The paper constructs new algebraic structures from Lie algebras and ternary Nambu-Lie algebras, leading to Yang-Baxter operators.
problem Constructing new algebraic structures from Lie algebras and ternary Nambu-Lie algebras.
method Using compositions of binary Lie algebras, 3-Lie algebras, and ternary Nambu-Lie algebras, the paper constructs ternary self-distributive objects and Yang-Baxter operators.
result The constructed Yang-Baxter operators are not gauge equivalent to the transposition operator and can be deformed to new solutions.
We extend the Nambu bracket to 1-forms. Following the Poisson-Lie case, we define Nambu-Lie groups as Lie groups endowed with a multiplicative Nambu structure. A Lie group G with a Nambu structure P is a Nambu-Lie group iff P=0 at the unit and the Nambu bracket of left (right) invariant forms is left (right) invariant.…
The paper extends ternary algebra concepts using cube roots of unity.
problem Extending algebraic structures from binary to ternary multiplication.
method Introducing ternary associator, commutator, and Lie algebra at cube roots of unity.
result Derived an identity for ternary commutator based on GA(1,5). We introduce a notion of ternary distributive algebraic structure, give examples, and relate it to the notion of a quandle. Classification is given for low order structures of this type. Constructions of such structures from ternary bialgebras are provided. We also describe ternary distributive algebraic structures com…
New algebraic structure for vector bundles with special properties.
problem Developing new algebraic structures for vector bundles.
method Introducing para-associative algebroids and showing local triviality conditions.
result Existence of a differential connection is necessary and sufficient for local triviality.
We study some properties of coisotropic submanifolds of a manifold with respect to a given multivector field. Using this notion, we generalize the results of Weinstein \cite{wein} from Poisson bivector field to Nambu-Poisson tensor or more generally to any multivector field. We also introduce the notion of Nambu-Lie gr…
We define homology of ternary algebras satisfying axioms derived from particle scattering or, equivalently, from the third Reidemeister move. We show that ternary quasigroups satisfying these axioms appear naturally in invariants of Reidemeister, Yoshikawa, and Roseman moves. Our homology has a degenerate subcomplex. T…
Lectures explore how differential methods improve understanding of algebraic group orbit spaces.
problem Understanding structure of invariants and orbit spaces of algebraic Lie groups.
method Combines algebraic and differential viewpoints to study orbit spaces.
result Differential approach provides deeper insights into invariants and orbit spaces.
We show that some ternary quasigroups appear naturally as invariants of classical links and links on surfaces. We also note how to obtain from them invariants of Yoshikawa moves. In our previous paper, we defined homology theory for algebras satisfying two axioms derived from the third Reidemeister move. In this paper,…
We introduce a way to color the regions of a classical knot diagram using ternary operations, so that the number of colorings is a knot invariant. By choosing appropriate substitutions in the algebras that we assign to diagrams, one obtains the relations from the knot group, and from the core group. Using the ternary o…
Quantum invariant derived from ternary cohomology of self-distributive structures.
problem Defining and proving a quantum invariant from ternary cohomology.
method Constructing a ribbon category from a TSD set, showing it coincides with the cocycle invariant.
result The ribbon cocycle invariant is a quantum invariant.
New braided Frobenius algebras created from specific Hopf algebras.
problem Creating new algebraic structures from Hopf algebras.
method Heap operation and Yang-Baxter operator on tensor product.
result Heap operation induces a braiding compatible with Frobenius operations.
This paper investigates higher order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that n-Lie algebroid structures correspond to n-ary generalization of Gerstenhaber algebras and are implied by n-ary generalization of linear Poisson structures on the dual bundle. A Nambu-…
We review supervised learning and deep neural network design for learning membership on algebraic varieties. We demonstrate that these trained artificial neural networks can predict the entanglement type for quantum states. We give examples for detecting degenerate states, as well as border rank classification for up t…
We introduce \textit{Niebrzydowski algebras}, algebraic structures with a ternary operation and a partially defined multiplication, with axioms motivated by the Reidemeister moves for Y-oriented trivalent spatial graphs and handlebody-links. As part of this definition, we identify generating sets of Y-oriented Reid…
We define a homology for ternary groups using both associativity and skew elements. We describe the odd-even construction which yields many examples of ternary groups. We define the ternary knot group, consider its homomorphisms into ternary groups, and discuss the applications.
