Nadir Matringe

Transcendental inductive invariants generation for non-linear differential and hybrid systems

We present the first verification methods that automatically generate bases of invariants expressed by multivariate formal power series and transcendental functions. We discuss the convergence of solutions generated over hybrid systems that exhibit non-linear models augmented with parameters. We reduce the invariant generation problem to linear algebraic matrix systems, from which one can provide effective methods for solving the original problem. We obtain very general sufficient conditions for the existence and the computation of formal power series invariants over multivariate polynomial continuous differential systems. The formal power series invariants generated are often composed by the expansion of some well-known transcendental functions like log or exp and have an analysable closed-form. This facilitates their use to verify safety properties. Moreover, we generate inequality and equality invariants. Our examples, dealing with non-linear continuous evolution similar to those present today in many critical hybrid embedded systems, show the strength of our results and prove that some of them are beyond the limits of other recent approaches.

Generating invariants for non-linear hybrid systems

We describe powerful computational techniques, relying on linear algebraic methods, for generating ideals of non-linear invariants of algebraic hybrid systems. We show that the preconditions for discrete transitions and the Lie-derivatives for continuous evolution can be viewed as morphisms, and so can be suitably represented by matrices. We reduce the non-trivial invariant generation problem to the computation of the associated eigenspaces or nullspaces by encoding the consecution requirements as specific morphisms represented by such matrices. Our methods are the first to establish very general sufficient conditions that show the existence and allow the computation of invariant ideals. Our approach also embodies a strategy to estimate certain degree bounds, leading to the discovery of rich classes of inductive invariants. By reducing the problem to related linear algebraic manipulations we are able to address various deficiencies of other state-of-the-art invariant generation methods, including the efficient treatment of non-linear hybrid systems. Our approach avoids first-order quantifier eliminations, Grobner basis computations or direct system resolutions, thereby circumventing difficulties met by other recent techniques. We handle non-linear hybrid systems extended with parameters.We extract the generator basis of a vector space of non-trivial invariants.The problem is reduced to the computation of associated eigenspaces or null spaces.Sufficient conditions for the existence and the computation of invariant ideals.

Generating invariants for non-linear loops by linear algebraic methods

We present new computational methods that can automate the discovery and the strengthening of non-linear interrelationships among the variables of programs containing non-linear loops, that is, that give rise to multivariate polynomial and fractional relationships. Our methods have complexities lower than the mathematical foundations of the previous approaches, which used Gröbner basis computations, quantifier eliminations or cylindrical algebraic decompositions. We show that the preconditions for discrete transitions can be viewed as morphisms over a vector space of degree bounded by polynomials. These morphisms can, thus, be suitably represented by matrices. We also introduce fractional and polynomial consecution, as more general forms for approximating consecution. The new relaxed consecution conditions are also encoded as morphisms represented by matrices. By so doing, we can reduce the non-linear loop invariant generation problem to the computation of eigenspaces of specific morphisms. Moreover, as one of the main results, we provide very general sufficient conditions allowing for the existence and computation of whole loop invariant ideals. As far as it is our knowledge, it is the first invariant generation methods that can handle multivariate fractional loops.

Test vectors for local periods

Abstract Let E / F {E/F} be a quadratic extension of non-Archimedean local fields of characteristic zero. An irreducible admissible representation π of GL ⁢ ( n , E ) {\mathrm{GL}(n,E)} is said to be distinguished with respect to GL ⁢ ( n , F ) {\mathrm{GL}(n,F)} if it admits a non-trivial linear form that is invariant under the action of GL ⁢ ( n , F ) {\mathrm{GL}(n,F)} . It is known that there is exactly one such invariant linear form up to multiplication by scalars, and an explicit linear form is given by integrating Whittaker functions over the F-points of the mirabolic subgroup when π is unitary and generic. In this paper, we prove that the essential vector of [14] is a test vector for this standard distinguishing linear form and that the value of this form at the essential vector is a local L-value. As an application we determine the value of a certain proportionality constant between two explicit distinguishing linear forms. We then extend all our results to the non-unitary generic case.

Shalika periods and parabolic induction for GL(n) over a non archimedean local field

Let

Conjectures about Distinction and Local Asai L-Functions

F

On the local Bump–Friedberg L-function

be a non archimedean local field, and

Linear and Shalika local periods for the mirabolic group, and some consequences

n_1

Distinguished Generic Representations of GL(n) over p-adic Fields

and

Distinction of some induced representations

n_2