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We observe a n-sample, the distribution of which is assumed to belong, or at least to be close enough, to a given mixture model. We propose an estimator of this distribution that belongs to our model and possesses some robustness properties with respect to a possible misspecification of it. We establish a non-asymptotic deviation bound for the Hellinger distance between the target distribution and its estimator when the model consists of a mixture of densities that belong to VC-subgraph classes. Under suitable assumptions and when the mixture model is well-specified, we derive risk bounds for the parameters of the mixture. Finally, we design a statistical procedure that allows us to select from the data the number of components as well as suitable models for each of the densities that are involved in the mixture. These models are chosen among a collection of candidate ones and we show that our selection rule combined with our estimation strategy result in an estimator which satisfies an oracle-type inequality.

We present a meshfree generalized finite difference method for solving Poisson's equation with a diffusion coefficient that contains jump discontinuities up to several orders of magnitude. To discretize the diffusion operator, we formulate a strong form method that uses a smearing of the discontinuity; and a conservative formulation based on locally computed Voronoi cells. Additionally, we propose a novel conservative formulation for enforcing Neumann boundary conditions that is compatible with the conservative formulation of the diffusion operator. Finally, we introduce a way to switch from the strong form to the conservative formulation to obtain a locally conservative and positivity preserving scheme. The presented numerical methods are benchmarked against four test cases of varying complexity and jump magnitude on point clouds with nodes that are not aligned to the discontinuity. Our results show that the new hybrid method that switches between the two formulations produces better results than the classical generalized finite difference approach for high jumps in diffusivity.

Let K be a number field which is multiquadratic or quartic cyclic. We prove several results about the Kummer extensions of K, namely concerning the intersection between the Kummer extensions and the cyclotomic extensions of K. For G a finitely generated subgroup of K ×, we consider the cyclotomic-Kummer extensions K ( ζ n t , G n ) / K ( ζ n t ) K\left( {{\zeta _{nt}},\root n \of G } \right)/K\left( {{\zeta _{nt}}} \right) for all positive integers n and t, and we describe an explicit finite procedure to compute at once the degree of all these extensions.

A classical result of Bondal-Orlov states that a standard flip in birational geometry gives rise to a fully faithful functor between derived categories of coherent sheaves. We complete their embedding into a semiorthogonal decomposition by describing the complement. As an application, we can lift the "quadratic Fano correspondence" (due to Galkin-Shinder) in the Grothendieck ring of varieties between a smooth cubic hypersurface, its Fano variety of lines, and its Hilbert square, to a semiorthogonal decomposition. We also show that the Hilbert square of a cubic hypersurface of dimension at least 3 is again a Fano variety, so in particular the Fano variety of lines on a cubic hypersurface is a Fano visitor. The most interesting case is that of a cubic fourfold, where this exhibits the first higher-dimensional hyperk\"ahler variety as a Fano visitor., Comment: 34 pages, nearly identical to published version

In this paper, we study sufficient conditions to improve the lower bound on the algebraic immunity of a direct sum of Boolean functions. We exhibit three properties on the component functions such that satisfying one of them is sufficient to ensure that the algebraic immunity of their direct sum exceeds the maximum of their algebraic immunities. These properties can be checked while computing the algebraic immunity and they allow to determine better the security provided by functions central in different cryptographic constructions such as stream ciphers, pseudorandom generators, and weak pseudorandom functions. We provide examples for each property and determine the exact algebraic immunity of candidate constructions.

Evaluating the quality of primary studies is a key step in meta-analytic reviews to reduce the risk of bias and establish the validity of the meta-analytic inferences. However, the extant body of research offers little guidance on how to represent and incorporate primary study quality (PSQ) in meta-analyses, and some common procedures, such as creating sum scores from a set of quality indicators, often lack the backing from measurement models. Addressing these issues, we present a tutorial that guides meta-analysts in their analytic decisions and approaches to represent and incorporate PSQ. Specifically, we describe, review, and illustrate approaches to (a) select or create quality indicators or scores a priori or as part of the meta-analytic model; (b) examine the possible moderator effects of PSQ; and (c) test the sensitivity of moderator effects to PSQ. We illustrate these approaches with three examples and present a step-by-step tutorial with analytic code for researchers’ guidance. Overall, we argue for representing PSQ model-based if multiple quality indicators are available, the testing of moderator effects of PSQ on the effect sizes and their heterogeneity, and performing moderator sensitivity analyses. 1

