Dimension Distortion by Sobolev Mappings in Foliated Metric Spaces

We quantify the extent to which a supercritical Sobolev mapping can increase the dimension of subsets of its domain, in the setting of metric measure spaces supporting a Poincaré inequality. We show that the set of mappings that distort the dimensions of sets by the maximum possible amount is a prevalent subset of the relevant function space. For foliations of a metric space X defined by a David–Semmes regular mapping Π : X → W, we quantitatively estimate, in terms of Hausdorff dimension in W, the size of the set of leaves of the foliation that are mapped onto sets of higher dimension. We discuss key examples of such foliations, including foliations of the Heisenberg group by left and right cosets of horizontal subgroups.

THE MULTIPLE LOCATION TIME WEIGHTED INDEX: USING PATIENT ACTIVITY SPACES TO CALCULATE PRIMARY CARE SERVICE AREAS

Geographic health planning analyses, such as service area calculations, are hampered by a lack of patient-specific geographic data. Using the limited patient address information in patient management systems, planners analyze patient origin based on home address. But activity space research done sparingly in public health and extensively in non-health related arenas uses multiple addresses per person when analyzing accessibility. Also, health care access research has shown that there are many non-geographic factors that influence choice of provider. Most planning methods, however, overlook non-geographic factors influencing choice of provider, and the limited data mean the analyses can only be related to home address. This research attempted to determine to what extent geography plays a part in patient choice of provider and to determine if activity space data can be used to calculate service areas for primary care providers. During Spring 2008, a convenience sample of 384 patients of a locally-funded Community Health Center in Houston, Texas, completed a survey that asked about what factors are important when he or she selects a health care provider. A subset of this group (336) also completed an activity space log that captured location and time data on the places where the patient regularly goes. Survey results indicate that for this patient population, geography plays a role in their choice of health care provider, but it is not the most important reason for choosing a provider. Other factors for choosing a health care provider such as the provider offering “free or low cost visits”, meeting “all of the patient’s health care needs”, and seeing “the patient quickly” were all ranked higher than geographic reasons. Analysis of the patient activity locations shows that activity spaces can be used to create service areas for a single primary care provider. Weighted activity-space-based service areas have the potential to include more patients in the service area since more than one location per patient is used. Further analysis of the logs shows that a reduced set of locations by time and type could be used for this methodology, facilitating ongoing data collection for activity-space-based planning efforts.

The multiple location time weighted index: Using patient activity spaces to calculate primary care service areas

Geographic health planning analyses, such as service area calculations, are hampered by a lack of patient-specific geographic data. Using the limited patient address information in patient management systems, planners analyze patient origin based on home address. But activity space research done sparingly in public health and extensively in non-health related arenas uses multiple addresses per person when analyzing accessibility. Also, health care access research has shown that there are many non-geographic factors that influence choice of provider. Most planning methods, however, overlook non-geographic factors influencing choice of provider, and the limited data mean the analyses can only be related to home address. This research attempted to determine to what extent geography plays a part in patient choice of provider and to determine if activity space data can be used to calculate service areas for primary care providers. ^ During Spring 2008, a convenience sample of 384 patients of a locally-funded Community Health Center in Houston, Texas, completed a survey that asked about what factors are important when he or she selects a health care provider. A subset of this group (336) also completed an activity space log that captured location and time data on the places where the patient regularly goes. ^ Survey results indicate that for this patient population, geography plays a role in their choice of health care provider, but it is not the most important reason for choosing a provider. Other factors for choosing a health care provider such as the provider offering "free or low cost visits", meeting "all of the patient's health care needs", and seeing "the patient quickly" were all ranked higher than geographic reasons. ^ Analysis of the patient activity locations shows that activity spaces can be used to create service areas for a single primary care provider. Weighted activity-space-based service areas have the potential to include more patients in the service area since more than one location per patient is used. Further analysis of the logs shows that a reduced set of locations by time and type could be used for this methodology, facilitating ongoing data collection for activity-space-based planning efforts. ^

Weighted composition operators on the predual and dual Banach spaces of Beurling algebras on Z

Let vv be a weight sequence on ZZ and let ψ,φψ,φ be complex-valued functions on ZZ such that φ(Z)⊂Zφ(Z)⊂Z. In this paper we study the boundedness, compactness and weak compactness of weighted composition operators Cψ,φCψ,φ on predual Banach spaces c0(Z,1/v)c0(Z,1/v) and dual Banach spaces ℓ∞(Z,1/v)ℓ∞(Z,1/v) of Beurling algebras ℓ1(Z,v)ℓ1(Z,v).

Determining the temperature from Cauchy data in corner domains

We consider the problem of reconstruction of the temperature from knowledge of the temperature and heat flux on a part of the boundary of a bounded planar domain containing corner points. An iterative method is proposed involving the solution of mixed boundary value problems for the heat equation (with time-dependent conductivity). These mixed problems are shown to be well-posed in a weighted Sobolev space.

Mixed Fractional Integration Operators in Mixed Weighted Hölder Spaces

MSC 2010: 26A33

Weighted composition followed by differentiation between weighted Banach spaces of holomorphic functions

2010 Mathematics Subject Classification: 47B33, 47B38.

The spectra of linear fractional composition operators on weighted Dirichlet spaces

We completely determine the spectra of composition operators induced by linear fractional self-maps of the unit disc acting on weighted Dirichlet spaces; extending earlier results by Higdon [8] and answering the open questions in this context.

