DSpace Community:
http://hdl.handle.net/2381/445
Mon, 23 Apr 2018 20:47:19 GMT2018-04-23T20:47:19ZDSpace Community:http://lra.le.ac.uk:80/retrieve/2392/mathsequation.jpg
http://hdl.handle.net/2381/445
Regression analysis: likelihood, error and entropy
http://hdl.handle.net/2381/41459
Title: Regression analysis: likelihood, error and entropy
Authors: Grechuk, Bogdan; Zabarankin, Michael
Abstract: In a regression with independent and identically distributed normal residuals, the log-likelihood function yields an empirical form of the L2L2-norm, whereas the normal distribution can be obtained as a solution of differential entropy maximization subject to a constraint on the L2L2-norm of a random variable. The L1L1-norm and the double exponential (Laplace) distribution are related in a similar way. These are examples of an “inter-regenerative” relationship. In fact, L2L2-norm and L1L1-norm are just particular cases of general error measures introduced by Rockafellar et al. (Finance Stoch 10(1):51–74, 2006) on a space of random variables. General error measures are not necessarily symmetric with respect to ups and downs of a random variable, which is a desired property in finance applications where gains and losses should be treated differently. This work identifies a set of all error measures, denoted by EE, and a set of all probability density functions (PDFs) that form “inter-regenerative” relationships (through log-likelihood and entropy maximization). It also shows that M-estimators, which arise in robust regression but, in general, are not error measures, form “inter-regenerative” relationships with all PDFs. In fact, the set of M-estimators, which are error measures, coincides with EE. On the other hand, M-estimators are a particular case of L-estimators that also arise in robust regression. A set of L-estimators which are error measures is identified—it contains EE and the so-called trimmed LpLp-norms.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Fri, 06 Apr 2018 09:00:07 GMThttp://hdl.handle.net/2381/414592018-04-06T09:00:07ZBlessing of dimensionality: mathematical foundations of the statistical physics of data
http://hdl.handle.net/2381/41439
Title: Blessing of dimensionality: mathematical foundations of the statistical physics of data
Authors: Gorban, A. N.; Tyukin, I. Y.
Abstract: The concentrations of measure phenomena were discovered as the mathematical background to statistical mechanics at the end of the nineteenth/beginning of the twentieth century and have been explored in mathematics ever since. At the beginning of the twenty-first century, it became clear that the proper utilization of these phenomena in machine learning might transform the curse of dimensionality into the blessing of dimensionality. This paper summarizes recently discovered phenomena of measure concentration which drastically simplify some machine learning problems in high dimension, and allow us to correct legacy artificial intelligence systems. The classical concentration of measure theorems state that i.i.d. random points are concentrated in a thin layer near a surface (a sphere or equators of a sphere, an average or median-level set of energy or another Lipschitz function, etc.). The new stochastic separation theorems describe the thin structure of these thin layers: the random points are not only concentrated in a thin layer but are all linearly separable from the rest of the set, even for exponentially large random sets. The linear functionals for separation of points can be selected in the form of the linear Fisher’s discriminant. All artificial intelligence systems make errors. Non-destructive correction requires separation of the situations (samples) with errors from the samples corresponding to correct behaviour by a simple and robust classifier. The stochastic separation theorems provide us with such classifiers and determine a non-iterative (one-shot) procedure for their construction.
This article is part of the theme issue ‘Hilbert’s sixth problem’.Thu, 05 Apr 2018 10:27:34 GMThttp://hdl.handle.net/2381/414392018-04-05T10:27:34ZNonparametric regression method with functional covariates and multivariate response
http://hdl.handle.net/2381/41347
Title: Nonparametric regression method with functional covariates and multivariate response
Authors: Wang, Bo; Omar, Kurdistan M. T.
Abstract: Nonparametric regression methods have been widely studied in functional regression analysis in the context of functional covariates and univariate response, but it is not the case for functional covariates with multivariate response. In this paper, we present two new solutions for the latter problem: the first is to directly extend the nonparametric method for univariate response to multivariate response, and in the second, the correlation among different responses is incorporated into the model. The asymptotic properties of the estimators are studied, and the effectiveness of the proposed methods is demonstrated through several simulation studies and a real data example.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Thu, 22 Mar 2018 09:49:53 GMThttp://hdl.handle.net/2381/413472018-03-22T09:49:53ZSensitivity Analysis in Applications with Deviation, Risk, Regret, and Error Measures
http://hdl.handle.net/2381/41299
Title: Sensitivity Analysis in Applications with Deviation, Risk, Regret, and Error Measures
Authors: Grechuk, Bogdan; Zabarankin, Michael
Abstract: The envelope formula is obtained for optimization problems with positively homogeneous convex functionals defined on a space of random variables. Those problems include linear regression with general error measures and optimal portfolio selection with the objective function being either a general deviation measure or a coherent risk measure subject to a constraint on the expected rate of return. The obtained results are believed to be novel even for Markowitz's mean-variance portfolio selection but are far more general and include explicit envelope relationships for the rates of return of portfolios that minimize lower semivariance, mean absolute deviation, deviation measures of ${\cal L}^p$-type and semi-${\cal L}^p$ type, and conditional value-at-risk. In each case, the envelope theorem yields explicit estimates for the absolute value of the difference between deviation/risk of optimal portfolios with the unperturbed and perturbed asset probability distributions in terms of a norm of the perturbation.Tue, 13 Mar 2018 15:48:19 GMThttp://hdl.handle.net/2381/412992018-03-13T15:48:19ZAdaptive discontinuous Galerkin methods for elliptic interface problems
http://hdl.handle.net/2381/41277
Title: Adaptive discontinuous Galerkin methods for elliptic interface problems
Authors: Cangiani, Andrea; Georgoulis, Emmanuil H.; Sabawi, Younis A.
Abstract: An interior-penalty discontinuous Galerkin (dG) method for an elliptic interface problem involving, possibly, curved interfaces, with flux-balancing interface conditions, e.g., modelling mass transfer of solutes through semi-permeable membranes, is considered. The method allows for extremely general curved element shapes employed to resolve the interface geometry exactly. A residual-type a posteriori error estimator for this dG method is proposed and upper and lower bounds of the error in the respective dG-energy norm are proven. The a posteriori error bounds are subsequently used to prove a basic a priori convergence result. The theory presented is complemented by a series of numerical experiments. The presented approach applies immediately to the case of curved domains with non-essential boundary conditions, too.Mon, 12 Mar 2018 09:48:43 GMThttp://hdl.handle.net/2381/412772018-03-12T09:48:43ZGeneration of mechanical force by grafted polyelectrolytes in an electric field: application to polyelectrolyte-based nano-devices
http://hdl.handle.net/2381/41270
Title: Generation of mechanical force by grafted polyelectrolytes in an electric field: application to polyelectrolyte-based nano-devices
Authors: Brilliantov, Nikolai V.; Budkov, Y. A.; Seidel, C.
Abstract: We analyse theoretically and by means of molecular dynamics (MD) simulations the generation of mechanical force by a polyelectrolyte (PE) chain grafted to a plane. The PE is exposed to an external electric field that favours its adsorption on the plane. The free end of the chain is linked to a deformable target body. By varying the field, one can alter the length of the non-adsorbed part of the chain. This entails variation of the deformation of the target body and hence variation of the force arising in the body. Our theoretical predictions for the generated force are in very good agreement with the MD data. Using the theory developed for the generated force, we study the effectiveness of possible PE-based nano-vices, composed of two clenching planes connected by PEs and exposed to an external electric field. We exploit the Cundall–Strack solid friction model to describe the friction between a particle and the clenching planes. We compute the diffusion coefficient of a clenched particle and show that it drastically decreases even in weak applied fields. This demonstrates the efficacy of the PE-based nano-vices, which may be a possible alternative to the existing nanotube nano-tweezers and optical tweezers.Wed, 07 Mar 2018 13:37:53 GMThttp://hdl.handle.net/2381/412702018-03-07T13:37:53ZDiscontinuous Galerkin timestepping for nonlinear parabolic problems
http://hdl.handle.net/2381/41216
Title: Discontinuous Galerkin timestepping for nonlinear parabolic problems
Authors: Sabawi, Mohammad Abd Moheemmeed
Abstract: We study space–time finite element methods for semilinear parabolic problems in (1 + d)–dimensions for d = 2, 3. The discretisation in time is based on the discontinuous Galerkin timestepping method with implicit treatment of the linear terms and either implicit or explicit multistep discretisation of the zeroth order nonlinear reaction terms. Conforming finite element methods are used for the space discretisation. For this implicit-explicit IMEX–dG family of methods, we derive a posteriori and a priori energy-type error bounds and we perform extended numerical experiments. We derive a novel hp–version a posteriori error bounds in the L∞(L2) and L2(H1) norms assuming an only locally Lipschitz growth condition for the nonlinear reactions and no monotonicity of the nonlinear terms. The analysis builds upon the recent work in [60], for the respective linear problem, which is in turn based on combining the elliptic and dG reconstructions in [83, 84] and continuation argument. The a posteriori error bounds appear to be of optimal order and efficient in a series of numerical experiments.
