arXiv subjects

Paul Martin

Publications and source records attributed to Paul Martin.

At least 19 recordsLinked to original sources

arXiv2026

Quantifying cell shape and density fluctuations in epithelial tissue in vivo

Controlling changes in cell shape are crucial for many biological processes, such as tissue development and wound healing. Tissues typically are heterogeneous with a variety of cell shapes and sizes. Of particular interest are local deviations from an average cell shape and size. These fluctuations may extend and transmit across tissues, potentially offering valuable insights into tissue characteristics such as variations in effective "stiffness" or rigidity. In this study, we present a theoretical framework that captures the dynamics of epithelial cell shapes within tissue, incorporating both their average behaviour and fluctuation patterns. We model cells as interacting soft ellipsoids of varying size and aspect ratio. Coarse-graining our model, we obtain a set of continuum stochastic differential equations from which we derive spatial-temporal correlation functions. These correlation functions fit with our experimental data from the developmental process of the \textit{Drosophila} pupal wing. From the correlation functions, critical parameters representing active cell shape changes and effective tissue "stiffness" can be determined.

cond-mat.stat-mech↗

arXiv2026

Spatially-resolved multiphoton photoemission from a lateral transition metal dichalcogenide heterostructure

Transition metal dichalcogenides (TMDs) in their monolayer form offer a premier platform for next-generation optoelectronics, particularly through the local manipulation of their robust excitonic states using nanoscale electric fields. These localized states can be dynamically controlled through spatial structuring as well as through ultrafast field modulations driven by tailored optical pulses. Characterizing the resulting rapid, nanoscale charge carrier dynamics requires a technique with exceptional spatial and temporal resolution. Here, we report on the spatio-temporally resolved investigation of ground state and excited state photoemission from a lateral heterostructure built of monolayers of WSe$_2$ and MoSe$_2$ using few-cycle light pulses with a photon energy of 0.62 eV. We utilize photoemission electron microscopy to spatially resolve the highly nonlinear photoemission from the monolayer structure with few tens of nanometer resolution. By varying the laser pulse energy, we extract the nonlinearity of the photoemission process and thus the dynamic binding energy of the photoelectrons before and after optical excitation with high spatial and temporal resolution.

physics.optics↗

arXiv2025

Experimental Sensitivity Enhancement of a Quantum Rydberg Atom-Based RF Receiver with a Metamaterial GRIN Lens

We experimentally demonstrate enhanced sensitivity of an atom-based Rydberg radio frequency (RF) receiver integrated with a gradient refractive index (GRIN) Luneburg-type metamaterial lens. By analyzing the electromagnetically induced transparency (EIT) effect in Cesium vapor, we compare receiver performance with and without the GRIN lens under a 2.2~GHz and a 3.6~GHz far-field excitation. Our measurements reveal a significant amplification of the EIT window when the lens is introduced, consistent with the theoretical prediction that the local E-field enhancement at the vapor cell reduces the minimum detectable electric field and improves the microwave electric field measurement sensitivity of the Rydberg atom-based RF receiver over an ultrawide bandwidth of the lens. This experimental validation demonstrates the potential of metamaterial-enhanced quantum RF sensing for a wide range of applications, such as electromagnetic compatibility (EMC) testing, quantum radar, and wireless communication.

quant-ph↗

arXiv2025

Dynamics of Wound Closure in Living Nematic Epithelia

We study theoretically the closure of a wound in a layer of epithelial cells in a living tissue after damage. Our analysis is informed by our recent experiments observing re-epithelialisation in vivo of Drosophila pupae. On time and length-scales such that the evolution of the epithelial tissue near the wound is well captured by that of a 2D active fluid with local nematic order, we consider the free-surface problem of a hole in a bounded region of tissue, and study the role that active stresses far from the hole play in the closure of the hole. For parallel anchored nematic order at the wound boundary (as we observe in our experiments), we find that closure is accelerated when the active stresses are contractile and slowed down when the stresses are extensile. Parallel anchoring also leads to the appearance of topological defects which annihilate upon wound closure.

cond-mat.soft↗

arXiv2025

I'm Sorry Dave: How the old world of personnel security can inform the new world of AI insider risk

Organisations are rapidly adopting artificial intelligence (AI) tools to perform tasks previously undertaken by people. The potential benefits are enormous. Separately, some organisations deploy personnel security measures to mitigate the security risks arising from trusted human insiders. Unfortunately, there is no meaningful interplay between the rapidly evolving domain of AI and the traditional world of personnel security. This is a problem. The complex risks from human insiders are hard enough to understand and manage, despite many decades of effort. The emerging security risks from AI insiders are even more opaque. Both sides need all the help they can get. Some of the concepts and approaches that have proved useful in dealing with human insiders are also applicable to the emerging risks from AI insiders.

