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International audience; In this work, we propose an algebraic method to study linear differential time-varying delay (DTVD) systems. Our goal is to give an effective construction of the ring of DTVD operators as an Ore extension, thanks to the concept of skew polynomial rings developed by Ore in the 30s. Some algebraic properties of the DTVD operators ring are analyzed, such as its Noetherianity, its homological and Krull dimensions, and the existence of Gröbner bases, all given in terms of the time-varying delay function. The algebraic analysis framework for linear systems theory allows us to study linear DTVD systems and essential properties such as the existence of autonomous elements, controllability, parametrizability, flatness, etc., through methods coming from module theory, homological algebra, and constructive algebra.

Inspired by recent work of Peter O'Sullivan (arXiv:2012.15703), we give a condition under which a faithful monoidal functor between abelian $\otimes$-categories is exact.

We show the coexistence of chaotic behaviors (positive metric entropy) and elliptic behaviors (intregrable KAM island) among analytic, symplectic diffeomorphism of any closed surface. In particilar this solves a problem by F. Przytycki (1982). Theorem A (Main result). For every analytic and closed, oriented surface (S, Leb), there exists a symplectic analytic map f ∈ Diff^ω_Leb (S) of any isotopy class, such that

We find the normal forms of hyperbolic logarithmic transseries with respect to parabolic logarithmic normalizing changes of variables. We provide a necessary and sufficient condition on such transseries for the normal form to be linear. The normalizing transformations are obtained via fixed point theorems, and are given algorithmically, as limits of Picard sequences in appropriate topologies., Comment: 33 pages, 0 figures

We consider the problem of computing the topology and describing the geometry of a parametric curve in $\mathbb{R}^n$. We present an algorithm, PTOPO, that constructs an abstract graph that is isotopic to the curve in the embedding space. Our method exploits the benefits of the parametric representation and does not resort to implicitization. Most importantly, we perform all computations in the parameter space and not in the implicit space. When the parametrization involves polynomials of degree at most $d$ and maximum bitsize of coefficients $\tau$, then the worst case bit complexity of PTOPO is $ \tilde{\mathcal{O}}_B(nd^6+nd^5\tau+d^4(n^2+n\tau)+d^3(n^2\tau+ n^3)+n^3d^2\tau)$. This bound matches the current record bound $\tilde{\mathcal{O}}_B(d^6+d^5\tau)$ for the problem of computing the topology of a plane algebraic curve given in implicit form. For plane and space curves, if $N = \max\{d, \tau \}$, the complexity of PTOPO becomes $\tilde{\mathcal{O}}_B(N^6)$, which improves the state-of-the-art result, due to Alc\'azar and D\'iaz-Toca [CAGD'10], by a factor of $N^{10}$. In the same time complexity, we obtain a graph whose straight-line embedding is isotopic to the curve. However, visualizing the curve on top of the abstract graph construction, increases the bound to $\tilde{\mathcal{O}}_B(N^7)$. For curves of general dimension, we can also distinguish between ordinary and non-ordinary real singularities and determine their multiplicities in the same expected complexity of PTOPO by employing the algorithm of Blasco and P\'erez-D\'iaz [CAGD'19]. We have implemented PTOPO in Maple for the case of plane and space curves. Our experiments illustrate its practical nature.

We present the quantitative method of the recent work arXiv:2209.09340 in a simple setting, together with a compactness argument that was not included in arXiv:2209.09340 and has interest per se. We are concerned with the exponential stabilization (spectral gap) for linear kinetic equations with degenerate thermalization, i.e. when the collision operator vanishes on parts of the spatial domain. The method in arXiv:2209.09340 covers both scattering and Fokker-Planck type operators, and deals with external potential and boundary conditions, but in these notes we present only its core argument and restrict ourselves to the kinetic Fokker-Planck in the periodic torus with unit velocities and a thermalization degeneracy. This equation is not covered by the previous results of Bernard and Salvarani (2013), Han-Kwan and L\'eautaud (2015), Evans and Moyano (arXiv:1907.12836)., Comment: 10 pages, 1 figure

International audience; We present algorithmic, complexity, and implementation results on the problem of sampling points from a spectrahedron, that is, the feasible region of a semidefinite program.Our main tool is geometric random walks. We analyze the arithmetic and bit complexity of certain primitive geometric operations that are based on the algebraic properties of spectrahedra and the polynomial eigenvalue problem. This study leads to the implementation of a broad collection of random walks for sampling from spectrahedra that experimentally show faster mixing times than methods currently employed either in theoretical studies or in applications, including the popular family of Hit-and-Run walks. The different random walks offer a variety of advantages, thus allowing us to efficiently sample from general probability distributions, for example the family of log-concave distributions which arise in numerous applications. We focus on two major applications of independent interest: (i) approximate the volume of a spectrahedron, and (ii) compute the expectation of functions coming from robust optimal control.We exploit efficient linear algebra algorithms and implementations to address the aforementioned computations in very high dimension. In particular, we provide a C++ open source implementation of our methods that scales efficiently, for the first time, up to dimension 200. We illustrate its efficiency on various data sets.