Secondly, we prove a novel hp–version a priori error bounds for the fully–discrete IMEX–dG timestepping schemes in the same setting in L∞(L2) and L2(H1) norms. These error bounds are explicit with respect to both the temporal and spatial meshsizes kn and h, respectively, and, where possible, with respect to the possibly varying temporal polynomial degree r. The a priori error estimates are derived using the elliptic projection technique with an inf-sup argument in time. Standard tools such as Grönwall inequality and discrete stability estimates for fully discrete semilinear parabolic problems with merely locally-Lipschitz continuous nonlinear reaction terms are used. The a priori analysis extends the applicability of the results from [52] to this setting with low regularity. The results are tested by an extensive set of numerical experiments.Tue, 20 Feb 2018 11:47:04 GMThttp://hdl.handle.net/2381/412162018-02-20T11:47:04ZExplicit Parameter-dependent Representations of Periodic Solutions for a Class of Nonlinear Systems
http://hdl.handle.net/2381/41208
Title: Explicit Parameter-dependent Representations of Periodic Solutions for a Class of Nonlinear Systems
Authors: Mohammed, J. Al-Ameri; Tyukin, I.
Abstract: We propose a method for deriving computationally efficient representations of periodic solutions of parameterized systems of nonlinear ordinary differential equations. These representations depend on parameters of the system explicitly, as quadratures of parameterized computable functions. The method applies to systems featuring both linear and nonlinear parametrization, and time-varying right-hand-side; it opens possibilities to invoke scalable parallel computations for numerical evaluation of solutions for various parameter values. Application of the method to parameter estimation problems is illustrated with constructing an algorithm for state and parameter estimation for the Morris-Lecar system.Mon, 19 Feb 2018 16:48:15 GMThttp://hdl.handle.net/2381/412082018-02-19T16:48:15ZSelf-organisation of small-world networks by adaptive rewiring in response to graph diffusion
http://hdl.handle.net/2381/41207
Title: Self-organisation of small-world networks by adaptive rewiring in response to graph diffusion
Authors: Jarman, Nicholas; Steur, Erik; Trengove, Chris; Tyukin, Ivan Y.; Van Leeuwen, Cees
Abstract: Complex networks emerging in natural and human-made systems tend to assume small-world structure. Is there a common mechanism underlying their self-organisation? Our computational simulations show that network diffusion (traffic flow or information transfer) steers network evolution towards emergence of complex network structures. The emergence is effectuated through adaptive rewiring: progressive adaptation of structure to use, creating short-cuts where network diffusion is intensive while annihilating underused connections. With adaptive rewiring as the engine of universal small-worldness, overall diffusion rate tunes the systems' adaptation, biasing local or global connectivity patterns. Whereas the former leads to modularity, the latter provides a preferential attachment regime. As the latter sets in, the resulting small-world structures undergo a critical shift from modular (decentralised) to centralised ones. At the transition point, network structure is hierarchical, balancing modularity and centrality - a characteristic feature found in, for instance, the human brain.
Description: Supplementary information accompanies this paper at https://doi.org/10.1038/s41598-017-12589-9Mon, 19 Feb 2018 16:38:51 GMThttp://hdl.handle.net/2381/412072018-02-19T16:38:51ZPatchy, not patchy, or how much patchy? Classification of spatial patterns appearing in a model of biological invasion
http://hdl.handle.net/2381/41157
Title: Patchy, not patchy, or how much patchy? Classification of spatial patterns appearing in a model of biological invasion
Authors: Petrovskaya, N.; Petrovskii, S.; Zhang, W.
Abstract: Good understanding of spatiotemporal patterns of species spread during biological invasion is needed for efficient monitoring and control of harmful alien pests. Various growth-dispersal-type models of population dynamics predict that invasive species spread can follow two qualitatively different scenarios such as the propagation of a continuous population front and the “no-front” patchy invasion. Distinguishing between these two patterns of spread is important, in particular because the patchy invasion poses a much greater challenge for monitoring and control. However, a mathematical theory of the patchy invasion is missing and it remains unclear what are the restrictions on parameter values and how much different this dynamical regime is from the continuous front propagation. In this paper, we address these issues in terms of a biologically meaningful mathematical model consisting of two coupled integral-difference equations. We show that the relevant domain of the parameter space has a complex intermittent structure. We also suggest a criterion that can be used to distinguish between the patchy invasion and the continuous front propagation: the patchy-invasion spatial pattern is shown to be much more sensitive to the cutoff at low densities.Tue, 13 Feb 2018 15:03:04 GMThttp://hdl.handle.net/2381/411572018-02-13T15:03:04ZOn the calibration of the Schwartz two-factor model to WTI crude oil options and the extended Kalman Filter
http://hdl.handle.net/2381/41156
Title: On the calibration of the Schwartz two-factor model to WTI crude oil options and the extended Kalman Filter
Authors: Ewald, Christian-Oliver; Zhang, Aihua; Zong, Zhe
Abstract: The Schwartz (J Finance 52(3):923–973, 1997) two factor model serves as a benchmark for pricing commodity contracts, futures and options. It is normally calibrated to fit the term-structure of a range of future contracts with varying maturities. In this paper, we investigate the effects on parameter estimates, if the model is fitted to prices of options, with varying maturities and strikes instead of futures, as is commonly done. The use of option prices rather than futures in the calibration leads to non-linearities, which the standard Kalman filter approach is unable to cope with. To overcome these issues, we use the extended Kalman Filter. We find that some parameters sensitively depend on the choice of strikes of the corresponding options, and are different from those estimates obtained from using futures prices. This effect is analogue to varying implied volatilities in the Black–Scholes model. This realization is important, as the use of ill-fitted models for pricing options in the Schwartz (1997) framework may cause traders to bear serious financial losses.Tue, 13 Feb 2018 14:52:38 GMThttp://hdl.handle.net/2381/411562018-02-13T14:52:38ZThe homology core of matchbox manifolds and invariant measures
http://hdl.handle.net/2381/41146
Title: The homology core of matchbox manifolds and invariant measures
Authors: Clark, Alex; Hunton, John
Abstract: x
Description: The file associated with this record is under embargo until publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Mon, 12 Feb 2018 17:07:06 GMThttp://hdl.handle.net/2381/411462018-02-12T17:07:06ZTowards developing a general framework for modelling vertical migration in zooplankton
http://hdl.handle.net/2381/41118
Title: Towards developing a general framework for modelling vertical migration in zooplankton
Authors: Morozov, Andrew Y.; Kuzenkov, Oleg A.
Abstract: Diel vertical migration (DVM) of zooplankton is a widespread phenomenon in both oceans and lakes, and
is generally considered to be the largest synchronized movement of biomass on Earth. Most existing
mathematical models of DVM are based on the assumption that animals maximize a certain criterion
such as the expected reproductive value, the venturous revenue, the ratio of energy gain/mortality or
some predator avoidance function when choosing their instantaneous depth. The major shortcoming of
this general point of view is that the predicted DVM may be strongly affected by a subjective choice of a
particular optimization criterion. Here we argue that the optimal strategy of DVM can be unambiguously
obtained as an outcome of selection in the underlying equations of genotype/traits frequency dynamics.