cs.CR↗

arXiv2024

On the spherical partition algebra

For $ k \in \mathbb{N}$ we introduce an idempotent subalgebra, the spherical partition algebra ${\mathcal{SP} }_{k}$, of the partition algebra ${\mathcal{P} }_{k}$, that we define using an embedding associated with the trivial representation of the symmetric group $\mathfrak{S}_k$. We determine a basis for ${\mathcal{SP} }_{k}$ and this provides a combinatorial interpretation of the dimension of $\mathcal{SP}_{k}$, involving bipartite partitions of $ k$. For $ t \in \mathbb{C} $ we consider the specialized algebra $\mathcal{SP}_{k}(t)$. For $ t = n \in \mathbb{N}$, we describe the structure of $\mathcal{SP}_{k}(n)$ by giving the permutation module decomposition of the $k$'th symmetric power of the defining module for the symmetric group algebra $ \mathbb{C} \mathfrak{S}_n $. In general, we show that $\mathcal{SP}_{k}(t)$ is quasi-hereditary over $ \mathbb{C}$ for all $ t \in \mathcal{C}$, except $ t=0$. We determine the decomposition numbers for $\mathcal{SP}_{k}(t)$ for every specialization $ t \in \mathbb{C} $ except $ t= 0 $, (which includes semisimple and non-semisimple cases). In particular we determine the structure of all indecomposable projective modules, and the indecomposable tilting modules.

math.RT↗

arXiv2023

Solutions to the constant Yang-Baxter equation: additive charge conservation in three dimensions

We find all solutions to the constant Yang--Baxter equation $R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$ in three dimensions, subject to an additive charge-conservation ansatz. This ansatz is a generalisation of (strict) charge-conservation, for which a complete classification in all dimensions was recently obtained. Additive charge-conservation introduces additional sector-coupling parameters -- in 3 dimensions there are $4$ such parameters. In the generic dimension 3 case, in which all of the $4$ parameters are nonzero, we find there is a single 3 parameter family of solutions. We give a complete analysis of this solution, giving the structure of the centraliser (symmetry) algebra in all orders. We also solve the remaining cases with three, two, or one nonzero sector-coupling parameter(s).

math.QA↗

arXiv2023

Classification of charge-conserving loop braid representations

Here a loop braid representation is a monoidal functor $\mathsf{F}$ from the loop braid category $\mathsf{L}$ to a suitable target category, and is $N$-charge-conserving if that target is the category $\mathsf{Match}^N$ of charge-conserving matrices (specifically $\mathsf{Match}^N$ is the same rank-$N$ charge-conserving monoidal subcategory of the monoidal category $\mathsf{Mat}$ used to classify braid representations in arXiv:2112.04533) with $\mathsf{F}$ strict, and surjective on $\mathbb{N}$, the object monoid. We classify and construct all such representations. In particular we prove that representations fall into varieties indexed by a set in bijection with the set of pairs of plane partitions of total degree $N$.

math.QA↗

arXiv2022

RTGNN: A Novel Approach to Model Stochastic Traffic Dynamics

Modeling stochastic traffic dynamics is critical to developing self-driving cars. Because it is difficult to develop first principle models of cars driven by humans, there is great potential for using data driven approaches in developing traffic dynamical models. While there is extensive literature on this subject, previous works mainly address the prediction accuracy of data-driven models. Moreover, it is often difficult to apply these models to common planning frameworks since they fail to meet the assumptions therein. In this work, we propose a new stochastic traffic model, Recurrent Traffic Graph Neural Network (RTGNN), by enforcing additional structures on the model so that the proposed model can be seamlessly integrated with existing motion planning algorithms. RTGNN is a Markovian model and is able to infer future traffic states conditioned on the motion of the ego vehicle. Specifically, RTGNN uses a definition of the traffic state that includes the state of all players in a local region and is therefore able to make joint predictions for all agents of interest. Meanwhile, we explicitly model the hidden states of agents, "intentions," as part of the traffic state to reflect the inherent partial observability of traffic dynamics. The above mentioned properties are critical for integrating RTGNN with motion planning algorithms coupling prediction and decision making. Despite the additional structures, we show that RTGNN is able to achieve state-of-the-art accuracy through comparisons with other similar works.

