Splitting Type Research Articles

Geometry of manifolds with densities

We study the geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry–Émery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtain some splitting type results by analyzing the Busemann functions. In particular, we show that a complete manifold with nonnegative Bakry–Émery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear growth weight functions.While some of our results are new even for the gradient Ricci solitons, the novelty here is that only a lower bound of the Bakry–Émery curvature is involved in our analysis.

On plane rational curves and the splitting of the tangent bundle

Given an immersion ' : P1 ! P2, we give new approaches to determining the splitting of the pullback of the cotangent bundle. We also give new bounds on the splitting type for immersions which factor as ' : P1 = D ⇢ X ! P2, where X ! P2 is obtained by blowing up r distinct points pi 2 P2. As applications in the case that the points pi are generic, we give a complete determination of the splitting types for such immersions when r  7. The case that D2 = 1 is of particular interest. For r  8 generic points, it is known that there are only finitely many inequivalent ' with D2 = 1, and all of them have balanced splitting. However, for r = 9 generic points we show that there are infinitely many inequivalent ' with D2 = 1 having unbalanced splitting (only two such examples were known previously). We show that these new examples are related to a semi-adjoint formula which we conjecture accounts for all occurrences of unbalanced splitting when D2 = 1 in the case of r = 9 generic points pi . In the last section we apply such results to the study of the resolution of fat point schemes. Mathematics Subject Classification (2010): 14C20 (primary); 13P10, 14J26, 14J60 (secondary).

Energy monitoring and analysis during deformation of bedded-sandstone: Use of acoustic emission

This paper investigates the mechanical behaviour and energy releasing characteristics of bedded-sandstone with bedding layers in different orientations, under uniaxial compression. Cylindrical sandstone specimens (54mm diameter and 108mm height) with bedding layers inclined at angles of 10°, 20°, 35°, 55°, and 83° to the minor principal stress direction, were produced to perform a series of Uniaxial Compressive Strength (UCS) tests. One of the two identical sample sets was fully-saturated with water before testing and the other set was tested under dry conditions. An acoustic emission system was employed in all the testing to monitor the acoustic energy release during the whole deformation process of specimens. From the test results, the critical joint orientation was observed as 55° for both dry and saturated samples and the peak-strength losses due to water were 15.56%, 20.06%, 13.5%, 13.2%, and 13.52% for the bedding orientations 10°, 20°, 35°, 55°, and 83°, respectively. The failure mechanisms for the specimens with bedding layers in 10°, 20° orientations showed splitting type failure, while the specimens with bedding layers in 55°, 83° orientations were failed by sliding along a weaker bedding layer. The failure mechanism for the specimens with bedding layers in 35° orientation showed a mixed failure mode of both splitting and sliding types. Analysis of the acoustic energy, captured from the acoustic emission detection system, revealed that the acoustic energy release is considerably higher in dry specimens than that of the saturated specimens at any bedding orientation. In addition, higher energy release was observed for specimens with bedding layers oriented in shallow angles (which were undergoing splitting type failures), whereas specimens with steeply oriented bedding layers (which were undergoing sliding type failures) showed a comparatively less energy release under both dry and saturated conditions. Moreover, a considerable amount of energy dissipation before the ultimate failure was observed for specimens with bedding layers oriented in shallow angles under both dry and saturated conditions. These results confirm that when rock having bedding layers inclined in shallow angles the failures could be more violent and devastative than the failures of rock with steeply oriented bedding layers.

Rational Curves on and K3 Surfaces

Let $(S,L)$ be a smooth primitively polarized K3 surface of genus $g$ and $f:X \rightarrow \mathbb{P}^1$ the fibration defined by a linear pencil in $|L|$. For $f$ general and $g \geq 7$, we work out the splitting type of the locally free sheaf $\Psi^{*}_f T_{{\overline{M}}_g}$, where $\Psi_f$ is the modular morphism associated to $f$. We show that this splitting type encodes the fundamental geometrical information attached to Mukai's projection map $\mathcal{P}_g \rightarrow \overline{\mathcal{M}}_g$, where $\mathcal{P}_g$ is the stack parameterizing pairs $(S,C)$ with $(S,L)$ as above and $C \in |L|$ a stable curve. Moreover, we work out conditions on a fibration $f$ to induce a modular morphism $\Psi_f$ such that the normal sheaf $N_{\Psi_f}$ is locally free.

