Authors:Jean-François Bertazzon Pages: 235 - 242 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Jean-François Bertazzon For each λ ∈ N ⁎ , we consider the integral equation: ∫ λ y λ x f ( t ) d t = f ( x ) − f ( y ) for every ( x , y ) ∈ R + 2 , where f is the concatenation of two continuous functions f a , f b : [ 0 , λ ] → R along a word u = u 0 u 1 ⋯ ∈ { a , b } N such that u = σ ( u ) , where σ is a λ-uniform substitution satisfying some combinatorial conditions. There exists some non-trivial solutions ([1]). We show in this work that the dimension of the set of solutions is at most two.

Authors:Stefanos Aivazidis; Thomas W. Müller Pages: 253 - 257 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Stefanos Aivazidis, Thomas W. Müller Following Isaacs (see [6, p. 94]), we call a normal subgroup N of a finite group G large, if C G ( N ) ≤ N , so that N has bounded index in G. Our principal aim here is to establish some general results for systematically producing large subgroups in finite groups (see Theorems A and C). We also consider the more specialised problems of finding large (non-abelian) nilpotent as well as abelian subgroups in soluble groups.

Authors:Thomas Gobet; Anne-Laure Thiel Pages: 258 - 263 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Thomas Gobet, Anne-Laure Thiel In this note, we compute the split Grothendieck ring of a generalized category of Soergel bimodules of type A 2 , where we take one generator for each reflection. We give a presentation by generators and relations of it and a parametrization of the indecomposable objects of the category, by realizing them as rings of regular functions on certain unions of graphs of group elements on a reflection faithful representation.

Authors:Pierre Bousquet; Augusto C. Ponce; Jean Van Schaftingen Pages: 264 - 271 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen We have recently introduced the trimming property for a complete Riemannian manifold N n as a necessary and sufficient condition for bounded maps to be strongly dense in W 1 , p ( B m ; N n ) when p ∈ { 1 , … , m } . We prove in this note that, even under a weaker notion of approximation, namely the weak sequential convergence, the trimming property remains necessary for the approximation in terms of bounded maps. The argument involves the construction of a Sobolev map having infinitely many analytical singularities going to infinity.

Authors:Ilgiz R. Kayumov; Saminathan Ponnusamy Pages: 272 - 277 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Ilgiz R. Kayumov, Saminathan Ponnusamy In this article, we prove several different improved versions of the classical Bohr's inequality. All the results are proved to be sharp.

Authors:Rakesh K. Parmar; Tibor K. Pogány; Ram K. Saxena Pages: 278 - 282 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Rakesh K. Parmar, Tibor K. Pogány, Ram K. Saxena We introduce the ( p , q ) -extended τ-hypergeometric and confluent hypergeometric functions along with their integral representations. We also present closed integral expressions for the Mathieu-type a -series and for the associated alternating versions whose terms contain the ( p , q ) -extended τ-hypergeometric functions with related contiguous functional relations.

Authors:Fatima Zahra Assila Pages: 283 - 287 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Fatima Zahra Assila We study the projective logarithmic potential G μ of a probability measure μ on the complex projective space P n . We prove that the range of the operator μ ⟶ G μ is contained in the (local) domain of definition of the complex Monge–Ampère operator acting on the class of quasi-plurisubharmonic functions on P n with respect to the Fubini–Study metric. Moreover, when the measure μ has no atom, we show that the complex Monge–Ampère measure of its logarithmic potential is an absolutely continuous measure with respect to the Fubini–Study volume form on P n .

Authors:Sourav Das Pages: 288 - 292 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Sourav Das In this work, multiple gamma functions of order n have been considered. The logarithmic derivative of the multiple gamma function is known as the multiple psi function. Subadditive, superadditive, and convexity properties of higher-order derivatives of the multiple psi function are derived. Some related inequalities for these functions and their ratios are also obtained.

Authors:Ann-Eva Christensen; Jon Johnsen Pages: 301 - 305 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Ann-Eva Christensen, Jon Johnsen We prove that a large class of parabolic final value problems is well posed. This results via explicit Hilbert spaces that characterise the data yielding existence, uniqueness and stability of solutions. This data space is the graph normed domain of an unbounded operator, which represents a new compatibility condition pertinent for final value problems. The framework is that of evolution equations for Lax–Milgram operators in vector distribution spaces. The final value heat equation on a smooth open set is also covered, and for non-zero Dirichlet data, a non-trivial extension of the compatibility condition is obtained by addition of an improper Bochner integral.