Composition operators on vector-valued BMOA and related function spaces

A composition operator is a linear operator between spaces of analytic or harmonic functions on the unit disk, which precomposes a function with a fixed self-map of the disk. A fundamental problem is to relate properties of a composition operator to the function-theoretic properties of the self-map. During the recent decades these operators have been very actively studied in connection with various function spaces. The study of composition operators lies in the intersection of two central fields of mathematical analysis; function theory and operator theory. This thesis consists of four research articles and an overview. In the first three articles the weak compactness of composition operators is studied on certain vector-valued function spaces. A vector-valued function takes its values in some complex Banach space. In the first and third article sufficient conditions are given for a composition operator to be weakly compact on different versions of vector-valued BMOA spaces. In the second article characterizations are given for the weak compactness of a composition operator on harmonic Hardy spaces and spaces of Cauchy transforms, provided the functions take values in a reflexive Banach space. Composition operators are also considered on certain weak versions of the above function spaces. In addition, the relationship of different vector-valued function spaces is analyzed. In the fourth article weighted composition operators are studied on the scalar-valued BMOA space and its subspace VMOA. A weighted composition operator is obtained by first applying a composition operator and then a pointwise multiplier. A complete characterization is given for the boundedness and compactness of a weighted composition operator on BMOA and VMOA. Moreover, the essential norm of a weighted composition operator on VMOA is estimated. These results generalize many previously known results about composition operators and pointwise multipliers on these spaces.

Sampling and learning the Mallows and Weighted Mallows models under the Hamming distance

[EN]In this paper we deal with distributions over permutation spaces. The Mallows model is the mode l in use. The associated distance for permutations is the Hamming distance.

Convergence Properties and Fixed Points of Two General Iterative Schemes with Composed Maps in Banach Spaces with Applications to Guaranteed Global Stability

This paper investigates the boundedness and convergence properties of two general iterative processes which involve sequences of self-mappings on either complete metric or Banach spaces. The sequences of self-mappings considered in the first iterative scheme are constructed by linear combinations of a set of self-mappings, each of them being a weighted version of a certain primary self-mapping on the same space. The sequences of self-mappings of the second iterative scheme are powers of an iteration-dependent scaled version of the primary self-mapping. Some applications are also given to the important problem of global stability of a class of extended nonlinear polytopic-type parameterizations of certain dynamic systems.

Hankel operators on Fock spaces

We study Hankel operators on the weighted Fock spaces Fp. The boundedness and compactness of these operators are characterized in terms of BMO and VMO, respectively. Along the way, we also study Berezin transform and harmonic conjugates on the plane. Our results are analogous to Zhu's characterization of bounded and compact Hankel operators on Bergman spaces of the unit disk.

Frequency of Sobolev and quasiconformal dimension distortion

We study Hausdorff and Minkowski dimension distortion for images of generic affine subspaces of Euclidean space under Sobolev and quasiconformal maps. For a supercritical Sobolev map f defined on a domain in RnRn, we estimate from above the Hausdorff dimension of the set of affine subspaces parallel to a fixed m-dimensional linear subspace, whose image under f has positive HαHα measure for some fixed α>mα>m. As a consequence, we obtain new dimension distortion and absolute continuity statements valid for almost every affine subspace. Our results hold for mappings taking values in arbitrary metric spaces, yet are new even for quasiconformal maps of the plane. We illustrate our results with numerous examples.

Ornstein-Uhlenbeck operator in convex domains of Banach spaces

Studiamo l'operatore di Ornstein-Uhlenbeck e il semigruppo di Ornstein-Uhlenbeck in un sottoinsieme aperto convesso $\Omega$ di uno spazio di Banach separabile $X$ dotato di una misura Gaussiana centrata non degnere $\gamma$. In particolare dimostriamo la disuguaglianza di Sobolev logaritmica e la disuguaglianza di Poincaré, e grazie a queste disuguaglianze deduciamo le proprietà spettrali dell'operatore di Ornstein-Uhlenbeck. Inoltre studiamo l'equazione ellittica $\lambdau+L^{\Omega}u=f$ in $\Omega$, dove $L^\Omega$ è l'operatore di Ornstein-Uhlenbeck. Dimostriamo che per $\lambda>0$ e $f\in L^2(\Omega,\gamma)$ la soluzione debole $u$ appartiene allo spazio di Sobolev $W^{2,2}(\Omega,\gamma)$. Inoltre dimostriamo che $u$ soddisfa la condizione di Neumann nel senso di tracce al bordo di $\Omega$. Questo viene fatto finita approssimazione dimensionale.

A procedure for the temperature reconstruction in corner domains from Cauchy data

An iterative procedure is proposed for the reconstruction of a stationary temperature field from Cauchy data given on a part of the boundary of a bounded plane domain where the boundary is smooth except for a finite number of corner points. In each step, a series of mixed well-posed boundary value problems are solved for the heat operator and its adjoint. Convergence is proved in a weighted L2-space. Numerical results are included which show that the procedure gives accurate and stable approximations in relatively few iterations.