Fractional powers of the Laplacian operator are important tools in the modeling and study of non-local phenomena. Several numerical challenges arise from the discretization of these operators due to their non-local nature. For example, a direct discretization via finite element methods can lead to dense and possibly large linear systems. One way to circumvent this density is by using a rational scheme combined with a finite element method. In this talk we describe a novel local a posteriori estimator for the finite element discretization error measured in the L2 norm that can be used to perform adaptive mesh refinement. This estimator is adapted from the strategy introduced by Bank and Weiser and can be used with any rational approx- imation scheme such as best uniform rational approximations or schemes based on the Dunford–Taylor formula. Especially, our estimator preserves the locality and robustness of the Bank–Weiser estimator and preserves the parallel nature of rational approximations. In addition, oour method can be combined with an estimator for the rational approximation error to obtain a more complete description of the discretization errors. Finally, we use an implementation in the FEniCSx finite element software to demonstrate the performances of our method on several numerical experiments including three–dimensional prob- lems.

In this paper we deal with Calabi-Yau structures associated with (differential graded versions of) deformed multiplicative preprojective algebras, of which we provide concrete algebraic descriptions. Along the way, we prove a general result that states the existence and uniqueness of negative cyclic lifts for non-degenerate relative Hochschild classes., 25 pages. Minor changes, final version, appeared in JNCG

Arithmetic billiards show a nice interplay between arithmetics and geometry. The billiard table is a rectangle with integer side lengths. A pointwise ball moves with constant speed along segments making a 45° angle with the sides and bounces on these. In the classical setting, the ball is shooted from a corner and lands in a corner. We allow the ball to start at any point with integer distances from the sides: either the ball lands in a corner or the trajectory is periodic. The length of the path and of certain segments in the path are precisely (up to the factor √2 or 2√2) the least common multiple and the greatest common divisor of the side lengths.

In this paper, we make a comprehensive study of two classes of Boolean functions whose interest originally comes from hybrid symmetric-FHE encryption (with stream ciphers like FiLIP), but which also present much interest for general stream ciphers. The functions in these two classes are cheap and easy to implement, and they allow the resistance to all classical attacks and to their guess and determine variants as well. We determine exactly all the main cryptographic parameters (algebraic degree, resiliency order, nonlinearity, algebraic immunity) for all functions in these two classes, and we give close bounds for the others (fast algebraic immunity, the dimension of the space of annihilators of minimal degree). This is the first time that this is done for all functions in large classes of cryptographic interest.

In the unoriented SL(3) foam theory, singular vertices are generic singularities of two-dimensional complexes. Singular vertices have neighbourhoods homeomorphic to cones over the one-skeleton of the tetrahedron, viewed as a trivalent graph on the two-sphere. In this paper we consider foams with singular vertices with neighbourhoods homeomorphic to cones over more general planar trivalent graphs. These graphs are subject to suitable conditions on their Kempe equivalence Tait coloring classes and include the dodecahedron graph. In this modification of the original homology theory it is straightforward to show that modules associated to the dodecahedron graph are free of rank 60, which is still an open problem for the original unoriented SL(3) foam theory., Comment: 25 pages, many figures

In this article, we develop a martingale approach to localized Bismut-type Hessian formulas for heat semigroups on Riemannian manifolds. Our approach extends the Hessian formulas established by Stroock (1996) and removes in particular the compact manifold restriction. To demonstrate the potential of these formulas, we give as application explicit quantitative local estimates for the Hessian of the heat semigroup, as well as for harmonic functions on regular domains in Riemannian manifolds., 18

Given a complex manifold endowed with a $\mathbb{C}^\times$-action and a DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action), we prove that if the $\mathbb{C}^\times$-action is free and proper, then the category of F-equivariant DQ-modules is equivalent to the category of modules over the sheaf of invariant sections of the DQ-algebra. As an application, we deduce the codimension three conjecture for formal microdifferential modules from the one for DQ-modules on a symplectic manifold., 34 pages. Comments are welcome, minor changes, several typos corrected

The classical parabolic induction functor is a fundamental tool on the representation theoretic side of the Langlands program. In this article, we study its derived version. It was shown by the second author that the derived category of smooth $G$-representations over $k$, $G$ a $p$-adic reductive group and $k$ a field of characteristic $p$, is equivalent to the derived category of a certain differential graded $k$-algebra $H_G^\bullet$, whose zeroth cohomology is a classical Hecke algebra. This equivalence predicts the existence of a derived parabolic induction functor on the dg Hecke algebra side, which we construct in this paper. This relies on the theory of six-functor formalisms for differential graded categories developed by O.\ Schn\"urer. We also discuss the adjoint functors of derived parabolic induction., Comment: comments welcome!

