Necessary Conditions for the Global Rigidity of Direction–Length Frameworks

It is an intriguing open problem to give a combinatorial characterisation or polynomial algorithm for determining when a graph is globally rigid in ℝ ^d . This means that any generic realisation is uniquely determined up to congruence when each edge represents a fixed length constraint. Hendrickson gave two natural necessary conditions, one involving connectivity and the other redundant rigidity. In general, these are not sufficient, but they do suffice in two dimensions, as shown by Jackson and Jordán. Our main result is an analogue of the redundant rigidity condition for frameworks that have both direction and length constraints. For any generic globally rigid direction-length framework in ℝ ^d with at least 2 length edges, we show that deleting any length edge results in a rigid framework. It seems harder to obtain a corresponding result when a direction edge is deleted: we can do this in two dimensions, under an additional hypothesis that we believe to be unnecessary. Our proofs use a lemma of independent interest, stating that a certain space parameterising equivalent frameworks is a smooth manifold. We prove this lemma using arguments from differential topology and the Tarski–Seidenberg theorem on semi-algebraic sets.     

Local Rings of Bounded Cohen–Macaulay Type

Let be a complete local Cohen–Macaulay (CM) ring of dimension one. It is known that R has finite CM type if and only if R is reduced and has bounded CM type. Here we study the one-dimensional rings of bounded but infinite CM type. We will classify these rings up to analytic isomorphism (under the additional hypothesis that the ring contains an infinite field). In the first section we deal with the complete case, and in the second we show that bounded CM type ascends to and descends from the completion. In the third section we study ascent and descent in higher dimensions and prove a Brauer–Thrall theorem for excellent rings. Presented by J. HerzogMathematics Subject Classifications (2000) 13C05, 13C14, 13H10.     

Bounded convergence theorem for abstract Kurzweil–Stieltjes integral

Monatshefte für Mathematik - In the theories of Lebesgue integration and of ordinary differential equations, the Lebesgue Dominated Convergence Theorem provides one of the most widely used...     

Higher Laplace–Beltrami Operators on Bounded Symmetric Domains

It was conjectured by the first author and Peetre that the higher Laplace–Beltrami operators generate the whole ring of invariant operators on bounded symmetric domains. We give a proof of the conjecture for domains of rank ≤ 6 by using a graph manipulation of K¨ahler curvature tensor. We also compute higher order terms in the asymptotic expansions of the Bergman kernels and the Berezin transform on bounded symmetric domain.     

A Two-on-One Linear Pursuit–Evasion Game with Bounded Controls

A linearized engagement with two pursuers versus a single evader is considered, in which the adversaries’ controls are bounded and have first-order dynamics and the pursuers’ intercept times are equal. Wishing to formulate the engagement as a zero-sum differential game, a suitable cost function is proposed and validated, and the resulting optimization problem and its solution are presented. Construction and analysis of the game space is shown, and the players’ closed-form optimal controls are derived for the case of two “strong” pursuers. The results are compared to those of a 1-on-1 engagement with a “strong” pursuer, and it is shown that the addition of a second pursuer enlarges the capture zone and introduces a new singular zone to the game space, in which the pursuers can guarantee equal misses, regardless of the evader’s actions. Additionally, it is concluded that in the regular zones the closed-form optimal pursuit strategies are unchanged compared to two 1-on-1 engagements, whereas the optimal evasion strategy is more complex. Several simulations are performed, illustrating the adversaries’ behavior in different regions of the game space.     

Fomenko–Zieschang Invariants of Topological Billiards Bounded by Confocal Parabolas

A topological (Liouville) classification of integrable billiards in locally flat compact domains bounded by arcs of confocal parabolas is obtained by methods of Fomenko–Zieschang invariants of integrable systems theory.     

Optimal Control of First-Order Hamilton–Jacobi Equations with Linearly Bounded Hamiltonian

We consider the optimal control of solutions of first order Hamilton–Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove existence of minimizers to this optimization problem as in a relaxed setting and characterize the minimizers as weak solutions to a mean field game type system of coupled partial differential equations. Furthermore, we prove existence and partial uniqueness of weak solutions to the PDE system. An interpretation in terms of mean field games is also discussed.     

Inexact Alternating Direction Methods of Multipliers with Logarithmic–Quadratic Proximal Regularization

In the literature, it was shown recently that the Douglas–Rachford alternating direction method of multipliers can be combined with the logarithmic-quadratic proximal regularization for solving a class of variational inequalities with separable structures. This paper studies the inexact version of this combination, where the resulting subproblems are allowed to be solved approximately subject to different inexactness criteria. We prove the global convergence and establish worst-case convergence rates for the derived inexact algorithms.     

The Nesterov–Todd Direction and Its Relation to Weighted Analytic Centers

The subject of this paper concerns differential-geometric properties of the Nesterov–Todd search direction for linear optimization over symmetric cones. In particular, we investigate the rescaled asymptotics of the associated flow near the central path. Our results imply that the Nesterov–Todd direction arises as the solution of a Newton system defined in terms of a certain transformation of the primal-dual feasible domain. This transformation has especially appealing properties which generalize the notion of weighted analytic centers for linear programming.     