We study a global invariant for path structures. The invariant is obtained as a secondary invariant from a Cartan connection on a canonical bundle associated to a path structure. It is computed in examples which are defined in terms of reductions of the path structure. In particular we give a formula for this global invariant for second order differential equations defined on a torus $T^2$.

In this article, we derive necessary and sufficient conditions for the existence of solutions to state-constrained continuity inclusions in Wasserstein spaces whose right-hand sides may be discontinuous in time. These latter are based on fine investigations of the infinitesimal behaviour of the underlying reachable sets, through which we show that up to a negligible set of times, every admissible velocity of the inclusion can be approximately realised as the metric derivative of a solution of the dynamics, and vice versa. Building on these results, we are able to establish necessary and sufficient geometric conditions for the viability and invariance of stationary and time-dependent constraints, which involve a suitable notion of contingent cones in Wasserstein spaces, presented in ascending order of generality. We then close the article by exhibiting two prototypical examples of constraints sets appearing in applications for which one can compute relevant subfamilies of contingent directions., Comment: 43 pages

In this paper we consider a punctured Riemann surface endowed with a Hermitian metric that equals the Poincaré metric near the punctures, and a holomorphic line bundle that polarizes the metric. We introduce a new method to compare the Bergman kernels of high tensor powers of the line bundle and of the Poincaré model near the singularity and show that their quotient tends to one uniformly on a neighborhood of the singularity up to arbitrary negative powers of the tensor power.

International audience; The formal integrability of systems of partial differential equations plays a fundamental role in different analysis and synthesis problems for both linear and nonlinear differential control systems. Following Spencer's theory, to test the formal integrability of a system of partial differential equations, we must study when the symbol of the system, namely, the top-order part of the linearization of the system, is 2-acyclic or involutive, i.e., when certain Spencer cohomology groups vanish. Combining the fact that Spencer cohomology is dual to Koszul homology and symbolic computation methods, we show how to effectively compute the homology modules defined by the Koszul complex of a finitely presented module over a commutative polynomial ring. These results are implemented using the OreMorphisms package. We then use these results to effectively characterize 2-acyclicity and involutivity of the symbol of a linear system of partial differential equations. Finally, we show explicit computations on two standard examples.

International audience; In this paper, the algebraic analysis approach to linear state-space systems is further developed using rings of integro-differential operators. The module structure of linear state-space systems is investigated over these rings. The module associated with a linear state-space system is shown to be the direct sum of the stably free module defined by the linear system without inputs and the free module defined by the inputs of the system.

In this article, we investigate some of the fine properties of the value function associated to an optimal control problem in the Wasserstein space of probability measures. Building on new interpolation and linearisation formulas for non-local flows, we prove semiconcavity estimates for the value function, and establish several variants of the so-called sensitivity relations which provide connections between its superdifferential and the adjoint curves stemming from the maximum principle. We subsequently make use of these results to study the propagation of regularity for the value function along optimal trajectories, as well as to investigate sufficient optimality conditions and optimal feedbacks for mean-field optimal control problems., Comment: 55 pages

We prove that the complex cobordism class of any hyper-Kähler manifold of dimension 2n is a unique combination with rational coefficients of classes of products of punctual Hilbert schemes of K3 surfaces. We also prove a similar result using the generalized Kummer varieties instead of punctual Hilbert schemes. As a key step, we establish a closed formula for the top Chern character of their tangent bundles.

