Abstract: Publication date: Available online 9 October 2017 Source:Comptes Rendus Mathematique Author(s): Martin Puchol, Yeping Zhang, Jialin Zhu We consider a compact manifold with a piece isometric to a (finite-length) cylinder. By making the length of the cylinder tend to infinity, we obtain an asymptotic gluing formula for the zeta determinant of the Hodge Laplacian and an asymptotic expansion of the L 2 torsion of the corresponding Mayer–Vietoris exact sequence. As an application, we give a purely analytic proof of the gluing formula for analytic torsion.

Abstract: Publication date: Available online 9 October 2017 Source:Comptes Rendus Mathematique Author(s): Rafał Czyż, Van Thien Nguyen In this note, we prove that the constant D ( p , m ) in the energy estimate, for m-subharmonic function with bounded p-energy, is strictly bigger than 1, for p > 0 , p ≠ 1 .

Abstract: Publication date: Available online 9 October 2017 Source:Comptes Rendus Mathematique Author(s): Anis S. Hoayek, Gilles R. Ducharme, Zaher Khraibani In a time series { X t , t ≥ 1 } , X j is said to be an upper record if X j > max { X 1 , … , X j − 1 } . Some popular models for records are the Yang–Nevzorov and the Linear Drift models. In this note, we introduce for these models the joint likelihood of the record sequence and the indicators of their occurrence. This likelihood can then be used to obtain estimators of the unknown parameters in the models. It can also be used to derive inferential procedures associated with the selection of a proper model for such data.

Abstract: Publication date: Available online 9 October 2017 Source:Comptes Rendus Mathematique Author(s): Abdullah Mir, Imtiaz Hussain Let P ( z ) be a polynomial of degree n and for any complex number α, let D α P ( z ) : = n P ( z ) + ( α − z ) P ′ ( z ) denote the polar derivative of P ( z ) with respect to α. In this paper, we present an integral inequality for the polar derivative of a polynomial. Our theorem includes as special cases several interesting generalisations and refinements of Erdöx–Lax theorem.

Abstract: Publication date: Available online 6 October 2017 Source:Comptes Rendus Mathematique Author(s): Xiaodong Cao, Bo Yang Motivated by a previous work by Zheng and the second-named author, we study pinching constants of compact Kähler manifolds with positive holomorphic sectional curvature. In particular, we prove a gap theorem following the work of Petersen and Tao on Riemannian manifolds with almost-quarter-pinched sectional curvature.

Abstract: Publication date: Available online 3 October 2017 Source:Comptes Rendus Mathematique Author(s): Paata Ivanisvili, Fedor Nazarov, Alexander Volberg In the present paper, we show that a correctly chosen Legendre transform of the Bellman functions of martingale problems give us the right tool to prove isoperimetric inequalities on Hamming cube independent of the dimension. We illustrate the power of this “dual function approach” by proving certain Poincaré inequalities on Hamming cube and by improving a particular inequality of Beckner on the Hamming cube.

Abstract: Publication date: Available online 29 September 2017 Source:Comptes Rendus Mathematique Author(s): Semyon Litvinov This article gives an affirmative solution to the problem whether the ergodic Cesáro averages generated by a positive Dunford–Schwartz operator in a noncommutative space L p ( M , τ ) , 1 ≤ p < ∞ , converge almost uniformly (in Egorov's sense). This problem goes back to the original paper of Yeadon [21], published in 1977, where bilaterally almost uniform convergence of these averages was established for p = 1 .

Abstract: Publication date: Available online 29 September 2017 Source:Comptes Rendus Mathematique Author(s): Aida Kh. Asgarova, Vugar E. Ismailov We give a necessary condition for the representation of the space of continuous functions by sums of its k closed subalgebras. A sufficient condition for this representation problem was first obtained by Sternfeld in 1978. In case of two subalgebras ( k = 2 ), our necessary condition turns out to be also sufficient. If k = 1 , our result coincides with a version of the classical Stone–Weierstrass theorem.

Abstract: Publication date: Available online 28 September 2017 Source:Comptes Rendus Mathematique Author(s): Hossein Hassani, Mahdi Kalantari, Masoud Yarmohammadi The Singular Spectrum Analysis (SSA) technique is a non-parametric powerful method in the field of time series analysis whose popularity has increased in recent years owing to its widespread applications. Recurrent forecasting is one of the important forecasting methods in SSA. In this paper, the forecasting accuracy of recurrent forecasts is improved via the introduction of a new recurrent forecasting algorithm. In the novel approach, the recurrent coefficients are generated from the filtered series which has less noise and thus enables one to achieve the better forecasts. The performance of the new method has been compared with the established recurrent forecasting method. The comparison involves applications to various real and simulated time series. The obtained results confirm that the new approach can provide more accurate forecasts.

