, that it is coordinatised by $ R \times R $ for some commutative ring $ R $., Comment: 42 pages

In this paper, we present a complete analytic probability based description of mobile-to-mobile uncorrelated scatter channels. The correlation based description introduced by Bello and Matz is thus complemented by the presented probabilistic description leading to a common theoretical description of uncorrelated scatter channels. Furthermore, we introduce novel twodimensional hybrid characteristic probability density functions, which remain a probability density in one of the variables and a characteristic function in the other variable. Such a probability based description allows us to derive a mathematical model, in which the attenuation of the scattering components is inherently included in these two-dimensional functions. Therefore, there is no need to determine the path loss exponent. Additionally, the Doppler probability density function with the inclusion of the path loss leads to a concave function of the Doppler spectrum, which is quite different from the Jakes and Doppler spectra and can be directly parameterized by the velocity vectors and geometry of the scattering plane. Thus, knowing those parameters permits the theoretical computation of the Doppler spectra and temporal characteristic functions. Finally, we present a comparison between the computed probability based theoretical results and measurement data for a generic mobile-to-mobile channel. The agreement between the two shows the usefulness of the probability based description and confirms new shapes of the Doppler power spectra.

Signatures of collectivity, including azimuthally anisotropic and radial flow as well as characteristic hadrochemical dependencies, have been observed since long in (ultra)relativistic nucleus-nucleus collisions. They underpin the interpretation of these collision systems in terms of QGP formation and close-to-perfect fluidity. Remarkably, however, essentially all these signatures of collectivity have been identified within the last decade in collision systems as small as pp and p-Pb, where collective phenomena had been assumed to be absent traditionally. Precursor phenomena may have been found even in ep and $e^+e^-$ collisions. This article provides a complete review of all data on small system collectivity. It reviews model simulations of these data where available. However, in the absence of a phenomenologically fully satisfactory description of collectivity across all system sizes, we focus in particular on the theoretical basis of all dynamical frameworks of collectivity invoked in heavy ion collisions, and their expected scaling with system size. Our discussion clarifies to what extent all dynamical explanations are challenged by the available data., Comment: Invited article submitted for consideration in World Scientific Annual Review of Particle Physics

We introduce a novel Image Quality Assessment (IQA) dataset comprising 6073 UHD-1 (4K) images, annotated at a fixed width of 3840 pixels. Contrary to existing No-Reference (NR) IQA datasets, ours focuses on highly aesthetic photos of high technical quality, filling a gap in the literature. The images, carefully curated to exclude synthetic content, are sufficiently diverse to train general NR-IQA models. Importantly, the dataset is annotated with perceptual quality ratings obtained through a crowdsourcing study. Ten expert raters, comprising photographers and graphics artists, assessed each image at least twice in multiple sessions spanning several days, resulting in 20 highly reliable ratings per image. Annotators were rigorously selected based on several metrics, including self-consistency, to ensure their reliability. The dataset includes rich metadata with user and machine-generated tags from over 5,000 categories and popularity indicators such as favorites, likes, downloads, and views. With its unique characteristics, such as its focus on high-quality images, reliable crowdsourced annotations, and high annotation resolution, our dataset opens up new opportunities for advancing perceptual image quality assessment research and developing practical NR-IQA models that apply to modern photos. Our dataset is available at https://database.mmsp-kn.de/uhd-iqa-benchmark-database.html

We devise a novel method for implied volatility smoothing based on neural operators. The goal of implied volatility smoothing is to construct a smooth surface that links the collection of prices observed at a specific instant on a given option market. Such price data arises highly dynamically in ever-changing spatial configurations, which poses a major limitation to foundational machine learning approaches using classical neural networks. While large models in language and image processing deliver breakthrough results on vast corpora of raw data, in financial engineering the generalization from big historical datasets has been hindered by the need for considerable data pre-processing. In particular, implied volatility smoothing has remained an instance-by-instance, hands-on process both for neural network-based and traditional parametric strategies. Our general operator deep smoothing approach, instead, directly maps observed data to smoothed surfaces. We adapt the graph neural operator architecture to do so with high accuracy on ten years of raw intraday S&P 500 options data, using a single set of weights. The trained operator adheres to critical no-arbitrage constraints and is robust with respect to subsampling of inputs (occurring in practice in the context of outlier removal). We provide extensive historical benchmarks and showcase the generalization capability of our approach in a comparison with SVI, an industry standard parametrization for implied volatility. The operator deep smoothing approach thus opens up the use of neural networks on large historical datasets in financial engineering.

