Density estimation and adaptive control of markov processes: Average and discounted criteria

We consider a class of discrete-time Markov control processes with Borel state and action spaces, and ^d i.i.d. disturbances with unknown distribution . Under mild semi-continuity and compactness conditions, and assuming that is absolutely continuous with respect to Lebesgue measure, we establish the existence of adaptive control policies which are (1) optimal for the average-reward criterion, and (2) asymptotically optimal in the discounted case. Our results are obtained by taking advantage of some well-known facts in the theory of density estimation. This approach allows us to avoid restrictive conditions on the state space and/or on the system's transition law imposed in recent works, and on the other hand, it clearly shows the way to other applications of nonparametric (density) estimation to adaptive control.Research partially supported by The Third World Academy of Sciences under Research Grant No. MP 898-152.     

A strange pigeon-hole principle

Using Ramsey theory, we establish the following pigeon-hole type principle: From a large number of random variables (functions, vectors, etc.) one can always select two, X and Y, such that P(X < Y) 1/2. We apply the principle for a poset problem.     

Existence of standing waves for dirac fields with singular nonlinearities

We prove the existence of stationary states for nonlinear Dirac equations of the form (E) $$i\sum\limits_{\mu = 0}^3 {\gamma ^\mu \partial _\mu \psi - M\psi + F\left( {\bar \psi \psi } \right)\psi = 0,} $$ whereM>0 andF is a singular self-interaction. In particular, in the model case whereF(s)=?s ^?α, for some 0<α<1, and for every ω>M, there exists a solution of (E) of the form ψ(t, x)=e ^iωt?(x), wherex ₀=t andx=(x ₁,x ₂,x ₃), such that ? has compact support. IF 0<α<1/3, then ? is of classC ¹. If 1/3<α<1, then ? is continuously differentiable, except on some sphere {|x|=R}, where |??| is infinite.     

Yttrium aluminate crystals for slabs

The methods of growing YAG:Nd crystals with a flat interface and YAP:Nd crystals with a sharp conical interface are described. The form of the interface was controlled by the He and H₂O content in the reducing atmosphere composed mainly of Ar and H₂ as well as by axial temperature gradient above the melt level. The crystals of 120–180 mm in length were machined to the high-quality slabs.     

Diffractive dissociation reaction np → Λ0K0p for incident neutron momenta between 30 and 70 GeV/c BIS-2 collaboration

We present results of the exclusive diffractive reaction np ⁰K⁰p at 0·08 < |t|< 0·53 GeV²/c² and momenta between 30–70 GeV/c at the Serpukhov neutron beam. Thet-dependence is exp (Bt) withB= 11·4± 2·3 (GeV/c)^–2 and it exhibits the diffractive minimum at ¦t¦ 0·28 GeV²/c² for ⁰K⁰ massesM < 1·8 GeV/c².B decreases with increasingM. Correlation with the beam direction is observed in the rest frame of the ⁰K⁰ system: the cross section rises for the forward-going ⁰. The rise is more pronounced for higherM. The cross section of reaction np ⁰K⁰p for 0<|t|< 0·53 GeV²/c² is 10·3 ± 2·5 ± 2·8 b.     

Finitely additive states and completeness of inner product spaces

For any unit vector in an inner product space S, we define a mapping on the system of all -closed subspaces of S, F(S), whose restriction on the system of all splitting subspaces of S, E(S), is always a finitely additive state. We show that S is complete iff at least one such mapping is a finitely additive state on F(S). Moreover, we give a completeness criterion via the existence of a regular finitely additive state on appropriate systems of subspaces. Finally, the result will be generalized to general inner product spaces.     

Radiative kinks and fermionic zero modes in a (λφ6)1+1 model

We consider bidimensional scalar models including kink solutions _k (x). Using the hidden supersymmetric properties of the Dirac equation, we describe a general method to find normalizable fermionic zero modes. In particular, we apply the technique to a ( ⁶)₁₊₁ model. Going to the one-loop order of the effective potential, the emergence of a radiative kink provides an interesting scalar background in order to discuss the Dirac equation.