local controllability of scalar input systems on 3 manifolds abstract local controllability of generic c k pairs of vector fields in a three dimensional connected manifold m is studied showing that in general a necessary condition is also sufficient on an open dense subset of the set of points where it is verified the results obtained are used to determine local controllability for scalar input affine control systems

transcendence of hecke operators in the big hecke algebra we prove a transcendence result of hecke operator t l over the ring generated by diamond operators over in a non cm component of the cyclotomic ordinary big p adic hecke algebra and we discuss its conjectural implication of a modulo p version to the vanishing of the cyclotomic invariant 2014

every ordinary symplectic isogeny class in positive characteristic is dense in the moduli we prove that any ordinary symplectic separable isogeny class in the moduli space of principally polarized abelian varieties over a field of positive characteristic is dense in the zariski topology

asymptotic stabilization of a generic class of three dimensional homogeneous quadratic systems abstract we give necessary and sufficient conditions for the asymptotic stabilizability of a generic class of single input three dimensional homogeneous quadratic systems

controllability of bilinear systems abstract the goal of this paper is to find conditions for the controllability of homogeneous in the state bilinear systems in state spaces of dimensions two and three the cases where the system matrices of such systems generate a lie algebra equal to the dimension of the state space are considered i e where the corresponding strictly bilinear system is controllable for the two dimensional case it turns out that there are only two situations when the strictly bilinear system is controllable but the homogeneous in the state bilinear system is not by checking the vector fields on the boundaries of the reachable sets obtained in these two situations one can easily determine whether the homogeneous in the state bilinear system is controllable or not provided the lie algebra of the system matrices has rank two this approach is extended to the three dimensional case and thus previously obtained results can be generalized

controllability of a class of underactuated mechanical systems with symmetry in this paper we develop results based on geometric mechanics to study the controllability of a class of controlled under actuated left invariant mechanical systems on lie groups we exploit the invariance of the controlled nonlinear dynamics to the group action symmetry to derive a set of reduced dynamics for the system we first present sufficient conditions for the controllability of the reduced dynamics we prove conditions boundedness of coadjoint orbits and existence of a radially unbounded lyapunov function under which the drift vector field of the reduced system is weakly positively poisson stable wpps the wpps nature of the drift vector field along with the lie algebra rank condition is used to show controllability of the reduced system we then extend these results to the unreduced dynamics considering separately the cases when the symmetry group is compact and noncompact in the noncompact case under further assumptions of equilibrium controllability sufficient conditions for controllability of the unreduced dynamics are derived jetpuck hovercraft and underwater vehicles are used as mechanical systems to motivate our work and illustrate our theoretical results

on the initial value formulation of classical electrodynamics we describe a seemingly un noticed feature of the text book maxwell lorentz system of classical electrodynamics which challenges its formulation in terms of an initial value problem for point charges even after appropriate renormalization we demonstrate that most of the generic initial data evolves to develop singularities in the electromagnetic fields along the light cones of the initial charge positions we provide explicit formulas for the corresponding fields demonstrate how this phenomenon renders the initial value problem ill posed and show how such bad initial data can be ruled out by extra conditions in addition to the maxwell constraints these extra conditions however require knowledge of the history of the solution and as we discuss effectively turn the maxwell lorentz system into a system of delay equations much like the fokker schwarzschild tetrode equations for extended charges such singular light fronts persist in a smoothened form and as we argue yield physically doubtful solutions our results also apply to some extent to expectation values of field operators in quantum field theory

controllability properties for finite dimensional quantum markovian master equations various notions from geometric control theory are used to characterize the behavior of the markovian master equation for n level quantum mechanical systems driven by unitary control and to describe the structure of the sets of reachable states it is shown that the system can be accessible but neither small time controllable nor controllable in finite time in particular if the generators of quantum dynamical semigroups are unital then the reachable sets admit easy characterizations as they monotonically grow in time the two level case is treated in detail

image of λ adic galois representations modulo p let p 5 be a prime if an irreducible component of the spectrum of the big ordinary hecke algebra does not have complex multiplication under mild assumptions we prove that the image of its mod p galois representation contains an open subgroup of open image in new window for the canonical weight variable t this fact appears to be deep as it is almost equivalent to the vanishing of the invariant of the kubota leopoldt p adic l function and the anticyclotomic katz p adic l function another key ingredient of the proof is the anticyclotomic main conjecture proven by rubin mazur tilouine

hecke fields of analytic families of modular forms we make finitenessconjectureson the composite of hecke fields of classicalmembers of a p adic analyticfamily of slope 0 elliptic modular forms in the verticalcase with fixed level varying weight in the horizontal case fixed weight varying p power level we prove the corresponding statements

