Formal methods verify continuous auctions at exchanges.
problem Ensuring fairness and correctness in continuous auctions.
method Formal specification, design, and verification of continuous double auctions.
result A verified algorithm satisfies natural properties of auctions.
Explores local structure of morphisms and formal submanifolds in formal manifolds theory.
problem Understanding the local structure of morphisms and formal submanifolds in formal manifolds.
method Study of formal manifolds, including local structure of constant rank morphisms and formal submanifolds.
result Developed the local structure of constant rank morphisms and formal submanifolds.
Formalizes vNM utility theorem using Lean 4, proving existence and uniqueness.
problem Formalizing and proving the von Neumann-Morgenstern utility theorem.
method Implement classical axioms in Lean 4, formalizing preference relations over lotteries.
result Machine-verified proofs of existence and uniqueness of utility representations.
Defines formal vertex laws related to Lie conformal algebras.
problem No specific problem stated; focuses on definitions and proofs.
method Definitions and proofs of vertex/conformal versions of classical Lie theory results.
result Proves vertex/conformal versions of important Lie theory results.
GPT-f uses language models to find new proofs in formal math.
problem Generating original mathematical terms for automated theorem proving.
method Transformer-based language model for automated theorem proving and proof assistant.
result GPT-f contributed new proofs to the Metamath library.
Transformed geometry into algebra to prove Pick's theorem efficiently.
problem Translating geometric Pick's theorem into formal algebraic proof.
method Formalized geometric Pick's theorem into algebraic proof using Lean.
result Efficient formal proof of Pick's theorem.
Lean 4 formalizes Stokes' theorem for smooth singular cubes.
problem Formalizing Stokes' theorem for singular cubes in arbitrary dimensions.
method Using true differential-form pullback via Frechet derivative, bridging to mathlib4's extDeriv.
result d^2=0 for singular cubical chains, chain-level Stokes extended.
In this paper, we consider formal series associated with events, profiles derived from events, and statistical models that make predictions about events. We prove theorems about realizations for these formal series using the language and tools of Hopf algebras.
This paper formalizes Q-learning and linear TD convergence using Lean 4.
problem Formalizing convergence properties of Q-learning and linear TD learning. method Formal verification using Lean 4 theorem prover and Mathlib library.
result Formalized almost sure convergence of Q-learning and linear TD learning. We extend the formality theorem of M. Kontsevich from deformations of the structure sheaf on a manifold to deformations of gerbes.
Formalizes synthetic differential geometry in Lean.
problem Formalizing synthetic differential geometry in a proof assistant.
method Formalization of synthetic differential geometry with Lean and mathlib.
result Proves a Taylor theorem for functions of several variables.
We present an original theorem in auction theory: it specifies general conditions under which the sum of the payments of all bidders is necessarily not identically zero, and more generally not constant. Moreover, it explicitly supplies a construction for a finite minimal set of possible bids on which such a sum is not …
We prove a Darboux theorem for formal deformations of Hamiltonian operators of hydrodynamic type (Dubrovin-Novikov). Not all deformations are equivalent to the original operator: there is a moduli 2-stack of normal forms. The paper utilizes three main concepts: 1) dg Lie algebras concentrated in degrees [-1,\infty) suc…
In this paper we study symmetries, Newtonoid vector fields, conservation laws, Noether's Theorem and its converse, in the framework of the k-symplectic formalism, using the Frölicher-Nijenhuis formalism on the space of k1-velocities of the configuration manifold. For the case k=1, it is well known that Cartan sy…
Kontsevich's formality the…
The paper extends the formal manifold theorem to higher dimensions and characterizes A∞-minimal models for certain differential graded algebras.
problem Characterizing formal differential graded algebras and their A∞-minimal models. method Expanding the formal manifold theorem and proving properties of A∞-minimal models for specific cases. result The de Rham complex of certain differential graded algebras has A∞-minimal models with specific non-trivial terms. The paper proves an index theorem for loop spaces of compact manifolds.
problem Defining an index theorem for loop spaces of compact manifolds.
method Formulated and proved an equivariant index theorem for non-compact manifolds with S1-actions, using a ring of formal power series. result Found an appropriate form of the index theorem for loop spaces.
