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Classifying Serre subcategories of finitely presented modules
Get PDFGiven a commutative coherent ring R, a bijective correspondence between the thick subcategories of perfect complexes D_{per}(R) and the Serre subcategories of finitely presented modules is established. To construct this correspondence, properties of the Ziegler and Zariski topologies on the set of (iso-classes for) indecomposable injective modules are essentially used
Singular value decomposition of multi-companion matrices
Get PDFWe obtain the singular value decomposition of multi-companion matrices. We completely
characterise the columns of the matrix and give a simple formula for obtaining the
columns of the other unitary matrix, , from the columns of . We also obtain necessary
and sufficient conditions for the related matrix polynomial to be hyperbolic
Level set reconstruction of conductivity and permittivity from boundary electrical measurements using experimental data
No full textA shape reconstruction method for electrical resistance and capacitance tomography is presented using a level set formulation. In this shape reconstruction approach, the conductivity (or permittivity) values of the inhomogeneous background and the obstacles are assumed to be (approximately) known, but the number, sizes, shapes, and locations of these obstacles have to be recovered from the data. A key point in this shape identification technique is to represent geometrical boundaries of the obstacles by using a level set function. This representation of the shapes has the advantage that the level set function automatically handles the splitting or merging of the objects during the reconstruction. Another key point of the algorithm is to solve the inverse problem of finding the interfaces between two materials using a narrow-band method, which not only decreases the number of unknowns and therefore the computational cost of the inversion, but also tends to improve the condition number of the discrete inverse problem compared to pixel (voxel)-based image reconstruction. Level set shape reconstruction results shown in this article are some of the first ones using experimental electrical tomography data. The experimental results also show some improvements in image quality compared with the pixel-based image reconstruction. The proposed technique is applied to 2D resistance and capacitance tomography for both simulated and experimental data. In addition, a full 3D inversion is performed on simulated 3D resistance tomography data
A GSVD formulation of a domain decomposition method for planar eigenvalue problems
Get PDFIn this article we present a modification of the domain
decomposition method of Descloux and Tolley for planar eigenvalue problems. Instead of formulating a generalized eigenvalue problem our method is based on the generalized singular value decomposition. This approach is robust and at the same time highly accurate. Furthermore, we give an improved convergence analysis based on results from complex approximation theory. Several examples show the effectiveness of our method
Scaling, Sensitivity and Stability in the Numerical Solution of Quadratic Eigenvalue Problems
Get PDFThe most common way of solving the quadratic eigenvalue problem (QEP)
(\l^2 M + \l D + K)x=0 is to convert it into a linear problem
(\l X + Y)z=0 of twice the dimension and solve the linear problem
by the QZ algorithm or a Krylov method.
In doing so, it is important to understand the influence of the linearization
process on the accuracy and stability of the computed solution.
We discuss these issues for three particular linearizations:
the standard companion linearization
and two linearizations that preserve symmetry in the problem.
For illustration we employ
a model QEP describing the motion of
a beam simply supported at both ends and damped at the midpoint.
We show that the above linearizations lead to poor numerical results
for the beam problem,
but that a two-parameter scaling proposed by Fan, Lin and Van Dooren
cures the instabilities.
We also show that half of the eigenvalues of the beam QEP are pure imaginary
and are eigenvalues of the undamped problem.
Our analysis makes use of recently developed theory explaining the sensitivity
and stability of linearizations,
the main conclusions of which are summarized.
As well as arguing that
scaling should routinely be used,
we give guidance on how to choose a linearization
and illustrate the practical value of condition numbers and backward errors
Word problems recognisable by deterministic blind monoid automata
No full textWe consider blind, deterministic, finite automata equipped with a register which stores an element of a given monoid, and which is modified by right multiplication by monoid elements. We show that, for monoids M drawn from a large class including groups, such an automaton accepts the word problem of a group H if and only if H has a finite index subgroup which embeds in the group of units of M. In the case that M is a group, this answers a question of Elston and Ostheimer
Pattern of growth and adiposity from infancy to adulthood in atopic dematitis
No full textBackground Impaired linear growth has been reported in children with atopic dermatitis (AD) but the pattern of growth in height and weight through childhood and adolescence has not been described.
Objectives To define the pattern of linear growth and adiposity in AD from early childhood through to adult life.
Patients and methods Growth measurements of 70 male and 40 female patients with AD followed through childhood and adolescence were studied retrospectively and compared with the 1990 U.K. normal values. Height, weight and body mass index (BMI) were converted to standard deviation scores (SDS). Regression analysis examined whether the mean trend was different from zero.
