Lattice Points on Similar Figures and Conics

Let us say that a plane figure F satisfies Steinhaus’ condition if for any positive integer n, there exists a figure F _n similar to F which satisfies the condition |F_n?\mathbb Z²|=n{|F_n\cap{\mathbb Z}^2|=n}. For example, the circular disc satisfies Steinhaus’ condition. We prove that every compact convex region in the plane \mathbb R²{\mathbb R^2} satisfies Steinhaus’ condition. As for plane curves, it is known that the circle satisfies Steinhaus’ condition. We consider Steinhaus’ condition for other conics, and present several results.     

High heat generation ability in AC magnetic field for nano-sized magnetic Y3Fe5O12 powder prepared by bead milling

Nano-sized magnetic Y₃Fe₅O₁₂ ferrite having a high heat generation ability in an AC magnetic field was prepared by bead milling. A commercial powder sample (non-milled sample) of ca. 2.9 μm in particle size did not show any temperature enhancement in the AC magnetic field. The heat generation ability in the AC magnetic field improved with a decrease in the average crystallite size for the bead-milled Y₃Fe₅O₁₂ ferrites. The highest heat ability in the AC magnetic field was for the fine Y₃Fe₅O₁₂ powder with a 15-nm crystallite size (the samples were milled for 4 h using 0.1 mm? beads). The heat generation ability of the excessively milled Y₃Fe₅O₁₂ samples decreased. The main reason for the high heat generation property of the milled samples was ascribed to an increase in the Néel relaxation of the superparamagnetic material. The heat generation ability was not influenced by the concentration of the ferrite powder. For the samples milled for 4 h using 0.1 mm? beads, the heat generation ability (W g⁻¹) was estimated using a 3.58×10⁻⁴ fH² frequency (f/kHz) and the magnetic field (H/kA m⁻¹), which is the highest reported value of superparamagnetic materials.     

Orthogonal projections of cubes and regular simplices into a plane

We show that for every \({k\ge 2}\) and \({n\ge k}\), there is an \({n}\)-dimensional unit cube in \({\mathbb{R}^n}\) which is mapped to a regular \({2k}\)-gon by an orthogonal projection in \({\mathbb{R}^n}\) onto a \({2}\)-dimensional subspace. Moreover, by increasing dimension \({n}\), arbitrary large regular \({2k}\)-gon can be obtained in such a way. On the other hand, for every \({m\ge 3}\) and \({n\ge m-1}\), there is an \({n}\)-dimensional regular simplex of unit edge in \({\mathbb{R}^n}\) which is mapped to a regular \({m}\)-gon by an orthogonal projection onto a plane. Moreover, contrary to the cube case, arbitrary small regular \({m}\)-gon can be obtained in such a way, by increasing dimension \({n}\).     

A few applications of negative- type inequalities

Applying the negative-type inequalities for square Euclidean distance, we present (1) a parallelatope theorem (a generalization of the parallelogram theorem), (2) a short proof of Rankin’s theorem for the maximum number of dispersed points on a sphere, and (3) a proof of impossibility of a certain geometric embedding for some graphs.     

Arranging Solid Balls to Represent a Graph

 By solid balls, we mean a set of balls in R ³ no two of which can penetrate each other. Every finite graph G can be represented by arranging solid balls in the following way: Put red balls in R ³, one for each vertex of G, and connect two red balls by a chain when they correspond to a pair of adjacent vertices of G, where a chain means a finite sequence of blue solid balls in which each consecutive balls are tangent. (We may omit the chain if the two red balls are already tangent.) The ball number b(G) of G is the minimum number of balls (red and blue) necessary to represent G. If we put the balls and chains on a table so that all balls sit on the table, then the minimum number of balls for G is denoted by b _T (G). Among other things, we prove that b(K ₆)=8,b(K ₇)=13 and b _T (K ₅)=8,b _T (K ₆)=14. We also prove that c ₁ n ³<b(K _n )<c ₂ n ³ log n, c ₃ n ⁴ log n<b _T (K _n )<c ₄ n ⁴. Received: March 29, 1999 Final version received: January 17, 2000     

Reflecting a triangle in the plane

We prove that if the three angles of a triangleT in the plane are different from (60°, 60°, 60°), (30°, 30°, 120°), (45°,45°,90°),(30°,60°,90°), then the set of vertices of those triangles which are obtained fromT by repeating ‘edge-reflection’ is everywhere dense in the plane.     

A fluorescent molecular switch driven by the input sequence of metal cations: an azamacrocyclic ligand containing bipolar anthracene fragments

An azamacrocyclic ligand (L) containing two anthracene (AN) fragments connected through two triethylenetetramine bridges has been synthesized, in which each of the bridges can coordinate with one metal cation. The effects of pH and metal cations (Zn2+ and Cd2+) on the emission properties of L were studied in water. Without metal cations, L does not show any emission at basic pH values. The addition of Zn2+ leads to the production of excimer emission, which is due to a static excimer formed by direct excitation of the intramolecular ground-state dimer of the bipolar AN fragments that approach each other by Zn2+ binding. In contrast, Cd2+ addition does not result in excimer emission because the Cd2+-AN pi complex, formed by donation of a pi electron of the AN fragments to the adjacent Cd2+, suppresses pi-stacking interactions of the AN fragments. The most notable feature is the appearance of excimer emission controlled by the input sequence of metal cations: Zn2+-->Cd2+ sequential addition (each one equivalent) allows excimer emission, whereas the reverse sequence (Cd2+-->Zn2+) does not. In the Zn2+-->Cd2+ sequence, Cd2+ coordination is structurally restricted by the first Zn2+ coordination with the other polyamine bridge, leading to the formation of a weak Cd2+-AN pi complex. In contrast, for the reverse sequence, the first Cd2+ coordination forms a stable Cd2+-AN pi complex, which is not weakened by sequential Zn2+ coordination, resulting in no excimer emission.