Authors:Marcelo M. Disconzi; Igor Kukavica Pages: 306 - 311 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Marcelo M. Disconzi, Igor Kukavica We consider the free boundary compressible and incompressible Euler equations with surface tension. In both cases, we provide a priori estimates for the local existence with the initial velocity in H 3 , with the H 3 condition on the density in the compressible case. An additional condition is required on the free boundary. Compared to the existing literature, both results lower the regularity of initial data for the Lagrangian Euler equation with surface tension.

Authors:Payman Eskandari Pages: 312 - 315 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Payman Eskandari Let X be a Riemann surface of positive genus. Denote by X ( n ) the configuration space of n distinct points on X. We use the Betti–de Rham comparison isomorphism on H 1 ( X ( n ) ) to define an integrable connection on the trivial vector bundle on X ( n ) with fiber the universal algebra of the Lie algebra associated with the descending central series of π 1 of X ( n ) . The construction is inspired by the Knizhnik–Zamolodchikov system in genus zero and its integrability follows from Riemann period relations.

Authors:Indranil Biswas; Sorin Dumitrescu Pages: 316 - 321 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Indranil Biswas, Sorin Dumitrescu In [6], it was asked whether all flat holomorphic Cartan geometries ( G , H ) on a complex torus are translation invariant. We answer this affirmatively under the assumption that the complex Lie group G is affine. More precisely, we show that every holomorphic Cartan geometry of type ( G , H ) , with G a complex affine Lie group, on any complex torus is translation invariant.

Authors:Oussama Hijazi; Sebastián Montiel; Simon Raulot Pages: 322 - 326 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Oussama Hijazi, Sebastián Montiel, Simon Raulot We give a spinorial proof of a Heintze–Karcher-type inequality in the hyperbolic space proved by Brendle [4]. The proof relies on a generalized Reilly formula on spinors recently obtained in [7].

Authors:Ursula Ludwig Pages: 327 - 332 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Ursula Ludwig The aim of this note is to extend a theorem by Cheeger and Müller to spaces with isolated conical singularities by generalising the proof of Bismut and Zhang to the singular setting. The main tools in this approach are the Witten deformation and local index techniques.

Authors:Heayong Shin; Young Wook Kim; Sung-Eun Koh; Hyung Yong Lee; Seong-Deog Yang Pages: 333 - 339 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Heayong Shin, Young Wook Kim, Sung-Eun Koh, Hyung Yong Lee, Seong-Deog Yang Choe and Soret [1] constructed infinitely many compact embedded minimal surfaces in S 3 by desingularizing Clifford tori which meet each other along a great circle at the angle of the same size. We show their method works with some modifications to construct compact embedded minimal surfaces in the Berger sphere as well.

Authors:Jan P. Boroński; Piotr Oprocha Pages: 340 - 344 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Jan P. Boroński, Piotr Oprocha We prove that the Sierpiński curve admits a homeomorphism with strong mixing properties. We also prove that the constructed example does not have Bowen's specification property.

Authors:Elena Bonetti; Michel Frémond Pages: 345 - 350 Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Elena Bonetti, Michel Frémond We study the motion of a visco-elastic solid with large deformations. We prove the existence of a local-in-time motion and of a non-negative pressure, which is a measure reaction to the incompressibility condition.

Authors:Amel Bendali-Braham; Noureddine Ikhlef-Eschouf; Mostafa Blidia Pages: 115 - 120 Abstract: Publication date: February 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 2 Author(s): Amel Bendali-Braham, Noureddine Ikhlef-Eschouf, Mostafa Blidia The b-chromatic number of a graph G is the largest integer k such that G admits a proper coloring with k colors for which each color class contains a vertex that has at least one neighbor in all the other k − 1 color classes. A graph G is called b e -critical if the contraction of any edge e of G decreases the b-chromatic number of G. The purpose of this paper is the characterization of all b e -critical trees.

Authors:Yong-Gao Chen Pages: 121 - 124 Abstract: Publication date: February 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 2 Author(s): Yong-Gao Chen Recently, motivated by Stanley's sequences, Kiss, Sándor, and Yang introduced a new type sequence: a sequence A of nonnegative integers is called an A P k -covering sequence if there exists an integer n 0 such that, if n > n 0 , then there exist a 1 ∈ A , … , a k − 1 ∈ A , a 1 < a 2 < ⋯ < a k − 1 < n such that a 1 , … , a k − 1 , n form a k-term arithmetic progression. They prove that there exists an A P 3 -covering sequence A such that lim sup n → ∞ A ( n ) / n ≤ 34 . In this note, we prove that there exists an A P 3 -covering sequence A such that lim sup n → ∞ A ( n ) / n = 15 .

