Defines invariants for reflection groups and connects them to Frobenius structures.
problem Understanding invariants for reflection groups and their relation to Frobenius structures.
method Defines good basic invariants and shows their connection to Frobenius structures.
result Good basic invariants for reflection groups lead to Frobenius structure constants.
New invariants show stronger virtual knot sets.
problem Classifying virtual knots using invariants.
method Defining F-order invariants using forbidden moves. result Set of F-order invariants is strictly stronger. The regulator is interested in proposing a capital adequacy test by specifying an acceptance set for firms' capital positions at the end of a given period. This set needs to be surplus-invariant, i.e., not to depend on the surplus of firms' shareholders, because the test means to protect firms' liability holders. We pr…
Paper defines invariants for elliptic Weyl groups and connects them to Frobenius structures.
problem Defining invariants for elliptic Weyl groups.
method Defines a set of good basic invariants and shows their connection to Frobenius structures.
result Good basic invariants give flat invariants and structure constants of Frobenius structures.
This paper presents relations between several types of closedness of a law-invariant convex set in a rearrangement invariant space X. In particular, we show that order closedness, σ(X,Xn∼)-closedness and σ(X,L∞)-closedness of a law-invariant convex set in $\mathc…
We construct a combinatorial invariant of 3-orbifolds with singular set a link that generalizes the Turaev torsion invariant of 3-manifolds. We give several gluing formulas from which we derive two consequences. The first is an understanding of how the components of the invariant change when we remove a curve from the …
Spray-invariant sets maintain geodesics on infinite-dimensional manifolds.
problem Geodesic preservation in infinite-dimensional manifolds.
method Definition of spray-invariant sets and analysis of their properties.
result Different geometric properties of spray-invariant sets based on their regularity.
Automates machine learning of correlations between knot invariants.
problem Discovering and validating new relationships between knot invariants.
method Trained a neural network on 200,000 sets of knot invariants to predict an output invariant.
result Found novel correlations not explained by known results in knot theory.
Homeomorphisms of hyperbolic 3-manifolds have invariant sets under certain conditions.
problem Understanding invariant sets for homeomorphisms on hyperbolic 3-manifolds.
method Shadowing techniques and analysis of invariant sets under positive speed with respect to a uniform foliation.
result Homeomorphisms of hyperbolic 3-manifolds are forced to have invariant sets, preventing them from being minimal.
When the signed weighted resolution set was defined as an invariant of pseudoknots, it was unknown whether this invariant was complete. Using the Gauss-diagrammatic invariants of pseudoknots introduced by Dorais et al, we show that the signed were-set cannot distinguish all non-equivalent pseudoknots. This goal is achi…
Maximal Laplacian algebras applied to invariant theory solved inverse problems.
problem Maximality of Laplacian algebras and their applications in invariant theory.
method Proof of maximality and applications to classical invariant theory.
result Introduction of generalized polarizations and if-and-only-if criterion.
For generic torus-invariant metrics, eigenspaces are 2D and nodal sets are connected hypersurfaces.
problem Understanding spectral multiplicity and nodal sets for generic torus-invariant metrics.
method Analyzing real Δg-eigenspaces and nodal sets for generic T-invariant metrics. result For generic T-invariant metrics, real Δg-eigenspaces are irreducible and have dimension at most 2, and nodal sets are connected hypersurfaces with specific properties. New tool: relative Hopf invariant for Poincaré surgery.
problem Non-simply connected Poincaré surgery.
method Relative Hopf invariant in equivariant setting.
result Established Poincaré embedding results in relative setting.
We find the complete set of fundamental invariants for systems of ordinary differential equations of order ≥4 under the group of point transformations generalizing similar results for contact invariants of a single ODE and point invariants of systems of the second and the third order. It turns out that starting fr…
Two complete knot invariants from diagrams, finite or infinite.
problem Classifying knots completely.
method Constructed two invariants from knot diagrams, finite or infinite.
result Finite set reveals knotting number.
Invariant Causal Set Covering Machines avoid spurious associations.
problem Learning algorithms for rule-based models are vulnerable to spurious associations.
method Building on invariant causal prediction, propose Invariant Causal Set Covering Machines for conjunctions/disjunctions of binary-valued rules.
result The method can identify causal parents of a variable of interest in polynomial time.