cs.LG↗

arXiv2022

Fluctuations of cell geometry and their non-equilibrium thermodynamics in living epithelial tissue

We measure different contributions to entropy production in a living functional epithelial tissue. We do this by extracting the functional dynamics of development while at the same time quantifying fluctuations. Using the translucent Drosophila melanogaster pupal epithelium as an ideal tissue for high resolution live imaging [1], we measure the entropy associated with the stochastic geometry of cells in the epithelium. This is done using a detailed analysis of the dynamics of the shape and orientation of individual cells which enables separation of local and global aspects of the tissue behaviour. We find intriguingly that we can observe irreversible dynamics in the cell geometries but without a change in the entropy associated with those degrees of freedom, showing that there is a flow of energy into those degrees of freedom. Hence the living system is controlling how the entropy is being produced and partitioned into its different parts.

cond-mat.soft↗

arXiv2021

Classification of spin-chain braid representations

A braid representation is a monoidal functor from the braid category $\mathsf{B}$, for example given by a solution to the constant Yang-Baxter equation. Given a monoidal category $\mathsf{C}$ with $ob(\mathsf{C})=\mathbb{N}$, a rank-$N$ charge-conserving representation (or spin-chain representation) is a strict monoidal functor $F$ from $\mathsf{C}$ to the category $\mathrm{Match}^N$ of rank-$N$ charge-conserving matrices that is natural in the sense that $F(1)=1$}. In this work we construct all spin-chain braid representations, and classify up to suitable notions of isomorphism.

math.QA↗

arXiv2020

Generalisations of Hecke algebras from Loop Braid Groups

We introduce a generalisation $LH_n$ of the ordinary Hecke algebras informed by the loop braid group $LB_n$ and the extension of the Burau representation thereto. The ordinary Hecke algebra has many remarkable arithmetic and representation theoretic properties, and many applications. We show that $LH_n$ has analogues of several of these properties. In particular we %introduce consider a class of local (tensor space/functor) representations of the braid group derived from a meld of the (non-functor) Burau representation and the (functor) Deguchi {\em et al}-Kauffman--Saleur-Rittenberg representations here called Burau-Rittenberg representations. In its most supersymmetric case somewhat mystical cancellations of anomalies occur so that the Burau-Rittenberg representation extends to a loop Burau-Rittenberg representation. And this factors through $LH_n$. Let $SP_n$ denote the corresponding quotient algebra, $k$ the ground ring, and $t \in k$ the loop-Hecke parameter. We prove the following: 1) $LH_n$ is finite dimensional over a field. 2) The natural inclusion $LB_n \rightarrow LB_{n+1}$ passes to an inclusion $SP_n \rightarrow SP_{n+1}$. 3) Over $k=\mathbb{C}$, $SP_n / rad $ is generically the sum of simple matrix algebras of dimension (and Bratteli diagram) given by Pascal's triangle. 4) We determine the other fundamental invariants of $SP_n$ representation theory: the Cartan decomposition matrix; and the quiver, which is of type-A. 5) The structure of $SP_n $ is independent of the parameter $t$, except for $t= 1$. \item For $t^2 \neq 1$ then $LH_n \cong SP_n$ at least up to rank$n=7$ (for $t=-1$ they are not isomorphic for $n>2$; for $t=1$ they are not isomorphic for $n>1$). Finally we discuss a number of other intriguing points arising from this construction in topology, representation theory and combinatorics.

math.GT↗

arXiv2020

Feedback Enhanced Motion Planning for Autonomous Vehicles

In this work, we address the motion planning problem for autonomous vehicles through a new lattice planning approach, called Feedback Enhanced Lattice Planner (FELP). Existing lattice planners have two major limitations, namely the high dimensionality of the lattice and the lack of modeling of agent vehicle behaviors. We propose to apply the Intelligent Driver Model (IDM) as a speed feedback policy to address both of these limitations. IDM both enables the responsive behavior of the agents, and uniquely determines the acceleration and speed profile of the ego vehicle on a given path. Therefore, only a spatial lattice is needed, while discretization of higher order dimensions is no longer required. Additionally, we propose a directed-graph map representation to support the implementation and execution of lattice planners. The map can reflect local geometric structure, embed the traffic rules adhering to the road, and is efficient to construct and update. We show that FELP is more efficient compared to other existing lattice planners through runtime complexity analysis, and we propose two variants of FELP to further reduce the complexity to polynomial time. We demonstrate the improvement by comparing FELP with an existing spatiotemporal lattice planner using simulations of a merging scenario and continuous highway traffic. We also study the performance of FELP under different traffic densities.

