Determining a Rotation of a Tetrahedron from a Projection

The following problem, arising from medical imaging, is addressed: Suppose that T is a known tetrahedron in ?³ with centroid at the origin. Also known is the orthogonal projection U of the vertices of the image ?T of T under an unknown rotation ? about the origin. Under what circumstances can ? be determined from T and U?     

Construction of a unique self-adjoint generator for a symmetric local semigroup

A definition is given of a symmetric local semigroup of (unbounded) operators P(t) (0 ? t ? T for some T > 0) on a Hilbert space $\text{H}$, such that P(t) is eventually densely defined as t → 0. It is shown that there exists a unique (unbounded below) self-adjoint operator H on $\text{H}$ such that P(t) is a restriction of e^?tH. As an application it is proven that H₀ + V is essentially self-adjoint, where e^?tH₀ is an L^p-contractive semigroup and V is multiplication by a real measurable function such that V ∈ L^{2 + ε} and e^?δV ∈ L¹ for some ε, δ > 0.     

The expansion of a semigroup and a Riesz basis criterion

Problems on the expansion of a semigroup and a criterion for being a Riesz basis are discussed in the present paper. Suppose that A is the generator of a C₀ semigroup on a Hilbert space and σ(A)=σ₁(A)∪σ₂(A) with σ₂(A) is consisted of isolated eigenvalues distributed in a vertical strip. It is proved that if σ₂(A) is separated and for each λ∈σ₂(A), the dimension of its root subspace is uniformly bounded, then the generalized eigenvectors associated with σ₂(A) form an L-basis. Under different conditions on the Riesz projection, the expansion of a semigroup is studied. In particular, a simple criterion for the generalized eigenvectors forming a Riesz basis is given. As an application, a heat exchanger problem with boundary feedback is investigated. It is proved that the heat exchanger system is a Riesz system in a suitable state Hilbert space.     

Extending a function on a graph

Let G be a graph with vertex set V, and let h be a function mapping a subset U of V into the real numbers R. If ? is a function from V to R, we define δ (?) to be the sum of ∥?(b)? ?(a)∥ over all edges {a, b} of G. A best extension of h is such a function ? with ?(x) = h(x) for X ∈ U and minimum δ (?). We show that such a best extension exists and derive an algorithm for obtaining such an extension. We also show that if instead we minimise the sum of (?(b)??(a))², there is generally a unique best extension, obtainable by solving a system of linear equations.     

The Cartesian product of a compactum and a space is a bifunctor in shape

In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution R(X,K) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X,K) consists of spaces having the homotopy type of polyhedra. In two subsequent papers the author proved that R(X,K) is a covariant functor in each of its variables X and K. In the present paper it is proved that R(X,K) is a bifunctor. Using this result, it is proved that the Cartesian product X×Z of a compact Hausdorff space X and a topological space Z is a bifunctor SSh(Cpt)×Sh(Top)→Sh(Top) from the product category of the strong shape category of compact Hausdorff spaces SSh(Cpt) and the shape category Sh(Top) of topological spaces to the category Sh(Top). This holds in spite of the fact that X×Z need not be a direct product in Sh(Top).     

Asymptotic stability for a free boundary problem arising in a tumor model

We consider a tumor model in which all cells are proliferating at a rate μ and their density is proportional to the nutrient concentration. The model consists of a coupled system of an elliptic equation and a parabolic equation, with the tumor boundary as a free boundary. It is known that for an appropriate choice of parameters, there exists a unique spherically symmetric stationary solution with radius RS which is independent of μ. It was recently proved that there is a function μ_∗(RS) such that the spherical stationary solution is linearly stable if μ<μ_∗(RS) and linearly unstable if μ>μ_∗(RS). In this paper we prove that the spherical stationary solution is nonlinearly stable (or, asymptotically stable) if μ<μ_∗(RS).     

