arXiv research

A formula for Rademacher symbols in triangle groups is provided.

Researchers solved a number-theoretic hypothesis to determine the spin parity of k-differentials.

We discuss algebraic properties for the symbols of geometric first order differential operators on almost Hermitian manifolds and Kähler manifolds. Through study on the universal enveloping algebra and higher Casimir elements, we know algebraic relations for the symbols like the Clifford algebra. From the relations, we…

The paper explores the pentagon relation and its algebraic forms.

We consider the following construction of quantization. For a Riemannian manifold $M$ the space of forms on $T^*M$ is made into a space of (full) symbols of operators acting on forms on $M$. This gives rise to the composition of symbols, which is a deformation of the (``super'')commutative multiplication of forms. The …

Study on opers over complex manifolds of dimension one.

SE-RRMs solve structured problems like Sudoku and ARC-AGI by enforcing symbol equivariance.

Paper introduces a new symbol map for differential symmetry breaking operators.

We use geometric algebra techniques to give a synthetic and computationally efficient approach to Fierz identities in arbitrary dimensions and signatures, thus generalizing previous work. Our approach leads to a formulation which displays the underlying real, complex or quaternionic structure in an explicit and concept…

The purpose of this paper is to revisit the Bianchi identities existing for the Riemann and Weyl tensors in the combined framework of the formal theory of systems of partial differential equations (Spencer cohomology, differential systems, formal integrability) and Algebraic Analysis (homological algebra, differential …

New algorithm learns mixtures of any constant number of Gaussians robustly.

EPSTE: A geometric token and deep learning approach to estimating transfer entropy in neuroimaging time series

We consider an elliptic self-adjoint first order differential operator acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of our operator is assumed to be trace-free. We study the spectral function which is the sum of squar…

Gradients are natural first order differential operators depending on Riemannian metrics. The principal symbols of them are related to the enveloping algebra and higher Casimir elements. We give certain relations in the enveloping algebra, which induce not only identities for higher Casimir elements but also all Bochne…

The paper models musical motif transformations in Beethoven's works.

The consequences for Berezin's quantization on symmetric spaces of the identity of the set of coherent vectors orthogonal to a fixed one with the cut locus are stated precisely. It is shown that functions expressing the coherent states, the covariant symbols of operators, the diastasis function, the characteristic and …

We provide evidence for the conjecture that the Wodzicki-Chern classes vanish for all bundles with the group Z of invertible zeroth order pseudodifferential operators as structure group. In particular, we prove this vanishing if the structure group reduces to pseudodifferential operators with leading order symbol the i…

We consider an elliptic self-adjoint first order pseudodifferential operator acting on columns of m complex-valued half-densities over a connected compact n-dimensional manifold without boundary. The eigenvalues of the principal symbol are assumed to be simple but no assumptions are made on their sign, so the operator …

Unified approach to geometric structure equivalence problem.

We review the representation theory of the quantum group $U_εsl_2\mathbb{C}$ at a root of unity $ε$ of odd order, focusing on geometric aspects related to the 3-dimensional quantum hyperbolic field theories (QHFT). Our analysis relies on the quantum coadjoint action of De Concini-Kac-Procesi, and the theory of Heisenbe…

A Semi-Hidden Markov Model (SHMM) for bursty error channels is defined by a state transition probability matrix $A$, a prior probability vector $Π$, and the state dependent output symbol error probability matrix $B$. Several processes are utilized for estimating $A$, $Π$ and $B$ from a given empirically obtained or sim…

Simplified Ricci curvature for spherical fluid dynamics models.

VaSST uses soft symbolic trees for probabilistic symbolic regression.

We describe dimensionally constrained symbolic regression which has been developed for mass measurement in certain classes of events in high-energy physics (HEP). With symbolic regression, we can derive equations that are well known in HEP. However, in problems with large number of variables, we find that by constraini…

Paper closes neural-symbolic learning loop with grammar model and back-search algorithm.

Reinforcement learning and symbolic planning have both been used to build intelligent autonomous agents. Reinforcement learning relies on learning from interactions with real world, which often requires an unfeasibly large amount of experience. Symbolic planning relies on manually crafted symbolic knowledge, which may …

For an arbitrary Riemannian manifold $X$ and Hermitian vector bundles $E$ and $F$ over $X$ we define the notion of the normal symbol of a pseudodifferential operator $P$ from $E$ to $F$. The normal symbol of $P$ is a certain smooth function from the cotangent bundle $T^*X$ to the homomorphism bundle $Hom (E,F)$ and dep…

Based on the ideas of Optimal Control, we introduce the new basic characteristic of a bracket generating distribution, the Jacobi symbol. In contrast to the classical Tanaka symbol, the set of Jacobi symbols is discrete and classifiable. We give an explicit and unified algebraic procedure for the construction of the ca…

Symbolic knowledge in neural models can inadvertently make them more vulnerable to adversarial attacks.

We introduce mod 3 triple Milnor invariants and triple cubic residue symbols for certain primes of the Eisenstein number field $\mathbb{Q}(\sqrt{-3})$, following the analogies between knots and primes. Our triple symbol generalizes both the cubic residue symbol and Rédei's triple symbol, and describes the decomposition…

The symbolic dynamics technique is well-known for low-dimensional dynamical systems and chaotic maps, and lies at the roots of the thermodynamic formalism of dynamical systems. Here we show that this technique can also be successfully applied to time series generated by complex systems of much higher dimensionality. Ou…

Complex $b$-$6j$ symbols relate to hyperbolic tetrahedron volumes and determinants.

The paper classifies symbols of differential operators on vector bundles.

NeSS combines neural and symbolic approaches for better compositional generalization.

Defines transverse symbols for foliated manifolds and proves their K-homology class.

A new method for spotting symbols in CAD images reduces annotation costs and improves accuracy.

Paper proposes a bijective approach for signal/symbol translation using variational auto-encoders.

The Wodzicki residue and the cut-off integral extend to classical symbol-valued forms. We show that they obey a Stokes' type property and that the extended Wodzicki residue can be interpreted as a complex residue like the ordinary one. In the case of cut-off integrals, Stokes' property (i.e. vanishing on exact forms) o…

Bayesian symbolic regression automates model discovery from data.

Symbolic regression finds simple formulas for implied volatility.

Study shows how transformers classify symbols without naming them, proving a margin-versus-collision criterion.

S2KAN integrates symbolic primitives into neural network activations for improved interpretability.

Symbolic LSTM improves time series forecasting by reducing hyperparameter sensitivity.

Introduces canonical connections for sub-Riemannian manifolds with constant symbol.

Meta-learning symbolic default hyperparameters from dataset properties.

ISR creates analytical relationships from data via invertible maps.

Bayesian symbolic regression uncovers missing physics from data with uncertainty quantification.

Quantum $6j$-symbols linked to tetrahedra angles and volumes.