Generic Finiteness Research Articles

Generic finiteness of 3-canonical maps of varieties of Albanese fiber dimension one

Generic finiteness of 3-canonical maps of varieties of Albanese fiber dimension one

Compactness and generic finiteness for free boundary minimal hypersurfaces, I

Given a compact Riemannian manifold with boundary, we prove that the space of embedded, which may be improper, free boundary minimal hypersurfaces with uniform area and Morse index upper bound is compact in the sense of smoothly graphical convergence away from finitely many points. We show that the limit of a sequence of such hypersurfaces always inherits a non-trivial Jacobi field when it has multiplicity one. In a forthcoming paper, we will construct Jacobi fields when the convergence has higher multiplicity.

A Generic Finite Element Framework on Parallel Tree-Based Adaptive Meshes

In this work we formally derive and prove the correctness of the algorithms and data structures in a parallel, distributed-memory, generic finite element framework that supports h-adaptivity on computational domains represented as forest-of-trees. The framework is grounded on a rich representation of the adaptive mesh suitable for generic finite elements that is built on top of a low-level, light-weight forest-of-trees data structure handled by a specialized, highly parallel adaptive meshing engine, for which we have identified the requirements it must fulfill to be coupled into our framework. Atop this two-layered mesh representation, we build the rest of data structures required for the numerical integration and assembly of the discrete system of linear equations. We consider algorithms that are suitable for both subassembled and fully-assembled distributed data layouts of linear system matrices. The proposed framework has been implemented within the FEMPAR scientific software library, using p4est as a practical forest-of-octrees demonstrator. A strong scaling study of this implementation when applied to Poisson and Maxwell problems reveals remarkable scalability up to 32.2K CPU cores and 482.2M degrees of freedom. Besides, a comparative performance study of FEMPAR and the state-of-the-art deal.ii finite element software shows at least comparative performance, and at most factor 2-3 improvements in the h-adaptive approximation of a Poisson problem with first- and second-order Lagrangian finite elements, respectively.

Wave interaction with nonlinear damage and generation of harmonics in composite structures

This work presents a generic Finite Element (FE) based computational scheme for quantifying guided wave interaction with Localised Nonlinear Structural Damage (LNSD) within structures of arbitrary layering and geometric complexity. The through-thickness mode-shape of the structure is obtained through a wave and finite element method. This is applied in a time domain FE simulation in order to generate time harmonic excitation for a specific wave mode. Interaction of the wave with LNSD within the system is computed through an element activation and deactivation iteration. The scheme is validated against experimental measurements and a WFE-FE methodology for calculating wave interaction with damage. Case studies for guided wave interaction with crack and delamination are presented to exhibit the robustness of the proposed method in classifying and identifying damage.

Generic Finiteness for a Class of Symmetric Planar Central Configurations of the Six-Body Problem and the Six-Vortex Problem

A symmetric planar central configuration of the Newtonian six-body problem $x$ is called cross central configuration if there are precisely four bodies on a symmetry line of $x$. We use complex algebraic geometry and Groebner basis theory to prove that for a generic choice of positive real masses $m_1,m_2,m_3,m_4,m_5=m_6$ there is a finite number of cross central configurations. We also show one explicit example of a configuration in this class. A part of our approach is based on relaxing the output of the Groebner basis computations. This procedure allows us to obtain upper bounds for the dimension of an algebraic variety. We get the same results considering cross central configurations of the six-vortex problem.

Generic finiteness of equilibrium distributions for bimatrix outcome game forms

We provide sufficient and necessary conditions for the generic finiteness of the number of distributions on outcomes, induced by the completely mixed Nash equilibria associated to a bimatrix outcome game form. These equivalent conditions are stated in terms of the ranks of two matrices constructed from the original game form.