A note on the splitting theorem for the weighted measure

In this paper we study complete manifolds equipped with smooth measures whose spectrum of the weighted Laplacian has an optimal positive lower bound and the $m$-dimensional Bakry-\'Emery Ricci curvature is bounded from below by some negative constant. In particular, we prove a splitting type theorem for complete smooth measure manifolds that have a finite weighted volume end. This result is regarded as a study of the equality case of an author's theorem (J. Math. Anal. Appl. 361 (2010) 10-18).

Rigidity of manifolds with Bakry–Émery Ricci curvature bounded below

Let M be a complete Riemannian manifold with Riemannian volume volg and f be a smooth function on M. A sharp upper bound estimate on the first eigenvalue of symmetric diffusion operator \({\Delta_f = \Delta- \nabla f \cdot \nabla}\) was given by Wu (J Math Anal Appl 361:10–18, 2010) and Wang (Ann Glob Anal Geom 37:393–402, 2010) under a condition that finite dimensional Bakry–Emery Ricci curvature is bounded below, independently. They propounded an open problem is whether there is some rigidity on the estimate. In this note, we will solve this problem to obtain a splitting type theorem, which generalizes Li–Wang’s result in Wang (J Differ Geom 58:501–534, 2001, J Differ Geom 62:143–162, 2002). For the case that infinite dimensional Bakry–Emery Ricci curvature of M is bounded below, we do not expect any upper bound estimate on the first eigenvalue of Δf without any additional assumption (see the example in Sect. 2). In this case, we will give a sharp upper bound estimate on the first eigenvalue of Δf under the additional assuption that \({|\nabla f|}\) is bounded. We also obtain the rigidity result on this estimate, as another Li–Wang type splitting theorem.

COMMON DIVISORS OF VALUES OF POLYNOMIALS AND COMMON FACTORS OF INDICES IN A NUMBER FIELD

Let K be a number field of degree n over ℚ, Â be the set of integers of K that are primitive over ℚ and let I(K) be its index. The prime factors of I(K) are called common factors of indices or inessential discriminant divisors. We show that these primes divide another index i(K) previously defined by Gunji and McQuillan as i(K) = lcm θ∈Âi(θ), where i(θ) = gcd x∈ℤFθ(x) and Fθ(x) is the characteristic polynomial of θ over ℚ. It is shown that there exists θ ∈ Â such that i(K) = i(θ) and an algorithm is given for the computation of such an integer. For any prime p|i(K), an integer ρK(p) defined as the number of [Formula: see text] such that p|i(θ) is investigated. It is shown that this integer determines in some cases the splitting type of p in K. Some open questions related to I(K), i(K) and ρK(p) are stated.

WEAKLY UNIFORM RANK TWO VECTOR BUNDLES ON MULTIPROJECTIVE SPACES

AbstractHere we classify the weakly uniform rank two vector bundles on multiprojective spaces. Moreover, we show that every rank r>2 weakly uniform vector bundle with splitting type a1,1=⋯=ar,s=0 is trivial and every rank r>2 uniform vector bundle with splitting type a1>⋯>ar splits.

Intussusceptive angiogenesis: pillars against the blood flow

Adaptation of vascular networks to functional demands needs vessel growth, vessel regression and vascular remodelling. Biomechanical forces resulting from blood flow play a key role in these processes. It is well-known that metabolic stimuli, mechanical forces and flow patterns can affect gene expression and remodelling of vascular networks in different ways. For instance, in the sprouting type of angiogenesis related to hypoxia, there is no blood flow in the rising capillary sprout. In contrast, it has been shown that an increase of wall shear stress initiates the splitting type of angiogenesis in skeletal muscle. Otherwise, during development, both sprouting and intussusception act in parallel in building the vascular network, although with differences in spatiotemporal distribution. Thereby, in addition to regulatory molecules, flow dynamics support the patterning and remodelling of the rising vascular tree. Herewith, we present an overview of angiogenic processes with respect to intussusceptive angiogenesis as related to local haemodynamics.

Combined methods for dynamic spatial auction market models

An equilibrium model for description of behavior of a system of auction markets joined by transmission lines subject to joint balance and capacity flows constraints for a certain time period is suggested. We treat this model as an extended primal-dual system of variational inequalities or a saddle point problem. Several splitting type methods are proposed to find its solution.

