Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions

In this paper we develop an integer-affine classification of three-dimensional multistory, completely empty convex marked pyramids. We apply it to obtain the complete lists of compact two-dimensional faces of multidimensional continued fractions lying in planes at integer distances 2, 3, 4, …  to the origin. The faces are considered up to the action of the group of integer-linear transformations.     

Pyroelectric effect in layered magnetoelectric PZT/Ni–Zn ferrite composites

The electric response of layered 2-2 connectivity magnetoelectric composites to the action of a modulated heat flow, which is detected by a dynamic method, is studied. The kinetics of the pyroelectric response of the composite material to a pulsed thermal action is calculated. The experimental results and calculation data suggest that a modulated heat action induces a signal due to both the pyroelectric and piezoelectric effects. The latter effect is caused by the mechanical interaction of the layers of lead zirconate titanate ceramic and nickel–zinc (Ni–Zn) ferrite.     

On Tori Triangulations Associated with Two-Dimensional Continued Fractions of Cubic Irrationalities

The notion of equivalence of multidimensional continued fractions is introduced. We consider some properties and state some conjectures related to the structure of the family of equivalence classes of two-dimensional periodic continued fractions. Our approach to the study of the family of equivalence classes of two-dimensional periodic continued fractions leads to revealing special subfamilies of continued fractions for which the triangulations of the torus (i.e., the combinatorics of their fundamental domains) are subjected to clear rules. Some of these subfamilies are studied in detail; the way to construct other similar subfamilies is indicated.     

Determination of periods of geometric continued fractions for two-dimensional algebraic hyperbolic operators

An explicit construction of a reduced hyperbolic integer operator from the group SL(2, ?) such that one of the periods of the corresponding geometric continued fraction in the sense of Klein coincides with a given sequence of positive integers is presented. An algorithm determining periods for any operator in SL(2, ?) (which is based on Gauss’ reduction theory) is experimentally studied.     

Three examples of three-dimensional continued fractions in the sense of Klein

The problem of the investigation of the simplest n-dimensional continued fraction in the sense of Klein for $n?2$ was posed by V. Arnold. The answer for the case $n=2$ can be found in the works of E. Korkina (1995) and G. Lachaud (1995). In present Note we study the case $n=3$. To cite this article: O. Karpenkov, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

On examples of difference operators for {0,1}-valued functions over finite sets

V.I. Arnold recently formulated a geometric concept of monads and used it to study difference operators on the sets of {0,1}-valued sequences of length n. We consider particular examples of these monads and indicate one question that arises. Partially supported by grants NWO-RFBR 047.011.2004.026 (RFBR 05-02-89000-NWO_a), RFBR SS-1972.2003.1, RFBR 05-01-02805-CNRSL_a, and RFBR 05-01-01012a.     

Magnetocaloric effect and magnetic phase transitions in nanocrystalline rare-earth metals: Tb, Dy, and Gd

The magnetocaloric effect of rare-earth metals (REMs) Gd, Tb, and Dy in the nanostructured state is investigated. The ranges of working temperatures and cool capacity for materials based on nanocrystalline REMs are calculated from the experimental data. These results enable us to evaluate in detail the magnetic properties of REMs in a nanocrystalline state. It is shown that nanocrystalline Dy possesses the largest cool capacity, making this material most effective for magnetic cooling in the temperature range of 82–134 K.