Divisor

About: Divisor is a research topic. Over the lifetime, 2462 publications have been published within this topic receiving 21394 citations. The topic is also known as: factor & submultiple.

$L^2$ and intersection cohomologies for the reductive representation of the fundamental groups of quasiprojective manifolds with unipotent local monodromy

Abstract: Let $X$ be a projective manifold, and $D$ be a normal crossing divisor of $X$. By Jost-Zuo's theorem that if we have a reductive representation $\rho$ of the fundamental group $\pi_{1}(X^{*})$ with unipotent local monodromy, where $X^*=X-D$, then there exists a tame pluriharmonic metric $h$ on the flat bundle $\mathcal V$ associated to the local system $\mathbb V$ obtain from $\rho$ over $X^*$. Therefore, we get a harmonic bundle $(E, \theta, h)$, where $\theta$ is the Higgs field, i.e. a holomorphic section of $End(E)\otimes\Omega^{1,0}_{X^*}$ satisfying $\theta^2=0$. In this paper, we study the harmonic bundle $(E,\theta,h)$ over $X^*$. We are going to prove that the intersection cohomology $IH^{k}(X; \mathbb V)$ is isomorphic to the $L^{2}$-cohomology $H^{k}(X, (\mathcal A_{(2)}^{\cdot}(X,\mathcal V), \mathbb D))$.

Cleanliness and log-characteristic cycles for vector bundles with flat connections

Abstract: Let \(X\) be a proper smooth algebraic variety over a field \(k\) of characteristic zero and let \(D\) be a divisor with simple normal crossings. Let \(M\) be a vector bundle over \(X-D\) equipped with a flat connection with possible irregular singularities along \(D\). We define a cleanliness condition which roughly says that the singularities of the connection are controlled by the singularities at the generic points of \(D\). When this condition is satisfied, we compute explicitly the associated log-characteristic cycle, and relate it to the so-called refined irregularities. As a corollary of a log-variant of Kashiwara–Dubson formula, we obtain the Euler characteristic of the de Rham cohomology of the vector bundle, under a mild technical hypothesis on \(M\).

Zerlegung total gerichteter Graphen in Kreise

Abstract: The following statements are valid: The complete directed graph ¯Kn, n≡1 (mod 2p), is decomposable into directed 2p-cycles. The complete directed bipartite graph ¯Km,n is decomposable into 2p-cycles if p is a divisor of m and n≧p. If p is a prime, then this condition is necessary, too. The complete directed graph ¯Kn, n≠12, is decomposable into 6-cycles if and only if 6

Substrate transportation method and substrate processing device

Abstract: PROBLEM TO BE SOLVED: To provide a technology capable of improving throughput of a substrate processing device.SOLUTION: A substrate transportation method includes a transportation process. A substrate transportation part is capable of simultaneously transporting N pieces of substrates specified by an integer N that is not a divisor of an integer M and is equal to or greater than 2, from a segment capable of simultaneously holding M pieces of substrates specified by the integer M that is equal to or greater than 2, or towards such a segment. In the transportation process, if N pieces of variables ik specified by an integer (k) from 1 to N are any arbitrary integers which are equal to or greater than 0 and equal to or smaller than (M/N) and satisfy a relation of M=N×i1+, ..., +1×iN, a substrate transportation step is implemented ik times in which (N-k-1) pieces of substrates are simultaneously transported from a segment or towards the segment by the substrate transportation part. The substrate transportation method iteratively executes a substrate transportation cycle that implements such a transportation process specifying the number of times of the substrate transportation step with each variable that is a natural number among the N pieces of variables ik.

Determinants of matrices associated with arithmetic functions on two quasi-coprime divisor chains

Abstract: For any integers xand y,we use(x,y)([x,y])to denote the greatest common divisor(the least common multiple)of xand y.Let fbe an arithmetic function and S= {x1,…,xn}be a set of ndistinct positive integers.By(f(S))=(f(xi,xj))((f[S])=(f[xi,xj])),we denote the n×n matrix having fevaluated at(xi,xj)([xi,xj])as its i,j-entry.The set Sis called a divisor chain if there is a permutationσof{1,2,…,n}such that xσ(1)|…|xσ(n).The set Sis called two quasi-coprime divisor chains if Scan be partitioned as S=SI ∪S2with all Si(1≤i≤2)being divisor chains and(max(S1),max(S2))=gcd(S).In this paper,we give the formulae for the determinants of the matrices(f(S))and(f[S])on two quasi-coprime divisor chains.