A new algorithm improves medical image grading accuracy.
problem Improving automatic grading of medical images for disease severity or risk score.
method Sparse range-constrained learning (SRCL) algorithm integrating sparse representation and grading.
result Improves accuracy in cup-to-disc ratio computation and cataract grading.
Theory of H-graded manifolds and coverings of supermanifolds.
problem Developing a theory for H-graded manifolds and coverings of supermanifolds. method Using tools from representation theory, we introduce and investigate H-graded coverings of supermanifolds. result Theory of H-graded coverings of supermanifolds introduced and investigated. The paper finds obstructions to Lie algebroid representations up to homotopy.
problem Obstacles to representations up to homotopy of Lie algebroids.
method Analyzes Pontryagin characters and vector bundles over Lie algebroids.
result Vanishing of Pontryagin characters implies no representation up to homotopy.
We generalise to the Z2-graded set-up a practical method for inspecting the (non)removability of parameters in zero-curvature representations for partial differential equations (PDEs) under the action of smooth families of gauge transformations. We illustrate the generation and elimination of parameters in …
This thesis generalizes structures on Q-manifolds and Lie n-algebroids.
problem Representation theory and linear structures of Q-manifolds and Lie n-algebroids. method Introduces differential graded modules and representations up to homotopy, defines Weil algebra, and studies VB-Lie n-algebroids. result Establishes an equivalence between VB-Lie n-algebroids and (n+1)-term representations up to homotopy of Lie n-algebroids. Defines a new 2+1-G-HQFT using graded skein modules.
problem Developing a new quantum field theory for groups.
method G-graded chromatic maps and skein modules.
result Recover modified Turaev-Viro invariants.
The paper explores dual structures in graded manifolds.
problem Understanding dual objects in graded manifolds.
method Developed dual objects for graded bundles and applied them to double vector bundles and graded bundles of degree 2.
result Elegant characterizations of double vector bundles and graded bundles of degree 2.
Develops higher representation theory for odd Khovanov homology and rewriting theory.
problem Quantum topology and higher algebraic structures.
method Higher representation theory and rewriting theory applied to Khovanov homology.
result Established a basis theorem for graded gl2-foams. We discuss generalizations of Ozsvath-Szabo's spectral sequence relating Khovanov homology and Heegaard Floer homology, focusing attention on an explicit relationship between natural Z (resp., 1/2 Z) gradings appearing in the two theories. These two gradings have simple representation-theoretic (resp., geometric) inter…
New formulas link knot invariants from DGA and satellite polynomials.
problem Establishing relationships between knot invariants.
method Introducing new polynomials and formulas linking DGA and satellite invariants.
result Arbitrary m-graded satellite ruling polynomials are determined by DGA of K.
GRADE models evolving graph dynamics by learning node and community representations.
problem Lack of tools to study temporal community dynamics in evolving graphs.
method GRADE is a probabilistic model that learns evolving node and community representations via a random walk prior and variational inference.
result GRADE outperforms baselines in dynamic link prediction and dynamic community detection.
RBM generates complex, graded data features.
problem Extracting complex features from high-dimensional data.
method Characterized structural conditions for RBM to generate compositional representations.
result RBM can operate in a compositional phase under specific conditions.
Graded Transformers embed algebraic structure in neural networks through graded transformations.
problem Efficiently modeling hierarchical and structured data in neural networks.
method Introduces Linearly Graded Transformer (LGT) and Exponentially Graded Transformer (EGT) with graded scaling operators.
result Establishes rigorous guarantees and improved efficiency for structured data.
Proposes grade-aware course recommendation methods to improve student GPA.
problem Helping students select courses that lead to timely graduation and good grades.
method Two approaches: ranking courses by expected GPA impact and combining grade predictions with course recommendations.
result Grade-aware methods recommend courses leading to better student performance.
Legendrian knot representations linked to colored Kauffman polynomial.
problem Relating Legendrian knot representations to colored Kauffman polynomial.
method Introducing ungraded n-colored ruling polynomial and relating it to the n-colored Kauffman polynomial. result Ungraded representation numbers of Legendrian knot DG-algebra agree with n-colored Kauffman polynomial specialization. The abstract discusses Lie 2-groupoids and their applications in representation theory.
problem Understanding symmetries of graded vector spaces and bundles.
method Construction of Lie 2-groupoids and exploration of their properties.
result Smooth pseudofunctors classify 2-term representations up to homotopy.
