School of Mathematics and Statistics - Research Publications

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    A statistical approach to knot confinement via persistent homology
    Celoria, D ; Mahler, BI (ROYAL SOC, 2022-05-25)
    In this paper, we study how randomly generated knots occupy a volume of space using topological methods. To this end, we consider the evolution of the first homology of an immersed metric neighbourhood of a knot's embedding for growing radii. Specifically, we extract features from the persistent homology (PH) of the Vietoris-Rips complexes built from point clouds associated with knots. Statistical analysis of our data shows the existence of increasing correlations between geometric quantities associated with the embedding and PH-based features, as a function of the knots' lengths. We further study the variation of these correlations for different knot types. Finally, this framework also allows us to define a simple notion of deviation from ideal configurations of knots.
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    Classical Length-5 Pattern-Avoiding Permutations
    Clisby, N ; Conway, AR ; Guttmann, AJ ; Inoue, Y (The Electronic Journal of Combinatorics, 2022-01-01)
    We have made a systematic numerical study of the 16 Wilf classes of length-5 classical pattern-avoiding permutations from their generating function coefficients. We have extended the number of known coefficients in fourteen of the sixteen classes. Careful analysis, including sequence extension, has allowed us to estimate the growth constant of all classes, and in some cases to estimate the sub-dominant power-law term associated with the exponential growth. In six of the sixteen classes we find the familiar power-law behaviour, so that the coefficients behave like $s_n \sim C \cdot \mu^n \cdot n^g,$ while in the remaining ten cases we find a stretched exponential as the most likely sub-dominant term, so that the coefficients behave like $s_n \sim C \cdot \mu^n \cdot \mu_1^{n^\sigma} \cdot n^g,$ where $0 < \sigma < 1.$ We have also classified the 120 possible permutations into the 16 distinct classes. We give compelling numerical evidence, and in one case a proof, that all 16 Wilf-class generating function coefficients can be represented as moments of a non-negative measure on $[0,\infty)$. Such sequences are known as Stieltjes moment sequences. They have a number of nice properties, such as log-convexity, which can be used to provide quite strong rigorous lower bounds. Stronger bounds still can be established under plausible monotonicity assumptions about the terms in the continued-fraction expansion of the generating functions implied by the Stieltjes property. In this way we provide strong (non-rigorous) lower bounds to the growth constants, which are sometimes within a few percent of the exact value.
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    Predicting Safe Regions Within Lava Flows Over Topography
    Saville, JM ; Hinton, EM ; Huppert, HE (AMER GEOPHYSICAL UNION, 2022-09-01)
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    A representation learning framework for detection and characterization of dead versus strain localization zones from pre-to post-failure
    Tordesillas, A ; Zhou, S ; Bailey, J ; Bondell, H (SPRINGER, 2022-08-01)
    Abstract Experiments have long shown that zones of near vanishing deformation, so-called “dead zones”, emerge and coexist with strain localization zones inside deforming granular media. To date, a method that can disentangle these dynamically coupled structures from each other, from pre- to post- failure, is lacking. Here we develop a framework that learns a new representation of the kinematic data, based on the complexity of a grain’s neighborhood structure in the kinematic-state-space, as measured by a recently introduced metric called s-LID. Dead zones (DZ) are first distinguished from strain localization zones (SZ) throughout loading history. Next the coupled dynamics of DZ and SZ are characterized using a range of discriminative features representing: local nonaffine deformation, contact topology and force transmission properties. Data came from discrete element simulations of biaxial compression tests. The deformation is found to be essentially dual in nature. DZ and SZ exhibit distinct yet coupled dynamics, with the separation in dynamics increasing in the lead up to failure. Force congestion and plastic deformation mainly concentrate in SZ. Although the 3-core of the contact network is highly prone to damage in SZ, it is robust to pre-failure microbands but is decimated in the shearband, leaving a fragmented 3-core in DZ at failure. We also show how loading condition and rolling resistance influence SZ and DZ differently, thus casting new light on controls on plasticity from the perspective of emergent deformation structures. Graphic abstract
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    Estimation of tumor cell total mRNA expression in 15 cancer types predicts disease progression.
    Cao, S ; Wang, JR ; Ji, S ; Yang, P ; Dai, Y ; Guo, S ; Montierth, MD ; Shen, JP ; Zhao, X ; Chen, J ; Lee, JJ ; Guerrero, PA ; Spetsieris, N ; Engedal, N ; Taavitsainen, S ; Yu, K ; Livingstone, J ; Bhandari, V ; Hubert, SM ; Daw, NC ; Futreal, PA ; Efstathiou, E ; Lim, B ; Viale, A ; Zhang, J ; Nykter, M ; Czerniak, BA ; Brown, PH ; Swanton, C ; Msaouel, P ; Maitra, A ; Kopetz, S ; Campbell, P ; Speed, TP ; Boutros, PC ; Zhu, H ; Urbanucci, A ; Demeulemeester, J ; Van Loo, P ; Wang, W (Springer Science and Business Media LLC, 2022-11)
    Single-cell RNA sequencing studies have suggested that total mRNA content correlates with tumor phenotypes. Technical and analytical challenges, however, have so far impeded at-scale pan-cancer examination of total mRNA content. Here we present a method to quantify tumor-specific total mRNA expression (TmS) from bulk sequencing data, taking into account tumor transcript proportion, purity and ploidy, which are estimated through transcriptomic/genomic deconvolution. We estimate and validate TmS in 6,590 patient tumors across 15 cancer types, identifying significant inter-tumor variability. Across cancers, high TmS is associated with increased risk of disease progression and death. TmS is influenced by cancer-specific patterns of gene alteration and intra-tumor genetic heterogeneity as well as by pan-cancer trends in metabolic dysregulation. Taken together, our results indicate that measuring cell-type-specific total mRNA expression in tumor cells predicts tumor phenotypes and clinical outcomes.
