In this paper, we consider flat epimorphisms of commutative rings [Formula: see text] such that, for every ideal which text], the quotient ring is semilocal Krull dimension zero. Under these assumptions, show that projective R-module U does not exceed 1. We also describe Geigle–Lenzing perpendicular subcategory in text]. Assuming additionally and all are perfect, R-modules U-strongly flat. Thus...

Bayesian optimization depends on solving a global optimization of a acquisition function. However, the acquisition function can be extremely sharp at high dimension having only a few peaks marooned in a large terrain of almost flat surface. Global optimization algorithms such as DIRECT are infeasible at higher dimensions and gradient-dependent methods cannot move if initialized in the flat terr...

We show that the recently introduced L1TV functional can be used to explicitly compute the flat norm for co-dimension one boundaries. While this observation alone is very useful, other important implications for image analysis and shape statistics include a method for denoising sets which are not boundaries or which have higher co-dimension and the fact that using the flat norm to compute dista...

It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs in view of an example of Lawson and Osserman. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. We also prove minimal graphs of dimension no greate...

The purpose of this paper is to study the second fundamental form of some submanifolds M in Euclidean spaces E" which have flat normal connection. As such, Theorem gives precise expressions for the (essentially 2) Weingarten maps of all 4-dimensional Emstem submanifolds in E6, which are specialized in Corollary 2 to the Rcciflat submanifolds. The main part ofthis paper deals with fiat submanifo...

This paper is a continuation of the papers J. Pure Appl. Algebra, 210 (2007), 437–445 and J. Algebra Appl., 8 (2009), 219–227. Namely, we introduce and study a doubly filtered set of classes of modules of finite Gorenstein projective dimension, which are called (n, m)-strongly Gorenstein projective ((n, m)-SG-projective for short) for integers n ≥ 1 and m ≥ 0. We are mainly interested in studyi...