Pairs of Complementary Variational Principles

Complementary variational principles can provide upper and lowerbounds for stationary values of generalized action functionalsassociated with certain differential equations. The bounds arealso helpful in assessing approximate solutions of the equations.By transforming a differential equation into an integral equation,it is possible to find a second pair of principles yieldingdifferent bounds for the same stationary value. The theory isillustrated by one-dimensional Liouville and diffusion equations.These examples indicate that the pair of integral equation boundsare closer to each other than are the corresponding differentialequation bounds obtained from the same trial solutions, providedthat these solutions are reasonably accurate. In the case of a linear differential equation, transformationcan also lead to bounds for a different quantity.     

Transparent Pairs

Let M be a closed orientable Riemannian surface. Consider an SO(3)-connection A and a Higgs field $\Phi:M\to \mathfrak{so}(3)$ . The pair (A,??) naturally induces a cocycle over the geodesic flow of M. We classify (up to gauge transformations) cohomologically trivial pairs (A,??) with finite Fourier series in terms of a suitable B?cklund transformation. In particular, if M is negatively curved we obtain a full classification of SO(3)-transparent pairs.     

Pairs of Projections: Geodesics,Fredholm and Compact Pairs

A pair \((P, Q)\) of orthogonal projections in a Hilbert space \( \mathcal{H} \) is called a Fredholm pair if $$\begin{aligned} QP : R(P) \rightarrow R(Q) \end{aligned}$$ is a Fredholm operator. Let \( \mathcal{F} \) be the set of all Fredholm pairs. A pair is called compact if \(P-Q\) is compact. Let \( \mathcal{C} \) be the set of all compact pairs. Clearly \( \mathcal{C} \subset \mathcal{F} \) properly. In this paper it is shown that both sets are differentiable manifolds, whose connected components are parametrized by the Fredholm index. In the process, pairs \(P, Q\) that can be joined by a geodesic (or equivalently, a minimal geodesic) of the Grassmannian of \( \mathcal{H} \) are characterized: this happens if and only if $$\begin{aligned} \dim (R(P)\cap N(Q))=\dim (R(Q)\cap N(P)). \end{aligned}$$      

Applications of Cotorsion Pairs

For an artin algebra , cotorsion pairs are studied for the categoryMod of finitely presented -modules and for the category Mod of all -modules. It is shown that every cotorsion pair forMod induces a cotorsion pair for Mod . This has some interestingapplications, even for the category of finitely presented modules.Another theme of the paper is the interplay between cotorsionand torsion pairs. This leads to a conjecture which is an analogueof the telescope conjecture in stable homotopy theory.     

Index Pairs Algorithms

We introduce some modifications and extensions of the concept of index pair in the Conley index theory. We then show how these concepts may be used to overcome some difficulties in obtaining efficient algorithms computing the Conley index. We also present examples of applications to computer assisted proofs in dynamics.     

Pairs of monotone operators

This note is an addendum to Sum theorems for monotone operators and convex functions. In it, we prove some new results on convex functions and monotone operators, and use them to show that several of the constraint qualifications considered in the preceding paper are, in fact, equivalent.

     

Arrays of compact Pairs

Summary The properties of direct limits ofCech homology groups for a non cofinal collection of compact pairs are explicitly stated with a view towards applications. The usual axioms including full excision are satisfied under obvious restrictions. Some of theMorse rank and span concepts are taken up in terms of these limit groups with the slight change of homology classes rather then cycles. A simple demonstration using the intuitive notions of the various cap types is presented for two basic exact sequences recently given byDeheuvel. To Mauro Picone on his 70^th birthday.     

Tilting Cotorsion Pairs

Let R be a ring and T a 1-tilting right R-module. Then T isof countable type. Moreover, T is of finite type in the casewhere R is a Prüfer domain. 2000 Mathematics Subject Classification16D90, 16D30, 13G05 (primary), 03E75, 20K40, 16G99 (secondary).     