We describe various properties and give several characterizations of ternary groups satisfying two axioms derived from the third Reidemeister move in knot theory. Using special attributes of such ternary groups, such as semi-commutativity, we construct a ternary invariant of curves immersed in compact surfaces, conside…
Novel ternary structures reveal new interpretations of linear connections.
problem Examining the ternary structure of Lie algebroid connections.
method Study of endomorphisms and explicit presentation of the endomorphism truss.
result Explicitly presented endomorphism truss of linear connections.
We prove that the quotients of the group algebra of the braid group introduced by L. Funar in Comm. Math. Phys., 1995, collapses in characteristic distinct from 2. In characteristic 2 we define several quotients of it, which are connected to the classical Hecke and Birman-Wenzl-Murakami quotients, but which admit in ad…
New cohomology theories for heaps and ternary operations linked to group cohomology.
problem Defining and studying cohomology theories for heaps and ternary operations.
method Introduced para-associative and heap cohomology theories, and ternary self-distributive cohomology with abelian heap coefficients.
result Heap cohomology is related to group cohomology via a long exact sequence, and injects into ternary self-distributive cohomology.
Motivated by large-scale Collaborative-Filtering applications, we present a Non-Commuting Latent Factor (NCLF) tensor-completion approach for modeling three-way arrays, which is diagonal like the standard PARAFAC, but wherein different terms distinguish different kinds of three-way relations of co-clusters, as determin…
Heap theory applied to framed links yields new invariants.
problem Developing invariants for framed links using heap theory.
method Introducing fundamental heap, defining cocycle invariant using ternary cohomology.
result Found cocycles and computed invariants for specific link families.
Introduces Lie semiheaps and their relation to Lie groups and bundles.
problem Defining and understanding Lie semiheaps and their properties.
method Introducing Lie semiheaps and proving their properties in relation to Lie groups and bundles.
result Established the existence of left-invariant vector fields on Lie semiheaps.
Ternary MobileNets improve efficiency and accuracy on constrained devices.
problem Efficiently compressing MobileNets for real-time applications on constrained devices.
method Per-layer hybrid filter banks for ternary quantization of MobileNets.
result 27.98% energy savings and 51.07% reduction in model size with comparable accuracy.
Smart Quantization adapts binary and ternary quantization for neural networks.
problem Resource constraints in deploying neural networks on devices with limited resources.
method Adaptive combination of binary and ternary quantization with a regularization function.
result Adapts quantization depth during training to maintain high model accuracy.
Study of SO(3)-irreducible geometry in complex 5D and ternary Pauli exclusion principle.
problem Exploring SO(3)-irreducible geometry in complex 5D.
method Defined a ternary skew-symmetric tensor, split the 10D space into irreducible SO(3) subspaces, found invariants and defined geometric structures.
result Defined a SO(3)-irreducible geometric structure on a 5D complex Hermitian manifold.
EC2T creates sparse and ternary neural networks for resource-constrained devices.
problem Deploying deep neural networks on resource-constrained devices.
method Entropy-Constrained Trained Ternarization (EC2T) framework.
result EC2T creates sparse and ternary neural networks that are efficient in terms of storage and computation.
Recurrent neural networks (RNNs) have shown excellent performance in processing sequence data. However, they are both complex and memory intensive due to their recursive nature. These limitations make RNNs difficult to embed on mobile devices requiring real-time processes with limited hardware resources. To address the…
The paper describes topological properties of arcs and crossings in knot theory.
problem Understanding the topological nature of arcs and crossings in knot theory.
method Topological description of arcs and crossings as isotopy classes of probes, homotopy classes of diagram elements.
result Sets of arcs and crossings are fundamental for algebraic objects like quandles, partial ternary quasigroups, biquandloids, and crossoids.
Unified cosmological and Einstein polytope theories.
problem Unified understanding of cosmological and Einstein polytope theories.
method Unified combinatorial perspective of cosmological and Einstein polytope theories.
result Unified construction of cosmological and Einstein polytope theories.
