a oscillation theorem for a class of even order neutral differential equations abstract the even order neutral differential equation 1 d n dt n x t h t x t f t x g t 0 is considered under the following conditions n 2 is even 0 h c r g c t 0 lim t g t f c t 0 r uf t u 0 for t u t 0 r and f t u is nondecreasing in u r for each fixed t t 0 it is shown that 1 is oscillatory if and only if the certain non neutral differential equation is oscillatory for the case where 0 h t h t

on the projective normality of double coverings over a rational surface abstract we study the projective normality of a minimal surface x which is a ramified double covering over a rational surface s with dim k s 1 in particular horikawa surfaces the minimal surfaces of general type with k x 2 2 p g x 4 are of this type up to resolution of singularities let be the covering map from x to s we show that the z 2 invariant adjoint divisors k x r a are normally generated where the integer r 3 and a is an ample divisor on s

a numerical criterion for uniruledness an n dimensional variety x over an algebraically closed field k is said to be uniruled if there exist an n 1 dimensional k variety w and a dominant rational map f p x w x x is called separably uniruled if f is a separable map i e k p x w is a finite separable extension of k x via f if x is uniniled then there exists a rational curve passing through a general k valued point of x and the converse is true by the countability of the components of the hilbert scheme provided that the ground field k is uncountable uniruled varieties are of importance in the classification of varieties it is conjectured that in characteristic 0 uniruledness is equivalent to k xc this article gives a numerical criterion for a projective variety to be uniruled

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

an algorithm to generate exactly once every tiling with lozenges of a domain we first show that the tilings of a domain d form a lattice using the same kind of arguments as in research report no 1999 25 which we then undertake to decompose and generate without any redundance to this end we study extensively the relatively simple case of hexagons and their deformations we show that general domains can be broken up into hexagon like parts finally we give an algorithm to generate exactly once every element in the lattice of the tilings of a general domain d

how quickly can we sample a uniform domino tiling of the 2l times 2l square via glauber dynamics the prototypical problem we study here is the following given a 2l times 2l square there are approximately exp 4kl 2 pi ways to tile it with dominos i e with horizontal or vertical 2 times 1 rectangles where k approx 0 916 is catalan s constant kasteleyn in physica 27 1209 1225 1961 temperley and fisher in phil mag 6 1061 1063 1997 a conceptually simple even if computationally not the most efficient way of sampling uniformly one among so many tilings is to introduce a markov chain algorithm glauber dynamics where with rate 1 two adjacent horizontal dominos are flipped to vertical dominos or vice versa the unique invariant measure is the uniform one and a classical question levin et al in am math soc 2009 luby et al siam j comput 31 167 192 2001 wilson in ann appl probab 14 274 325 2004 is to estimate the time t mathrm mix it takes to approach equilibrium i e the running time of the algorithm luby et al in siam j comput 31 167 192 2001 and randall and tetali in j math phys 41 3 1598 1614 2000 fast mixing was proven t mathrm mix o l c for some finite c here we go much beyond and show that c l 2 le t mathrm mix le l 2 o 1 our result applies to rather general domain shapes not just the 2l times 2l square provided that the typical height function associated to the tiling is macroscopically planar in the large l limit under the uniform measure this is the case for instance for the temperley type boundary conditions considered in kenyon ann probab 28 759 795 2000 also our method extends to some other types of tilings of the plane for instance the tilings associated to dimer coverings of the hexagon or square hexagon lattices

height fluctuations in the honeycomb dimer model we study a model of random surfaces arising in the dimer model on the honeycomb lattice for a fixed wire frame boundary condition as the lattice spacing 0 cohn kenyon and propp 3 showed the almost sure convergence of a random surface to a non random limit shape 0 in 12 okounkov and the author showed how to parametrize the limit shapes in terms of analytic functions in particular constructing a natural conformal structure on them we show here that when 0 has no facets for a family of boundary conditions approximating the wire frame the large scale surface fluctuations height fluctuations about 0 converge as 0 to a gaussian free field for the above conformal structure we also show that the local statistics of the fluctuations near a given point x are as conjectured in 3 given by the unique ergodic gibbs measure on plane configurations whose slope is the slope of the tangent plane of 0 at x

on the oscillation of an n th order nonlinear neutral delay differential equation abstract the authors consider the n th order nonlinear neutral delay differential equation y t p t y g t n q t f y h t 0 where n 1 q t 0 and 1 they give sufficient conditions for solutions to either oscillate or converge to zero

