integral calculus

This work proposes a novel class of Lancaster copulas by integrating, for the first time, the orthogonal expansion of continuous Lancaster probabilities with copula theory. By constructing their infinite series representation and associated density expressions, and by systematically analyzing truncation effects, the authors achieve efficient approximations of both the target copula and its density. Theoretical analysis and numerical experiments demonstrate that high-accuracy approximation can be attained using only low-order truncations, offering both rigorous theoretical guarantees and computational tractability. This approach provides a powerful new tool for modeling complex dependence structures in multivariate data.

Matrix Lie group operations in robotics, computer vision, and graphics traditionally rely on cumbersome infinite series expansions, hindering analytical tractability, interpretability, and computational efficiency. Method: This paper introduces a novel analytical paradigm based on closed-form integral representations. It systematically replaces power series with compact integral expressions; embeds minimal polynomials of the Lie algebra early in derivation to preserve expression conciseness; and exploits recursive structures and algebraic relationships among integral kernels. Contribution/Results: The approach successfully recovers classical results—including the Euler–Rodrigues formula—and significantly simplifies derivations of analytical solutions in rigid-body kinematics, dynamics, and related domains. It enhances symbolic computation in terms of interpretability, consistency, and efficiency. By unifying treatment across matrix Lie groups, the method provides a more concise, general, and theoretically grounded framework for analytical Lie group modeling.

This work addresses the challenge of verifying mathematical proofs generated by large language models by formally encoding, for the first time, an entire advanced undergraduate probability textbook—including its measure-theoretic foundations—into Lean. To bridge the semantic gap between the textbook’s exposition and the abstract formalism of the Mathlib library, the authors introduce an “interface lemma” strategy. Combined with structured proof engineering and formalization techniques specific to measure theory, this approach yields a reusable, machine-verifiable infrastructure spanning fourteen textbook chapters. The resulting formalization not only provides rigorous verification of all stated theorems and explicit articulation of their assumptions but also establishes a robust foundation for reliable AI-assisted mathematics, educational applications, and future formalization efforts in probability theory.

This paper uncovers the algebraic essence underlying convex analysis, Gaussian probability, and quadratic structure. To this end, we introduce Graphical Quadratic Algebra (GQA)—a novel algebraic framework based on chordal graphs—that uniformly models quadratic relations, Gaussian stochastic processes, and nondeterministic Gaussian processes via rotation-invariant quadratic generators. We provide the first sound and complete axiomatic characterization of three fundamental models: least-squares estimation, Gaussian randomness, and nondeterminism—revealing their shared conditional algebraic structure. Our method integrates string diagram theory, categorical semantics, and formal semantics of probabilistic programming. Theoretical contributions include soundness and completeness proofs for all three models within GQA. Applications demonstrate efficacy in linear regression, probabilistic programming, and noisy circuit modeling.

This work addresses the longstanding limitation in conditional density estimation—namely, the absence of closed-form solutions for multivariate conditional densities under non-Gaussian assumptions. We propose a generative conditional density estimation framework grounded in copula modeling and analytic conditionalization in latent space. Methodologically, we first establish the inheritability of “conditional stability” under mixture and transformation operations, thereby extending analytically tractable conditional families to non-Gaussian, nonlinear, and cross-dimensional settings. The core components include a Gaussian Mixture Copula Model (GMCM), an explicit latent-space conditionalization mechanism, and joint copula modeling. Experiments on synthetic and real-world datasets demonstrate substantial improvements in conditional density estimation accuracy and robustness to missing data imputation. Crucially, our approach enables efficient, differentiable, and sampling-free deterministic conditional inference.

This work presents the first formalization of the multivariable Taylor’s theorem in synthetic differential geometry within a constructive mathematical framework, using the Lean proof assistant and its mathlib library. To meet the analytical demands of infinitesimal neighborhoods, the authors develop a novel proof technique compatible with constructive logic, successfully achieving a machine-verified formalization of the Taylor expansion. This contribution not only fills a notable gap in mathlib regarding synthetic differential geometry but also demonstrates the feasibility and potential of interactive theorem proving systems to support constructive infinitesimal analysis. The result underscores the capacity of modern proof assistants to handle sophisticated geometric and analytic concepts while adhering to the constraints of constructive mathematics.

We introduce Exhaustive Symbolic Integration (ESI), a method that enumerates all symbolic functions up to a given complexity $k$ within a specified operator basis and determines which admit closed-form antiderivatives within the same class. This allows us to compute the "integrability fraction" $ρ(k)$ (the fraction of functions whose derivatives lie within the same class), which we do for five operator bases including combinations of rational functions, powers, exponentials, logarithms and trigonometric functions. We find that $ρ(k)$ declines at high complexity and that the operator basis has a dramatic effect -- in particular, adding the logarithm boosts $ρ(k)$ by a factor of $\sim$3 and produces or exacerbates a clear peak at $k=6$. We also deploy ESI as a novel integration algorithm, identifying three integrals that resist SymPy, Mathematica, RUBI, FriCAS, Maxima and Giac under all tested strategies. When an antiderivative can be found by multiple methods, ESI often returns the simplest form. These results reveal that the landscape of symbolic integrability is shaped primarily by the choice of operators, and that exhaustive enumeration can systematically discover integrable forms -- including novel ones -- that elude computer albegra systems.

This work investigates the creative space of mathematical proofs under constraints, with a particular focus on the impact of non-constructive reasoning. We introduce a strategy ablation methodology that integrates our custom-built Meno automated formalization tool with Goedel Prover embeddings to systematically explore both formal and informal proof spaces for foundational theorems from *Analysis I* within the Lean theorem prover. Our experiments successfully generate a novel class of machine-produced proofs, revealing that these proofs cluster along low-dimensional submanifolds in a high-dimensional representation space and significantly diverge from human-constructed proof trajectories. This study provides the first quantitative characterization of the structural differences between machine-generated and human proofs.

This study addresses the absence of efficient and accurate analytical approximations for VIX option implied volatility, which has traditionally necessitated time-consuming numerical root-finding in model calibration. Building upon forward variance models—including the standard, rough Bergomi, and hybrid specifications—the authors derive, for the first time, closed-form asymptotic expansions of implied volatility with explicit correction terms by leveraging weak approximation and asymptotic expansion techniques. This approach entirely circumvents numerical root-finding and demonstrates high accuracy and exceptional computational efficiency across multiple model settings. Consequently, it substantially enhances both the speed and numerical stability of VIX option calibration.