Diameters of sections and coverings of convex bodies

We study the diameters of sections of convex bodies in R^N determined by a random N×n matrix Γ, either as kernels of Γ^* or as images of Γ. Entries of Γ are independent random variables satisfying some boundedness conditions, and typical examples are matrices with Gaussian or Bernoulli random variables. We show that if a symmetric convex body K in R^N has one well bounded k-codimensional section, then for any m>ck random sections of K of codimension m are also well bounded, where c?1 is an absolute constant. It is noteworthy that in the Gaussian case, when Γ determines randomness in sense of the Haar measure on the Grassmann manifold, we can take c=1.     

Stopping for two-dimensional stochastic processes

The aim of this paper is to introduce some techniques that can be used in the study of stochastic processes which have as parameter set the positive quadrant of the plane R²₊. We define stopping lines and derive an interesting property of measurability for them. The notion of predictability is developed, and we show the connection between predictable processes, fields associated with stopping lines, and predictable stopping lines. We also give a theorem of section for predictable sets. Extension to processes indexed by any partially ordered set with some regularity assumptions can be carried out quite easily with the same techniques.     

On transitive Lie bialgebroids and Poisson groupoids

We prove that, for any transitive Lie bialgebroid (A, A^∗), the differential associated to the Lie algebroid structure on A^∗ has the form d_∗=A[Λ,⋅]+Ω, where Λ is a section of ^∧2A and Ω is a Lie algebroid 1-cocycle for the adjoint representation of A. Globally, for any transitive Poisson groupoid (Γ,Π), the Poisson structure has the form , where ΠF is a bivector field on Γ associated to a Lie groupoid 1-cocycle.     

Convexity and log convexity for the spectral radius

The starting point of this paper is a theorem by J. F. C. Kingman which asserts that if the entries of a nonnegative matrix are log convex functions of a variable then so is the spectral radius of the matrix. A related result of J. Cohen asserts that the spectral radius of a nonnegative matrix is a convex function of the diagonal elements. The first section of this paper gives a new, unified proof of these results and also analyzes exactly when one has strict convexity. The second section gives some very simple proofs of results of Friedland and Karlin concerning “min-max” characterizations of the spectral radius of nonnegative matrices. These arguments also yield, as will be shown in another paper, min-max characterizations of the principal eigenvalue of second order elliptic boundary value problems on bounded domains. The third section considers the cone K of nonnegative vectors in Rⁿ and continuous maps f: K → K which are homogeneous of degree one and preserve the partial order induced by K. The (cone) spectral radius of such maps is defined and a direct generalization of Kingman's theorem to a subclass of such nonlinear maps is given. The final section of this paper treats a problem that arises in population biology. If K₀ denotes the interior of K and f is as above, when can one say that f has a unique eigenvector (to within normalization) in K₀? A subtle point to be noted is that f may have other eigenvectors in the boundary of K. If u ϵ K₀ is an eigenvector of f, |u| = 1, and g(x) = f(x)/|f(x)|, when can one say that for any x ϵ K₀, g^p(x), the pth iterate of g acting on x, converges geometrically to u? The fourth section provides answers to these questions that are adequate for many of the population biology problems.     

Uniform attractors for the closed process and applications to the reaction-diffusion equation with dynamical boundary condition

In this paper, we first introduce the concept of a closed process in a Banach space, and we obtain the structure of a uniform attractor of the closed process by constructing a skew product-flow on the extended phase space. Then, the properties of the kernel section of closed process are investigated. Moreover, we prove the existence and structure of the uniform attractor for the reaction-diffusion equation with a dynamical boundary condition in L^p without any restriction on the growth order of the nonlinear term.     

Flexure of beams with certain curvilinear cross sections

Special complex variables techniques are used to obtain the six flexure functions (one of which is the torsion function) of a certain isotropic cylinder under flexure. The cross section is bounded by the closed curver=a sin⁴(θ/4) (?π<θ?π). The torsional rigidity, moment integrals, and the associated twist are also evaluated for this beam. It is worthy to mention that these techniques work successfully when they are applied to any of the cross sections bounded byr=a∥sin(θ/n)∥ ⁿ ·(?π<θ?π), wheren is a positive integer (n>1).     

The Conley conjecture for Hamiltonian systems on the cotangent bundle and its analogue for Lagrangian systems

In this paper, the Conley conjecture, which was recently proved by Franks and Handel [J. Franks, M. Handel, Periodic points of Hamiltonian surface diffeomorphism, Geom. Topol. 7 (2003) 713-756] (for surfaces of positive genus), Hingston [N. Hingston, Subharmonic solutions of Hamiltonian equations on tori, Ann. Math., in press] (for tori) and Ginzburg [V.L. Ginzburg, The Conley conjecture, arXiv: math.SG/0610956v1] (for closed symplectically aspherical manifolds), is proved for C¹-Hamiltonian systems on the cotangent bundle of a C³-smooth compact manifold M without boundary, of a time 1-periodic C²-smooth Hamiltonian H:R×T^*M→R which is strongly convex and has quadratic growth on the fibers. Namely, we show that such a Hamiltonian system has an infinite sequence of contractible integral periodic solutions such that any one of them cannot be obtained from others by iterations. If H also satisfies H(−t,q,−p)=H(t,q,p) for any (t,q,p)∈R×T^*M, it is shown that the time-1-map of the Hamiltonian system (if exists) has infinitely many periodic points siting in the zero section of T^*M. If M is C⁵-smooth and dimM>1, H is of C⁴ class and independent of time t, then for any τ>0 the corresponding system has an infinite sequence of contractible periodic solutions of periods of integral multiple of τ such that any one of them cannot be obtained from others by iterations or rotations. These results are obtained by proving similar results for the Lagrangian system of the Fenchel transform of H, L:R×TM→R, which is proved to be strongly convex and to have quadratic growth in the velocities yet.     

