 School of Mathematics and Statistics  Research Publications
School of Mathematics and Statistics  Research Publications
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ItemA statistical approach to knot confinement via persistent homologyCeloria, D ; Mahler, BI (ROYAL SOC, 20220525)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 VietorisRips 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 PHbased 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.

ItemClassical Length5 PatternAvoiding PermutationsClisby, N ; Conway, AR ; Guttmann, AJ ; Inoue, Y (The Electronic Journal of Combinatorics, 20220101)We have made a systematic numerical study of the 16 Wilf classes of length5 classical patternavoiding 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 subdominant powerlaw term associated with the exponential growth. In six of the sixteen classes we find the familiar powerlaw 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 subdominant 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 Wilfclass generating function coefficients can be represented as moments of a nonnegative measure on $[0,\infty)$. Such sequences are known as Stieltjes moment sequences. They have a number of nice properties, such as logconvexity, 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 continuedfraction expansion of the generating functions implied by the Stieltjes property. In this way we provide strong (nonrigorous) lower bounds to the growth constants, which are sometimes within a few percent of the exact value.

ItemPredicting Safe Regions Within Lava Flows Over TopographySaville, JM ; Hinton, EM ; Huppert, HE (AMER GEOPHYSICAL UNION, 20220901)

ItemA representation learning framework for detection and characterization of dead versus strain localization zones from preto postfailureTordesillas, A ; Zhou, S ; Bailey, J ; Bondell, H (SPRINGER, 20220801)Abstract Experiments have long shown that zones of near vanishing deformation, socalled “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 kinematicstatespace, as measured by a recently introduced metric called sLID. 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 3core of the contact network is highly prone to damage in SZ, it is robust to prefailure microbands but is decimated in the shearband, leaving a fragmented 3core 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

ItemEstimation 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, 202211)Singlecell RNA sequencing studies have suggested that total mRNA content correlates with tumor phenotypes. Technical and analytical challenges, however, have so far impeded atscale pancancer examination of total mRNA content. Here we present a method to quantify tumorspecific 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 intertumor variability. Across cancers, high TmS is associated with increased risk of disease progression and death. TmS is influenced by cancerspecific patterns of gene alteration and intratumor genetic heterogeneity as well as by pancancer trends in metabolic dysregulation. Taken together, our results indicate that measuring celltypespecific total mRNA expression in tumor cells predicts tumor phenotypes and clinical outcomes.

ItemSpinRuijsenaars, qDeformed HaldaneShastry and Macdonald PolynomialsLamers, J ; Pasquier, V ; Serban, D (SPRINGER, 20220513)Abstract We study the analogue of the Haldane–Shastry model, a partially isotropic (xxzlike) longrange spin chain that by construction enjoys quantumaffine (really: quantumloop) 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 spinMacdonald operators of the (trigonometric) spinRuijsenaars 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 quantumloop action, can be reformulated in terms of polynomials. Our main tools are the Yoperators 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) Yoperators in N variables reduce, upon evaluation as above, to Yoperators in M variables with parameters at the quantum zonal spherical point.

ItemNONPERVERSE PARITY SHEAVES ON THE FLAG VARIETYMcNamara, PJ (CAMBRIDGE UNIV PRESS, 20220823)Abstract We give examples of nonperverse parity sheaves on Schubert varieties for all primes.

ItemInferring rheology from freesurface observationsHinton, EM (CAMBRIDGE UNIV PRESS, 20220303)We develop direct inversion methods for inferring the rheology of a fluid from observations of its shallow flow. First, the evolution equation for the freesurface flow of an inertialess 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 freesurface 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 freesurface flows demonstrating that a wide range of constitutive laws can be directly obtained. For flows in which the freesurface velocity is known, we derive a different method, which circumvents the need to calculate the flux.

ItemDistributed forwardbackward methods for ring networksAragónArtacho, FJ ; Malitsky, Y ; Tam, MK ; TorregrosaBelén, D (Springer Science and Business Media LLC, 20220101)Abstract In this work, we propose and analyse forwardbackwardtype 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 setvalued 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.

ItemCritical scaling of lattice polymers confined to a box without endpoint restrictionBradly, CJ ; Owczarek, AL (SPRINGER, 20220821)Abstract We study selfavoiding 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.