This paper was written in response to the following question: what conditions are sufficient to guarantee that if a compact subset A of a topological linear space L^3 is not convex, then for every point x belonging to the complement of A relative to the convex hull of A there exists a line segment yz such that x belongs to yz and y belongs to A and z belongs to A? Restated in the terminology of this paper the question bay be given as follow: what conditions may be imposed upon a compact subset A of L^3 to insure that A is braced?

The purpose of this paper is to investigate some of the properties of convex sets in the plane through synthetic geometry.

A method that can be used to detect damage in structures is developed. This method is based on the convexity of the geometric interpretation of the eigenvalue problem for undamped positive definite systems. The damage detection scheme establishes various damage scenarios which are used as failure sets. These scenarios are then compared to the structure's actual response by measuring the natural frequencies of the structure and using a Euclideian norm. Mathematical models were developed for application of the method on a cantilever beam. Damage occurring at a single location or in multiple locations was estalished and studied. Experimental verification was performed on serval prismatic beams in which the method provided adequate results. The exact location and extent of damage for several cases was predicted. When the method failed the prediction was very close to the actual condition in the structure. This method is easy to use and does not require a rigorous amount of instrumentation for obtaining the experimental data required in the detection scheme. / Ph. D.

Given a connected oriented graph D, we say that a subset S of V(D) is convex in D if, for every pair of vertices x, y in S, the vertex set of every x-y geodesic (x-y shortest dipath) and y-x geodesic in D is contained in S. The convexity number con (D) of a nontrivial connected oriented graph D is the maximum cardinality of a proper convex set of D. Let S_{C}(K_{n})={con(D)|D is an orientation of K_{n}} and S_{SC}(K_{n})={con(D)|D is a strong orientation of K_{n}}. We show that S_{C}(K_{3})={1,2} and S_{C}(K_{n})={1,3,4,...,n-1} if n >= 4. We also have that S_{SC}(K_{3})={1} and S_{SC}(K_{n})={1,3,4,...,n-2} if n >= 4 . We also show that every triple n, m, k of integers with n >= 5, 3 <= k <= n-2, and n+1 <= m <= n(n-1)/2, there exists a strong connected oriented graph D of order n with |E(D)|=m and con (D)=k.

Some problems concerning holomorphic continuation of the class of bounded holo­morphic functions from bounded domains in Cn that are domains of holomorphy are solved. A bounded domain of holomorphy Ω in C2 with nonschlicht H°°-envelope of holomorphy is constructed and it is shown that there is a point in D for which Glea­son’s Problem for H°°(Ω) cannot be solved. Furthermore a proof of the existence of a bounded domain of holomorphy in C2 for which the volume of the H°°-envelope of holomorphy is infinite is given. The idea of the proof is to put a family of so-called ”Sibony domains” into the unit bidisk by a packing procedure and patch them together by thin neighbourhoods of suitably chosen curves. If H°°(Ω) is the Banach algebra of bounded holomorphic functions on a bounded domain Ω in Cn and if p is a point in Ω, then the following problem is known as Gleason’s Problem for Hoo(Ω) : Is the maximal ideal in H°°(Ω) consisting of functions vanishing at p generated by (z1 -p1) , ... ,   (zn - pn) ? A sufficient condition for solving Gleason’s Problem for 77°° (Ω) for all points in Ω is given. In particular, this condition is fulfilled by a convex domain Ω with Lip1+e boundary (0 &lt; e &lt; 1) and thus generalizes a theorem of S.L.Leibenson. It is also proved that Gleason’s Problem can be solved for all points in certain unions of two polydisks in C2. One of the ideas in the methods of proof is integration along specific polygonal lines. Certain properties of some open sets defined by global plurisubharmonic func­tions in Cn are studied. More precisely, the sets Du = {z e Cn : u(z) &lt; 0} and Eh = {{z,w) e Cn X C : h(z,w) &lt; 1} are considered where ti is a plurisubharmonic function of minimal growth and h≠0 is a non-negative homogeneous plurisubharmonic function. (That is, the functions u and h belong to the classes L(Cn) and H+(Cn x C) respectively.) It is examined how the fact that Eh and the connected components of Du are H°°-domains of holomorphy is related to the structure of the set of disconti­nuity points of the global defining functions and to polynomial convexity. Moreover it is studied whether these notions are preserved under a certain bijective mapping from L(Cn) to H+(Cn x C). Two counterexamples are given which show that polynomial convexity is not preserved under this bijection. It is also proved, for example, that if Du is bounded and if the set of discontinuity points of u is pluripolar then Du is of type H°°. A survey paper on general properties of envelopes of holomorphy is included. In particular, the paper treats aspects of the theory for the bounded holomorphic functions. The results for the bounded holomorphic functions are compared with the corresponding ones for the holomorphic functions. / digitalisering@umu.se