In the Euclidean case, it is well-known, by Malgrange and Ehrenpreis, that linear differential operators with constant coefficients are solvable. However, what happens, if we genuinely extend this situation and consider systems of linear invariant differential operators, is still solvable? In the case of $\mathbb{R}^n$ (for some positive integer $n$), the question has been proved mainly by Hörmander. We will show that this remains still true for Riemannian symmetric spaces of non-compact type $X=G/K$, in particular for hyperbolic planes. More precisely, we will present a possible strategy to solve this problem by using the Fourier transform and its Paley-Wiener(-Schwartz) theorem for (distributional) sections of vector bundles over $\mathbb{H}^2=SL(2, \mathbb{R})/SO(2)$. This work was part of my doctoral dissertation supervised by Martin Olbrich.

Transferability is the property of adversarial examples to be misclassified by other models than the surrogate model for which they were crafted. Previous research has shown that transferability is substantially increased when the training of the surrogate model has been early stopped. A common hypothesis to explain this is that the later training epochs are when models learn the non-robust features that adversarial attacks exploit. Hence, an early stopped model is more robust (hence, a better surrogate) than fully trained models. We demonstrate that the reasons why early stopping improves transferability lie in the side effects it has on the learning dynamics of the model. We first show that early stopping benefits transferability even on models learning from data with non-robust features. We then establish links between transferability and the exploration of the loss landscape in the parameter space, on which early stopping has an inherent effect. More precisely, we observe that transferability peaks when the learning rate decays, which is also the time at which the sharpness of the loss significantly drops. This leads us to propose RFN, a new approach for transferability that minimizes loss sharpness during training in order to maximize transferability. We show that by searching for large flat neighborhoods, RFN always improves over early stopping (by up to 47 points of transferability rate) and is competitive to (if not better than) strong state-of-the-art baselines.

Wavelet-type random series representations of the well-known Fractional Brownian Motion (FBM) and many other related stochastic processes and fields have started to be introduced since more than two decades. Such representations provide natural frameworks for approximating almost surely and uniformly rough sample paths at different scales and for study of various aspects of their complex erratic behavior. Hermite process of an arbitrary integer order $d$, which extends FBM, is a paradigmatic example of a stochastic process belonging to the $d$th Wiener chaos. It was introduced very long time ago, yet many of its properties are still unknown when $d\ge 3$. In a paper published in 2004, Pipiras raised the problem to know whether wavelet-type random series representations with a well-localized smooth scaling function, reminiscent to those for FBM due to Meyer, Sellan and Taqqu, can be obtained for a Hermite process of any order $d$. He solved it in this same paper in the particular case $d=2$ in which the Hermite process is called the Rosenblatt process. Yet, the problem remains unsolved in the general case $d\ge 3$. The main goal of our article is to solve it, not only for usual Hermite processes but also for generalizations of them. Another important goal of our article is to derive almost sure uniform estimates of the errors related with approximations of such processes by scaling functions parts of their wavelet-type random series representations.

Let X be a projective, equidimensional, singular scheme over an al- gebraically closed field. Then the existence of a geometric smoothing (i.e. a fam- ily of deformations of X over a smooth base curve whose generic fibre is smooth) implies the existence of a formal smoothing as defined by Tziolas. In this paper we address the reverse question giving sufficient conditions on X that guaran- tee the converse, i.e. formal smoothability implies geometric smoothability. This is useful in light of Tziolas’ results giving sufficient criteria for the existence of formal smoothings.

We study the pointwise regularity of the Multifractional Brownian Motion and in particular, we get the existence of slow points. It shows that a non self-similar process can still enjoy this property. We also consider various extensions of our results in the aim of requesting a weaker regularity assumption for the Hurst function without altering the regularity of the process.

