Remarks on the integer Talmud solution for integer bankruptcy problems

In Fragnelli et al. (TOP 22:892–933, 2014; TOP 24:88–130, 2016), we considered a bankruptcy problem with the additional constraint that the estate has to be assigned in integer unities, allowing for non-integer claims; we dealt with the extension to our setting of the constrained equal losses solution and of the constrained equal awards solution. Here, we analyze the possibilities of extending the Talmud solution to the integer situation, starting from the existing approaches for the non-integer case; some of these approaches are compatible with the non-integer claims, but in order to comply with as much as possible of the approaches it is necessary to switch to integer claims.     

2D constrained Navier–Stokes equations

We study 2D Navier–Stokes equations with a constraint forcing the conservation of the energy of the solution. We prove the existence and uniqueness of a global solution for the constrained Navier–Stokes equation on ${\mathbb{R}}^2$ and ${\mathbb{T}}^2$, by a fixed point argument. We also show that the solution of the constrained equation converges to the solution of the Euler equation as the viscosity ν vanishes.     

On solutions of a boundary value problem for the biharmonic equation

We study the uniqueness of the solution of a boundary value problem for the biharmonic equation in unbounded domains under the assumption that the generalized solution of this problem has a bounded Dirichlet integral with weight |x|^a. Depending on the value of the parameter a, we prove uniqueness theorems or present exact formulas for the dimension of the solution space of this problem in the exterior of a compact set and in a half-space.     

Approximation of Solutions of Fractional-Order Delayed Cellular Neural Network on $$\varvec{[0,\infty )}$$

In this paper, we study a class of fractional-order cellular neural network containing delay. We prove the existence and uniqueness of the equilibrium solution followed by boundedness. Based on the theory of fractional calculus, we approximate the solution of the corresponding neural network model over the interval \([0,\infty )\) using discretization method with piecewise constant arguments and variation of constants formula for fractional differential equations. Furthermore, we conclude that the solution of the fractional-delayed system can be approximated for large t by the solution of the equation with piecewise constant arguments, if the corresponding linear system is exponentially stable. At the end, we give two numerical examples to validate our theoretical findings.     

Oil displacement by water and gas in a porous medium: the Riemann problem

In this work we present the construction of the Riemann solution for a system of two conservation laws representing displacement in immiscible three-phase flow. The porousmedium is initially filled with oil and small amounts of water and gas; then a fixed proportion of water and gas is injected. We use the wave curve method to determine the wave sequences in the Riemann solution for arbitrary initial and injection data in the above mentioned class. We show the L_Loc¹-stability of the Riemann solution with variation of data. We do not verify uniqueness of the Riemann solution, but we believe that it is valid.     

Uniqueness of almost periodic-in-time solutions to Navier�CStokes equations in unbounded domains

We present a uniqueness theorem for almost periodic-in-time solutions to the Navier?CStokes equations in 3-dimensional unbounded domains. Thus far, uniqueness of almost periodic-in-time solutions to the Navier?CStokes equations in unbounded domain, roughly speaking, is known only for a small almost periodic-in-time solution in ${BC(\mathbb {R};L^{3}_w)}$ within the class of solutions that have sufficiently small ${L^{\infty}(L^{3}_w)}$ -norm. In this paper, we show that a small almost periodic-in-time solution in ${BC(\mathbb {R};L^{3}_w\cap L^{6,2})}$ is unique within the class of all almost periodic-in-time solutions in ${BC(\mathbb {R};L^{3}_w\cap L^{6,2})}$ . The proof of the present uniqueness theorem is based on the method of dual equations.     

On the asymptotic analysis of the Boltzmann equation tending towards the Stokes equations

This work deals with the analysis of the asymptotic limit for the Boltzmann equation tending towards the linearized Navier–Stokes equations when the Knudsen number $\epsilon$ tends to zero. Global existence and uniqueness theorems are proven for regular initial fluctuations. As $\epsilon$ tends to zero, the solution converges strongly to the solution of the linearized Navier–Stokes systems.     

Nonlocal Dezin’s problem for Lavrent’ev–Bitsadze equation

Using the method of spectral analysis, for the mixed type equation u_xx + (sgny)u_yy = 0 in a rectangular domain we establish a criterion of uniqueness of its solution satisfying periodicity conditions by the variable x, a nonlocal condition, and a boundary condition. The solution is constructed as the sum of a series in eigenfunctions for the corresponding one-dimensional spectral problem. At the investigation of convergence of the series, the problem of small denominators occurs. Under certain restrictions on the parameters of the problem and the functions, included in the boundary conditions, we prove uniform convergence of the constructed series and stability of the solution under perturbations of these functions.     

