We consider spherical data Xi noised by a random rotation εi ∈ SO(3) so that only the sample Zi = εiXi, i = 1, . . . , N is observed. We define a nonparametric test procedure to distinguish H0 : ”the density f of Xi is the uniform density f0 on the sphere” and H1 : ”‖f − f0‖2 ≥ CψN and f is in a Sobolev space with smoothness s”. For a noise density fε with smoothness index ν, we show that an ad...

We consider a nonparametric model E, generated by independent observations Xi, i = 1, ..., n, with densities p(x, θi), i = 1, ..., n, the parameters of which θi = f(i/n) ∈ Θ are driven by the values of an unknown function f : [0, 1]→ Θ in a smoothness class. The main result of the paper is that, under regularity assumptions, this model can be approximated, in the sense of the Le Cam deficiency ...

Starting with a paper of Jacobson form the 1960s, many authors became interested in characterizing all algebraic number fields in which each integer is the sum of pairwise distinct units. Although there exist many partial results for number fields of low degree, a full characterization of these number fields is still not available. Narkiewicz and Jarden posed an analogous question for sums of u...

The possibility of reconciliation between canonical probability distributions obtained from the q-maximum entropy principle with predictions from the law of large numbers when empirical samples are held to the same constraints, is investigated into. Canonical probability distributions are constrained by both: (i) the additive duality of generalized statistics and (ii) normal averages expectatio...

We consider the magnetic NLS equation (−εi∇+A(x)) u+ V (x)u = K(x) |u| u, x ∈ R , where N ≥ 3, 2 < p < 2∗ := 2N/(N − 2), A : R → R is a magnetic potential and V : R → R, K : R → R are bounded positive potentials. We consider a group G of orthogonal transformations of R and we assume that A is G-equivariant and V , K are G-invariant. Given a group homomorphism τ : G → S into the unit complex num...

The antibiotics time machine is an optimization question posed by Mira et al. on the design of antibiotic treatment plans to minimize antibiotic resistance. The problem is a variation of the Markov decision process. These authors asked if the problem can be solved efficiently. In this paper, we show that this problem is NP-hard in general. 1. Problem statement and results Resistance to antibiot...

Limit distribution is studied for the integrated squared error of the projection regression estimator (2) constructed on the basis of independent observations (1). By means of the obtained limit theorems, a test is given for verifying the hypothesis on the regression, and the power of this test is calculated in the case of Pitman alternatives. Let observations Y1, Y2, . . . , Yn be represented ...

Starting from the well-known quantum Miura transformation for the Lie algebra An, we compute explicitly the OPEs for n = 3 and 4. The primary fields with spin 3, 4 and 5 are found (for general n). By using these primary fields and the OPEs from quantum Miura transformation, we derive the complete structure of the nonlinear W4 and W5 algebras. It is known that the quantum Miura transformation fo...

Weconsidermultiple resonance scatteringwith complete frequency redistribution (CFR) in a semi-infinite conservative atmosphere (photon destruction probability εI = 0) with the sources at infinite depth. The polarization arising in resonance scattering is completely accounted for. The problemweconsider is the resonance-scattering counterpart of the ChandrasekharSobolev problem of Rayleigh scatte...

The nonparametric regression originated in mathematical statistics offers an original approach to signal processing problems (e.g., [1], [2]). It basically results in linear Þltering with the linear Þlters designed using some moving window local approximations. In many applications like speech recognition or image denoising, nonlinear or locally adaptive methods have been shown to be more effic...