Solving Algebraic Problems with Geometry Diagrams Using Syntax-Semantics Diagram Understanding

Solving Algebraic Problems with Geometry Diagrams(APGDs)poses a significant challenge in artificial intelligence due to the complex and diverse geometric relations among geometric objects.Problems typically involve both textual descriptions and geometry diagrams,requiring a joint understanding of these modalities.Although considerable progress has been made in solving math word problems,research on solving APGDs still cannot discover implicit geometry knowledge for solving APGDs,which limits their ability to effectively solve problems.In this study,a systematic and modular three-phase scheme is proposed to design an algorithm for solving APGDs that involve textual and diagrammatic information.The three-phase scheme begins with the application of the statetransformer paradigm,modeling the problem-solving process and effectively representing the intermediate states and transformations during the process.Next,a generalized APGD-solving approach is introduced to effectively extract geometric knowledge from the problem’s textual descriptions and diagrams.Finally,a specific algorithm is designed focusing on diagram understanding,which utilizes the vectorized syntax-semantics model to extract basic geometric relations from the diagram.A method for generating derived relations,which are essential for solving APGDs,is also introduced.Experiments on real-world datasets,including geometry calculation problems and shaded area problems,demonstrate that the proposed diagram understanding method significantly improves problem-solving accuracy compared to methods relying solely on simple diagram parsing.

Algebra-Geometry of Piecewise Algebraic Varieties

Algebraic variety is the most important subject in classical algebraic geometry. As the zero set of multivariate splines, the piecewise algebraic variety is a kind generalization of the classical algebraic variety. This paper studies the correspondence between spline ideals and piecewise algebraic varieties based on the knowledge of algebraic geometry and multivariate splines.

ON THE FOUNDATION OF ALGEBRAIC DIFFERENTIAL GEOMETRY

An algebraic differential variety is defined as the zero-set of a differentialpolynomial set,and algebraic differential geometry is devoted to the study of such varieties.We give various decomposition formulas for the structures of such zero-sets which imply inparticular,the unique decomposition of an algebraic differential variety into its irreduciblecomponents.These formulas will find applications in various directions including mechanicaltheorem-proving of differential geometries.

PIECEWISE SEMIALGEBRAIC SETS

Semialgebraic sets are objects which are truly a special feature of real algebraic geometry. This paper presents the piecewise semialgebraic set, which is the subset of Rn satisfying a boolean combination of multivariate spline equations and inequalities with real coefficients. Moreover, the stability under projection and the dimension of C^μ piecewise semialgebraic sets are also discussed.

Some Researches on Real Piecewise Algebraic Curves

The piecewise algebraic curve,defined by a bivariate spline,is a generalization of the classical algebraic curve.In this paper,we present some researches on real piecewise algebraic curves using elementary algebra.A real piecewise algebraic curve is studied according to the fact that a real spline for the curve is indefinite,definite or semidefinite(nondefinite).Moreover, the isolated points of a real piecewise algebraic curve is also discussed.

Hyperbolic geometry with geometric algebra

WE use the geometric algebra in refs. [1, 2] to study hyperbolic geometry. The n-dimensional hyperbolic space Hn is taken to be the unit sphere of G1 (I-n,1) with antipodal points identified. We study typically H2: the dualities between generalized point and generalized line, between generalized triangle and imaginary triangle; convex generalized triangles; Lorentz transformations; generalized circles and double-cycles, etc. Below we list some of the results.

Asymptotic bound on binary self-orthogonal codes

We present two constructions for binary self-orthogonal codes. It turns out that our constructions yield a constructive bound on binary self-orthogonal codes. In particular, when the in-formation rate R = 1/2, by our constructive lower bound, the relative minimum distance δ≈ 0.0595 (for GV bound, δ≈ 0.110). Moreover, we have proved that the binary self-orthogonal codes asymptotically achieve the Gilbert-Varshamov bound.

Rigid Analytic p-Adic Simpson Correspondence for Line Bundles

The p-adic Simpson correspondence due to Faltings(Adv Math 198(2):847-862,2005)is a p-adic analogue of non-abelian Hodge theory.The following is the main result of this article:The correspondence for line bundles can be enhanced to a rigid analytic morphism of moduli spaces under certain smallness conditions.In the complex setting,Simpson shows that there is a complex analytic morphism from the moduli space for the vector bundles with integrable connection to the moduli space of representations of a finitely generated group as algebraic varieties.We give a p-adic analogue of Simpson’s result.