A Krasnosel’skii-type result for planar sets starshaped via orthogonally convex paths

A Krasnosel’skii-type theorem for compact sets that are starshaped via staircase paths may be extended to compact sets that are starshaped via orthogonally convex paths: Let S be a nonempty compact planar set having connected complement. If every two points of S are visible via orthogonally convex paths from a common point of S, then S is starshaped via orthogonally convex paths. Moreover, the associated kernel Ker S has the expected property that every two of its points are joined in Ker S by an orthogonally convex path. If S is an arbitrary nonempty planar set that is starshaped via orthogonally convex paths, then for each component C of Ker S, every two of points of C are joined in C by an orthogonally convex path. Communicated by Imre Bárány     

On Intersection of Maximal Orthogonally k-Starshaped Polygons

Let F be a simply connected orthogonal polygon in R ² and let P denote the intersection of all maximal orthogonally k-starshaped polygons in F for any fixed integer k,k2. If P and for every x,y P which are joined in F by a staircase path having two segments there is a similar staircase path from x to y in P, then there exists a maximal orthogonally k-starshaped polygon Q in F such that the staircase k-kernel of Q is a subset of the staircase k-kernel of P. In particular, F is either an orthogonally k-starshaped simply connected polygon in F or empty.     

Unions of orthogonally convex or orthogonally starshaped polygons

LetT be a simply connected orthogonal polygon having the property that for every three points ofT, at least two of these points see each other via staircases inT. ThenT is a union of three orthogonally convex polygons. The number three is best possible.ForT a simply connected orthogonal polygon,T is a union of two orthogonally convex polygons if and only if for every sequencev ₁,...,v _n,v _n+1 =v ₁ inT, n odd, at least one consecutive pairv _i ,v _i+1 sees each other via staircase paths inT, 1 i n. An analogous result says thatT is a union of two orthogonal polygons which are starshaped via staircase paths if and only if for every odd sequence inT, at least one consecutive pair sees a common point via staircases inT.Supported in part by NSF grants DMS-8908717 and DMS-9207019.     

Simply connected orthogonal polygons as unions of two orthogonally starshaped sets

Let S be a simply connected orthogonal polygon in the plane. The set S is a union of two sets which are starshaped via staircase paths (i.e., orthogonally starshaped) if and only if for every three points of S, at least two of these points see (via staircase paths) a common point of S. Moreover, the simple connectedness condition cannot be deleted.     

The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms

The orthogonal Procrustes problem involves finding an orthogonal matrix which transforms one given matrix into another in the least-squares sense, and thus it requires the minimization of the Frobenius matrix norm. We consider, the solution of this problem for a family of orthogonally invariant norms which includes the Frobenius norm as a special case.     

Structure of closed finitely starshaped sets

A set is finitely starshaped if any finite subset of is totally visible from some point of . It is well known that in a finite-dimensional linear space, a closed finitely starshaped set which is not starshaped must be unbounded. It is proved here that such a set must admit at least one direction of recession. This fact clarifies the structure of such sets and allows the study of properties of their visibility elements, well known in the case of starshaped sets. A characterization of planar finitely starshaped sets by means of its convex components is obtained. Some plausible conjectures are disproved by means of counterexamples.     

Intersection properties of maximal directed cuts in digraphs

If $D$ is a finite digraph, a directed cut is a subset of arcs in $D$ having tail in some subset $X?V\left(D\right)$ and head in $V\left(D\right)?X$. In this paper we prove two general results concerning intersections between maximal paths, cycles and maximal directed cuts in $D$. As a direct consequence of these results, we deduce that there is a path, or a cycle, in $D$ that crosses each maximal directed cut.     

A Helly theorem for intersections of orthogonally starshaped sets

Let $\cal{F}$ be a finite family of simply connected orthogonal polygons in the plane. If every three (not necessarily distinct) members of $\cal{F}$ have a nonempty intersection which is starshaped via staircase paths, then the intersection $\cap \{F : F\; \hbox{in}\; \cal{F}\}$ is a (nonempty) simply connected orthogonal polygon which is starshaped via staircase paths. Moreover, the number three is best possible, even with the additional requirement that the intersection in question be nonempty. The result fails without the simple connectedness condition.     

Detection of multivariate outliers with location slippage or scale inflation in left orthogonally invariant or elliptically contoured distributions

This paper is concerned with two kinds of multiple outlier problems in multivariate regression. One is a multiple location-slippage problem and the other is a multiple scale-inflation problem. A multi-decision rule is proposed. Its optimality is shown for the first problem in a class of left orthogonally invariant distributions and is also shown for the second problem in a class of elliptically contoured distributions. Thus the decision rule is robust against departures from normality. Further the null robustness of the decision statistic which the rule is based on is pointed out in each problem.     

On a conjecture of Wan about limiting Newton polygons

We show that for a monic polynomial $f(x)$ over a number field K containing a global permutation polynomial of degree >1 as its composition factor, the Newton Polygon of $f\mathrm{mod}\mathfrak{p}$ does not converge for $\mathfrak{p}$ passing through all finite places of K. In the rational number field case, our result is the “only if” part of a conjecture of Wan about limiting Newton polygons.     

Elliptic operators with unbounded coefficients: construction of a maximal dissipative extension

We prove maximal dissipativity of some dissipative systems in where is an invariant measure. Received May 23, 2000; accepted June 10, 2000.     

The Intersection of a Maximal Intransitive Subgroup with a Maximal Imprimitive Subgroup

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Starshapedness of the superlevel sets of solutions to equations involving the fractional Laplacian in starshaped rings

We study solutions of the problem (0.1) where are open sets such that , , and f is a nonlinearity. Under different assumptions on f we prove that, if D₀ and D₁ are starshaped with respect to the same point , then the same occurs for every superlevel set of u.     

Characterizing Starshaped Sets by Maximal Visibility

Let S be a nonempty set in a real topological linear space L. p S is a point of maximal visibility of S if and only if it admits a neighbourhood N in L such that Sq Sp for every point q S N, where Sx = { s S : x is visible from s via S }. For S being either open and connected or the closure of its connected interior, it is shown that the kernel of S is the set of all maximal visibility points of S. Planar examples reveal that the topological assumptions on S are necessary. This substantially strengthens a recent result of Toranzos and Forte Cunto.     

Compatible spreads of symmetry in near polygons

In De Bruyn [7] it was shown that spreads of symmetry of near polygons give rise to many other near polygons, the so-called glued near polygons. In the present paper we will study spreads of symmetry in product and glued near polygons. Spreads of symmetry in product near polygons do not lead to new glued near polygons. The study of spreads of symmetry in glued near polygons gives rise to the notion of ‘compatible spreads of symmetry'. We will classify all pairs of compatible spreads of symmetry for the known classes of dense near polygons. All these pairs of spreads can be used to construct new glued near polygons. Postdoctoral Fellow of the Research Foundation-Flanders.