Projections in a normed linear space and a generalization of the pseudo-inverse

The concept of a "projection function" in a finite-dimensional real or complex normed linear space H (the function P_M which carries every element into the closest element of a given subspace M) is set forth and examined.

If dim M = dim H - 1, then P_M is linear. If P_N is linear for all k-dimensional subspaces N, where 1 ≤ k < dim M, then P_M is linear.

The projective bound Q, defined to be the supremum of the operator norm of P_M for all subspaces, is in the range 1 ≤ Q < 2, and these limits are the best possible. For norms with Q = 1, P_M is always linear, and a characterization of those norms is given.

If H also has an inner product (defined independently of the norm), so that a dual norm can be defined, then when P_M is linear its adjoint P_M^H is the projection on (kernel P_M)^⊥ by the dual norm. The projective bounds of a norm and its dual are equal.

The notion of a pseudo-inverse F⁺ of a linear transformation F is extended to non-Euclidean norms. The distance from F to the set of linear transformations G of lower rank (in the sense of the operator norm ∥F - G∥) is c/∥F⁺∥, where c = 1 if the range of F fills its space, and 1 ≤ c < Q otherwise. The norms on both domain and range spaces have Q = 1 if and only if (F⁺)⁺ = F for every F. This condition is also sufficient to prove that we have (F⁺)^H = (F^H)⁺, where the latter pseudo-inverse is taken using dual norms.

In all results, the real and complex cases are handled in a completely parallel fashion.

A pseudo-spin model for the wall of cytoskeletal microtubule

A pseudo-spin model is intended to describe the physical dynamics of unbound electrons in the wall of cytoskeletal microtubule (MT). Due to the inherent symmetry of the structure and the electric properties in the MT, one may treat it as a one-dimensional ferroelectric system, and describe the nonlinear dynamics of dimer electric dipoles in one protofilament of the MT by virtue of the double-well potential. Consequently, the physical problem has been mapped onto the pseudo-spin system, and the mean-field approximation has been taken to get some physical results.

Optimizing the Curie temperature of pseudo-binary RxR2-x ' Fe-17 (R, R ' = rare earth) for magnetic refrigeration

Several pseudo-binary RxR2-x'Fe-17 alloys (with R = Y, Ce, Pr, Gd and Dy) were synthesized with rhombohedral Th2Zn17-type crystal structure determined from x-ray and neutron powder diffraction. The choice of compositions was done with the aim of tuning the Curie temperature (T-C) in the 270 +/- 20 K temperature range, in order to obtain the maximum magneto-caloric effect around room temperature. The investigated compounds exhibit broad isothermal magnetic entropy changes, Delta S-M(T), with moderate values of the refrigerant capacity, even though the values of Delta S-M(Peak) are relatively low compared with those of the R2Fe17 compounds with R = Pr or Nd. The reduction on the Delta S-M(Peak) is explained in terms of the diminution in the saturation magnetization value. Furthermore, the Delta S-M(T) curves exhibit a similar caret-like behavior, suggesting that the magneto-caloric effect is mainly governed by the Fe-sublattice. A single master curve for Delta S-M/Delta S-M(Peak)(T) under different values of the magnetic field change are obtained for each compound by rescaling of the temperature axis.