Calculation of guided wave interaction with nonlinearities and generation of harmonics in composite structures through a wave finite element method

The extensive usage of composite materials in modern industrial applications implies a great range of possible structural failure modes for which the structure has to be frequently and thoroughly inspected. Nonlinear guided wave inspection techniques have been continuously gaining attention during the last decade. This is primarily due to their sensitivity to very small sizes of localised damage. A number of complex transformation phenomena take place when an elastic wave impinges on a nonlinear segment, including the generation of higher and sub-harmonics. Moreover, the transmission and reflection coefficients of each wave type become amplitude dependent. In this work, a generic Finite Element (FE) based computational scheme is presented for quantifying guided wave interaction effects with Localised Structural Nonlinearities (LSN) within complex composite structures. Amplitude dependent guided wave reflection, transmission and conversion is computed through a Wave and Finite Element (WFE) method. The scheme couples wave propagation properties within linear structural waveguides to a LSN and is able to compute the generation of higher and sub-harmonics through a harmonic balance projection. A Newton-like iteration scheme is employed for solving the system of nonlinear differential equations. Numerical case studies are presented for waveguides coupled through a joint exhibiting nonlinear mechanical behaviour.

Generic finiteness of minimal surfaces with bounded Morse index

Given a compact 3-manifold N without boundary, we prove that for a bumpy metric of positive scalar curvature the space of minimal surfaces having a uniform upper bound on the Morse index is always finite unless the manifold itself contains an embedded minimal RP^2. In particular, we derive a generic finiteness result whenever N does not contain a copy of RP^3 in its prime decomposition. We discuss the obstructions to any further generalization of such a result. When the metric g is required to be (scalar positive and) strongly bumpy (meaning that all closed, immersed minimal surfaces do not have Jacobi fields, a notion recently proved to be generic by B. White) the same conclusion holds true for any closed 3-manifold.

New equations for central configurations and generic finiteness

We consider the finiteness problem for central configurations of the n n -body problem. We prove that, for n ≥ 4 n\geq 4 , there exists a (Zariski) closed subset B B in the mass space R n \mathbb {R}^{n} , such that if ( m 1 , … , m n ) ∈ R n ∖ B (m_1,\dots ,m_n) \in \mathbb {R}^n\setminus B , then there is a finite number of corresponding classes of ( n − 2 ) (n-2) -dimensional central configurations for potential associated to a semi-integer exponent. Also, we obtain trilinear homogeneous polynomial equations of degree 3 3 for central configurations of fixed dimension and, for each integer k ≥ 1 k \geq 1 , we show that the set of mutual distances associated to a k k -dimensional central configuration is contained in a determinantal algebraic set.

Generic Finiteness of Equilibrium Distributions for Bimatrix Outcome Game Forms

We provide sufficient and necessary conditions for the generic finiteness of the number of distributions on outcomes, induced by the completely mixed Nash equilibria associated to a bimatrix outcome game form. These equivalent conditions are stated in terms of the ranks of two matrices constructed from the original game form.

Determinacy of equilibrium in outcome game forms

We show the generic finiteness of the number of probability distributions on outcomes induced by Nash equilibria for two-person game forms such that either (i) one of the players has no more than two strategies or (ii) both of the players have three strategies, and (iii) for outcome game forms with three players, each with at most two strategies. Finally, we exhibit an example of a game form with three outcomes and three players for which the Nash equilibria of the associated game induce a continuum of payoffs for an open non-empty set of utility profiles.

Infinite computations and the generic finite

This paper introduces the concept of a generic finite set and points out that a consistent and significant interpretation of the grossone, ① notation of Sergeyev is that ① takes the role of a generic natural number. This means that ① is not itself a natural number, yet it can be treated as one and used in the generic expression of finite sets and finite formulas, giving a new power to algebra and algorithms that embody this usage. In this view,N={1,2,3,…,①-2,①-1,①}is not an infinite set, it is a symbolic structure representing a generic finite set. We further consider the concept of infinity in categories. An object A in a given category C is infinite relative to that category if and only if there is a injection J:A⟶A in C that is not a surjection. In the category of sets this recovers the usual notion of infinity. In other categories, an object may be non-infinite (finite) while its underlying set (if it has one) is infinite. The computational methodology due to Sergeyev for executing numerical calculations with infinities and infinitesimals is considered from this categorical point of view.