In the vanishing viscosity limit from the Navier-Stokes to Euler equations on domains with boundaries, a main difficulty comes from the mismatch of boundary conditions and, consequently, the possible formation of a boundary layer. Within a purely interior framework, Constantin and Vicol showed that the two-dimensional viscosity limit is justified for any arbitrary but finite time under the assumption that on each compactly contained subset of the domain, the enstrophies are bounded uniformly along the viscosity sequence. Within this framework, we upgrade to local strong convergence of the vorticities under a similar assumption on the $p$-enstrophies, $p>2$. The key novel idea is the analysis of the evolution of the weak convergence defect.

Rydberg atoms have large transition electric dipole moments and high sensitivity to electric fields. We describe a new method for microwave field sensing in a vapor cell consisting of separate excitation, quantum evolution between two Rydberg levels in the dark and state-dependent detection with probe laser transmission. Using microwave pulse techniques we study homogeneous and inhomogeneous relaxation of the coherence between Rydberg levels and demonstrate detection of a 10 GHz signal with electric field sensitivity of 10 nV/cm/Hz$^{1/2}$., Comment: 5 pages, 4 figures

We consider the homogenisation of a diffusion equation in a porous medium. The microstructure is time-dependent and oscillating on a small time scale. This oscillation causes a novel advection in the homogenised equations. Allowing for a locally varying geometry, the oscillating microstructure demonstrates the ability to generate arbitrary and locally varying advection velocity fields., Comment: 33 pages, 2 figures

This work discusses a novel method for estimating the location of a gas source based on spatially distributed concentration measurements taken, e.g., by a mobile robot or flying platform that follows a predefined trajectory to collect samples. The proposed approach uses a Physics-Guided Neural Network to approximate the gas dispersion with the source location as an additional network input. After an initial offline training phase, the neural network can be used to efficiently solve the inverse problem of localizing the gas source based on measurements. The proposed approach allows avoiding rather costly numerical simulations of gas physics needed for solving inverse problems. Our experiments show that the method localizes the source well, even when dealing with measurements affected by noise., Comment: Submission to the International Symposium of Electronic Nose (ISOEN)

The lack of cycling infrastructure in urban environments hinders the adoption of cycling as a viable mode for commuting, despite the evident benefits of (e-)bikes as sustainable, efficient, and health-promoting transportation modes. Bike network planning is a tedious process, relying on heuristic computational methods that frequently overlook the broader implications of introducing new cycling infrastructure, in particular the necessity to repurpose car lanes. In this work, we call for optimizing the trade-off between bike and car networks, effectively pushing for Pareto optimality. This shift in perspective gives rise to a novel linear programming formulation towards optimal bike network allocation. Our experiments, conducted using both real-world and synthetic data, testify the effectiveness and superiority of this optimization approach compared to heuristic methods. In particular, the framework provides stakeholders with a range of lane reallocation scenarios, illustrating potential bike network enhancements and their implications for car infrastructure. Crucially, our approach is adaptable to various bikeability and car accessibility evaluation criteria, making our tool a highly flexible and scalable resource for urban planning. This paper presents an advanced decision-support framework that can significantly aid urban planners in making informed decisions on cycling infrastructure development.

We consider the Cauchy problem for the 3D incompressible axisymmetric swirl-free Euler equations. The convex integration method developed by De Lellis and Sz\'ekelyhidi rules out the possibility that the Euler equations admit unique admissible weak solutions. It had remained conceivable, though, that axisymmetry of the solution might serve as a selection criterion. Using a surprising link to the 2D isentropic compressible Euler equations, we will show that this is not the case: There exists initial data for which there are infinetely many admissible swirl-free axisymmetric weak solutions of the 3D incompressible Euler equations. Moreover, somewhat conversely, we show that there exists an axisymmetric swirl-free initial velocity for which the axisymmetry breaks down instantaneously.