For $$s \in (0,1)$$ s ∈ ( 0 , 1 ) , $$N > 2s$$ N > 2 s , and a bounded open set $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N with $$C^2$$ C 2 boundary, we study the fractional Brezis–Nirenberg type minimization problem of finding $$\begin{aligned} S(a):= \inf \frac{\int _{{\mathbb {R}}^N} |(-\Delta )^{s/2} u|^2 + \int _\Omega a u^2}{\left( \int _\Omega u^\frac{2N}{N-2s} \right) ^\frac{N-2s}{N}}, \end{aligned}$$ S ( a ) : = inf ∫ R N | ( - Δ ) s / 2 u | 2 + ∫ Ω a u 2 ∫ Ω u 2 N N - 2 s N - 2 s N , where the infimum is taken over all functions $$u \in H^s({\mathbb {R}}^N)$$ u ∈ H s ( R N ) that vanish outside $$\Omega $$ Ω . The function a is assumed to be critical in the sense of Hebey and Vaugon. For low dimensions $$N \in (2\,s, 4\,s)$$ N ∈ ( 2 s , 4 s ) , we prove that the Robin function $$\phi _a$$ ϕ a satisfies $$\inf _{x \in \Omega } \phi _a(x) = 0$$ inf x ∈ Ω ϕ a ( x ) = 0 , which extends a result obtained by Druet for $$s = 1$$ s = 1 . In dimensions $$N \in (8s/3, 4s)$$ N ∈ ( 8 s / 3 , 4 s ) , we then study the asymptotics of the fractional Brezis–Nirenberg energy $$S(a + \varepsilon V)$$ S ( a + ε V ) for some $$V \in L^\infty (\Omega )$$ V ∈ L ∞ ( Ω ) as $$\varepsilon \rightarrow 0+$$ ε → 0 + . We give a precise description of the blow-up profile of (almost) minimizing sequences and characterize the concentration speed and the location of concentration points.

Finding roots of univariate polynomials is one of the fundamental tasks of numerics, and there is still a wide gap between root finders that are well understood in theory and those that perform well in practice. We investigate the root-finding method of Weierstrass, also known as the Durand–Kerner-method: this is a root finder that tries to approximate all roots of a given polynomial in parallel. This method has been introduced 130 years ago and has since then a good reputation for finding all roots in practice except in obvious cases of symmetry. Nonetheless, very little is known about its global dynamics and convergence properties. We show that the Weierstrass method, like the well-known Newton method, is not generally convergent: there are open sets of polynomials p p of every degree d ≥ 3 d \ge 3 such that the dynamics of the Weierstrass method applied to p p exhibits attracting periodic orbits. Specifically, all polynomials sufficiently close to Z 3 + Z + 175 Z^3 + Z + 175 have attracting cycles of period 4 4 . Here, period 4 4 is minimal: we show that for cubic polynomials, there are no periodic orbits of length 2 2 or 3 3 that attract open sets of starting points. We also establish another convergence problem for the Weierstrass method: for almost every polynomial of degree d ≥ 3 d\ge 3 there are orbits that are defined for all iterates but converge to ∞ \infty ; this is a problem that does not occur for Newton’s method. Our results are obtained by first interpreting the original problem coming from numerical mathematics in terms of higher-dimensional complex dynamics, then phrasing the question in algebraic terms in such a way that we could finally answer it by applying methods from computer algebra. The main innovation here is the translation into an algebraic question, which is amenable to (exact) computational methods close to the limits of current computer algebra systems.

Belinskaya's theorem states that given an ergodic measure-preserving transformation, any other transformation with the same orbits and an $\mathrm{L}^1$ cocycle must be flip-conjugate to it. Our main result shows that this theorem is optimal: for all $p, Some more fixes in the appendix; added a reference to M. Joseph's thesis in Remark 3.4. Final final version matching the online first version in Fundamenta Mathematicae

In this article, we extend the foundations of the theory of differential inclusions in the space of probability measures with compact support, laid down recently in one of our previous work, to the setting of general Wasserstein spaces. Anchoring our analysis on novel estimates for solutions of continuity equations, we propose a new existence result ``\`a la Peano'' for this class of dynamics, under mere Carath\'eodory regularity assumptions. The latter is based on a set-valued generalisation of the semi-discrete Euler scheme proposed by Filippov to study ordinary differential equations with measurable right-hand sides. We also bring substantial improvements to the earlier versions of the Filippov theorem, compactness and relaxation properties of the solution sets of continuity inclusions which are derived in the more restrictive Cauchy-Lipschitz setting.

International audience; A recent work of Sipasseuth, Plantard and Susilo proposed to accelerate lattice-based signature verifications and compress public key storage at the cost of a precomputation on a public key. This first approach, which focused on a restricted type of key, did not include most NIST candidates or most lattice representations in general. In this work, we first present a way to improve even further both their verification speed and their public key compression capability by using a generator of numbers that better suit the method needs. We then also generalize their framework to apply to q-ary lattice schemes as well as classical lattices using Hermite Normal Form, improving their security and applicable scope, thus exhibiting potential trade-offs to accelerate lattice-based signature verification in general and compression of the public key on the verifier side for unstructured lattices.