Using this general paradigm, we explore the optimal strategy for the migration across different depths by
zooplankton grazers throughout the day. To illustrate our ideas we consider four generic DVM models,
each making different assumptions on the population dynamics of zooplankton, and demonstrate that in
each model we need to maximize a particular functional to find the optimal strategy. Surprisingly,
patterns of DVM obtained for different models greatly differ in terms of their parameters dependence.
We then show that the infinite dimensional trait space of different zooplankton trajectories can be
projected onto a low dimensional space of generalized parameters and the genotype evolution dynamics
can be easily followed using this low-dimensional space. Using this space of generalized parameters we
explore the influence of mutagenesis on evolution of DVM, and we show that strong mutagenesis allows
the coexistence of an infinitely large number of strategies whereas for weak mutagenesis the selection
results in the extinction of most strategies, with the surviving strategies all staying close to the optimal
strategy in the corresponding mutagenesis-free systemThu, 08 Feb 2018 09:49:13 GMThttp://hdl.handle.net/2381/411182018-02-08T09:49:13ZSize distribution of particles in Saturn's rings from aggregation and fragmentation
http://hdl.handle.net/2381/41102
Title: Size distribution of particles in Saturn's rings from aggregation and fragmentation
Authors: Brilliantov, Nikolai; Krapivsky, P. L.; Bodrova, Anna; Spahn, Frank; Hayakawa, Hisao; Stadnichuk, Vladimir; Schmidt, Jurgen
Abstract: Saturn's rings consist of a huge number of water ice particles, with a tiny addition of rocky material. They form a flat disk, as the result of an interplay of angular momentum conservation and the steady loss of energy in dissipative interparticle collisions. For particles in the size range from a few centimeters to a few meters, a power-law distribution of radii, ~r(-q) with q ≈ 3, has been inferred; for larger sizes, the distribution has a steep cutoff. It has been suggested that this size distribution may arise from a balance between aggregation and fragmentation of ring particles, yet neither the power-law dependence nor the upper size cutoff have been established on theoretical grounds. Here we propose a model for the particle size distribution that quantitatively explains the observations. In accordance with data, our model predicts the exponent q to be constrained to the interval 2.75 ≤ q ≤ 3.5. Also an exponential cutoff for larger particle sizes establishes naturally with the cutoff radius being set by the relative frequency of aggregating and disruptive collisions. This cutoff is much smaller than the typical scale of microstructures seen in Saturn's rings.Tue, 06 Feb 2018 12:09:55 GMThttp://hdl.handle.net/2381/411022018-02-06T12:09:55ZMechanism of Chain Collapse of Strongly Charged Polyelectrolytes
http://hdl.handle.net/2381/41101
Title: Mechanism of Chain Collapse of Strongly Charged Polyelectrolytes
Authors: Tom, Anvy Moly; Vemparala, Satyavani; Rajesh, R.; Brilliantov, Nikolai V.
Abstract: We perform extensive molecular dynamics simulations of a charged polymer in a good solvent in the
regime where the chain is collapsed. We analyze the dependence of the gyration radius Rg on the reduced
Bjerrum length lB and find two different regimes. In the first one, called a weak electrostatic regime,
Rg ∼ l−1=2
B , which is consistent only with the predictions of the counterion-fluctuation theory. In the second
one, called a strong electrostatic regime, we find Rg ∼ l−1=5
B . To explain the novel regime we modify the
counterion-fluctuation theory.Tue, 06 Feb 2018 11:56:20 GMThttp://hdl.handle.net/2381/411012018-02-06T11:56:20ZConway groupoids and completely transitive codes
http://hdl.handle.net/2381/41056
Title: Conway groupoids and completely transitive codes
Authors: Gill, Nick; Gillespie, Neil I.; Semeraro, Jason
Abstract: To each supersimple 2−(n,4,λ) design D one associates a ‘Conway groupoid’, which may
be thought of as a natural generalisation of Conway’s Mathieu groupoid M13 which is
constructed from P3.
We show that Sp2m(2) and 22m.Sp2m(2) naturally occur as Conway groupoids associated
to certain designs. It is shown that the incidence matrix associated to one of these
designs generates a new family of completely transitive F2-linear codes with minimum
distance 4 and covering radius 3, whereas the incidence matrix of the other design gives
an alternative construction of a previously known family of completely transitive codes.
We also give a new characterization of M13 and prove that, for a fixed λ > 0, there
are finitely many Conway groupoids for which the set of morphisms does not contain all
elements of the full alternating group.
Description: Mathematics Subject Classification (2010): 20B15, 20B25, 05B05; The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Mon, 29 Jan 2018 17:35:23 GMThttp://hdl.handle.net/2381/410562018-01-29T17:35:23ZTournaments, 4-uniform hypergraphs, and an exact extremal result
http://hdl.handle.net/2381/41055
Title: Tournaments, 4-uniform hypergraphs, and an exact extremal result
Authors: Gunderson, Karen; Semeraro, Jason
Abstract: We consider 4-uniform hypergraphs with the maximum number of hyperedges subject to the condition that every set of 5 vertices spans either 0 or exactly 2 hyperedges and give a construction, using quadratic residues, for an infinite family of such hypergraphs with the maximum number of hyperedges. Baber has previously given an asymptotically best-possible result using random tournaments. We give a connection between Baber's result and our construction via Paley tournaments and investigate a ‘switching’ operation on tournaments that preserves hypergraphs arising from this construction.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Mon, 29 Jan 2018 17:31:24 GMThttp://hdl.handle.net/2381/410552018-01-29T17:31:24ZCharacterization of melting properties of several Fe-C model potentials
http://hdl.handle.net/2381/41040
Title: Characterization of melting properties of several Fe-C model potentials
Authors: Melnykov, Mykhailo; Davidchack, Ruslan L.
Abstract: We use the coexisting phases approach to calculate melting phase diagrams of several Fe-C interaction potentials, such as Embedded Atom Method (EAM) potential of Lau et al. [Phys. Rev. Lett. 98 (2007) 215501], EAM potential of Hepburn and Ackland [Phys. Rev. B 78 (2008) 165115] , and two flavours of the Analytic Bond Order potential (ABOP) of Henriksson and Nordlund [Phys. Rev. B 79 (2009) 144107]. Melting of both bcc (ferrite) and fcc (austenite) crystals is investigated with C concentrations up to 5 wt%. The results are compared with the experimental data and suggest that the potential of Hepburn and Ackland is the most accurate in reproducing the melting phase diagram of the ferrite, although the austenite cannot be stabilized at any C concentration for this potential. The potential of Lau et al. yields the best qualitative agreement with the real phase diagram in that the ferrite-liquid coexistence at low C concentrations is replaced by the austenite-liquid coexistence at higher C concentrations. However, the crossover C concentration is much larger and the ferrite melting temperature is much higher than in the real Fe-C alloy. The ABOP of Henriksson and Nordlund without the Ziegler-Biersack-Littmark (ZBL) correction correctly predicts the relative stability of ferrite and austenite at melting, but significantly underestimates the solubility of C in the solid phases, while the same potential with the ZBL correction predicts the austenite to be more stable compared to the ferrite at all C concentrations near the melting transition.
