Study uses geometric algebra to analyze credit cycles, revealing dangerous feedback loops.
problem Understanding and predicting dangerous feedback loops in credit cycles.
method Represent economic states as multi-vectors in Clifford algebra, focusing on bivector elements for rotational coupling.
result Geometric relationship between unemployment and credit contraction shifts from simple correlation to dangerous rotational dynamics during crises.
New algebraic framework for Jacobi manifolds connects geometric mechanics and dimensional analysis.
problem Lack of clear algebraic interpretation for Jacobi manifolds.
method Developed a dimensioned algebra approach to capture algebraic counterparts of Jacobi manifolds.
result Poly-Jacobi manifolds provide a new connection between geometric mechanics and dimensional analysis.
New tools for analyzing Kähler manifolds, proving operator algebra and asymptotic kernel.
problem Analyzing Berezin-Toeplitz operators on Kähler manifolds.
method Introducing new tools for analytic microlocal analysis.
result Space of analytic Berezin-Toeplitz operators is an algebra.
Study non-integer power-law potentials for Schrödinger operators using Lie-Rinehart algebras.
problem Analyzing Schrödinger operators with non-integer power-law potentials.
method Using Lie-Rinehart algebras and microlocal analysis.
result Microlocal analysis can be applied to Schrödinger operators with non-integer power-law potentials.
New proof found for Khovanov's assertion about Frobenius algebra twists.
problem Proving the isomorphism between chain complexes of twisted Frobenius algebras and link diagrams.
method Detailed analysis of configurations of circles in each state.
result A new proof of Khovanov's assertion with a detailed analysis of configurations.
Hilbert(ian) A-modules over finite von Neumann algebras A with a faithful normal trace state (from global analysis) and Hilbert W*-modules over A (from operator algebra theory) are compared, and a categorical equivalence is established. The correspondence between these two structures sheds new light on basic results in…
Geometric AD framework simplifies derivative computation in JAX.
problem Efficient and accurate automatic differentiation.
method Jet functors and Weil algebras for geometric analysis.
result Unified view of derivative propagation with algebraic exactness.
The paper proposes a method to compute higher infinitesimals in numerical and symbolic analysis.
problem Computing higher-order derivatives with higher infinitesimals.
method Automatic differentiation in terms of C-infinity rings and Weil algebras.
result A unifying theoretical framework for multivariate higher-order derivatives.
Efficiently solves inverse PDE problems with Gaussian processes.
problem Solving inverse problems in linear PDEs with noisy data.
method Gaussian process regression with algebraic priors.
result High accuracy and computational efficiency achieved.
New abelian quotient found in symplectic derivation Lie algebra.
problem Understanding the structure of symplectic derivation Lie algebra.
method Computational approach to abelianization of the weight 12 part.
result 1-dimensional weight 12 part for g≥8. We classify the algebraic curvature tensors which are both Osserman and complex Osserman in all but a finite number of exceptional dimensions.Information concerning the possible eigenvalue structures, which is provided by methods of algebraic topology, plays a central role in the analysis.
Survey on positivity in vector bundles, inspired by Brunn-Minkowski theorems.
problem Positivity of vector bundles in complex analysis, Kähler geometry, and algebraic geometry.
method Inspiration from Brunn-Minkowski theorems.
result Survey results on positivity in vector bundles.
Isomorphic algebra connects Toeplitz to Heisenberg group.
problem Connecting Toeplitz algebra to Heisenberg group.
method Isomorphism between Toeplitz algebra and Heisenberg group ideal.
result Found isomorphism between algebra and Heisenberg group ideal.
This paper offers a new algebraic perspective of GCCA using subspace intersection.
problem Finding common variables across multiple feature representations.
method Subspace intersection approach based on a (bi-)linear generative model.
result GCCA is equivalent to subspace intersection, with conditions for identifiable common subspace.
