On the deviation and the type of a Cohen-Macaulay ring

In this note we discuss the question, which pairs of integers (d,r) can occur as the deviation d and the type r of a Cohen-Macaulay domain.     

A Krull-Schmidt theorem for one-dimensional rings of finite Cohen-Macaulay type

This paper determines when the Krull-Schmidt property holds for all finitely generated modules and for maximal Cohen-Macaulay modules over one-dimensional local rings with finite Cohen-Macaulay type. We classify all maximal Cohen-Macaulay modules over these rings, beginning with the complete rings where the Krull-Schmidt property is known to hold. We are then able to determine when the Krull-Schmidt property holds over the non-complete local rings and when we have the weaker property that any two representations of a maximal Cohen-Macaulay module as a direct sum of indecomposables have the same number of indecomposable summands.     

The diagonal subring and the Cohen-Macaulay property of a multigraded ring

Let be a multigraded ring defined over a local ring . This paper deals with the question how the Cohen-Macaulay property of is related to that of its diagonal subring . In the bigraded case we are able to give necessary and sufficient conditions for the Cohen-Macaulayness of . If are ideals of positive height, we can then compare the Cohen-Macaulay property of the multi-Rees algebra with the Cohen-Macaulay property of the usual Rees algebra . We also obtain a bound for the joint reduction numbers of two -primary ideals in the case the corresponding multi-Rees algebra is Cohen-Macaulay.

     

A note on the quasi-ergodic distribution of one-dimensional diffusions

In this note, we study quasi-ergodicity for one-dimensional diffusions on $(0,\infty )$, where 0 is an exit boundary and +∞ is an entrance boundary. Our main aim is to improve some results obtained by He and Zhang (2016) [3]. In simple terms, the same main results of the above paper are obtained with more relaxed conditions.     

A note on the multiplicity in factor ring

Let (R,m) = k[x ₁,..., x _n ](x ₁,...,x _n ) be a local polynomial ring (k being an algebraically closed field), and Q:= (F ₁,..., F _r )R be a primary ideal in R with respect to a maximal ideal m ⊂ R. In this short note we give a formula for the multiplicity e ₀ (QR/(F ₁)R, R/(F ₁)R). The author was supported by the grant No. 1/0262/03) of the Slovak Ministry of Education.     

Gorenstein property and symmetry for one-dimensional local Cohen-Macaulay rings

Supported by Acciones Integradas and DAAD. The authors thank the Universities of Paderborn and Valladolid for their hospitality.     

A note on invariant regions in one-dimensional nonlinear viscoelasticity

We consider a nonlinear parabolic system arising as a viscosity regularization of the hyperbolic functional equation [001]-For such a system, the existence of invariant regions is proved on condition that the kernel K is dissipative in a certain sense and б satisfies some growth conditions     

Every local ring is dominated by a one-dimensional local ring

Let be a local (Noetherian) ring. The main result of this paper asserts the existence of a local extension ring of such that (i) dominates , (ii) the residue field of is a finite purely transcendental extension of , (iii) every associated prime of (0) in contracts in to an associated prime of (0), and (iv) . In addition, it is shown that can be obtained so that either is the maximal ideal of or is a localization of a finitely generated -algebra.

     

A note on fractional moments for the one-dimensional continuum Anderson model

We give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the one-dimensional continuum Anderson model via the fractional moments method. This follows via exponential decay of fractional moments of the Green function, which is shown to hold at arbitrary energy and for any single-site distribution with bounded, compactly supported density.     

A note on finite deflections of circular ring plates

We solve the problem of the symmetrically deforming edge loaded polar-orthotropic circular ring plate for the case of large deformations of a radially rigid plate, and compare the results with the corresponding solutions in accordance with the small finite deformation equations of von Karman.

---

Zusammenfassung Wir lösen das Problem der rotationssymmetrischen Biegung von polar-orthotropen Kreisringplatten für den Fall von grossen Durchbiegungen von Platten welche starr in radialer Richtung sind. Wir vergleichen unsere Resultate mit den entsprechenden Lösungen welche aus den von Karman'schen Gleichungen für kleine endliche Durchbiegungen folgen.

---

---

A report on work supported by the Office of Naval Research.     

A note on asymptotic formulae for one-dimensional network flow problems

This note develops asymptotic formulae for single-commodity network flow problems with random inputs. The transportation linear programming problem (TLP) where N points lie in a region of R¹ is one example. It is found that the average distance traveled by an item in the TLP increases with N^1/2; i.e., the unit cost is unbounded when N and the length of the region are increased in a fixed ratio. Further, the optimum distance does not converge in probability to the average value. These one-dimensional results are a useful stepping stone toward a network theory for two and higher dimensions. Research supported in part by the University of California Transportation Center.