Description: Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.commatsci.2017.12.052.Fri, 26 Jan 2018 17:06:43 GMThttp://hdl.handle.net/2381/410402018-01-26T17:06:43ZNew Findings on Key Factors Influencing the UK's Referendum on Leaving the EU
http://hdl.handle.net/2381/40932
Title: New Findings on Key Factors Influencing the UK's Referendum on Leaving the EU
Authors: Zhang, Aihua
Abstract: The UK’s EU in/out referendum raised significant debate and speculation of the intention of the electorate
and its motivations in voting; much of this debate was informed by simple data analysis examining individual
factors, in isolation, and using opinion polling data. This, in the case of the EU referendum where
multiple factors influence the decision simultaneously, failed to predict the eventual outcome. On June
23, 2016, Britain’s vote to leave the EU came as a surprise to most observers, with a bigger voter turnout
than that of any UK general election in the past decade. In this research, we apply multivariate regression
analysis and a Logit Model to real voting data to identify statistically significant factors influencing the EU
referendum voting preference simultaneously as well as the odd ratio in favor of Leave. Visualizations of
the key findings are also provided with heat maps and graphs. We find that higher education is the predominant factor dividing the nation, with a marginal effect on the referendum decision being stronger than any other factors particularly in England and Wales, where most Leave voters reside. An increase of about 3% in the proportion of British adults accessing to higher education in England and Wales could have reversed the referendum result in the UK. We also find that areas in England and Wales with a lower unemployment rate tend to have a higher turnout to support
Leave while areas in Scotland and Northern Ireland with a higher proportion of university-educated
British adults have a higher turnout to support Remain. Further we find that areas with high proportions
of British male adults show a higher percentage of Leave votes. A higher proportion of elderly British contributes
to a higher percentage of Leave votes, but does not lead to Leave outcomes on their own.
Description: The file associated with this record is under embargo until 24 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Wed, 17 Jan 2018 13:11:27 GMThttp://hdl.handle.net/2381/409322018-01-17T13:11:27ZWedderburn-Malcev decomposition of one-sided ideals of finite dimensional algebras
http://hdl.handle.net/2381/40922
Title: Wedderburn-Malcev decomposition of one-sided ideals of finite dimensional algebras
Authors: Baranov, A. A.; Mudrov, A.; Shlaka, H. M.
Abstract: Let $A$ be a finite dimensional associative algebra over a perfect field and let $R$ be the radical of $A$. We show that for every one-sided ideal I of A there is a semisimple subalgebra $S$ of $A$ such that $I=I_S\oplus I_R$ where $I_S=I\cap S$ and $I_R=I\cap R$.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Wed, 17 Jan 2018 11:33:53 GMThttp://hdl.handle.net/2381/409222018-01-17T11:33:53ZMultistability, synchronization, and self-organization in networks of nonlinear systems with changing graph topologies
http://hdl.handle.net/2381/40913
Title: Multistability, synchronization, and self-organization in networks of nonlinear systems with changing graph topologies
Authors: Jarman, Nicholas J.
Abstract: Complex network structures appear in myriad contexts. From social networks to computer networks, protein and transport networks, and neuronal networks of the mammalian brain. Furthermore, many of these networks share common structural properties. Are there general underlying mechanisms for the emergence of certain complex network structures? One such shared principle is the mutual relationship between structure and function in self-organising networks. Understanding their emergence can be decomposed into two simpler problems: (1) How does structure effect dynamics? (2) How do dynamics shape the structure? Concepts of nonlinear systems theory provide a tool-set for stability analysis of dynamics on a network. Here, stability analysis is applied to the problem of how a small change in network structure effects the dynamics. Two connectivity configurations are considered; the directed chain and the directed cycle, distinguished by a single edge. Their linear stability is first analysed, followed by the stability of interconnected nonlinear oscillators. Stability analysis reveals radical changes in the patterns of dynamics; while the directed chain possess only one stable solution (synchronization), the directed cycle possesses multistabiliy (synchronization and rotating waves). This capacity for multistability is realised by the extremal properties of the directed cycle; the slow decal of oscillations in the coupling dynamics resonates with the dynamics of the individual oscillators. This result is generalised to networks that contain modular structures and heterogeneous dynamics. For applications of evolving network structures, systems theory is limited. Computational modelling, on the other hand, provides an effcacious alternative. A well-established driving mechanism for network structure evolution is adaptive rewiring; adaptation of structure to function. Computational modelling reveals a synergy between spatial organisation and adaptive rewiring. Emergence of modular small-world network structures are more pronounced, and evolution more robust, than in models without spatial organisation. However, studies employing adaptive rewiring have been frustrated by the need to explicitly specify dynamics. To address this, explicit dynamics are replaced by an abstract representation of network diffusion (information transfer or traffic flow): shortcuts are created where traffic flow is intense, while annihilating underused connections - like pedestrians define walkways in parks. The resulting networks are a family of small-world structures; networks may be modular or centralised. Moreover, at the critical point of phase transition of network structure, hierarchical structures emerge - like those found in the brain. This thesis therefore serves to help bridge the gap between dynamical systems theory and computational modelling in the field of complex network theory; the importance of connectivity on dynamics, as detailed in systems theory, is captured using graph diffusion, and applied in the context of computational modelling. This successfully reduces the highly complex problem of complex network emergence to a much simpler one, namely, patterns of connectivity. In doing so, the generality of this machinery provides a more lucid understanding for the self-organisation of complex network structures across a broad range of contexts.Wed, 17 Jan 2018 10:27:28 GMThttp://hdl.handle.net/2381/409132018-01-17T10:27:28ZAnalysis of discontinuous Galerkin methods using mesh-dependent norms and applications to problems with rough data
http://hdl.handle.net/2381/40909
Title: Analysis of discontinuous Galerkin methods using mesh-dependent norms and applications to problems with rough data
Authors: Georgoulis, Emmanuil H.; Pryer, Tristan
Abstract: We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these norms. As an application we examine some problems with rough source term where the solution can not be characterised as a weak solution and show quasi-optimal error control.Wed, 17 Jan 2018 10:10:28 GMThttp://hdl.handle.net/2381/409092018-01-17T10:10:28ZTheoretical and numerical analysis of nano-actuators based on grafted polyelectrolytes in an electric field
http://hdl.handle.net/2381/40847
Title: Theoretical and numerical analysis of nano-actuators based on grafted polyelectrolytes in an electric field
Authors: Brilliantov, Nikolai V.; Budkov, Y. A.; Seidel, C.
Abstract: We analyze, theoretically and by means of molecular dynamics (MD) simulations, the generation of mechanical force by a polyelectrolyte (PE) chain grafted to a plane and exposed to an external electric field; the free end of the chain is linked to a deformable target body. Varying the field, one can alter the length of the non-adsorbed (bulk) part of the chain and hence the deformation of the target body and the arising force. We focus on the impact of added salt on the magnitude of the generated force, which is especially important for applications. In particular, we develop a simple variational theory for the double layer formed near electrodes to compute the electric field acting on the bulk part of the chain. Our theoretical predictions agree well with the MD simulations. Next, we study the effectiveness of possible PE-based nano-vices, comprised of two clenching planes connected by PEs exposed to an external electric field. We analyze a novel phenomenon – two-dimensional diffusion of a nano-particle, clenched between two planes, and introduce a quantitative criterion for clenching efficiency, the clenching coefficient. It is defined as a logarithm of the ratio of the diffusion coefficients of a free and clenched particle. Using first a microscopic counterpart of the Coulomb friction model, and then a novel microscopic model based on surface phonons, with the vibration direction normal to the surface, we calculate the clenching coefficient as a function of the external electric field. Our results demonstrate a dramatic decrease of the diffusion coefficient of a clenched nano-particle for the range of parameters relevant for applications; this proves the effectiveness of the PE-based nano-vices.Thu, 11 Jan 2018 13:51:17 GMThttp://hdl.handle.net/2381/408472018-01-11T13:51:17ZOrthogonal basis for the Shapovalov form on U-q (sl(n+1))
http://hdl.handle.net/2381/40845
Title: Orthogonal basis for the Shapovalov form on U-q (sl(n+1))
Authors: Mudrov, Andrey
Abstract: Let U be either the classical or quantized universal enveloping algebra of the Lie algebra sl(n + 1) extended over the field of fractions of the Cartan subalgebra. We suggest a PBW basis in U over the extended Cartan subalgebra diagonalizing the contravariant Shapovalov form on generic Verma module. The matrix coefficients of the form are calculated and the inverse form is explicitly constructed.Thu, 11 Jan 2018 12:32:41 GMThttp://hdl.handle.net/2381/408452018-01-11T12:32:41ZHilbert's 6th Problem: exact and approximate hydrodynamic manifolds for kinetic equations
http://hdl.handle.net/2381/40822
Title: Hilbert's 6th Problem: exact and approximate hydrodynamic manifolds for kinetic equations
Authors: Gorban, Alexander N; Karlin, Ilya
Abstract: The problem of the derivation of hydrodynamics from the Boltzmann
equation and related dissipative systems is formulated as the problem
of a slow invariant manifold in the space of distributions. We review a few
instances where such hydrodynamic manifolds were found analytically both as
the result of summation of the Chapman–Enskog asymptotic expansion and by
the direct solution of the invariance equation. These model cases, comprising
Grad’s moment systems, both linear and nonlinear, are studied in depth in
order to gain understanding of what can be expected for the Boltzmann equation.
