Tangent Hyperplane Research Articles

Biharmonic Hypersurfaces of LP-Sasakian Manifolds

Biharmonic Hypersurfaces of LP-Sasakian ManifoldsIn this paper the biharmonic hypersurfaces of Lorentzian para-Sasakian manifolds are studied. We firstly find the biharmonic equation for a hypersurface which admits the characteristic vector field of the Lorentzian para-Sasakian as the normal vector field. We show that a biharmonic spacelike hypersurface of a Lorentzian para-Sasakian manifold with constant mean curvature is minimal. The biharmonicity condition for a hypersurface of a Lorentzian para-Sasakian manifold is investigated when the characteristic vector field belongs to the tangent hyperplane of the hypersurface. We find some necessary and sufficient conditions for a timelike hypersurface of a Lorentzian para-Sasakian manifold to be proper biharmonic. The nonexistence of proper biharmonic timelike hypersurfaces with constant mean curvature in a Ricci flat Lorentzian para-Sasakian manifold is proved.

The cardinality of the separated vertex set of a multidimensional cube

An n-dimensional cube and the sphere inscribed into it are considered. The conjecture of A. Ben-Tal, A. Nemirovski, and C. Roos states that each tangent hyperplane to the sphere strictly separates not more than 2n−2 cube vertices. In this paper this conjecture is proved for n ≤ 6. New examples of hyperplanes separating exactly 2n−2 cube vertices are constructed for any n. It is proved that hyperplanes orthogonal to radius vectors of cube vertices separate less than 2n−2 cube vertices for n ≥ 3.

PROJECT Method for Multiobjective Optimization Based on Gradient Projection and Reference Points

In this paper, we propose a new interactive method for multiobjective programming (MOP) called the PROJECT method. Interactive methods in MOP are techniques that can help the decision maker (DM) to generate the most preferred solution from a set of efficient solutions. An interactive method should be capable of capturing the preferences of the DM in a pragmatic and comprehensive way. In certain decision situations, it may be easier and more reliable for DMs to follow an interactive process for providing local tradeoffs than other kinds of preferential information like aspiration levels, objective function classification, etc. The proposed PROJECT method belongs to the class of interactive local tradeoff methods. It is based on the projection of utility function gradients onto the tangent hyperplane of an efficient set and on a new local search procedure that inherits the advantages of the reference-point method to search for the best compromise solution within a local region. Most of the interactive methods based on local tradeoffs assume convexity conditions in a MOP problem, which is too restrictive in many real-life applications. The use of a reference-point procedure makes it possible to generate any efficient solutions, even the nonsupported solutions or efficient solutions located in the nonconvex part of the efficient frontier of a nonconvex MOP problem. The convergence of the proposed method is investigated. A nonlinear example is examined using the new method, as well as a case study on efficiency analysis with value judgements. The proposed PROJECT method is coded in Microsoft Visual C++ and incorporated into the software PROMOIN (Interactive MOP).

Rectifiability of Sets of Finite Perimeter in Carnot Groups: Existence of a Tangent Hyperplane

We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G then, for almost every x∈G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups: they show in Math. Ann. 321, 479–531, 2001 and J. Geom. Anal. 13, 421–466, 2003 that, for almost every x, E has a unique tangent at x, and this tangent is a vertical halfspace.

Analysis of the constraint proposal method for two-party negotiations

In the constraint proposal method a mediator locates points at which the two decision makers have joint tangent hyperplanes. We give conditions under which these points are Pareto optimal and we prove that under these conditions the mediator’s problem has a solution. In practice, the mediator adjusts a hyperplane going through a reference point until the decision makers’ most preferred alternatives on the hyperplane coincide. We give local convergence conditions for fixed-point iteration as an adjustment process. We also discuss the relationship of exchange economies and the constraint proposal method, and the possible ways of using the method.