We define and study root graded groups, that is, groups graded by finite root systems. This notion generalises several existing concepts in the literature, including in particular Jacques Tits' notion of RGD-systems. The most prominent examples of root graded groups are Chevalley groups over commutative associative rings. Our main result is that every root graded group of rank at least 3 is coordinatised by some algebraic structure satisfying a variation of the Chevalley commutator formula. This result can be regarded as a generalisation of Tits' classification of thick irreducible spherical buildings of rank at least 3 to the case of non-division algebraic structures. All coordinatisation results in this book are proven in a characteristic-free way. This is made possible by a new computational method that we call the blueprint technique., Comment: 327 pages

We consider the homogenisation of a coupled Stokes flow and advection-reaction-diffusion problem in a perforated domain with an evolving microstructure of size $\varepsilon$. Reactions at the boundaries of the microscopic interfaces lead to the formation of a solid layer having a variable, a priori unknown thickness. This results in a growth or shrinkage of the solid phase and, thus, the domain evolution is not known a priori but induced by the advection-reaction-diffusion process. The achievements of this work are the existence and uniqueness of a weak microscopic solution and the rigorous derivation of an effective model for $\varepsilon \to 0$, based on $\varepsilon$-uniform a priori estimates. As a result of the limit passage, the processes on the macroscale are described by an advection-reaction-diffusion problem coupled to Darcy's equation with effective coefficients (porosity, diffusivity and permeability) depending on local cell problems. These local problems are formulated on cells, which depend on the macroscopic position and evolve in time. In particular, the evolution of these cells depends on the macroscopic concentration. Thus, the cell problems (respectively the effective coefficients) are coupled to the macroscopic unknowns and vice versa, leading to a strongly coupled micro-macro model. For pure reactive-diffusive transport coupled with microscopic domain evolution but without advective transport, homogenisation results have recently been presented. We extend these models by advective transport which is driven by the Stokes equation in the a priori unknown evolving pore domain., Comment: 44 pages

We consider the homogenisation of the instationary Stokes equations in a porous medium with an a-priori given evolving microstructure. In order to pass to the homogenisation limit, we transform the Stokes equations to a domain with a fixed periodic microstructure. The homogenisation result is a Darcy-type equation with memory term and has the form of an integro-differential equation. The evolving microstructure leads to a time and space dependent permeability coefficient and the local change of the porosity causes an additional source term for the pressure., Comment: 34 pages

Genome rearrangement is a common model for molecular evolution. In this paper, we consider the Pairwise Rearrangement problem, which takes as input two genomes and asks for the number of minimum-length sequences of permissible operations transforming the first genome into the second. In the Single Cut-and-Join model (Bergeron, Medvedev, & Stoye, J. Comput. Biol. 2010), Pairwise Rearrangement is $\#\textsf{P}$-complete (Bailey, et. al., COCOON 2023), which implies that exact sampling is intractable. In order to cope with this intractability, we investigate the parameterized complexity of this problem. We exhibit a fixed-parameter tractable algorithm with respect to the number of components in the adjacency graph that are not cycles of length $2$ or paths of length $1$. As a consequence, we obtain that Pairwise Rearrangement in the Single Cut-and-Join model is fixed-parameter tractable by distance. Our results suggest that the number of nontrivial components in the adjacency graph serves as the key obstacle for efficient sampling., Comment: Full version of paper to appear in SWAT 2024; arXiv admin note: text overlap with arXiv:2305.01851