Closure in a Hilbert space of a prehilbert space Chebyshev set

Let E denote the real inner product space that is the union of all finite dimensional Euclidean spaces. There is a bounded nonconvex set S, that is a subset of E, such that each point of E has a unique nearest point in S. Let H denote the separable Hilbert space that is the completion of space E. A condition is given in order that a point in H have a unique nearest point in the closure of S. We shall also provide an example where the condition fails.     

A linear algorithm for a core of a tree

A core of a graph G is a path P in G that is central with respect to the property of minimizing d(P) = Σ_υ?V(G)d(υ, P), where d(υ, P) is the distance from vertex υ to path P. We present a linear algorithm for finding a core of a tree. Since the core of a graph is not necessarily unique, we also output a list of all the vertices which are in some core.     

The prime dicompletion of a di-uniformity on a plain texture

Working within a plain texture (S,S), the authors construct a completion of a dicovering uniformity υ on (S,S) in terms of prime S-filters. In case υ is separated, a separated completion is then obtained using the T₀-quotient, and it is shown that this construction produces a reflector. For a totally bounded di-uniformity it is verified that these constructions lead to dicompactifications of the uniform ditopology. A condition is given under which complementation is preserved on passing to these completions, and an example on the real texture (R,R,ρ) is presented.     

Transforming a graph into a 1-balanced graph

Let G be a non-trivial, loopless graph and for each non-trivial subgraph H of G, let . The graph G is 1-balanced if γ(G), the maximum among g(H), taken over all non-trivial subgraphs H of G, is attained when H=G. This quantity γ(G) is called the fractional arboricity of the graph G. The value γ(G) appears in a paper by Picard and Queyranne and has been studied extensively by Catlin, Grossman, Hobbs and Lai. The quantity γ(G)−g(G) measures how much a given graph G differs from being 1-balanced. In this paper, we describe a systematic method of modifying a given graph to obtain a 1-balanced graph on the same number of vertices and edges. We obtain this by a sequence of iterations; each iteration re-defining one end-vertex of an edge in the given graph. After each iteration, either the value γ of the new graph formed is less than that of the graph from the previous iteration or the size of the maximal γ-achieving subgraph of the new graph is smaller than that of the graph in the previous iteration. We show that our algorithm is polynomial in time complexity. Further ways to decrease the number of iterations are also discussed.     

Fuzzy sets as a basis for a theory of possibility

The theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable. More specifically, if F is a fuzzy subset of a universe of discourse U = {u} which is characterized by its membership function μ_F, then a proposition of the form $\text{“X}\text{is}\text{F”}$, where X is a variable taking values in U, induces a possibility distribution $\text{t}\text{?}{}_{\mathrm{x}}$ which equates the possibility of X taking the value u to μ_F(u)—the compatibility of u with F. In this way, X becomes a fuzzy variable which is associated with the possibility distribution $\text{t}\text{?}{}_{\mathrm{x}}$ in much the same way as a random variable is associated with a probability distribution. In general, a variable may be associated both with a possibility distribution and a probability distribution, with the weak connection between the two expressed as the possibility/probability consistency principle.A thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is, in the main, possibilistic rather than probabilistic in nature. Thus, by employing the concept of a possibility distribution, a proposition, p, in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p. Several types of conditional translation rules are discussed and, in particular, a translation rule for propositions of the form $\text{“X}\text{is}\text{F}$ is α-possible”, where α is a number in the interval [0,1], is formulated and illustrated by examples.     

Harmonic measures for a point may form a square

Let X be a Green domain in Rd, d?2, x∈X, and let Mx(P(X)) denote the compact convex set of all representing measures for x. Recently it has been proven that the set of harmonic measures , U open in X, x∈U, which is contained in the set of extreme points of Mx(P(X)), is dense in Mx(P(X)). In this paper, it is shown that Mx(P(X)) is not a simplex (and hence not a Poulsen simplex). This is achieved by constructing open neighborhoods U₀, U₁, U₂, U₃ of x such that the harmonic measures are pairwise different and . In fact, these measures form a square with respect to a natural L²-structure. Since the construction is mainly based on having certain symmetries, it can be carried out just as well for Riesz potentials, the Heisenberg group (or any stratified Lie algebra), and the heat equation (or more general parabolic situations).     