Study on Lie algebras from Clifford modules, focusing on specific types.
problem Characterizing Lie algebras from Clifford modules and their graded structures.
method Analysis of pseudo H-type Lie algebras and their representations through Clifford algebras. result Different types of Lie algebras have varying possibilities of containing pseudo H-type Lie algebras in their negative part. Examples of SL(2, Z) actions on differential graded categories are defined and explored.
We conjecture the existence of four independent gradings in the colored HOMFLY homology. We describe these gradings explicitly for the rectangular colored homology of torus knots and make qualitative predictions of various interesting structures and symmetries in the colored homology of general knots. We also give a si…
We show that the exterior powers of the matrix valued random walk invariant of string links, introduced by Lin, Tian, and Wang, are isomorphic to the graded components of the tangle functor associated to the Alexander Polynomial by Ohtsuki divided by the zero graded invariant of the functor. Several resulting propertie…
We rewrite the recently proposed differential expansion formula for HOMFLY polynomials of the knot 41 in arbitrary rectangular representation R=[rs] as a sum over all Young sub-diagrams λ of R with extraordinary simple coefficients Dλtr(r)⋅Dλ(s) in front of the Z-factors. Somewhat miraculously…
The paper connects knot homology, quantum 6j-symbols, and complements of knots.
problem Investigating the relationship between knot homology, quantum 6j-symbols, and knot complements.
method Developed a grading rule for HOMFLY-PT and Kauffman homology, found relationships between A-polynomials, and conjectured closed-form expressions for quantum 6j-symbols and knot complements.
result Closed-form expressions for SO(N) quantum 6j-symbols and conjectured expressions for (a,t)-deformed F_K for knot complements.
The paper studies Lie n-algebroids and their representations up to homotopy.
problem Understanding Lie n-algebroids and their representations.
method Analyzes differential graded modules and representations up to homotopy, describes adjoint and coadjoint modules, and computes the Weil algebra.
result Alternative characterisation of non-degeneracy of higher Poisson structures.
Context-aware CNN improves cancer grading accuracy.
problem Grading colorectal cancer histology images accurately.
method Proposes a context-aware neural network for 1,792x1,792 pixel images.
result Outperforms traditional methods by 3.61%.
The pull back of a flat bundle E→X along the evaluation map π:LX→X from the free loop space LX to X comes equipped with a canonical automorphism given by the holonomies of E. This construction naturally generalizes to flat Z-graded connections on X. Our main …
Study spider evaluations and web group representations using algebraic topology.
problem Evaluate and represent the topology of web groups using algebraic methods.
method Analyze the fundamental groups of planar webs, associate irreducible components and graded subalgebras, and use Poincaré polynomials.
result Spider evaluations correspond to symmetrized Poincaré polynomials of associated graded subalgebras.
New dg-algebras link graph colorings to sheaves.
problem Linking graph colorings to sheaves for Legendrian surfaces.
method Generalized Casals-Murphy dg-algebra to non-commutative coefficients and computed Legendrian contact dg-algebra.
result Rank r representations of dg-algebras correspond to colorings of faces in Grassmannian.
Study prolongations of nilpotent Lie algebras with specific structural subalgebras.
problem Understanding prolongations of nilpotent Lie algebras with specific structural subalgebras.
method Analyzing finite dimensional almost and quasi-effective prolongations of nilpotent Z-graded Lie algebras, focusing on those with decomposable reductive structural subalgebras.
result Obtained Levi-Malčev and Levi-Chevalley decompositions and precise properties of prolongations.
A notion of n-Lie algebra introduced by V.T. Filippov can be viewed as a generalization of a concept of binary Lie algebra to the algebras with n-ary multiplication law. A notion of Lie algebra can be extended to Z_2-graded structures giving a notion of Lie superalgebra. Analogously a notion of n-Lie algebra can be ext…
The abstract discusses homological stability in topological moduli spaces.
problem Homological stability in topological moduli spaces.
method Constructing canonical resolutions and introducing coefficient systems.
result Homological stability for various moduli spaces.
The paper studies traceless SU(2) representations and instanton gradings for specific knot types.
problem Analyzing traceless SU(2) representations and instanton gradings for two-bridge and (3,n)-torus knots. method Representation-theoretic data assembly and verification; explicit computation of character varieties and spectral-flow gradings.
result Explicit characterization of traceless representations and instanton gradings for specific knot types.
The abstract discusses cohomologies and deformations of Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures.
problem Characterizing and studying deformations and cohomologies of relative Rota-Baxter operators.
method Constructing graded Lie algebras and studying their Maurer-Cartan elements, cohomology, and deformations.
result Homomorphisms between cohomology groups of relative Rota-Baxter operators and deformation cohomology groups of left-symmetric algebroids.