Particularly, the dispersive dominance and saturation of dissipation rate
of the exact hydrodynamics in the short-wave limit and the viscosity modification
at high divergence of the flow velocity are indicated as severe obstacles to
the resolution of Hilbert’s 6th Problem. Furthermore, we review the derivation
of the approximate hydrodynamic manifold for the Boltzmann equation using
Newton’s iteration and avoiding smallness parameters, and compare this to
the exact solutions. Additionally, we discuss the problem of projection of the
Boltzmann equation onto the approximate hydrodynamic invariant manifold
using entropy concepts. Finally, a set of hypotheses is put forward where we
describe open questions and set a horizon for what can be derived exactly or
proven about the hydrodynamic manifolds for the Boltzmann equation in the
future.Thu, 11 Jan 2018 09:10:47 GMThttp://hdl.handle.net/2381/408222018-01-11T09:10:47ZDirect data-based decision making under uncertainty
http://hdl.handle.net/2381/40793
Title: Direct data-based decision making under uncertainty
Authors: Grechuk, Bogdan; Zabarankin, Michael
Abstract: In a typical one-period decision making model under uncertainty, unknown consequences are modeled as random variables. However, accurately estimating probability distributions of the involved random variables from historical data is rarely possible. As a result, decisions made may be suboptimal or even unacceptable in the future. Also, an agent may not view data occurred at different time moments, e.g. yesterday and one year ago, as equally probable. The agent may apply a so-called “time” profile (weights) to historical data. To address these issues, an axiomatic framework for decision making based directly on historical time series is presented. It is used for constructing data-based analogues of mean-variance and maxmin utility approaches to optimal portfolio selection.
Description: The file associated with this record is under embargo until 24 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Wed, 10 Jan 2018 09:54:13 GMThttp://hdl.handle.net/2381/407932018-01-10T09:54:13ZGeometric integrator for Langevin systems with quaternion-based rotational degrees of freedom and hydrodynamic interactions.
http://hdl.handle.net/2381/40791
Title: Geometric integrator for Langevin systems with quaternion-based rotational degrees of freedom and hydrodynamic interactions.
Authors: Davidchack, R. L.; Ouldridge, T. E.; Tretyakov, M. V.
Abstract: We introduce new Langevin-type equations describing the rotational and translational motion of rigid bodies interacting through conservative and non-conservative forces and hydrodynamic coupling. In the absence of non-conservative forces, the Langevin-type equations sample from the canonical ensemble. The rotational degrees of freedom are described using quaternions, the lengths of which are exactly preserved by the stochastic dynamics. For the proposed Langevin-type equations, we construct a weak 2nd order geometric integrator that preserves the main geometric features of the continuous dynamics. The integrator uses Verlet-type splitting for the deterministic part of Langevin equations appropriately combined with an exactly integrated Ornstein-Uhlenbeck process. Numerical experiments are presented to illustrate both the new Langevin model and the numerical method for it, as well as to demonstrate how inertia and the coupling of rotational and translational motion can introduce qualitatively distinct behaviours.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Wed, 10 Jan 2018 09:43:04 GMThttp://hdl.handle.net/2381/407912018-01-10T09:43:04ZPseudo-outcrop Visualization of Borehole Images and Core Scans
http://hdl.handle.net/2381/40768
Title: Pseudo-outcrop Visualization of Borehole Images and Core Scans
Authors: Mirkes, Evgeny M.; Gorban, Alexander N.; Levesley, Jeremy; Elkington, Peter A. S.; Whetton, James A.
Abstract: A pseudo-outcrop visualization is demonstrated for borehole and full-diameter rock core images to augment the ubiquitous unwrapped cylinder view and thereby assist nonspecialist interpreters. The pseudo-outcrop visualization is equivalent to a nonlinear projection of the image from borehole to earth frame of reference that creates a solid volume sliced longitudinally to reveal two or more faces in which the orientations of geological features indicate what is observed in the subsurface. A proxy for grain size is used to modulate the external dimensions of the plot to mimic profiles seen in real outcrops. The volume is created from a mixture of geological boundary elements and texture, the latter being the residue after the sum of boundary elements is subtracted from the original data. In the case of measurements from wireline microresistivity tools, whose circumferential coverage is substantially <100 %, the missing circumferential data are first inpainted using multiscale directional transforms, which decompose the image into its elemental building structures, before reconstructing the full image. The pseudo-outcrop view enables direct observation of the angular relationships between features and aids visual comparison between borehole and core images, especially for the interested nonspecialist.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Tue, 09 Jan 2018 11:49:11 GMThttp://hdl.handle.net/2381/407682018-01-09T11:49:11ZPatterns of invasive species spread in a landscape with a complex geometry
http://hdl.handle.net/2381/40746
Title: Patterns of invasive species spread in a landscape with a complex geometry
Authors: Alharbi, Weam; Petrovskii, Sergei
Abstract: Patterns and rates of invasive species spread have been a focus of attention for several decades. Majority of studies focused on the species proliferation in a relatively uniform "open space" thus leaving aside the effects of the landscape geometry as given by size and shape of inaccessible areas. In this paper, we address this issue by considering the spatiotemporal dynamics of an alien species in a domain where two large uniform habitats are connected by a narrow corridor. We consider the case where the species is originally introduced into one of the habitats but not to the other. The alien species is assumed to be affected by a predator, so that mathematically our system consists of two coupled diffusion-reaction equations. We show that the corridor tends to slow down the spread: it takes the alien population an extra time to penetrate through the corridor, and this delay time can be significant in the case of patchy spread. We also show that a sufficiently narrow corridor blocks the spread; simple analytical estimates for the critical width of the corridor are obtained. Finally, we show that the corridor can become a refuge for the alien population. If considered on a longer timescale that includes species adaptation and/or climate change, the corridor may then become a source of a secondary invasion.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Mon, 08 Jan 2018 16:34:27 GMThttp://hdl.handle.net/2381/407462018-01-08T16:34:27ZDecomposition spaces, incidence algebras and Möbius inversion III: the decomposition space of Möbius intervals
http://hdl.handle.net/2381/40684
Title: Decomposition spaces, incidence algebras and Möbius inversion III: the decomposition space of Möbius intervals
Authors: Gálvez-Carrillo, Imma; Kock, Joachim; Tonks, Andrew
Abstract: Decomposition spaces are simplicial ∞-groupoids subject to a certain
exactness condition, needed to induce a coalgebra structure on the space of arrows.
Conservative ULF functors (CULF) between decomposition spaces induce
coalgebra homomorphisms. Suitable added finiteness conditions define the notion
of Möbius decomposition space, a far-reaching generalisation of the notion of
Möbius category of Leroux. In this paper, we show that the Lawvere–Menni Hopf
algebra of Möbius intervals, which contains the universal Möbius function (but is
not induced by a Möbius category), can be realised as the homotopy cardinality
of a Möbius decomposition space U of all Möbius intervals, and that in a certain
sense U is universal for Möbius decomposition spaces and CULF functors.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Mon, 18 Dec 2017 14:36:54 GMThttp://hdl.handle.net/2381/406842017-12-18T14:36:54ZA posteriori error estimates for the virtual element method
http://hdl.handle.net/2381/40683
Title: A posteriori error estimates for the virtual element method
Authors: Cangiani, Andrea; Georgoulis, Emmanuil H.; Pryer, Tristan; Sutton, Oliver J.