The Grünwald–Letnikov Fractional-Order Derivative with Fixed Memory Length

Contrary to integer-order derivative, the fractional-order derivative of a non-constant periodic function is not a periodic function with the same period. As a consequence of this property, the time-invariant fractional-order systems do not have any non-constant periodic solution unless the lower terminal of the derivative is ±∞, which is not practical. This property limits the applicability of the fractional derivative and makes it unfavorable, for a wide range of periodic real phenomena. Therefore, enlarging the applicability of fractional systems to such periodic real phenomena is an important research topic. In this paper, we give a solution for the above problem by imposing a simple modification on the Grünwald–Letnikov definition of fractional derivative. This modification consists of fixing the memory length and varying the lower terminal of the derivative. It is shown that the new proposed definition of fractional derivative preserves the periodicity.     

Direction of Hopf bifurcation in a delayed Lotka–Volterra competition diffusion system

This paper is concerned with a delayed Lotka–Volterra two species competition diffusion system with a single discrete delay and subject to homogeneous Dirichlet boundary conditions. The main purpose is to investigate the direction of Hopf bifurcation resulting from the increase of delay. By applying the implicit function theorem, it is shown that the system under consideration can undergo a supercritical Hopf bifurcation near the spatially inhomogeneous positive stationary solution when the delay crosses through a sequence of critical values.     

Local Convergence Properties of Douglas–Rachford and Alternating Direction Method of Multipliers

The Douglas–Rachford and alternating direction method of multipliers are two proximal splitting algorithms designed to minimize the sum of two proper lower semi-continuous convex functions whose proximity operators are easy to compute. The goal of this work is to understand the local linear convergence behaviour of Douglas–Rachford (resp. alternating direction method of multipliers) when the involved functions (resp. their Legendre–Fenchel conjugates) are moreover partly smooth. More precisely, when the two functions (resp. their conjugates) are partly smooth relative to their respective smooth submanifolds, we show that Douglas–Rachford (resp. alternating direction method of multipliers) (i) identifies these manifolds in finite time; (ii) enters a local linear convergence regime. When both functions are locally polyhedral, we show that the optimal convergence radius is given in terms of the cosine of the Friedrichs angle between the tangent spaces of the identified submanifolds. Under polyhedrality of both functions, we also provide conditions sufficient for finite convergence. The obtained results are illustrated by several concrete examples and supported by numerical experiments.     

Bounded Extremal and Cauchy–Laplace Problems on the Sphere and Shell

In this work, we develop a theory of approximating general vector fields on subsets of the sphere in ℝ ⁿ by harmonic gradients from the Hardy space H ^p of the ball, 1<p<∞. This theory is constructive for p=2, enabling us to solve approximate recovery problems for harmonic functions from incomplete boundary values. An application is given to Dirichlet–Neumann inverse problems for n=3, which are of practical importance in medical engineering. The method is illustrated by two numerical examples.     

Bounded immune response in immunotherapy described by the deterministic delay Kirschner–Panetta model

We describe restrictions on the parameters of the delay Kirschner–Panetta model which has bounded immune system behavior.     

Γ-inverses of Bounded Linear Operators

Let B(H) be the algebra of all the bounded linear operators on a Hilbert space H.For A,P and Q in B(H),if there exists an operator X∈ B(H) such thatAP X QA=A,X QAP X=X,(QAP X)*=QAP X and(X QAP)*=X QAP,then X is said to be the Γ-inverse of A associated with P and Q,and denoted by AP,Q+.In this note,we present some necessary and su?cient conditions for which A+P,Qexists,and give an explicit representation of AP,Q+(if AP,Q+exists).     

New Complexity Analysis of the Primal–Dual Method for Semidefinite Optimization Based on the Nesterov–Todd Direction

Interior-point methods for semidefinite optimization have been studied intensively in recent times, due to their polynomial complexity and practical efficiency. In this paper, first we present some technical results about symmetric matrices. Then, we apply these results to give a unified analysis for both large update and small update interior-point methods for SDP based on the Nesterov–Todd (NT) direction.     

Bounded differential forms,generalized Milnor–Wood inequality and an application to deformation rigidity

We establish sufficient conditions for a cohomology class of a discrete subgroup Γ of a connected semisimple Lie group with finite center to be representable by a bounded differential form on the quotient by Γ of the associated symmetric space; furthermore if \(\rho : \Gamma\to\mathrm{PU}(1,q)\) is any representation of any discrete subgroup Γ of SU (1, p), we give an explicit closed bounded differential form on the quotient by Γ of complex hyperbolic space which is a representative for the pullback via ρ of the Kähler class of PU(1,q). If G,G′ are Lie groups of Hermitian type, we generalize to representations \(\rho : \Gamma\to G'\) of lattices Γ < G the invariant defined in [Burger, M., Iozzi, A.: Bounded cohomology and representation variates in PU (1,n). Preprint announcement, April 2000] for which we establish a Milnor–Wood type inequality. As an application we study maximal representations into PU(1, q) of lattices in SU(1,1).     

Équation Périodique de Navier–Stokes sans Viscosité dans une Direction

ABSTRACT

We study the periodic Navier–Stokes equation when the vertical viscosity vanishes, in a critical space (invariant by scaling). We shall prove local-in-time existence of the solution. When the tridimensional part of the initial data is small compared with the bidimesional part and with the horizontal viscosity, we shall show global existence of solutions.     

Bounded Turning in Möbius Structures

Siberian Mathematical Journal - We study a&nbsp;Möbius-invariant generalization, called BTR, of the classical property of bounded turning in a&nbsp;metric space which was introduced by...