Pre-trained language models (PLM) based on transformer neural networks developed in the field of natural language processing (NLP) offer great opportunities to improve automatic content analysis in communication science, especially for the coding of complex semantic categories in large datasets via supervised machine learning. However, three characteristics so far impeded the widespread adoption of the methods in the applying disciplines: the dominance of English language models in NLP research, the necessary computing resources, and the effort required to produce training data to fine-tune PLMs. In this study, we address these challenges by using a multilingual transformer model in combination with the adapter extension to transformers, and few-shot learning methods. We test our approach on a realistic use case from communication science to automatically detect claims and arguments together with their stance in the German news debate on arms deliveries to Ukraine. In three experiments, we evaluate (1) data preprocessing strategies and model variants for this task, (2) the performance of different few-shot learning methods, and (3) how well the best setup performs on varying training set sizes in terms of validity, reliability, replicability and reproducibility of the results. We find that our proposed combination of transformer adapters with pattern exploiting training provides a parameter-efficient and easily shareable alternative to fully fine-tuning PLMs. It performs on par in terms of validity, while overall, provides better properties for application in communication studies. The results also show that pre-fine-tuning for a task on a near-domain dataset leads to substantial improvement, in particular in the few-shot setting. Further, the results indicate that it is useful to bias the dataset away from the viewpoints of specific prominent individuals., Comment: Accepted for Studies in Communication and Media

A swarm of robots has advantages over a single robot, since it can explore larger areas much faster and is more robust to single-point failures. Accurate relative positioning is necessary to successfully carry out a collaborative mission without collisions. When Visual Simultaneous Localization and Mapping (VSLAM) is used to estimate the poses of each robot, inter-agent loop closing is widely applied to reduce the relative positioning errors. This technique can mitigate errors using the feature points commonly observed by different robots. However, it requires significant computing and communication capabilities to detect inter-agent loops, and to process the data transmitted by multiple agents. In this paper, we propose Collaborative SLAM using Visual Odometry and Range measurements (CoVOR-SLAM) to overcome this challenge. In the framework of CoVOR-SLAM, robots only need to exchange pose estimates, covariances (uncertainty) of the estimates, and range measurements between robots. Since CoVOR-SLAM does not require to associate visual features and map points observed by different agents, the computational and communication loads are significantly reduced. The required range measurements can be obtained using pilot signals of the communication system, without requiring complex additional infrastructure. We tested CoVOR-SLAM using real images as well as real ultra-wideband-based ranges obtained with two rovers. In addition, CoVOR-SLAM is evaluated with a larger scale multi-agent setup exploiting public image datasets and ranges generated using a realistic simulation. The results show that CoVOR-SLAM can accurately estimate the robots' poses, requiring much less computational power and communication capabilities than the inter-agent loop closing technique., Comment: Submitted to the IEEE Transactions on Intelligent Transportation Systems

The proliferation of car sharing services in recent years presents a promising avenue for advancing sustainable transportation. Beyond merely reducing car ownership rates, these systems can play a pivotal role in bolstering grid stability through the provision of ancillary services via vehicle-to-grid (V2G) technologies - a facet that has received limited attention in previous research. In this study, we analyze the potential of V2G in car sharing by designing future scenarios for a national-scale service in Switzerland. We propose an agent-based simulation pipeline that considers population changes as well as different business strategies of the car sharing service, and we demonstrate its successful application for simulating scenarios for 2030. To imitate car sharing user behavior, we develop a data-driven mode choice model. Our analysis reveals important differences in the examined scenarios, such as higher vehicle utilization rates for a reduced fleet size as well as in a scenario featuring new car sharing stations. These disparities translate into variations in the power flexibility of the fleet available for ancillary services, ranging from 12 to 50 MW, depending on the scenario and the time of the day. Furthermore, we conduct a case study involving a subset of the car sharing fleet, incorporating real-world electricity pricing data. The case study substantiates the existence of a sweet spot involving monetary gains for both power grid operators and fleet owners. Our findings provide guidelines to decision makers and underscore the pressing need for regulatory enhancements concerning power trading within the realm of car sharing.

Cryogenic electron tomography is a technique for imaging biological samples in 3D. A microscope collects a series of 2D projections of the sample, and the goal is to reconstruct the 3D density of the sample called the tomogram. Reconstruction is difficult as the 2D projections are noisy and can not be recorded from all directions, resulting in a missing wedge of information. Tomograms conventionally reconstructed with filtered back-projection suffer from noise and strong artifacts due to the missing wedge. Here, we propose a deep-learning approach for simultaneous denoising and missing wedge reconstruction called DeepDeWedge. The algorithm requires no ground truth data and is based on fitting a neural network to the 2D projections using a self-supervised loss. DeepDeWedge is simpler than current state-of-the-art approaches for denoising and missing wedge reconstruction, performs competitively and produces more denoised tomograms with higher overall contrast.