Baer cotorsion Pairs

LetR be a unital associative ring and two classes of leftR-modules. In [St3] the notion of a ( ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses is called a ( ) pair if it is maximal with respect to the classes and the condition Ext _R ¹ (V, W)=0 for all . In this paper we study pairs whereR = ℤ and is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every pair is singly cognerated underV=L. The author was supported by a DFG grant.     

Higher Order Coherent Pairs

In this paper, we study necessary and sufficient conditions for the relation $$\begin{array}{@{}l}P_n^{{[r]}}(x) + a_{n-1,r} P_{n-1}^{{[r]}}(x)= R_{n-r}(x) + b_{n-1,r} R_{n-r-1}(x),\\[5pt]\quad a_{n-1,r}\neq0,\ n\geq r+1,\end{array}$$ where {P _n (x)} _n??0 and {R _n (x)} _n??0 are two sequences of monic orthogonal polynomials with respect to the quasi-definite linear functionals $\mathcal{U},\mathcal{V}$ , respectively, or associated with two positive Borel measures ?? ₀,?? ₁ supported on the real line. We deduce the connection with Sobolev orthogonal polynomials, the relations between these functionals as well as their corresponding formal Stieltjes series. As sake of example, we find the coherent pairs when one of the linear functionals is classical.     

Expanded Degenerations and Pairs

We provide a universal approach to the moduli of Jun Li's expanded pairs and expanded degenerations [20 Li , J. ( 2001 ). Stable morphisms to singular schemes and relative stable morphisms . J. Differential Geom. 57 : 509 – 578 .[Web of Science ®] , [Google Scholar]]. This enables us to prove algebraicity results, compare with Li's approach and with the approach of Graber and Vakil [16 Graber , T. , Vakil , R. ( 2005 ). Relative virtual localization and vanishing of tautological classes on moduli spaces of curves . Duke Math. J. 130 : 1 – 37 .[Crossref], [Web of Science ®] , [Google Scholar]], and generalize to the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneration formulas.     

Characterising Pairs of Refinable Splines

A complete characterisation is given, in terms of Fourier transforms, of pairs of refinable univariate spline functions, with knots at the integers, whose integer translates form a Riesz basis.     

The Generalized Pairs Plot

This article develops a generalization of the scatterplot matrix based on the recognition that most datasets include both categorical and quantitative information. Traditional grids of scatterplots often obscure important features of the data when one or more variables are categorical but coded as numerical. The generalized pairs plot offers a range of displays of paired combinations of categorical and quantitative variables. A mosaic plot, fluctuation diagram, or faceted bar chart may be used to display two categorical variables. A side-by-side boxplot, stripplot, faceted histogram, or density plot helps visualize a categorical and a quantitative variable. A traditional scatterplot is suitable for displaying a pair of numerical variables, but options also support density contours or annotating summary statistics such as the correlation and number of missing values, for example. By combining these, the generalized pairs plot may help to reveal structure in multivariate data that otherwise might go unnoticed in the process of exploratory data analysis. Two different R packages provide implementations of the generalized pairs plot, gpairs and GGally. Supplementary materials for this article are available online on the journal web site.     

Pairs of II1 factors

A class of objects—that are best described as being actions ofgroup-like objects of von Neumann algebras—is axiomatised and it is shown that there exists a bijective correspondence between isomorphism classes of suchcovariant systems and isomorphism classes of pairs of II₁ factors (M, N) satisfyingN⊂M, [M:N]<∞ andM∮N′=C.     

Pairs of edge-disjoint Hamiltonian circuits

Aequationes mathematicae -     

Pairs of non-intersecting random flats

Summary Let A _r,A _s,1r+sn-1, be independent isotropic uniform random r- resp. s-flats meeting a n-dimensional convex body K. It is shown that the probability that the points realizing the distance of A _rand A _sbelong to K is maximal if and only if K is a ball.