Sufficient conditions for a partially hyperbolic attractor to be a homoclinic class

We prove that a partially hyperbolic attractor with two-dimensional central direction Λ is a homoclinic class if it exhibits a hyperbolic periodic orbit O and a Lorenz-like singularity σ with Wu(σ)∩Ws(O)≠∅ such that Ws(σ) is dense in Λ.     

Choosing a sheltered middle path

We say that point x∈R² is sheltered by a continuum S⊂R² if x does not belong to the unbounded component of R²\S. Suppose that points a and b are the endpoints of each of three arcs A₀, A₁ and A₂ contained in R². We prove that there is an arc B⊂A₀∪A₁∪A₂ with its endpoints a and b such that each point of B is sheltered by the union of each two of the arcs A₀, A₁ and A₂.     

Covering a Polish group by translates of a nowhere dense set

We show that, for every nonlocally compact Polish group G with a left-invariant complete metric ρ, we have covG=cov(M). Here, covG is the minimal number of translates of a fixed closed nowhere dense subset of G, which is needed to cover G, and cov(M) is the minimal cardinality of a cover of the real line R by meagre sets.     

Existence of a solution to a nonlinear equation

Equation (−Δ+k²)u+f(u)=0 in D, u_|∂D=0, where k=const>0 and D⊂R³ is a bounded domain, has a solution if is a continuous function in the region |u|?a, piecewise-continuous in the region |u|?a, with finitely many discontinuity points uj such that f(uj±0) exist, and uf(y)?0 for |u|?a, where a?0 is an arbitrary fixed number.     

Every vertex a king

A king in a tournament is a vertex which can reach every other vertex via a 1-path or 2-path. A new inductive proof is given for the existence of an n-tournament with exactly k kings for all integers n ? k ? 1 with the following exceptions: k = 2 with n arbitrary, and n = k = 4 (in which cases no such n-tournament exists). Also, given an n-tournament T, the smallest order m is determined so that there exists an m-tournament W which contains T as a subtournament and so that every vertex of W is a king. Bounds are obtained in a similar problem in which the kings of W are exactly the vertices of T.     

Exponent of a finite group with a fixed-point-free four-group of automorphisms

Let e be a positive integer and G a finite group acted on by the four-group V in such a manner that C _G (V) = 1. Suppose that V contains an element v such that the centralizer C _G (v) has exponent e. Then the exponent of G″, the second derived group of G, is bounded in terms of e only.     

Graphic sequences with a realization containing a complete multipartite subgraph

A nonincreasing sequence of nonnegative integers π=(d₁,d₂,…,d_n) is graphic if there is a (simple) graph G of order n having degree sequence π. In this case, G is said to realizeπ. For a given graph H, a graphic sequence π is potentiallyH-graphic if there is some realization of π containing H as a (weak) subgraph. Let σ(π) denote the sum of the terms of π. For a graph H and n∈Z⁺, σ(H,n) is defined as the smallest even integer m so that every n-term graphic sequence π with σ(π)≥m is potentially H-graphic. Let denote the complete t partite graph such that each partite set has exactly s vertices. We show that and obtain the exact value of σ(K_j+K_s,s,n) for n sufficiently large. Consequently, we obtain the exact value of for n sufficiently large.     

Constructing a covering triangulation by means of a nowhere-zero dual flow

Tutte conjectured that every graph with no isthmus can be provided with an integral nowhere-zero flow with no absolute value greater than k=5. As yet the result is established for k=6, and it is used for proving that the existence of a triangular imbedding of a graph G in a surface S implies the existence of a triangular imbedding of G_(m) in a surface $\text{S}\text{?}$ with the same orientability characteristic as S. G_(m) stands for the composition of G by an independent set of m vertices.