The Verlinde bundles and the semihomogeneous Wirtinger duality

We determine the splitting type of the Verlinde vector bundles in higher genus in terms of simple semihomogeneous factors. In agreement with strange duality, the simple factors are interchanged by the Fourier–Mukai transform, and their spaces of sections are naturally dual.

SPLITTING TYPE, GLOBAL SECTIONS AND CHERN CLASSES FOR TORSION FREE SHEAVES ON PN

In this paper we compare a torsion free sheaf F on <TEX>$P^N$</TEX> and the free vector bundle <TEX>$\oplus^n_{i=1}O_{P^N}(b_i)$</TEX> having same rank and splitting type. We show that the first one has always "less" global sections, while it has a higher second Chern class. In both cases bounds for the difference are found in terms of the maximal free subsheaves of F. As a consequence we obtain a direct, easy and more general proof of the "Horrocks' splitting criterion", also holding for torsion free sheaves, and lower bounds for the Chern classes <TEX>$c_i$</TEX>(F(t)) of twists of F, only depending on some numerical invariants of F. Especially, we prove for rank n torsion free sheaves on <TEX>$P^N$</TEX>, whose splitting type has no gap (i.e., <TEX>$b_i{\geq}b_{i+1}{\geq}b_i-1$</TEX> 1 for every i = 1,<TEX>$\ldots$</TEX>,n-1), the following formula for the discriminant: <TEX>$$\Delta(F):=2_{nc_2}-(n-1)c^2_1\geq-\frac{1}{12}n^2(n^2-1)$$</TEX>. Finally in the case of rank n reflexive sheaves we obtain polynomial upper bounds for the absolute value of the higher Chern classes <TEX>$c_3$</TEX>(F(t)),<TEX>$\ldots$</TEX>,<TEX>$c_n$</TEX>(F(t)) for the dimension of the cohomology modules <TEX>$H^iF(t)$</TEX> and for the Castelnuovo-Mumford regularity of F; these polynomial bounds only depend only on <TEX>$c_1(F)$</TEX>, <TEX>$c_2(F)$</TEX>, the splitting type of F and t.

개량 스킨팀버의 압축 강도에 관한 연구

기존의 중공재에 비해 다량의 목질부를 제거하여 천공한 스킨 팀버는 짧은 건조시간과 건조 결함 발생이 적은 점, 상당히 낮은 수준(MC 8~9%)까지 함수율을 떨어트릴 수 있다는 이점, 약제 주입처리의 용이함, 한옥을 포함한 중목구조의 대단면 경량 구조부재로서의 사용 가능성, 다양한 가구나 생활용품으로의 가능성 등 많은 장점을 가지고 있는 재료이다. 하지만 고부가가치 재료로 사용되기 위해서, 그리고 일반 소비자들에게 구조적 신뢰감을 제공하기 위해서는 스킨 팀버를 하이브리드 구조부재로 만들 필요가 있다. 따라서 본 연구에서는 가능한 얇은 철판을 사용하여 소나무와 낙엽송 개량 스킨 팀버를 제작하고 압축 성능을 분석하였다. 이에 따른 본 연구의 결론은 다음과 같다: 1. 소나무와 낙엽송 개량 스킨 팀버의 경우 무처리 스킨 팀버에 비해 재료 성능의 균일성을 가져올 수 있었다. 2. 소나무나 낙엽송 개량 스킨 팀버 모두 철판두께별 압축 성능의 유의차가 존재하지 않아 두꺼운 철판을 사용 할 필요성이 없다고 판단되었다. 소나무 개량 스킨 팀버와 낙엽송 개량 스킨 팀버간의 압축 성능 역시 유의차가 존재하지 않아 사용자의 기호에 따라 사용할 수 있을 것으로 판단되었다. 4. 소나무 개량 스킨 팀버의 경우 다양한 파괴 형태를 보였지만 낙엽송 스킨 팀버의 경우 전체적으로 splitting의 파괴 모드를 보이는 것이 특징이었다. As compared with existing center-boring timber, skin timber which be hollowed out of its considerable inner parts has some merits as like short drying time, less developed checks during drying, a advantage of lower MC (8~%), more easy injection of chemicals, a possibility of using as a lighter structural heavy timber including Hanok and heavy timber construction, a possibility for the various living necessaries and furniture materials. However, development of hybrid skin timber is required for using as a value-added materials and giving a confidence for the structural safety of skin timber to general user. Thus, improved pine skin timber (IPST) and improved larch skin timber (ILST) were manufactured using the lighter steel plate possible. And compressive capacity of improved skin timber was analyzed. From the results of this study, the following conclusions have been made: 1. Both of IPST and ILST can give a uniformity of material capacity compared with non-treated skin timber. 2. Both IPST and ILST, there was not statistical significancy among the thickness of steel plate. Therefore, it concluded that it was not necessary to use thicker steel plate. 3. There was also not statistical significancy between IPSR and ILST, so it need not to be selective about the species of improved skin timber. 4. IPST showed various failure types, but most failure types of ILST is a splitting type.