controllability of spacecraft systems in a central gravitational field the configuration space for rigid spacecraft systems in a central gravitational field can be modeled by so 3 spl times ir sup 3 where the special orthogonal group so 3 represents the attitude dynamics and ir sup 3 is for the orbital motion the attitude dynamics of the spacecraft system is affected by the orbital elements through the well known gravity gradient torque on the other hand the effects of attitude orbit coupling can also possibly be used to alter orbital motions by controlling the attitude this controllability property is the primary issue of this paper the control systems for spacecraft with either reaction wheels or gas jets being used as attitude controllers are proven in this study to be controllable rigorously establishing these results necessitates starting with the formal definitions of controllability and poisson stability it is then shown that if the drift vector field of the system is weakly positively poisson stable and the lie algebra rank condition is satisfied controllability can be concluded the hamiltonian structure of the spacecraft model provides a natural route of verifying the property of weakly positive poisson stability accordingly the controllability is obtained after confirming the lie algebra rank condition developing a methodology in deriving lie brackets in the tangent space of t so 3 spl times ir sup 3 i e the second tangent bundle is thus deemed necessary a general formula is offered for the computation of lie brackets of second tangent vector fields in tt so 3 sup m spl times ir sup n in light of its importance in the fields of mechanics robotics optimal control and nonlinear control among others with these tools the controllability results can be proved the analysis in this paper gives some insight into the attitude orbit coupling effects and may potentially lead towards new techniques in designing controllers for large spacecraft systems

a finiteness property of abelian varieties with potentially ordinary good reduction a g dimensional abelian variety a f over a number field f is of gl 2 type if end a f end a f z q contains a field e of degree g we call such a field e an endomorphism field of a we say that an f simple abelian variety a f over a number field f of dimension g is non cm if end a f f q does not contain any semi simple commutative algebra of degree 2g over q cf acm if an abelian variety a f of gl 2 type is f simple d end a f is a division algebra with a positive involution since d has a maximal commutative subfield stable under we may assume that its endomorphism field e is totally real or a cm field the galois action on the tate module of an f simple abelian variety a f of gl 2 type with endomorphism field e produces a two dimensional strictly compatible system of galois representations a gal q f gl2 e indexed by primes of e thus we have its l function l s a two f simple abelian varieties a and b a are twist equivalent if l s b l s a for a finite order character gal q f q note that the dimension is possibly unbounded over a twist equivalent class since tate s conjecture has been proven by faltings for abelian varieties one could formulate this equivalence by insisting that the two abelian varieties share a simple component over an abelian extension of f in this more geometric context q simple abelian varieties have been studied in depth as q simple factors of modular jacobians by ribet for example see r and his papers quoted there however we adopt an analytic definition of twistequivalence using their l function as it can also be applied to rank 2 q motives for which the tate conjecture is still unknown here the identity of two l functions

on the controllability of quadratic systems in r 3 in this paper we consider quadratic systems in r3 that is systems of the type x t q x t ub x t r3 u r b r3 where q is a map from r3 in r3 all whose components are homogeneous quadratic forms we give a necessary and sufficient algebraic condition of complete controllability for these systems

the iwasawa mu invariant of p adic hecke l functions for an odd prime p we compute the invariant of the anticyclotomic katz p adic l function of a p ordinary cm field if the conductor of the branch character is a product of primes split over the maximal real subfield except for rare cases where the root number of the p adic functional equation is congruent to 1 modulo p the invariant vanishes

on the synge equations in a three dimensional two body problem of classical electrodynamics abstract our goal is to obtain an existence result for the synge equations describing the motion of two charged particles within the framework of classical electrodynamics the one dimensional case was recognized by r d driver as a system of functions differential equations whose delays depend on the unknown trajectories of the charged particles while in the one dimensional case the equations of motion form a retarded functional differential system in the three dimensional case they form a neutral functional differential system the main difficulty arises in the existence theory of neutral systems here we propound a fixed point approach to solving the problem based on the theory of densifying mappings in locally convex spaces the fixed point theorem obtained includes as a particular case a corresponding theorem of b n sadovskii

electrodynamic two body problem for prescribed initial data on a straight line abstract electrodynamic interaction between point charges can be described by a system of odes involving advanced and retarded delays the so called fokker schwarzschild tetrode fst equations in special situations approximate equations can be derived which are purely retarded upon omission of the terms describing radiation friction these are called synge equations in both cases few mathematical results are available on existence and uniqueness of solutions we investigate the situation of two like point charges in 3 1 space time dimensions restricted to motion on a straight line we give a priori estimates on the asymptotic motion and using a leray schauder argument prove 1 existence of solutions to the fst equations on the future or past half line given finite trajectory segments 2 global existence of the synge equations for cauchy data 3 global existence of a fst toy model furthermore we give a sufficient criterion that uniquely distinguishes solutions by means of finite trajectory segments