The paper defines cyclic sets from ribbon string links and connects them to quantum invariants.
problem Defining and relating cyclic sets from ribbon string links.
method Endowing ribbon string links with cyclic and cocyclic structures, relating to coend of a ribbon category via quantum invariants.
result Established a relationship between ribbon string links and quantum invariants.
Proposes a method to learn stable invariant sets in dynamical systems.
problem Learning stable invariant sets in general dynamical systems.
method Generalizes Manek and Kolter's approach by introducing projection onto latent space shapes and using invertible neural networks.
result Validates the method and shows its usefulness for long-term prediction.
We construct polynomial conformal invariants, the vanishing of which is necessary and sufficient for an n-dimensional suitably generic (pseudo-)Riemannian manifold to be conformal to an Einstein manifold. We also construct invariants which give necessary and sufficient conditions for a metric to be conformally relate…
New invariants for virtual knots and links defined via quiver representations.
problem Defining new invariants for virtual knots and links.
method Quiver representations associated to virtual biquandles and rings.
result New polynomial invariants for virtual knots and links.
Given a group action, known by its infinitesimal generators, we exhibit a complete set of syzygies on a generating set of differential invariants. For that we elaborate on the reinterpretation of Cartan's moving frame by Fels and Olver (1999). This provides constructive tools for exploring algebras of differential inva…
We relate the L^2 cohomology of a complete hyperbolic manifold to the invariant currents on its limit set.
Identifies conditions for multiple invariant probabilities in Markov kernels.
problem Global irreducibility and recurrence do not guarantee uniqueness of invariant probabilities.
method Uses Jordan decomposition of the difference of two invariant probabilities.
result A Markov kernel has more than one invariant probability if and only if it admits a visible absorbing decomposition.
We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.
For germs of subanalytic sets, we define two finite sequences of new numerical invariants. The first one is obtained by localizing the classical Lipschitz-Killing curvatures, the second one is the real analogue of the evanescent characteristics introduced by M. Kashiwara. We show that each invariant of one sequence is …
Define quiver representation-valued invariants for classical and virtual knots
problem Define quiver representation-valued invariants for classical and virtual knots
method Define an infinite family of quiver representation-valued invariants of classical and virtual knots associated to a choice of data vector consisting of a biquandle, abelian group, set of biquandle arrows weights with values in the abelian group, coefficient ring and set of biquandle endomorphisms.
result Extract four new polynomial invariants as decategorifications
In this thesis, we give a unification of the quantum WRT invariants. Given a rational homology 3-sphere M and a link L inside, we define the unified invariants, such that the evaluation of these invariants at a root of unity equals the corresponding quantum WRT invariant. In the SU(2) case, we assume the order of the f…
Abstract: Proves relative versions of group splitting results.
problem Group splittings and invariant sets.
method Relative versions of earlier results about almost invariant sets and splittings of groups.
result Existence and uniqueness of relative versions of algebraic regular neighbourhoods and JSJ decompositions.
The article discusses invariant measures outside homogeneous dynamics.
problem Understanding invariant measures in non-homogeneous settings.
method Comparing two recent results on rigidity and invariant measures.
result The importance of invariant and partially invariant measures in rigidity questions.
The paper studies invariant convex sets in representations with nontrivial copolarity.
problem Understanding the face structure of invariant convex sets in representations with nontrivial copolarity.
method Proves that the face structure of an invariant convex set is determined by its intersection with a fat section, and that a face is exposed if and only if the corresponding face of the intersection is exposed.
result The face structure of invariant convex sets is completely determined by their intersections with fat sections, and exposed faces are preserved.
Researchers address the generation of differential invariants for geometric structures.
problem Finite generation of differential algebra of relative differential invariants.
method Investigation of algebraic and differential properties, localization, weight analysis.
result Localization on a finite set of relative invariants makes the differential algebra finitely generated.
SGD tends to favor simpler subnetworks, improving generalization.
problem SGD's tendency to favor simpler subnetworks over complex ones.
method Identifying invariant sets and analyzing SGD's behavior around them.
result SGD collapses networks to simpler subnetworks, improving generalization.