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

semistable sheaves on projective varieties and their restriction to curves let x be a nonsingular projective variety of dimension n over an algebraically closed field k let h be a very ample line bundle on x if v is a torsion free coherent sheaf on x we define deg v to be cl v c h and v deg v rk v we call v sernistable resp stable if for all proper subsheaves w of v we have w v resp w v cf 7 14 in this paper we prove that if v is semistable on x then its restriction to a general complete intersection curve of sufficiently high degree is semistable theorem 6 1 to give an idea of the proof assume x is a surface and v a vector bundle of rank 2 the restriction of v to a general curve c of degree m is not semistable if and only if it is not semistable on the generic curve ym defined over the function field of iph h let s be the line bundle on y contradicting the semistability of vi y cf sects 4 1 and 4 2 first we show that l extends uniquely to a line bundle l m on x proposition 2 1 if we can get l as a subbundle of v we are through for then l would contradict the semistability of v so we would like the restriction map h hom lm v h hom l m v to be surjective now for fixed l it follows from the lemma of enriques severi proposition 3 2 6 corollary 7 8 that h horn l v h horn l v is surjective for large m therefore it is enough if the l remain the same line bundle l for infinitely many m to prove that l l we construct a degenerating family of curves d f s x x s 3 d p x such that the generic fibre is a curve ct 1 of degree 2 1 and the special fibre is a reduced curve with two nonsingular components ci of degree 2 cf sect 5 let m denote 2 extending the subbundle lt ct to a subsheaf of p v on d and restricting the extension to ci gives a lower bound for the maximal degree of a line subbundle of v cim in terms of that for v ctm i proposition 4 3 this implies that degl is bounded lemma 6 5 1 so that for an infinite subsequence of m degl is constant if degltm r deglt by refining the above argument with the degenerating family one can prove that lt ci

oscillation of second order nonlinear neutral differential equations abstract in this paper some sufficient conditions for oscillation and nonoscillation are obtained for the second order nonlinear neutral differential equation x t p t x t q t f x t 0 t 0 where 0 p t 1 q t 0 0

oscillation and existence of positive solutions in a class of higher order neutral equations abstract consider the higher order neutral differential equation x t x t n q t x t 0 t t 0 with q t continuous 0 0 and n odd we establish several new sufficient conditions for the oscillation of all solutions and the existence of a positive solution by an associated ordinary differential equation some of these conditions are sharp and improve the known results in the literature which include the newest one by 8

a sharp lower bound for the canonical volume of 3 folds of general type let v be a smooth projective 3 fold of general type denote by k 3 a rational number the self intersection of the canonical sheaf of any minimal model of v one defines k 3 as the canonical volume of v assume p g v 2 we show that k 3 ge frac 1 3 which is a sharp lower bound then we classify those v with small volume we also give some new examples with p g 2 which have maximal canonical stability index finally we give an application to 4 folds of general type

an o n log n algorithm for finding a domino tiling of a plane picture whose number of holes is bounded we consider the problem of tiling a plane picture with dominoes this picture can be with holes or without holes we give a necessary and sufficient condition for the existence of such a tiling and then we deduce an algorithm to decide whether a picture is tileable or not and if it is to determine a tiling this algorithm runs in o kn n log n time where n and k are the respective numbers of cells and holes we also propose a simplified version of this algorithm which is linear in the area enclosed by the outer boundary provided the picture has a bounded number of holes

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

tiling with dominoes pictures of plane figures without holes graphic basis of the thurston algorithm parallelization uniqueness and decomposition resume le desormais classique algorithme de thurston 1990 pour le pavage des figures planes finies et sans trous par des dominos peut etre retrouve par la condition de hall sur les couplages dans les graphes bipartis et la consideration de plus courts chemins dans un graphe associe a la figure a paver ce point de vue nouveau permet une parallelisation en temps o log n avec n 3 processeurs sur un modele pram c est a dire de facon efficace classe nc ce meme algorithme resout la question de l unicite d un pavage et de facon plus generale donne une decomposition canonique de la figure

oscillation criteria for nth order nonlinear neutral differential equations abstract sufficient conditions are established for the oscillation of nth order neutral differential equations of the form r t x t x t 1 p t x t n 1 q t f x t 0 t t 0 where n 2 is even integer 1 p q c t 0 r f c r r the results obtained extend some of the known results