Paper formalizes multi-dimensional FSD using geometric methods.
problem Complex measure theory and calculus barriers to formalization in proof assistants.
method Geometric framework for first-order stochastic dominance in N dimensions.
result Geometric approach bypasses complex integration theory for direct comparison of survival probabilities.
Formal framework verifies fairness, uniformity, and rationality in financial market trades.
problem Ensuring fairness, uniformity, and rationality in financial market trades.
method Formal definition and proof of properties in a theorem prover (Coq).
result Formal verification of double-sided auctions in financial markets.
Paper formalizes Simon's satisficing through FFSD, proving its equivalence to expected utility theory.
problem Formalizing Herbert Simon's bounded rationality concept in economic decision-making.
method Developed FFSD framework using Lean 4 theorem prover, proving equivalence to expected utility theory.
result Equivalence theorem linking FFSD to expected utility maximization for approximate indicator functions.
We give a proof of Kontsevich's formality theorem for a general manifold using Fedosov resolutions of algebras of polydifferential operators and polyvector fields. The main advantage of our construction of the formality quasi-isomorphism is that it is based on the use of covariant tensors unlike Kontsevich's original p…
Develops thermodynamic formalism for quasimorphisms on negatively curved spaces.
problem Analyzing quasimorphisms on negatively curved spaces.
method Thermodynamic formalism framework, Banach isomorphism, weak Livšic cohomology.
result Establishes Central Limit Theorem and invariance principle for unbounded quasimorphisms.
This paper contains a generalization of the convex ideal case of the Thurston-Andreev theorem when the genus is greater than 1. The heart of the paper concerns taking formal angle data on a surface and ``conformally flowing'' this formal angle data to uniquely associated uniform angle data. This flow turns out to be th…
Formalizes integral curves on Banach manifolds in Lean.
problem Existence and uniqueness of integral curves on Banach manifolds.
method Formalized differential equations on Banach spaces, then generalized to Banach manifolds.
result Established theorems for integral curves on Banach manifolds.
The paper studies graded manifolds and their functorial relationship.
problem Understanding the functor between two categories of graded manifolds.
method Examines polynomial filtrations and homogeneity structures, applying the Batchelor-Gawedzki theorem and Borel-Whitney theorem.
result The functor is full and surjective on objects between the categories of graded vector bundles and manifolds.
This paper deals with a semi-classical limit (Theorem 1) by using traditional mathematical methods, and shows a Hopf theorem as a corollary. A formal discussion of it may be found in [7].
This paper autoformalizes Euclidean geometry using LLMs and theorem provers.
problem Challenges in formalizing Euclidean geometry due to reliance on diagrams.
method Combines neuro-symbolic framework, SMT solvers, and LLMs to fill in diagrammatic gaps.
result Demonstrates the capability and limitations of LLMs on autoformalizing geometry problems.
Two impossibility theorems show formal alignment certification is impossible for AI systems.
problem Formal certification of AI alignment over open-ended domains is impossible.
method Two independent impossibility theorems: Semantic and Statistical barriers.
result No procedure can simultaneously satisfy soundness, completeness, and tractability.
Graphs improve theorem proving in higher-order logic.
problem Challenges in converting higher-order logic formulas into graph-based representations.
method Used graph neural networks (GNNs) to represent and search higher-order logic.
result GNNs outperform state-of-the-art methods in higher-order theorem proving.
Method proves ellipticity of vacuum spacetime boundary problems.
problem Proving ellipticity of boundary value problems for stationary vacuum spacetimes.
method Developed a general method using the projection formalism.
result Proved manifold theorem for moduli space of stationary vacuum spacetimes.
We review the geometric formulation of the second Noether's theorem in time-dependent mechanics. The commutation relations between the dynamics on the final constraint manifold and the infinitesimal generator of a symmetry are studied. We show an algorithm for determining a gauge symmetry which is closely related to th…
We prove Tsygan's formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold M using the Fedosov resolutions proposed in math.QA/0307212 and the formality quasi-isomorphism for Hochschild chains of R[[y_1, ..., y_d]] proposed in paper math.QA/0010321 by Shoikhet. This result …
Formalism for superfield theory problems via Poincaré-Cartan form.
problem Formalism for first-order Berezinian variational problems in superfield theory.
method Intrinsic description of Hamilton-Cartan formalism through Poincaré-Cartan form.
result Noether theorem and examples from superfield theory and supermechanics discussed.
