In this paper, we propose a general existence theorem for a multi-valued control problem. The proof of the theorem is based on a decomposition result of the weak ⋆ closure of the set containing all the multi-valued controls and the bathtub principle. We also obtain the optimality condition for the optimal control.

de recherche Algebra, Representation theory / Algèbre, Théorie des représentations Calabi-Yau structures on Drinfeld quotients and Amiot's conjecture Structures Calabi-Yau sur les quotients de Drinfeld et la conjecture d'Amiot

We provide the first solution to the problem of counting rooted 3-connected bipartite planar maps. Our starting point is the enumeration of bicoloured planar maps according to the number of edges and monochromatic edges, following Bernardi and Bousquet-Mélou [ J. Comb. Theory Ser. B , 101 (2011), 315–377]. The decomposition of a map into 2- and 3-connected components allows us to obtain the generating functions of 2- and 3-connected bicoloured maps. Setting to zero the variable marking monochromatic edges we obtain the generating function of 3-connected bipartite maps, which is algebraic of degree 26. We deduce from it an asymptotic estimate for the number of 3-connected bipartite planar maps of the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mspace width="0.166667em"/> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>5</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:msup> <mml:mi>γ</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>≈</mml:mo> <mml:mn>2</mml:mn> <mml:mo>.</mml:mo> <mml:mn>40958</mml:mn> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ρ</mml:mi> <mml:mo>≈</mml:mo> <mml:mn>0</mml:mn> <mml:mo>.</mml:mo> <mml:mn>41501</mml:mn> </mml:mrow> </mml:math> is an algebraic number of degree <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>10</mml:mn> </mml:math> .

Let A be a nonempty finite set of integers. For a real number m, the set m·A={ma:a∈A} denotes the set of m-dilates of A. In 2008, Bukh initiated an interesting problem of finding a lower bound for the sumset of dilated sets, i.e., a lower bound for |λ 1 ·A+λ 2 ·A+⋯+λ h ·A|, where λ 1 ,λ 2 ,⋯,λ h are integers and A be a subset of integers. In particular, for nonempty finite subsets A and B, the problem of dilates of A and B is defined as A+k·B={a+kb:a∈A and b∈B}. In this article, we obtain the lower bound for the cardinality of A+k·B with k≥3 and describe sets for which equality holds. We also derive an extended inverse result with some conditions for the sumset A+3·B.

Let A and H be nonempty finite sets of integers and positive integers, respectively. The generalized H -fold sumset, denoted by H (r ) A, is the union of the sumsets h (r ) A for h ∈ H where, the sumset h (r ) A is the set of all integers that can be represented as a sum of h elements from A with no summand in the representation appearing more than r times. In this paper, we find the optimal lower bound for the cardinality of H (r ) A, i.e., for |H (r ) A| and the structure of the underlying sets A and H when |H (r ) A| is equal to the optimal lower bound in the cases A contains only positive integers and A contains only nonnegative integers. This generalizes recent results of Bhanja. Furthermore, with a particular set H , since H (r ) A generalizes subsequence sum and hence subset sum, we get several results of subsequence sums and subset sums as special cases.

We prove that a transversely product component of the singular set of a holomorphic foliation on P n is necessarily a Kupka component.

Thanks to a finite element method, we solve numerically parabolic partial differential equations on complex domains by avoiding the mesh generation, using a regular background mesh, not fitting the domain and its real boundary exactly. Our technique follows the ϕ-FEM paradigm, which supposes that the domain is given by a level-set function. In this paper, we prove a priori error estimates in l 2 (H 1 ) and l ∞ (L 2 ) norms for an implicit Euler discretization in time. We give numerical illustrations to highlight the performances of ϕ-FEM, which combines optimal convergence accuracy, easy implementation process and fastness.

We study nonlocal conservation laws with a discontinuous flux function of regularity L ∞ (ℝ) in the spatial variable and show existence and uniqueness of weak solutions in C[0,T] ; L loc 1 , as well as related maximum principles. We achieve this well-posedness by a proper reformulation in terms of a fixed-point problem. This fixed-point problem itself necessitates the study of existence, uniqueness and stability of a class of discontinuous ordinary differential equations. On the ODE level, we compare the solution type defined here with the well-known Carathéodory and Filippov solutions.

We prove that the spectrum of the two-photon quantum Rabi Hamiltonian consists of two eigenvalue sequences (E m + ) m=0 ∞ , (E m - ) m=0 ∞ satisfying a three-term asymptotic formula with the remainder estimate O(m -1 lnm) when m tends to infinity. By analogy to the one-photon quantum Rabi model, the leading three terms of this asymptotic formula, describe a generalized rotating-wave approximation for large eigenvalues of the two-photon quantum Rabi model.

We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits of points in a generalised Hecke orbit in terms of this height function, assuming the "weakly adelic Mumford-Tate hypothesis" and prove the generalised André-Pink-Zannier conjecture (a special case of the Zilber-Pink conjecture) under this assumption using the Pila-Zannier strategy. Résumé. On introduit et étudie la notion d'orbite de Hecke généralisée dans une variété de Shimura. On définit une notion de hauteur sur une telle orbite et étudie ses properiétés. On obtient une borne inférieure pour la taille des orbites Galoisiennes de points dans une orbite de Hecke généralisee en termes de cette fonction hauteur en admettant « la conjecture de Mumford-Tate faiblement adélique » et on démontre la conjecture de André-Pink-Zannier généralisée (un cas particulier de la conjecture de Zilber-Pink) en utilisant la stratégie de Pila-Zannier.