Abstract: Publication date: Available online 28 September 2017 Source:Comptes Rendus Mathematique Author(s): Vitali Milman, Liran Rotem Do we have enough examples of convex bodies that we truly understand' Is out standard set of examples diverse enough to understand convexity' In this note, we will dramatically increase our set of examples. More specifically, we will present several new constructions of convex bodies: the geometric mean of two convex bodies, the power function K α (which in general exists only for α ≤ 1 ), and even the logarithm log K .

Abstract: Publication date: Available online 28 September 2017 Source:Comptes Rendus Mathematique Author(s): Augusto C. Ponce, Daniel Spector We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of De Giorgi. Our motivation comes from a new fractional Boxing inequality that relates the fractional perimeter and the Hausdorff content and implies several known inequalities involving the Gagliardo seminorm of the Sobolev spaces W α , 1 of order 0 < α < 1 .

Abstract: Publication date: Available online 27 September 2017 Source:Comptes Rendus Mathematique Author(s): Amod Agashe, Lydia Eldredge A theorem of Matsushima–Shimura shows that the space of harmonic differential forms on the quotient of products of upper half planes under the action of certain groups, when the quotient is compact, is the direct sum of two subspaces called the universal and cuspidal subspaces. We generalize this result to compact quotients of products of upper half planes and upper half spaces (hyperbolic three spaces) under the action of certain groups to obtain a similar decomposition.

Abstract: Publication date: Available online 27 September 2017 Source:Comptes Rendus Mathematique Author(s): Giampiero Chiaselotti, Tommaso Gentile, Federico Infusino In this paper, we develop in a more general mathematical context the notion of indistinguishability, which in graph theory has recently been investigated as a symmetry relation with respect to a fixed vertex subset. The starting point of our analysis is to consider a set Ω of functions defined on a universe set U and to define an equivalence relation ≡ A on U for any subset A ⊆ Ω in the following way: u ≡ A u ′ if a ( u ) = a ( u ′ ) for any function a ∈ A . By means of this family of relations, we introduce the indistinguishability relation ≈ on the power set P ( Ω ) as follows: for A , A ′ ∈ P ( Ω ) , we set A ≈ A ′ if the relations ≡ A and ≡ A ′ coincide. We use then the indistinguishability relation ≈ to introduce several set families on Ω that have interesting order, matroidal and combinatorial properties. We call the above set families the indistinguishability structures of the function system ( U , Ω ) . Furthermore, we obtain a closure system and an abstract simplicial complex interacting each other by means of three hypergraphs having relevance in both theoretical computer science and graph theory. The first part of this paper is devoted to investigate the basic mathematical properties of the indistinguishability structures for arbitrary function systems. The second part deals with some specific cases of study derived from simple undirected graphs and the usual Euclidean real line.

Abstract: Publication date: Available online 25 September 2017 Source:Comptes Rendus Mathematique Author(s): Cédric Dion, Antonio Lei We give a new description of Pollack's plus and minus p-adic logarithms log p ± in terms of distributions. In particular, if μ ± denote the pre-images of log p ± under the Amice transform, we give explicit formulae for the values μ ± ( a + p n Z p ) for all a ∈ Z p and all integers n ≥ 1 . Our formulae imply that the distribution μ − agrees with a distribution studied by Koblitz in 1977. Furthermore, we show that a similar description exists for Loeffler's two-variable analogues of these plus and minus logarithms.

Abstract: Publication date: Available online 21 September 2017 Source:Comptes Rendus Mathematique Author(s): Nguyen Anh Dao, Quoc-Hung Nguyen We establish a sufficient condition for the existence of solutions to the incompressible Navier–Stokes equations, with singular time-dependent external forces defined in terms of capacity Cap H 1 , 2 ( E ) .

Abstract: Publication date: Available online 20 September 2017 Source:Comptes Rendus Mathematique Author(s): Kiryong Chung, Han-Bom Moon We study birational geometry of the moduli space of stable sheaves on a quadric surface with Hilbert polynomial 5 m + 1 and c 1 = ( 2 , 3 ) . We describe a birational map between the moduli space and a projective bundle over a Grassmannian as a composition of smooth blow-ups/downs.

