On splitting the area under curves II

This paper continues the author's work [3, S. Minsker, J. Differential Equations, 26, No. 3 (1977), 443–457.] on an area-splitting problem leading to the functional differential equation $\text{a′(a(x)) =}\text{a(x)}\text{x}$. This equation is dealt with by transforming it into the linear equation ψ′(x) = ψ(x + c), for which positive solutions on (?∞, ?c) are sought.     

On the double curves of least area tori

The double curves of least area immersions of the torus into closed, orientable, irreducible 3-manifolds are simple in the torus. A related result for other least area surfaces is given.

     

The linear combinations of biomarkers which maximize the partial area under the ROC curves

As biotechnology has made remarkable progress nowadays, there has also been a great improvement on data collection with lower cost and higher quality outcomes. More often than not investigators can obtain the measurements of many disease-related features simultaneously. When multiple potential biomarkers are available for constructing a diagnostic tool of a disease, an effective approach is to combine these biomarkers to build one single indicator. For continuous-scaled variables, the use of linear combinations is popular due to its easy interpretation. Su and Liu (J Ame Stat Assoc 88(424):1350–1355, 1993) derived the best linear combination under the criterion of the area under the receiver operating characteristic (ROC) curve, when the joint normality of biomarkers is assumed. However, in many investigations, the emphases are placed only on a limited extent of clinical relevancy, instead of the whole ROC curve. The goal of this study is to find the linear combination that maximizes the partial area under a ROC curve (pAUC) for a pre-specified range. In order to find an analytic solution, the first derivative of the pAUC under normal assumption is derived. The explicit form is so complicated, that a further validation on the Hessian matrix is difficult. On the other hand, we find that the pAUC maximizer may not be unique and local maximizers do exist in some cases. Consequently, the existing algorithms find an initial-point dependent solution and are inadequate to serve our needs. Hence, we propose a new algorithm by adopting several initial points at one time. Intensive numerical studies have been performed to show the adequacy of the proposed algorithm. Real examples are also provided for illustration.     

On the space curves with the same image under the gauss maps

From an irreducible complete immersed curveX in a projective space ? other than a line, one obtains a curveX ^′ in a Graasmann manifoldG of lines in ? that is the image ofX under the Gauss map, which is defined by the embedded tangents ofX. The main result of this article clarifies in case of positive characteristic what curvesX have the sameX′: It is shown thatX is uniquely determined byX′ ifX, or equivalentlyX′, has geometric genus at least two, and that for curvesX ₁ andX ₂ withX ₁ ≠X ₂ in ?, ifX′₁ =X′₂ inG and eitherX ₁ orX ₂ is reflexive, then bothX ₁ andX ₂ are rational or supersingular elliptic; moreover, examples of smoothX ₁ andX ₂ in that case are given.     

On the greatest splitting topology

It is well known that (see, for example, [H. Render, Nonstandard topology on function spaces with applications to hyperspaces, Trans. Amer. Math. Soc. 336 (1) (1993) 101-119; M. Escardo, J. Lawson, A. Simpson, Comparing cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143 (2004) 105-145; D.N. Georgiou, S.D. Iliadis, F. Mynard, in: Elliott Pearl (Ed.), Function Space Topologies, Open Problems in Topology, vol. 2, Elsevier, 2007, pp. 15-22]) the intersection of all admissible topologies on the set C(Y,Z) of all continuous maps of an arbitrary space Y into an arbitrary space Z, is always the greatest splitting topology. However, this intersection maybe not admissible. In the case, where Y is a locally compact Hausdorff space the compact-open topology on the set C(Y,Z) is splitting and admissible (see [R.H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945) 429-432; R. Arens, A topology for spaces of transformations, Ann. of Math. 47 (1946) 480-495; R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31]), which means that the intersection of all admissible topologies on C(Y,Z) is admissible. In [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31] an example of a non-locally compact Hausdorff space Y is given having the same property for the case, where Z=[0,1], that is on the set C(Y,[0,1]) the compact-open topology is splitting and admissible. This space Y is the set [0,1] with a topology τ, whose semi-regular reduction coincides with the usual topology on [0,1]. Also, in [R. Arens, J. Dugundji, Topologies for function spaces, Pacific J. Math. 1 (1951) 5-31, Theorem 5.3] another example of a non-locally compact space Y is given such that the compact-open topology on the set C(Y,[0,1]) is distinct from the greatest splitting topology.In this paper first we construct non-locally compact Hausdorff spaces Y such that the intersection of all admissible topologies on the set C(Y,Z), where Z is an arbitrary regular space, is admissible. Furthermore, for a Hausdorff splitting topology t on C(Y,Z) we find sufficient conditions in order that t to be distinct from the greatest splitting topology. Using this result, we construct some concrete non-locally compact spaces Y such that the compact-open topology on C(Y,Z), where Z is a Hausdorff space, is distinct from the greatest splitting topology. Finally, we give some open problems.     

On the equitangential curves

Monatshefte für Mathematik -     

On the splitting problem for selections

We investigate when does the Repovš-Semenov splitting problem for selections have an affirmative solution for continuous set-valued mappings in finite-dimensional Banach spaces. We prove that this happens when images of set-valued mappings or even their graphs are P-sets (in the sense of Balashov) or strictly convex sets. We also consider an example which shows that there is no affirmative solution of this problem even in the simplest case in R³. We also obtain affirmative solution of the approximate splitting problem for Lipschitz continuous selections in the Hilbert space.     

On the cofinality of the splitting number

The splitting number $\mathfrak{s}$ can be singular. The key method is to construct a forcing poset with finite support matrix iterations of ccc posets introduced by Blass and Shelah (1989).     

On universal elliptic curves over Igusa curves

Summary The purpose of this note is to introduce the arithmetic, study of the universal elliptic curve over Igusa curves. Specifically, its Hasse-WeilL-function is computed in terms of modular forms and is shown to have interesting zeros. Explicit examples are presented for which the Birch and Swinnerton-Dyer conjecture is verified.This paper summarizes part of the author's Ph.D. thesis. He wishes to thank the Sloan Foundation for financial support in the form of a Doctoral Dissertation Fellowship and his advisor, Dick Gross, for mathematical guidance and inspirational enthusiasm.To my parents in their 50th year     

On the behavior of F-signatures,splitting primes,and test modules under finite covers

We give a comprehensive treatment on how F-signatures, splitting primes, splitting ratios, and test modules behave under finite covers. To this end, we expand on the notion of transposability along a section of the relative canonical module as first introduced by K. Schwede and K. Tucker.     

On the singularities of polar curves

Let be a given equisingularity class of irreducible algebroid plane curves with just one characteristic exponent. We first obtain the equisingularity type of the general polar of a curve of with general moduli and we also determine the Newton-Cramer polygon of the general polar of all curves in .     

On the order of projective curves

Here we prove the existence of several componentsW of the Hilbert scheme of curves inP ⁿ such that the generalC W has Hartshorne-Rao module with order equal to its diameter.     

On the gonality of graph curves

A genus g graph curve in the sense of Bayer and Eisenbud is a genus g stable curve (hence nodal) with 2g − 2 irreducible components, each of then smooth and rational and intersecting exactly 3 other components. Here, we study the existence of spanned or very ample non-special line bundles on X whose restriction to each irreducible component of X has degree 1.