In this work we develop new finite element discretisations of the shear-deformable Reissner--Mindlin plate problem based on the Hellinger-Reissner principle of symmetric stresses. Specifically, we use conforming Hu-Zhang elements to discretise the bending moments in the space of symmetric square integrable fields with a square integrable divergence $\boldsymbol{M} \in \mathcal{HZ} \subset H^{\mathrm{sym}}(\mathrm{Div})$. The latter results in highly accurate approximations of the bending moments $\boldsymbol{M}$ and in the rotation field being in the discontinuous Lebesgue space $\boldsymbol{\phi} \in [L]^2$, such that the Kirchhoff-Love constraint can be satisfied for $t \to 0$. In order to preserve optimal convergence rates across all variables for the case $t \to 0$, we present an extension of the formulation using Raviart-Thomas elements for the shear stress $\mathbf{q} \in \mathcal{RT} \subset H(\mathrm{div})$. We prove existence and uniqueness in the continuous setting and rely on exact complexes for inheritance of well-posedness in the discrete setting. This work introduces an efficient construction of the Hu-Zhang base functions on the reference element via the polytopal template methodology and Legendre polynomials, making it applicable to hp-FEM. The base functions on the reference element are then mapped to the physical element using novel polytopal transformations, which are suitable also for curved geometries. The robustness of the formulations and the construction of the Hu-Zhang element are tested for shear-locking, curved geometries and an L-shaped domain with a singularity in the bending moments $\boldsymbol{M}$. Further, we compare the performance of the novel formulations with the primal-, MITC- and recently introduced TDNNS methods., Comment: Additional implementation material in: https://github.com/Askys/NGSolve_HuZhang_Element

We define multifractional Hermite processes which generalize and extend both multifractional Brownian motion and Hermite processes. It is done by substituting the Hurst parameter in the definition of Hermite processes as a multiple Wiener-It\^o integral by a Hurst function. Then, we study the pointwise regularity of these processes, their local asymptotic self-similarity and some fractal dimensions of their graph. Our results show that the fundamental properties of multifractional Hermite processes are, as desired, governed by the Hurst function. Complements are given in the second order Wiener chaos, using facts from Malliavin calculus.

We study the minor relation for algebra homomorphims in finitely generated quasivarieties that admit a logarithmic natural duality. We characterize the minor homomorphism posets of finite algebras in terms of disjoint unions of dual partition lattices and investigate reconstruction problems for homomorphisms.

Let $K$ be a number field and $G$ a finitely generated torsion-free subgroup of $K^\times$. Given a prime $\mathfrak p$ of $K$ we denote by ${\rm ind}_{\mathfrak p}(G)$ the index of the subgroup $(G\bmod\mathfrak p)$ of the multiplicative group of the residue field at $\mathfrak p$. Under the Generalized Riemann Hypothesis we determine the natural density of primes of $K$ for which this index is in a prescribed set $S$ and has prescribed Frobenius in a finite Galois extension $F$ of $K$. We study in detail the natural density in case $S$ is an arithmetic progression, in particular its positivity., Comment: 15 pages, comments are welcome

Das Projekt ist eine Zusammenarbeit mit Studierenden und Museen. Die Idee ist einfach: Mit der Zeit können Lehramtsstudierende Material hoher Qualität produzieren, das allen frei zur Verfügung steht.

The chain gravity properad introduced earlier by the author acts on the cyclic Hochschild of any cyclic A∞ algebra equipped with a scalar product of degree −d. In particular, it acts on the cyclic Hochschild complex of any Poincare duality algebra of degree d, and that action factors through a quotient dg properad ST3−d of ribbon graphs which is in focus of this paper. We show that its cohomology properad H∙(ST3−d) is highly non-trivial and that it acts canonically on the reduced equivariant homology H¯S1∙(LM) of the loop space LM of any simply connected d-dimensional closed manifold M. By its very construction, the string topology properad H∙(ST3−d) comes equipped with a morphism from the gravity properad which is fully determined by the compactly supported cohomology of the moduli spaces Mg,n of stable algebraic curves of genus g with marked points. This result gives rise to new universal operations in string topology as well as reproduces in a unified way several known constructions: we show that (i) H∙(ST3−d) is also a properad under the properad of involutive Lie bialgebras in degree 3−d whose induced action on H¯S1∙(LM) agrees precisely with the famous purely geometric construction of M. Chas and D. Sullivan, (ii) H∙(ST3−d) is a properad under the properad of homotopy involutive Lie bialgebras in degree 2−d; (iii) E. Getzler's gravity operad injects into H∙(ST3−d) implying a purely algebraic counterpart of the geometric construction of C. Westerland establishing an action of the gravity operad on H¯S1∙(LM).