Unified framework for complex, split-complex, and dual numbers.
problem Analytic and geometric scope of real-analytic functions.
method Generalized Cauchy-Riemann structure and unified real algebra family.
result Milnor-Le type fibration theorem for nondegenerate algebras.
Novel analysis of neural networks using geometric algebra and convex optimization.
problem Understanding the inner workings of deep neural networks.
method Geometric (Clifford) algebra and convex optimization.
result Optimal weights are given by the wedge product of training samples.
It is shown that the non-trivial cocycles on simple Lie algebras may be used to introduce antisymmetric multibrackets which lead to higher-order Lie algebras, the definition of which is given. Their generalised Jacobi identities turn out to be satisfied by the antisymmetric tensors (or higher-order `structure constants…
Develops method to construct Lie algebra weight system kernel using Vogel algebra.
problem Detecting correlators and distinguishing knots in 3D Chern-Simons theory.
method Uses Vogel's Λ algebra and Jacobi diagrams.
result Explicitly provides Jacobi diagrams in the kernel of sl_N weight system.
The problem of determining the volume of a tubular neighbourhood has a long and rich history. Bounds on the volume of neighbourhoods of algebraic sets have turned out to play an important role in the probabilistic analysis of condition numbers in numerical analysis. We present a self-contained derivation of bounds on t…
The paper analyzes privacy leakage in federated learning using linear algebra and optimization theory.
problem Privacy leakage in federated learning despite its promise for data privacy.
method Theoretical analysis from linear algebra and optimization theory perspectives.
result Derives sufficient conditions to prevent data reconstruction attacks and establishes an upper bound on privacy leakage.
We give a global picture of the Ricci flow on the space of three-dimensional, unimodular, nonabelian metric Lie algebras considered up to isometry and scaling. The Ricci flow is viewed as a two-dimensional dynamical system for the evolution of structure constants of the metric Lie algebra with respect to an evolving or…
Study local and global aspects of complex plane curve embeddings.
problem Local and global problems of complex plane curve embeddings.
method Braid monodromy, local and global analysis.
result Historical progress in understanding complex plane curve embeddings.
Smooth algebra analysis for one-dimensional singular foliations.
problem Analyzing smooth algebras of one-dimensional singular foliations.
method Analyzing natural ideals and using Dixmier-Malliavin theorem.
result Smooth algebras of one-dimensional singular foliations are pairwise nonisomorphic.
Geometric analysis on real analytic manifolds using seminorms.
problem Characterizing operations on real analytic manifolds and vector bundles.
method Using seminorms and geometric decompositions of jet bundles.
result New characterizations of real analytic mappings and operations.
Survey on symmetry in manifold structures.
problem Classifying manifolds with differential-geometric structures.
method Algebra, dynamics, and analysis techniques.
result Illustration of various techniques in action.
Witt algebra acts on Khovanov-Rozansky homology of links.
problem Understanding algebraic structures in knot theory.
method Construction of Witt algebra action on Khovanov-Rozansky homology.
result Induced maps between twists of the homology via link cobordisms.
We discuss and develop some connections between analysis on singular spaces and operator algebras, as presented in my sequence of four lectures at the conference "Noncommutative geometry and applications," Frascati, Italy, June 16-21, 2014. Therefore this paper is mostly a survey paper, but the presentation is new, and…
Algebraic definition classifies Markowitz markets simplifying portfolio optimization.
problem Complexity in Markowitz market classification.
method Algebraic definition and isomorphism classification.
result Classification shows little beyond portfolio optimization insights.
Researchers solve a 25-year-old conjecture about vector fields.
problem Proving a 25-year-old conjecture about divergence-free vector fields.
method Analysis of a Leibniz algebra underlying these vector fields.
result Construction of the universal central extension for divergence-free vector fields and diffeomorphisms.
We solve a long-standing question about the monodromy of certain complex surfaces.
problem When is the monodromy group of an algebraic family of complex varieties arithmetic?
method Topological analysis of the 'geometric' monodromy, valued in the mapping class group of the fiber.
result We resolve the question affirmatively for Atiyah-Kodaira manifolds.
