Investigates neural codes and their embeddings, proving conjectures and introducing new code types.
problem Analyzing neural codes and their embedding dimensions.
method Combinatorial, topological, and algebraic analysis; proving conjectures; introducing new neural code types.
result Proves conjectures about neural codes and their embeddings, introduces new code types.
Ring-reservoir networks simplify graph embeddings efficiently.
problem Efficient graph embeddings using deep neural networks.
method Progressive simplification of Reservoir Computing models to ring topology.
result Ring-reservoir networks show consistent advantages in predictive performance.
T-Basis represents neural network tensors with fewer parameters.
problem Efficiently representing neural network tensors with fewer parameters.
method T-Basis uses Tensor Rings to represent tensors in a neural network, parameterizing them with a small number of coefficients.
result T-Basis achieves high compression rates with minimal performance loss.
Flexible VHDL design for multiple neural networks on FPGAs.
problem Inflexible neural network designs for FPGAs.
method Proposes a flexible VHDL structure with multiple processor groups.
result Allows training and testing of multiple neural networks on multiple FPGAs.
Deep neural networks have demonstrated state-of-the-art performance in a variety of real-world applications. In order to obtain performance gains, these networks have grown larger and deeper, containing millions or even billions of parameters and over a thousand layers. The trade-off is that these large architectures r…
A new tensor ring mixture model improves density estimation efficiency.
problem Efficient probability density estimation in statistical machine learning.
method Tensor ring decomposition with mixture model for adaptive weights.
result Enhanced expressive capability and flexibility in density estimation.
The paper explores zero-divisors and idempotents in quandle rings, proving their absence in certain cases.
problem Understanding zero-divisors and idempotents in quandle rings.
method Development of quandle rings theory, definition of orderability, computation of idempotents, and analysis of automorphism groups.
result Quandle rings of left or right orderable quandles with semi-latin structure have no zero-divisors.
We employ techniques of machine-learning, exemplified by support vector machines and neural classifiers, to initiate the study of whether AI can "learn" algebraic structures. Using finite groups and finite rings as a concrete playground, we find that questions such as identification of simple groups by "looking" at the…
It is inevitable to train large deep learning models on a large-scale cluster equipped with accelerators system. Deep gradient compression would highly increase the bandwidth utilization and speed up the training process but hard to implement on ring structure. In this paper, we find that redundant gradient and gradien…
Lie-Rinehart algebras over C∞-rings defined and studied.
problem Defining and studying Lie-Rinehart algebras over C∞-rings. method Defining Lie-Rinehart algebras over C∞-rings and showing their relationship with Poisson C∞-rings. result A natural Poisson bracket on the C∞-ring associated with a Lie-Rinehart algebra over a C∞-ring. A new invariant of Poisson manifolds, a Poisson K-ring, is introduced. Hypothetically, this invariant is more tractable than such invariants as Poisson (co)homology. A version of this invariant is also defined for arbitrary algebroids. Basic properties of the Poisson K-ring are proved and the Poisson K-rings are calcul…
Investigates differential smoothness of 3D skew polynomial rings.
problem Differential smoothness of 3D skew polynomial rings.
method Analyzes Bell and Smith's characterization of 3D skew polynomial rings.
result Provides insights into the differential smoothness of these rings.
We define a notion of stability for chiral ring of four dimensional N=1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N=1 field theory derived from D3 branes probing …
A classical theorem due to Quillen (1969) identifies the unitary bordism ring with the Lazard ring, which classifies the universal one-dimensional commutative formal group law. We prove an equivariant generalization of this result by identifying the homotopy theoretic Z/2-equivariant unitary bordism ring, in…
The paper examines differential smoothness in skew PBW extensions over polynomial rings.
problem Differential smoothness in skew PBW extensions over polynomial rings.
method Investigation of skew PBW extensions over commutative polynomial rings.
result Results on differential smoothness for skew PBW extensions over polynomial rings.
Criteria for smoothness of ambiskew polynomial rings.
problem Smoothness of ambiskew polynomial rings.
method Determined sufficient criteria for differential smoothness.
result Criteria for differential smoothness of ambiskew polynomial rings.