Newlander-Nirenberg theorem extended to complex b-manifolds.
problem Characterizing complex b-manifolds.
method Involutive splitting of b-tangent bundle, formal local invariants, singular coordinate change.
result Complex b-manifolds have a single local model.
Generalizes Quillen's theorem to equivariant settings.
problem Classifying commutative formal group laws in equivariant bordism.
method Combines computations and detailed investigations of equivariant Lazard rings.
result Identifies the Z/2-equivariant unitary bordism ring with the Z/2-equivariant Lazard ring. We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the…
Formalizes the Fundamental Theorem of Asset Pricing in Lean 4.
problem Formalizing the Fundamental Theorem of Asset Pricing in a proof assistant.
method Formalization in Lean 4 over Mathlib, covering three market settings.
result Constructs the equivalent martingale measure explicitly and proves its properties.
Let (M, π ) be a Poisson manifold. A Poisson submanifold P∈M gives rise to an algebroid AP→P, to which we associate certain chomology groups which control formal deformations of π around P . Assuming that these groups vanish, we prove that π is formally rigid around P , i.e. any other Poisson struct…
The paper defines and studies the category of Z-graded manifolds, including their intrinsic structure and formal properties.
problem Understanding the categorical properties and intrinsic structure of Z-graded manifolds.
method Describing local models, explaining formality, and formulating analogues of theorems.
result Proper definitions of objects and morphisms in the category of Z-graded manifolds, and formulation of Batchelor's theorem.
The paper extends Newlander-Nirenberg theorem to domains with C2 boundary.
problem Extending Newlander-Nirenberg theorem to domains with C2 boundary. method Analyzing formally integrable complex structures on domains with C2 boundary. result Existence of global holomorphic coordinate systems on the closure of a bounded strictly pseudoconvex domain.
Using the concept of s-formality we are able to extend the bounds of a Theorem of Miller and show that a compact k-connected 4k+3- or 4k+4-manifold with b_{k+1}=1 is formal. We study k connected n-manifolds, n= 4k+3, 4k+4, with a hard Lefschetz-like property and prove that in this case if b_{k+1}=2, then the manifold i…
The paper proves a quadratic formality for Sasakian manifolds' representation varieties.
problem Analyzing the variety of representations of fundamental groups of Sasakian manifolds.
method Proving almost-formality of de Rham complex and vanishing cup product theorem.
result Quadratic formality of analytic germs of representation varieties.
Paper develops a weighted linearization approach for vector fields.
problem Linearizability of vector fields under weighted conditions.
method Formal Moser trick applied to power series, addressing weighted non-resonance condition.
result Formal Moser trick works over any field of characteristic zero.
Formally verifies fairness and uniformity in financial market trades.
problem Ensuring fairness and uniformity in automated trading systems.
method Formal definition and verification in Coq proof assistant.
result Properties of double-sided auction mechanisms verified.
Clock theorem extended to knotoids and linkoids.
problem Generalizing the Clock Theorem to knotoids and linkoids.
method Extending the Clock Theorem to knotoids and linkoids.
result Clock states of knotoid diagrams form a lattice under transpositions.
The projective metrizability problem can be formulated as follows: under what conditions the geodesics of a given spray coincide with the geodesics of some Finsler space, as oriented curves. In Theorem 3.8 we reformulate the projective metrizability problem for a spray in terms of a first-order partial differential ope…
We exploit the Fedosov-Weinstein-Xu (FWX) resolution proposed in q-alg/9709043 to establish an isomorphism between the ring of Hochschild cohomology of the quantum algebra of functions on a symplectic manifold M and the ring H(M, C((h))) of De Rham cohomology of M with the coefficient field C((h)) without making use of…
Paper formalizes analogy between data sets and models using Hoare logic.
problem Lack of formal criteria for transferring machine learning models between data domains.
method Formalization of analogy using first-order logic and Hoare logic, rigorous theorem proving.
result Rigorous formalization of analogy in knowledge transfer between machine learning models.