Characteristic vector fields of generic distributions of corank 2

We study generic distributions D⊂TM$D\subset TM$ of corank 2 on manifolds M   of dimension n?5$n?5$. We describe singular curves of such distributions, also called abnormal curves. For n   even the singular directions (tangent to singular curves) are discrete lines in D(x)$D(x)$, while for n   odd they form a Veronese curve in a projectivized subspace of D(x)$D(x)$, at generic x∈M$x\in M$. We show that singular curves of a generic distribution determine the distribution on the subset of M where they generate at least two different directions. In particular, this happens on the whole of M if n is odd. The distribution is determined by characteristic vector fields and their Lie brackets of appropriate order. We characterize pairs of vector fields which can appear as characteristic vector fields of a generic corank 2 distribution, when n is even.     

Successional stability of vector fields in dimension three

A topological invariant, the community transition graph, is introduced for dissipative vector fields that preserve the skeleton of the positive orthant. A vector field is defined to be successionally stable if it lies in an open set of vector fields with the same community transition graph. In dimension three, it is shown that vector fields for which the origin is a connected component of the chain recurrent set can be approximated in the Whitney topology by a successionally stable vector field.

     

Geometric modeling and validation of twist drills with a generic point profile

Traditionally, twist drills with a few specific point geometry, such as planar, conical, cylindrical, ellipsoidal or hyperboloidal, have been designed and adapted for specific applications. Using CAD, the point geometry can be given a generic definition which will enhance the freedom to design drills with different point profiles and optimize them for multiple objectives. Such a definition can also be used for several downstream applications. This paper presents a methodology to model the twist drills with generic point geometry using NURBS. To begin with, a detailed basic model for a fluted twist drill with sectional geometry made up of arcs and straight lines has been presented in terms of bi-parametric surface patches. The coordinates of cutting lips and chisel edge of the drill have been obtained as solution to a surface-curve intersection problem using optimization algorithm. Subsequently, the model has been generalized by employing NURBS to represent the curves whereby the cutting edges and angles can be altered simply by changing the control points or their respective weights. Using this methodology, the generic definitions of the conventional angles on the drill point have been derived and presented. The proposed model has been illustrated in MATLAB environment and validated experimentally for a conical and an arbitrary point geometry. The experiments show a good conformity with the theoretical evaluations.     

Three-Dimensional Topological Loops with Solvable Multiplication Groups

We prove that each 3-dimensional connected topological loop L having a solvable Lie group of dimension ≤5 as the multiplication group of L is centrally nilpotent of class 2. Moreover, we classify the solvable non-nilpotent Lie groups G which are multiplication groups for 3-dimensional simply connected topological loops L and dim G ≤ 5. These groups are direct products of proper connected Lie groups and have dimension 5. We find also the inner mapping groups of L.     

Asymptotically good towers of function fields with small p-rank

Over any quadratic finite field we construct function fields of large genus that have simultaneously many rational places, small p-rank, and many automorphisms.     

Minimal unit vector fields with respect to Riemannian natural metrics

Let $(M,g)$ be a Riemannian manifold. We denote by $\stackrel{?}{G}$ an arbitrary Riemannian g-natural metric on the unit tangent sphere bundle $T_{1}M$, such metric depends on four real parameters satisfying some inequalities. The Sasaki metric, the Cheeger–Gromoll metric and the Kaluza–Klein metrics are special Riemannian g-natural metrics. In literature, minimal unit vector fields have been already investigated, considering $T_{1}M$ equipped with the Sasaki metric ${\stackrel{?}{G}}_S$ [12]. In this paper we extend such characterization to an arbitrary Riemannian g-natural metric $\stackrel{?}{G}$. In particular, the minimality condition with respect to the Sasaki metric ${\stackrel{?}{G}}_S$ is invariant under a two-parameters deformation of the Sasaki metric. Moreover, we show that a minimal unit vector field (with respect to $\stackrel{?}{G}$) corresponds to a minimal submanifold. Then, we give examples of minimal unit vector fields (with respect to $\stackrel{?}{G}$). In particular, we get that the Hopf vector fields of the unit sphere, the Reeb vector field of a K-contact manifold, and the Hopf vector field of a quasi-umbilical hypersurface with constant principal curvatures in a Kähler manifold, are minimal unit vector fields (with respect to $\stackrel{?}{G}$).     

On a certain sum-product estimate in fields of prime order

Let Fp be the finite field of p elements with p prime.If A is a subset of Fp and g is an element of F*p with order ν,then max{|A + g·A|,|A·A|} (ν/(ν + |A|2) )1/12|A|13/12.     