Computation of probabilities of survival/failure of technical, economic systems/structures by means of piecewise linearization of the performance function

The computation of the probability of survival/failure of technical/economic structures and systems is based on an appropriate performance or so-called (limit) state function separating the safe and unsafe states in the space of random model parameters. Starting with the survival conditions, hence, the state equation and the condition for the admissibility of states, an optimizational representation of the state function can be given in terms of the minimum value function of a closely related minimization problem. Selecting a certain number of boundary points of the safe/unsafe domain, hence, on the limit state surface, the safe/unsafe domain is approximated by a convex polyhedron. This convex polyhedron is defined by the intersection of the half spaces in the parameter space generated by the tangent hyperplanes to the safe/unsafe domain at the selected boundary points on the limit state surface. The approximative probability functions are then defined by means of the resulting probabilistic linear constraints in the parameter space. After an appropriate transformation, the probability distribution of the parameter vector can be assumed to be normal with zero mean vector and unit covariance matrix. Working with separate linear constraints, approximation formulas for the probability of survival of the structure are obtained immediately. More exact approximations are obtained by considering joint probability constraints. In a second approximation step, these approximations can be evaluated by using probability inequalities and/or discretizations of the underlying probability distribution.

Reliability Analysis of Technical Systems/Structures by means of Polyhedral Approximation of the Safe/Unsafe Domain

AbstractReliability analysis of technical structures and systems is based on an appropriate (limit) state function separating the safe and unsafe/states in the space of random parameters. Starting with the survival conditions, hence, the state equation and the condition for the admissibility of states, an optimizational representation of the state function can be given in terms of the minimum function of a certainminimization problem. Selecting a certain number of boundary points of the safe/unsafe domain, hence, on the limit state surface, the safe/unsafe domain is approximated by a convex polyhedron defined by the intersection of the half spaces in the parameter space generated by the tangent hyperplanes to the safe/unsafe domain at the selected boundary points on the limit state surface. The resulting approximative probability functions are then defined by means of probabilistic linear constraints in the parameter space, where, after an appropriate transformation, the probability distribution of the parameter vector can be assumed to be normal with zero mean vector and unit covariance matrix. Working with separate linear constraints, approximation formulas for the probability of survival of the structure are obtained immediately. More exact approximations are obtained by considering joint probability constraints, which, in a second approximation step, can be evaluated by using probability inequalities and/or discretization of the underlying probability distribution. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An Exit Probability Approach to Solving High Dimensional Dirichlet Problems

We present an approach to solving high dimensional Dirichlet problems in bounded domains which is based on a variant of the Feynman--Kac formula that connects solutions to elliptic partial differential equations and functional integration. We integrate the resulting system of stochastic differential equations numerically with the Euler scheme. To correct for possible intermediate excursions of the simulated paths and to find good approximations for first exit times, we extend to higher dimensions a strategy which we introduced in [F. M. Buchmann, J. Comput. Phys., 202 (2005), pp. 446-462] for stopping problems in one dimension. In addition, for Dirichlet problems, good approximations of the first exit points are needed to evaluate the boundary condition. To this end, we sample a random point on the tangent hyperplane of the boundary of the domain once a first exit is estimated. To detect excursions and adequately compute first exit times, we apply the same local half-space approximation. We therefore assume that the domain is smooth enough such that these approximations are possible. Numerical experiments in dimensions up to 128 show that resulting approximations are of very high quality. In particular, we observe that first order convergence behavior of the Euler scheme can be maintained.

Corank 1 Singularities of Stable Smooth Maps and Special Tangent Hyperplanes to a Space Curve

Let γ be a smooth generic curve in ℝP3. Denote by C the number of its flattening points, and by T the number of planes tangent to γ at three distinct points. Consider the osculating planes to γ at the flattening points. Let N denote the total number of points where γ intersects these osculating plane transversally. Then T ≡ [N + θ(γ)C]/2 (mod 2), where θ(γ) is the number of noncontractible components of γ. This congruence generalizes the well-known Freedman theorem, which states that if a smooth connected closed generic curve in ℝ3 has no flattening points, then the number of its triple tangent planes is even. We also give multidimensional analogs of this formula and show that these results follow from certain general facts about the topology of codimension 1 singularities of stable maps between manifolds having the same dimension.

