Forecasting using belief functions: An application to marketing econometrics

A method is proposed to quantify uncertainty on statistical forecasts using the formalism of belief functions. The approach is based on two steps. In the estimation step, a belief function on the parameter space is constructed from the normalized likelihood given the observed data. In the prediction step, the variable Y to be forecasted is written as a function of the parameter θ and an auxiliary random variable Z with known distribution not depending on the parameter, a model initially proposed by Dempster for statistical inference. Propagating beliefs about θ and Z through this model yields a predictive belief function on Y. The method is demonstrated on the problem of forecasting innovation diffusion using the Bass model, yielding a belief function on the number of adopters of an innovation in some future time period, based on past adoption data.     

Approximations of stable actions on $ \Bbb {R} $-trees

This article shows how to approximate a stable action of a finitely presented group on an -tree by a simplicial one while keeping control over arc stabilizers. For instance, every small action of a hyperbolic group on an -tree can be approximated by a small action of the same group on a simplicial tree. The techniques we use highly rely on Rips's study of stable actions on -trees and on the dynamical study of exotic components by D. Gaboriau. Received: 22 October, 1997     

Fuzzy dynamical systems

A fuzzy dynamical system on an underlying complete, locally compact metric state space X is defined axiomatically in terms of a fuzzy attainability set mapping on X. This definition includes as special cases crisp single and multivalued dynamical systems on X. It is shown that the support of such a fuzzy dynamical system on X is a crisp multivalued dynamical system on X, and that such a fuzzy dynamical system can be considered as a crisp dynamical system on a state space of nonempty compact fuzzy subsets of X. In addition fuzzy trajectories are defined, their existence established and various properties investigated.     

On the de Rham cohomology of differential and algebraic stacks

We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids.Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack determines the flat groupoid up to étale equivalence.We show how a cofoliation on a stack gives rise to a refinement of the Hodge to De Rham spectral sequence, where the E₁-term consists entirely of vector bundle valued cohomology groups.Our theory works for differentiable, holomorphic and algebraic stacks.     

Markov processes associated with L p -resolvents and applications to stochastic differential equations on Hilbert space

We give general conditions on a generator of a C₀-semigroup (resp. of a C₀-resolvent) on L^p(E,μ), p ≥ 1, where E is an arbitrary (Lusin) topological space and μ a σ-finite measure on its Borel σ-algebra, so that it generates a sufficiently regular Markov process on E. We present a general method how these conditions can be checked in many situations. Applications to solve stochastic differential equations on Hilbert space in the sense of a martingale problem are given. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

The Denjoy extension of the Riemann and Mcshane integrals

In this paper we study the Denjoy-Riemann and Denjoy-McShane integrals of functions mapping an interval [a, b] into a Banach space X. It is shown that a Denjoy-Bochner integrable function on [a, b] is Denjoy-Riemann integrable on [a, b], that a Denjoy-Riemann integrable function on [a, b] is Denjoy-McShane integrable on [a, b] and that a Denjoy-McShane integrable function on [a, b] is Denjoy-Pettis integrable on [a, b]. In addition, it is shown that for spaces that do not contain a copy of c ₀, a measurable Denjoy-McShane integrable function on [a, b] is McShane integrable on some subinterval of [a, b]. Some examples of functions that are integrable in one sense but not another are included.     

Lifespan Theorem for simple constrained surface diffusion flows

We consider closed immersed hypersurfaces in R³ and R⁴ evolving by a special class of constrained surface diffusion flows. This class of constrained flows includes the classical surface diffusion flow. In this paper we present a Lifespan Theorem for these flows, which gives a positive lower bound on the time for which a smooth solution exists, and a small upper bound on the total curvature during this time. The hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature. This turns out to be a deep property of the initial manifold, as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in L² for M² immersed in R³ and additionally on the concentration in L³ for M³ immersed in R⁴. This is stronger than a previous result on a different class of constrained surface diffusion flows, as here we obtain an improved lower bound on maximal time, a better estimate during this period, and eliminate any assumption on the area of the evolving hypersurface.     

A Decomposition Theorem for Lévy Processes on Local Fields

The study of Lévy processes on local fields has been initiated by Albeverio et al. (1985)–(1998) and Evans (1989)–(1998). In this paper, a decomposition theorem for Lévy processes on local fields is given in terms of a structure result for measures on local fields and a Lévy–Khinchine representation. It is shown that a measure on a local field can be decomposed into three parts: a spherically symmetric measure, a totally non-spherically symmetric measure and a singular measure. We show that if the Radon–Nikodym derivative of the absolutely continuous part of a Lévy measure on a local field is locally constant, the Lévy process is the sum of a spherically symmetric random walk, a finite or countable set of totally non, spherically symmetric Lévy processes with single balls as support of their Lévy measure, end a singular Lévy process. These processes are independent. Explicit formulae for the transition function are obtained.     

Maximum Orbit Weights in the σ-game and Lit-only σ-game on Grids and Graphs

Let G be a graph in which each vertex can be in one of two states: on or off. In the σ-game, when you “push” a vertex v you change the state of all of its neighbors, while in the σ⁺-game you change the state of v as well. Given a starting configuration of on vertices, the object of both games is to reduce it, by a sequence of pushes, to the smallest possible number of on vertices. We show that any starting configuration in a graph with no isolated vertices can, by a sequence of pushes, be reduced to at most half on, and we characterize those graphs for which you cannot do better. The proofs use techniques from coding theory. In the lit-only versions of these two games, you can only push vertices which are on. We obtain some results on the minimum number of on vertices one can obtain in grid graphs in the regular and lit-only versions of both games.     