Authors:Byungchan Kim; Ji Young Kim; Chong Gyu Lee; Poo-Sung Park Pages: 125 - 128 Abstract: Publication date: Available online 9 January 2018 Source:Comptes Rendus Mathematique Author(s): Byungchan Kim, Ji Young Kim, Chong Gyu Lee, Poo-Sung Park We prove that the set GP of all nonzero generalized pentagonal numbers is an additive uniqueness set; if a multiplicative function f satisfies the equation f ( a + b ) = f ( a ) + f ( b ) , for all a , b ∈ G P , then f is the identity function.

Authors:Bing-Ling Wu; Yong-Gao Chen Pages: 129 - 132 Abstract: Publication date: February 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 2 Author(s): Bing-Ling Wu, Yong-Gao Chen Let H n be the n-th harmonic number and let v n be its denominator. It is well known that v n is even for every integer n ≥ 2 . In this paper, we study the properties of v n . One of our results is: the set of positive integers n such that v n is divisible by the least common multiple of 1 , 2 , ⋯ , ⌊ n 1 / 4 ⌋ has density one. In particular, for any positive integer m, the set of positive integers n such that v n is divisible by m has density one.

Authors:Jongho Yang Pages: 133 - 137 Abstract: Publication date: Available online 5 January 2018 Source:Comptes Rendus Mathematique Author(s): Jongho Yang Let N be the linear space of functions ∑ k = 1 n a k ρ ( θ k / x ) with a condition ∑ k = 1 n a k θ k = 0 for 0 < θ k ≤ 1 . Here ρ ( x ) denotes the fractional part of x. Beurling pointed out that the problem of how well a constant function can be approximated by functions in N is closely related to the zero-free region of the Riemann zeta function. More precisely, Báez-Duarte gave a zero-free region related to a L p -norm estimation of a constant function by using the Dirichlet series for the zeta function. In this paper, we consider the L ∞ -norm estimation of a constant function and give a wider zero-free region than that of the Báez-Duarte result.

Authors:Mozhgan Rezakhanlou; Mohammad Reza Darafsheh Pages: 138 - 140 Abstract: Publication date: February 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 2 Author(s): Mozhgan Rezakhanlou, Mohammad Reza Darafsheh A finite group G is called a Q 1 -group if all of its non-linear irreducible characters are rational valued. In this paper, we will find the general structure of a metabelian Q 1 -group.

Authors:Cyril Lacoste Pages: 141 - 145 Abstract: Publication date: February 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 2 Author(s): Cyril Lacoste We prove that the set of symplectic lattices in the Siegel space h g whose systoles generate a subspace of dimension at least 3 in R 2 g does not contain any Sp ( 2 g , Z ) -equivariant deformation retract of h g .

Authors:Carlos Matheus; Carlos Gustavo Moreira; Jacob Palis Pages: 146 - 149 Abstract: Publication date: Available online 9 January 2018 Source:Comptes Rendus Mathematique Author(s): Carlos Matheus, Carlos Gustavo Moreira, Jacob Palis We show that the non-uniformly hyperbolic horseshoes of Palis and Yoccoz occur in the standard family of area-preserving diffeomorphisms of the two-torus.

Authors:Dan Yang; JinRong Wang; Donal O'Regan Pages: 150 - 171 Abstract: Publication date: February 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 2 Author(s): Dan Yang, JinRong Wang, Donal O'Regan In this article, we investigate the orbital Hausdorff continuous dependence of the solutions to integer order and fractional nonlinear non-instantaneous differential equations. The concept of orbital Hausdorff continuous dependence is used to characterize the relations of solutions corresponding to the impulsive points and junction points in the sense of the Hausdorff distance. Then, we establish sufficient conditions to guarantee this specific continuous dependence on their respective trajectories. Finally, two examples are given to illustrate our theoretical results.

Authors:Paolo Albano; Piermarco Cannarsa; Teresa Scarinci Pages: 172 - 176 Abstract: Publication date: February 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 2 Author(s): Paolo Albano, Piermarco Cannarsa, Teresa Scarinci On a bounded domain Ω in the Euclidean space R n , we study the homogeneous Dirichlet problem for the eikonal equation associated with a system of smooth vector fields, which satisfies Hörmander's bracket generating condition. We prove that the solution is smooth in the complement of a closed set of Lebesgue measure zero.

