Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity

We define a function, called s-multiplicity, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert-Samuel multiplicity for small values of s and is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it has an Associativity Formula generalizing the Associativity Formulas for Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of closures such that if two ideals have the same s-closure then they have the same s-multiplicity, and the converse holds under mild conditions. We describe the s-multiplicity of monomial ideals in toric rings as a certain volume in real spaceComment: 19 page

On Lower Bounds for ss-multiplicities

A recent continuous family of multiplicity functions on local rings was introduced by Taylor interpolating between Hilbert-Samuel and Hilbert-Kunz multiplicities. The obvious goal is to use this as a tool for deforming results from one to the other. The values in this family which do not match these classic variants however are not known yet to be well-behaved. This article explores lower bounds for these intermediate multiplicities as well as gives evidence for analogies of the Watanabe-Yoshida minimality conjectures for unmixed singular rings.Comment: 10 page

The s-multiplicity function of 2x2-determinantal rings

This article generalizes joint work of the first author and I. Swanson to the $s$-multiplicity recently introduced by the second author. For $k$ a field and $X = [ x_{i,j}]$ a $m \times n$-matrix of variables, we utilize Gr\"obner bases to give a closed form the length $\lambda( k[X] / (I_2(X) + \mathfrak{m}^{ \lceil sq \rceil} + \mathfrak{m}^{[q]} ))$ where $s \in \mathbf{Z}[p^{-1}]$, $q$ is a sufficiently large power of $p$, and $\mathfrak{m}$ is the homogeneous maximal ideal of $k[X]$. This shows this length is always eventually a {\it polynomial} function of $q$ for all $s$.Comment: 9 pages, Errors fixe

Sediment management for Southern California mountains, coastal plains and shoreline

The Environmental Quality Laboratory at Caltech and the Shore Processes Laboratory at Scripps Institution of Oceanography have jointly undertaken a study of regional sediment balance problems in coastal southern California (see map in Figure 1). The overall objective in this study is to define specific alternatives in sediment management that may be implemented to alleviate a) existing sediment imbalance problems (e.g. inland debris disposal, local shoreline erosion) and b) probable future problems that have not yet manifested themselves. These alternatives will be identified through a consideration of economic, legal, and institutional issues as well as an analysis of governing physical processes and engineering constraints. The first part of this study (Phase I), which is currently under way, involves a compilation and analysis of all available data in an effort to obtain an accurate definition of the inland/coastal regional sediment balance under natural conditions, and specific quantitative effects man-made controls have on the overall natural process. During FY77, substantial progress was made at EQL and SPL in achieving the objectives of the initial Planning and Assessment Phase of the CIT/SIO Sediment Management Project. Financial support came from Los Angeles County, U.S. Geological Survey, Orange County, U.S. Army Corps of Engineers, and discretionary funding provided by a grant from the Ford Foundation. The current timetable for completion of this phase is Fall 1978. This report briefly describes the project status, including general administration, special activities, and research work as of January 1978

Fundamental Results on ss-Closures

This paper establishes the fundamental properties of the $s$-closures, a recently introduced family of closure operations on ideals of rings of positive characteristic. The behavior of the $s$-closure of homogeneous ideals in graded rings is studied, and criteria are given for when the $s$-closure of an ideal can be described exactly in terms of its tight closure and rational powers. Sufficient conditions are established for the weak $s$-closure to equal to the $s$-closure. A generalization of the Briancon-Skoda theorem is given which compares any two different $s$-closures applied to powers of the same ideal.Comment: 10 page

Interpolating Between Multiplicities and F-thresholds

We define a family of functions, called s-multiplicity for each s\u3e0, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert-Samuel multiplicity for small values of s and is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it has an associativity formula generalizing the associativity formulas for Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of closures, called s-closures, such that if two ideals have the same s-closure then they have the same s-multiplicity, and the converse holds under mild conditions. We describe methods for computing the $F$-threshold, the $s$-multiplicity, and the $s$-closure of monomial ideals in toric rings using the geometry of the cone defining the ring

RURAL ECONOMIC DEVELOPMENT: NEW OPPORTUNITIES AND CHALLENGES FOR COMMERCIAL BANKERS

The report discusses commercial banks' role in supporting economic development in rural America. It details demographic and economic trends in rural America. It discusses a number of economic development programs available to commercial bankers and to private sector/public sector partnerships. Finally, the report proposes a set of new tools for commercial bankers to further strengthen their participation in rural community economic development. Note: Figures are not included in the machine readable copy--contact the authors for more information.demographic trends, economic trends, rural America, role of commercial banks, economic development programs, new tools for bankers, Community/Rural/Urban Development, Financial Economics,

Grain Shipments on the Mississippi River System: A Long-Term Projection

The costs of delays for shipping commodities on the Mississippi River are important and adversely impact growth in shipments. Lock and dam expansion requires substantial capital investment and an extended time period to complete. This study analyzes delay costs and the competitive position of grain shipments on the Mississippi River system. A spatial optimization model of the world grain trade was developed. Results indicated that without expansion in barge capacity, delay costs in 2020 would increase on each reach, with some up to $1.08/mt. Expansion results in reduced delay costs. Barge demand is also impacted by rail capacity. Finally, expanding the locks would result in a re-allocation of shipments among modes, reaches, and ports, notwithstanding minor adjustments in production.Agribusiness, International Relations/Trade,