Abstract: An posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing.Mon, 18 Dec 2017 14:26:58 GMThttp://hdl.handle.net/2381/406832017-12-18T14:26:58ZHow priors of initial hyperparameters affect Gaussian process regression models
http://hdl.handle.net/2381/40631
Title: How priors of initial hyperparameters affect Gaussian process regression models
Authors: Chen, Zexun; Wang, Bo
Abstract: The hyperparameters in Gaussian process regression (GPR) model with a specified kernel are often estimated from the data via the maximum marginal likelihood. Due to the non-convexity of marginal likelihood with respect to the hyperparameters, the optimisation may not converge to the global maxima. A common approach to tackle this issue is to use multiple starting points randomly selected from a specific prior distribution. As a result the choice of prior distribution may play a vital role in the predictability of this approach. However, there exists little research in the literature to study the impact of the prior distributions on the hyperparameter estimation and the performance of GPR. In this paper, we provide the first empirical study on this problem using simulated and real data experiments. We consider different types of priors for the initial values of hyperparameters for some commonly used kernels and investigate the influence of the priors on the predictability of GPR models. The results reveal that, once a kernel is chosen, different priors for the initial hyperparameters have no significant impact on the performance of GPR prediction, despite that the estimates of the hyperparameters are very different to the true values in some cases.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Tue, 28 Nov 2017 14:12:55 GMThttp://hdl.handle.net/2381/406312017-11-28T14:12:55ZAugmented Homotopical Algebraic Geometry
http://hdl.handle.net/2381/40623
Title: Augmented Homotopical Algebraic Geometry
Authors: Balchin, Scott Lewis
Abstract: In this thesis we are interested in extending the theory of homotopical algebraic geometry, which itself is a homotopification of classical algebraic geometry. We introduce the concept of augmentation categories, which are a class of generalised Reedy categories. An augmentation category is a category which has enough structure that we can mirror the simplicial constructions which make up the theory of homotopical algebraic geometry. In particular, we construct a Quillen model structure on their presheaf categories, and introduce the concept of augmented hypercovers to define a local model structure on augmented presheaves.
As an application, we show that a crossed simplicial group is an example of an augmentation category. The resulting augmented geometric theory can be thought of as being equivariant. Using this, we define equivariant cohomology theories as special mapping spaces in the category of equivariant stacks. We also define the SO(2)-equivariant derived stack of local systems by using a twisted nerve construction. Moreover, we prove that the category of planar rooted trees appearing in the theory of dendroidal sets is also an augmentation category. The augmented geometry over this setting should be thought of as being stable in the spectral sense of the word. Finally, we show that we can combine the two main examples presented using a categorical amalgamation construction.Tue, 28 Nov 2017 11:41:53 GMThttp://hdl.handle.net/2381/406232017-11-28T11:41:53ZA random acceleration model of individual animal movement allowing for diffusive, superdiffusive and superballistic regimes
http://hdl.handle.net/2381/40612
Title: A random acceleration model of individual animal movement allowing for diffusive, superdiffusive and superballistic regimes
Authors: Tilles, Paulo F. C.; Petrovskii, Sergei V.; Natti, Paulo L.
Abstract: Patterns of individual animal movement attracted considerable attention over the last two decades. In particular, question as to whether animal movement is predominantly diffusive or superdiffusive has been a focus of discussion and controversy. We consider this problem using a theory of stochastic motion based on the Langevin equation with non-Wiener stochastic forcing that originates in animal's response to environmental noise. We show that diffusive and superdiffusive types of motion are inherent parts of the same general movement process that arises as interplay between the force exerted by animals (essentially, by animal's muscles) and the environmental drag. The movement is superballistic with the mean square displacement growing with time as 〈x 2 (t)〉 ∼ t 4 at the beginning and eventually slowing down to the diffusive spread 〈x 2 (t)〉 ∼ t. We show that the duration of the superballistic and superdiffusive stages can be long depending on the properties of the environmental noise and the intensity of drag. Our findings demonstrate theoretically how the movement pattern that includes diffusive and superdiffusive/superballistic motion arises naturally as a result of the interplay between the dissipative properties of the environment and the animal's biological traits such as the body mass, typical movement velocity and the typical duration of uninterrupted movement.
Description: Supplementary information accompanies this paper at https://doi.org/10.1038/s41598-017-14511-9.Mon, 27 Nov 2017 15:27:41 GMThttp://hdl.handle.net/2381/406122017-11-27T15:27:41ZMultilevel sparse grids collocation for linear partial differential equations, with tensor product smooth basis functions
http://hdl.handle.net/2381/40561
Title: Multilevel sparse grids collocation for linear partial differential equations, with tensor product smooth basis functions
Authors: Zhao, Yangzhang; Zhang, Qi; Levesley, Jeremy
Abstract: Radial basis functions have become a popular tool for approximation and solution of partial differential equations (PDEs). The recently proposed multilevel sparse interpolation with kernels (MuSIK) algorithm proposed in \cite{Georgoulis} shows good convergence. In this paper we use a sparse kernel basis for the solution of PDEs by collocation. We will use the form of approximation proposed and developed by Kansa \cite{Kansa1986}. We will give numerical examples using a tensor product basis with the multiquadric (MQ) and Gaussian basis functions. This paper is novel in that we consider space-time PDEs in four dimensions using an easy-to-implement algorithm, with smooth approximations. The accuracy observed numerically is as good, with respect to the number of data points used, as other methods in the literature; see \cite{Langer1,Wang1}.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Tue, 21 Nov 2017 09:55:58 GMThttp://hdl.handle.net/2381/405612017-11-21T09:55:58ZMultilevel quasi-interpolation on a sparse grid with the Gaussian
http://hdl.handle.net/2381/40549
Title: Multilevel quasi-interpolation on a sparse grid with the Gaussian
Authors: Usta, Fuat; Levesley, Jeremy
Abstract: Motivated by the recent multilevel sparse kernel-based interpolation (MuSIK) algorithm proposed in Georgoulis et al. (SIAM J. Sci. Comput. 35, 815–832, 2013), we introduce the new quasi-multilevel sparse interpolation with kernels (Q-MuSIK) via the combination technique. The Q-MuSIK scheme achieves better convergence and run time when compared with classical quasi-interpolation. Also, the Q-MuSIK algorithm is generally superior to the MuSIK methods in terms of run time in particular in high-dimensional interpolation problems, since there is no need to solve large algebraic systems. We subsequently propose a fast, low complexity, high-dimensional positive-weight quadrature formula based on Q-MuSIKSapproximation of the integrand. We present the results of numerical experimentation for both quasi-interpolation and quadrature in high dimensions.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Fri, 17 Nov 2017 12:35:30 GMThttp://hdl.handle.net/2381/405492017-11-17T12:35:30ZHomotopy Linear Algebra
http://hdl.handle.net/2381/40545
Title: Homotopy Linear Algebra
Authors: Tonks, Andrew P.; Kock, Joachim; Gálvez-Carrillo, Imma
Abstract: By homotopy linear algebra we mean the study of linear functors between slices of the ∞-category of ∞-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices into ∞-categories to model the duality between vector spaces and profinite-dimensional vector spaces, and set up a global notion of homotopy cardinality à la Baez, Hoffnung and Walker compatible with this duality. We needed these results to support our work on incidence algebras and Möbius inversion over ∞-groupoids; we hope that they can also be of independent interest.
Description: The file associated with this record is under embargo until 6 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Fri, 17 Nov 2017 10:41:36 GMThttp://hdl.handle.net/2381/405452017-11-17T10:41:36ZOn the Module Category of Symmetric Special Multiserial Algebras
http://hdl.handle.net/2381/40505
Title: On the Module Category of Symmetric Special Multiserial Algebras
Authors: Duffield, Drew Damien
Abstract: The module category of an algebra is a major source of study for representation theorists. The indecomposable modules over an algebra and the morphisms between them are of tremendous importance, since these essentially determine the finitely generated module category over the algebra. The Auslander-Reiten quiver is a means of presenting this information.
In this thesis, we focus on the class of symmetric special multiserial algebras. These are a broad class of algebras that include the well-studied subclass of symmetric special biserial algebras. A useful property of these algebras is that they have a decorated hypergraph (with orientation) associated to them, called a Brauer configuration. As well as offering a pictorial presentation of the algebra, many aspects of the representation theory are encoded in the combinatorial data of the hypergraph.
