Related papers: Fast Computation of Secondary Invariants
Recently the algebraic structure of gauge-invariant operators in multi-matrix quantum mechanics has been clarified: this space forms a module over a freely generated ring. The ring is generated by a set of primary invariants, while the…
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three…
This paper introduces two forms of modular inverses and proves their reciprocity formulas respectively. These formulas are then applied to formulate new and generalized algorithm for computing these modular inverses. The same algorithm is…
Complexity bounds for many problems on matrices with univariate polynomial entries have been improved in the last few years. Still, for most related algorithms, efficient implementations are not available, which leaves open the question of…
Loop invariants are properties of a program loop that hold before and after each iteration of the loop. They are often employed to verify programs and ensure that algorithms consistently produce correct results during execution.…
Computing topological invariants of 3-manifolds is generally intractable, yet specialized algebraic structures can enable efficient algorithms. For Witten-Reshetikhin-Turaev (WRT) invariants of torus bundles, we exploit the non-commutative…
The computation of triangular decompositions are based on two fundamental operations: polynomial GCDs modulo regular chains and regularity test modulo saturated ideals. We propose new algorithms for these core operations relying on modular…
The paper presents a new algorithmic construction of a finite generating set of rational invariants for the rational action of an algebraic group on the affine space. The construction provides an algebraic counterpart of the moving frame…
We address the problem of improving the performance and in particular the sample complexity of deep neural networks by enforcing and guaranteeing invariances to symmetry transformations rather than learning them from data. Group-equivariant…
The purpose of this survey paper is to bring to a large mathematical audience (containing also non-algebraists) some topics of invariant theory both in the classical commutative and the recent noncommutative case. We have included only…
We show that the word problem for an amalgam $[S_1,S_2;U,\omega_1,\omega_2]$ of inverse semigroups may be undecidable even if we assume $S_1$ and $S_2$ (and therefore $U$) to have finite $\mathcal{R}$-classes and $\omega_1,\omega_2$ to be…
This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some output variables are also input variables, linked by a linear dependency. Fundamental examples include the…
Spingarn's method of partial inverses has found many applications in nonlinear analysis and in optimization. We show that it can be employed to solve composite monotone inclusions in duality, thus opening a new range of applications for the…
Leveraging prior knowledge on intraclass variance due to transformations is a powerful method to improve the sample complexity of deep neural networks. This makes them applicable to practically important use-cases where training data is…
Beginning with the projectively invariant method for linear programming, interior point methods have led to powerful algorithms for many difficult computing problems, in combinatorial optimization, logic, number theory and non-convex…
This paper addresses the problem of checking invariant properties for a large class of symbolic transition systems, defined by a combination of SMT theories and quantifiers. State variables can be functions from an uninterpreted sort…
The pseudoinverse of a matrix, a generalized notion of the inverse, is of fundamental importance in linear algebra and, thereby, in many different fields. Despite its proven existence, an algorithmic approach is typically necessary to…
Modular integer arithmetic occurs in many algorithms for computer algebra, cryptography, and error correcting codes. Although recent microprocessors typically offer a wide range of highly optimized arithmetic functions, modular integer…
Group invariants are used in high energy physics to define quantum field theory interactions. In this paper, we are presenting the parallel algebraic computation of special invariants called symplectic and even focusing on one particular…
We study some aspects of the invariant pair problem for matrix polynomials, as introduced by Betcke and Kressner and by Beyn and Thuemmler. Invariant pairs extend the notion of eigenvalue-eigenvector pairs, providing a counterpart of…