Algorithmic Differentiation (AD) provides the analytic derivatives of functions given as programs. Adjoint AD, which computes gradients, is similar to Back Propagation for Machine Learning. AD researchers study strategies to overcome the difficulties of adjoint AD, to get closer to its theoretical efficiency. To promote fruitful exchanges between Back Propagation and adjoint AD, we present thre...

The non self-adjointness of the radial momentum operator has been noted before by several authors, but the various proofs are incorrect. We give a rigorous proof that the n-dimensional radial momentum operator is not self-adjoint and has no selfadjoint extensions. The main idea of the proof is to show that this operator is unitarily equivalent to the momentum operator on L[(0,∞), dr] which is n...

The focus of CFD applications has shifted to aerodynamic design. This shift has been mainly motivated by the availability of high performance computing platforms and by the development of new and efficient analysis and design algorithms. In particular automatic design procedures, which use CFD combined with gradient-based optimization techniques, have had a significant impact on the design proc...

Introduction. In classical algebraic geometry the adjoint curves to an irreducible plane curve are an essential tool in the study of the geometry on the curve. In this paper we shall give an algebro-arithmetic development of the theory of adjoint curves, and shall extend the classical results to irreducible plane curves with arbitrary singularities defined over arbitrary ground fields. Our defi...

New formulas on the inverse problem for the continuous skewself-adjoint Dirac type system are obtained. For the discrete skewself-adjoint Dirac type system the solution of a general type inverse spectral problem is also derived in terms of the Weyl functions. The description of the Weyl functions on the interval is given. BorgMarchenko type uniqueness theorems are derived for both discrete and ...

One-dimensional δ ′ -function potential is discussed in the framework of Green’s function formalism without invoking perturbation expansion. It is shown that the energy-dependent Green’s function for this case is crucially dependent on the boundary conditions which are provided by self-adjoint extension method. The most general Green’s function which contains four real self-adjoint extension pa...

For cylindrically symmetric functions dyadically supported on the paraboloid, we obtain a family of sharp linear and bilinear adjoint restriction estimates. As corollaries, we first extend the ranges of exponents for the classical linear or bilinear adjoint restriction conjectures for such functions and verify the linear adjoint restriction conjecture for the paraboloid. We also interpret the r...

For any even positive integer 2n and any positive integermwe construct a class of regular self-adjoint and non-self-adjoint boundary value problems whose spectrum consists of at most (2n − 1)m + 1 eigenvalues. Our main result reduces to previously known results for the cases n = 1 and n = 2. In the self-adjoint case with separated boundary conditions this upper bound can be improved to n(m + 1)...

S U M M A R Y We compare finite-frequency phase and amplitude sensitivity kernels calculated based on frequency-domain surface wave mode summation and a time-domain adjoint method. The adjoint calculations involve a forward wavefield generated by an earthquake and an adjoint wavefield generated at a seismic receiver. We determine adjoint sources corresponding to frequency-dependent phase and am...

A differential operator on a directed graph with weighted edges is characterized as a system of ordinary differential operators. A class of local operators is introduced to clarify which operators should be considered as defined on the graph. When the edge lengths have a positive lower bound, all local self-adjoint extensions of the minimal symmetric operator may be classified by boundary condi...