Dually Degenerate Varieties and the Generalization of a Theorem of Griffiths?Harris

The dual variety X * for a smooth n-dimensional variety X of the projective space PN is the set of tangent hyperplanes to X. In the general case, the variety X * is a hypersurface in the dual space (PN)*. If dim X *<N−1, then the variety X is called dually degenerate. The authors refine these definitions for a variety X⊂PN with a degenerate Gauss map of rankr. For such a variety, in the general case, the dimension of its dual variety X * is N−l−1, where l=n−r, and X is dually degenerate if dim X *<N−l−1. In 1979 Griffiths and Harris proved that a smooth variety X⊂PN is dually degenerate if and only if all its second fundamental forms are singular. The authors generalize this theorem for a variety X⊂PN with a degenerate Gauss map of rankr.

Singularities of centre symmetry sets

The center symmetry set (CSS) of a smooth hypersurface S in an affine space Rn is the envelope of lines joining pairs of points where S has parallel tangent hyperplanes. The idea stems from a definition of Janeczko, in an alternative version due to Giblin and Holtom. For n = 2 the envelope is always real, while for n > 3 the existence of a real envelope depends on the geometry of the hypersurface. In this paper we make a local study of the CSS, some results applying to n ⩽ 5 and others to the cases n = 2,3. The method is to construct a generating function whose bifurcation set contains the CSS and possibly some other redundant components. Focal sets of smooth hypersurfaces are a special case of the construction, but the CSS is an affine and not a euclidean invariant. Besides the familiar local forms of focal sets there are other local forms corresponding to boundary singularities, and yet others which do not appear to have arisen elsewhere in a geometrical context. There are connections with Finsler geometry. This paper concentrates on the theory and the proof of the local normal forms for the CSS. 2000 Mathematics Subject Classification 57R45, 58K40, 32S25, 58B20.

Remarks on the Segre cubic

The Igusa quartic 3-fold is the moduli space of Abelian surfaces with the level two structure, and it is known that the tangent hyperplanes cut out Kummer surfaces. In this note, we show similar results for the Segre cubic.

Asymptotic Shape of the Erlang Capacity Region of a Multiservice Shared Resource

We consider a loss model of an unbuffered resource having C channels, which are shared by several different types of service connections. Connections of each type arrive in a Poisson stream and request a number of channels, which depends on the type. An arriving connection is blocked and lost if there are not enough free channels. Otherwise, the channels are held for the duration of the connection, and the holding period is generally distributed. It is assumed that C and the traffic intensities are proportionately large. The admission control problem is considered for specified upper bounds on the blocking probabilities, and the boundary of the admissible set is investigated asymptotically. The results are derived by investigating the local behavior with respect to the tangent hyperplane at a point on the boundary of the admissible set. The lowest order results that hold in the asymptotic limit $C \to \infty$ are given first. Importantly, the boundary is linear for the critically loaded and overloaded reg...

On the range of the derivatives of a smooth function between Banach spaces

We study the size of the range of the derivatives of a smooth function between Banach spaces. We establish conditions on a pair of Banach spacesX andY to ensure the existence of a C p smooth (Fr echet smooth or a continuous G^ ateaux smooth) function f from X onto Y such that f vanishes outside a bounded set and all the derivatives off are surjections. In particular we deduce the following results. For the G^ ateaux case, whenX andY are separable andX is innite-dimensional, there exists a continuous G^ ateaux smooth functionf fromX toY , with bounded support, so that f 0 (X )= L(X;Y ). In the Fr echet case, we get that if a Banach spaceX has aF r echet smooth bump and densX = dens L(X;Y ), then there is a Fr echet smooth function f:X! Y with bounded support so that f 0 (X )= L(X;Y ). Moreover, we see that if X has a C p smooth bump with bounded derivatives and densX = dens L m (X;Y ) then there exists another C p smooth function f:X! Y so that f (k) (X )= L k(X;Y ) for all k =0 ; 1;:::;m. As an application, we show that every bounded starlike body on a separable Banach space X with a (Fr echet or G^ ateaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes ll the dual space X. In the non-separable case, we prove that X has such property if X has smooth partitions of unity.