Convexity in graphs has been widely discussed in graph theory and G. Chartrand et al. introduced the convexity number of oriented graphs in 2002. In this thesis, we follow the notions addressed by them and develop an extension in some related topics of convexity in directed graphs. Let D be a connected oriented graph. A set S subseteq V(D) is convex in D if, for every pair of vertices x, yin S, the vertex set of every x-y geodesic (x-y shortest directed path) and y-x geodesic in D is contained in S. The convexity number con(D) of a nontrivial oriented graph D is the maximum cardinality of a proper convex set of D. We show that for every possible triple n, m, k of integers except for k=4, there exists a strongly connected digraph D of order n, size m, and con(D)=k. Let G be a graph. We define the convexity spectrum S_{C}(G)={con(D): D is an orientation of G} and the strong convexity spectrum S_{SC}(G)={con(D): D is a strongly connected orientation of G}. Then S_{SC}(G) ⊆ S_{C}(G). We show that for any n ¡Ú 4, 1 ≤ a ≤ n-2 and a n ¡Ú 2, there exists a 2-connected graph G with n vertices such that S_C(G)=S_{SC}(G)={a,n-1}, and there is no connected graph G of order n ≥ 3 with S_{SC}(G)={n-1}. We also characterizes the convexity spectrum and the strong convexity spectrum of complete graphs, complete bipartite graphs, and wheel graphs. Those convexity spectra are of different kinds. Let the difference of convexity spectra D_{CS}(G)=S_{C}(G)- S_{SC}(G) and the difference number of convexity spectra dcs(G) be the cardinality of D_{CS}(G) for a graph G. We show that 0 ≤ dcs(G) ≤ ⌊|V(G)|/2⌋, dcs(K_{r,s})=⌊(r+s)/2⌋-2 for 4 ≤ r ≤ s, and dcs(W_{1,n-1})= 0 for n ≥ 5. The convexity spectrum ratio of a sequence of simple graphs G_n of order n is r_C(G_n)= liminflimits_{n to infty} frac{|S_{C}(G_n)|}{n-1}. We show that for every even integer t, there exists a sequence of graphs G_n such that r_C(G_n)=1/t and a formula for r_C(G) in subdivisions of G.

Some problems concerning the algebra of bounded holomorphic functions from bounded domains in Cn are solved. A bounded domain of holomorphy Q in C2 with nonschlicht i7°°- envelope of holomorphy is constructed and it is shown that there is a point in Q for which Gleason’s Problem for H°°(Q) cannot be solved. If A(f2) is the Banach algebra of functions holomorphic in the bounded domain Q in Cn and continuous on the boundary and if p is a point in Q, then the following problem is known as Gleason’s Problem for A(Q) : Is the maximal ideal in A(Q) consisting of functions vanishing at p generated by (Zl ~Pl) , ■■■ , (Zn - Pn) ? A sufficient condition for solving Gleason’s Problem for A(Q) for all points in Q is given. In particular, this condition is fulfilled by a convex domain Q with Lipi+£-boundary (0 &lt; e &lt; 1) and thus generalizes a theorem of S.L.Leibenzon. One of the ideas in the methods of proof is integration along specific polygonal lines. If Gleason’s Problem can be solved in a point it can be solved also in a neighbourhood of the point. It is shown, that the coefficients in this case depends holomorphically on the points. Defining a projection from the spectrum of a uniform algebra of holomorphic functions to Cn, one defines the fiber in the spectrum over a point as the elements in the spectrum that projects on that point. Defining a kind of maximum modulus property for domains in Cn, some problems concerning the fibers and the number of elements in the fibers in certain algebras of bounded holomorphic functions are solved. It is, for example, shown that the set of points, over which the fibers contain more than one element is closed. A consequence is also that a starshaped domain with the maximum modulus property has schlicht /y°°-envelope of holomorphy. These kind of problems are also connected with Gleason’s problem. A survey paper on general properties of algebras of bounded holomorphic functions of several variables is included. The paper, in particular, treats aspects connecting iy°°-envelopes of holomorphy and some areas in the theory of uniform algebras. / <p>Diss. (sammanfattning) Umeå : Umeå universitet, 1994, härtill 6 uppsatser</p> / digitalisering@umu