We observe $n$ independent pairs of random variables $(W_{i}, Y_{i})$ for which the conditional distribution of $Y_{i}$ given $W_{i}=w_{i}$ belongs to a one-parameter exponential family with parameter ${\mathbf{\gamma}}^{*}(w_{i})\in{\mathbb{R}}$ and our aim is to estimate the regression function ${\mathbf{\gamma}}^{*}$. Our estimation strategy is as follows. We start with an arbitrary collection of piecewise constant candidate estimators based on our observations and by means of the same observations, we select an estimator among the collection. Our approach is agnostic to the dependencies of the candidate estimators with respect to the data and can therefore be unknown. From this point of view, our procedure contrasts with other alternative selection methods based on data splitting, cross validation, hold-out etc. To illustrate its theoretical performance, we establish a non-asymptotic risk bound for the selected estimator. We then explain how to apply our procedure to the changepoint detection problem in exponential families. The practical performance of the proposed algorithm is illustrated by a comparative simulation study under different scenarios and on two real datasets from the copy numbers of DNA and British coal disasters records.

In this paper, we study $L^p$-boundedness ($1, Comment: 24 pages

Since Hooley's seminal 1967 resolution of Artin's primitive root conjecture under the Generalized Riemann Hypothesis, numerous variations of the conjecture have been considered. We present a framework generalizing and unifying many previously considered variants, and prove results in this full generality (under GRH)., Comment: 16 pages. Improved exposition, fixed typos. To appear in Research in Number Theory

We distinguish between faint, weak, strong and strict localizations of categories at morphism families and show that this framework captures the different types of derived functors that are considered in the literature. More precisely, we show that Kan and faint derived functors coincide when we use the classical Kan homotopy category, and when we use the Quillen homotopy category, Kan and strong derived functors coincide. Our comparison results are based on the fact that the Kan homotopy category is a weak localization and that the Quillen homotopy category is a strict localization.

We establish new explicit bounds on the Gaussian approximation of Poisson functionals based on novel estimates of moments of Skorohod integrals. Combining these with the Malliavin-Stein method, we derive bounds in the Wasserstein and Kolmogorov distances whose application requires minimal moment assumptions on add-one cost operators $\unicode{x2014}$ thereby extending the results from (Last, Peccati and Schulte, 2016). Our applications include a CLT for the Online Nearest Neighbour graph, whose validity was conjectured in (Wade, 2009; Penrose and Wade, 2009). We also apply our techniques to derive quantitative CLTs for edge functionals of the Gilbert graph, of the $k$-Nearest Neighbour graph and of the Radial Spanning Tree, both in cases where qualitative CLTs are known and unknown.

For a complete connected Riemannian manifold M let V∊ C^2(M) be such that µ(dx)=exp(-V(x))vol(dx) is a probability measure on M. Taking µ as reference measure, we derive inequalities for probability measures on M linking relative entropy, Fisher information, Stein discrepancy and Wasserstein distance. These inequalities strengthen in particular the famous log-Sobolev and transportation-cost inequality and extend the so-called Entropy/Stein-discrepancy/Information (HSI) inequality established by Ledoux, Nourdin and Peccati (2015) for the standard Gaussian measure on Euclidean space to the setting of Riemannian manifolds.

The COVID-19 pandemic has left its marks in the sports world, forcing the full-stop of all sports-related activities in the first half of 2020. Football leagues were suddenly stopped and each country was hesitating between a relaunch of the competition and a premature ending. Some opted for the latter option, and took as the final standing of the season the ranking from the moment the competition got interrupted. This decision has been perceived as unfair, especially by those teams who had remaining matches against easier opponents. In this paper, we introduce a tool to calculate in a fairer way the final standings of domestic leagues that have to stop prematurely: our Probabilistic Final Standing Calculator (PFSC). It is based on a stochastic model taking into account the results of the matches played and simulating the remaining matches, yielding the probabilities for the various possible final rankings. We have compared our PFSC with state-of-the-art prediction models, using previous seasons which we pretend to stop at different points in time. We illustrate our PFSC by showing how a probabilistic ranking of the French Ligue 1 in the stopped 2019-2020 season could have led to alternative, potentially fairer, decisions on the final standing., Comment: 4 tables, 2 figures