This work preserves linear invariants in ensemble filters for non-Gaussian data assimilation.
problem Maintaining critical invariants like mass, stoichiometric balance, and charge in non-Gaussian data assimilation.
method Introducing a novel class of nonlinear ensemble filters using measure transport theory.
result Recovery of a constrained Kalman filter for Gaussian settings and combination with regularization techniques.
This study simplifies verification of invariants in oriented virtual knots.
problem Verifying invariants of oriented virtual knots is complex and time-consuming.
method Identifying a minimal generating set of oriented virtual Reidemeister moves.
result A four-element subset serves as a generating set for oriented virtual Reidemeister moves.
The paper proves a relation between four types of invariants.
problem Proving a precise relation between four types of invariants.
method Analyzing pseudo-Anosov homeomorphisms and cusped hyperbolic 3-manifolds at roots of unity.
result A precise relation between the Baseilhac-Benedetti invariants and the Bonahon-Liu-Wong-Yang invariants.
Interpolation can prevent classifiers from having desired invariance properties.
problem Invariance properties in over-parameterized models are often ineffective.
method Theoretical analysis and algorithm design for non-interpolating classifiers.
result Interpolating classifiers cannot satisfy desired invariance properties.
Unified invariant for immersed surface-links using biquandle cocycles.
problem Invariants for immersed surface-links in 4-space.
method Biquandle cohomology, Roseman moves, singular biquandle 3-cocycles.
result Invariance of state-sum invariant under all generating moves, including singular move (h).
The theory of acceptance sets and their associated risk measures plays a key role in the design of capital adequacy tests. The objective of this paper is to investigate, in the context of bounded financial positions, the class of surplus-invariant acceptance sets. These are characterized by the fact that acceptability …
We continue the work of [10], studying properties of digital images determined by fixed point invariants. We introduce pointed versions of invariants that were introduced in [10]. We introduce freezing sets and cold sets to show how the existence of a fixed point set for a continuous self-map restricts the map on the c…
Study geometric inequalities for CR-submanifolds using curvature invariants.
problem Geometric inequalities for CR-submanifolds in almost Hermitian spaces.
method Comparing mutual curvature invariants with Chen-type invariants and proving geometric inequalities.
result Proved geometric inequalities with intermediate mean curvature squared for CR-submanifolds.
Let f be the gluing map of a Heegaard splitting of a 3-manifold W. The goal of this paper is to determine the information about W contained in the image of f under the symplectic representation of the mapping class group. We prove three main results. First, we show that the first homology group of the three man…
A homology theory is developed for set-theoretic Yang-Baxter equations, and knot invariants are constructed by generalized colorings by biquandles and Yang-Baxter cocycles.
Let Fn be the free group on n generators and Γg the surface group of genus g. We consider two particular generating sets: the set of all primitive elements in Fn and the set of all simple loops in Γg. We give a complete characterization of distorted and undistorted elements in the corresponding Aut-in…
In this short survey article we collect the current state of the art in the nascent field of \textit{quantum enhancements}, a type of knot invariant defined by collecting values of quantum invariants of knots with colorings by various algebraic objects over the set of such colorings. This class of invariants includes c…
We analyze in this paper a random feature map based on a theory of invariance I-theory introduced recently. More specifically, a group invariant signal signature is obtained through cumulative distributions of group transformed random projections. Our analysis bridges invariant feature learning with kernel methods, as …
Proposes an alternative invariance penalty to address domain generalization issues.
problem Addressing domain generalization problems by finding invariant representations.
method Revisits the Gramian matrix of the data representation to propose an alternative invariance penalty.
result The proposed approach guarantees recovery of an invariant representation under mild conditions.
Optimal smooth subspaces approximate large data sets efficiently.
problem Approximating large data sets with invariant subspaces.
method Smooth functions under lattice translations or crystallographic groups, with optimal selection of Paley-Wiener space.
result Optimal lattice selection enhances approximation efficiency.
Study topological Iwasawa invariants for 3-sphere links, proving density results.
problem Detect and analyze Iwasawa invariants for 3-sphere links.
method Explicit criteria for detecting Iwasawa invariants, statistical analysis of 2-bridge links.
result Density of 2-bridge links with specific Iwasawa invariants.