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    Spin-Ruijsenaars, q-Deformed Haldane-Shastry and Macdonald Polynomials
    Lamers, J ; Pasquier, V ; Serban, D (SPRINGER, 2022-05-13)
    Abstract We study the -analogue of the Haldane–Shastry model, a partially isotropic (xxz-like) long-range spin chain that by construction enjoys quantum-affine (really: quantum-loop) symmetries at finite system size. We derive the pairwise form of the Hamiltonian, found by one of us building on work of D. Uglov, via ‘freezing’ from the affine Hecke algebra. To this end we first obtain explicit expressions for the spin-Macdonald operators of the (trigonometric) spin-Ruijsenaars model. Through freezing these give rise to the higher Hamiltonians of the spin chain, including another Hamiltonian of the opposite ‘chirality’. The sum of the two chiral Hamiltonians has a real spectrum also when $$|\mathsf {q}|=1$$ | q | = 1 , so in particular when is a root of unity. For generic $$\mathsf {q}$$ q the eigenspaces are known to be labelled by ‘motifs’. We clarify the relation between these patterns and the corresponding degeneracies (multiplicities) in the crystal limit $$\textsf {q}\rightarrow \infty $$ q → ∞ . For each motif we obtain an explicit expression for the exact eigenvector, valid for generic , that has (‘pseudo’ or ‘l-’) highest weight in the sense that, in terms of the operators from the monodromy matrix, it is an eigenvector of A and D and annihilated by C. It has a simple component featuring the ‘symmetric square’ of the -Vandermonde polynomial times a Macdonald polynomial—or more precisely its quantum spherical zonal special case. All other components of the eigenvector are obtained from this through the action of the Hecke algebra, followed by ‘evaluation’ of the variables to roots of unity. We prove that our vectors have highest weight upon evaluation. Our description of the exact spectrum is complete. The entire model, including the quantum-loop action, can be reformulated in terms of polynomials. Our main tools are the Y-operators from the affine Hecke algebra. From a more mathematical perspective the key step in our diagonalisation is as follows. We show that on a subspace of suitable polynomials the first M ‘classical’ (i.e. no difference part) Y-operators in N variables reduce, upon evaluation as above, to Y-operators in M variables with parameters at the quantum zonal spherical point.
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    NON-PERVERSE PARITY SHEAVES ON THE FLAG VARIETY
    McNamara, PJ (CAMBRIDGE UNIV PRESS, 2022-08-23)
    Abstract We give examples of non-perverse parity sheaves on Schubert varieties for all primes.
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    Inferring rheology from free-surface observations
    Hinton, EM (CAMBRIDGE UNIV PRESS, 2022-03-03)
    We develop direct inversion methods for inferring the rheology of a fluid from observations of its shallow flow. First, the evolution equation for the free-surface flow of an inertia-less current with general constitutive law is derived. The relationship between the volume flux of fluid and the basal stress, $\tau _b$ , is encapsulated by a single function $F(\tau _b)$ , which depends only on the constitutive law. The inversion method consists of (i) determining the flux and basal stress from the free-surface evolution, (ii) comparing the flux with the basal stress to constrain $F$ and (iii) inferring the constitutive law from $F$ . Examples are presented for both steady and transient free-surface flows demonstrating that a wide range of constitutive laws can be directly obtained. For flows in which the free-surface velocity is known, we derive a different method, which circumvents the need to calculate the flux.
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    Distributed forward-backward methods for ring networks
    Aragón-Artacho, FJ ; Malitsky, Y ; Tam, MK ; Torregrosa-Belén, D (Springer Science and Business Media LLC, 2022-01-01)
    Abstract In this work, we propose and analyse forward-backward-type algorithms for finding a zero of the sum of finitely many monotone operators, which are not based on reduction to a two operator inclusion in the product space. Each iteration of the studied algorithms requires one resolvent evaluation per set-valued operator, one forward evaluation per cocoercive operator, and two forward evaluations per monotone operator. Unlike existing methods, the structure of the proposed algorithms are suitable for distributed, decentralised implementation in ring networks without needing global summation to enforce consensus between nodes.
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    Critical scaling of lattice polymers confined to a box without endpoint restriction
    Bradly, CJ ; Owczarek, AL (SPRINGER, 2022-08-21)
    Abstract We study self-avoiding walks on the square lattice restricted to a square box of side L weighted by a length fugacity without restriction of their end points. This is a natural model of a confined polymer in dilute solution such as polymers in mesoscopic pores. The model admits a phase transition between an ‘empty’ phase, where the average length of walks are finite and the density inside large boxes goes to zero, to a ‘dense’ phase, where there is a finite positive density. We prove various bounds on the free energy and develop a scaling theory for the phase transition based on the standard theory for unconstrained polymers. We compare this model to unrestricted walks and walks that whose endpoints are fixed at the opposite corners of a box, as well as Hamiltonian walks. We use Monte Carlo simulations to verify predicted values for three key exponents: the density exponent $$\alpha =1/2$$ α = 1 / 2 , the finite size crossover exponent $$1/\nu =4/3$$ 1 / ν = 4 / 3 and the critical partition function exponent $$2-\eta =43/24$$ 2 - η = 43 / 24 . This implies that the theoretical framework relating them to the unconstrained SAW problem is valid.