A subset B of a group G is called a difference basis of G if each element g ∈ G can be written as the difference g = ab−1 of some elements a, b ∈ B. The smallest cardinality |B| of a difference basis B ⊂ G is called the difference size of G and is denoted by ∆[G]. The fraction ð[G] := ∆[G]/ √ |G| is called the difference characteristic of G. We prove that for every n ∈ N the dihedral group D2n ...

Recently, Rogers and Crocker (1) proposed a method to predict the interaction between colloids coated with two kinds of ssDNA, A and B. A key step in ref. 1 was to estimate the average number of DNA bonds, 〈N〉, assuming local chemical equilibrium (LCE) between hybridized and unhybridized sticky end concentrations: hNi 1⁄4 Ð dr CAðrÞCBðrÞ exp1⁄2− βΔG0 ρ0 1⁄4 Ð dr1⁄2CAðrÞ−CABðrÞ 1⁄2CBðrÞ−CABðrÞ e...

|̂ ß ý l û G þ c v þ A K ý l ô o A ë A q A ð õ A Ñ o ô } | ø B ÿ G þ k o k ÿ ì õ Â Ï þ A u Q Þ ú k o A è Û B ÿ G þ k o k ÿ G ú O ñ ù B ü þ ü B G ú 3⁄4 õ o R Þ í ß þ ø í p A û G B u B ü p O ß ñ ý à | ø B ÿ G ý ù õ y þ k o Æ I A ð v B ð þ ô k A ì L r y ß þ o A ü Y A u Q . A ü ò o ô } G þ k o k ÿ k o G v ý B o ÿ A q W p A c þ | ø B ÿ ì d õ Æ ú y ß í þ , ð õ A c þ K ...

We analyzed transaction logs containing 51,473 queries posed by 18,113 users of Excite, a major Internet search service. We provide data on: (i) sessions Ð changes in queries during a session, number of pages viewed, and use of relevance feedback; (ii) queries Ð the number of search terms, and the use of logic and modi®ers; and (iii) terms Ð their rank/frequency distribution and the most highly...

We study the max-plus equation P þ cðxÞ 1⁄4 max xpspM ðHðsÞ þ cðgðsÞÞÞ; xA1⁄20;M ; ð*Þ where H : 1⁄20;M -ð N;NÞ and g : 1⁄20;M -1⁄20;M are given functions. The function c : 1⁄20;M -1⁄2 N;NÞ and the quantity P are unknown, and are, respectively, an eigenfunction and additive eigenvalue. Eigensolutions c are known to describe the asymptotics of certain solutions of singularly perturbed differenti...

We investigate oscillatory properties of the second order half-linear differential equation ðrðtÞFðy 0ÞÞ 0 þ cðtÞFðyÞ 1⁄4 0; FðsÞ :1⁄4 jsj s; p > 1; ð*Þ viewed as a perturbation of a nonoscillatory equation of the same form ðrðtÞFðy 0ÞÞ 0 þ ~ cðtÞFðyÞ 1⁄4 0: Conditions on the di¤erence cðtÞ ~ cðtÞ are given which guarantee that equation ð*Þ becomes oscillatory (remains nonoscillatory).

For Jacobi matrices with an 1⁄4 1þ ð 1Þan g; bn 1⁄4 ð 1Þbn g; we study bound states and the Szeg + o condition. We provide a new proof of Nevai’s result that if g4 2 ; the Szeg + o condition holds, which works also if one replaces ð 1Þ by cos ðmnÞ: We show that if a 1⁄4 0; ba0; and go1 2 ; the Szeg + o condition fails. We also show that if g 1⁄4 1; a and b are small enough (b þ 8a2o 1 24 will d...

This paper shows that the projective and inductive limits, lim ←Ð Hs and lim Ð→ Hs, of the Sobolev spaces are flat modules over the ring C[∂1, . . . , ∂n] of constant coefficient differential operators. This can be interpreted as a GAGA type result for the affine space Rn, and as a response to a remark of V.I.Danilov [in Algebraic Geometry II: Cohomology of Algebraic Varieties, Encyclopaedia of...