arXiv subjects

Marcelo P. Santos

Publications and source records attributed to Marcelo P. Santos.

5 recordsLinked to original sources

arXiv2026

Point-group filtering in unitary coupled-cluster ansätze: exact criticality under the Abelian filter, freeness measured near equilibrium under the full group

Filtering a unitary coupled-cluster ansatz by molecular point-group symmetry has no settled variational cost for groups with degenerate irreducible representations. We prove that the symmetry-adapted subvariety of the amplitude space is a critical subvariety of the energy, for irreducible representations of any dimension. The gradient along every removed direction vanishes there, so a quasi-Newton optimisation with exact gradients started at the reference determinant cannot distinguish the filtered from the unfiltered ansatz. Every operator invariant under the full point group already lies in the span of the pool retained by an Abelian subgroup, so the filtered pool reaches at every geometry an energy no higher than the fully invariant one reaches. A Hamiltonian-informed pool lies inside the set the equivariant pool touches, and in a symmetry-adapted basis inside the symmetry-filtered pool, which makes a published pair of parameter counts a certificate for the orbital basis. Across six molecules and two basis sets, the full-group filter changes the classical coupled-cluster correlation energy by at most $3.4 \times 10^{-12}$ millihartree in the larger basis where the point group is finite. The parameter count of the unitary ansatz falls from 75 to 30 for ammonia and from 65 to 21 for methane, measured against the Abelian filter current practice uses. The freeness measured for the full-group filter holds near equilibrium and degrades into the static-correlation regime.

quant-ph↗

arXiv2026

Parallelisation of Discrete Exterior Calculus via Representation Theory on Curved and Three-Dimensional Meshes

We establish a universal block-diagonalization framework for Discrete Exterior Calculus (DEC) operators on symmetric meshes, enabling embarrassingly parallel solvers with provable FLOP reductions. We prove that the two fundamental DEC operators, the discrete exterior derivative $d$ and the Hodge star $\star$, are equivariant under isometric finite group actions on simplicial complexes. The proof exploits the permutation representation induced on cochain spaces by the group action. As a consequence, any operator assembled from $d$ and $\star$ (including the Hodge Laplacian, the codifferential, Maxwell-type operators, and elasticity operators) inherits a block-diagonal structure in a single symmetry-adapted basis, which is computed only once per mesh. Unlike spectral methods restricted to flat Platonic domains, the framework applies natively to curved manifolds and is applicable in principle to computational electromagnetism and geometric fluid simulation on symmetric domains. Numerical experiments on a geodesic sphere ($I_h$ symmetry) and a hexagonal torus ($D_{6h}$ symmetry) yield FLOP-based parallel speedups, relative to a dense direct factorization, of up to $62\times$ and $182\times$, respectively. A further experiment on a body-centred-cubic (BCC) tessellation of the flat 3-torus $T^3$ with $T_d$ symmetry confirms equivariance of the exterior derivative, Hodge star, and Hodge Laplacian at machine precision for form degrees $k=0,1,2$ across three mesh resolutions. The FLOP-based sequential speedup approaches its theoretical asymptote of $\approx 9.07\times$, which a standard Schur-multiplicity reduction deepens by a further factor of order $|G|$. These results show that a single symmetry-adapted basis reduces the linear-solve cost of structure-preserving DEC computations on curved and three-dimensional meshes.

math.NA↗

arXiv2025

PySymmetry: A Sage/Python Framework for the Symmetry Reduction of Linear G-Equivariant Systems

Despite the prevalence of symmetry in scientific linear systems, these structural properties are often underutilized by standard computational software. This paper introduces PySymmetry, an open-source Sage/Python framework that implements classical representation theory to simplify G-equivariant linear systems. PySymmetry uses projection operators to generate symmetry-adapted bases, transforming equivariant operators into a more efficient block-diagonal form. Its functionalities include defining and reducing representations, calculating multiplicities, and obtaining the explicit block structure. We demonstrate PySymmetry's versatility through three case studies: a chemistry application, a numerical benchmark on the non-Hermitian Schrödinger equation that achieved a performance increase of over 17x compared to standard methods, and a symbolic investigation that enabled the first complete analytical classification of a challenging problem in celestial mechanics. Designed for seamless integration with libraries like NumPy and SciPy, PySymmetry offers a powerful, user-friendly tool for exploring symmetries in theoretical and applied contexts. ```

math.GR↗

arXiv2025

Decomposition of Symmetrical Classes of Central Configurations

We study central configurations when the set of positions is symmetric. We use a theorem from representation theory of finite groups to explore the symmetry properties of equations for central configurations. This approach simplifies equations for central configurations by considering arbitrary numbers of bodies, symmetry groups, and dimensions. We discuss how to use this theorem to obtain a more refined decomposition of the equations than that given before. The decomposition presented here uses the symmetry-adapted basis method. As an application, we give a complete description of the existence and which masses are possible for central configurations of two nested regular tetrahedrons, two nested regular octahedrons, and two nested regular cubes. To do this, we employ some methods of rational parameterizations and isolation of zeros of multivariate polynomials. The decomposition obtained allows symbolic calculations to be used to study the expressions. This way, we summarize previous discussions and extend them by completing the analysis on the cube case, for both the inverse and direct problems.

math.DS↗

arXiv2017

The Inverse Problem for Nested Polygonal Relative Equilibria

We prove that for some potentials (including the Newtonian one, and the potential of Helmholtz vortices in the plane) relative equilibria consisting of two homothetic regular polygons of arbitrary size can only occur if the masses at each polygon are equal. The same result is true for many regular polygons as long as the ratio between the radii of the polygons are sufficient large. Moreover, under these hypotheses, the relative equilibrium always exist.

math.DS↗