In the first half of this thesis, we show that the Auslander-Reiten quiver of a symmetric special biserial algebra is completely determined by its associated Brauer configuration. Specifically, we can determine the indecomposable modules and the irreducible morphisms belonging to any component of the Auslander-Reiten quiver using only information from the Brauer configuration. We also show the number of certain components and their precise size and shape is entirely determined by the Green walks along the Brauer configuration.
The second half of this thesis, comprising of the last two chapters, is a study on the representation type of symmetric special multiserial algebras. Unlike in the biserial case, not all of these algebras are tame. It is important to know if an algebra is tame or wild, since if it is wild, a classification of the indecomposable modules is considered to be hopeless. In this section of the thesis, we describe which symmetric special multiserial algebras are wild, which we present in terms of the Brauer configuration.Tue, 07 Nov 2017 09:27:51 GMThttp://hdl.handle.net/2381/405052017-11-07T09:27:51ZGaussian process regression methods and extensions for stock market prediction
http://hdl.handle.net/2381/40502
Title: Gaussian process regression methods and extensions for stock market prediction
Authors: Chen, Zexun
Abstract: Gaussian process regression (GPR) is a kernel-based nonparametric method that has been proved to be effective and powerful in many areas, including time series prediction. In this thesis, we focus on GPR and its extensions and then apply them to financial time series prediction. We first review GPR, followed by a detailed discussion about model structure, mean functions, kernels and hyper-parameter estimations. After that, we study the sensitivity of hyper-parameter and performance of GPR to the prior distribution for the initial values, and find that the initial hyper-parameters’ estimates depend on the choice of the specific kernels, with the priors having little influence on the performance of GPR in terms of predictability. Furthermore, GPR with Student-t process (GPRT) and Student-t process regression (TPR), are introduced. All the above models as well as autoregressive moving average (ARMA) model are applied to predict equity indices.
We find that GPR and TPR shows relatively considerable capability of predicting equity indices so that both of them are extended to state-space GPR (SSGPR) and state-space TPR (SSTPR) models, respectively. The overall results are that SSTPR outperforms SSGPR for the equity index prediction. Based on the detailed results, a brief market efficiency analysis confirms that the developed markets are unpredictable on the whole. Finally, we propose and test the multivariate GPR (MV-GPR) and multivariate TPR (MV-TPR) for multi-output prediction, where the model settings, derivations and computations are all directly performed in matrix form, rather than vectorising the matrices involved in the existing method of GPR for multi-output prediction. The effectiveness of the proposed methods is illustrated through a simulated example. The proposed methods are then applied to stock market modelling in which the Buy&Sell strategies generated by our proposed methods are shown to be profitable in the equity investment.Tue, 07 Nov 2017 09:00:39 GMThttp://hdl.handle.net/2381/405022017-11-07T09:00:39ZTowards characterisation of chaotic attractors in terms of embedded coherent structures
http://hdl.handle.net/2381/40289
Title: Towards characterisation of chaotic attractors in terms of embedded coherent structures
Authors: Crane, Daniel Lewis
Abstract: The central theme of this thesis is the development of general methods for the modelling of the dynamics on chaotic attractors by a coarse-grained representation constructed through the use of embedded periodic orbits & other coherent structures. Our aim is to develop tools for constructing two types of reduced representations of chaotic attractors: Markov-type models, and symbolic dynamics. For Markov models, we present construction of a minimal cover of chaotic attractors of maps and high-dimensional flows by embedded coherent structures such as periodic orbits from which a Markov chain of the dynamics can be constructed. For the symbolic dynamics, we investigate the utility of unstable periodic orbits for the construction of an approximate generating partition of a chaotic attractor.
In the first section of Part 1 we present an original method by which chaotic attractors of discrete-time dynamical systems can be covered using a small set of unstable periodic orbits (UPOs) following an iterative selection algorithm that only chooses those UPOs that provide additional covering of the attractor to be included into the cover. We then show how this representation can be used to represent trajectories in the system as a series of transition between cover elements, using which as a basis for the construction of a Markov chain representation of the dynamics. In the second section we extend this method to continuous-time dynamical systems, introducing methods by which covers of high-dimensional attractors can be constructed in low dimensional projections with as little information loss as possible, and also giving an example of how group symmetries of the system can be dealt with.
In Part 2 we change our focus to the construction of symbolic dynamics of discrete-time systems, presenting an extension to an existing method for the computational construction of approximate generating partitions that increases the applicability of the method to a wider range of systems, and also significantly improving the results for more complex maps.Wed, 30 Aug 2017 14:44:44 GMThttp://hdl.handle.net/2381/402892017-08-30T14:44:44ZComputational diagnosis and risk evaluation for canine lymphoma.
http://hdl.handle.net/2381/40283
Title: Computational diagnosis and risk evaluation for canine lymphoma.
Authors: Mirkes, E. M.; Alexandrakis, I.; Slater, K.; Tuli, R.; Gorban, A. N.
Abstract: The canine lymphoma blood test detects the levels of two biomarkers, the acute phase proteins (C-Reactive Protein and Haptoglobin). This test can be used for diagnostics, for screening, and for remission monitoring as well. We analyze clinical data, test various machine learning methods and select the best approach to these oblems. Three families of methods, decision trees, kNN (including advanced and adaptive kNN) and probability density evaluation with radial basis functions, are used for classification and risk estimation. Several pre-processing approaches were implemented and compared. The best of them are used to create the diagnostic system. For the differential diagnosis the best solution gives the sensitivity and specificity of 83.5% and 77%, respectively (using three input features, CRP, Haptoglobin and standard clinical symptom). For the screening task, the decision tree method provides the best result, with sensitivity and specificity of 81.4% and >99%, respectively (using the same input features). If the clinical symptoms (Lymphadenopathy) are considered as unknown then a decision tree with CRP and Hapt only provides sensitivity 69% and specificity 83.5%. The lymphoma risk evaluation problem is formulated and solved. The best models are selected as the system for computational lymphoma diagnosis and evaluation of the risk of lymphoma as well. These methods are implemented into a special web-accessed software and are applied to the problem of monitoring dogs with lymphoma after treatment. It detects recurrence of lymphoma up to two months prior to the appearance of clinical signs. The risk map visualization provides a friendly tool for exploratory data analysis.Tue, 29 Aug 2017 13:10:13 GMThttp://hdl.handle.net/2381/402832017-08-29T13:10:13ZWhat could have tipped the EU referendum result in favour of Remain
http://hdl.handle.net/2381/40225
Title: What could have tipped the EU referendum result in favour of Remain
Authors: Zhang, Aihua
Abstract: [First paragraphs] Much of the analysis about why the UK voted to leave the European Union in June 2016 has been done by looking at individual factors in isolation, or using opinion poll data from both before and after the vote.
In a new research paper, I applied two statistical analyses to the actual referendum voting data obtained from the Electoral Commission and the UK’s latest census data. I found that while voters’ level of higher education was the most important factor, the gender of voters and the turnout level also had parts to play in the victory for the Leave campaign.
Description: This article was written for The Conversation by invitationWed, 23 Aug 2017 09:08:40 GMThttp://hdl.handle.net/2381/402252017-08-23T09:08:40ZThe Lusternik-Schnirelmann Category for a Differentiable Stack
http://hdl.handle.net/2381/40216
Title: The Lusternik-Schnirelmann Category for a Differentiable Stack
Authors: Alsulami, Samirah; Colman, Hellen; Neumann, Frank
Abstract: We introduce the notion of Lusternik-Schnirelmann category for differentiable stacks and establish its relation with the groupoid Lusternik-Schnirelmann category for Lie groupoids. This extends the notion of Lusternik-Schnirelmann category for smooth manifolds and orbifolds.Tue, 22 Aug 2017 12:17:53 GMThttp://hdl.handle.net/2381/402162017-08-22T12:17:53ZLong and short range multi-locus QTL interactions in a complex trait of yeast
http://hdl.handle.net/2381/40203
Title: Long and short range multi-locus QTL interactions in a complex trait of yeast
Authors: Mirkes, Evgeny M.; Walsh, Thomas; Louis, Edward J.; Gorban, Alexander N.