Rank-determining sets of metric graphs

A metric graph is a geometric realization of a finite graph by identifying each edge with a real interval. A divisor on a metric graph Γ is an element of the free abelian group on Γ. The rank of a divisor on a metric graph is a concept appearing in the Riemann-Roch theorem for metric graphs (or tropical curves) due to Gathmann and Kerber, and Mikhalkin and Zharkov. We define a rank-determining set of a metric graph Γ to be a subset A of Γ such that the rank of a divisor D on Γ is always equal to the rank of D restricted on A. We show constructively in this paper that there exist finite rank-determining sets. In addition, we investigate the properties of rank-determining sets in general and formulate a criterion for rank-determining sets. Our analysis is based on an algorithm to derive the v₀-reduced divisor from any effective divisor in the same linear system.     

Good Banach spaces for piecewise hyperbolic maps via interpolation

We introduce a weak transversality condition for piecewise C^1+α and piecewise hyperbolic maps which admit a C^1+α stable distribution. We show bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel–Lizorkin type which are better than previously known estimates (when our assumption on the stable distribution holds). In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) but gives a new result on bounded multipliers. Our method applies also to piecewise expanding maps and to Anosov diffeomorphisms, giving a unifying picture of several previous results on a simpler scale of Banach spaces.     

The combinatorial game Nofil played on Steiner triple systems

We introduce an impartial combinatorial game on Steiner triple systems called Next One to Fill Is the Loser (Nofil ). Players move alternately, choosing points of the triple system. If a player is forced to fill a block on their turn, they lose. By computing nim-values, we determine optimal strategies for Nofil on all Steiner triple systems up to order 15 and a sampling for orders 19, 21 and 25. The game Nofil can be thought of in terms of play on a corresponding hypergraph which will become a graph during play. At that point Nofil is equivalent to playing the game Node Kayles on the graph. We prove necessary conditions and sufficient conditions for a graph to reached playing Nofil. We conclude that the complexity of determining the outcome of the game Nofil on Steiner triple systems is PSPACE-complete for randomized reductions.     

On (strong) α-favorability of the Wijsman hyperspace

The Banach-Mazur game as well as the strong Choquet game are investigated on the Wijsman hyperspace from the nonempty player's (i.e. α's) perspective. For the strong Choquet game we show that if X is a locally separable metrizable space, then α has a (stationary) winning strategy on X iff it has a (stationary) winning strategy on the Wijsman hyperspace for each compatible metric on X. The analogous result for the Banach-Mazur game does not hold, not even if X is separable, as we show that α may have a (stationary) winning strategy on the Wijsman hyperspace for each compatible metric on X, and not have one on X. We also show that there exists a separable 1st category metric space such that α has a (stationary) winning strategy on its Wijsman hyperspace. This answers a question of Cao and Junnila (2010) [6].     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups. (Received 13 June 1998; in revised form 31 March 1999)     

Invariant Metrics on G-Spaces

Let X be a G-space such that the orbit space X/G is metrizable. Suppose a family of slices is given at each point of X. We study a construction which associates, under some conditions on the family of slices, with any metric on X/G an invariant metric on X. We show also that a family of slices with the required properties exists for any action of a countable group on a locally compact and locally connected metric space.     

Powell–Sabin splines with boundary conditions for polygonal and non-polygonal domains

Powell–Sabin splines are piecewise quadratic polynomials with a global C¹$C^1$-continuity, defined on conforming triangulations. Imposing boundary conditions on such a spline leads to a set of constraints on the spline coefficients. First, we discuss boundary conditions defined on a polygonal domain, before we treat boundary conditions on a general curved domain boundary. We consider Dirichlet and Neumann conditions, and we show that a particular choice of the PS-triangles at the boundary can greatly simplify the corresponding constraints. Finally, we consider an application where the techniques developed in this paper are used: the numerical solution of a partial differential equation by the Galerkin and collocation method.     

Contractive maps on normed linear spaces and their applications to nonlinear matrix equations

In this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space. These conditions are already well known for maps on intervals in R. Using the conditions and Banach’s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space. The fixed point theorem will be applied to the matrix equation X = I_n + A^∗f(X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, ∞). This will give conditions on A and f under which the equation has a unique solution in a certain set. We will consider two examples of f in detail. In one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on A.     

Examples for cross curvature flow on 3-manifolds

Recently, B. Chow and R.S. Hamilton [3] introduced the cross curvature flow on 3-manifolds. In this paper, we analyze two interesting examples for this new flow. One is on a square torus bundle over a circle, and the other is on a S² bundle over a circle. We show that the global flow exists in both cases. However, on the former the flow diverges at time infinity, and on the latter the flow converges at time infinity. Mathematics Subject Classification (1991) 53C44     

An algorithm for the difference between set covers

A set cover for a set S is a collection C of special subsets whose union is S. Given covers A and B for two sets, the set-cover difference problem is to construct a new cover for the elements covered by A but not B. Applications include testing equivalence of set covers and maintaining a set cover dynamically. In this paper, we solve the set-cover difference problem by defining a difference operation A-B, which turns out to be a pseudocomplement on a distributive lattice. We give an algorithm for constructing this difference, and show how to implement the algorithm for two examples with applications in computer science: face covers on a hypercube, and rectangle covers on a grid. We derive an upper bound on the time complexity of the algorithm, and give upper and lower bounds on complexity for face covers and rectangle covers.     

Higher order Hessian structures on manifolds

In this paper we define nth order Hessian structures on manifolds and study them. In particular, whenn = 3, we make a detailed study and establish a one-to-one correspondence betweenthird-order Hessian structures and acertain class of connections on the second-order tangent bundle of a manifold. Further, we show that a connection on the tangent bundle of a manifold induces a connection on the second-order tangent bundle. Also we define second-order geodesics of special second-order connection which gives a geometric characterization of symmetric third-order Hessian structures.