Effect of the magnetic field from the accelerating system on the beam dynamics in the cyclotron

Magnetic components of the HF field in a cyclotron can change the transverse beam size and the phase width of bunches during acceleration. As a consequence, the phase portraits of the beam at the entrance to the electrostatic deflector and thus the efficiency of the extraction system can change. Expressions are obtained for the components of the magnetic field from the HF system in the vicinity of the cyclotron median plane. The effect of this field on the beam parameters during acceleration and at the entrance to the electrostatic deflector is shown by numerical simulation. For the C235 proton cyclotron, the beam parameter variation is as large as 50%.     

Global Optimization: Local Minima and Transition Points

We consider the minimization of smooth functions of the Euclidean space with a finite number of stationary points having moderate asymptotic behavior at infinity. The crucial role of transition points of first order (i.e., saddle points of index 1) is emphasized. It is shown that (generically) any two local minima can be connected via an alternating sequence of local minima and transition points of first order. In particular, the graph with local minima as its nodes and first order transition points representing the edges turns out to be connected (Theorem A). On the other hand, any connected (finite) graph can be realized in the above sense by means of a smooth function of three variables having a minimal number of stationary points (Theorem B).     

Criteria for efficiency in vector optimization

We consider unconstrained finite dimensional multi-criteria optimization problems, where the objective functions are continuously differentiable. Motivated by previous work of Brosowski and da Silva (1994), we suggest a number of tests (TEST 1–4) to detect, whether a certain point is a locally (weakly) efficient solution for the underlying vector optimization problem or not. Our aim is to show: the points, at which none of the TESTs 1–4 can be applied, form a nowhere dense set in the state space. TESTs 1 and 2 are exactly those proposed by Brosowski and da Silva. TEST 3 deals with a local constant behavior of at least one of the objective functions. TEST 4 includes some conditions on the gradients of objective functions satisfied locally around the point of interest. It is formulated as a Conjecture. It is proven under additional assumptions on the objective functions, such as linear independence of the gradients, convexity or directional monotonicity. This work was partially supported by grant 55681 of the CONACyT.     

Zero-field incommensurate spin-Peierls phase with interchain frustration in TiOCl

We report on the magnetic, thermodynamic, and optical properties of the quasi-one-dimensional quantum antiferromagnets TiOCl and TiOBr, which have been discussed as spin-Peierls compounds. The observed deviations from canonical spin-Peierls behavior, e.g., the existence of two distinct phase transitions, have been attributed previously to strong orbital fluctuations. This can be ruled out by our optical data of the orbital excitations. We show that the frustration of the interchain interactions in the bilayer structure gives rise to incommensurate order with a subsequent lock-in transition to a commensurate dimerized state. In this way, a single driving force, the spin-Peierls mechanism, induces two separate transitions.