We study several explicit finite index subgroups in the known complex hyperbolic lattice triangle groups, and show some of them are neat, some of them have positive first Betti number, some of them have a homomorphisms onto a non-Abelian free group. For some lattice triangle groups, we determine the minimal index of a neat subgroup. Finally, we answer a question raised by Stover and describe an infinite tower of neat ball quotients all with a single cusp., Minor typos corrected, including the Magma command to produce a 1-cusped neat ball quotient

28 pages; International audience; This paper shows how to construct coherent presentations (presentations by generators, relations and relations among relations) of monoids admitting a right-noetherian Garside family. Thereby, it resolves the question of finding a unifying generalisation of the following two distinct extensions of construction of coherent presentations for spherical Artin-Tits monoids: to general Artin-Tits monoids, and to Garside monoids. The result is applied to some monoids which are neither Artin-Tits nor Garside.

We consider degenerate Monge-Amp\`ere equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Amp\`ere operators. We then use the Perron method to solve Monge-Amp\`ere equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delano\"e, Caffarelli-Viaclovsky and Hultgren-\"Onnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow conjecture.

In numerical linear algebra, a well-established practice is to choose a norm that exploits the structure of the problem at hand in order to optimize accuracy or computational complexity. In numerical polynomial algebra, a single norm (attributed to Weyl) dominates the literature. This article initiates the use of $L_p$ norms for numerical algebraic geometry, with an emphasis on $L_{\infty}$. This classical idea yields strong improvements in the analysis of the number of steps performed by numerous iterative algorithms. In particular, we exhibit three algorithms where, despite the complexity of computing $L_{\infty}$-norm, the use of $L_p$-norms substantially reduces computational complexity: a subdivision-based algorithm in real algebraic geometry for computing the homology of semialgebraic sets, a well-known meshing algorithm in computational geometry, and the computation of zeros of systems of complex quadratic polynomials (a particular case of Smale's 17th problem)., 54 pages

Let X denote a purely d-dimensional reduced complex analytic space. If it has singularities, it has no tangent bundle, which makes many classical and fundamental constructions impossible directly. However, there is a unique proper map ν X : N X Ñ X which has the property that it is an isomorphism over the non-singular part X 0 of X and the tangent bundle T X 0 lifted to N X by this isomorphism extends uniquely to a vector bundle on N X. For x P X, the set-theoretical fiber |ν ´1 X pxq| is the set of limit directions of tangent spaces to X 0 at points approaching x. The space N X is reduced and equidimensional, but in general singular. If X is a closed analytic subspace of an open set U of C N , the space N X is a closed analytic subspace of X ˆGpd, Nq, where Gpd, Nq denotes the Grassmannian of d-dimensional vector subspaces of C N. The rich geometry of the Grassmannian makes it complicated to study the geometry of the map ν X using intersection theory. There is an analogous construction where tangent spaces are replaced by tangent hyperplanes, and the map ν X is replaced by the conormal map κ X : CpXq Ñ X, where CpXq denotes the conormal space, which is a subspace of X ˆP N ´1, where PN´1 is the space of hyperplanes of P N , the dual projective space, so that the intersection theory is simpler. This paper is devoted to these two constructions, their applications to stratification theory in the sense of Whitney and to a general Plücker type formula for projective varieties.

Given a Baumslag-Solitar group, we study its space of subgroups from a topological and dynamical perspective. We first determine its perfect kernel (the largest closed subset without isolated points). We then bring to light a natural partition of the space of subgroups into one closed subset and countably many open subsets that are invariant under the action by conjugation. One of our main results is that the restriction of the action to each piece is topologically transitive. This partition is described by an arithmetically defined function, that we call the phenotype, with values in the positive integers or infinity. We eventually study the closure of each open piece and also the closure of their union. We moreover identify in each phenotype a (the) maximal compact invariant subspace., Comment: 60 pages, companion webpage available at https://doi.org/10.5281/zenodo.7225585 . Comments welcome!

