Automatic linearity detection

Given a function, or more generally an operator, the question "Is it linear?" seems simple to answer. In many applications of scientific computing it might be worth determining the answer to this question in an automated way; some functionality, such as operator exponentiation, is only defined for linear operators, and in other problems, time saving is available if it is known that the problem being solved is linear. Linearity detection is closely connected to sparsity detection of Hessians, so for large-scale applications, memory savings can be made if linearity information is known. However, implementing such an automated detection is not as straightforward as one might expect. This paper describes how automatic linearity detection can be implemented in combination with automatic differentiation, both for standard scientific computing software, and within the Chebfun software system. The key ingredients for the method are the observation that linear operators have constant derivatives, and the propagation of two logical vectors, $\ell$ and $c$, as computations are carried out. The values of $\ell$ and $c$ are determined by whether output variables have constant derivatives and constant values with respect to each input variable. The propagation of their values through an evaluation trace of an operator yields the desired information about the linearity of that operator

Linearity of stability conditions

We study different concepts of stability for modules over a finite dimensional algebra: linear stability, given by a "central charge", and nonlinear stability given by the wall-crossing sequence of a "green path". Two other concepts, finite Harder-Narasimhan stratification of the module category and maximal forward hom-orthogonal sequences of Schurian modules, which are always equivalent to each other, are shown to be equivalent to nonlinear stability and to a maximal green sequence, defined using Fomin-Zelevinsky quiver mutation, in the case the algebra is hereditary. This is the first of a series of three papers whose purpose is to determine all maximal green sequences of maximal length for quivers of affine type $\tilde A$ and determine which are linear. The complete answer will be given in the final paper [1].Comment: 24 pages, 3 figure

Testing for linearity

linearitytesting

On Linearity of Nonclassical Differentiation

We introduce a real vector space composed of set-valued maps on an open set X and note it by S. It is a complete metric space and a complete lattice. The set of continuous functions on X is dense in S as in a metric space and as in a lattice. Thus the constructed space plays the same role for the space of continuous functions with uniform convergence as the field of reals plays for the field of rationals. The classical gradient may be extended in the space S as a close operator. If a function f belongs to its domain then f is locally lipschitzian and the values of our gradient coincide with the values of Clarke's gradient. However, unlike Clarke's gradient, our gradient is a linear operator.Comment: Sorry, this article is being rewritten. Please email the author to be informed about its availabilit

Lower bounds for adaptive linearity tests

Linearity tests are randomized algorithms which have oracle access to the truth table of some function f, and are supposed to distinguish between linear functions and functions which are far from linear. Linearity tests were first introduced by (Blum, Luby and Rubenfeld, 1993), and were later used in the PCP theorem, among other applications. The quality of a linearity test is described by its correctness c - the probability it accepts linear functions, its soundness s - the probability it accepts functions far from linear, and its query complexity q - the number of queries it makes. Linearity tests were studied in order to decrease the soundness of linearity tests, while keeping the query complexity small (for one reason, to improve PCP constructions). Samorodnitsky and Trevisan (Samorodnitsky and Trevisan 2000) constructed the Complete Graph Test, and prove that no Hyper Graph Test can perform better than the Complete Graph Test. Later in (Samorodnitsky and Trevisan 2006) they prove, among other results, that no non-adaptive linearity test can perform better than the Complete Graph Test. Their proof uses the algebraic machinery of the Gowers Norm. A result by (Ben-Sasson, Harsha and Raskhodnikova 2005) allows to generalize this lower bound also to adaptive linearity tests. We also prove the same optimal lower bound for adaptive linearity test, but our proof technique is arguably simpler and more direct than the one used in (Samorodnitsky and Trevisan 2006). We also study, like (Samorodnitsky and Trevisan 2006), the behavior of linearity tests on quadratic functions. However, instead of analyzing the Gowers Norm of certain functions, we provide a more direct combinatorial proof, studying the behavior of linearity tests on random quadratic functions..

Optimizations of sub-100 nm Si/SiGe MODFETs for high linearity RF applications

Based on careful calibration in respect of 70 nm n-type strained Si channel S/SiGe modulation doped FETs (MODFETs) fabricated by Daimler Chrysler, numerical simulations have been used to study the impact of the device geometry and various doping strategies on device performance and linearity. The device geometry is sensitive to both RF performance and device linearity. Doped channel devices are found to be promising for high linearity applications. Trade-off design strategies are required for reconciling the demands of high device performance and high linearity simultaneously. The simulations also suggest that gate length scaling helps to achieve higher RF performance, but decreases the linearity