Concerns about data privacy are omnipresent, given the increasing usage of digital applications and their underlying business model that includes selling user data. Location data is particularly sensitive since they allow us to infer activity patterns and interests of users, e.g., by categorizing visited locations based on nearby points of interest (POI). On top of that, machine learning methods provide new powerful tools to interpret big data. In light of these considerations, we raise the following question: What is the actual risk that realistic, machine learning based privacy attacks can obtain meaningful semantic information from raw location data, subject to inaccuracies in the data? In response, we present a systematic analysis of two attack scenarios, namely location categorization and user profiling. Experiments on the Foursquare dataset and tracking data demonstrate the potential for abuse of high-quality spatial information, leading to a significant privacy loss even with location inaccuracy of up to 200m. With location obfuscation of more than 1 km, spatial information hardly adds any value, but a high privacy risk solely from temporal information remains. The availability of public context data such as POIs plays a key role in inference based on spatial information. Our findings point out the risks of ever-growing databases of tracking data and spatial context data, which policymakers should consider for privacy regulations, and which could guide individuals in their personal location protection measures.

The Geometric Brownian Motion (GBM) is a standard model in quantitative finance, but the potential function of its stochastic differential equation (SDE) cannot include stable nonzero prices. This article generalises the GBM to an SDE with polynomial drift of order q and shows via model selection that q=2 is most frequently the optimal model to describe the data. Moreover, Markov chain Monte Carlo ensembles of the accompanying potential functions show a clear and pronounced potential well, indicating the existence of a stable price., Comment: 9 pages, 5 figures

Despite the strong predictive performance of deep learning models for traffic prediction, their widespread deployment in real-world intelligent transportation systems has been restrained by a lack of interpretability. Uncertainty quantification (UQ) methods provide an approach to induce probabilistic reasoning, improve decision-making and enhance model deployment potential. To gain a comprehensive picture of the usefulness of existing UQ methods for traffic prediction and the relation between obtained uncertainties and city-wide traffic dynamics, we investigate their application to a large-scale image-based traffic dataset spanning multiple cities and time periods. We compare two epistemic and two aleatoric UQ methods on both temporal and spatio-temporal transfer tasks, and find that meaningful uncertainty estimates can be recovered. We further demonstrate how uncertainty estimates can be employed for unsupervised outlier detection on changes in city traffic dynamics. We find that our approach can capture both temporal and spatial effects on traffic behaviour in a representative case study for the city of Moscow. Our work presents a further step towards boosting uncertainty awareness in traffic prediction tasks, and aims to highlight the value contribution of UQ methods to a better understanding of city traffic dynamics., Comment: 39 pages, 22 figures. Code publicly available at: https://github.com/alextimans/traffic4cast-uncertainty

Based on the characterization of the polyconvex envelope of isotropic functions by their signed singular value representations, we propose a simple algorithm for the numerical approximation of the polyconvex envelope. Instead of operating on the $d^2$-dimensional space of matrices, the algorithm requires only the computation of the convex envelope of a function on a $d$-dimensional manifold, which is easily realized by standard algorithms. The significant speedup associated with the dimensional reduction from $d^2$ to $d$ is demonstrated in a series of numerical experiments., Comment: 17 pages, 7 figures

The just noticeable difference (JND) is the minimal difference between stimuli that can be detected by a person. The picture-wise just noticeable difference (PJND) for a given reference image and a compression algorithm represents the minimal level of compression that causes noticeable differences in the reconstruction. These differences can only be observed in some specific regions within the image, dubbed as JND-critical regions. Identifying these regions can improve the development of image compression algorithms. Due to the fact that visual perception varies among individuals, determining the PJND values and JND-critical regions for a target population of consumers requires subjective assessment experiments involving a sufficiently large number of observers. In this paper, we propose a novel framework for conducting such experiments using crowdsourcing. By applying this framework, we created a novel PJND dataset, KonJND++, consisting of 300 source images, compressed versions thereof under JPEG or BPG compression, and an average of 43 ratings of PJND and 129 self-reported locations of JND-critical regions for each source image. Our experiments demonstrate the effectiveness and reliability of our proposed framework, which is easy to be adapted for collecting a large-scale dataset. The source code and dataset are available at https://github.com/angchen-dev/LocJND.