We propose an algorithm for quickly evaluating polynomials. It pre-conditions a complex polynomial $P$ of degree $d$ in time $O(d\log d)$, with a low multiplicative constant independent of the precision. Subsequent evaluations of $P$ computed with a fixed precision of $p$ bits are performed in average arithmetic complexity $O\big(\sqrt{d(p+\log d)}\big)$ and memory $O(dp)$. The average complexity is computed with respect to points $z \in \mathbb{C}$, weighted by the spherical area of $\overline{\mathbb{C}}$. The worst case does not exceed the complexity of H{\"o}rner's scheme. In particular, our algorithm performs asymptotically as $O(\sqrt{d\log d})$ per evaluation. For many classes of polynomials, in particular those with random coefficients in a bounded region of $\mathbb{C}$, or for sparse polynomials, our algorithm performs much better than this upper bound, without any modification or parameterization.The article contains a detailed analysis of the complexity and a full error analysis, which guarantees that the algorithm performs as well as H\''orner's scheme, only faster. Our algorithm is implemented in a companion library, written in standard C and released as an open-source project [MV22].Our claims regarding complexity and accuracy are confirmed in practice by a set of comprehensive benchmarks.

National audience; Utilisation des assistants de preuves pour l'enseignement en L1 Utilisation des assistants de preuves pour l'enseignement en L1 Retours d'expériences Nous rendons compte de cinq expériences récentes de l'enseignement de la démonstration utilisant les assistants de preuve Coq, D∃∀DUCTION, Edukera et Lean.

This article contains a complete proof of Gabrielov's rank Theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung Theorem. We include, furthermore, new extensions of the rank Theorem (concerning the Zariski main Theorem and elimination theory) to commutative algebra.

Given a K\"ahler fiber space $p:X\to Y$ whose generic fiber is of general type, we prove that the fiberwise singular K\"ahler-Einstein metric induces a semipositively curved metric on the relative canonical bundle $K_{X/Y}$ of $p$. We also propose a conjectural generalization of this result for relative twisted K\"ahler-Einstein metrics. Then we show that our conjecture holds true if the Lelong numbers of the twisting current are zero. Finally, we explain the relevance of our conjecture for the study of fiber-wise Song-Tian metrics (which represent the analogue of KE metrics for fiber spaces whose generic fiber has positive but not necessarily maximal Kodaira dimension)., Comment: v2: Title changed as the initial text is now split into two separate articles (authored by the same persons). The main results we obtain here are less general than their respective counterparts in the previous version, due to a gap in our arguments

We adapt a proof of Lascar in order to show the simplicity of the group of automorphisms fixing pointwise all non-generic elements for a class of uncountable models of suitable theories of fields with operators.

International audience; In this paper, we initiate a new algebraic analysis approach to linear differential systems based on rings of integro-differential operators. Within this algebraic analysis approach, we first interpret the method of variations of constants as an operator identity. Using this result, we show that the module associated with a state-space representation of a linear system is the same as the one associated with its standard convolution representation. This finitely presented module over the ring of integro-differential operators is proved to be stably free. Finally, we show how the reachability property can be expressed within this algebraic analysis approach.

In this article, we prove a general viability theorem for continuity inclusions in Wasserstein spaces, and provide an application thereof to the existence of exponentially stable trajectories obtained via the second method of Lyapunov.

International audience; Residue Number Systems (RNS) are proven to be effective in speeding up computations involving additions and products. For these representations, there exists efficient modular reduction algorithms that can be used in the context of arithmetic over finite fields or modulo large numbers, especially when used in the context of cryptographic engineering. Their independence allows random draws of bases, which also makes it possible to protect against side-channel attacks, or even to detect them using redundancy. These systems are easily scalable, however the existence of large bases for some specific uses remains a difficult question. In this paper, we present four techniques to extract RNS bases from specific sets of integers, giving better performance and flexibility to previous works in the litterature. While our techniques do not allow to solve efficiently every possible case, we provide techniques to provably and efficiently find the largest possible available RNS bases in several cases, improving the state-of-the-art on various works of the recent literature.

We solve the equivariant generalized Nash problem for any non-rational normal variety with torus action of complexity one. Namely, we give an explicit combinatorial description of the Nash order on the set of equivariant divisorial valuations on any such variety. Using this description, we positively solve the classical Nash problem in this setting, showing that every essential valuation is a Nash valuation. We also describe terminal valuations and use our results to answer negatively a question of de Fernex and Docampo by constructing examples of Nash valuations which are neither minimal nor terminal, thus illustrating a striking new feature of the class of singularities under consideration., Comment: proofs in Section 6 completed

A positive correlation inequality is established for circular-invariant plurisubharmonic functions, with respect to complex Gaussian measures. The main ingredients of the proofs are the Ornstein-Uhlenbeck semigroup, and another natural semigroup associated to the Gaussian $\partial$-Laplacian.