Functions of bounded variation on “good” metric spaces

In this paper we give a natural definition of Banach space valued BV functions defined on complete metric spaces endowed with a doubling measure (for the sake of simplicity we will say doubling metric spaces) supporting a Poincaré inequality (see Definition 2.5 below). The definition is given starting from Lipschitz functions and taking closure with respect to a suitable convergence; more precisely, we define a total variation functional for every Lipschitz function; then we take the lower semicontinuous envelope with respect to the L¹ topology and define the BV space as the domain of finiteness of the envelope. The main problem of this definition is the proof that the total variation of any BV function is a measure; the techniques used to prove this fact are typical of Γ-convergence and relaxation. In Section 4 we define the sets of finite perimeter, obtaining a Coarea formula and an Isoperimetric inequality. In the last section of this paper we also compare our definition of BV functions with some definitions already existing in particular classes of doubling metric spaces, such as Weighted spaces, Ahlfors-regular spaces and Carnot–Carathéodory spaces.     

Maskit combinations of Poincaré-Einstein metrics

We establish a boundary connected sum theorem for asymptotically hyperbolic Einstein metrics, and also show that if the two metrics have scalar positive conformal infinities, then the same is true for this boundary join. This construction is also extended to spaces with a finite number of interior conic singularities, and as a result we show that any 3-manifold which is a finite connected sum of quotients of S³ and S²×S¹ bounds such a space (with conic singularities); putatively, any 3-manifold admitting a metric of positive scalar curvature is of this form.     

The tracing procedure: A Bayesian approach to defining a solution forn-person noncooperative games

The paper proposes a Bayesian approach to selecting a particular equilibrium points ^* of any given finiten-person noncooperative game Γ as solution for Γ. It is assumed that each playeri starts his analysis of the game situation by assigning a subjective prior probability distributionp _j to the set of all pure strategies available to each other playerj. (The prior distributionsp _j used by all other playersi in assessing the likely strategy choice of any given playerj will be identical, because all these playersi will compute this prior distributionp _j from the basic parameters of game Γ in the same way.) Then, the players are assumed to modify their subjective probability distributionsp _j over each other's pure strategies systematically in a continuous manner until all of these probability distributions will converge, in an appropriate sense, to a specific equilibrium points ^* of Γ, which, then, will be accepted as solution. A mathematical procedure, to be called thetracing procedure, is proposed to provide a mathematical representation for this intellectual process of convergent expectations. Two variants of this procedure are described. One, to be called thelinear tracing procedure, is shown to define a unique solution in “almost all” cases but not quite in all cases. The other variant, to be called thelogarithmic tracing procedure, always defines a unique solution in all possible cases. Moreover, in all cases where the linear procedure yields a unique solution at all, both procedures always yield the same solution. For any given game Γ, the solution obtained in this way heavily depends on the prior probability distributionsp ₁,...,p _n used as a starting point for the tracing procedure. In the last section, the results of the tracing procedure are given for a simple class of two-person variable-sum games, in numerical detail.     

Centroids,Representations, and Submodular Flows

The notion of centroid of a tree is generalized to apply to an arbitrary intersecting family of sets. Centroids are used to construct a compact representation for any intersecting family of sets, as well as any crossing family. The size of the representation for a family on n elements is O(n²), compared to size O(n³) for previous representations. Efficient algorithms to construct the representation are given. For example on a network of n vertices and m edges, the representation of all minimum cuts uses O(m log(n²/m)) space; it is constructed in O(nm log(n²/m)) time (this is the best-known time for finding one minimum cut). The representation is used to improve several submodular flow algorithms. For example a minimum-cost dijoin is found in time O(n²m); as a result a minimum-cost planar feedback are set is found in time O(n³). The previous best-known time bounds for these two problems are both a factor n larger.     

On defining sets of vertices of the hypercube by linear inequalities

This paper shows that for any subset S of vertices of the n-dimensional hypercube, ind(S)≤2^n?1, where ind(S) is the minimum number of linear inequalities needed to define S. Furthermore, for any k in the range 1≤k≤2^n?1, there is an S with ind(S) = k, with the defining inequalities taken as canonical cuts. Other related results are included, and all are proven by explicit constructions of the sets S or explicit definitions of such sets by linear inequalities.The paper is aimed at researchers in bivalent programming, since it provides upper bounds on the performance of algorithms which combine several linear constraints into one, even when the given constraints have a particularly simple form.     

