change of variables

This paper addresses the symbolic reduction problem for multivariate holomorphic integrals—particularly parametric ones—in the D-module framework. We propose the first Griffiths–Dwork-type reduction algorithm applicable to general holomorphic systems, circumventing the strong D-finiteness requirement of traditional approaches. The method integrates D-module theory, differential algebra, and geometric reduction techniques, and is implemented efficiently in Julia for symbolic computation. Our key contribution is the derivation of a previously inaccessible linear differential equation satisfied by the generating function for 8-regular graphs—a result unattainable via any existing method. This breakthrough provides a new computational tool for deriving differential equations of generating functions, with direct applications in enumerative combinatorics and algebraic geometry.

This work addresses the injectivity testing problem for univariate polynomial maps over finite fields, motivated by algebraic decomposition and controllable design in discrete dynamical systems. Methodologically, it integrates algebraic dynamics theory, finite-field polynomial analysis, and synchronous/alternating execution modeling. The main contribution is the first coefficient-wise complete algebraic characterization of injective polynomials—explicit structural conditions on coefficients are derived. Based on this characterization, the paper proposes the first polynomial-time injectivity test, with time complexity $O(n^2)$, markedly improving upon exponential brute-force enumeration. This resolves a long-standing fundamental decision problem and provides a computationally tractable tool for algebraically structured modeling and control of dynamical systems.

This work addresses the efficient computation of Jacobian matrices for chains of sequentially differentiable subroutines. We formulate Jacobian chain multiplication as a matrix chain multiplication scheduling problem with precedence constraints—a novel modeling approach. Our method integrates dynamic programming with scheduling-aware heuristics, achieving near-theoretical-optimal solution quality while substantially improving parallel scalability. By unifying automatic differentiation, matrix chain optimization, and task scheduling theory, it eliminates the serial bottlenecks inherent in conventional chain multiplication strategies. In benchmark evaluations, our algorithm attains solution quality comparable to that of branch-and-bound—yielding near-global optima—while demonstrating superior computational efficiency and parallel speedup over state-of-the-art methods. This establishes a new paradigm for high-performance Jacobian computation in large-scale differentiable programs.

Existing directional statistics tools are seldom adopted in engineering and computer science due to terminological barriers and lack of practical interfaces for modeling orientation data—such as angles, unit vectors, rotation matrices, and quaternions—in applications ranging from robotics to 3D vision. Method: We introduce the first comprehensive, practitioner-oriented reference guide for probability distributions over multi-degree-of-freedom orientation domains (1D–3D), employing a unified, engineering-friendly notation. The guide systematically presents density functions, maximum-likelihood parameter estimation procedures, and inverse-transform or rejection-sampling algorithms for six canonical directional distributions. Contribution/Results: We release an open-source Python library (built on NumPy/SciPy) supporting distribution fitting and random sampling. Empirical validation on robot pose calibration and 3D point cloud normal estimation demonstrates its practical efficacy, substantially bridging the gap between theoretical directional statistics and real-world engineering deployment.

This work addresses the challenge of reconstructing analytic ordinary differential equation (ODE) vector fields from limited discrete trajectory data. Methodologically, it introduces a novel approximation framework centered on the push-forward operator—employed here for the first time as the core modeling tool—combined with the Fourier–Borel transform and Fock space theory to construct finite-dimensional operator approximations within a local analytic functional space. Theoretically, it establishes rigorous convergence guarantees with explicit rates, proving that truncated least-squares polynomials achieve superior approximation both inside and outside their support domain. Experimentally, the method accurately recovers vector fields induced by analytic flow maps, exhibits strong extrapolation capability, and maintains numerical stability. Overall, it provides a new paradigm for analytic dynamical system modeling from sparse data.

This work addresses the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.

This work addresses optimization problems defined over products of simplices, such as low-rank learning of discrete multivariate probability distributions and function data registration based on the Square-Root Velocity Function (SRVF) representation. To tackle the inherent constraints, the authors propose a smooth reparameterization that is strictly convex element-wise, transforming the constrained problem into an unconstrained optimization over a Riemannian manifold. The resulting problem is solved via Riemannian gradient descent (RGD). Theoretical analysis shows that this reparameterization maps second-order KKT points on the manifold to weak second-order KKT points of the original problem, ensuring theoretical soundness while enhancing computational efficiency. Experiments demonstrate that RGD significantly outperforms projected gradient descent (PGD), achieving more accurate shape-preserving registration in functional data and efficiently solving probability tensor decomposition tasks.

This work addresses the long-standing issue of inconsistency in generalized matrix inverses under nonsingular diagonal transformations—a problem particularly critical in robotics, tracking, and control systems where physical unit sensitivity is paramount. The paper introduces a novel generalized inverse that, for the first time, achieves invariance under arbitrary nonsingular diagonal transformations, thereby rigorously preserving the physical units of state-space variables. This new inverse, together with the Moore–Penrose and Drazin inverses, forms a complete triad encompassing the principal linear system transformations. The framework is further extended to unit-consistent matrix factorizations. Grounded in matrix analysis theory, the authors develop a new algebraic construction method, successfully applied across multiple engineering domains, providing a robust mathematical foundation for unit-preserving modeling and significantly advancing the theoretical completeness of generalized inverses.

This work addresses the lack of efficient, end-to-end differentiable solvers for differential-algebraic equations (DAEs) arising from multiphysics systems, which hinders parameter inversion and optimal control. We present the first natively JAX-based differentiable DAE solver that unifies forward simulation with reverse-mode automatic differentiation. The solver integrates adaptive BDF, Radau, and Rosenbrock methods, combined with Pantelides index reduction and dummy derivative techniques. By freezing the forward time-step grid and re-solving a variable-step BDF-2 scheme on this fixed mesh, gradient propagation becomes highly efficient, enabling DAEs to serve as differentiable primitives. A single call to jax.grad computes full gradients, and wall-clock time remains nearly constant when scaling batched parameter sweeps from 1 to 1,000, dramatically improving differentiable computation efficiency for multiphysics DAE systems.

This work addresses the absence of a comprehensive machine-verified foundation for the extended complex plane, Möbius transformations, and their fundamental invariant—the cross-ratio—in existing formal mathematics. Building upon the Mathlib library in Lean 4, the authors construct the extended complex plane using the Option type and present the first formalization of the group structure of Möbius transformations, the uniqueness of such transformations determined by three points, and the invariance of the cross-ratio. The development, grounded in dependent type theory, comprises approximately 6,000 lines of code, including 40 definitions and 150 theorems, thereby establishing the first formally verified basis for conformal geometry, hyperbolic models, and mathematical physics.