scholarly journals Extensions and Smooth Approximations of Definable Functions in O-minimal Structures

2018 ◽  
Vol 24 (4) ◽  
pp. 459-460
Author(s):  
Athipat Thamrongthanyalak
Keyword(s):  
Definable Functions ◽  
Smooth Approximations

scholarly journals Aperiodic String Transducers

2018 ◽  
Vol 29 (05) ◽  
pp. 801-824
Author(s):  
Luc Dartois ◽  
Ismaël Jecker ◽  
Pierre-Alain Reynier
Keyword(s):  
Regular Languages ◽  
Definable Functions

Regular string-to-string functions enjoy a nice triple characterization through deterministic two-way transducers (2DFT), streaming string transducers (SST) and MSO definable functions. This result has recently been lifted to FO definable functions, with equivalent representations by means of aperiodic 2DFT and aperiodic 1-bounded SST, extending a well-known result on regular languages. In this paper, we give three direct transformations: [Formula: see text] from 1-bounded SST to 2DFT, [Formula: see text] from 2DFT to copyless SST, and [Formula: see text] from [Formula: see text]-bounded to [Formula: see text]-bounded SST. We give the complexity of each construction and also prove that they preserve the aperiodicity of transducers. As corollaries, we obtain that FO definable string-to-string functions are equivalent to SST whose transition monoid is finite and aperiodic, and to aperiodic copyless SST.

Fundamental Solutions of PDEs as Radial Basis Functions in Multivariate Interpolation

2007 ◽  
Vol 7 (4) ◽  
pp. 321-340
Author(s):  
A. Masjukov
Keyword(s):  
Radial Basis Functions ◽  
Interpolation Method ◽  
A Priori ◽  
Fundamental Solutions ◽  
Basis Functions ◽  
Scale Parameters ◽  
Radial Basis ◽  
Undecidable Problem ◽  
The Right ◽  
Smooth Approximations

AbstractFor bivariate and trivariate interpolation we propose in this paper a set of integrable radial basis functions (RBFs). These RBFs are found as fundamental solutions of appropriate PDEs and they are optimal in a special sense. The condition number of the interpolation matrices as well as the order of convergence of the inter- polation are estimated. Moreover, the proposed RBFs provide smooth approximations and approximate fulfillment of the interpolation conditions. This property allows us to avoid the undecidable problem of choosing the right scale parameter for the RBFs. Instead we propose an iterative procedure in which a sequence of improving approx- imations is obtained by means of a decreasing sequence of scale parameters in an a priori given range. The paper provides a few clear examples of the advantage of the proposed interpolation method.

Smooth approximations by continuous choice-functions

2021 ◽  
Author(s):  
Mihai Prunescu
Keyword(s):  
Error Bound ◽  
Common Sense ◽  
Choice Function ◽  
Continuous Functions ◽  
Ordered Sets ◽  
Choice Functions ◽  
Continuous Choice ◽  
Functions Of Two Variables ◽  
Smooth Approximations

Abstract We explore the existence of rational-valued approximation processes by continuous functions of two variables, such that the output continuously depends of the imposed error-bound. To this sake we prove that the theory of densely ordered sets with generic predicates is ℵ0- categorical. A model of the theory and a particular continuous choice-function are constructed. This function transfers to all other models by the respective isomorphisms. If some common-sense conditions are fulfilled, the processes are computable. As a byproduct, other functions with surprising properties can be constructed.

scholarly journals Zeno breaking, the Contact effect and sensitive behaviour in piecewise-linear systems

2018 ◽  
Vol 29 (5) ◽  
pp. 826-844
Author(s):  
R. EDWARDS
Keyword(s):  
Vector Fields ◽  
Infinite Sequence ◽  
Piecewise Linear ◽  
Need For Care ◽  
Time Intervals ◽  
Finite Time Convergence ◽  
Contact Effect ◽  
Qualitative Models ◽  
Pass Through ◽  
Smooth Approximations

Non-smooth approximations of steep sigmoidal switching networks, such as those used as qualitative models of gene regulation, lead to analytic and computational challenges that arise as a result of the discontinuities in the vector fields. In order to highlight the need for care in dealing with such systems, several particular phenomena are presented here through illustrative examples, including ‘Zeno breaking’, or computing beyond the finite time convergence of an infinite sequence of threshold transitions; the ‘Contact’ effect, in which in the discontinuous limit, trajectories can pass through a ‘saddle point’ without stopping, though these solutions are not unique and other solutions stop for arbitrary time intervals; and sensitive behaviour that arises from exotic dynamics within switching regions.

On regularized distance and related functions

1979 ◽  
Vol 83 (1-2) ◽  
pp. 115-122 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Continuous Function ◽  
Closed Subset ◽  
Parameter Family ◽  
Lipschitz Continuous Function ◽  
Partial Derivatives ◽  
Lipschitz Continuous ◽  
Order Of Magnitude ◽  
Derivatives Of ◽  
Smooth Approximations

SynopsisLetFbe any closed subset of ℝN. Stein's regularized distance is a smooth (C∞) function, defined on the complementcF, that approximates the distance fromFof any pointx ∈cFin the manner shown by the inequalities (*) in the Introduction below. In this paper we use a method different from Stein's to construct a one-parameter family of smooth approximations to any positive Lipschitz continuous function, with the effect that the constants in (*) can be made arbitrarily close to 1. It is shown that partial derivatives of order two or more, while necessarily unbounded, are best possible in order of magnitude.

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