Focal Loci of Algebraic Hypersurfaces: A General Theory

The focal locus is traditionally defined for a differentiable submanifold of R ⁿ. However, since it depends essentially only on the notion of orthogonality, a focal locus can be also associated to an algebraic subvariety of the space , once we have chosen an orthogonal structure on this space. In this paper, we establish somebasic results in the theory of focal loci of algebraichypersurfaces in . Our main results concern the irreducibility of the ramification divisor of the end-point map and the dimension of the singular locus of this divisor, the birationality of the focal map and the degree of the focal locus of an algebraic hypersurface.     

The Quantum Cohomology Ring of Flag Varieties

We describe the small quantum cohomology ring of complete flag varieties by algebro-geometric methods, as presented in our previous work Quantum cohomology of flag varieties (Internat. Math. Res. Notices, no. 6 (1995), 263-277). We also give a geometric proof of the quantum Monk formula.

     

STUDY OF THREE-SHEAR STRESS UNIFIED STRENGTH THEORY

材料强度理论研究是高等材料力学的一个重要研究领域,200多年来,各国学者提出了许多强度理论及试验研究结果.这些理论都是从不同的假设和力学模型出发,推导出不同的数学表达式,但一般只适用于某一类特定的材料.各种强度理论之间是否有联系?是否可能建立一个广泛适用的统一强度理论?自19世纪末以来,世界各国学者都在努力寻求建立统一强度理论,但一直没有成功.本文应用理论研究和试验验证相结合的方法,提出并验证了三剪应力统一强度理论。认为当作用于菱形十二面单元体上的三个主剪应力及其作用面上的三个正应力的函数达到某一极限值时,材料发生破坏.三剪应力统一强度理论是全应力理论,它用一个统一的线性表达式包含或逼近了现有的和其他新的各种单一和统一、线性和非线性、外凸和非凸的强度理论,形成了以单剪强度理论为下限、而以三剪应力强度理论为上限的一系列强度理论的新体系,实现了外凸强度理论和非凸强度理论的高度统一,使强度理论从适用于某一类材料、某种应力状态的单一强度理论发展为可以适用于各种材料及不同应力状态的统一强度理论,并能更大程度地发挥材料的强度潜力.通过与大量试验结果的对比分析表明：三剪应力统一强度理论可以广泛适用于各种材料及其不同的应力状态.     

Conjugate and cut loci of a two-sphere of revolution with application to optimal control

The objective of this article is to present a sharp result to determine when the cut locus for a class of metrics on a two-sphere of revolution is reduced to a single branch. This work is motivated by optimal control problems in space and quantum dynamics and gives global optimal results in orbital transfer and for Lindblad equations in quantum control.     

Algebraic jump loci for rank and Betti numbers over Laurent polynomial rings

Let C be a chain complex of finitely generated free modules over a commutative Laurent polynomial ring $L_s$ in s indeterminates. Given a group homomorphism $p:{\mathbb{Z}}^s?{\mathbb{Z}}^t$ we let $p_{!}(C)=C{?}_{L_s}{L}_t$ denote the resulting induced complex over the Laurent polynomial ring $L_t$ in t indeterminates. We prove that the Betti number jump loci, that is, the sets of those homomorphisms p such that $b_k(p_{!}(C))>b_k(C)$, have a surprisingly simple structure. We allow non-unital commutative rings of coefficients, and work with a notion of Betti numbers that generalises both the usual one for integral domains, and the analogous concept involving McCoy ranks in case of unital commutative rings.     

Isosingular loci of algebraic varieties

We define the notion of isosingular loci of algebraic varieties, following the analytic case first studied by Ephraim. These are subsets of points where the variety has a prescribed formal singularity type. We show that the isosingular loci of an algebraic variety are locally closed in the Zariski topology and the associated reduced subschemes are smooth. Moreover, assuming characteristic 0, we prove the existence of a decomposition of the formal neighborhoods at closed points into a product of the respective isosingular locus at that point and a smooth factor. One of the main obstructions in the positive characteristic case is the non-separability of the orbit map associated to the contact group, as first observed by Greuel and Pham for isolated singularities.     

Threefolds in ℙ5 with One Apparent Quadruple Point

Abstract

In this article we classify all the smooth threefolds in ?⁵ with an apparent quadruple point provided that the family of its 4-secant lines is an irreducible (first order) congruence. This is sufficient to conclude the classification of all the smooth codimension two varieties in ? ⁿ with one apparent (n ? 1)-point and with irreducible family of (n ? 1)-secant lines.     

A study of auditory responses to transitions of signal frequencies

I.IntrQductionThecomplexsigna1swithcertainmodesoffrcquencyvariation,suchaschirpsignal,andfrcquencycodingsignal,havebecncxtensivelyusedinm0dcrnactivesonardetcc-hons,andthroughcorrespondingmatched-fllteringthcoptima1receptiont0echoescanbepcrformed.l"'lPractical1yasonar0pcratorncedst0direct1ylistcntotheinputechoesinthemeantimcwatchingthescreent0disp1aytheech0esafterprocessing,andthenaconfidentjudgementfortargetappearancccanbemade-Fromtheviewpointofauditorydetection,aninterestingproblemiswhatthed…     

Partial Exclusion of the Local Effect from the Assessment of Recognized Illuminant Using Depth Separation II: Influence of Depth Separation under Chromatic Illumination

In the previous work we investigated the influence of depth separation on exclusion of the local effect from the assessment of the recognized illuminant. Under whitish illumination, only a portion of the local effect was excluded, while this study, we extended the range of illumination from whitish to chromatic illuminant. We evaluated how the depth separation influences the amount of local effect under the chromatic illumination. Subjects did perceptual achromatic setting under various combinations of local surround and illumination conditions. We found that under chromatic illumination, depth separation is more effective to exclude the local effect even though the local effect was smaller than under whitish chromatic illumination.     

Parameter space for families of parabolic-like mappings

In this paper we study families of degree 2 parabolic-like mappings _(fλ)λ∈Λ${(f_{\lambda })}_{\lambda \in \Lambda }$ (as defined in [4]). We prove that the hybrid conjugacies between a nice analytic family of degree 2 parabolic-like mappings and members of the family _Per1(1)${\mathit{Per}}_1(1)$ induce a continuous map χ:Λ→C$\chi :\Lambda \to \mathbb{C}$, which under suitable conditions restricts to a ramified covering from the connectedness locus of _(fλ)λ∈Λ${(f_{\lambda })}_{\lambda \in \Lambda }$ to the connectedness locus _M1?{1}$M_1?\{1\}$ of _Per1(1)${\mathit{Per}}_1(1)$. As an application, we prove that the connectedness locus of the family _Ca(z)=z+az²+z³$C_a(z)=z+a{z}^2+z^3$, a∈C$a\in \mathbb{C}$ presents baby _M1$M_1$.