cs.RO↗

arXiv2019

On the number of principal ideals in d-tonal partition monoids

For a positive integer $d$, a non-negative integer $n$ and a non-negative integer $h\leq n$, we study the number $C_{n}^{(d)}$ of principal ideals; and the number $C_{n,h}^{(d)}$ of principal ideals generated by an element of rank $h$, in the $d$-tonal partition monoid on $n$ elements. We compute closed forms for the first family, as partial cumulative sums of known sequences. The second gives an infinite family of new integral sequences. We discuss their connections to certain integral lattices as well as to combinatorics of partitions.

math.CO↗

arXiv2019

Tonal partition algebras: fundamental and geometrical aspects of representation theory

For $l,n \in \mathbb{N}$ we define tonal partition algebra $P^l_n$ over $\mathbb{Z}[δ]$. We construct modules $\{ Δ_{\underlineμ} \}_{\underlineμ}$ for $P^l_n$ over $\mathbb{Z}[δ]$, and hence over any integral domain containing $\mathbb{Z}[δ]$ that is a $\mathbb{Z}[δ]$-algebra (such as $\mathbb{C}[δ]$), that pass to a complete set of irreducible modules over the field of fractions. We show that $P^l_n$ is semisimple there. That is, we construct for the tonal partition algebras a modular system in the sense of Brauer [6]. (The aim is to investigate the non-semisimple structure of the tonal partition algebras over suitable quotient fields of the natural ground ring, from a geometric perspective.) Using a `geometrical' index set for the $Δ$-modules, we give an order with respect to which the decomposition matrix over $\mathbb{C}$ (with $δ\in \mathbb{C}^{\times}$) is upper-unitriangular. We establish several crucial properties of the $Δ$-modules. These include a tower property, with respect to $n$, in the sense of Green [20, §6] and Cox $\textit{ et al}$ [8]; contravariant forms with respect to a natural involutive antiautomorphism; a highest weight category property; and branching rules.

math.RT↗

arXiv2019

Indecomposable tilting modules for the blob algebra

The blob algebra is a finite-dimensional quotient of the Hecke algebra of type $B$ which is almost always quasi-hereditary. We construct the indecomposable tilting modules for the blob algebra over a field of characteristic $0$ in the doubly critical case. Every indecomposable tilting module of maximal highest weight is either a projective module or an extension of a simple module by a projective module. Moreover, every indecomposable tilting module is a submodule of an indecomposable tilting module of maximal highest weight. We conclude that the graded Weyl multiplicities of the indecomposable tilting modules in this case are given by inverse Kazhdan-Lusztig polynomials of type $\tilde{A}_1$.

math.RT↗

arXiv2018

Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory

We show that representations of the loop braid group arise from Aharonov-Bohm like effects in finite 2-group (3+1)-dimensional topological higher gauge theory. For this we introduce a minimal categorification of biracks, which we call W-bikoids (welded bikoids). Our main example of W-bikoids arises from finite 2-groups, realised as crossed modules of groups. Given a W-bikoid, and hence a groupoid of symmetries, we construct a family of unitary representations of the loop braid group derived from representations of the groupoid algebra. We thus give a candidate for higher Bais' flux metamorphosis, and hence also a version of a `higher quantum group'.

math-ph↗

arXiv2018

Representations of the Necklace Braid Group: Topological and Combinatorial Approaches

The necklace braid group $\mathcal{NB}_n$ is the motion group of the $n+1$ component necklace link $\mathcal{L}_n$ in Euclidean $\mathbb{R}^3$. Here $\mathcal{L}_n$ consists of $n$ pairwise unlinked Euclidean circles each linked to an auxiliary circle. Partially motivated by physical considerations, we study representations of the necklace braid group $\mathcal{NB}_n$, especially those obtained as extensions of representations of the braid group $\mathcal{B}_n$ and the loop braid group $\mathcal{LB}_n$. We show that any irreducible $\mathcal{B}_n$ representation extends to $\mathcal{NB}_n$ in a standard way. We also find some non-standard extensions of several well-known $\mathcal{B}_n$-representations such as the Burau and LKB representations. Moreover, we prove that any local representation of $\mathcal{B}_n$ (i.e. coming from a braided vector space) can be extended to $\mathcal{NB}_n$, in contrast to the situation with $\mathcal{LB}_n$. We also discuss some directions for future study from categorical and physical perspectives.

math.QA↗