Smallest maximal snakes of translates of convex domains

LetK be a convex domain. A maximal snake of sizen is a set of non-overlapping translatesK ₁, ...,K _N ofK, whereK _i touchesK _j if and only if |i –j|=1 and no translate ofK can touchK ₁ orK _n without intersecting an additionalK _i ,i=1, ...,n. The size of the smallest maximal snake is proved to be 11 ifK is a parallelogram and to be 10 otherwise.     

Primitive Inner Illuminating Systems for Convex Bodies

A set X of boundary points of a (possibly unbounded) convex body KE ^d illuminating K from within is called primitive if no proper subset of X still illuminates K from within. We prove that for such a primitive set X of an unbounded, convex set KE ^d (distinct from a cone) one has X=2 if d=2, X6 if d=3, and that there is no upper bound for X if d4.     

Sections and projections of homothetic convex bodies

For a pair of convex bodies K₁ and K₂ in Euclidean space , n ≥ 3, possibly unbounded, we show that K₁ is a translate of K₂ if either of the following conditions holds: (i) the orthogonal projections of K₁ on 2-dimensional planes are translates of the respective orthogonal projections of K₂, (ii) there are points p₁ ∈K₁ and p₂ ∈K₂ such that for every pair of parallel 2-dimensional planesL₁and L₂ through p₁ and p₂, respectively, the section K₁ ∩ L₁is a translate of K₂ ∩ L₂.     

On Grünbaum's Conjecture about Inner Illumination of Convex Bodies

In 1964 Grünbaum conjectured that any primitive set illuminating from within a convex body in E ^d , d ≥ 3 , has at most 2 ^d points. This was confirmed by V. Soltan in 1995 for the case d = 3 . Here we give a negative answer to Grünbaum's conjecture for all d ≥ 4 , by constructing a convex body K ⊂ E ^d with primitive illuminating sets of an arbitrarily large cardinality. Received December 1, 1997, and in revised form January 21, 1999.     

In vitro selection and characterization of deoxyribonucleic acid aptamers for digoxin

The low therapeutic index of digoxin necessitates careful monitoring of its serum levels. Most of digoxin immunoassays suffer from interferences with digoxin-like immunoreactive substances. Since aptamers have been shown to be highly specific for their targets, the aim of this study was to develop DNA aptamers for this widely used cardiac glycoside. Digoxin was coated onto the surface of streptavidin magnetic beads. DNA aptamers against digoxin were designed using Systematic Evolution of Ligands by Exponential enrichment method (SELEX) by 11 iterative rounds of incubation of digoxin-coated streptavidin magnetic beads with synthetic DNA library, DNA elution, electrophoresis and PCR amplification. The PCR product was cloned and sequenced. Binding affinity was determined using digoxin–BSA conjugate, coated onto ELISA plate. Inhibitory effect of anti-digoxin aptamer was conducted using isolated guinea-pig atrium. Three aptamers (D1, D2 and D3) were identified. Binding studies of fluorescein-labeled truncated (without primer binding region) D1 and D2 and full length D1 anti-digoxin aptamers were performed and their corresponding dissociation constants values were 8.2 × 10⁻⁹, 44.0 × 10⁻⁹ and 17.8 × 10⁻⁹ M, respectively. This is comparable to what other workers have obtained for interaction of monoclonal antibodies raised against digoxin. There was little difference in binding affinity between full length and truncated anti-digoxin D1 aptamer. D1 anti-digoxin aptamer also inhibited the effects of digoxin on the isolated guinea-pig atrium. D1 anti-digoxin aptamer distinguished between digoxin and ouabain in both tissue study and binding experiments. Our finding indicated that D1 anti-digoxin aptamer can selectively bind to digoxin. Further studies might show its suitability for use in digoxin assays and as a therapeutic agent in life-threatening digoxin toxicity.     

The stability of limit cycles in nonlinear systems

In this paper, a necessary condition is first presented for the existence of limit cycles in nonlinear systems, then four theorems are presented for the stability, instability, and semistabilities of limit cycles in second order nonlinear systems. Necessary and sufficient conditions are given in terms of the signs of first and second derivatives of a continuously differentiable positive function at the vicinity of the limit cycle. Two examples considering nonlinear systems with familiar limit cycles are presented to illustrate the theorems.     

The Characteristic Intersection Property of Line-Free Choquet Simplices in <Emphasis Type="Italic">E</Emphasis><Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

   Abstract. Generalizing the characteristic intersection property of Choquet simplices, it is proved that for line-free convex bodies B ₁ and B ₂ in E ^d , the following conditions are equivalent: (i) there is a line-free convex body B ⊂ E ^d such that every nonempty intersection B ₁ ∩ (v + B ₂ ) , v ∈ E ^d , is a homothetic copy of B , (ii) both B ₁ and B ₂ are Choquet simplices and the nonempty intersections B ₁ ∩ (v + B ₂ ) , v ∈ E ^d , are homothetic copies of a Choquet simplex B . All such triplets B ₁ ,B ₂ ,B are described.     

A solution of Hadwiger's covering problem for zonoids

For a convex body M ⁿ byb(M) the least integerp is denoted, such that there are bodiesM ₁, ...,M _p each of which is homothetic toM with a positive ratiok<1 andM ₁...M _p M. H. Martini has proved [7] thatb(M)<-3·2 ^n–2 for every zonotope M ⁿ , which is not a parallelotope.In the paper this Martini's result is extended to zonoids. In the proof some notions and facts of real functions theory are used (points of density, approximative continuity).