On the time to first failure in multicomponent exponential reliability systems

Consider an n-component reliability system having the property that at any time each of its components is either up (i.e., working) or down (i.e., being repaired). Each component acts independently and we suppose that each time the i^th component goes up it remains up for an exponentially distributed time having mean μ_i, and each time it goes down it remains down for an exponentially distributed time having mean υ_i. We further suppose that whether or not the system itself is up at any time depends only on which components are up at that time. We are interested in the distribution of the time of first system failure when all components are initially up at time zero. In section 2 we show that this distribution has the NBU (i.e., new better than used) property, and in Section 3 we make use of this and other results to obtain a lower bound to the mean time until first system failure.     

Properties of P-sets and trapped compact convex sets

New properties of P-sets, which constitute a large class of convex compact sets in ? ⁿ that contains all convex polyhedra and strictly convex compact sets, are obtained. It is shown that the intersection of a P-set with an affine subspace is continuous in the Hausdorff metric. In this theorem, no assumption of interior nonemptiness is made, unlike in other known intersection continuity theorems for set-valued maps. It is also shown that if the graph of a set-valued map is a P-set, then this map is continuous on its entire effective set rather than only on the interior of this set. Properties of the so-called trapped sets are also studied; well-known Jung’s theorem on the existence of a minimal ball containing a given compact set in ? ⁿ is generalized. As is known, any compact set contains n + 1 (or fewer) points such that any translation by a nonzero vector takes at least one of them outside the minimal ball. This means that any compact set is trapped in the minimal ball. Compact sets trapped in any convex compact sets, rather than only in norm bodies, are considered. It is shown that, for any compact set A trapped in a P-set M ? ? ⁿ , there exists a set A ⁰ ? A trapped in M and containing at most 2n elements. An example of a convex compact set M ? ? ⁿ for which such a finite set A ⁰ ? A does not exist is given.     

The topological type of the α-sections of convex sets

An α=(α₁,…,αk)(0?αi?1) section of a family {K₁,…,Kk} of convex bodies in Rd is a transversal halfspace H⁺ for which Vold(Ki∩H⁺)=αi⋅Vold(Ki) for every 1?i?k. Our main result is that for any well-separated family of strictly convex sets, the space of α-sections is diffeomorphic to Sd−k.     

Derivations of non-separable C1-algebras

The question of which C¹-algebras have only inner derivations has been considered by a number of authors for 25 years. The separable case is completely solved, so this paper deals only with the non-separable case. In particular, we show that the C¹-tensor product of a von Neumann algebra and an abelian C¹-algebra has only inner derivations. Other special types of C¹-algebras are shown to have only inner derivations as well such as the C¹-tensor product of L(H) (all bounded operators on separable Hilbert space) and any separable C¹-algebra having only inner derivations. Derivations from a smaller C¹-algebra into a larger one are also considered, and this concept is generalized to include derivations between C¹-algebras connected by a ¹-homomorphism. Finally, we consider the general problem of a sequence of linear functionals on a C¹-algebra which converges to zero (in norm) when restricted to any abelian C¹-subalgebra. Does such a sequence converge to zero in norm? The answer is “yes” for normal functionals on L(H), but unknown in general.     

Monochromatic simplices of any volume

We give a very short proof of the following result of Graham from 1980: For any finite coloring of R^d, d≥2, and for any α>0, there is a monochromatic (d+1)-tuple that spans a simplex of volume α. Our proof also yields new estimates on the number A=A(r) defined as the minimum positive value A such that, in any r-coloring of the grid points Z² of the plane, there is a monochromatic triangle of area exactly A.     

Quasi-median hulls in Hamming space are Steiner hulls

A Hamming space Λⁿ consists of all sequences of length n over an alphabet Λ and is endowed with the Hamming distance. In particular, any set of aligned DNA sequences of fixed length constitutes a subspace of a Hamming space with respect to mismatch distance. The quasi-median operation returns for any three sequences u,v,w the sequence which in each coordinate attains either the majority coordinate from u,v,w or else (in the case of a tie) the coordinate of the first entry, u; for a subset of Λⁿ the iterative application of this operation stabilizes in its quasi-median hull. We show that for every finite tree interconnecting a given subset X of Λⁿ there exists a shortest realization within Λⁿ for which all interior nodes belong to the quasi-median hull of X. Hence the quasi-median hull serves as a Steiner hull for the Steiner problem in Hamming space.     

The method of boundary integral equations for a mixed scattering problem

In this paper we consider the scattering of an electromagnetic time-harmonic plane wave by an infinite cylinder having an open arc and a bounded domain in R² as cross section. To this end, we solve a scattering problem for the Helmholtz equation in R² where the scattering object is a combination of a crack Γ and a bounded obstacle D, and we have Dirichlet-impedance type boundary condition on Γ and Dirichlet boundary condition on ∂D (∂D∈C²). Applying potential theory, the problem can be reformulated as a boundary integral system. We establish the existence and uniqueness of a solution to the system by using the Fredholm theory.