The paper extends Riemann-Hilbert correspondence to foliations.
problem Understanding representations of Lie algebroids and groupoids in foliated settings.
method Establishing an A∞ de Rham theorem and constructing an integration functor. result An equivalence between ∞-representations of L∞-algebroids and ∞-representations of Lie ∞-groupoids for foliations. Study compactifies representations space of hyperbolic surfaces.
problem Compactify the space of maximal representations of hyperbolic surfaces.
method Vectorial length compactification, geometric interpretation, dual tree-graded space.
result Identify boundary with sphere of measured geodesic laminations.
New tools study curvature measures of convex bodies, revealing structured spaces.
problem Investigate translation invariant curvature measures of convex bodies.
method Introduce new tools to study curvature measures, proving conjectures about their structure.
result Space of curvature measures has length at most 2 as a representation of the general linear group in degrees 0 and n-2.
Colored knot polynomials possess a peculiar Z-expansion in certain combinations of differentials, which depends on the representation. The coefficients of this expansion are functions of the three variables (A,q,t) and can be considered as new distinguished coordinates on the space of knot polynomials, analogous to the…
We argue that some classical local geometries are of infinity origin, i.e. their smooth formal germs are (homotopy) representations of cofibrant (di)operads in spaces concentrated in degree zero. In particular, they admit natural infinity generalizations when one considers homotopy representations of that (di)operads i…
We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m, n) torus knot from the unique finite dimensional simple representation of the rational DAHA of type A, rank n - 1, and central character m/n. The conjectural differentials of Gukov, Dunfield and the third author receive an explicit algebra…
We observe that the determinant of the representation provides a little restriction for the structure of the graded quotients GkQF introduced in both [Algebr. Geom. Topol. 1 (2001) 39-55] and [J. Knot Theory Ramifications 13 (2004) 297-306] that any one of them does not contain the trivial 1-dimensional…
Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings. We study the variety of unitary representations of the fundamental group of U with certain restrictions related to the divisor. We show that the possible singularities of this variety as well as of the corresponding modul…
The paper proves an asymptotic development for weighted sums of multiplicity functions on a torus-manifold.
problem Proving an asymptotic development for weighted sums of multiplicity functions on a torus-manifold.
method Using equivariant Riemann-Roch theorem and graded Todd class, the paper proves an asymptotic development for weighted sums of multiplicity functions on a torus-manifold.
result The weighted sum of multiplicity functions has an asymptotic development in terms of the twisted Duistermaat-Heckman distributions associated to the graded Todd class of M.
Dual-edge spatial Jacobian image graph for interpretable diabetic retinopathy grading
problem Automated diabetic retinopathy grading from color fundus photographs
method Dual-edge spatial-Jacobian image graph
result 0.8076 accuracy, 0.8312 quadratic weighted kappa, 0.5915 macro-F1, 0.9330 adjacent-grade accuracy
Study Legendrian links using representations and sheaves.
problem Understanding Legendrian links through algebraic and geometric representations.
method Investigate an A∞ category of n-dimensional representations and conjecture equivalence to sheaves. result Established cohomological equivalence for Legendrian (2,m) torus links. A new model calculates colored Jones polynomials using homology.
problem Calculating colored Jones polynomials for different colors.
method Using a homological model based on the Lawrence representation and Kohno's result.
result Colored Jones polynomials are described as a graded intersection pairing of homology classes.
Classifies real trivectors in 9D using Galois cohomology.
problem Classifying real trivectors in R^9.
method Galois cohomology, theta-representations, centralizers computation.
result Classification of real trivectors into nilpotent, semisimple, and mixed types.
A formula connects two algebraic structures derived from a category.
problem Connecting two algebraic structures derived from a category.
method Using differential graded modular functors and Calabi-Yau structures.
result The action of a specific mapping class group element transforms one algebraic structure into another.
Defines Lie and Courant algebroids over Lie groupoids using homological vector fields.
problem Provides a Morita invariant definition of Lie and Courant algebroids over Lie groupoids.
method Views vector fields as Maurer-Cartan elements in a differential graded Lie algebra and as functors and natural transformations.
result Obtains a unifying conceptual framework for studying various algebraic structures.
Homotopy actions of Lie algebroids defined as L∞-algebra morphisms.
problem Defining and studying homotopy actions of Lie algebroids.
method Constructing homological vector fields on the semi-direct product and proving bijection.
result The construction is a bijection between homotopy actions and homological vector fields.