Authors:Wei-Guo Foo; Joël Merker; The-Anh Ta Pages: 214 - 221 Abstract: Publication date: February 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 2 Author(s): Wei-Guo Foo, Joël Merker, The-Anh Ta For every real numbers a ⩾ 1 , b ⩾ 1 with ( a , b ) ≠ ( 1 , 1 ) , the curve parametrized by θ ∈ R valued in C 2 ≅ R 4 γ : θ ⟼ ( x ( θ ) + − 1 y ( θ ) , u ( θ ) + − 1 v ( θ ) ) with components: x ( θ ) : = a − 1 a ( a b − 1 ) cos θ , y ( θ ) : = b ( a − 1 ) a b − 1 sin θ , u ( θ ) : = b − 1 b ( a b − 1 ) sin θ , v ( θ ) : = − a ( b − 1 ) a b − 1 cos θ , has image contained in the CR-umbilical locus: γ ( R ) ⊂ UmbCR ( E a , b ) ⊂ E a , b of the ellipsoid E a , b ⊂ C 2 PubDate: 2018-02-26T04:55:22Z DOI: 10.1016/j.crma.2017.11.019

Authors:Mattia Pujia; Luigi Vezzoni Pages: 222 - 226 Abstract: Publication date: February 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 2 Author(s): Mattia Pujia, Luigi Vezzoni In this note, we observe that, on a 2-step nilpotent Lie group equipped with a left-invariant SKT structure, the ( 1 , 1 ) -part of the Bismut–Ricci form is seminegative definite. As an application, we give a simplified proof of the non-existence of invariant SKT static metrics on 2-step nilmanifolds and of the existence of a long-time solution to the pluriclosed flow in 2-step nilmanifolds.

Authors:Philippe G. Ciarlet; Cristinel Mardare Pages: 227 - 234 Abstract: Publication date: Available online 5 January 2018 Source:Comptes Rendus Mathematique Author(s): Philippe G. Ciarlet, Cristinel Mardare We define a new two-dimensional nonlinear shell model “of Koiter's type” that can be used for the modeling of any type of shell and boundary conditions and for which we establish an existence theorem. The model uses a specific three-dimensional stored energy function of Ogden's type that satisfies all the assumptions of John Ball's fundamental existence theorem in three-dimensional nonlinear elasticity and that is adapted here to the modeling of thin nonlinearly elastic shells by means of specific deformations that are quadratic with respect to the transverse variable.

Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Ce Xu Recently, Dil and Boyadzhiev [10] proved an explicit formula for the sum of multiple harmonic numbers whose indices are the sequence ( { 0 } r , 1 ) . In this paper, we show that the sums of multiple harmonic numbers whose indices are the sequence ( { 0 } r , 1 ; { 1 } k − 1 ) can be expressed in terms of (multiple) zeta values, (multiple) harmonic numbers, and Stirling numbers of the first kind, and give an explicit formula.

Abstract: Publication date: March 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 3 Author(s): Müfit Şan The first aim of this work is to establish a Peano-type existence theorem for an initial value problem involving a complex fractional derivative, and then, as a consequence of this theorem, to give a partial answer for the local existence of the continuous solution to the initial value problem: { D x q u ( x ) = f ( x , u ( x ) ) , u ( 0 ) = b , ( b ≠ 0 ) . Moreover, for some special cases of the problem, we investigate the corresponding geometric properties of the solutions.

Authors:Mohammad F. Tehrani; Mark McLean; Aleksey Zinger Abstract: Publication date: Available online 23 February 2018 Source:Comptes Rendus Mathematique Author(s): Mohammad F. Tehrani, Mark McLean, Aleksey Zinger We overview our work [7–11,6] defining and studying normal crossings varieties and subvarieties in symplectic topology. This work answers a question of Gromov on the feasibility of introducing singular (sub)varieties into symplectic topology in the case of normal crossings singularities. It also provides a necessary and sufficient condition for smoothing normal crossings symplectic varieties. In addition, we explain some connections with other areas of mathematics and discuss a few directions for further research.

Authors:Shimon Garti Abstract: Publication date: Available online 23 February 2018 Source:Comptes Rendus Mathematique Author(s): Shimon Garti We prove the consistency of ♣ with the negation of Galvin's property. On the other hand, we show that superclub implies Galvin's property. We also prove the consistency of ♣ κ + with s κ > κ + for a supercompact cardinal κ.