Results While dermatitis was the predominant atopic problem in all 110 patients, 92 had a history of asthma which was mild in 85 of 92. Regression analyses showed that the trends in height, weight and BMI SDS for AD patients were significantly different from zero and also different between males and females. Both sexes were short and relatively overweight from early childhood, a trend that was more pronounced in males than females. At 5 years (school entry), the 50th centile BMI of male (but not female) patients was 0·44 kg m-2 higher than the reference population but height and weight were lower. The age at adiposity rebound in AD males and females was 0·8 year and 0·7 year later than the U.K. population (6·2 years vs. 5·4 years and 6·2 years vs. 5·3 years, respectively). AD patients attained peak height velocity later than the 1990 U.K. population (males 16·0 years vs. 13·5 years, P = 0·0002; females 13·4 years vs. 11·0 years, P = 0·008). In addition, males had greater mean gain in height during late adolescence (12·2 vs. 8·8 cm, P = 0·03) and were shorter as young adults (170·9 vs. 177·6 cm, P = 0·0005).
Conclusions Our patients with AD were relatively overweight very early but had a later adiposity rebound, were short in childhood and had a delayed adolescent growth spurt. Serial growth measurements should be done on all children with troublesome AD and can be helpful in counselling about the growth prognosis
Optimal scaling for partially updating MCMC algorithms
Get PDFIn this paper we shall consider optimal scaling problems for high-dimensional Metropolis–Hastings algorithms where updates can be chosen to be lower dimensional than the target density itself. We find that the optimal scaling rule for the Metropolis algorithm, which tunes the overall algorithm acceptance rate to be 0.234, holds for the so-called Metropolis-within-Gibbs algorithm as well. Furthermore, the optimal efficiency obtainable is independent of the dimensionality of the update rule. This has important implications for the MCMC practitioner since high-dimensional updates are generally computationally more demanding, so that lower-dimensional updates are therefore to be preferred. Similar results with rather different conclusions are given for so-called Langevin updates. In this case, it is found that high-dimensional updates are frequently most efficient, even taking into account computing costs
Rational Cherednik algebras and Hilbert schemes. II: representations and sheaves
No full textLet H_c be the rational Cherednik algebra of type A_{n-1} with spherical subalgebra U_c = e H_c e. Then U_c is filtered by order of differential operators with associated graded ring gr U_c = \mathbb{C} [ \mathfrac{h} ⊕ \mathfrac{h}^* ]^W, where is the n-th symmetric group. Using the Z-algebra construction from [GS], it is also possible to associate to a filtered H_c - or U_c - module \hat{Φ}(M) on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of U_c and H_c, and we relate it to Hilb(n) and to the resolution of singularities τ : Hilb(n) → \mathfrac{h} ⊕ \mathfrac{h}^* / W. For example, we prove the following.
• If c=1/n so that L_c(triv) is the unique one-dimensional simple H_c-module, then \hat{Φ}(e L_c(triv)) ≅ \mathcal{O}_{Z_n}, where Z_n = τ^{-1}(0) is the punctual Hilbert scheme.
• If c = 1/n+k for k \in \mathbb{N}, then under a canonical filtration on the finite-dimensional module L_c(triv), gr e L_c(triv) has a natural bigraded structure that coincides with that on H^0( Z_n, \mathscr{L}^k), where \mathscr{L} ≅ \mathcal{O}_{Hilb(n)}(1); this confirms conjectures of Berest, Etingof, and Ginzburg [BEG2, Conjectures 7.2, 7.3].
• Under mild restrictions on c, the characteristic cycle of \hat{Φ}(e Δ_c(μ)) equals \sum_λ K_{μλ}[Z_λ], where K_{μλ} are Kostka numbers and the Z_λ are (known) irreducible components of τ^{-1}(\mathfrak{h}/W
Modelling Heterogeneous Covariances in the Growth Curve Models
Get PDFThe growth curve models (GCM) are widely used in longitudinal studies
and repeated measures. Most existing approaches for statistical inference in the GCM
assume a specific structure on the within-subject covariances, for example, compound
symmetry, AR(1) and unstructured covariances. The specification, however, may select a
suboptimal or even wrong model, which in turn may affect the estimates of regression coef-
¯cients and/or bias standard errors of the estimates. Accordingly, statistical inferences of
the models may be severely influenced by misspecification of covariance structures. Within
the framework of the GCM in this paper we propose a data-driven approach for modelling
the within-subject covariance structures, investigate the effects of misspecification of co-
variance structures on statistical inferences, and study the heterogeneity of covariances
between different treatment groups