In this thesis we investigate the sequence of Mahler measures of a family of bivariate regular exact polynomials, called Pd := P0≤i+j≤d xiyj , unbounded in both degree and the genus of the algebraic curve. We obtain a closed formula for the Mahler measure of Pd in termsof special values of the Bloch–Wigner dilogarithm. We approximate m(Pd), for 1 ≤ d ≤ 1000,with arbitrary precision using SageMath. Using 3 different methods we prove that the limitof the sequence of the Mahler measure of this family converges to 92π2 ζ(3). Moreover, we compute the asymptotic expansion of the Mahler measure of Pd which implies that the rate of the convergence is O(log dd2 ). We also prove a generalization of the theorem of the Boyd-Lawton which asserts that the multivariate Mahler measures can be approximated using the lower dimensional Mahler measures. Finally, we prove that the Mahler measure of Pd, for arbitrary d can be written as a linear combination of L-functions associated with an odd primitive Dirichlet character. In addition, we compute explicitly the representation of the Mahler measure of Pd in terms of L-functions, for 1 ≤ d ≤ 6., Dans cette thèse, nous étudions la suite de mesures de Mahler d’une famille de polynômes à deux variables exacts et réguliers, que nous notons Pd := P0≤i+j≤d xiyj . Elle n’est bornée ni en volume, ni en genre de la courbe algébrique sous-jacente. Nous obtenons une expression pour la mesure de Mahler de Pd comme somme finie de valeurs spéciales du dilogarithme de Bloch-Wigner. Nous utilisons SageMath pour approximer m(Pd) pour 1 ≤ d ≤ 1000. En recourant à trois méthodes différentes, nous prouvons que la limite de la suite de mesures de Mahler de cette famille converge vers 92π2 ζ(3). De plus, nous calculons le développement asymptotique de la mesure de Mahler de Pd et prouvons que sa vitesse de convergence est de O(log dd2 ). Nous démontrons également une généralisation du théorème de Boyd-Lawton, affirmant que les mesures de Mahler multivariées peuvent être approximéess en utilisant les mesures de Mahler de dimension inférieure. Enfin, nous prouvons que la mesure de Mahler de Pd pour d arbitraire peut être écrite comme une combinaison linéaire de fonctions L associées à un caractère de Dirichlet primitif impair. Nous calculons finalement explicitement la représentation de la mesure de Mahler de Pd en termes de fonctions L, pour 1 ≤ d ≤ 6.

Given a parametric polynomial curve $\gamma:[a,b]\rightarrow \mathbb{R}^n$, how can we sample a random point $\mathfrak{x}\in \mathrm{im}(\gamma)$ in such a way that it is distributed uniformly with respect to the arc-length? Unfortunately, we cannot sample exactly such a point-even assuming we can perform exact arithmetic operations. So we end up with the following question: how does the method we choose affect the quality of the approximate sample we obtain? In practice, there are many answers. However, in theory, there are still gaps in our understanding. In this paper, we address this question from the point of view of complexity theory, providing bounds in terms of the size of the desired error., Comment: 10 pages, 5 figures, 1 table. 2nd version: New title, major changes

Vukičević and Gasperov introduced the concept of 148 discrete Adriatic indices in 2010. These indices showed good predictive properties against the testing sets of the International Academy of Mathematical Chemistry. Among these indices, twenty indices were taken as beneficial predictors of physicochemical properties. The inverse sum indeg index denoted by ISI G k of G k is a notable predictor of total surface area for octane isomers and is presented as ISI G k = ∑ g k g k ′ ∈ E G k d G k g k d G k g k ′ / d G k g k + d G k g k ′ , where d G k g k represents the degree of g k ∈ V G k . In this paper, we determine sharp bounds for ISI index of graph operations, including the Cartesian product, tensor product, strong product, composition, disjunction, symmetric difference, corona product, Indu–Bala product, union of graphs, double graph, and strong double graph.

International audience; Recently Navarro proposed a strengthening of the unsolved McKay conjecture using Galois automorphisms. We prove that the Navarro conjecture and its blockwise version hold for the alternating groups.

Let $\varrho$ be a complex number and let $f$ be a multiplicative arithmetic function whose Dirichlet series takes the form $\zeta(s)^\varrho G(s)$, where $G$ is associated to a multiplicative function $g$. The classical Selberg-Delange method furnishes asymptotic estimates for averages of $f$ under assumptions of either analytic continuation for $G$, or absolute convergence of a finite number of derivatives of $G(s)$ at $s=1$. We consider different set of hypotheses, not directly comparable to the previous ones, and investigate how they can yield sharp asymptotic estimates for the averages of~$f$.