Abstract: We analyse interactions of Quantitative Trait Loci (QTL) in heat selected yeast by comparing them to an unselected pool of random individuals. Here we re-examine data on individual F12 progeny selected for heat tolerance, which have been genotyped at 25 locations identified by sequencing a selected pool [Parts, L., Cubillos, F. A., Warringer, J., Jain, K., Salinas, F., Bumpstead, S. J., Molin, M., Zia, A., Simpson, J. T., Quail, M. A., Moses, A., Louis, E. J., Durbin, R., and Liti, G. (2011). Genome research, 21(7), 1131-1138]. 960 individuals were genotyped at these locations and multi-locus genotype frequencies were compared to 172 sequenced individuals from the original unselected pool (a control group). Various non-random associations were found across the genome, both within chromosomes and between chromosomes. Some of the non-random associations are likely due to retention of linkage disequilibrium in the F12 population, however many, including the inter-chromosomal interactions, must be due to genetic interactions in heat tolerance. One region of particular interest involves 3 linked loci on chromosome IV where the central variant responsible for heat tolerance is antagonistic, coming from the heat sensitive parent and the flanking ones are from the more heat tolerant parent. The 3-locus haplotypes in the selected individuals represent a highly biased sample of the population haplotypes with rare double recombinants in high frequency. These were missed in the original analysis and would never be seen without the multigenerational approach. We show that a statistical analysis of entropy and information gain in genotypes of a selected population can reveal further interactions than previously seen. Importantly this must be done in comparison to the unselected population's genotypes to account for inherent biases in the original population.Mon, 21 Aug 2017 10:03:01 GMThttp://hdl.handle.net/2381/402032017-08-21T10:03:01ZStochastic Separation Theorems
http://hdl.handle.net/2381/40202
Title: Stochastic Separation Theorems
Authors: Gorban, A. N.; Tyukin, I. Y.
Abstract: The problem of non-iterative one-shot and non-destructive correction of unavoidable mistakes arises in all Artificial Intelligence applications in the real world. Its solution requires robust separation of samples with errors from samples where the system works properly. We demonstrate that in (moderately) high dimension this separation could be achieved with probability close to one by linear discriminants. Surprisingly, separation of a new image from a very large set of known images is almost always possible even in moderately high dimensions by linear functionals, and coefficients of these functionals can be found explicitly. Based on fundamental properties of measure concentration, we show that for $M<a\exp(b{n})$ random $M$-element sets in $\mathbb{R}^n$ are linearly separable with probability $p$, $p>1-\vartheta$, where $1>\vartheta>0$ is a given small constant. Exact values of $a,b>0$ depend on the probability distribution that determines how the random $M$-element sets are drawn, and on the constant $\vartheta$. These {\em stochastic separation theorems} provide a new instrument for the development, analysis, and assessment of machine learning methods and algorithms in high dimension. Theoretical statements are illustrated with numerical examples.
Description: The file associated with this record is under embargo until 12 months after publication, in accordance with the publisher's self-archiving policy. The full text may be available through the publisher links provided above.Mon, 21 Aug 2017 09:52:59 GMThttp://hdl.handle.net/2381/402022017-08-21T09:52:59ZMaximal zero product subrings and inner ideals of simple rings
http://hdl.handle.net/2381/40198
Title: Maximal zero product subrings and inner ideals of simple rings
Authors: Baranov, Alexander; Fernández López, Antonio
Abstract: Let Q be a (non-unital) simple ring. A nonempty subset S of Q is said to have zero product if S^2=0. We classify all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the maximal inner ideals of its associated Lie algebra.
Description: MSC classes: 16D30, 17B60Fri, 18 Aug 2017 09:54:27 GMThttp://hdl.handle.net/2381/401982017-08-18T09:54:27ZA posteriori error estimates for leap-frog and cosine methods for second order evolution problems
http://hdl.handle.net/2381/40197
Title: A posteriori error estimates for leap-frog and cosine methods for second order evolution problems
Authors: Georgoulis, Emmanuil H.; Lakkis, Omar; Makridakis, Charalambos G.; Virtanen, Juha M.
Abstract: We consider second order explicit and implicit two-step time-discrete schemes for wave-type equations. We derive optimal order a posteriori estimates controlling the time discretization error. Our analysis has been motivated by the need to provide a posteriori estimates for the popular leap-frog method (also known as Verlet's method in the molecular dynamics literature); it is extended, however, to general cosine-type second order methods. The estimators are based on a novel reconstruction of the time-dependent component of the approximation. Numerical experiments confirm similarity of the convergence rates of the proposed estimators and the theoretical convergence rate of the true error.
Description: AMS Subject Headings 35L05, 37M05, 37M15, 65M60, 65N50Thu, 17 Aug 2017 10:27:51 GMThttp://hdl.handle.net/2381/401972017-08-17T10:27:51ZModelling Share Prices as a Random Walk on a Markov Chain
http://hdl.handle.net/2381/40129
Title: Modelling Share Prices as a Random Walk on a Markov Chain
Authors: Samci Karadeniz, Rukiye
Abstract: In the financial area, a simple but also realistic means of modelling real data is very important. Several approaches are considered to model and analyse the data presented herein. We start by considering a random walk on an additive functional of a discrete time Markov chain perturbed by Gaussian noise as a model for the data as working with a continuous time model is more convenient for option prices. Therefore, we consider the renowned (and open) embedding problem for Markov chains: not every discrete time Markov chain has an underlying continuous time Markov chain. One of the main goals of this research is to analyse whether the discrete time model permits extension or embedding to the continuous time model. In addition, the volatility of share price data is estimated and analysed by the same procedure as for share price processes. This part of the research is an extensive case study on the embedding problem for financial data and its volatility.
Another approach to modelling share price data is to consider a random walk on the lamplighter group. Specifically, we model data as a Markov chain with a hidden random walk on the lamplighter group Z3 and on the tensor product of groups Z2 ⊗ Z2. The lamplighter group has a specific structure where the hidden information is actually explicit. We assume that the positions of the lamplighters are known, but we do not know the status of the lamps. This is referred to as a hidden random walk on the lamplighter group. A biased random walk is constructed to fit the data. Monte Carlo simulations are used to find the best fit for smallest trace norm difference of the transition matrices for the tensor product of the original transition matrices from the (appropriately split) data.
In addition, splitting data is a key method for both our first and second models. The tensor product structure comes from the split of the data. This requires us to deal with the missing data. We apply a variety of statistical techniques such as Expectation- Maximization Algorithm and Machine Learning Algorithm (C4.5).
In this work we also analyse the quantum data and compute option prices for the binomial model via quantum data.Wed, 02 Aug 2017 14:10:44 GMThttp://hdl.handle.net/2381/401292017-08-02T14:10:44ZAlgebras and varieties
http://hdl.handle.net/2381/40125
Title: Algebras and varieties
Authors: Green, Edward L.; Hille, Lutz; Schroll, Sibylle
Abstract: In this paper we introduce new affine algebraic varieties whose points correspond to quotients of paths algebras. We show that the algebras within a variety share many important homological properties. The case of finite dimensional algebras as well as that of graded algebras arise as classes of subvarieties of the varieties we define.
Description: 20 pagesWed, 02 Aug 2017 13:25:47 GMThttp://hdl.handle.net/2381/401252017-08-02T13:25:47ZOn extensions for gentle algebras
http://hdl.handle.net/2381/40124
Title: On extensions for gentle algebras
Authors: Canakci, Ilke; Pauksztello, David; Schroll, Sibylle
Abstract: We develop an algorithmic method for determining the cohomology of homotopy string and band complexes in the derived category of a gentle algebra. We then use this to give a complete description of a basis of the extensions between string and quasi-simple band modules in the module category of a gentle algebra.
Description: 32 pages, comments welcomeWed, 02 Aug 2017 13:21:14 GMThttp://hdl.handle.net/2381/401242017-08-02T13:21:14Z