Relative Riemann-Zariski spaces

In this paper we study relative Riemann-Zariski spaces attached to a morphism of schemes and generalizing the classical Riemann-Zariski space of a field. We prove that similarly to the classical RZ spaces, the relative ones can be described either as projective limits of schemes in the category of locally ringed spaces or as certain spaces of valuations. We apply these spaces to prove the following two new results: a strong version of stable modification theorem for relative curves; a decomposition theorem which asserts that any separated morphism between quasi-compact and quasi-separated schemes factors as a composition of an affine morphism and a proper morphism. (In particular, we obtain a new proof of Nagata's compactification theorem.)Comment: 30 pages, the final version, to appear in Israel J. of Mat

Deception in Undercover Investigations: Conduct-Based vs. Status-Based Ethical Analysis

As part of the public school system, online schools “have a responsibility to provide equal access to [their] educational opportunities[,] and restricting access to these opportunities can be problematic, if not illegal.” Given the rapid growth of online education in Washington, legislators must examine whether online schools that receive public education funding are benefiting the entire public or are benefiting merely a select group of students. Part II of this Comment briefly discusses the history of online schools in Washington, including how they receive funding within Washington\u27s unique statutory and regulatory framework. Part III then examines how online schools discriminate against minority, lowerincome, special education, and transitional bilingual students. Part IV argues that online schools that are not equally accessible to all students should not receive public education funding. Lastly, Part V advocates for creation of a state-led task force to continually monitor the growth and development of online schools to ensure that policymakers stay informed and have a voice in this rapidly evolving industry

Deception in Undercover Investigations: Conduct-Based vs. Status-Based Ethical Analysis

As part of the public school system, online schools “have a responsibility to provide equal access to [their] educational opportunities[,] and restricting access to these opportunities can be problematic, if not illegal.” Given the rapid growth of online education in Washington, legislators must examine whether online schools that receive public education funding are benefiting the entire public or are benefiting merely a select group of students. Part II of this Comment briefly discusses the history of online schools in Washington, including how they receive funding within Washington\u27s unique statutory and regulatory framework. Part III then examines how online schools discriminate against minority, lowerincome, special education, and transitional bilingual students. Part IV argues that online schools that are not equally accessible to all students should not receive public education funding. Lastly, Part V advocates for creation of a state-led task force to continually monitor the growth and development of online schools to ensure that policymakers stay informed and have a voice in this rapidly evolving industry

Onsager approach to 1D solidification problem and its relation to phase field description

We give a general phenomenological description of the steady state 1D front propagation problem in two cases: the solidification of a pure material and the isothermal solidification of two component dilute alloys. The solidification of a pure material is controlled by the heat transport in the bulk and the interface kinetics. The isothermal solidification of two component alloys is controlled by the diffusion in the bulk and the interface kinetics. We find that the condition of positive-definiteness of the symmetric Onsager matrix of interface kinetic coefficients still allows an arbitrary sign of the slope of the velocity-concentration line near the solidus in the alloy problem or of the velocity-temperature line in the case of solidification of a pure material. This result offers a very simple and elegant way to describe the interesting phenomenon of a possible non-single-value behavior of velocity versus concentration which has previously been discussed by different approaches. We also discuss the relation of this Onsager approach to the thin interface limit of the phase field description.Comment: 5 pages, 1 figure, submitted to Physical Review