Let $A$ be a $n\times n$ real matrix. The piecewise linear equation system $z-A\vert z\vert =b$ is called an absolute value equation (AVE). It is well-known to be equivalent to the linear complementarity problem. Unique solvability of the AVE is known to be characterized in terms of a generalized Perron root called the sign-real spectral radius of $A$. For mere, possibly non-unique, solvability no such characterization exists. We narrow this gap in the theory. That is, we define the concept of the aligned spectrum of $A$ and prove, under some mild genericity assumptions on $A$, that the mapping degree of the piecewise linear function $F_A:\mathbb{R}^n\to\mathbb{R}^n\,, z\mapsto z-A\lvert z\rvert$ is congruent to $(k+1)\mod 2$, where $k$ is the number of aligned values of $A$ which are larger than $1$. We also derive an exact--but more technical--formula for the degree of $F_A$ in terms of the aligned spectrum. Finally, we derive the analogous quantities and results for the LCP., Comment: 20 pages, 2 figures

We show results on the completeness and incompleteness of some closed manifolds carrying a nil-affine flat structure, said with rays. Those ray geometries appear at the boundary of symmetric spaces and in various contexts as affine geometry or contact geometry. We show that incomplete manifolds have a developing map that covers its image. It allows to show new cases of the Markus conjecture on flat affine closed manifolds having parallel volume. We then study the representations of the census of Falbel-Koseleff-Rouillier of fundamental groups of 3-manifolds in the isometry group of the hyperbolic complex plane. We propose a numerical computation of their limit sets and isolate those that are fractal and thus have experimentally the quality of being a discrete representation. We show that representations that seem discrete often factorize through a complex hyperbolic triangle group. It generalizes a phenomenon first noted by Deraux in a larger and more systematic framework.; Nous montrons des résultats de complétude et d’incomplétude de certaines variétés fermées portant une structure nil-affine plate, dite à rayons. Ces géométries à rayons apparaissent aux bords des espaces symétriques et dans des contextes variés comme la géométrie affine ou de contact. Nous montrons que les variétés incomplètes ont une développante qui revêt son image. Cela permet de montrer de nouveaux cas de la conjecture de Markus sur les variétés affines fermées à volume parallèle. Nous étudions ensuite les représentations du census de Falbel-Koseleff-Rouillier de groupes fondamentaux de 3-variétés dans le groupe des isométries du plan hyperbolique complexe. Nous proposons le calcul numérique de leurs ensembles limites et désignons ceux qui sont fractals et présentent donc expérimentalement la qualité d’être discret. Nous montrons que les représentations qui semble discrètes grâce à leur ensemble limite se factorisent la plupart du temps par un groupe d’un triangle hyperbolique complexe. Cela généralise un phénomène constaté par Deraux dans un cadre plus large et systématique.

We consider here the $3$-sphere $\mathbf S^3$ seen as the boundary at infinity of the complex hyperbolic plane $\mathbf{H}^2_{\mathbf C}$. It comes equipped with a contact structure and two classes of special curves. First $\mathbf R$-circles are boundaries at infinity of totally real totally geodesic subspaces and are tangent to the contact distribution. Second, $\mathbf C$-circles, which are boundaries of complex totally geodesic subspaces and are transverse to the contact distribution. We define a quantitative notion, called slimness, that measures to what extent a continuous path in the sphere $\mathbf S^3$ is near to be an $\mathbf R$-circle. We analyze the classical foliation of the complement of an $\mathbf R$-circle by arcs of $\mathbf C$-circles. Next, we consider deformations of this situation where the $\mathbf R$-circle becomes a slim curve. We apply these concepts to the particular case where the slim curve is the limit set of a quasi-Fuchsian subgroup of $\mathrm{PU}(2,1)$. As a consequence, we describe a class of spherical CR uniformizations of certain cusped $3$-manifolds.

A measure-scaling quasi-isometry between two connected graphs is a quasi-isometry that is quasi-$\kappa$-to-one in a natural sense for some $\kappa>0$. For non-amenable graphs, all quasi-isometries are quasi-$\kappa$-to-one for any $\kappa>0$, while for amenable ones there exists at most one possible such $\kappa$. For an amenable graph $X$, we show that the set of possible $\kappa$ forms a subgroup of $\mathbb{R}_{>0}$ that we call the (measure-)scaling group of $X$. This group is invariant under measure-scaling quasi-isometries. In the context of Cayley graphs, this implies for instance that two uniform lattices in a given locally compact group have same scaling groups. We compute the scaling group in a number of cases. For instance it is all of $\mathbb{R}_{>0}$ for lattices in Carnot groups, SOL or solvable Baumslag Solitar groups, but is a (strict) subgroup $\mathbb{Q}_{>0}$ for lamplighter groups over finitely presented amenable groups., Comment: 19 pages, 1 figure. Comments are welcome