A splitting type algorithm for multi-valued complementarity problems

We consider a generalized complementarity problem whose cost mapping is multi-valued and is the sum of upper Z and antitone mappings. We suggest a simple splitting type algorithm which utilizes an extended Jacobi iteration. Its convergence is proved under mild assumptions. Preliminary results of numerical experiments confirm efficiency of the algorithm presented.

Coordinate relaxation methods for multivalued complementarity problems

Methods of the Jacobi and Gauss-Seidel type with underrelaxation and a combined method of the splitting type are proposed for complementarity problems with multivalued mappings. The convergence of these methods to the solution is proved under the conditions that the basic mapping is upper off-diagonal antitone and the feasible set is nonempty. The numerical results obtained for test examples are presented.

Stability of operator splitting methods for systems with indefinite operators: Advection–diffusion–reaction systems

This brief paper presents an A-stability result for operator splitting type time integration methods applied to advection–diffusion–reaction equations with possibly indefinite source terms. These results extend our earlier work on diffusion–reaction systems [D.L. Ropp, J.N. Shadid, Stability of operator splitting methods for systems with indefinite operators: reaction–diffusion systems, J. Comput. Phys. 203 (2) (2005) 449–466]. The A-stability result presents sufficient conditions that control both low and high wave number instabilities. A corollary shows that if L-stable methods are used for the diffusion term the high wave number instability will be controlled more easily. Numerical results are presented that verify second-order convergence for the operator splitting methods and demonstrate control of instabilities on a chemotaxis problem by use of an L-stable diffusion integrator.

Dynamic Equilibrium Model for Spatial Auction Markets

Spatial equilibrium models are utilized for description of behavior and for management of complex systems containing many interacting elements, with taking into account their spatial distribution. We consider such a system of auction markets joined by transmission lines subject to joint balance and capacity flows constraints for a certain time period propose a primal-dual system of variational inequalities whose solutions yield an equilibrium trajectory of this system. A splitting type method is proposed to find a solution.

Decomposition Approaches for Constrained Spatial Auction Market Problems

We consider several equilibrium formulations for the problem of managing spatially distributed auction markets of a homogeneous commodity, which are joined by transmission lines in a network. At each market, traders and buyers are determined by their price functions and choose their offer/bid values. We present equivalent variational inequality, optimization, and saddle point formulations of this problem. The corresponding models possess a special structure of constraint and cost functions and lead to different decomposition schemes. We propose proximal and splitting type methods and discuss their properties and preliminary computational results.

Betti numbers for fat point ideals in the plane: A geometric approach

We consider the open problem of determining the graded Betti numbers for fat point subschemes Z Z supported at general points of P 2 \mathbf {P}^2 . We relate this problem to the open geometric problem of determining the splitting type of the pullback of Ω P 2 \Omega _{\mathbf {P}^2} to the normalization of certain rational plane curves. We give a conjecture for the graded Betti numbers which would determine them in all degrees but one for every fat point subscheme supported at general points of P 2 \mathbf {P}^2 . We also prove our Betti number conjecture in a broad range of cases. An appendix discusses many more cases in which our conjecture has been verified computationally and provides a new and more efficient computational approach for computing graded Betti numbers in certain degrees. It also demonstrates how to derive explicit conjectural values for the Betti numbers and how to compute splitting types.

Spectrum of the Laplacian on manifolds with Spin(9) holonomy

We consider noncompact complete manifolds with Spin(9) holonomy and proved an one end result and a splitting type theorem under different conditions on the bottom of the spectrum. We proved that any harmonic functions with finite Dirichlet integral must be Cayley-harmonic, which allowed us to conclude an one end result. In the second part, we established a splitting type theorem by utilizing the Busemann function.