Paper ranks stocks by compression risk, not volatility.
problem Investment risk not correlated with stock price volatility.
method Binary-ternary compressive coding of price change time series.
result Compression risk is a better indicator of stock investment risk.
New inequality for ternary variables improves on existing measures.
problem Analyzing excess losses and weighted majority votes with ternary random variables.
method Developed a split-kl inequality and its PAC-Bayes extension.
result Outperforms existing inequalities in certain regimes.
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…
Compressed sensing (CS) is a sampling theory that allows reconstruction of sparse (or compressible) signals from an incomplete number of measurements, using of a sensing mechanism implemented by an appropriate projection matrix. The CS theory is based on random Gaussian projection matrices, which satisfy recovery guara…
Algorithm identifies Copeland winners in dueling bandits with ternary feedback.
problem Identifying Copeland winners in dueling bandits with indifferences.
method Proposed POCOWISTA algorithm with a sample complexity close to lower bound.
result Algorithm shows excellent performance, even for conventional dueling bandits.
Selective classification improves trading strategies by abstaining from predictions.
problem Designing effective trading strategies using selective classification.
method Extends binary or multi-class classifiers to allow abstaining from predictions, evaluates across different feature sets and classifiers.
result Selective classifiers can improve trading performance by avoiding poor predictions.
Study examines null vector fields on Lorentzian manifolds.
problem Understanding the structure of null vector fields on Lorentzian manifolds.
method Investigates the bundle structure and ternary product of nowhere vanishing null vector fields.
result Null tangent bundle is a non-polynomial graded bundle with a para-associative ternary product.
The group of C1-diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson's groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin's higher dimensional generalizations nV of Thompson's group V arise…
FTTQ optimizes quantized networks in federated learning, reducing communication costs.
problem Redundant parameters in full-precision models lead to excessive communication costs in federated learning.
method FTTQ algorithm that optimizes quantized networks on clients through self-learning quantization factors.
result FTTQ reduces communication costs and can achieve slightly better performance on non-IID data.
The use of deep neural networks in edge computing devices hinges on the balance between accuracy and complexity of computations. Ternary Connect (TC) \cite{lin2015neural} addresses this issue by restricting the parameters to three levels −1,0, and +1, thus eliminating multiplications in the forward pass of the net…
Study curvature loci of 3-manifolds in R^6 and R^5.
problem Characterize curvature loci of 3-manifolds in different dimensions.
method Refine affine classification of real nets of quadrics, study singularities, and analyze systems of ternary cubics.
result Obtain generic curvature loci and singularities of 3-manifolds.
GOCPD detects change points by maximizing the probability of two independent models.
problem Large false discovery rates in online change point detection methods.
method GOCPD uses ternary search to find change points by maximizing the probability of two independent models.
result GOCPD accelerates CPD with logarithmic complexity for single change point detection.
Recent breakthroughs in computer vision make use of large deep neural networks, utilizing the substantial speedup offered by GPUs. For applications running on limited hardware, however, high precision real-time processing can still be a challenge. One approach to solving this problem is training networks with binary or…
Machine learning-based applications are increasingly prevalent in IoT devices. The power and storage constraints of these devices make it particularly challenging to run modern neural networks, limiting the number of new applications that can be deployed on an IoT system. A number of compression techniques have been pr…
TRF uses ternary random features to improve ML performance without extra computation.
problem Improving ML performance with less computation and storage.
method Proposes Ternary Random Features (TRF) for random features compression.
result TRF asymptotically yields the same limiting kernel as original matrices, with improved efficiency.
We inductively define layers of colorings of knot and knotted surface diagrams using ternary quasigroups. Homological invariants from such systems of colorings use shorter differentials and of higher degree than the standard homology differentials, and give access to typically more complex homology groups.
The paper is based on relations between a ternary symmetric form defining the SO(3) geometry in dimension five and Cartan's works on isoparametric hypersurfaces in spheres. As observed by Bryant such a ternary form exists only in dimensions n_k=3k+2, where k=1,2,4,8. In these dimensions it reduces the orthogonal group …
Proposes volumization for neural networks to control bias-variance tradeoff.
problem Improving generalization and preventing memorization in neural networks.
method Defines a physical volume for weights, interpolating between L2 and L∞ regularization.
result Volumization interpolates between weight decay and clipping, improving generalization.