Determinacy of equilibrium outcome distributions for zero sum and common utility games

We show the generic finiteness of probability distributions induced on outcomes by the Nash equilibria in two player zero sum and common interest outcome games.

Finiteness of spatial central configurations in the five-body problem

We strengthen a generic finiteness result due to Moeckel by showing that the number of spatial central configurations of the Newtonian five-body problem with positive masses is finite, apart from some explicitly given special cases of mass values.

Generic finiteness of outcome distributions for two-person game forms with three outcomes

A two-person game form is given by nonempty finite sets S 1 , S 2 of pure strategies, a nonempty set Ω of outcomes, and a function θ : S 1 × S 2 → Δ ( Ω ) , where Δ ( Ω ) is the set of probability measures on Ω . We prove that if the set of outcomes contains just three elements, generically, there are finitely many distributions on Ω induced by Nash equilibria.

Generic finiteness of equilibrium payoffs for bimatrix games

It is shown that in any affine space of payoff matrices the equilibrium payoffs of bimatrix games are generically finite.

On the generic finiteness of equilibrium outcome distributions in bimatrix game forms

We provide an example of an outcome game form with two players for which there is an open set of utilities for both players such that, in each of the associated games, the set of Nash equilibria induces a continuum of outcome distributions.

Equilibria without the survival assumption

It is well known that an equilibrium in the Arrow–Debreu model may fail to exist if a very restrictive condition called the survival assumption is not satisfied. We study two approaches that allow for the relaxation of this condition. Danilov and Sotskov [Danilov, V.I., Sotskov, A.I., 1990. A generalized economic equilibrium. Journal of Mathematical Economics 19, 341–356], and Florig [Florig, M., 2001. Hierarchic competitive equilibria. Journal of Mathematical Economics 35, 515–546] developed a concept of a generalized equilibrium based on a notion of hierarchic prices. Marakulin [Marakulin, V., 1988. An equilibrium with nonstandard prices and its properties in mathematical models of economy. Discussion Paper No. 18. Institute of Mathematics, Siberian Branch of the USSR Academy of Sciences, Novosibirsk, 51 pp. (in Russian); Marakulin, V., 1990. Equilibrium with nonstandard prices in exchange economies. In: Quandt, R., Triska, D. (Eds.), Optimal Decisions in Market and Planned Economies. Westview Press, London, pp. 268–282] proposed a concept of an equilibrium with non-standard prices. In this paper, we establish the equivalence between non-standard and hierarchic equilibria. Furthermore, we show that for any specified system of dividends the set of such equilibria is generically finite. As a consequence, we have generic finiteness of Mas-Colell’s equilibria with slack, uniform dividend equilibria, and other special cases of our concept.

The generic finiteness of the m-canonical map for 3-folds of general type

Let be a projective minimal threefold of general type with only Q-factorial terminal singularities. We study the generic finiteness of t he -canonial map for such 3-folds. Suppose ( ) 2 and ( ) 3. We prove that the -canonical map is generically finite for 3, which is a supplement to Koll´ ar’s result. Suppose ( ) 5. We prove that the 3-canonical map is generically finite, whic h improves Meng Chen’s result.

Minimal Planes in Hyperbolic Space

We show a generic finiteness result for least area planes in 3-dimensional hyperbolic space. Moreover, we prove that the space of minimal immersions of disk into hyperbolic space is a submanifold of a product bundle over a space of immersions of circle into sphere at infinity. The bundle projection map when restricted to this submanifold is Fredholm of index zero. By using this result, we also show that the space of minimal planes with smooth boundary curve at infinity is a manifold.