Abstract: Publication date: Available online 19 September 2017 Source:Comptes Rendus Mathematique Author(s): Tomasz Kania, Martin Rmoutil We employ the theory of elementary submodels to improve a recent result by Aron, Jaramillo and Le Donne (2017) [1] concerning restricting uniformly open, continuous surjections to smaller subspaces where they remain surjective. To wit, suppose that X and Y are metric spaces and let f : X → Y be a continuous surjection. If X is complete and f is uniformly open, then X contains a closed subspace Z with the same density as Y such that f restricted to Z is still uniformly open and surjective. Moreover, if X is a Banach space, then Z may be taken to be a closed linear subspace. A counterpart of this theorem for uniform spaces is also established.

Abstract: Publication date: Available online 19 September 2017 Source:Comptes Rendus Mathematique Author(s): Ming Xu, Shaoqiang Deng We prove some rigidity results on geodesic orbit Finsler spaces with non-positive curvature. In particular, we show that a geodesic orbit Finsler space with strictly negative flag curvature must be a non-compact Riemannian symmetric space of rank one.

Abstract: Publication date: Available online 19 September 2017 Source:Comptes Rendus Mathematique Author(s): Juan Fernández Sánchez, Wolfgang Trutschnig We point out a mistake in the proof of the main theorem in a recent article on a family of generalized Minkowski's question-mark functions, saying that each member of the family is a singular homeomorphism, and provide two alternative proofs, one based on the ergodicity of the Gauss map G and the α-Lüroth map L α , and another one focusing more on classical properties of continued fraction expansions.

Abstract: Publication date: Available online 12 September 2017 Source:Comptes Rendus Mathematique Author(s): Kuldeep Kumar Kataria, Palaniappan Vellaisamy We obtain some recurrence relationships among the partition vectors of the partial exponential Bell polynomials. On using such results, the n-th Adomian polynomial for any nonlinear operator can be expressed explicitly in terms of the partial exponential Bell polynomials. Some new identities for the partial exponential Bell polynomials are obtained by solving certain ordinary differential equations using the Adomian decomposition method.

Abstract: Publication date: August 2017 Source:Comptes Rendus Mathematique, Volume 355, Issue 8 Author(s): Laura Escobar, Alexander Yong Symmetric Grothendieck polynomials are inhomogeneous versions of Schur polynomials that arise in combinatorial K-theory. A polynomial has saturated Newton polytope (SNP) if every lattice point in the polytope is an exponent vector. We show that the Newton polytopes of these Grothendieck polynomials and their homogeneous components have SNP. Moreover, the Newton polytope of each homogeneous component is a permutahedron. This addresses recent conjectures of C. Monical–N. Tokcan–A. Yong and of A. Fink–K. Mészáros–A. St. Dizier in this special case.

Abstract: Publication date: Available online 6 September 2017 Source:Comptes Rendus Mathematique Author(s): Albert Fathi The following fact seems to have been unnoticed until now: Let F be a closed subset of the (finite-dimensional) connected manifold M. If f : F → M is a proper continuous map which is the identity on the boundary ∂F of F in M, then either f ( F ) ⊃ F or f ( F ) ⊃ M ∖ F . The proof is elementary and simple using degree theory. The statement has many deep consequences.

Abstract: Publication date: Available online 6 September 2017 Source:Comptes Rendus Mathematique Author(s): Moritz Gerlach, Jochen Glück We give a new and very short proof of a theorem of Greiner asserting that a positive and contractive C 0 -semigroup on an L p -space is strongly convergent in case it has a strictly positive fixed point and contains an integral operator. Our proof is a streamlined version of a much more general approach to the asymptotic theory of positive semigroups developed recently by the authors. Under the assumptions of Greiner's theorem, this approach becomes particularly elegant and simple. We also give an outlook on several generalisations of this result.

Abstract: Publication date: Available online 7 August 2017 Source:Comptes Rendus Mathematique Author(s): Guanlong Bao, Nihat Gökhan Göğüş, Stamatis Pouliasis In 1991, S. Richter introduced harmonically weighted Dirichlet spaces D ( μ ) , motivated by his study of cyclic analytic two-isometries. In this paper, we consider ⋂ μ ∈ P D ( μ ) , the intersection of D ( μ ) spaces, where P is the family of Borel probability measures. Several function-theoretic characterizations of the Banach space ⋂ μ ∈ P D ( μ ) are given. We also show that ⋂ μ ∈ P D ( μ ) is located strictly between some classical analytic Lipschitz spaces and ⋂ μ ∈ P D ( μ ) can be regarded as the endpoint case of analytic Morrey spaces in some sense.