Impulsive control systems with commutative vector fields

We consider variational problems with control laws given by systems of ordinary differential equations whose vector fields depend linearly on the time derivativeu=(u ¹,...,u ^m ) of the controlu=(u ¹,...,u ^m ). The presence of the derivativeu, which is motivated by recent applications in Lagrangian mechanics, causes an impulsive dynamics: at any jump of the control, one expects a jump of the state.The main assumption of this paper is the commutativity of the vector fields that multiply theu . This hypothesis allows us to associate our impulsive systems and the corresponding adjoint systems to suitable nonimpulsive control systems, to which standard techniques can be applied. In particular, we prove a maximum principle, which extends Pontryagin's maximum principle to impulsive commutative systems.     

Sharp Hardy-Leray inequality for curl-free fields with a remainder term

In this paper, we give a new and a simpler approach to the result in [8] concerning the best constant of Hardy-Leray inequality for curl-free fields. As a by-product, we obtain an improved inequality with a remainder term. The non-attainability of the best constant is an easy consequence of the new inequality. The proof is based on a decomposition of curl-free fields into radial and spherical parts.     

On the index of finite determinacy of vector fields with resonances

In this paper we study normal forms for a class of germs of 1-resonant vector fields on R^n with mutually different eigenvalues which may admit extraneous resonance relations. We give an estimation on the index of finite determinacy from above as well as the essentially simplified polynomial normal forms for such vector fields. In the case that a vector field has a zero eigenvalue, the result leads to an interesting corollary, a linear dependence of the derivatives of the hyperbolic variables on the central variable.     

On vector fields with simply connected trajectories and one invariant line

We study polynomial vector fields X on ${\mathbb{C}}^2$ which have simply connected trajectories and satisfy $dP(X)=a?P$, for a constant $a\in {\mathbb{C}}^{?}$ and a primitive polynomial $P\in \mathbb{C}[x,y]$. We determine X, up to an algebraic change of coordinates. In particular, we obtain that X is complete.     

Measure solutions for impulsive evolution equations with measurable vector fields

In this paper we present some results on the question of existence of generalized or measure valued solutions for semilinear impulsive evolution equations on Banach spaces with the nonlinear parts being merely measurable and bounded on bounded sets. This is a far reaching generalization of the previous results of the author and others. It may be interesting to consider extension of the results of this paper to cover impulsive differential inclusions and their potential applications to control theory and uncertain dynamics.     

A nullstellensatz and a positivstellensatz for ordered differential fields

We use the model completeness and axiomatisation of the theory of closed ordered differential fields to give a differential version of Dubois, Krivine and Risler's nullstellensatz and Stengle's positivstellensatz for ordered fields.     

Geometry of quasilinear hyperbolic systems with characteristic fields of constant multiplicity

In this paper, we study the regularity of the eigenvalues and eigenvectors and the existence of normalized coordinates for quasilinear hyperbolic systems with characteristic fields of constant multiplicity. We prove that the eigenvalues and eigenvectors of the system have the same regularity as the coefficients of the system. On the other hand, we show that, for the quasilinear hyperbolic system of conservation laws with characteristic fields of constant multiplicity, the normalized coordinates exist on the domain under consideration.     

A sequence of one-point codes from a tower of function fields

We construct a sequence of one-point codes from a tower of function fields whose relative minimum distances have a positive limit. Our tower is characterized by principal divisors. We determine completely the minimum distance of the codes from the first field of our tower. These results extend those of Stichtenoth [IEEE Trans Inform Theory (1988), 34(15):1345–1348], Yang and Kumar [Lecture Notes in Mathematics, 1518, (1991), Springer-Verlag, Berlin Hidelberg New York, pp. 99–107], and Garcia [Comm. Algebra, 20(12): 3683–3689]. As an application, we show that the minimum distance corresponds to the Feng–Rao bound.     

Topological Duality for Boolean Algebras with a Normal <Emphasis Type="Italic">n</Emphasis>-ary Monotonic Operator

In this paper we shall give a topological duality for Boolean algebras endowed with an n-ary monotonic operator (BAMOs). The dual spaces of BAMOs are structures of the form , such that is a Boolean space, and R is a relation between X and a finite sequences of non-empty closed subsets of X. By means of this duality we shall characterize the equivalence relations of the dual space of a BAMO A that correspond biunivocally to subalgebras of A. We shall prove that there exist bijective correspondences between the lattice of congruences, the lattice of closed filters, and the lattice of certain closed subsets of the dual space of a BAMO. These correspondences are used to study the simple and the subdirectly irreducible algebras.      

Fourth-order problems with fully nonlinear boundary conditions

Differential inequality method, bounding function method and topological degree are applied to obtain the existence criterions of at least one solution for the general fourth-order differential equations under nonlinear boundary conditions, and many existing results are complemented.     

Homology of the Lie algebra of vector fields on the line with coefficients in symmetric powers of its adjoint representation

We compute the homology of the Lie algebra W ₁ of (polynomial) vector fields on the line with coefficients in symmetric powers of its adjoint representation. We also list the results obtained so far for the homology with coefficients in tensor powers and, in turn, use them for partially computing the homology of the Lie algebra of W ₁-valued currents on the line.