We say that two abelian varieties $A$ and $A'$ defined over a field $F$ are polyquadratic twists if they are isogenous over a Galois extension of $F$ whose Galois group has exponent dividing $2$. Let $A$ and $A'$ be abelian varieties defined over a number field $K$ of dimension $g\geq 1$. In this article we prove that, if $g\leq 2$, then $A$ and $A'$ are polyquadratic twists if and only if for almost all primes $\p$ of $K$ their reductions modulo $\p$ are polyquadratic twists. We exhibit a counterexample to this local-global principle for $g=3$. This work builds on a geometric analogue by Khare and Larsen, and on a similar criterion for quadratic twists established by Fit\'e, relying itself on the works by Rajan and Ramakrishnan., Comment: 16 pages

Dirichlet’s theorem on primes in arithmetic progressions states that for any positive integer q and any coprime integer a, there are infinitely many primes in the arithmetic progression a + nq (n ∈ N). Unfortunately, the theorem does not predict the gap between those primes. Several renowned open questions such as the size of Linnik’s constant try to say more about the distribution of such primes. This manuscript postulates a weak, but explicit, generalization of Linnik’s theorem, namely that each geometric interval [q^t, q^(t+1)] contains a prime from each coprime congruence class modulo q ∈ N≥2. Subsequently, it proves the conjecture theoretically for all sufficiently large t, as well as computationally for all sufficiently small q. Finally, the impact of this conjecture on a result of Pomerance related to Carmichael’s conjecture is outlined.

This dissertation is a collection of four research articles devoted tothe study of Kummer theory for commutative algebraic groups. In numbertheory, Kummer theory refers to the study of field extensions generatedby n-th roots of some base field. Its generalization to commutativealgebraic groups involves fields generated by the division points of afixed algebraic group, such as an elliptic curve or a higher dimensionalabelian variety. Of particular interest in this dissertation is the degreeof such field extensions. In the first two chapter, classical results forelliptic curves are improved by providing explicitly computable bounds anduniform and explicit bounds over the field of rational numbers. In thelast two chapters a general framework for the study of similar problemsis developed.

We study vertex-like operators built from the Brownian loop soup in the limit as the loop soup intensity tends to infinity. More precisely, following Camia, Gandolfi and Kleban (Nuclear Physics B 902, 2016), we take a Brownian loop soup in a planar domain and assign a random sign to each loop. We then consider random fields defined by taking, at every point of the domain, the exponential of a purely imaginary constant times the sum of the signs associated to the loops that wind around that point. As smaller loops are included in the count, that sum diverges logarithmically with the diameter of the loops, but we show that a suitable renormalization procedure allows to define the fields in an appropriate Sobolev space. Subsequently, we let the intensity of the loop soup tend to infinity and prove that these vertex-like fields tend to a conformally covariant random field which can be expressed as an explicit functional of the imaginary Gaussian multiplicative chaos with covariance kernel given by the Brownian loop measure. Besides using properties of the Brownian loop soup and the Brownian loop measure, a main tool in our analysis is an explicit Wiener-It\^{o} chaos expansion of linear functionals of vertex-like fields., Comment: 67 pages, no figures. V2: We have expanded the introduction, improved Theorem 1, added a new appendix (A.6), corrected several typos, made other minor changes

We use recent tools from stochastic analysis (such as Stein's method and Malliavin calculus) to study the asymptotic behaviour of some functionals of a Gaussien Field.

We present prevalent results concerning generalized versions of the $T_p^\alpha$ spaces, initially introduced by Calderón and Zygmund. We notably show that the logarithmic correction appearing in the quasi-characterization of such spaces is mandatory for almost every function; it is in particular true for the Hölder spaces, for which the existence of the correction was showed necessary for a specific function. We also show that almost every function from $T_p^α (x0 )$ has α as generalized Hölder exponent at $x_0$.

In the Euclidean case, it is well-known, by Malgrange and Ehrenpreis, that linear differential operators with constant coefficients are solvable. However, what happens, if we genuinely extend this situation and consider systems of linear invariant differential operators, is still solvable? In case of Rn (for some positive integer n), the question has been proved mainly by Hörmander. We will show that this remains still true for Riemannian symmetric spaces of non-compact type X = G/K. More precisely, we will present a possible strategy to solve this problem by using the Fourier trans- form and its Paley-Wiener(-Schwartz) theorem for (distributional) sections of vector bundles over X. We will get complete solvability for the hyperbolic plane H2 = SL(2, R)/SO(2) and beyond. This work was part of my doctoral dissertation supervised by Martin Olbrich.