Global dynamics of delayed reaction–diffusion equations in unbounded domains

We consider a nonlocal delayed reaction–diffusion equation in an unbounded domain that includes some special cases arising from population dynamics. Due to the non-compactness of the spatial domain, the solution semiflow is not compact. We first show that, with respect to the compact open topology for the natural phase space, the solutions induce a compact and continuous semiflow ${\Phi}$ on a bounded and positively invariant set Y in C ₊?=?C([?1, 0], X ₊) that attracts every solution of the equation, where X ₊ is the set of all bounded and uniformly continuous functions from ${\mathbb{R}}$ to [0, ∞). Then, to overcome the difficulty in describing the global dynamics, we establish a priori estimate for nontrivial solutions after describing the delicate asymptotic properties of the nonlocal delayed effect and the diffusion operator. The estimate enables us to show the permanence of the equation with respect to the compact open topology. With the help of the permanence, we can employ standard dynamical system theoretical arguments to establish the global attractivity of the nontrivial equilibrium. The main results are illustrated with the diffusive Nicholson’s blowfly equation and the diffusive Mackey–Glass equation.     

Construction of inclusions with vanishing generalized polarization tensors by imperfect interfaces

We investigate the problem of planar conductivity inclusion with imperfect interface conditions. We assume that the inclusion is simply connected. The presence of the inclusion causes a perturbation in the incident background field. This perturbation admits a multipole expansion of which coefficients we call as the generalized polarization tensors (GPTs), extending the previous terminology for inclusions with perfect interfaces. We derive explicit matrix expressions for the GPTs in terms of the incident field, material parameters, and geometry of the inclusion. As an application, we construct GPT-vanishing structures of general shape that result in negligible perturbations for all uniform incident fields. The structure consists of a simply connected core with an imperfect interface. We provide numerical examples of GPT-vanishing structures obtained by our proposed scheme.     

On the contrapositive of countable choice

We show that in elementary analysis (EL) the contrapositive of countable choice is equivalent to double negation elimination for ${\Sigma_{2}^{0}}$ -formulas. By also proving a recursive adaptation of this equivalence in Heyting arithmetic (HA), we give an instance of the conservativity of EL over HA with respect to recursive functions and predicates. As a complement, we prove in HA enriched with the (extended) Church thesis that every decidable predicate is recursive.     

Identity-based non-interactive key distribution with forward security

Identity-based non-interactive key distribution (ID-NIKD) is a cryptographic primitive that enables two users to establish a common secret key without exchanging messages. All users of the system have access to public system parameters and a private key, obtained through the help of a trusted key generation center. In this contribution, we discuss how to capture an intuitive form of forward security for ID-NIKD schemes in a security model. Building on results of Sakai et?al. as well as of Paterson and Srinivasan, we discuss how the proposed notion of forward security can be achieved in the random oracle model, using a Bilinear Diffie-Hellman assumption in combination with a forward-secure pseudorandom bit generator. We also show how a forward-secure ID-NIKD scheme can be used to realize forward-secure identity-based encryption.     

On the numerical approximations of stiff convection–diffusion equations in a circle

Our aim in this article is to study the numerical solutions of singularly perturbed convection–diffusion problems in a circular domain and provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a $P_1$ classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasi-uniform mesh.     

Approximation of analytical functions by k-positive linear operators in the closed domain

This work treats the problem of convergence for the sequences of linear \(k\) -positive operators on a space of functions that are analytic in a closed domain. By convergence in this space, we mean a uniform convergence in a closed domain that contains the original domain strictly inside itself, while the linear \(k\) -positive operators are naturally associated with Faber polynomials related to the considered domain. Until now, this problem has been solved in the space of functions analytic in an open bounded domain with the topology of compact convergence.     

The stable trees are nested

We show that we can construct simultaneously all the stable trees as a nested family. More precisely, if $1 < \alpha < \alpha ^{\prime } \le 2$ we prove that hidden inside any $\alpha $ -stable tree we can find a version of an $\alpha ^{\prime }$ -stable tree rescaled by an independent Mittag-Leffler type distribution. This tree can be explicitly constructed by a pruning procedure of the underlying stable tree or by a modification of the fragmentation associated with it. Our proofs are based on a recursive construction due to Marchal which is proved to converge almost surely towards a stable tree.     

Objective Bayesian analysis for a capture–recapture model

In this paper, we study a special capture–recapture model, the $M_t$ model, using objective Bayesian methods. The challenge is to find a justified objective prior for an unknown population size $N$ . We develop an asymptotic objective prior for the discrete parameter $N$ and the Jeffreys’ prior for the capture probabilities $\varvec{\theta }$ . Simulation studies are conducted and the results show that the reference prior has advantages over ad-hoc non-informative priors. In the end, two real data examples are presented.     