Some natural hidden symmetries in the Verma modules over the Virasoro algebra are constructed in terms of geometric quantization. Their differential geometric meaning is established and their expression via qR-conformal symmetries in the Verma modules over the Lie algebra sl(2,C) is found. The analysis and the unr…
The paper explores geometric and algebraic structures on Lie groups.
problem Investigating F-manifolds and Fextman-algebras on Lie groups. method Constructing a canonical connection and analyzing curvature and holonomy.
result Established the integrability of a Poisson-algebra distribution.
Geometrically connects toric varieties to normed spaces.
problem Connecting algebraic geometry with convex analysis.
method Establishes a 1-1 correspondence using topological models.
result Toric varieties correspond to horofunction compactifications of polyhedral norms.
Study differential and integral calculus on noncommutative C*-algebras.
problem Develop calculus on noncommutative spaces.
method Formal smooth structure on nonpure states of C*-algebras.
result Prove Stokes' theorem in both commutative and noncommutative settings.
We develop an algebraic framework for the description and analysis of financial behaviours, that is, behaviours that consist of transferring certain amounts of money at planned times. To a large extent, analysis of financial products amounts to analysis of such behaviours. We formalize the cumulative interest compliant…
We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the…
New geometric proofs and interpretations of scattering diagrams and theta functions.
problem Analyzing the asymptotic behavior of Maurer-Cartan elements for differential graded Lie algebras.
method Asymptotic analytic approach and differential geometric proofs.
result Alternative proofs of consistent completion of scattering diagrams and geometric interpretations of theta functions.
We solved the Schr{ö}dinger equation for a particle in a uniform magnetic field in the n-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus T^n = R^n / Λ is defined as a quotient of the Euclidean space R^n by an arbitrary n-dimensional lattice Λ. The lattice is not neces…
Machine learning accelerates Lie algebra computations.
problem Computing tensor products and branching rules of Lie algebras.
method Machine learning for Lie algebra computations.
result Achieves significant speed-ups in Lie algebra computations.
We classify superintegrable systems in the Euclidean plane using algebraic geometry.
problem Classifying superintegrable systems in the Euclidean plane.
method Derived and solved a system of algebraic equations to classify systems.
result Associated a unique line triple arrangement to each superintegrable system.
Paper analyzes stability of discrete-time hypercomplex-valued Hopfield neural networks.
problem Stability of discrete-time hypercomplex-valued Hopfield neural networks.
method Introduces real-part associative hypercomplex number systems and B-projection functions to ensure stability. result Stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks confirmed.
New transforms improve signal classification and data analysis.
problem Improving signal classification and data analysis.
method Algebraic generative models and transport transforms.
result Classes of signals are transformed into convex sets, simplifying classification.
We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition of the plane. Compared to …
New feature map for topological data analysis improves classification performance.
problem Lack of effective feature maps for topological data analysis.
method Realize barcodes as paths in a vector space, compute path signature, resulting in a feature map.
result Achieves state-of-the-art results on classification benchmarks.
We explore the graded and filtered formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how thes…
Study of skein invariants on tori for various groups and quantum parameters.
problem Analysis of G-skein theory invariants on tori for different groups and parameters. method Combinatorial and algebraic methods, including DAHA and skein relations.
result Isomorphisms and homomorphisms between skein algebras and DAHA, proving equivalence of tangles.
Empirical moment matrix reveals properties of point clouds.
problem Uncovering properties of point clouds, especially those with singular support.
method Combining statistics, real algebraic geometry, and approximation theory.
result The empirical moment matrix provides insights into data analysis.
This work speeds up fHMM analysis by tensor algebra.
problem Scalability issues in analyzing factorial hidden Markov models.
method Tensorized algorithms and scalable filtering methods.
result Significant improvement in computational performance.