In this issue, we feature two very different articles. The first, “Estimation of Subspace Arrange... more In this issue, we feature two very different articles. The first, “Estimation of Subspace Arrangements with Applications in Modeling and Segmenting Mixed Data” by Yi Ma, Allen Yang, Harm Derksen, and Robert Fossum, is an excellent example of interdisciplinary mathematical research. The work shows how sophisticated mathematical ideas can be used to address a problem which arises in many applications. Moreover, it points to new mathematical research directions. The basic problem the article addresses is a very important problem in analysis of high dimensional data. Suppose you have a cloud of points in a large dimensional space; the points could represent consumer preferences, for example. The question is to find patterns in the data by identifying a set of hyperplanes which “fit” the data. The challenge is that the number of hyperplanes and their dimensions are to be learned from the data. Thus the task involves “grouping the points,” i.e., segmentation, and determining the best appr...
This concluding chapter is concerned mainly with the behavior of the divisor class group under ba... more This concluding chapter is concerned mainly with the behavior of the divisor class group under base change to the formal power series ring. It has been seen that Cl(A) → Cl(A[[T]]) is not always a bijection, even when A is factorial. Using the good functorial properties of the Picard group and its relation to the divisor class group, Danilov has studied the class of rings for which the homomorphism is a bijection. A fundamental concept is the Picard group of an open subscheme of Spec A. For example, the set of points in Spec A at which an (divisorial) ideal is invertible is open, and on this subscheme, the class of the ideal belongs to the Picard group. Using this simple property, it is possible to show that there is a natural splitting of the injection Cl(A) → Cl(A[[T]]).
In this chapter the divisor class group of a Krull domain is introduced and its functorial proper... more In this chapter the divisor class group of a Krull domain is introduced and its functorial properties are investigated. Two applications of the functorial properties are made to polynomial extensions and subintersections (cf. Proposition 1.2d). Power series extensions are considered in a later section.
As anyone who resided in Denmark at the time can attest, the centenary of the birth of the Danish... more As anyone who resided in Denmark at the time can attest, the centenary of the birth of the Danish physicist Niels Bohr was celebrated in the Fall of 1985. The centenary was marked by a symposium held at the Niels Bohr Institute; Danish television programs on the man, his physics, and his philosophy; numerous newspaper articles; and the publication of biographies, essays, and remembrances, among them the two books under review. NB was born on 7 October 1885, the son of Christian Bohr, a professor of physiology at the University of Copenhagen. Bohr studied physics at the University of Copenhagen and defended his doctoral thesis, on the electron theory of metals, in 1911. In 1912 he traveled to England where he visited J. J. Thompson in Cambridge and Rutherford in Manchester. In Manchester he learned of Rutherford's atomic model theory. In the summer of 1912 he returned to Denmark, married Margrethe Norland, and assumed his duties as a teaching assistant at the University of Copenh...
Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1, 2005
Page 1. Hilbert Functions and Applications to the Estimation of Subspace Arrangements ∗ Allen Y. ... more Page 1. Hilbert Functions and Applications to the Estimation of Subspace Arrangements ∗ Allen Y. Yang, Shankar Rao, Andrew Wagner, Yi Ma Coordinated Science Lab, UIUC 1308 W. Main, Urbana, Illinois 61801 {yangyang,srrao,awagner,yima}@uiuc.edu ...
ABSTRACT This paper examines the existence of apolar covariants of order greater than or equal to... more ABSTRACT This paper examines the existence of apolar covariants of order greater than or equal to \(n.\) These are shown to exist for all \(n>2.\)
In this issue, we feature two very different articles. The first, “Estimation of Subspace Arrange... more In this issue, we feature two very different articles. The first, “Estimation of Subspace Arrangements with Applications in Modeling and Segmenting Mixed Data” by Yi Ma, Allen Yang, Harm Derksen, and Robert Fossum, is an excellent example of interdisciplinary mathematical research. The work shows how sophisticated mathematical ideas can be used to address a problem which arises in many applications. Moreover, it points to new mathematical research directions. The basic problem the article addresses is a very important problem in analysis of high dimensional data. Suppose you have a cloud of points in a large dimensional space; the points could represent consumer preferences, for example. The question is to find patterns in the data by identifying a set of hyperplanes which “fit” the data. The challenge is that the number of hyperplanes and their dimensions are to be learned from the data. Thus the task involves “grouping the points,” i.e., segmentation, and determining the best appr...
This concluding chapter is concerned mainly with the behavior of the divisor class group under ba... more This concluding chapter is concerned mainly with the behavior of the divisor class group under base change to the formal power series ring. It has been seen that Cl(A) → Cl(A[[T]]) is not always a bijection, even when A is factorial. Using the good functorial properties of the Picard group and its relation to the divisor class group, Danilov has studied the class of rings for which the homomorphism is a bijection. A fundamental concept is the Picard group of an open subscheme of Spec A. For example, the set of points in Spec A at which an (divisorial) ideal is invertible is open, and on this subscheme, the class of the ideal belongs to the Picard group. Using this simple property, it is possible to show that there is a natural splitting of the injection Cl(A) → Cl(A[[T]]).
In this chapter the divisor class group of a Krull domain is introduced and its functorial proper... more In this chapter the divisor class group of a Krull domain is introduced and its functorial properties are investigated. Two applications of the functorial properties are made to polynomial extensions and subintersections (cf. Proposition 1.2d). Power series extensions are considered in a later section.
As anyone who resided in Denmark at the time can attest, the centenary of the birth of the Danish... more As anyone who resided in Denmark at the time can attest, the centenary of the birth of the Danish physicist Niels Bohr was celebrated in the Fall of 1985. The centenary was marked by a symposium held at the Niels Bohr Institute; Danish television programs on the man, his physics, and his philosophy; numerous newspaper articles; and the publication of biographies, essays, and remembrances, among them the two books under review. NB was born on 7 October 1885, the son of Christian Bohr, a professor of physiology at the University of Copenhagen. Bohr studied physics at the University of Copenhagen and defended his doctoral thesis, on the electron theory of metals, in 1911. In 1912 he traveled to England where he visited J. J. Thompson in Cambridge and Rutherford in Manchester. In Manchester he learned of Rutherford's atomic model theory. In the summer of 1912 he returned to Denmark, married Margrethe Norland, and assumed his duties as a teaching assistant at the University of Copenh...
Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1, 2005
Page 1. Hilbert Functions and Applications to the Estimation of Subspace Arrangements ∗ Allen Y. ... more Page 1. Hilbert Functions and Applications to the Estimation of Subspace Arrangements ∗ Allen Y. Yang, Shankar Rao, Andrew Wagner, Yi Ma Coordinated Science Lab, UIUC 1308 W. Main, Urbana, Illinois 61801 {yangyang,srrao,awagner,yima}@uiuc.edu ...
ABSTRACT This paper examines the existence of apolar covariants of order greater than or equal to... more ABSTRACT This paper examines the existence of apolar covariants of order greater than or equal to \(n.\) These are shown to exist for all \(n>2.\)
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