Abstract: Publication date: Available online 7 August 2017 Source:Comptes Rendus Mathematique Author(s): Nicolas Libedinsky, Geordie Williamson A basic question concerning indecomposable Soergel bimodules is to understand their endomorphism rings. In characteristic zero all degree-zero endomorphisms are isomorphisms (a fact proved by Elias and the second author) which implies the Kazhdan–Lusztig conjectures. More recently, many examples in positive characteristic have been discovered with larger degree zero endomorphisms. These give counter-examples to expected bounds in Lusztig's conjecture. Here we prove the existence of indecomposable Soergel bimodules in type A having non-zero endomorphisms of negative degree. This gives the existence of a non-perverse parity sheaf in type A.

Abstract: Publication date: Available online 7 August 2017 Source:Comptes Rendus Mathematique Author(s): Kangwei Li In this note, we study the sharp weighted estimate involving one supremum. In particular, we give a positive answer to an open question raised by Lerner and Moen. We also extend the result to rough homogeneous singular integral operators.

Abstract: Publication date: Available online 4 August 2017 Source:Comptes Rendus Mathematique Author(s): Alexandru Aleman, Bartosz Malman We prove that functions analytic in the unit disk and continuous up to the boundary are dense in the de Branges–Rovnyak spaces induced by the extreme points of the unit ball of H ∞ . Together with previous theorems, it follows that this class of functions is dense in any de Branges–Rovnyak space.

Abstract: Publication date: Available online 2 August 2017 Source:Comptes Rendus Mathematique Author(s): Teng Huang In this note, we prove an L n 2 -energy gap result for Yang–Mills connections on a principal G-bundle over a compact manifold without using the Lojasiewicz–Simon gradient inequality ([2] Theorem 1.1).

Abstract: Publication date: Available online 31 July 2017 Source:Comptes Rendus Mathematique Author(s): Marie-Claude Arnaud, Patrice Le Calvez We introduce a notion of Denjoy sub-system that generalizes that of the Aubry–Mather set. For such systems, we prove a result similar to Denjoy theorem (non-existence of C 2 Denjoy sub-systems), and study their Lyapunov exponents.

Abstract: Publication date: Available online 31 July 2017 Source:Comptes Rendus Mathematique Author(s): Pierre Cantin An extension of the well-posedness analysis of the scalar and the vector advection–reaction problem in Banach graph spaces of power p ∈ ( 1 , ∞ ) is proposed. This analysis is based on the sign of the associated Friedrichs tensor, taking positive, null or reasonably negative values.

Abstract: Publication date: Available online 31 July 2017 Source:Comptes Rendus Mathematique Author(s): Aihua Fan We prove that fully oscillating sequences are orthogonal to multiple ergodic realizations of affine maps of zero entropy on compact Abelian groups. It is more than what Sarnak's conjecture requires for these dynamical systems.

Abstract: Publication date: Available online 27 July 2017 Source:Comptes Rendus Mathematique Author(s): Saulius Norvidas We discuss a uniqueness property of the characteristic function of an absolutely continuous probability measure. Our study is initiated by the question posed by N.G. Ushakov: is it true that, for any interval [ a , b ] ⊂ R , 0 ∉ [ a , b ] , there exists a characteristic function f such that f ≢ e − t 2 / 2 , but f ( t ) = e − t 2 / 2 for all t ∈ [ a , b ] '

Abstract: Publication date: Available online 21 July 2017 Source:Comptes Rendus Mathematique Author(s): Tushar Das, Lior Fishman, David Simmons, Mariusz Urbański We establish a new connection between metric Diophantine approximation and the parametric geometry of numbers by proving a variational principle facilitating the computation of the Hausdorff and packing dimensions of many sets of interest in Diophantine approximation. In particular, we show that the Hausdorff and packing dimensions of the set of singular m × n matrices are both equal to m n ( 1 − 1 m + n ) , thus proving a conjecture of Kadyrov, Kleinbock, Lindenstrauss, and Margulis as well as answering a question of Bugeaud, Cheung, and Chevallier. Other applications include computing the dimensions of the sets of points witnessing conjectures of Starkov and Schmidt.

Abstract: Publication date: Available online 21 July 2017 Source:Comptes Rendus Mathematique Author(s): Mário Bessa In this note, we prove the flowbox theorem for divergence-free Lipschitz vector fields.

