This paper explores Wittgenstein＇s early work as it relates to emerging philosophical problems in ecological modeling. Here I use his thought to structure a logical framework from which to discuss ecological simulati...This paper explores Wittgenstein＇s early work as it relates to emerging philosophical problems in ecological modeling. Here I use his thought to structure a logical framework from which to discuss ecological simulation models in a way that captures how these dynamic representations describe a world from which we can draw logical inferences about real-world ecological processes. I argue that Wittgenstein＇s Tractatus Logico-Philosophicus provides a way of reading problems that arise in using simulation as a way to make inferences about the world. Conversely, ecological simulation provides an illustration of a Tractarian system, because the digital world it creates completely describes and is defined by the programing language. This reading is a novel, but productive, way that notes that the language used in modeling requires a hermeneutical approach to make inferences about modeling/real-world relationships.展开更多
Let R, S be rings, U a flat right .R-rnodule and V a flat right S-module. We show in this paper that (N, (U, V))-lc.dim(R(?) S) = sup((N, U)-lc.dimR, (N, V)-lc.dimS).
During the last decade, a great deal of activity has been devoted to the calculation of the HilbertPoincar′e series of unitary highest weight representations and related modules in algebraic geometry. However,uniform...During the last decade, a great deal of activity has been devoted to the calculation of the HilbertPoincar′e series of unitary highest weight representations and related modules in algebraic geometry. However,uniform formulas remain elusive—even for more basic invariants such as the Gelfand-Kirillov dimension or the Bernstein degree, and are usually limited to families of representations in a dual pair setting. We use earlier work by Joseph to provide an elementary and intrinsic proof of a uniform formula for the Gelfand-Kirillov dimension of an arbitrary unitary highest weight module in terms of its highest weight. The formula generalizes a result of Enright and Willenbring(in the dual pair setting) and is inspired by Wang's formula for the dimension of a minimal nilpotent orbit.展开更多
For a commutative ring R and a faithfully fiat R-algebra S we prove, under mild extra assumptions, that an R-module M is Gorenstein flat if and only if the left S-module S R M is Gorenstein flat, and that an R-module ...For a commutative ring R and a faithfully fiat R-algebra S we prove, under mild extra assumptions, that an R-module M is Gorenstein flat if and only if the left S-module S R M is Gorenstein flat, and that an R-module N is Gorenstein injective if and only if it is cotorsion and the left S-module Homn（S, N） is Gorenstein injective. We apply these results to the study of Gorenstein homological dimensions of unbounded complexes. In particular, we prove two theorems on stability of these dimensions under faithfully flat （co-）base change.展开更多
文摘This paper explores Wittgenstein＇s early work as it relates to emerging philosophical problems in ecological modeling. Here I use his thought to structure a logical framework from which to discuss ecological simulation models in a way that captures how these dynamic representations describe a world from which we can draw logical inferences about real-world ecological processes. I argue that Wittgenstein＇s Tractatus Logico-Philosophicus provides a way of reading problems that arise in using simulation as a way to make inferences about the world. Conversely, ecological simulation provides an illustration of a Tractarian system, because the digital world it creates completely describes and is defined by the programing language. This reading is a novel, but productive, way that notes that the language used in modeling requires a hermeneutical approach to make inferences about modeling/real-world relationships.
基金National Natural Science Foundation of China(10171082)by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE.
文摘Let R, S be rings, U a flat right .R-rnodule and V a flat right S-module. We show in this paper that (N, (U, V))-lc.dim(R(?) S) = sup((N, U)-lc.dimR, (N, V)-lc.dimS).
基金supported by National Natural Science Foundation of China(Grant No.11171324)the Hong Kong Research Grants Council under RGC Project(Grant No.60311)the Hong Kong University of Science and Technology under DAG S09/10.SC02.
文摘During the last decade, a great deal of activity has been devoted to the calculation of the HilbertPoincar′e series of unitary highest weight representations and related modules in algebraic geometry. However,uniform formulas remain elusive—even for more basic invariants such as the Gelfand-Kirillov dimension or the Bernstein degree, and are usually limited to families of representations in a dual pair setting. We use earlier work by Joseph to provide an elementary and intrinsic proof of a uniform formula for the Gelfand-Kirillov dimension of an arbitrary unitary highest weight module in terms of its highest weight. The formula generalizes a result of Enright and Willenbring(in the dual pair setting) and is inspired by Wang's formula for the dimension of a minimal nilpotent orbit.
基金supported by the National Security Agency (Grant No. H98230-140140)National Natural Science Foundation of China (Grant Nos. 11301240 and 11371187)the Scientific Research Foundation for the Returned Overseas Chinese Scholars (State Education Ministry)
文摘For a commutative ring R and a faithfully fiat R-algebra S we prove, under mild extra assumptions, that an R-module M is Gorenstein flat if and only if the left S-module S R M is Gorenstein flat, and that an R-module N is Gorenstein injective if and only if it is cotorsion and the left S-module Homn（S, N） is Gorenstein injective. We apply these results to the study of Gorenstein homological dimensions of unbounded complexes. In particular, we prove two theorems on stability of these dimensions under faithfully flat （co-）base change.