The Helmholtz equation with impedance in a half-space

In this Note we obtain existence and uniqueness results for the Helmholtz equation in the half-space ${\mathbb{R}}_{+}^3$ with an impedance or Robin boundary condition. Basically, we follow the procedure we have already used in the bi-dimensional case (the half-plane). Thus, we compute the associated Green's function with the help of a double Fourier transform and we analyze its far field in order to obtain radiation conditions that allow us to prove the uniqueness of an outgoing solution. Again, these radiation conditions are somewhat unusual due to the appearance of a surface wave guided by the boundary. An integral representation of the solution is presented by means of the Green's function and the boundary data. To cite this article: M. Durán et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Tsirel'son's equation in discrete time

Summary Motivated by Tsirel'son's equation in continuous time, a similar stochastic equation indexed by discrete negative time is discussed in full generality, in terms of the law of a discrete time noise. When uniqueness in law holds, the unique solution (in law) is not strong; moreover, when there exists a strong solution, there are several strong solution. In general, for any time,n, the -field generated by the past of a solution up to timen is shown to be equal, up to negligible sets, to the -field generated by the 3 following components: the infinitely remote past of the solution, the past to the noise up to timen, together with an adequate independent complement.     

Constrained and unconstrained rearrangement minimization problems related to the <Emphasis Type="Italic">p</Emphasis>-Laplace operator

In this paper we consider an unconstrained and a constrained minimization problem related to the boundary value problem

$$ - {\Delta _p}u = f{\text{ in }}D,{\text{ }}u = 0{\text{ on }}\partial D$$

. In the unconstrained problem we minimize an energy functional relative to a rearrangement class, and prove existence of a unique solution. We also consider the case when D is a planar disk and show that the minimizer is radial and increasing. In the constrained problem we minimize the energy functional relative to the intersection of a rearrangement class with an affine subspace of codimension one in an appropriate function space. We briefly discuss our motivation for studying the constrained minimization problem.

     

On solutions of the mixed Dirichlet-Steklov problem for the biharmonic equation in exterior domains

We study the unique solvability of the mixed Dirichlet-Steklov problem for the biharmonic equation in exterior domains under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight |x| ^a . Depending on the value of the parameter a, we prove uniqueness theorem or present exact formulas for the dimension of the solution space of the mixed Dirichlet-Steklov problem in the exterior of a compact set.     

Backward stochastic differential equations with reflection and weak assumptions on the coefficients

In this paper, we study reflected BSDE’s with one continuous barrier, under monotonicity and general increasing conditions in y$y$ and non-Lipschitz conditions in z$z$. We prove the existence and uniqueness of a solution by an approximation method.     

On a Nonstationary Model of a Catalytic Process in a Fluidized Bed

In the half-strip 0 ≤ x ≤ h, t ≤ 0 we consider a mixed problem for an almost linear system of three first order PDEs, one of which does not involve derivatives with respect to t. We prove the existence and uniqueness of a generalized Holder continuous solution and generalized piecewise smooth and smooth solutions. For the piecewise smooth solution we prove the stabilization of some functionals as t → ∞.     

Global modified Hamiltonian for constrained symplectic integrators

Summary. We prove that the numerical solution of partitioned Runge-Kutta methods applied to constrained Hamiltonian systems (e.g., the Rattle algorithm or the Lobatto IIIA–IIIB pair) is formally equal to the exact solution of a constrained Hamiltonian system with a globally defined modified Hamiltonian. This property is essential for a better understanding of their longtime behaviour. As an illustration, the equations of motion of an unsymmetric top are solved using a parameterization with Euler parameters. Mathematics Subject Classification (2000):65L06, 65L80, 65P10     

A constrained optimization reformulation and a feasible descent direction method for L_{1/2} regularization

In this paper, we first propose a constrained optimization reformulation to the \(L_{1/2}\) regularization problem. The constrained problem is to minimize a smooth function subject to some quadratic constraints and nonnegative constraints. A good property of the constrained problem is that at any feasible point, the set of all feasible directions coincides with the set of all linearized feasible directions. Consequently, the KKT point always exists. Moreover, we will show that the KKT points are the same as the stationary points of the \(L_{1/2}\) regularization problem. Based on the constrained optimization reformulation, we propose a feasible descent direction method called feasible steepest descent method for solving the unconstrained \(L_{1/2}\) regularization problem. It is an extension of the steepest descent method for solving smooth unconstrained optimization problem. The feasible steepest descent direction has an explicit expression and the method is easy to implement. Under very mild conditions, we show that the proposed method is globally convergent. We apply the proposed method to solve some practical problems arising from compressed sensing. The results show its efficiency.