We introduce a notion of reduction on resource vectors, i.e. infinite linear combinations of resource λ-terms. The latter form the multilinear fragment of the differential λ-calculus introduced by Ehrhard and Regnier, and resource vectors are the target of the Taylor expansion of λ-terms. We show that the reduction of resource vectors contains the image, through Taylor expansion, of β-reduction...

Nonlinear optimization problems in complex variables are frequently encountered in applied mathematics and engineering applications such as control theory, signal processing and electrical engineering. Optimization of these problems often requires a firstor second-order approximation of the objective function to generate a new step or descent direction. However, such methods cannot be applied t...

We present recent lattice results on QCD thermodynamics at nonvanishing baryon number density obtained from a 6th order Taylor expansion in the chemical potential. Results for bulk thermodynamic observables, in particular for fluctuations in the baryon number density, are found to be well described by a hadron resonance gas model at low temperature and an ideal quark gluon gas at high temperatu...

Taylor series methods compute a solution to an initial value problem in ordinary differential equations by expanding each component of the solution in a long Taylor series. The series terms are generated recursively using the techniques of automatic differentiation. The ATOMFT system includes a translator to transform statements of the system of ODES into a FORTRAN 77 object program that is com...

In this paper, we propose wavelet Taylor–Galerkin schemes for parabolic and hyperbolic PDEs taking full advantage of the compression properties of wavelet basis. The discretization in time is performed before the spatial discretization by introducing high-order generalization of the standard time-stepping schemes with the help of Taylor series expansion in time step. Then, we present numerical ...

We introduce the concept of three-step wavelet-Galerkin method based on the Taylor series expansion in time. Unlike the Taylor–Galerkin methods, the present scheme does not contain any new higher-order derivatives which makes it suitable for solving nonlinear problems. Numerical schemes taking advantage of the wavelet bases capabilities to compress the operators and sparse representation of fun...

A three-step wavelet Galerkin method based on Taylor series expansion in time is proposed. The scheme is third-order accurate in time and O(2−jp) accurate in space. Unlike Taylor–Galerkin methods, the present scheme does not contain any new higher-order derivatives which makes it suitable for solving non-linear problems. The compactly supported orthogonal wavelet bases D6 developed by Daubechie...

In this paper, an R-analytical function and the sequence of its Taylor polynomials (which are Lyapunov functions different from those of Vanelli & Vidyasagar (1985, Automatica, 21(1):6 9–80)) is presented, in order to determine and approximate the domain of attraction of the exponentially asymptotically stable zero steady state of an autonomous, R-analytical system of differential equations. Th...

We provide results for redesign of Lyapunov function based continuous time controllers for sampled-data implementation, using a particular form of the redesigned controller and the Taylor expansion of the sampled-data Lyapunov difference. We develop two types of redesigned controllers that (i) make the lower order terms (in T ) in the series expansion of the Lyapunov difference with the redesig...

We construct an invariant JM of integral homology spheres M with values in a completion Ẑ[q] of the polynomial ring Z[q] such that the evaluation at each root of unity ζ gives the the SU(2) Witten-ReshetikhinTuraev invariant τζ(M) of M at ζ. Thus JM unifies all the SU(2) WittenReshetikhin-Turaev invariants of M . As a consequence, τζ(M) is an algebraic integer. Moreover, it follows that τζ(M) a...