On Estimation of Temperature Uncertainty Using the Second Order Adjoint Problem

The uncertainty of temperature prediction from the heat flux error is estimated using first and second order adjoint equations. The adjoint codes developed for the inverse heat transfer problems provide the uncertainty estimation for the corresponding forward problems. Numerical tests corroborate the feasibility of fast uncertainty estimation using Hessian maximum eigenvalue obtained via second order adjoint equations.     

Multi-adjoint algebras versus non-commutative residuated structures

Adjoint triples and pairs are basic operators used in several domains, since they increase the flexibility in the framework in which they are considered. This paper introduces multi-adjoint algebras and several properties; also, we will show that an adjoint triple and its “dual” cannot be considered in the same framework.Moreover, a comparison among general algebraic structures used in different frameworks, which reduce the considered mathematical requirements, such as the implicative extended-order algebras, implicative structures, the residuated algebras given by sup-preserving aggregations and the conjunctive algebras given by semi-uninorms and u-norms, is presented. This comparison shows that multi-adjoint algebras generalize these structures in domains which require residuated implications, such as in formal concept analysis, fuzzy rough sets, fuzzy relation equations and fuzzy logic.     

Relationship between different approaches to derive weighting functions related to atmospheric remote sensing problems

The paper is devoted to the investigation of the relationship between different methods used to derive weighting functions required to solve numerous inverse problems related to the remote sensing of the Earth's atmosphere by means of scattered solar light observations. The first method commonly referred to as the forward-adjoint approach is based on a joint solution of the forward and adjoint radiative transfer equations and the second one requires the linearized forward radiative transfer equation to be solved. In the framework of the forward-adjoint method we consider two approaches commonly used to derive the weighting functions. These approaches are referenced as the “response function” and the “formal solution” techniques, respectively. We demonstrate here that the weighting functions derived employing the formal solution technique can also be obtained substituting the analytical representations for the direct forward and direct adjoint intensities into corresponding expressions obtained in the framework of the response function technique. The advantages and disadvantages of different techniques are discussed.     

Adjoints of rationally induced composition operators

We give an elementary proof of a formula recently obtained by Hammond, Moorhouse, and Robbins for the adjoint of a rationally induced composition operator on the Hardy space H² [Christopher Hammond, Jennifer Moorhouse, Marian E. Robbins, Adjoints of composition operators with rational symbol, J. Math. Anal. Appl. 341 (2008) 626-639]. We discuss some variants and implications of this formula, and use it to provide a sufficient condition for a rationally induced composition operator adjoint to be a compact perturbation of a weighted composition operator.     

Convergence of Distributed Optimal Controls on the Internal Energy in Mixed Elliptic Problems when the Heat Transfer Coefficient Goes to Infinity

We consider a steady-state heat conduction problem P _α with mixed boundary conditions for the Poisson equation depending on a positive parameter α , which represents the heat transfer coefficient on a portion Γ ₁ of the boundary of a given bounded domain in R ⁿ . We formulate distributed optimal control problems over the internal energy g for each α . We prove that the optimal control g_ op _α and its corresponding system u_ g_ op _α α and adjoint p_ g_ op _α α states for each α are strongly convergent to g _op , u_ g _op and p _ g _op , respectively, in adequate functional spaces. We also prove that these limit functions are respectively the optimal control, and the system and adjoint states corresponding to another distributed optimal control problem for the same Poisson equation with a different boundary condition on the portion Γ ₁ . We use the fixed point and elliptic variational inequality theories.     

Refractive index imaging from radiative transfer equation-based reconstruction algorithm: Fundamentals

We present the first reconstruction algorithm for refractive index imaging, which is based on the radiative transfer equation (RTE). An objective function is iteratively minimized to find a solution to the problem of inversion of the refractive index field. The function describes the discrepancies of the emerging light measurements on the surface of the sample to be probed with predicted data from the corresponding numerical model. The unknown refractive index field is updated within each reconstruction iteration according to a search direction on the index distribution given by the adjoint model to the RTE. In this paper, emphasis is placed on the theoretical aspects. Preliminary tests are demonstrated on generic phantoms.     

Monotonicity of input-output mappings in inverse coefficient and source problems for parabolic equations

This article presents a mathematical analysis of input-output mappings in inverse coefficient and source problems for the linear parabolic equation ut=(kx(x)ux)+F(x,t), (x,t)∈ΩT:=(0,1)×(0,T]. The most experimentally feasible boundary measured data, the Neumann output (flux) data f(t):=−k(0)ux(0,t), is used at the boundary x=0. For each inverse problems structure of the input-output mappings is analyzed based on maximum principle and corresponding adjoint problems. Derived integral identities between the solutions of forward problems and corresponding adjoint problems, permit one to prove the monotonicity and invertibility of the input-output mappings. Some numerical applications are presented.     

Broer-Kaup系统的达布变换及其孤子解

根据Broer-Kaup系统的Lax对, 借助Broer-Kaup系统的谱问题的规范变换, 一个包含多参数的达布变换被构造出. 以一个平凡解作为种子解, 利用达布变换, 可以求得Broer-Kaup系统的非平凡解的一般表达式. 并且讨论了N=1和N=2两种孤子解的情形. 这是一种与2X2谱问题有关的孤子碰撞图像的新类型.     

Adjoint polynomials of bridge–path and bridge–cycle graphs and Chebyshev polynomials

The chromatic polynomial of a simple graph G with n>0 vertices is a polynomial of degree n, where α_k(G) is the number of k-independent partitions of G for all k. The adjoint polynomial of G is defined to be , where is the complement of G. We find explicit formulas for the adjoint polynomials of the bridge–path and bridge–cycle graphs. Consequence, we find the zeros of the adjoint polynomials of several families of graphs.     

A bijection between the adjoint and coadjoint orbits of a semidirect product

We prove that there exists a geometric bijection between the sets of adjoint and coadjoint orbits of a semidirect product, provided a similar bijection holds for particular subgroups. We also show that under certain conditions the homotopy types of any two orbits in bijection with each other are the same. We apply our theory to the examples of the affine group and the Poincaré group, and discuss the limitations and extent of this result to other groups.