Abstract: Publication date: Available online 21 July 2017 Source:Comptes Rendus Mathematique Author(s): Samuel Le Fourn We present results on the integral points of some modular varieties. These results are based on a generalisation of the so-called Runge's method to higher dimensions, which will be explained first. In particular, we obtain an explicit result for the Siegel modular variety A 2 ( 2 ) .

Abstract: Publication date: Available online 10 July 2017 Source:Comptes Rendus Mathematique Author(s): Christos Sourdis We prove that entire, complex valued solutions to the Ginzburg–Landau system with positive real and imaginary parts are constant in any spatial dimension. This property was shown very recently only in the planar case.

Abstract: Publication date: Available online 10 July 2017 Source:Comptes Rendus Mathematique Author(s): Jean-François Bony, Setsuro Fujiié, Thierry Ramond, Maher Zerzeri In the framework of semiclassical resonances, we make more precise the link between the polynomial estimates of the extension of the resolvent and the propagation of the singularities through the trapped set. This approach makes it possible to eliminate infinity and to concentrate the study near the trapped set. It has allowed us in previous papers to obtain the asymptotic of resonances in various geometric situations.

Abstract: Publication date: Available online 28 June 2017 Source:Comptes Rendus Mathematique Author(s): Bao-Wei Wang, Jun Wu, Jian Xu In this note, we consider the metric theory of the dynamical covering problems on the triadic Cantor set K . More precisely, let T x = 3 x ( mod 1 ) be the natural map on K , μ the standard Cantor measure and x 0 ∈ K a given point. We consider the size of the set of points in K which can be well approximated by the orbit { T n x 0 } n ≥ 1 of x 0 , namely the set D ( x 0 , φ ) : = { y ∈ K : T n x 0 − y < φ ( n ) for infinitely many n ∈ N } , where φ is a positive function defined on N . It is shown that for μ almost all x 0 ∈ K , the Hausdorff measure of D ( x 0 , φ ) is either zero or full depending upon the convergence or divergence of a certain series. Among the proof, as a byproduct, we obtain an inhomogeneous counterpart of Levesley, Salp and Velani's work on a Mahler's question about the Diophantine approximation on the Cantor set K .

Abstract: Publication date: Available online 28 June 2017 Source:Comptes Rendus Mathematique Author(s): Liangang Ma For a real x ∈ ( 0 , 1 ) ∖ Q , let x = [ a 1 ( x ) , a 2 ( x ) , ⋯ ] be its continued fraction expansion. Denote by T n ( x ) : = max { a k ( x ) : 1 ≤ k ≤ n } the maximum partial quotient up to n. For any real α ∈ ( 0 , ∞ ) , γ ∈ ( 0 , ∞ ) , let F ( γ , α ) : = { x ∈ ( 0 , 1 ) ∖ Q : lim n → ∞ T n ( x ) e n γ = α } . For a set E ⊂ ( 0 , 1 ) ∖ Q , let dim H E be its Hausdorff dimension. Recently, Lingmin Liao and Michal Rams showed that dim H F ( γ , α ) = { 1 i f γ ∈ ( 0 , 1 / 2 ) 1 / 2 i f γ ∈ ( 1 / 2 , ∞ ) for any α ∈ ( 0 , ∞ ) . In this paper, we show that dim H F ( 1 / 2 , α ) = 1 / 2 for any α ∈ ( 0 , ∞ ) following Liao and Rams' method, which supplements their result.

Abstract: Publication date: Available online 16 June 2017 Source:Comptes Rendus Mathematique Author(s): Stéphane Ballet, Nicolas Baudru, Alexis Bonnecaze, Mila Tukumuli The Chudnovsky algorithm for the multiplication in extensions of finite fields provides a bilinear complexity uniformly linear with respect to the degree of the extension. Recently, Randriambololona has generalized the method, allowing asymmetry in the interpolation procedure and leading to new upper bounds on the bilinear complexity. In this note, we describe the construction of this asymmetric method without derived evaluation. To do this, we translate this generalization into the language of algebraic function fields and we give a strategy of construction and implementation.

Abstract: Publication date: Available online 12 June 2017 Source:Comptes Rendus Mathematique Author(s): Sylvain Carpentier It has been proved by Sokolov that Krichever–Novikov equation's hierarchy is hamiltonian for the Hamiltonian operator H 0 = u x ∂ − 1 u x and possesses two weakly non-local recursion operators of degrees 4 and 6, L 4 and L 6 . We show here that H 0 , L 4 H 0 and L 6 H 0 are compatible Hamiltonians operators for which the Krichever–Novikov equation's hierarchy is hamiltonian.