From local to global behavior in competitive Lotka-Volterra systems

In this paper we exploit the linear, quadratic, monotone and geometric structures of competitive Lotka-Volterra systems of arbitrary dimension to give geometric, algebraic and computational hypotheses for ruling out non-trivial recurrence. We thus deduce the global dynamics of a system from its local dynamics. The geometric hypotheses rely on the introduction of a split Liapunov function. We show that if a system has a fixed point p ∈ int ⁡ R + n p\in \operatorname {int}{{\mathbf R}^n_+} and the carrying simplex of the system lies to one side of its tangent hyperplane at p p , then there is no nontrivial recurrence, and the global dynamics are known. We translate the geometric hypotheses into algebraic hypotheses in terms of the definiteness of a certain quadratic function on the tangent hyperplane. Finally, we derive a computational algorithm for checking the algebraic hypotheses, and we compare this algorithm with the classical Volterra-Liapunov stability theorem for Lotka-Volterra systems.

Asymptotic shape of the Erlang capacity region of a multi-service shared resource

We consider a loss model of an unbuffered resource having C channels, which are shared by several different types of service connections. Connections of each type arrive in a Poisson stream and request a number of channels, which depends on the type. An arriving connection is blocked and lost if there are not enough free channels. Otherwise, the channels are held for the duration of the connection, and the holding period is generally distributed. It is assumed that C and the traffic intensities are proportionately large. The admission control problem is considered for specified upper bounds on the blocking probabilities, and the boundary of the admissible set is investigated asymptotically. Characterization of admissible sets is extremely useful, not only for connection-level admission control, which is the context in which this topic has typically been considered in the past, but also for higher level objectives, such as network economics, network design, and network control. The asymptotic view of the admissible set is particularly appropriate for the higher level objectives, where the fine details are not as important as the qualitative properties of the shape of the set and tractability of the numerical calculations for large systems. Our results are derived by investigating the local behavior with respect to the tangent hyperplane at a point on the boundary of the admissible set. The lowest order results that hold in the asymptotic limit C→∞ are given first. Importantly, the boundary is linear for the key critically loaded and also for the overloaded regimes, and weakly convex for the underloaded regime. Next, refined results that hold for C≫1 are given, which indicate that the boundary is not convex, although only slightly so.

Singularities of Families of Chords

A chord is a straight line joining two points of a pair of hypersurfaces in an affine space such that the tangent hyperplanes at these points are parallel. We classify the singularities of envelopes of the families of chords determined by generic pairs of plane curves and surfaces in three-space. The list contains all bifurcation diagrams of simple boundary singularities (of the corresponding multiplicity).

On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space

We study the size of the sets of gradients of bump functions on the Hilbert space ℓ 2 , and the related question as to how small the set of tangent hyperplanes to a smooth bounded starlike body in ℓ 2 can be. We find that those sets can be quite small. On the one hand, the usual norm of the Hilbert space ℓ 2 can be uniformly approximated by C 1 smooth Lipschitz functions ψ so that the cones generated by the ranges of its derivatives ψ ' (ℓ 2 ) have empty interior. This implies that there are C 1 smooth Lipschitz bumps in ℓ 2 so that the cones generated by their sets of gradients have empty interior. On the other hand, we construct C 1 -smooth bounded starlike bodies A⊂ℓ 2 , which approximate the unit ball, so that the cones generated by the hyperplanes which are tangent to A have empty interior as well. We also explain why this is the best answer to the above questions that one can expect.

Boolean Representation of Manifolds and Functions

It is shown that a smooth n dimensional manifold with a boundary in R^n admits a Boolean representation in terms of closed half spaces defined by the tangent hyperplanes at the points on its boundary. A similar result is established for smooth functions on closed convex domains in R^n. The latter result is considered as a form of the Legendre transformation.

Weakly defective varieties

A projective variety X X is ‘ k k -weakly defective’ when its intersection with a general ( k + 1 ) (k+1) -tangent hyperplane has no isolated singularities at the k + 1 k+1 points of tangency. If X X is k k -defective, i.e. if the k k -secant variety of X X has dimension smaller than expected, then X X is also k k -weakly defective. The converse does not hold in general. A classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension. We start this classification here, describing all weakly defective irreducible surfaces. Our method also provides a new proof of the classical Terracini’s classification of k k -defective surfaces.