For a finite extension $F$ of ${\mathbf Q}_p$, Drinfeld defined a tower of coverings of ${\mathbb P}^1\setminus {\mathbb P}^1(F)$ (the Drinfeld half-plane). For $F = {\mathbf Q}_p$, we describe a decomposition of the $p$-adic geometric \'etale cohomology of this tower analogous to Emerton's decomposition of completed cohomology of the tower of modular curves. A crucial ingredient is a finitness theorem for the arithmetic \'etale cohomology modulo $p$ which is shown by first proving, via a computation of nearby cycles, that this cohomology has finite presentation. This last result holds for all $F$; for $F\neq {\mathbf Q}_p$, it implies that the representations of ${\rm GL}_2(F)$ obtained from the cohomology of the Drinfeld tower are not admissible contrary to the case $F = {\mathbf Q}_p$., Comment: in French. Final version. To appear in Forum of Mathematics, Pi

We use a ${\mathcal B}$-adic completion and the $p$-adic local Langlands correspondence for ${\mathrm {GL}}_2({\mathbf Q}_p )$ to give a construction of Kisin's rings and the attached universal Galois representations (in dimension 2 and for ${\mathbf Q}_p$) directly from the classical Langlands correspondence. This gives, in particular, a uniform proof of the geometric Breuil-M\'ezard conjecture in the supercuspidal case., Comment: in French language. First version. Comments are welcomeinal version. To appear in Acta Arithmetica

In this paper we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. More precisely, given a totally nonholonomic analytic distribution of rank 2 on a 3-dimensional analytic manifold, we investigate the size of the set of points that can be reached by singular horizontal paths starting from a given point and prove that it has Hausdorff dimension at most 1. In fact, provided that the lengths of the singular curves under consideration are bounded with respect to a given complete Riemannian metric, we demonstrate that such a set is a semianalytic curve. As a consequence, combining our techniques with recent developments on the regularity of sub-Riemannian minimizing geodesics, we prove that minimizing sub-Riemannian geodesics in 3-dimensional analytic manifolds are always of class C-1, and actually they are analytic outside of a finite set of points., Inventiones mathematicae, 229, ISSN:0020-9910, ISSN:1432-1297

International audience; Expansion of real numbers is a basic research topic in number theory. Usually we expand real numbers in one given base. In this paper, we begin to systematically study expansions in multiple given bases in a reasonable way, which is a generalization in the sense that if all the bases are taken to be the same, we return to the classical expansions in one base. In particular, we focus on greedy, quasi-greedy, lazy, quasi-lazy and unique expansions in multiple bases.

Given an integer $$q\ge 2$$ and $$\theta _1,\ldots ,\theta _{q-1}\in \{0,1\}$$ , let $$(\theta _n)_{n\ge 0}$$ be the generalized Thue–Morse sequence, defined to be the unique fixed point of the morphism $$\begin{aligned} 0\mapsto & {} 0\theta _1\cdots \theta _{q-1}\\ 1\mapsto & {} 1\overline{\theta }_1\cdots \overline{\theta }_{q-1} \end{aligned}$$ beginning with $$\theta _0:=0$$ , where $$\overline{0}:=1$$ and $$\overline{1}:=0$$ . For ad hoc rational functions R, we evaluate infinite products of the forms $$\begin{aligned} \prod _{n=1}^\infty \Big (R(n)\Big )^{(-1)^{\theta _n}}\quad \text {and}\quad \prod _{n=1}^\infty \Big (R(n)\Big )^{\theta _n}.\end{aligned}$$ This generalizes relevant results given by Allouche, Riasat and Shallit in 2019 on infinite products related to the famous Thue–Morse sequence $$(t_n)_{n\ge 0}$$ of the forms $$\begin{aligned} \prod _{n=1}^\infty \Big (R(n)\Big )^{(-1)^{t_n}}\quad \text {and}\quad \prod _{n=1}^\infty \Big (R(n)\Big )^{t_n}. \end{aligned}$$

We prove that horospheres, hyperspheres, and hyperplanes in a hyperbolic space $${\mathbb {H}}^n,\,n\ge 3$$ , admit no perturbations with compact support which increase their mean curvature. This is an extension of the analogous result in the Euclidean spaces, due to M. Gromov, which states that a hyperplane in a Euclidean space $${\mathbb {R}}^n$$ admits no mean convex perturbations with compact supports.