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Modelling acidosis and the cell cycle in multicellular tumour spheroids
Get PDFA partial differential equation model is developed to understand the effect that nutrient and acidosis have on the distribution of proliferating and quiescent cells and dead cell material (necrotic and apoptotic) within a multicellular tumour spheroid. The rates of cell quiescence and necrosis depend upon the local nutrient and acid concentrations and quiescent cells are assumed to consume less nutrient and produce less acid than proliferating cells. Analysis of the differences in nutrient consumption and acid production by quiescent and proliferating cells shows low nutrient levels do not necessarily lead to increased acid concentration via anaerobic metabolism. Rather, it is the balance between proliferating and quiescent cells within the tumour which is important; decreased nutrient levels lead to more quiescent cells, which produce less acid than proliferating cells. We examine this effect via a sensitivity analysis which also includes a quantification of the effect that nutrient and acid concentrations have on the rates of cell quiescence and necrosis
A systematic survey of the response of a model NF-κB signalling pathway to TNFα stimulation
Get PDFWhite's lab established that strong, continuous stimulation with tumour necrosis factor-α (TNFα) can induce sustained oscillations in the subcellular localisation of the transcription factor nuclear factor κB (NF-κB). But the intensity of the TNFα signal varies substantially, from picomolar in the blood plasma of healthy organisms to nanomolar in diseased states. We report on a systematic survey using computational bifurcation theory to explore the relationship between the intensity of TNFα stimulation and the existence of sustained NF-κB oscillations. Using a deterministic model developed by Ashall et al. in 2009, we find that the system's responses to TNFα are characterised by a supercritical Hopf bifurcation point: above a critical intensity of TNFα the system exhibits sustained oscillations in NF-κB localisation. For TNFα below this critical value, damped oscillations are observed. This picture depends, however, on the values of the model's other parameters. When the values of certain reaction rates are altered the response of the signalling pathway to TNFα stimulation changes: in addition to the sustained oscillations induced by high-dose stimulation, a second oscillatory regime appears at much lower doses. Finally, we define scores to quantify the sensitivity of the dynamics of the system to variation in its parameters and use these scores to establish that the qualitative dynamics are most sensitive to the details of NF-κB mediated gene transcription
A Recursive Blocked Schur Algorithm for Computing the Matrix Square Root
Get PDFThe Schur method for computing a matrix square root reduces the matrix to the Schur triangular form and then computes a square root of the triangular matrix. We show that by using a recursive blocking technique the computation of the square root of the triangular matrix can be made rich in matrix multiplication. Numerical experiments making appropriate use of level 3 BLAS show significant speedups over the point algorithm, both in the square root phase and in the algorithm as a whole. The excellent numerical stability of the point algorithm is shown to be preserved by recursive blocking. These results are extended to the real Schur method. Recursive blocking is also shown to be effective for multiplying triangular matrices
An Assessment of sustainable housing affordability using a multiple criteria decision making method
Get PDFHousing affordability is a complex issue that must not only be assessed in terms economic viability. In order to increase quality of life and community sustainability the environmental and social sustainability of housing must also be taken into consideration.
The paper considers the application of a methodology that can be applied to assess the affordability of different housing locations in a sustainable manner, taking into account a range of economic, environmental and social criteria. The COPRAS method of multi-criteria decision making (MCDM) is selected and applied to three residential areas as an example of how sustainable housing affordability can be assessed using a MCDM method. The outcome of the study reveals that considering a range of social and environmental criteria can greatly affect the calculation of an areas affordability, in comparison to focusing solely on financial attributes. COPRAS was found to be an effective method for the assessment and could be applied in other regions or internationally
Structured Matrix Nearness Problems: Theory and Algorithms
Get PDFIn many areas of science one often has a given matrix, representing for example a measured data set and is required to find a matrix that is closest in a suitable norm to the matrix and possesses additionally a structure,inherited from the model used or coming from the application. We call these problems structured matrix nearness problems. We look at three different groups of these problems that come from real applications, analyze the properties of the corresponding matrix structure, and propose algorithms to solve them efficiently.The first part of this thesis concerns the nearness problem of finding the nearest factor correlation matrix C(X) =\diag(I_n -XX^T)+XX^T to a given symmetric matrix, subject to natural nonlinear constraints on the elements of the matrix , where distance is measured in the Frobenius norm.Such problems arise, for example, when one is investigating factor models of collateralized debt obligations (CDOs) or multivariate time series. We examine several algorithms for solving the nearness problem that differ in whether or not they can take account of the nonlinear constraints and in their convergence properties. Our numerical experiments show that the performance of the methods depends strongly on the problem, but that, among our tested methods, the spectral projected gradient method is the clear winner.In the second part we look at two two-sided optimization problems where the matrix of unknowns lies in the Stiefel manifold. These two problems come from an application in atomic chemistry where one is looking for atomic orbitals with prescribed occupation numbers. We analyze these two problems, propose an analytic optimal solution of the first and show that an optimal solution of the second problem can be found by solving a convex quadratic programming problem with box constraints and unknowns. We prove that the latter problem can be solved by the active-set method in at most iterations. Subsequently, we analyze the set of optimal solutions of the first problem for symmetric and diagonal and find that a slight modification of it is a Riemannian manifold. We derive the geometric objects required to make an optimization over this manifold possible. We propose an augmented Lagrangian-based algorithm that uses these geometric tools and allows us to optimize an arbitrary smooth function over . This algorithm can be used to select a particular solution out of the latter set by posing a new optimization problem. We compare it numerically with a similar algorithm that,however, does not apply these geometric tools and find that our algorithm yields better performance.The third part is devoted to low rank nearness problems in the -norm, where the matrix of interest is additionally of linear structure, meaning it lies in the set spanned by predefined matrices . These problems are often associated with model reduction, for example in speech encoding, filter design, or latent semantic indexing. We investigate three approaches that support any linear structure and examine further the geometric reformulation by Schuermans et al.\ (2003). We improve their algorithm in terms of reliability by applying the augmented Lagrangian method and show in our numerical tests that the resulting algorithm yields better performance than other existing methods
An Algorithm For Finding the Optimal Embedding of a Symmetric Matrix into the Set of Diagonal Matrices
Get PDFWe investigate two two-sided optimization problems that have their application in atomic chemistry and
whose matrix of unknowns () lies in the Stiefel manifold. We propose an
analytic optimal solution of the first problem, and show that an optimal solution
of the second problem can be found by solving a convex quadratic programming
problem with box constraints and unknowns.