Authors:Mamoru Nunokawa; Janusz Sokół; Derek K. Thomas Abstract: Publication date: Available online 23 February 2018 Source:Comptes Rendus Mathematique Author(s): Mamoru Nunokawa, Janusz Sokół, Derek K. Thomas Let f be analytic in a convex domain D ⊂ C . A well-known theorem of Ozaki states that if f is analytic in D, and is given by f ( z ) = z p + ∑ n = p + 1 ∞ a n z n for z ∈ D , and Re { e i α f ( p ) ( z ) } > 0 , ( z ∈ D ) , for some real α, then f is at most p-valent in D. Ozaki's condition is a generalization of the well-known Noshiro–Warschawski univalence condition. The purpose of this paper is to provide some related sufficient conditions for functions analytic in the unit disk D = { z ∈ C : z < 1 } to be p-valent in D , and to give an improvement to Ozaki's sufficient condition for p-valence when z ∈ D .

Authors:Wojciech Abstract: Publication date: February 2018 Source:Comptes Rendus Mathematique, Volume 356, Issue 2 Author(s): Wojciech S. Ożański We extend the generalised comparison principle for the Monge–Ampère equation due to Rauch & Taylor (1977) [15] to nonconvex domains. From the generalised comparison principle, we deduce bounds (from above and below) on solutions to the Monge–Ampère equation with sign-changing right-hand side. As a consequence, if the right-hand side is nonpositive (and does not vanish almost everywhere), then the equation equipped with a constant boundary condition has no solutions. In particular, due to a connection between the two-dimensional Navier–Stokes equations and the Monge–Ampère equation, the pressure p in 2D Navier–Stokes equations on a bounded domain cannot satisfy Δ p ≤ 0 in Ω unless Δ p ≡ 0 (at any fixed time). As a result, at any time t > 0 there exists z ∈ Ω such that Δ p ( z , t ) = 0 .

Authors:Brezis Abstract: Publication date: Available online 5 January 2018 Source:Comptes Rendus Mathematique Author(s): Haïm Brezis In Optimal Transport theory, three quantities play a central role: the minimal cost of transport, originally introduced by Monge, its relaxed version introduced by Kantorovich, and a dual formulation also due to Kantorovich. The goal of this Note is to publicize a very elementary, self-contained argument extracted from [9], which shows that all three quantities coincide in the discrete case.

Authors:Quan-Hui Yang; Qing-Qing Zhao Abstract: Publication date: Available online 6 December 2017 Source:Comptes Rendus Mathematique Author(s): Quan-Hui Yang, Qing-Qing Zhao Let q be a positive integer. Recently, Niu and Liu proved that, if n ≥ max { q , 1198 − q } , then the product ( 1 3 + q 3 ) ( 2 3 + q 3 ) ⋯ ( n 3 + q 3 ) is not a powerful number. In this note, we prove (1) that, for any odd prime power ℓ and n ≥ max { q , 11 − q } , the product ( 1 ℓ + q ℓ ) ( 2 ℓ + q ℓ ) ⋯ ( n ℓ + q ℓ ) is not a powerful number, and (2) that, for any positive odd integer ℓ, there exists an integer N q , ℓ such that, for any positive integer n ≥ N q , ℓ , the product ( 1 ℓ + q ℓ ) ( 2 ℓ + q ℓ ) ⋯ ( n ℓ + q ℓ ) is not a powerful number.

Authors:Anshul Adve; Alexander Yong Abstract: Publication date: Available online 6 December 2017 Source:Comptes Rendus Mathematique Author(s): Anshul Adve, Alexander Yong We discuss implications of the following statement about representation theory of symmetric groups: every integer appears infinitely often as an irreducible character evaluation and every nonnegative integer appears infinitely often as a Littlewood–Richardson coefficient and as a Kronecker coefficient.

Authors:Ivica Martinjak; Riste Abstract: Publication date: Available online 6 December 2017 Source:Comptes Rendus Mathematique Author(s): Ivica Martinjak, Riste Škrekovski We provide a combinatorial interpretation of Lah numbers by means of planar networks. Henceforth, as a consequence of Lindström's lemma, we conclude that the related Lah matrix possesses a remarkable property of total non-negativity.

Authors:Cornelius Greither; Toufik Abstract: Publication date: Available online 6 December 2017 Source:Comptes Rendus Mathematique Author(s): Cornelius Greither, Toufik Zaïmi Let K be a noncyclotomic CM field. We show that the field K ∩ R has a reciprocal unit-primitive element when K does. Also, we prove some related conditions that make the converse of this assertion true.