Self-adjoint curl operators

We study the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on a 3D bounded domain D. We aim to identify boundary conditions that render this operator self-adjoint. By the symplectic version of the Glazman-Krein-Naimark theorem, this amounts to identifying complete Lagrangian subspaces of the trace space of H(curl, D) equipped with a symplectic pairing arising from the ${\wedge}$ -product of 1-forms on ${\partial D}$ . Substantially generalizing earlier results, we characterize Lagrangian subspaces associated with closed and co-closed traces. In the case of non-trivial topology of the domain, different contributions from co-homology spaces also distinguish different self-adjoint extensions. Finally, all self-adjoint extensions discussed in the paper are shown to possess a discrete point spectrum, and their relationship with curl curl-operators is discussed.     

On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments

In this paper, we present a numerical scheme for a first-order hyperbolic equation of nonlinear type perturbed by a multiplicative noise. The problem is set in a bounded domain D of ${\mathbb{R}^{d}}$ and with homogeneous Dirichlet boundary condition. Using a time-splitting method, we are able to show the existence of an approximate solution. The result of convergence of such a sequence is based on the work of Bauzet–Vallet–Wittbold (J Funct Anal, 2013), where the authors used the concept of measure-valued solution and Kruzhkov’s entropy formulation to show the existence and uniqueness of the stochastic weak entropy solution. Then, we propose numerical experiments by applying this scheme to the stochastic Burgers’ equation in the one-dimensional case.     

A Stochastic Recursive Optimal Control Problem Under the G-expectation Framework

In this paper, we study a stochastic recursive optimal control problem in which the objective functional is described by the solution of a backward stochastic differential equation driven by \(G\) -Brownian motion. Under standard assumptions, we establish the dynamic programming principle and the related Hamilton–Jacobi–Bellman (HJB) equation in the framework of \(G\) -expectation. Finally, we show that the value function is the viscosity solution of the obtained HJB equation.     

Real quadrics in C n , complex manifolds and convex polytopes

In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics C ⁿ which are invariant with respect to the natural action of the real torus (S ¹) ⁿ onto C ⁿ . The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non-Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.     

Compactification de Chabauty de l’espace des sous-groupes de Cartan de {\text{ SL}}_n(\mathbb{R })

Let $G$ be a real semisimple Lie group with finite center, with a finite number of connected components and without compact factor. We are interested in the homogeneous space of Cartan subgroups of $G$ , which can be also seen as the space of maximal flats of the symmetric space of $G$ . We define its Chabauty compactification as the closure in the space of closed subgroups of $G$ , endowed with the Chabauty topology. We show that when the real rank of $G$ is 1, or when $G={\text{ SL}}_3(\mathbb{R })$ or ${\text{ SL}}_4(\mathbb{R })$ , this compactification is the set of all closed connected abelian subgroups of dimension the real rank of $G$ , with real spectrum. And in the case of ${\text{ SL}}_3(\mathbb{R })$ , we study its topology more closely and we show that it is simply connected.     

On the cohomology of moduli spaces of (weighted) stable rational curves

We give a recursive algorithm for computing the character of the cohomology of the moduli space ${\overline{M}}_{0,n}$ of stable $n$ -pointed genus zero curves as a representation of the symmetric group $\mathbb{S }_n$ on $n$ letters. Using the algorithm we can show a formula for the maximum length of this character. Our main tool is connected to the moduli spaces of weighted stable curves introduced by Hassett.     

A symmetry problem with applications to finance

In this paper we will consider an overdetermined problem in an unknown ring-shaped domain, and we prove that the domain has to be spherical ring. We will apply this problem to an \(n\) -dimensional toy model in the mean field game theory introduced by Lasry-Lions too. We show that if in Lasry-Lions model, the additional extra data ”the boundary is a level set” is assumed, then the region, where the solution is harmonic, has to have spherical symmetry.     

On Multigraded Generalizations of Kirillov–Reshetikhin Modules

We study the category of $\mathbb Z^\ell$ -graded modules with finite-dimensional graded pieces for certain $\mathbb Z^\ell_+$ -graded Lie algebras. We also consider certain Serre subcategories with finitely many isomorphism classes of simple objects. We construct projective resolutions for the simple modules in these categories and compute the Ext groups between simple modules. We show that the projective covers of the simple modules in these Serre subcategories can be regarded as multigraded generalizations of Kirillov–Reshetikhin modules and give a recursive formula for computing their graded characters.