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With the rapid advancement of metasurfaces and the increasing demand for programmable metasurfaces to simplify information systems, wave-based computation using metasurfaces has emerged as an attractive research topic. To facilitate the…
We establish an operator--theoretic correspondence between periodic Bernoulli kernels and Hermite polynomials, framed through the umbral calculus and a quantum analogy. Starting from the analytic master function $F^\ast$, the periodic…
We develop a generating-function formulation for the symbolic reduction of multi-loop Feynman integrals. In this framework, integration-by-parts identities are rewritten as differential equations for sector-wise generating functions, so the…
We examine basis functions on momentum space for the three dimensional Euclidean Snyder algebra. We argue that the momentum space is isomorphic to the SO(3) group manifold, and that the basis functions span either one of two Hilbert spaces.…
A key module in neural transformer-based deep architectures is positional encoding. This module enables a suitable way to encode positional information as input for transformer neural layers. This success has been rooted in the use of…
Binary semirings such as the tropical, log, and probability semirings form a core algebraic tool in classical and modern neural inference systems, supporting tasks like Viterbi decoding, dynamic programming, and probabilistic reasoning.…
We present GRAPE (Group Representational Position Encoding), a unified framework for positional encoding based on group actions. GRAPE unifies two families of mechanisms: (i) multiplicative rotations (Multiplicative GRAPE) in…
A finite number of rational functions are compatible if they satisfy the compatibility conditions of a first-order linear functional system involving differential, shift and q-shift operators. We present a theorem that describes the…
Continuous vector representations of words and objects appear to carry surprisingly rich semantic content. In this paper, we advance both the conceptual and theoretical understanding of word embeddings in three ways. First, we ground…
Mapping words into a fixed-dimensional vector space is the backbone of modern NLP. While most word embedding methods successfully encode semantic information, they overlook phonetic information that is crucial for many tasks. We develop…
We introduce a superspace analogue of combinatorial Hopf algebras (Aguiar-Bergeron-Sottile, 2006), and show that the Hopf superalgebra of quasi-symmetric (resp. symmetric) functions in superspace (Fishel-Lapointe-Pinto, 2019) is a terminal…
Despite recent advances in the lattice representation theory of (generalized) symmetries, many simple quantum spin chains of physical interest are not included in the rigid framework of fusion categories and weak Hopf algebras. We…
Given a pool of observations selected from a sensor stream, input data can be robustly represented, via a multiscale process, in terms of invariant concepts, and themes. Applying this to episodic natural language data, one may obtain a…
The dispute of how the human brain represents conceptual knowledge has been argued in many scientific fields. Brain imaging studies have shown that the spatial patterns of neural activation in the brain are correlated with thinking about…
We present an algebraic characterization of the complexity classes Logspace and NLogspace, using an algebra with a composition law based on unification. This new bridge between unification and complexity classes is inspired from proof…
The integration of structured hierarchical embeddings into transformer-based architectures introduces a refined approach to lexical representation, ensuring that multi-scale semantic relationships are preserved without compromising…
Like the letters in the alphabet forming words, reusing components of a heterogeneous mixture is an efficient strategy for assembling a large number of target structures. Examples range from synthetic DNA origami to proteins self-assembling…
We introduce a Recursive INsertion-based Encoder (RINE), a novel approach for semantic parsing in task-oriented dialog. Our model consists of an encoder network that incrementally builds the semantic parse tree by predicting the…
This research started with an algebra for reasoning about rely/guarantee concurrency for a shared memory model. The approach taken led to a more abstract algebra of atomic steps, in which atomic steps synchronise (rather than interleave)…
We present a theorem which elucidates the connection between self-duality of Markov processes and representation theory of Lie algebras. In particular, we identify sufficient conditions such that the intertwining function between two…