This paper calculates the skein algebra of the Borromean rings complement.
problem Calculating the skein algebra of the Borromean rings complement.
method Using the skein algebra definition and character variety, the polynomial ring quotient is determined.
result An explicit formula for the skein algebra of the Borromean rings complement is provided.
The paper explores idempotents in quandle rings and their connections to quandle coverings.
problem Understanding idempotents in quandle rings and their relation to quandle coverings.
method Investigation of idempotents in quandle rings, proving properties of idempotents in free products and unions of quandles.
result Integral quandle rings of quandles of finite type that are non-trivial coverings over nice base quandles admit infinitely many non-trivial idempotents.
Defines vector fields and differential forms on local C-infinity-ringed spaces.
problem No specific problem stated; focuses on mathematical definitions.
method Defines tangent sheaf, contractions, Lie derivatives, and proves Cartan equations.
result Standard Cartan calculus equations hold for local C-infinity-ringed spaces.
Researchers found only one hyperbolic structure for Borromean rings.
problem Characterizing hyperbolic structures in knot complements.
method Analyzing the fundamental group of the Borromean rings' complement and its representations in PSL(2,C).
result Borromean rings admit exactly one hyperbolic structure.
Global group laws connect equivariant bordism rings to formal group laws.
problem Establishing connections between equivariant bordism rings and formal group laws.
method Global homotopy theory framework; proving isomorphisms and universal properties.
result Equivariant bordism rings are isomorphic to Lazard rings for abelian Lie groups.
New argument for 3-manifold cohomology with F2 coefficients.
problem Characterization of 3-manifold cohomology rings with F2 coefficients. method New argument based on Postnikov's 1948 characterization using intersection rings.
result A new proof for the characterization of 3-manifold cohomology rings.
Paper computes hyperbolic structure of Borromean rings complement.
problem Computing hyperbolic structures for link complements.
method Classical construction of Thurston's sense.
result Exact computation of hyperbolic structure for Borromean rings.
New hyperbolic manifolds found with same trace ring.
problem Finding non-commensurable hyperbolic manifolds with identical trace rings.
method Proved existence of infinitely many non-commensurable manifolds with same ambient group and trace ring.
result Infinitely many non-commensurable hyperbolic manifolds with the same ambient group and trace ring.
We calculate the intersection ring of three-dimensional graph manifolds with rational coefficients and give an algebraic characterization of these rings when the manifold's underlying graph is a tree. We are able to use this characterization to show that the intersection ring obstructs arbitrary three-manifolds from be…
Homological algebra used to study local equivalence of complex rings.
problem Local equivalence of bounded complexes over polynomial rings.
method Homological algebra approach
result Results have been proved in many places in the literature.
In this paper we compute a presentation for the group of ring motions of the split union of a Hopf link with Euclidean components and a Euclidean circle. A key part of this work is the study of a short exact sequence of groups of ring motions of general ring links in R3. This sequence allowed us to build th…
Study introduces dynamical ideals for non-commutative rings and classifies knots and links.
problem Classifying surface knots and links in smooth 4-manifolds.
method Introduced dynamical analog of prime ideals for non-commutative rings and proved a factorization theorem.
result Classified surface knots and links in smooth 4-manifolds.
We show that solutions of Thurston equation on triangulated 3-manifolds in a commutative ring carry topological information. We also introduce a homogeneous Thurston equation and a commutative ring associated to triangulated 3-manifolds.
We build extensions of the arc rings, relate their centers to the cohomology rings of the Springer varieties, and categorify all level two representations of quantum sl(N).
Study Coxeter groups over fusion rings and their geometric realisations.
problem Understanding Coxeter groups and their embeddings.
method Investigate faithful realisations and Vinberg systems.
result Induce embeddings of hyperplane complements.
Differential K-theory gets a λ-ring structure.
problem Establishing a λ-ring structure in differential K-theory. method Splitting principle for differential K-theory, Adams operations construction.
result Differential K0-ring admits a λ-ring structure. Division algorithm for surface group rings yields standard complexes and cohomological dimensions.
problem Understanding cohomological dimensions of surface group actions.
method Division algorithm for group rings of surface groups.
result Some 2-complexes with surface fundamental groups are standard.