We prove that the latter problem can be solved by the active-set method in at most iterations. Subsequently, we analyze the set of the optimal solutions
of both problems, which is of the form of for and diagonal and we address the problem how an arbitrary smooth function over can be minimized. We find that a slight modification of is a Riemannian
manifold for which geometric objects can be derived that are required to make an optimization over this
manifold possible. By using these geometric tools we propose then an augmented
Lagrangian-based algorithm that minimizes an arbitrary smooth function over and
guarantees global convergence to a stationary point. Latter is shown by investigating when the LICQ (Linear Independence Constraint Qualification) is
satisfied. The algorithm can be used to select a particular solution out of the set by posing a new optimization problem. Finally we compare
this algorithm numerically with a similar algorithm that,
however, does not apply these geometric tools and that is to our knowledge not guaranteed to converge.
Our results show that our algorithm yields a significantly better performance
Automorphisms of finite -groups admitting a partition
Get PDFFor a finite -group the following three conditions are equivalent:
(a) to have a (proper) partition, that is, to be the union of some proper subgroups with trivial
pairwise intersections; (b) to have a proper subgroup outside of which all elements have order ;
(c) to be a semidirect product P=P_1\rtimes\langle \f\rangle where is a subgroup of index and
\f is a splitting automorphism of order of . It is proved that if a
finite -group with a partition admits a soluble group of automorphisms
of coprime order such that
the fixed-point subgroup is soluble of derived length , then
has a maximal subgroup that is nilpotent of class bounded
in terms of , , and . The proof is based on a similar result of
the author and Shumyatsky
for the case where has exponent and on the method of ``elimination of
automorphisms by nilpotency'', which was earlier developed by the author,
in particular, for studying finite -groups with a partition.
It is also proved that if a finite -group with a partition admits a
group of automorphisms that acts faithfully on , then
the exponent of is bounded in terms of the exponent of . The
proof of this result is based on the author's
positive solution of the analogue of Restricted Burnside Problem for finite
-groups with a splitting automorphism of order .
Both theorems yield corollaries on finite groups admitting a
Frobenius group of automorphisms whose kernel is generated by
a splitting automorphism of prime order
Triangularizing matrix polynomials
Get PDFFor an algebraically closed field \F, we show that any matrix polynomial P(\lambda)\in \F[\lambda]^{\nbym}, , can be reduced to triangular form, preserving the degree and the finite and infinite elementary divisors.
We also characterize the real matrix polynomials that are
triangularizable over the real numbers and show that those that are not triangularizable are quasi-triangularizable
with diagonal blocks of sizes and . The proofs we present solve the structured inverse problem of building up triangular matrix polynomials starting from lists of elementary divisors
Avoiding communication through a multilevel LU factorization
Get PDFDue to the evolution of massively parallel computers towards deeper levels of parallelism and memory hierarchy, and due to the exponentially increasing ratio of the time required to transfer data, either through the memory hierarchy or between different compute units, to the time required to compute floating point operations, the algorithms are confronted with two challenges. They need not only to be able to exploit multiple levels of parallelism, but also to reduce the communication between the compute units at each level of the hierarchy of parallelism and between the different levels of the memory hierarchy.
In this paper we present an algorithm for performing the LU factorization of dense matrices that is suitable for computer systems with two levels of parallelism. This algorithm is able to minimize both the volume of communication and the number of messages transferred at every level of the two-level hierarchy of parallelism. We present its implementation for a cluster of multicore processors based on MPI and Pthreads. We show that this implementation leads to a better performance than routines implementing the LU factorization in well-known numerical libraries. For matrices that are tall and skinny, that is they have many more rows than columns, our algorithm outperforms the corresponding algorithm from ScaLAPACK by a factor of 4.5 on a cluster of 32 nodes, each node having two quad-core Intel Xeon EMT64 processors
Optimal Staffing Policy: A Service System with Stochastic Travel Times
No full textPrivate sector operators of response services such as ambulance, fire or police etc. are often regulated with
targets on the distribution of response times, which may result in inefficient over staffing to ensure those targets
are met. In this paper, we use a network chain of M=M=K queues to model the arrival and completion of jobs
on the system so that quantities such as the expected total time waiting for all jobs can be calculated. The
Markov nature enables us to evoke the Hamilton Jacobi Bellman equation (HJB) principle to optimize the
required number of staff whilst still meeting targets