Modern wind turbines gather a wealth of data with Supervisory Control And Data Acquisition (SCADA) systems. We study the short-term mutual dependencies of a variety of observables by evaluating Pearson correlation matrices on a moving time window. Using clustering on these matrices, we identify multiple stable operational states, which characterize the non-stationarity of mutual dependencies at a single turbine. They represent different turbine operational settings. Moreover, we combine the clustering analysis with a construction of a stochastic process to study the switching dynamics of those states in more detail. Calculating the distances between correlation matrices we obtain a time series that describes the behavior of the complex system in a collective way. Assuming this time series to be governed by a Langevin equation, we estimate the deterministic (drift) and stochastic (diffusion) components of the dynamics to understand the underlying non-stationarity. After adapting our method to specific features of our data, we are able to study the dynamics of operational states and their transitions as well as to resolve hysteresis effects., Comment: 13 pages, 9 figures

We experimentally demonstrate optoacoustic cooling via stimulated Brillouin-Mandelstam scattering in a 50 cm-long tapered photonic crystal fiber. For a 7.38 GHz acoustic mode, a cooling rate of 219 K from room temperature has been achieved. As anti-Stokes and Stokes Brillouin processes naturally break the symmetry of phonon cooling and heating, resolved sideband schemes are not necessary. The experiments pave the way to explore the classical to quantum transition for macroscopic objects and could enable new quantum technologies in terms of storage and repeater schemes., Comment: 5 pages, 3 figures, L. B. M., P. W. and C. Z. contributed equally to this work

In the last few years, the dynamical characterization of the power output of a wind turbine by means of a Langevin equation has been well established. For this approach, temporally highly resolved measurements of wind speed and power output are used to obtain the drift and diffusion coefficients of the energy conversion process. These coefficients fully determine a Langevin stochastic differential equation with Gaussian white noise. The drift term specifies the deterministic behavior of the system whereas the diffusion term describes the stochastic behavior of the system. A precise estimation of these coefficients is essential to understand the dynamics of the power conversion process of the wind turbine. We show that the dynamics of the power output of a wind turbine have a hidden dependency on turbine's different operational states. Here, we use an approach based on clustering Pearson correlation matrices for different observables on a moving time window to identify different operational states. We have identified five operational states in total, for example the state of rated power. Those different operational states distinguish non-stationary behavior in the mutual dependencies and represent different turbine control settings. As a next step, we condition our Langevin analysis on these different states to reveal distinctly different behaviors of the power conversion process for each operational state. Moreover, in our new representation hysteresis effects which have typically appeared in the Langevin dynamics of wind turbines seem to be resolved. We assign these typically observed hysteresis effects clearly to the change of the wind energy system between our estimated different operational states. In this contribution, we discuss further consequences for the meaning of hysteric switching and detection of malbehaviors in wind turbines.

In this paper, we examine the computational complexity of enumeration in certain genome rearrangement models. We first show that the Pairwise Rearrangement problem in the Single Cut-and-Join model (Bergeron, Medvedev, & Stoye, J. Comput. Biol. 2010) is $\#\textsf{P}$-complete under polynomial-time Turing reductions. Next, we show that in the Single Cut or Join model (Feijao & Meidanis, IEEE ACM Trans. Comp. Biol. Bioinf. 2011), the problem of enumerating all medians ($\#$Median) is logspace-computable ($\textsf{FL}$), improving upon the previous polynomial-time ($\textsf{FP}$) bound of Mikl\'os & Smith (RECOMB 2015)., Comment: Full version of paper that appeared in COCOON 2023: https://doi.org/10.1007/978-3-031-49190-0_1

We present a singular value polyconvex conjugation. Employing this conjugation, we derive a necessary and sufficient criterion for polyconvexity of isotropic functions by means of the convexity of a function with respect to the signed singular values. Moreover, we present a new criterion for polyconvexity of isotropic functions by means of matrix invariants.