New Frobenius manifold structures found on Dicyclic group orbits.
problem Finding Frobenius manifold structures on orbits spaces of Dicyclic groups.
method Applying Dubrovin's method to Dicyclic groups.
result Dicyclic group orbits spaces acquire two Frobenius manifold structures.
We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are available at regular primes, but we seek more conceptual answers in terms of lo…
The paper explains the topological origin of the distinction between incidence theorems over division rings and fields.
problem Understanding the distinction between incidence theorems over division rings and fields.
method Extending the surface-graph approach to noncommutative settings, the paper analyzes the topological properties of graphs embedded on surfaces of different genera.
result Theorems associated with graphs on the sphere hold over any division ring, while those on surfaces of positive genus typically hold only if the ground ring is a field.
A C-infinity ring is a set equipped with n-ary operations corresponding to smooth n-ary functions on the real line (satisfying natural axioms). We prove that the cosimplicial abelian group associated to the de Rham complex of Euclidean space has the structure of a cosimplicial C-infinity ring. We also analyse the notio…
Explains the Borromean rings, icosahedron, and Poincaré homology sphere.
problem Exploring the relationship between Borromean rings, icosahedron, and Poincaré homology sphere.
method Introduction of topological concepts and geometric construction of icosahedral compound of octahedra.
result Proofs about the orientation-preserving symmetry group of an icosahedron and the linked nature of Borromean rings.
Generalizes cohomology ring result for combinatorial line arrangements.
problem Cohomology ring of boundary manifold for combinatorial line arrangements.
method Introduced boundary manifold, constructed homology cycles, computed cohomology ring.
result Cohomology ring of boundary manifold is isomorphic to double of Orlik-Solomon algebra.
New method uses quandle rings to distinguish knots and their mirrors.
problem Distinguishing knots and their mirror images.
method Combining idempotents in quandle rings with state sum invariants.
result Improved knot and mirror image distinction using fewer quandles.
Constructs differential forms on C∞-ringed spaces.
problem Developing a theory of differential forms for non-manifold spaces.
method Functorial construction of differential forms on local C∞-ringed spaces. result Stokes' theorem holds for integrated forms on simplices.
We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartments. Assuming the generalized Riemann hypothesis, this shows that the Steinberg module of a number ring is generated by integral apartments if and only if the ri…
L. Kauffman conjectured that a particular solution of the Chinese Rings puzzle is the simplest possible. We prove his conjecture by using low-dimensional topology and group theory. We notice also a surprising connection between the Chinese Rings and Habiro moves (related to Vassiliev invariants).
We present a deRham model for Chen-Ruan cohomology ring of abelian orbifolds. We introduce the notion of \emph{twist factors} so that formally the stringy cohomology ring can be defined without going through pseudo-holomorphic orbifold curves. Thus our model can be viewed as the classical description of Chen-Ruan cohom…
The ideal of relations in the (small) quantum cohomology ring of the generalized flag manifold G/B has been determined by B. Kim. We are going to point out a limited number of properties that, if they are satisfied by an R[q1,...,ql]-bilinear product ∘ on H∗(G/B)⊗R[q1,...,ql, then the ring $(H^*…
Study cohomology rings of Grassmannians using Clifford algebras and symmetric spaces.
problem Understanding cohomology rings of Grassmannians over different fields.
method Explicit generators and relations for de Rham cohomology rings, filtered deformations related to Clifford algebras.
result Explicit generators and relations for the de Rham cohomology rings of Grassmannians.
The abstract defines and studies a Tits building for commutative rings and proves a Solomon-Tits theorem under certain conditions.
problem Defining and studying a Tits building for commutative rings.
method Proving a Solomon-Tits theorem for commutative rings under specific conditions, defining Steinberg modules, and computing ranks and lengths.
result Proves a Solomon-Tits theorem for commutative rings satisfying certain conditions.
Study calculates ring structure in instanton homology for a surface with points.
problem Calculating the ring structure in instanton homology for a surface with points.
method Used singular instanton Floer homology and excision formula.
result Proved an excision formula for instanton Floer homology when n=1.